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abstract: 'We report on Keck and [*Hubble Space Telescope*]{} optical observations of the eclipsing binary pulsar system , in the direction of the globular cluster . These reveal a faint star ($m_{\rm{}F702W}=25.21\pm0.07$; Vega system) within the pulsar’s 05 radius positional error circle. This may be the companion. If it is a main-sequence star in the cluster, it has radius $\rcomp\simeq0.3\,\rsun$, temperature $\teff\simeq3600\,$K, and mass $\mcomp\simeq0.3\,\msun$. In many formation models, however, the pulsar (spun up by accretion or newly formed) and its companion are initially in an eccentric orbit. If so, for tidal circularization to have produced the present-day highly circular orbit, a large stellar radius is required, i.e., the star must be bloated. Using constraints on the radius and temperature from the Roche and Hayashi limits, we infer from our observations that $\rcomp\simlt0.44\,\rsun$ and $\teff\simgt3300\,$K. Even for the largest radii, the required efficiency of tidal dissipation is larger than expected for some prescriptions.'
author:
- 'M. H. van Kerkwijk, V. M. Kaspi, A. R. Klemola, S. R. Kulkarni, A. G. Lyne,'
- 'D. Van Buren'
title: 'Optical Observations of the Binary Pulsar System : Implications for Tidal Circularization'
---
\#1[§\[sec:\#1\]]{} \#1[Fig. \[fig:\#1\]]{}
Introduction {#sec:intro}
============
is a 1-s radio pulsar in the direction of the globular cluster ([@lbhb93]). It is in a 6.2h circular binary orbit with a companion that has mass $\mcomp\geq0.11\,$ (assuming a pulsar mass $\mpsr\simeq1.35\,\msun$). At radio frequencies around 400–600MHz, irregular eclipses of the pulsar occur around superior conjunction, indicating the presence of material around the companion. The eclipsing material must be tenuous since the eclipses are absent at higher frequencies. The properties of the pulsar itself are not special; from the spin period and its derivative, one infers a dipole magnetic field strength $B\simeq3.2\times10^{19}(P\dot{P})^{1/2}\simeq1.5\times10^{12}$G and a spin-down age $\tau_{\rm{}sd}=P/2\dot{P}=10$Myr.
The system appears peculiar for several reasons ([@lbhb93]): (i) supernova activity in should have ceased long ago, hence one does not expect it to contain an apparently recently formed pulsar; (ii) all other known binary pulsars in globular clusters have millisecond periods and low magnetic fields, evidence that they were “recycled” or spun up by accretion from a binary companion, the field presumably having decayed in the process (for a review, see [@vdh95]); (iii) unlike in other eclipsing pulsars, the rotational energy loss of is many orders of magnitude below the critical flux necessary to drive a wind from the companion, so the origin of the eclipsing material is unclear.
Earlier papers on this system ([@lbhb93]; [@erg93]; [@wp93]; [@zwi93]; [@bk94]; [@esg96]) have focused on the three peculiarities discussed above, mostly in the context of the two favored formation scenarios: forming a new pulsar by accretion-induced collapse of a white dwarf, and recycling an old pulsar in a close encounter with other stars in the core of the globular cluster. We believe, however, another important issue is that (iv) the current orbit is near-circular ($e\simlt0.005$), while the formation should have left the pulsar and its companion in an eccentric orbit. An initially eccentric orbit is not only expected in all formation models, but also indicated by the fact that the system currently is not in the core of the cluster, as would be expected given its mass, but offset by 24; apparently, it received a kick, which should have made the orbit eccentric as well.
It has been noted by Verbunt () that if circularization occurred, the energy dissipated in the companion would have been of order the binding energy of the star. Thus, the star might have become bloated or even have been partially destroyed. Little attention, however, has been given to the question of whether the tidal circularization efficiency is sufficient for circularization to have happened within the spin-down age.
The above puzzles motivated us to try to identify the counterpart of . In this article, we report on Jodrell Bank, Very Large Array (VLA), Keck and [*Hubble Space Telescope*]{} ([*HST*]{}) observations of . The improved position of the pulsar from the radio observations and the results of the optical imaging observations are reported and summarized in . Given the crowded field, establishing a precise tie between the radio and optical observations is crucial. A detailed account of our astrometry is given in . The photometry of a candidate object is also reported in this section. The constraints on the basic parameters of the companion derived from optical and pulsar timing observations are discussed in . In , we discuss the implications of our results, focusing in particular on the circularization.
Observations {#sec:obs}
============
The position of the pulsar was first determined from radio timing measurements using the 76-m Lovell telescope at Jodrell Bank ([@lbhb93]). We derived an improved timing position from data obtained between 1994.6 and 1998.8: $\rm\alpha_{J2000}=17^h21^m01\fs53$ and $\rm\delta_{J2000}=-19\arcdeg36\arcmin36\arcsec$, with uncertainties of $0\fs04$ and $6\arcsec$, respectively. This position has a relatively large uncertainty in declination because of the source’s proximity to the plane of the ecliptic. An image of the field was obtained on 11 November 1992 using the VLA in the A-configuration. It reveals a source within the timing error box with a flux density of 0.3mJy, consistent with the flux of the pulsar at the observing frequency of around 1400MHz. The position of this source is $\rm\alpha_{J2000}=17^h21^m01\fs54$ and $\rm\delta_{J2000}=-19\arcdeg36\arcmin36\farcs6$, with an uncertainty of $0\farcs2$ in each coordinate.
0.9
The field was observed on 4 June 1995 using the Low-Resolution Imaging Spectrometer (LRIS; [@occ+95]) on the 10-m Keck telescope. Two series of exposures with integration times ranging from 10s to 240s were taken through several filters. The seeing was $\sim\!0\farcs9$ during the first series, and $\sim\!1\farcs2$ during the second. Our best-seeing I-band ($\sim\!0.8\mu{}m$) image is shown in . These images showed that there was no object in the VLA error box, and that with 95% confidence $I>24.0\,$mag. Furthermore, they showed that a brighter star was so close that it would be hard to make any further progress from the ground.
The Wide Field Planetary Camera 2 (WFPC2) on board the [*HST*]{} was used to observe the field through the F702W filter in four contiguous [*HST*]{} orbits, from 14:39 to 20:07 [ut]{} on 8 March 1997. In each spacecraft orbit, three 700-s exposures were taken, offset in both X and Y by 0, 3, and 6 PC pixels in order to mitigate the effects of hot pixels and imperfect flat fields. The pointing was such that the field around the pulsar is on a clean spot on the CCD of the Planetary Camera (PC).
The reduction and analysis started from the pipeline-calibrated PC images ([@hhc+95]). We first obtained median images for each of the three observing positions. The median image for the middle offset was used for the astrometry described below. Next, for each of the three sets, we found pixels hit by cosmic rays in the individual images by comparison with the median image; pixels with values more than $5\sigma$ above the median were replaced with the median (here, $\sigma$ is an estimate of the expected uncertainty based on the median value). Finally, the images were registered by applying integer pixel shifts and co-added to form a grand average. This average is free of cosmic rays, but not of hot pixels and other chip defects. We used it for the photometry nonetheless, as our candidate (see below) and the other stars we selected were on clean parts of the chip. For display purposes, however, we have used the median of all registered images in .
Astrometry and Photometry {#sec:astro}
=========================
The astrometry of the field was carried out in four stages. First, 23 stars from the ACT Reference Catalog ([@ucw98]) were used to derive an astrometric solution for a plate taken at epoch 1988.37 on Kodak 103a-G emulsion using the yellow lens of the 0.5-m Carnegie double astrograph at Lick Observatory. The model employed five terms in each coordinate (proportional to 1, $x$, $y$, $xy$, and $(x^2,y^2)$ in $(x,y)$). The inferred rms error for a single star is 016 in each coordinate, and the zero-point uncertainty in the solution 004. We note that the errors are larger by about 50% than expected based on the measurement and ACT coordinate uncertainties. Probably, this is because some stars are blended with fainter objects; thus, it should induce no systematic error.
Second, the plate solution was used to calculate right ascension and declination for 30 fainter, relatively isolated stars in common with the short LRIS R-band exposure. For these stars, positions were also measured on the LRIS frame, and corrected for instrumental distortion using a bi-cubic transformation determined by J. Cohen (1997, private communication). Solving for offset, scale and rotation, the inferred rms single-star error is 026 in each coordinate, consistent with the expected measurement errors on the astrograph plate for these relatively faint stars. The zero-point uncertainty in the solution is 005.
Third, using 23 fainter stars on a smaller part of the LRIS image around the VLA position of , the solution was transferred to one of the 240s LRIS R-band images, solving only for the offset between the two exposures. The inferred rms error is $0\farcs011$ in each coordinate, and the zero-point uncertainty in the solution 0002.
Finally, 37 stars were used to tie the astrometry to the WFPC2 PC image (14 of these were used in the previous stage as well). The WFPC2 positions were corrected for instrumental distortion using the cubic transformation given by Holtzman et al. (). Offset, scale and rotation were left free in the solution[^1]. The inferred rms error was 0017 in each coordinate, and the zero-point uncertainty in the solution 0003.
From the numbers above, the total uncertainty in the zero points of positions due to measurement errors is 006 in each coordinate. We expect systematic effects to be smaller than this; e.g., no large systematic offset as a function of magnitude is seen on yellow astrograph plates used for other applications. A possible additional uncertainty is the extent to which our ACT-based solution is on the same system as the VLA position of , i.e., the International Celestial Reference System (ICRS). The ACT combines the Tycho ([@esa97]) and AC2000 ([@ucw+98]) catalogs to determine positions and accurate proper motions. The Tycho positions are on the ICRS to within 0.6mas ([@esa97]), and any offset from the ICRS at the plate epoch, which is close to the Tycho epoch (1991.25), should be small as well.
In summary, we expect the uncertainties in the astrometry to be dominated by measurement errors, 006 in each coordinate for the optical position, and 02 for the radio position. Combining the two in quadrature, the 95% confidence error radius is $[-2\log(1-0.95)(0\farcs2^2+0\farcs06^2)]^{1/2}=0\farcs5$. Within this radius, there is one faint object (), at $\rm\alpha_{J2000}=17^h21^m01\fs549\pm0\fs004$ and $\rm\delta_{J2000}=-19\arcdeg36\arcmin36\farcs76\pm0\farcs06$. This may be the optical counterpart of .
We measured magnitudes for the candidate counterpart of and for a number of other stars in the field following the prescriptions of Holtzman et al. () and Baggett et al. (). We performed aperture photometry for a range of different radii, and used some thirty brighter stars in the frame to determine aperture corrections relative to the standard 05 (11pix) radius aperture. For the candidate, the best signal-to-noise ratio is for relatively small apertures, with radii between 1.5 and 3pix. From these, we infer a count rate for the 05 radius aperture of $0.070\pm0.005{\rm\,DN\,s^{-1}}$ (we find consistent values for the larger apertures; here, $1{\rm\,DN}$ corresponds to about 7 electrons, given the gain used for our observations). This count rate corresponds to a magnitude $m_{\rm{}F702W}=25.21\pm0.07$ in the Vega system (using $m_{\rm{}F702W}=22.428$ for a count rate of $1{\rm\,DN\,s^{-1}}$ in the PC, and applying a $0.10\,$mag aperture correction from 05 radius to “nominal infinity”; [@bcgr97]). Magnitudes for the individual frames show a standard deviation of 0.25mag around the average, similar to what is found for other stars at this brightness level. We do not find modulation at the 62 orbital period of (which is well covered by our observations), although the limit on the modulation amplitude is not very restrictive: $<\!0.3\,$mag at 95% confidence.
At or above the brightness level of the candidate counterpart, there is about one object per four square arcseconds in this field. Thus, there is a probability of about one in five of finding an object within the 95% confidence error radius by chance. If it is a chance coincidence, the real counterpart must be substantially fainter. We derive a 95% confidence limit of $m_{\rm{}F702W}=27.2$ for any other object in the error circle (using the observed noise in the sky near the VLA position, of $\sigma_{\rm{}sky}=5.6\times10^{-4}{\rm\,DN\,pix^{-1}\,s^{-1}}$, and the fact that within the 05 radius error circle there are about 400 resolution elements).
Observational Constraints {#sec:constraints}
=========================
We now evaluate the constraints set on the system by the observations. We first summarize the constraints on the age of the system and the mass of the companion set by radio observations, and then discuss the constraints on the companion radius and temperature from our [*HST*]{} detection and Keck limit. We will assume that we detected the companion, and that the system is located in .
System Age {#sec:age}
----------
Clearly, something happened to the system recently (as compared to the cluster age). One indication is the short inferred spin-down age of the pulsar, $\tau_{\rm{}sd}=P/2\dot{P}=10$Myr. This is the time required for a dipole rotating [*in vacuo*]{} to spin down to the present-day spin period from an infinitely fast rotation rate. It is thus an upper limit to the true age, unless other mechanisms influenced the spin period (e.g., transient spin-up by accretion or a braking index very different from that predicted by dipole emission).
Another indication of a recent event is that the system is offset from the cluster core. Unperturbed, a system as massive as this should have settled in the core long ago due to mass segregation (in $\sim\!0.5\,$Gyr given its present position). Indeed, the progenitor(s) of this system, which must have been massive as well, should have resided in the core. It seems natural to argue that it was a single event that brought the system to its present state and kicked it out of the core.
We note that it would require fine-tuning for the kick to have resulted in the system remaining in a cluster orbit as wide as is indicated by the 24 offset (the half-mass radius is 09; for a discussion, see, e.g., [@phi92]). If instead the system is unbound, the kick needs to have happened $\simlt\!1\,$Myr ago, which would imply that the event left the pulsar with a spin period only slightly shorter than the present one. This would be consistent with scenarios in which the pulsar was spun up by accretion: for Eddington-limited accretion, the equilibrium spin period is close to 1s ([@lbhb93]). We conclude that the system has been in its present state for 10Myr at most.
Companion Mass {#sec:compm}
--------------
Radio pulse timing of has yielded the mass function, $f(\mcomp,\mpsr)=\mcomp^3\sin^3i/(\mcomp+\mpsr)^2=0.000706\,\msun$, where $i$ is the inclination of the binary orbit ([@lbhb93]). From the mass function, assuming $\mpsr\simeq1.35\,\msun$ ([@tc99]), one infers $\mcomp\geq0.11\,\msun$; furthermore, there is a 95% [*a priori*]{} probability that $i>18\arcdeg$ and hence that $\mcomp<0.43\,\msun$.
The radio eclipses allow a direct constraint on $i$, albeit in a model-dependent way. Burderi & King () and Thorsett () calculated the expected attenuation of the radio flux at different frequencies for a simple constant-velocity, spherically symmetric wind, and found that they could reproduce the observations of Lyne et al. () for $i\simgt30\arcdeg$. This would correspond to $\mcomp\simlt0.25\,\msun$. Burderi & King argue that almost certainly $i>20\arcdeg$, implying $\mcomp<0.35\,\msun$. In summary, most likely $0.11\simlt\mcomp\simlt0.35\,\msun$.
Companion Radius and Temperature {#sec:randt}
--------------------------------
The interpretation of the apparent F702W magnitude depends on the companion radius , effective temperature , and (to a lesser extent) surface gravity $\log{}g$, as well as the distance and reddening to . To determine the constraint set by our measurement, we used parameters for from the May 1997 edition of the catalog of globular clusters ([@har96]): reddening $E_{B-V}=0.44$, distance modulus $(m-M)_V=16.15$, and metallicity relative to solar ${\rm[Fe/H]}=-0.65$. The distance scale used in the catalog is similar to the Hipparcos-based one (e.g., that of Chaboyer et al. \[\] would give $(m-M)_V=16.22$). The reddening corresponds to $A_V=1.36$, $A_R=1.02$, $A_I=0.65$ (using the extinction curve of [@mat90]), and one infers $(m-M)_0=14.79$ and $d=9.1\,$kpc. We also used the evolutionary tracks for $\rm[Fe/H]=-0.5$ stars of Baraffe et al. (), to relate temperatures and radii to absolute magnitudes in various standard bands.
With these data in hand, we proceeded as follows. First, we looked up temperatures and corresponding radii for the main sequence (at age 10Gyr, appropriate for a metal-rich globular cluster; [@sw98]). Second, we used $M_V$ and $M_R$ for these stars to calculate $V_{\rm{}ms}$ and $R_{\rm{}ms}$ for the distance and reddening of . Third, for each $(V-R)_{\rm{}ms}$, we used the calibration of Holtzman et al. (; Eq. 9, Table 10) to infer the expected $R_{\rm{}F702W}$ magnitude corresponding to the observed F702W count rate. Fourth, we derived the radii required to match the observations from the difference $R_{\rm{}F702W}-R_{\rm{}ms}$. To estimate the uncertainty, we assumed a total uncertainty of 0.3mag in the magnitude difference, which we regard as a 95% confidence estimate (it is dominated by the estimated uncertainties in distance and reddening). Strictly speaking, one should redden the F702W magnitude and then calculate $R_{\rm{}F702W}$ using $(V-R)_0$. Furthermore, one should consider the influence of $\log{}g$ on the star’s colors, and take into account the slight difference in metallicity. None of these corrections, however, is important at the present level of accuracy.
Another constraint is set by our I-band non-detection. This corresponds to an upper limit to the $m_{\rm{}F702W}-I$ color, and thus a lower limit to the effective temperature. We find $\teff>3150\,$K for $m_{\rm{}F702W}-I>1.4$ (here, we have decreased the limit to the magnitude difference by 0.2mag to account for uncertainties in $m_{\rm{}F702W}$, in the color transformation, in the effects of changes in $\log{}g$, and in the reddening).
The observational constraints can be summarized graphically in a diagram of versus the effective temperature of the companion (see ). To make further progress we need to have some knowledge of the companion’s nature. We consider two cases: the companion is an ordinary main sequence star; and the companion is a bloated star.
0.95
### A Main Sequence Companion {#sec:mscomp}
In , the locus of radius versus effective temperature for main sequence stars of different masses is indicated (from [@bcah98]). From the intersection with the region allowed by our observation, one infers that if the companion were a main sequence star, it would need to have $0.26\simlt\rcomp\simlt0.33\,\rsun$ and $3580\simlt\teff\simlt3680\,$K. Its mass would be $0.26\simlt\mcomp\simlt0.34\,\msun$, consistent with the constraints inferred dynamically and from eclipse modeling (§\[sec:compm\]).
### A Bloated Companion {#sec:bloat}
The companion does not necessarily have to be on the main sequence. Indeed, bloating might be expected: most formation mechanisms produce a binary that has substantial eccentricity initially, and the amount of energy that needs to be dissipated in order to circularize the orbit is a substantial fraction of the binding energy of a low-mass star[^2] ([@ver94]). Bloating may also be required, as alluded to in and discussed in , for tidal dissipation to have circularized the orbit in the short time since formation.
We are not aware of detailed calculations of bloating due to tidal dissipation for almost completely convective, low-mass stars[^3]. Regardless of the expansion process, however, for a given mass an upper limit to the radius is set by the Roche radius, above which mass transfer would occur. Furthermore, for given mass and radius, a lower limit to the temperature is set by the Hayashi () limit, below which a star cannot be in hydrostatic equilibrium.
To delineate the constraint on the temperature, we can use pre-main-sequence tracks, since these follow the Hayashi limit closely. Indeed, it seems not unlikely that the companion is currently contracting along a similar track, since all processes that could have induced bloating (irradiation, tidal heating) should have ceased to be operative (the pulsar’s spin-down luminosity being very low and tidal dissipation having ceased as the orbit became circular). Of course, it is not clear that there has been enough time for the companion to adapt to a quasi–pre main sequence configuration. On the other hand, if the bloating is due to tidal heating, and if the energy was dissipated somewhere in the convective regions, convection would have distributed the energy throughout the star and expansion may well have been along a quasi–pre main sequence track as well.
In , dotted lines indicate the pre-main-sequence tracks of Baraffe et al. (), and the long-dashed line shows the Roche radii for the whole range of masses. Under our assumptions, the companion must lie between the latter line and the main sequence, and at any given radius and temperature, an upper limit to its mass is set by the mass for which the pre-main-sequence track passes through that radius and temperatures. In addition, it must reproduce the observed F702W flux and be consistent with the $m_{\rm{}F702W}-I$ limit. The allowed ranges are $3300\simlt\teff\simlt3680\,$K and $0.44\simgt\rcomp\simgt0.26\,\rsun$. The inferred masses are $0.11\simlt\mcomp\simlt0.34\,\msun$, consistent with the constraints inferred dynamically and from eclipse modeling (§\[sec:compm\]) except for the very low-mass end ($\mcomp\simlt0.12\,\msun$), at which total eclipses would be expected (for a 1.35 neutron star), which are not observed.
Discussion {#sec:disc}
==========
We now discuss the implications of our results. The assumption we continue to make is that the system is associated with and that we have detected the counterpart. The constraints we regard as most important are: (i) the system was brought to its present state $\simlt\!10\,$Myr ago; (ii) the current orbit is nearly circular ($e\simlt0.005$); and (iii) the companion has $0.26\simlt\rcomp\simlt0.44\,\rsun$ and $3680\simgt\teff\simgt3300\,$K. In these ranges, the left-hand limits correspond to the case where the companion is a $\sim\!0.3\,\msun$ main-sequence star, while the right-hand limits correspond to the case where the companion is a $\sim\!0.13\,\msun$ star which has been maximally bloated and is currently contracting along a pre-main-sequence track.
Below, we first briefly review the formation models that have been suggested, in order to set the stage for a discussion of how the system could have been circularized. We end by noting briefly what constraints one could set if the star we detected is not the counterpart of , or if the system is not in .
Formation Models {#sec:form}
----------------
Two models have been suggested to explain the origin of the system. One is that the neutron star was formed early in the life of the globular cluster ([@lbhb93]; [@wp93]; [@zwi93]; Ergma et al. ). It had stopped being a radio pulsar and was dormant until it had a close interaction with a star or binary, $\sim\!10\,$Myr ago. During that interaction, some mass was lost from a normal star, part of which was accreted by the neutron star. As a consequence, it was spun up sufficiently for the radio mechanism to become active again, yet did not undergo a long phase of mass transfer and thus kept a large magnetic field. Similar close encounters have been invoked to explain the presence of two other long-period (but single) pulsars in globular clusters: PSR B1745$-$20 in NGC 6440 and PSR B1820$-$30B in NGC 6624 ([@lmd96]).
To give the system the velocity required by the present location far outside the cluster core and conserve momentum, the putative interaction must have involved at least one other object. The system’s initial orbit is expected to have been highly eccentric, $e\simgt0.8$ (see, e.g, [@phi92]).
The second model that has been considered for the origin of the system is that the neutron star was formed via accretion-induced collapse (AIC) of a white dwarf ([@lbhb93]; [@erg93]; [@wp93]; Ergma et al. ). The kick imparted to the neutron star during AIC, whether intrinsic or due to mass loss, must have been relatively small, since otherwise the systemic velocity would have been so large that the system would have left the cluster long ago. For a kick of order the $20{\rm\,km\,s^{-1}}$ escape velocity from the core ([@web85]), an initial eccentricity of $\sim\!0.2$ is expected.
Given such a kick, the increase in the orbital separation should have been small, about a factor 1.25. Thus, the companion, which must have filled its Roche lobe before AIC for mass transfer to occur, should still be close to filling its Roche lobe. This implies that it cannot have been a Roche-lobe filling main-sequence star, as this would require a mass of $\sim\!0.7\,\msun$ and hence a much brighter optical counterpart than we observe (see ). It has been suggested that the star was bloated already before AIC due to irradiation from the primary during the mass transfer phase ([@erg93]; Ergma et al. ). It is not clear whether strong bloating is possible by irradiating only one side of a star (e.g., [@kfkr96]), but if it happened, the star should still be bloated, because the thermal time scale is much longer than the pulsar characteristic age.
Circularization {#sec:circ}
---------------
In both formation scenarios that have been suggested, the initial binary orbit is expected to be eccentric. Indeed, in general any event that imparted a systemic velocity large enough for the system to (almost) escape the cluster will most likely have left the binary orbit eccentric. The current tiny eccentricity therefore requires explanation. Since eccentricity likely decays exponentially, the current low $e$, even for an initial eccentricity as small as $e\simeq0.1$, requires a circularization time $t_{\rm{}circ}\simlt\tau_{\rm sd}/4=2.5\,$Myr.
It is not straightforward to estimate whether such a short $t_{\rm
circ}$ is possible. This is because the companion is probably completely convective, with convective turnover time scales far longer than the 6.2-h orbital period (see below). This makes the energy transfer less efficient, and thus circularization time scales longer, but it is not clear to what extent. We will first use the prescription of Zahn (), in which the efficiency is assumed to decrease linearly with the ratio of the convective to orbital timescale, and then discuss the prescription of Goldreich & Nicholson () and Goodman & Oh (), in which the efficiency decreases almost quadratically with the timescale ratio. We should stress that at present it is not clear that either prescription is reliable; see Goodman & Oh for a discussion.
Following the formalism of Zahn (; Eq. \[21\]), we write[^4] $$\frac{1}{t_{\rm circ}}=-\frac{1}{e}\frac{{\rm d}e}{{\rm d}t} =
21\frac{\lambda_{\rm circ}}{t_{\rm f}}q(1+q)\left(\frac{\rcomp}{a}\right)^8,$$ where $t_{\rm{}circ}$ is the circularization timescale, $e$ the eccentricity, $\lambda_{\rm{}circ}$ a dimensionless average of the turbulent viscosity weighted by the square of the tidal shear, $t_{\rm{}f}=(\mcomp\rcomp^2/\lcomp)^{1/3}$ the convective friction time, $q=\mpsr/\mcomp$ the mass ratio, and $a$ the orbital separation. Here, all uncertainty is hidden in the parameter $\lambda_{\rm{}circ}$. In the prescription of Zahn (), it can be approximated by $\lambda_{\rm{}circ}\simeq0.019\alpha^{4/3}(1+\eta^2/320)^{-1/2}$, where $\alpha$ is the mixing length parameter and $\eta=2t_{\rm{}f}/P_{\rm{}orb}$ a measure of the timescale mismatch.
For a main-sequence star companion with parameters in the ranges listed above, we find $t_{\rm{}f}\simeq0.5\,$yr and $\eta\simeq1500$. Assuming $\mpsr=1.4\,\msun$ (i.e., $a\simeq2\,\rsun$) and taking $\alpha=2$ (as in Zahn \[\]; consistent with the observational constraint derived by Verbunt & Phinney \[\] for small $\eta$), one infers $\lambda_{\rm{}circ}\simeq6\times10^{-4}$, and a circularization time in the range $5\simlt{}t_{\rm{}circ}\simlt17\,$Myr, too long to understand the current small eccentricity. If the star is bloated, circularization is much faster, because of the very strong dependence on the ratio $\rcomp/a$: we find $t_{\rm{}circ}\simeq0.07\,$Myr for a maximally bloated companion.
The situation is different for the prescription of Goodman & Oh (), in which $\lambda_{\rm{}circ}\propto\eta^{-2}$. Extrapolating in their Fig. 2, we infer $\lambda_{\rm{}circ}\simeq4\times10^{-6}$ (note their slightly different definition of $\eta$). Thus, the inferred circularization times are two orders of magnitude longer than those estimated with the prescription of Zahn (); if correct, it may be difficult to understand how the orbit could have been circularized even if the companion was maximally bloated.
As mentioned in , the star could have become bloated due to the energy dissipated by the circularization proper ([@ver94]). It is difficult to estimate, however, by how much, as it is not clear where, how, and on what time scale the tidal energy is dissipated, especially in such a low-mass star. For high eccentricity, the tides excite low-order pulsation modes in the star, which will be dissipated, either by direct viscous damping ([@koc92]), or, perhaps more likely, by non-linear coupling to higher-degree modes and damping of these ([@kg96]). For either case, it appears that for a low-mass, (almost) completely convective star ($\simlt\!0.5\msun$), most of the energy will be dumped in the outermost layers. These will be heated and may expand, which would lead to stronger tidal coupling.
If the expansion involved the whole star, or a substantial fraction of it, most likely the star would still be bloated, as the thermal time would be much longer than the pulsar characteristic age. Also if only the outer layers of the star expanded, however, it seems likely the star would still be bloated, as otherwise it would be difficult to reduce the eccentricity sufficiently. This is because the tidal luminosity will decrease rapidly with decreasing eccentricity and increasing periastron distance; if the expanded layers shrunk too quickly in response, the circularization time would have become long again while the eccentricity was still substantial. One way to verify whether the companion is still bloated is to measure its temperature.
Caveat: Fainter Companion {#sec:caveat}
-------------------------
There is a probability of about one in five of a chance coincidence between and the object in our HST images (). If so, the companion has to be substantially fainter, with $m_{\rm{}F702W}>27.2$ (). One can go through a similar exercise as in to constrain the companion properties for this case. The result is that the companion needs to have mass $\simlt\!0.15\,\msun$, close to the minimum allowed by pulse timing. In this case, the circularization time scale problem is exacerbated.
Another possibility is that is not associated with . Indeed, the dispersion measure is only $71{\rm\,cm^{-3}\,pc}$, while $130{\rm\,cm^{-3}\,pc}$ is expected for ([@tc93]). Taken at face value, a distance of only 3kpc is implied. If were in the foreground, again the companion would be less luminous and thus less massive, even more problematic for circularization.
Conclusions {#sec:conclusions}
===========
In summary, using [HST]{} observations, we have detected a faint star at the position of the unusual eclipsing binary radio pulsar . This faint star most likely is the pulsar’s companion. We have shown that it is difficult to explain the highly circular present-day orbit if the companion is a $\mcomp\simeq0.3\,\msun$, $\rcomp\simeq0.3\,\rsun$ main-sequence star (unless circularization was not by tidal interaction). If it is a bloated, $\rcomp\simgt0.4\,\rsun$ star, circularization may be sufficiently rapid, depending on the extent to which the efficiency of tidal dissipation is suppressed by the orbital period being far shorter than the convective turnover time in the star. The system thus provides an interesting test-case for tidal-interaction theory.
A measurement of the color of the companion would be the best way to determine whether it is bloated or not. For a main-sequence star with $\teff=3640\,$K and a bloated star with $\teff=3300\,$K, one expects $(R-I,I-J,J-H,H-K)_0=(1.0,1.1,0.6,0.2)$ and $(1.3,1.4,0.6,0.2)$, respectively. Radio and optical proper-motion studies could settle association of our candidate counterpart with , and of the system with .
We are indebted to the referee, R. Webbink, for his thoughtful report, and, in particular, for pointing out the importance of the Hayashi limit for constraining the temperature of a bloated companion. We also thank him for the estimate of the relaxation time of at its present position in . We thank M. Goss for help with the VLA observations, and P. Goldreich, V. Kalogera, F. Rasio, S. Sigurdsson, C. Tout, M. van den Berg, F. Verbunt, and R. Wijers for useful discussions. The observations were obtained at the W. M. Keck Observatory on Mauna Kea, Hawaii, which is operated by the California Association for Research in Astronomy, and with the NASA/ESA Hubble Space Telescope at STScI, which is operated by AURA. The reduction of the optical data was done using the Munich Image Data Analysis System (MIDAS), which is developed and maintained by the European Southern Observatory. This research made use of the SIMBAD data base. We acknowledge support of a NASA Guest Observer grant (GO-06769.01-95A), a fellowship of the Royal Netherlands Academy of Arts and Sciences (MHvK), a NSF grant (95-30632; ARK), a Sloan Research Fellowship (VMK), and visitor grants of the Netherlands Organisation for Scientific Research NWO (VMK) and the Leids Kerkhoven-Bosscha Fonds (MHvK, VMK). MHvK thanks MIT and Caltech for hospitality, VMK the Aspen Center for Physics and Utrecht University.
Baggett, S., Casertano, S., Gonzaga, S., & Ritchie, C., 1997, WFPC2 Instrument Science Report 97-10 (Baltimore: STScI) Baraffe, I., Chabrier, G., Allard, F., Hauschildt, P. H. 1998, A&A, 337, 403 Burderi, L., & King, A. R. 1994, ApJ, 430, L57 Chaboyer, B., Demarque, P., Kernan, P. J., & Krauss, L. M. 1998, ApJ, 494, 96 Ergma, E. 1993, A&A, 273, L38 Ergma, E., Sarna, M. J., & Giersz, M. 1996, A&A, 307, 768 ESA 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200 Goldreich, P., & Nicholson, P. D. 1977, Icarus, 30, 301 Goodman, J., & Oh, S. P. 1997, ApJ, 486, 403 Harris, W. E. 1996, AJ, 112, 1487 Hayashi, C. 1961, PASJ, 13, 450 Holtzman, J. A., et al. 1995a, PASP, 107, 156 Holtzman, J. A., Burrows, C. J., Casertano, S., Hester, J. J., Trauger, J. T., Watson, A. M., & Worthey, G. 1995b, PASP, 107, 1065 King, A. R., Frank, J., Kolb, U., & Ritter, H. 1996, ApJ, 467, 761 Kochanek, C. 1992, ApJ, 385, 604 Kumar, P., & Goodman, J. 1996, ApJ, 466, 946 Lyne, A. G., Biggs, J. D., Harrison, P. A., & Bailes, M. 1993, Nature, 361, 47 Lyne, A. G., Manchester, R. N., & D’Amico, N. 1996, ApJ, 460, L41 Mathis, J. S. 1990, ARA&A 28, 37 Oke, J. B., et al. 1995, PASP, 107, 375 Phinney, E. S. 1992, Phil. Trans. R. Soc. Lond. A, 341, 39 Podsiadlowski, P. 1996, MNRAS, 279, 1104 Salaris, M., & Weiss, A. 1998, A&A, 335, 953 Taylor, J. H., & Cordes, J. M. 1993, ApJ, 411, 674 Thorsett, S. E. 1995, in Millisecond Pulsars: A Decade of Surprise, eds. A. S. Fruchter, M. Tavani, & D. C. Backer, ASP Conf. Ser. 72, 253 Thorsett, S. E., & Chakrabarty, D. 1999, ApJ, 512, 299 Urban, S. E., Corbin, T. E., Wycoff, G. L., Martin, J. C., Jackson, E. S., Zacharias, M. I. & Hall, D. M. 1998a, AJ, 115, 1212 Urban, S. E., Corbin, T. E., & Wycoff, G. L. 1998b, AJ, 115, 2161 van den Heuvel, E. P. J. 1995, J. Astrophys. Astron., 16, 255 Verbunt, F. 1994, A&A, 285, L21 Verbunt, F., & Phinney, E. S. 1995, A&A, 296, 709 Webbink, R. F. 1985, in IAU Symp. 113, Dynamics of Stellar Clusters, ed. J. Goodman & P. Hut (Dordrecht: Kluwer), 541 Wijers, R. A. M. J. & Paczynski, B. 1993, ApJ, 415, L115 Zahn, J.-P. 1989, A&A, 220, 112 Zwitter, T. 1993, MNRAS, 264, L3
[^1]: The fitted scale is close to the value given by Holtzman et al.(), as is the offset between the fitted position angle and the position angle of the [*HST*]{} V3 axis listed in the image header. If we fix the values, the inferred position of our proposed counterpart changes by 0006 in RA and $-0\farcs005$ in Dec. Thus, a perhaps more conservative estimate of the zero-point uncertainty is about 001 in each coordinate.
[^2]: The star cannot be large because of having evolved off the main-sequence, as a star at or past the turn-off mass would have a luminosity much higher than the limit implied by our observations. Even if mass were lost, the thermal timescale for such a star likely is too long for the luminosity to have changed substantially.
[^3]: See Podsiadlowski () for calculations for somewhat more massive stars, for which only the outer layers are convective.
[^4]: See Phinney () for a pedestrian derivation.
|
---
abstract: |
Accurate segmentation of brain tissue in magnetic resonance images (MRI) is a diffcult task due to different types of brain abnormalities. Using information and features from multimodal MRI including T1, T1-weighted inversion recovery (T1-IR) and T2-FLAIR and differential geometric features including the Jacobian determinant(JD) and the curl vector(CV) derived from T1 modality can result in a more accurate analysis of brain images. In this paper, we use the differential geometric information including JD and CV as image characteristics to measure the differences between different MRI images, which represent local size changes and local rotations of the brain image, and we can use them as one CNN channel with other three modalities (T1-weighted, T1-IR and T2-FLAIR) to get more accurate results of brain segmentation. We test this method on two datasets including IBSR dataset and MRBrainS datasets based on the deep voxelwise residual network, namely VoxResNet, and obtain excellent improvement over single modality or three modalities and increases average DSC(Cerebrospinal Fluid (CSF), Gray Matter (GM) and White Matter (WM)) by about 1.5% on the well-known MRBrainS18 dataset and about 2.5% on the IBSR dataset. Moreover, we discuss that one modality combined with its JD or CV information can replace the segmentation effect of three modalities, which can provide medical conveniences for doctor to diagnose because only to extract T1-modality MRI image of patients. Finally, we also compare the segmentation performance of our method in two networks, VoxResNet and U-Net network. The results show VoxResNet has a better performance than U-Net network with our method in brain MRI segmentation. We believe the proposed method can advance the performance in brain segmentation and clinical diagnosis.\
**Keywords:** magnetic resonance images (MRI), differential geometric features, multimodal MRI, Jacobian determinant, curl vector, VoxResNet, U-Net
author:
- |
Yongpei Zhu$^{1}$, Zicong Zhou$^{2}$, Guojun Liao$^{2}$, Qianxi Yang$^{1}$, Kehong Yuan$^{1*}$\
$^{1}$Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China.\
$^{2}$The University of Texas at Arlington, Arlington 76019, USA.\
\*Corresponding author: Kehong Yuan (e-mail: yuankh@sz.tsinghua.edu.cn)\
`zhuyp17@mails.tsinghua.edu.cn`\
bibliography:
- 'D3D\_DRL.bib'
title: '**The Method of Multimodal MRI Brain Image Segmentation Based on Differential Geometric Features**'
---
Introduction
============
Magnetic resonance imaging (MRI) is usually the preferred method of structural brain analysis, as it provides high-contrast and high-spatial resolution images of soft tissue with no known health risks, which is the most popular choice to analyze the brain and we will focus on MRI in this work. Quantitative analysis of brain MR images is routine for many neurological diseases and conditions. Segmentation, i.e., labeling of pixels in 2D (voxels in 3D), is a critical component of quantitative analysis. Brain tissue segmentation generally refers to the separation of the brain into three functional components, namely, cerebrospinal fluid (CSF), grey matter (GM) and white matter (WM). There is a need for automated segmentation methods to provide accuracy close to manual segmentation with a high consistency.\
Deep learning techniques are gaining popularity in many areas of medical image analysis [@lin2016Neural], different from traditional machine learning algorithm, it is a new and popular machine learning technique, which can extract complex feature levels from images. Some of the known deep learning algorithms are stacked auto-encoders, deep Boltzmann machines, deep neural networks, and convolutional neural networks (CNNs). CNNs are the most commonly applied to image segmentation and classification.\
At present, there are three main CNN architecture styles for brain MRI image segmentation [@Akkus2017Deep]: (1) Patch-Wise CNN Architecture. This is an easy way to train the CNN segmentation algorithm. Fixed size patches around each pixel were extracted from the given image, and then the training model was trained on these patches with pixel labels in the patch center, such as normal brain and tumor. The disadvantage of this method is that it is computationally intensive and difficult to train. (2) Semantic-Wise CNN Architecture [@Long2015Fully][@Ronneberger2015U]. This architecture predicts each pixel of the entire input image, and the network only needs one forward inference. This structure includes the encoder part that extracts features and the decoder part that combines lower features from the encoder part to form abstract features. The input image is mapped to the segmentation labels in a way that minimizes a loss function. (3) Cascaded CNN Architecture [@Dou2016Automatic]. This type of architecture combines two CNN architectures. The first CNN is used for the preliminary prediction of the training model, and the second CNN is used to further adjust the prediction of the first network.\
Nowadays, the application of differential geometry in deep learning is more and more extensive, especially in the field of medical image. Based on the manifold characteristics of differential geometry, the success of deep learning is attributed to the inherent laws of the data itself. High-dimensional data are distributed near low-dimensional manifolds, which have a specific probability distribution, and it is also attributed to the strong ability of deep learning network to approximate nonlinear mapping. Deep learning technology can extract manifold structure from a kind of data and express the global prior knowledge with manifold, specifically, encoding and decoding mapping, which is implied in the weight of neurons. In the field of medical image analysis, doctors can determine whether the organs are abnormal by precisely comparing the geometry of the organs. By analyzing the geometric features of the tumor, we can judge the benign and malignant nature of the tumor. It can be attributed to the registration and analysis of medical images. Also the deep learning method based on differential geometry plays an important role in medical image registration.
Based on calculus of variation and optimization, we proposed a new variational method with prescribed Jacobian determinant and curl vector to construct diffeomorphisms of MRI brain images. Since the Jacobian determinant has a direct physical meaning in grid generation, i.e. the grid cell size changes, and the curl-vector represents the grid cell rotations, the deformation method was applied successfully to grid generation and adaptation problems.
Related Work
============
Accurate automated segmentation of brain structures such as white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) in MRI is important for studying early brain development in infants and precise assessment of the brain tissue. In recent years, CNNs have been used in the segmentation of brain tissue, avoiding a clear definition of spatial and strength characteristics and providing better performance than the classical approach, which we will describe below.\
Zhang et al. [@ZHANG2015214] proposed a 2D patch-wise CNN method to segment gray matter, white matter and cerebrospinal fluid from multimodal MR images of infants, and outperforms the traditional methods and machine learning algorithms; Nie et al.[@Nie2015FULLY] proposed a semantic-wise full convolution networks method and obtained improved results than Zhang’s method. Their overall DSC were 85.5%(CSF),87.3%(GM), and 88.7%(WM) vs. 83.5%(CSF),85.2%(GM), and 86.4%(WM) by[@ZHANG2015214]; Moeskops et al. [@Moeskops2016Automatic] proposed a multi-scale (25$^{2}$,51$^{2}$,75$^{2}$pixels) patch-wise CNN method to segment brain images of infants and young adults with overall DSC=73.53% vs. 72.5% by [@Br2015Deep] in MICCAI challenge; De Brebisson et al.[@Br2015Deep] presented a 2D and 3D patch-wise CNN approach to segment human brain and they achieved competitive results(DSC=72.5%$\mp$16.3%) in MICCAI 2012 challenge. Bao et al. [@Bao2015Multi] also proposed a multi-scale patch-wise CNN method together with dynamic random walker with decay region of interest to obtain smooth segmentation of subcortical structures in IBSR (developed by the Centre for Morphometric Analysis at Massachusetts General Hospital-available at [^1] to download) and LPBA40 datasets; Chen et al. [@Chen2016VoxResNet] proposed deep voxelwise residual networks for volumetric brain segmentation, which borrows the spirit of deep residual learning in 2D image recognition tasks, and is extended into a 3D variant for handling volumetric data. Olaf Ronneberger et al.[@Ronneberger2015U] presents an architecture named U-Net which consists of a contracting path to capture context and a symmetric expanding path that enables precise localization, and it outperforms the prior best method(a sliding-window convolutional network) on the ISBI challenge for segmentation of neuronal structures.\
In this paper, we use the differential geometric information including JD and CV, which represent the change rate of the area or volume of the brain image, and we can use them as one CNN channel with other three modalities to get more accurate results of brain segmentation. We test this method on three datasets including IBSR dataset, MRBrainS13 dataset and MRBrainS18 datasets based on the deep voxelwise residual network VoxResNet[@Chen2016VoxResNet].
Method
======
Deformation Method
------------------
The deformation method [@Liu1998An],[@Liao2000Level],[@Chen2015New] is derived from differential geometry. Consider $\mathrm{\Omega}$ and $\mathrm{\Omega}_t \subset \mathbb R^{2,3}$ with $0\leq{t}\leq{1}$, be moving (includes fixed) domains. Let $\pmb{v}(\pmb{x},t)$ be the velocity field on $\partial\mathrm{\Omega_t}$, where $\pmb{v}(\pmb{x},t)\cdot{\pmb{\mathrm{n}}}=0$ on any part of $\partial\mathrm{\Omega}_t$ with slippery-wall boundary conditions where $\pmb{\mathrm{n}}$ is the outward normal vector of $\partial\mathrm{\Omega}_t$. Given diffeomorphism $\pmb{\varphi}_0:\mathrm{\Omega}\rightarrow\mathrm{\Omega}_0$ and scalar function $f(\pmb{x},t)>0 \in C^1(\pmb{x},t)$ on the domain $\mathrm{\Omega}_t \times [0,1]$, such that
$$\label{General1}
\begin{aligned}
&f(\pmb{x},0)=J(\pmb{\varphi}_0)\\
&\int_{\mathrm{\Omega}_t} \dfrac{1}{f(\pmb{x},t)}d\pmb{x} = |\mathrm{\Omega}_0|.
\end{aligned}$$
A new (differ from $\pmb{\varphi}_0$) diffeomorphism $\pmb{\phi}(\pmb{\xi},t):\mathrm{\Omega}_0\rightarrow\mathrm{\Omega}_t$, such that $J(\pmb{\phi}(\pmb{\xi},t)) =\text{det}\nabla(\pmb{\phi}(\pmb{\xi},t)) = f(\pmb{\phi}(\pmb{\xi},t),t)$, $\forall t \in [0,1]$, can be constructed the following two steps:
- First, determine $\pmb{u}(\pmb{x},t)$ on $\mathrm{\Omega}_t$ by solving $$\label{General2}
\left\{
\begin{aligned}
\text{div } \pmb{u}(\pmb{x},t)& = -\frac{\partial}{\partial t}(\dfrac{1}{f(\pmb{x},t)}) \\
\text{curl } \pmb{u}(\pmb{x},t)& = 0\\
\pmb{u}(\pmb{x},t)& = \dfrac{\pmb{v}(\pmb{x},t)}{f(\pmb{x},t)} \text{, on } \partial\mathrm{\Omega}_t
\end{aligned}\right.$$
- Second, determine $\pmb{\phi}(\pmb{\xi},t)$ on $\mathrm{\Omega}_0$ by solving $$\label{General3}
\left\{
\begin{aligned}
\frac{\partial \pmb{\phi}(\pmb{\xi},t)}{\partial t}& = f(\pmb{\phi}(\pmb{\xi},t),t) \pmb{u}(\pmb{\phi}(\pmb{\xi},t),t), \\
\pmb{\phi}(\pmb{\xi},0)& = \pmb{\varphi}_{0}(\pmb{\xi})
\end{aligned}\right.$$
For computational simplicity system (\[General2\]) is modified into a Poisson equation as follows. Let $\pmb{u}(\pmb{x},t)= \nabla \pmb{w}(\pmb{x},t)$, then $$\label{General4}
\Delta \pmb{w}(\pmb{x},t) = \text{div } \nabla \pmb{w}(\pmb{x},t) = \text{div } \pmb{u}(\pmb{x},t) = -\frac{\partial}{\partial t}(\dfrac{1}{f(\pmb{x},t)})$$ Hence, determination of $\pmb{u}(\pmb{x},t)$ is depended on $\pmb{w}(\pmb{x},t)$, which $\pmb{w}(\pmb{x},t)$ is cheaper to find.
Experiments of Recovering Transformations
-----------------------------------------
Here we demonstrate some simulations of the deformation method. With both the Jacobian and curl term here in our algorithm, we can recover a transformation, which proves the algorithm is reliable and accurate in grid generation.
### 2D Example {#d-example .unnumbered}
We design a recovering experiment to test the accuracy of our method and discover more details insides. We give the grid image of the map of (a)Guangdong, China, (b)Texas, USA and (c)a dog. All images are given a nonlinear transformation $\mathbf{T_0}$ from the square($65\times65$)(phi1 and phi2).
### 3D Example {#d-example-1 .unnumbered}
We also design a recovering experiment to brain MRI images and display them in three dimensions. Given a nonlinear transformation $\mathbf{T_0}$ from the cube $[1,176]\times[1,256]\times[1,256]$ (phi1, phi2 and phi3) to itself, we want to reconstruct $\mathbf{T_0}$ from its jacobian determinant and curl. We show the grid image from full scale view, half cut on x, y, z-axis view respectively.
Experiments of Grid Images Based on Brightness and Gradient
-----------------------------------------------------------
In this experiment, we apply the method in 3.1 and 3.2 based on the brightness and its gradient of medical images. We generated corresponding grid images by calculating the brightness and gradient of brightness of brain MRI images, which further demonstrated that different brain MRI images could be differentiated by differential geometry. Figure 3(a) shows the original image(size 256$\times$256$\times$3).Figure 3(b)(c)(d) represent the grid images based on only gradient of brightness, only brightness(intensity) and both brightness and its gradient respectively.We can see that they reflect the morphological characteristics of CSF, GM, WM in MRI brain image. We used the differential geometric features of JD and CV as a method to measure the differences between different MRI images, and use this as CNN input to get more accurate results of brain segmentation.\
{width="0.8\linewidth"}
Experiments of generated images based on JD and CV
--------------------------------------------------
Based on the above finite difference and differential geometry theory, we use MATLAB to generate the grid images based on JD and CV of brain MRI images as 3.3 discribes.And we extract the images formed by JD and CV information from the grid images above, and saved it as nii image format. The following figures show in order are the original T1 image, the image formed by JD and the image formed by CV, which present geometric deformation features of the image, especially highlight the change of morphological features of CSF, GM, WM.\
{width="1.\linewidth"}
Multi-modalities Information
----------------------------
In medical image analysis, 3D volumetric data is usually obtained in a variety of ways for robust detection of different organizational structures. For example, three modalities including T1, T1-weighted inversion recovery (T1-IR) and T2-FLAIR are usually available in brain structure segmentation task [@Mendrik2015MRBrainS]. T1 image has good anatomical structure and T2 image can show good tissue lesions, T1-IR image has strong T1 contrast characteristics, and T2-FLAIR is often used to inhibit CSF issues. The main reason for obtaining multimodal images is that the information of multimodal data sets can supplement each other and provide robust diagnostic results. Thus, we concatenate these multi-modality data with JD and CV data as input, then in the process of network training, complementary information is combined and fused in an implicit way, which is more consistent than any single training method. The following figure shows each modality of brain image and the image formed by its JD and CV information.\
{width="1.\linewidth"}
Proposed Framework
------------------
Figure 6 shows the process framework of our proposed method. All modalities should be skull stripped and we can extract differential geometric features including the Jacobian determinant(JD) and the curl vector(CV) derived from T1 modality. After all modalities, including image labels(ground truth) have preprocessed, three modalities including T1-weighted, T1-IR and T2-FLAIR images with JD or CV image will be concatenated together as a new multi-modality. And it will be used as input of VoxResNet network. After training and testing, the predicted result with Ground Truth will be calculated for each tissue type(CSF, GM and WM), respectively, including DSC (Dice coefficient), HD ( Hausdorff distance) and AVD (Absolute Volume Difference) as evaluation criteria of segmentation. We will perform all the experiments based on this process.\
{width="1.2\linewidth"}
Results
=======
Datasets and Pre-processing
---------------------------
### Datasets
**$\mathbf{IBSR}$** The Internet Brain Segmentation Repository (IBSR) provides manually-guided expert segmentation results along with magnetic resonance brain image data. Its purpose is to encourage the evaluation and development of segmentation methods. The dataset consists of 18 MRI volumes and the corresponding ground truth (GT) is provided. The data can be downloaded from [^2].\
**$\mathbf{MRBrainS}$** The aim of MRBrainS challenge is to segment brain into four-class structures, namely background, cerebrospinal fluid (CSF), gray matter (GM) and white matter (WM). Multi-sequence 3T MRI brain scans, including T1-weighted, T1-IR and T2-FLAIR are provided for each subject. Five brain MRI scans with manual segmentations are provided for training and 15 only MRI scans are provided for testing. The data can be downloaded from [^3] for MRBrainS18 dataset and [^4] for MRBrainS13 dataset.\
### Image Preprocessing
Typical preprocessing steps for structural brain MRI include the following key steps [@Akkus2017Deep]: registration, skull stripping, bias field correction, intensity normalization and noise reduction. All the datasets we use are already skull stripped. In this paper, we subtract Gaussian smoothed image, and apply intensity normalization(z-scores) and Contrast-Limited Adaptive Histogram Equalization (CLAHE) for enhancing local contrast by [@Stollenga2015Parallel] in the pre-processing step. Then multiple input volumes pre-processed were used as input data in our experiments.\
Segmentation Evaluation Criteria
--------------------------------
In this paper, three metrics are used to evaluate the segmentation result [@Deng2018A] [@Sun2018A] : DSC (Dice coefficient), HD ( Hausdorff distance) and AVD (Absolute Volume Difference), which are calculated for each tissue type(CSF, GM and WM), respectively. First, DSC is the most widely used metric in the evaluation of medical volume segmentations. In addition to the direct comparison between automatic and ground truth segmentations, it is common to use the DSC to measure reproducibility (repeatability) [@Taha2015Metrics]. DSC is computed by:
$$DSC=\frac{2\times{TP}}{2\times{TP}+FP+FN}$$
Where TP, FP and FN are the subjects of true positive, false positive and false negative predictions for the considered class.\
Second, the distance between crisp volumes (HD) between two finite point sets A and B is defined by:
$$H(A,B)=max(h(A,B),h(B,A))$$
Where $h(A,B)$ is called the directed HD and given by: $$\begin{gathered}
h(A,B)=max(a \in A)min(b \in B)\parallel a-b \parallel \\
h(B,A)=max(b \in B)min(a \in A)\parallel b-a \parallel\end{gathered}$$
Finally, the AVD is defined by:
$$AVD(A,B)=\frac{\parallel A-B \parallel}{\Sigma A}$$
Where A is ground truth and B is predicted volume of one class.
Experimental Results
--------------------
### Experiments on MRBrainS
Regarding the evaluation of testing data, we compared our method with other modalities based on the network VoxResNet, including single modality(T1-weighted), three modalities(T1-weighted, T1-IR and T2-FLAIR), three modalities+JD modality, three modalities+CV modality and three modalities+JD+CV modality. In this experiment , we use subject 4,5,7,14,070 of MRBrainS18 as training set and subject 1,148 as testing set. We can see that combining the multi-modality information can improve the segmentation performance than that of single image modality. Three modalities gain a little improvement over single modality. While three modalities combined with JD, CV or both have excellent improvement over single or three modalities and three modalities with JD increases average Dice by about 1.5%(0.8529 and 0.8677), especially gain much higher improvement by about 3.7% increase (0.8490 and 0.8860) in WM tissue. This also can be seen from the improvement of HD and AVD(the smaller, the better) from the Table 2.\
[|\*[10]{}[c|]{}]{} & & &\
& CSF & GM & WM & CSF & GM & WM & CSF & GM & WM\
Single M(T1) & 0.8778 & 0.8138 & 0.8461 & 1 & 1.2071 & 1.9319 & **0.0373** & 0.0392 & 0.0751\
Three M & 0.8926 & 0.8172 & 0.8490 & 1 & 1.2071 & 1.8251 & 0.0458 & 0.0411 & 0.0937\
Three M+JD & **0.8939** & 0.8232 & $\textbf{0.8860}$ & $\textbf{1}$ & $\textbf{1}$ & $\textbf{1.2071}$ & 0.0457 & $\textbf{0.0331}$ & 0.0484\
Three M+CV & 0.8920 & 0.8217 & 0.8859 & 1 & 1 & 1.4142 & 0.0455 & 0.0373 & $\textbf{0.0457}$\
Three M+JV & 0.8914 & $\textbf{0.8329}$ & 0.8701 & 1 & 1 & 1.5 & 0.0430 & 0.0508 & 0.0511\
From the Table 2, we can see the subject 1(Dice:$\mathbf{0.8760}$, HD:$\mathbf{1.0829}$, AVD:$\mathbf{0.0294}$) of testing set obtain higher improvement than subject 148 in the performance in brain segmentation. And the experiment of three modalities+JD gains the best result with average Dice, HD, AVD by 0.8677,1.0690 and 0.0424.But three modalities combined with both JD and CV have a little decrease under three modalities combined with either JD or CV, maybe because of overfitting.\
[|\*[10]{}[c|]{}]{} & & &\
& 1 & 148 & average & 1 & 148 & average & 1 & 148 & average\
Single M(T1) & 0.8720 & 0.8198 & 0.8459 & 1.1381 & 1.6212 & 1.3797 & $\textbf{0.0156}$ & 0.0855 & 0.0505\
Three M & $\textbf{0.8796}$ & 0.8263 & 0.8529 & 1.1381 & 1.5501 & 1.3441 & 0.0273 & 0.0931 & 0.0602\
Three M+JD & 0.8784 & 0.8570 & $\textbf{0.8677}$ & $\textbf{1}$ & $\textbf{1.1381}$ & $\textbf{1.0690}$ & 0.0333 & 0.0515 & $\textbf{0.0424}$\
Three M+CV & 0.8742 & $\textbf{0.8589}$ & 0.8665 & 1.1381 & $\textbf{1.1381}$ & 1.1381 & 0.0366 & $\textbf{0.0491}$ & 0.0428\
Three M+JV & 0.8756 & 0.8540 & 0.8648 & $\textbf{1}$ & 1.3333 & 1.1667 & 0.0341 & 0.0626 & 0.0483\
average & $\textbf{0.8760}$ & 0.8432 & 0.8596 & $\textbf{1.0829}$ & 1.3562 & 1.2195 & $\textbf{0.0294}$ & 0.0684 & 0.0488\
Moreover, We use subject 1,3,4 of MRBrainS13 as training set and subject 2,5 as testing set and its results are shown in Table 3 and Table 4. We can get the same results as the MRBrainS18 dataset. The experiment of three modalities+CV gain the best Dice by 0.8272 and increase 1.7% over only three modalities(Dice:0.8100).The subject 5 gain higher average Dice(0.8310) and HD(1.6697) but lower AVD(0.0607) than subject 2. We also compare the predicted results of subject 5 from the MRBrainS13 dataset and can be seen in Figure 8. We can get the same results as we discuss above. Three modalities combined with JD, CV or both have excellent improvement over only single or three modalities.\
[|\*[10]{}[c|]{}]{} & & &\
& CSF & GM & WM & CSF & GM & WM & CSF & GM & WM\
Single M(T1) & 0.8206 & 0.7855 & 0.7907 & 1.7071 & 1.5731 & 2.5 & 0.1072 & 0.1204 & 0.0751\
Three M & 0.8167 & 0.8115 & 0.8018 & $\textbf{1.4142}$ & $\textbf{1.4142}$ & $\textbf{2.4142}$ & 0.0574 & 0.0509 & 0.0679\
Three M+JD & 0.8255 & 0.7984 & $\textbf{0.8078}$ & 1.7321 & 1.5731 & 2.5 & 0.1042 & 0.1067 & $\textbf{0.0084}$\
Three M+CV & 0.8394 & $\textbf{0.8367}$ & 0.7985 & $\textbf{1.4142}$ & $\textbf{1.4142}$ & 2.5 & $\textbf{0.0369}$ & $\textbf{0.0098}$ & 0.0112\
Three M+JV & $\textbf{0.8406}$ & 0.8343 & 0.8068 & $\textbf{1.4142}$ & $\textbf{1.4142}$ & 2.5811 & 0.0573 & 0.0310 & 0.0355\
[|\*[10]{}[c|]{}]{} & & &\
& 2 & 5 & average & 2 & 5 & average & 2 & 5 & average\
Single M(T1) & 0.7803 & 0.8176 & 0.7989 & 2.0488 & 1.8047 & 1.9267 & 0.0962 & 0.1057 & 0.1009\
Three M & 0.7975 & 0.8224 & 0.8100 & $\textbf{1.8856}$ & $\textbf{1.6095}$ & $\textbf{1.7475}$ & 0.0536 & 0.0639 & 0.0587\
Three M+JD & 0.7919 & 0.8292 & 0.8106 & 2.1547 & 1.7154 & 1.9351 & 0.0715 & 0.0747 & 0.0731\
Three M+CV & $\textbf{0.8078}$ & 0.8419 & 0.8249 & 1.9428 & 1.6095 & 1.7761 & $\textbf{0.0154}$ & $\textbf{0.0233}$ & $\textbf{0.0193}$\
Three M+JV & 0.8072 & $\textbf{0.8437}$ & $\textbf{0.8272}$ & 1.9969 & 1.6095 & 1.8032 & 0.0465 & 0.0360 & 0.0413\
average & 0.7969 & $\textbf{0.8310}$ & 0.8143 & 2.0058 & $\textbf{1.6697}$ & 1.8377 & $\textbf{0.0566}$ & 0.0607 & 0.0587\
From the red box in the Figure 7, We can see that (b)Single modality and (d)Three modalities+JD have a better result in WM and GM issue segmentation than other experiments. Other predicted results comparison of different experiments also can be seen in the Figure 7.
{width="1.\linewidth"}
### Experiments on IBSR
In this experiment, we use subject 10,11,12,13,14 of IBSR as training set and subject 1-9 and 15-18 as testing set. we also compared our method with other modalities based on the network VoxResNet, including single modality(T1-weighted), single modality+JD modality, single modality+CV modality and single modalities+JD+CV modality. We can see that combining single modality with JD or CV information can improve the segmentation performance than that of single modality.\
[|\*[10]{}[c|]{}]{} & & &\
& CSF & GM & WM & CSF & GM & WM & CSF & GM & WM\
Single M(T1) & 0.7158 & 0.9137 & 0.9023 & 19.1106 & $\textbf{1.7693}$ & 1.5159 & 0.1877 & 0.0384 & 0.0461\
Single M+JD & $\textbf{0.7926}$ & $\textbf{0.9140}$ & 0.9008 & $\textbf{4.6863}$ & 1.7904 & 1.6379 & $\textbf{0.1584}$ & $\textbf{0.0296}$ & $\textbf{0.0450}$\
Single M+CV & 0.7571 & 0.9113 & $\textbf{0.9024}$ & 6.8682 & 1.8951 & $\textbf{1.5101}$ & 0.1973 & 0.0327 & 0.0466\
Single M+JV & 0.7612 & 0.9139 & 0.8888 & 5.0406 & 2.0204 & 1.8391 & 0.1879 & 0.0449 & 0.0652\
Single modality combined with JD, CV or both have improvement over single modality and single modality with JD increases average Dice by about 2.5%($\mathbf{0.8439}$ and $\mathbf{0.8691}$), especially gain much higher improvement by about 7.7% increase ($\mathbf{0.7158}$ and $\mathbf{0.7926}$) in CSF tissue. This also can be seen from the improvement of HD($\mathbf{19.1106}$ and $\mathbf{4.6863}$)from the Table 5. But three modalities combined with both JD and CV also have a little decrease under three modalities combined with either JD or CV. From the Table 6, single modality with JD gets the best result with $\mathbf{0.8691}$,$\mathbf{2.7049}$,$\mathbf{0.0777}$ for average Dice, HD, AVD respectively.
[|\*[4]{}[c|]{}]{} & & &\
Single modality(T1) & 0.8439 & 7.4653 & 0.0907\
Single modality+JD & $\textbf{0.8691}$ & $\textbf{2.7049}$ & $\textbf{0.0777}$\
Single modality+CV & 0.8569 & 3.4245 & 0.0922\
Single modality+JV & 0.8546 & 2.9667 & 0.0993\
### Comparison with Other Experiments on MRBrainS18
We also implement an additional experiment on the MRBrainS18 dataset. And we use subject 4,5,148,070 of MRBrainS18 as training set and subject 1,7,14 as testing set. Single modality with CV has the best results(Dice:$\mathbf{0.8710}$, HD:$\mathbf{1.0460}$, AVD:$\mathbf{0.0308}$), we can also see that combining single modality with JD or CV information can improve the segmentation performance than that of single modality. Single modality combined with JD, CV or both have a little improvement or not worse than three modalities(0.8619,0.8608,0.8710), which means one modality combined with its JD or CV information can replace the segmentation effect of three modalities, and patients only need to extract MRI images of T1 modality for diagnosis, which provides medical conveniences for doctor to diagnose.\
[|\*[4]{}[c|]{}]{} & & &\
Single modality(T1) & 0.8592 & 1.0920 & 0.0546\
Three modalities & 0.8619 & 1.1381 & 0.0630\
Single modality+JD & 0.8608 & 1.0920 & 0.0517\
Single modality+CV & $\textbf{0.8710}$ & $\textbf{1.0460}$ & $\textbf{0.0308}$\
Single modality+JV &0.8634 & 1.1381 & 0.0333\
In Table 8, the subject 14 of testing set has better improvement results with average result (Dice:$\mathbf{0.8750}$, HD:$\mathbf{1.0552}$, AVD:$\mathbf{0.0418}$)than subject 1 and 7.
[|\*[10]{}[c|]{}]{} & & &\
& 1 & 7 & 14 & 1 & 7 & 14 & 1 & 7 & 14\
Single M(T1) & 0.8543 & 0.8445 & $\textbf{0.8788}$ & 1.1381 & 1.1381 & $\textbf{1}$ & 0.0494 & 0.0711 & 0.0432\
Three M & 0.8514 & 0.8584 & 0.8759 & 1.1381 & 1.1381 & 1.1381 & 0.0890 & 0.0558 & 0.0442\
Single M+JD & 0.8682 & 0.8467 & 0.8674 & 1.1381 & 1.1381 & $\textbf{1}$ & 0.0295 & 0.0625 & 0.0631\
Single M+CV & $\textbf{0.8736}$ & $\textbf{0.8627}$ & 0.8766 & $\textbf{1}$ & $\textbf{1.1381}$ & $\textbf{1}$ & $\textbf{0.0191}$ & 0.0367 & 0.0366\
Single M+JV & 0.8618 & 0.8522 & 0.8762 & 1.1381 & 1.1381 & 1.1381 & 0.0442 & $\textbf{0.0337}$ & $\textbf{0.0220}$\
average & 0.8619 & 0.8529 & $\textbf{0.8750}$ & 1.1105 & 1.1381 & $\textbf{1.0552}$ & 0.0462 & 0.0520 & $\textbf{0.0418}$\
### Comparison between VoxResNet and U-Net on MRBrainS
In the experiment, we compare the segmentation performance of our method in two networks, VoxResNet and U-Net network. We also use subject 4,5,7,14,070 of MRBrainS18 as training set and subject 1,148 as testing set. The results show VoxResNet has a little better performance with average results(Dice:$\mathbf{0.8596}$, HD:$\mathbf{1.2195}$, AVD:$\mathbf{0.0488}$)than U-Net network with our method in brain MRI segmentation and increase average Dice by about 1% and this result can be also seen from HD and AVD from Table 9, especially in the case of three modalities with JD, CV or both. While VoxResNet has weaker performance in the case of single or three modalities.\
[|\*[7]{}[c|]{}]{} & & &\
& VoxResNet & U-Net & VoxResNet & U-Net & VoxResNet & U-Net\
Single modality(T1) & 0.8459 & $\textbf{0.8669}$ & $\textbf{1.3797}$ & 1.9917 & 0.0505 & $\textbf{0.0397}$\
Three modalities & 0.8529 & $\textbf{0.8548}$ & $\textbf{1.3441}$ & 2.2270 & 0.0602 & $\textbf{0.0479}$\
Three modalities+JD & $\textbf{0.8677}$ & 0.8526 & $\textbf{1.0690}$ & 2.1578 & $\textbf{0.0424}$ & 0.0464\
Three modalities+CV & $\textbf{0.8665}$ & 0.8547 & $\textbf{1.1381}$ & 2.1075 & $\textbf{0.0428}$ & 0.0506\
Three modalities+JV & $\textbf{0.8648}$ & 0.8566 & $\textbf{1.1667}$ & 2.0191 & 0.0483 & $\textbf{0.0468}$\
average & $\textbf{0.8596}$ & 0.8571 & $\textbf{1.2195}$ & 2.1006 & 0.0488 & $\textbf{0.0463}$\
We also get the same results based on MRBrainS13 by using subject 1,3,4 of MRBrainS13 as training set and subject 2,5 as testing set from Table 9.But VoxResNet has much higher performance with average results(Dice:$\mathbf{0.8143}$, HD:$\mathbf{1.8377}$, AVD:$\mathbf{0.0587}$)than U-Net network(Dice:$\mathbf{0.5635}$, HD:$\mathbf{4.7994}$, AVD:$\mathbf{0.2385}$) with our method in MRBrainS13.
[|\*[7]{}[c|]{}]{} & & &\
& VoxResNet & U-Net & VoxResNet & U-Net & VoxResNet & U-Net\
Single modality(T1) & $\textbf{0.7989}$ & 0.6469 & $\textbf{1.9267}$ & 3.9520 & $\textbf{0.1009}$ & 0.1297\
Three modalities & $\textbf{0.8100}$ & 0.5448 & $\textbf{1.7475}$ & 4.8834 & $\textbf{0.0587}$ & 0.2530\
Three modalities+JD & $\textbf{0.8106}$ & 0.5438 & $\textbf{1.9351}$ & 4.9356 & $\textbf{0.0731}$ & 0.2575\
Three modalities+CV & $\textbf{0.8249}$ & 0.5220 & $\textbf{1.7761}$ & 5.2546 & $\textbf{0.0193}$ & 0.2899\
Three modalities+JV & $\textbf{0.8272}$ & 0.5601 & $\textbf{1.8032}$ & 4.9716 & $\textbf{0.0413}$ & 0.2626\
average & $\textbf{0.8143}$ & 0.5635 & $\textbf{1.8377}$ & 4.7994 & $\textbf{0.0587}$ & 0.2385\
Implementation Details
======================
Our method was implemented using MATLAB and a flexible framework neural networks named Chainer in Python. we used MATLAB to generate the images formed by JD and CV information based on brain MRI images, and saved it as nii image format. It took about 8 hours to train the network while less than 3 minutes for processing each test volume(size $240\times240\times48$) using one NVIDIA Quadro P2000 GPU. Due to the limited GPU memory, we cropped volumetric regions(size $80\times80\times80\times{m}$, m is the number of image modalities and set as 2,6,8,8,10 for single modality, three modalities, three modalities+JD, three modalities+CV, three modalities+JD+CV respectively in our experiments) for the input into the network. This was implemented in an on-the-fly way during the training and the probability map of whole volume was generated in an overlap-tiling strategy for stitching the sub-volume results.\
Conclusions
===========
In this paper, we use the differential geometric information including JD and CV derived from T1 modality as MRI image features, and use them as one CNN channel with other three modalities (T1-weighted, T1-IR and T2-FLAIR) to get more accurate results of brain segmentation. We test this method on two datasets including IBSR dataset and MRBrainS datasets based on VoxResNet network, and obtain excellent improvement on the two datasets. Moreover, we discuss that one modality combined with its JD or CV information can replace the segmentation effect of three modalities, which can provide medical conveniences for doctor to diagnose. Finally, we compare the segmentation performance of our method in two networks, VoxResNet and U-Net network. We believe the proposed method can advance the performance in brain segmentation and clinical diagnosis. In the future, we will investigate the performance of our method on more object detection and segmentation tasks from 3D volumetric data.\
Acknowledgments
===============
The authors would like to thank Yang Deng and Yao Sun of Graduate School at Shenzhen, Tsinghua University, Mingwang Zhu of Beijing Sanbo Brain Hospital for their technical support.\
Appendix: Other Experimental Results on MRBrainS
================================================
We change the training set and testing set of MRBrainS18 dataset and the results in Table 11 and 12 show our method has the same improvement as previous experiments in 4.3.1. The best results of different experiments are colored by black. The experiment of three modalities with JD has the best result with average Dice, HD, AVD by 0.8779, 1, 0.0297 respectively(not shown in Table 11).
[|\*[10]{}[c|]{}]{} & & &\
& CSF & GM & WM & CSF & GM & WM & CSF & GM & WM\
Single M(T1) & 0.8745 & 0.8298 & 0.8733 & 1 & 1 & 1.2761 & 0.0459 & 0.0330 & 0.0849\
Three M & 0.8796 & 0.8436 & 0.8624 & 1 & 1 & 1.4142 & 0.0580 & 0.0766 & 0.0543\
Three M+JD & $\textbf{0.8934}$ & $\textbf{0.8531}$ & $\textbf{0.8872}$ & $\textbf{1}$ & $\textbf{1}$ & $\textbf{1}$ & 0.0413 & $\textbf{0.0224}$ & $\textbf{0.0255}$\
Three M+CV & 0.8927 & 0.8371 & 0.8764 & 1 & 1 & 1.2761 & $\textbf{0.0408}$ & 0.0362 & 0.0684\
Three M+JV & 0.8911 & 0.8481 & 0.8854 & 1 & 1 & 1.1381 & 0.0442 & 0.0245 & 0.0476\
[|\*[10]{}[c|]{}]{} & & &\
& 1 & 7 & 14 & 1 & 7 & 14 & 1 & 7 & 14\
Single M(T1) & 0.8543 & 0.8445 & 0.8788 & 1.1381 & 1.1381 & $\textbf{1}$ & 0.0494 & 0.0711 & 0.0432\
Three M & 0.8514 & 0.8584 & 0.8759 & 1.1381 & 1.1381 & 1.1381 & 0.0890 & 0.0558 & 0.0442\
Three M+JD & $\textbf{0.8775}$ & $\textbf{0.8724}$ & 0.8837 & $\textbf{1}$ & $\textbf{1}$ & $\textbf{1}$ & $\textbf{0.0279}$ & $\textbf{0.0291}$ & 0.0323\
Three M+CV & 0.8664 & 0.8573 & 0.8824 & 1.1381 & 1.1381 & $\textbf{1}$ & 0.0373 & 0.0586 & 0.0495\
Three M+JV & 0.8721 & 0.8623 & $\textbf{0.8902}$ & $\textbf{1}$ & 1.1381 & $\textbf{1}$ & 0.0416 & 0.0453 & $\textbf{0.0294}$\
average & 0.8643 & 0.8590 & $\textbf{0.8822}$ & 1.0829 & 1.1105 & $\textbf{1.0276}$ & 0.0490 & 0.0520 & $\textbf{0.0397}$\
Table 13 and Table 14 show Single modality combined with JD, CV or both have a little improvement or not worse than three modalities of MRBrainS13 dataset as 4.3.3 describes. And the subject 5 of testing set has better improvement results than subject 1 with average Dice, HD, AVD by 0.8493, 1.5640, 0.0385 respectively from Table 14.
[|\*[4]{}[c|]{}]{} & & &\
Single modality(T1) & 0.7840 & 2.1979 & 0.0571\
Three modalities & 0.7856 & 2.0426 & 0.0590\
Single modality+JD & $\textbf{0.7889}$ & 2.0242 & 0.0713\
Single modality+CV & 0.7870 & $\textbf{1.9538}$ & $\textbf{0.0512}$\
Single modality+JV & 0.7805 & 2.1343 & 0.0734\
[|\*[7]{}[c|]{}]{} & & &\
& 1 & 5 & 1 & 5 & 1 & 5\
Single modality(T1) & 0.7148 & $\textbf{0.8531}$ & 2.7864 & 1.6095 & 0.0748 & 0.0393\
Three modalities & 0.7183 & 0.8529 & 2.6139 & $\textbf{1.4714}$ & 0.1056 & $\textbf{0.0123}$\
Single modality+JD & 0.7258 & 0.8521 & 2.4389 & 1.6095 & 0.1014 & 0.0412\
Single modality+CV & $\textbf{0.7282}$ & 0.8459 & $\textbf{2.3874}$ & 1.5202 & $\textbf{0.0492}$ & 0.0533\
Single modality+JV & 0.7184 & 0.8425 & 2.6593 & 1.6095 & 0.1005 & 0.0463\
average & 0.7211 & $\textbf{0.8493}$ & 2.5772 & $\textbf{1.5640}$ & 0.0863 & $\textbf{0.0385}$\
[^1]: <https://www.nitrc.org/projects/ibsr>
[^2]: <https://www.nitrc.org/frs/?group_id=48>
[^3]: <http://mrbrains18.isi.uu.nl/>
[^4]: <http://mrbrains13.isi.uu.nl/>
|
---
abstract: 'In random matrix theory, the spacing distribution functions $p^{(n)}(s)$ are well fitted by the Wigner surmise and its generalizations. In this approximation the spacing functions are completely described by the behavior of the exact functions in the limits $s\rightarrow0$ and $s\rightarrow\infty$. Most non equilibrium systems do not have analytical solutions for the spacing distribution and correlation functions. Because of that, we explore the possibility to use the Wigner surmise approximation in these systems. We found that this approximation provides a first approach to the statistical behavior of complex systems, in particular we use it to find an analytical approximation to the nearest neighbor distribution of the annihilation random walk.'
author:
- |
Diego Luis González[^1]\
Gabriel Téllez[^2]\
Departamento de Física, Universidad de Los Andes\
A. A. 4976 Bogotá, Colombia.
title: Wigner Surmise For Domain Systems
---
[**Keywords:**]{} Systems out of equilibrium, random matrices, Wigner surmise.\
Introduction
============
In random matrix theory the analytic expressions for the spacing distribution functions of eigenvalues $p^{(n)}(s)$ in the circular and Gaussian orthogonal ensembles (COE and GOE respectively) in the limit of large matrices are given in terms of the eigenvalues $\mu_i$ and eigenfunctions $f_i(x)$ of the following integral equation, see Ref. [@mehta]:
$$\mu_i f_i(x)=\int^{1}_{-1}e^{i\pi x y s/2}f_i(y)dy.$$
The spacing distributions are calculated explicitly by using
$$E(2r,s)=\prod^{\infty}_{i=0}(1-\lambda_{2i})\sum_{0\leq j_1<j_2<\cdots<j_r}\prod^{r}_{i=1}\left(\frac{\lambda_{j_i}}{1-\lambda_{j_i}}\right)\times\left[1-(b_{j_1}+\cdots+b_{j_r})\right],$$
$$E(2r-1,s)=\prod^{\infty}_{i=0}(1-\lambda_{2i})\sum_{0\leq j_1<j_2<\cdots<j_r}\prod^{r}_{i=1}\left(\frac{\lambda_{j_i}}{1-\lambda_{j_i}}\right)\times(b_{j_1}+\cdots+b_{j_r}),$$
where
$$b_j=f_{2j}(1)\int^{1}_{-1}f_{2j}(x)dx/\int^{1}_{-1}f^{2}_{2j}(x)dx,$$
$$\lambda_{j}=s \left|\mu_j\right|^2/4,$$
and
$$p^{(n)}(s)=\frac{d^2}{ds^2}\sum^{n}_{j=0}(n-j+1)E(j,s).$$
These expressions are difficult to manage, however in Ref. [@abdul], the authors find an excellent approximation for spacing distributions $p^{(n)}(s)$ from their well-known behavior in the limits $s\rightarrow 0$ and $s\rightarrow\infty$. This approximation is easy to use and provide an excellent fit to the exact distributions. We will use this approximation many times in this paper, because of that, we summarize now its most important aspects.
By definition, $p^{(n)}(s)$ is the probability density that an interval of length $s$ which starts at a level contains exactly $n$ levels and the next, the $n+1$ level, is in $[s,s+ds]$. In the same way, let $F^{(n)}(s)$ be the probability that an interval of length $s$ which starts at a level, contains $n$ levels. By using this definition we can write
$$F^{(n)}(s)=\int^{\infty}_{s}\left(p^{(n)}(s')-p^{(n-1)}(s')\right)ds'.$$
Additionally, let $r^{(n)}(s)$ be the probability density that an interval $[0,s]$ which starts at a level at $s=0$ is limited by a level on its right side, under the condition that there are exactly $n$ levels in the interval $\left(0,s\right)$, i.e., $r^{(n)}(s)$ is the conditional probability $$r^{(n)}(s)=\frac{p^{(n)}(s)}{F^{(n)}(s)},$$ this probability is called level repulsion function. Following Ref. [@abdul], in the limit $s\rightarrow 0$, this equation can be written as
$$\label{pitera}
p^{(n)}(s)=r^{(n)}(s)\int^{s}_{0}p^{(n-1)}(s')ds'.$$
In the GOE ensemble the matrix elements are chosen using a Gaussian distribution, this fact suggest that $p^{(n)}(s)$ decays as Gaussian function. The appropriate function for fit is$$\label{surmise}
p^{(n)}(s)=A_{n}s^{\alpha_{n}}e^{-B_{n}s^2},$$ under the surmise $r^{(n)}(s)\rightarrow s^{n+1}$ with $s\rightarrow 0$. Additionally, the functions $p^{(n)}(s)$ satisfy the normalization conditions
$$\label{pcond1}
\int^{\infty}_{0}p^{(n)}(s)ds=1,$$
and $$\label{pcond2}
\int^{\infty}_{0}s p^{(n)}(s)ds=1.$$
By using the surmise for the level repulsion and the normalization conditions, is straightforward to find [@abdul]
$$\label{a}
A_{n}=2\frac{B_{n}^{(\alpha_{n}+1)/2}}{\Gamma\left(\frac{\alpha_{n}+1}{2}\right)},$$
$$\label{b}
B_{n}=\left[\frac{\Gamma\left(\frac{\alpha_{n}}{2}+1\right)}{(n+1)\Gamma\left(\frac{\alpha_{n}+1}{2}\right)}\right]^2,$$
where
$$\alpha_{n}=n+\frac{(n+1)(n+2)}{2}.$$
Then, the approximate spacing distribution functions $p^{(n)}(s)$ are given explicitly by $$\label{pwignersurmise}
p^{(n)}(s)=\left[\frac{\Gamma\left(\frac{\alpha_{n}}{2}+1\right)}{(n+1)}\right]^{\alpha_{n}+1}\frac{2 s^{\alpha_{n}}}{\Gamma\left(\frac{\alpha_{n}+1}{2}\right)^{\alpha_{n}+2}}e^{-\left[\frac{\Gamma\left(\frac{\alpha_{n}}{2}+1\right)}
{(n+1)\Gamma\left(\frac{\alpha_{n}+1}{2}\right)}\right]^2 s^2}.$$
The result obtained for $\alpha_{n}$ coincides with the results obtained by using the exact expression for the spacing distribution functions, see Ref. [@mehta]. Notice that the approximate spacing distributions functions are characterized by the level repulsion, normalization condition, scaling condition for the average spacing and Gaussian decay. This approximation is called generalized Wigner surmise and provides a very good approximation for $p^{(n)}(s)$, because it reproduce not only the distributions behavior in the limits $s\rightarrow 0$ and $s\rightarrow \infty$, but also reproduce their global behavior, as we can see in figure \[coe\]. In particular the function with $n=0$ is called Wigner distribution. This fit allow us calculate also the approximate pair correlation distribution $g(r)$. For this purpose we use
$$\label{pkycdepares1}
g(r)=\sum^{\infty}_{n=0}p^{\left(n\right)}(r)
\,,$$
then
$$\label{gcoeap}
g(r)=2\sum^{\infty}_{n=0}\left[\frac{\Gamma\left(\frac{\alpha_{n}}{2}+1\right)}{(n+1)}\right]^{\alpha_{n}+1}\frac{r^{\alpha_{n}}}{\Gamma\left(\frac{\alpha_{n}+1}{2}\right)^{\alpha_{n}+2}}e^{-\left[\frac{\Gamma\left(\frac{\alpha_{n}}{2}+1\right)}{(n+1)\Gamma\left(\frac{\alpha_{n}+1}{2}\right)}\right]^2 r^2}
\,.$$
In figure \[coe\] we can see that this is a good approximation for $g(r)$, however, it is not as useful as the Wigner surmise for $p^{(0)}(s)$ because the exact expression for $g(r)$ is well known and easy to use, see Ref. [@mehta].
In Ref. [@gonzalez] the authors study the statistical behavior of several out of equilibrium domain systems which evolve with formation of domains which grow in time. For intermediate times where the size of the domains is much smaller than the total size $L$ of the system, the domain size distribution exhibit a dynamic scaling. The authors studied the statistical properties of these domains in the scaling regime. They found that the statistical behavior of those is similar to the one in random matrices, for example, the nearest neighbor distribution $p^{(0)}(s)$ of several out of equilibrium domain systems is well fitted by the Wigner surmise which also describe closely the distribution $p^{(0)}(s)$ in the case of the circular and Gaussian orthogonal ensembles in random matrix theory (actually this distribution is exact in the case of $2\times2$ matrices). However, the next distributions $\left(n>0\right)$ for these systems are different from their counterpart in random matrix theory. Another important aspect is the pair correlation function $g(r)$ which, in COE and GOE ensembles and the coalescing random walk and interacting random walk does not have any oscillation but in other systems $g(r)$ describe one oscillation near to $r=1$. For more information see Refs. [@gonzalez; @ben; @cornell; @mettetal]. In most of the non equilibrium domain systems, the main problem is the absence of analytical expressions for the spacing and correlation functions. Then, the question is: can the generalized Wigner surmise provide a good approximation for $p^{(n)}(s)$ and $g(r)$ in the domain systems as it happens with the random matrix ensembles?
\[!htp\]
![Comparison between the generalized Wigner surmise and the COE ensemble. In the simulation we took $20000$ matrices of size $200\times200$.[]{data-label="coe"}](coe.eps)
Wigner surmise for domains systems
==================================
For all systems considered in this paper $p^{(0)}(s)$ is well described by the Wigner distribution, because of that and following the method used in the random matrix theory we propose the next model $$\label{alfagen}
\alpha_n=\left\{
\begin{tabular}{cc}
1 & for $n=0$\\
$h(n)$ & for $n\geq1$\\
\end{tabular}
\right.$$ with $h(n)$ is a function to determine. The spacing distribution functions in this model are given by $$\label{pgen}
p^{(n)}(s)=\left\{
\begin{tabular}{cc}
$\frac{\pi}{2}se^{-\frac{\pi}{4} x^2}$ & if $n=0$\\
$A_n s^{\alpha_n}e^{-B_n x^{\beta_n}}$ & if $n\geq1$\\
\end{tabular}
\right.$$ using (\[pcond1\]) and (\[pcond2\]), we find $$\label{agen}
A_n=\frac{\beta_n B_n^{\frac{1+\alpha_n}{\beta_n}}}{\Gamma(\frac{1+\alpha_n}{\beta_n})},$$ and $$\label{bgen}
B_n=\left(\frac{\Gamma(\frac{2+\alpha_n}{\beta_n})}{(1+n)\Gamma(\frac{1+\alpha_n}{\beta_n})}\right)^{\beta_n}.$$
Independent interval approximation model (IIA)
----------------------------------------------
The independent intervals are used as an approximate solution in many equilibrium and non equilibrium systems [@gonzalez; @salsburg; @alemany] in order to find analytical results. In this approximation, $p^{\left(n\right)}(s)$ is given by the convolution product of $n+1$ nearest neighbor distribution factors, because of that, the spacing distribution functions can be calculated by using the Laplace transformation, see Ref. [@salsburg]. In particular, in Ref. [@gonzalez] the IIA is used to find an approximate model for the statistical behavior of two non equilibrium systems which will be explained in next sections.
### Independent interval model for small values of $s$
In Ref. [@gonzalez] the authors choose $p^{(0)}(s)$ equal to the Wigner distribution. In order to apply the method of the last section, we need to know the behavior of $p^{(n)}(s)$ for small and large values of $s$. For the first region we expand the Wigner distribution in power series $$p^{(0)}(s)=\frac{\pi}{2}s e^{-\frac{\pi}{4}s^2}=\frac{\pi}{2}s\left(1-\frac{\pi}{4}s^{2}+\cdots\right),$$ then, to the first order, the nearest neighbor distribution $p^{(0)}(s)$ has a lineal behavior, given by $$p^{(0)}(s)\propto s.$$
In the same limit $s\rightarrow0$, by using the independent interval approximation for arbitrary values of $n$, we have $$p^{(n)}(s)\propto \int_{0<x_1<x_2\cdots<x_n<s} x_1(x_2-x_1)\cdots(s-x_n)dx_1\cdots dx_n,$$ which can be evaluated by using the Laplace transform $$\widetilde{p}^{(n)}(t)\propto \frac{1}{t^{2(n+1)}},$$ and then, taking its inverse $$\label{exp}
p^{(n)}(s)\propto s^{2n+1}.$$
As consequence, in the IIA case the exponent $\alpha_n$ depends linearly on $n$ $$\label{alfaida}
\alpha_{n}=2n+1.$$
By using this result it is possible to determine the behavior of the level repulsion function for $s\rightarrow 0$. Following Ref. [@abdul] we have $$r^{(n)}(s)\propto s^{f(n)},$$ where $f(n)$ is the function to determine. By using equation (\[pitera\]), we can write
$$p^{(n)}(s)\propto s^{f(n)}\int^{s}_{0}p^{(n-1)}(s')ds',$$
then $$\label{fn}
p^{(n)}(s)\propto s^{f(n)+\cdots +f(0)+n}.$$ By comparing (\[exp\]) with (\[fn\]) is straightforward to find $$f(n)=1,$$ for all $n\geq 0$, as a consequence $$\label{alfaida2}
r^{(n)}(s)\propto s,\quad s\to 0\,,$$ then, the level repulsion does not depend on $n$ as it happens in the COE/GOE case.
### Independent interval model for large values of $s$
Now, we need the behavior of $p^{(n)}(s)$ for large values of $s$. The exact expression for $p^{(n)}(s)$ is $$p^{(n)}(s)=\int_{0<x_1<x_2\cdots<x_n<s} p^{(0)}(x_1)p^{(0)}(x_2-x_1)\cdots p^{(0)}(s-x_n) dx_1\cdots dx_n\,.$$ In our case $p^{(0)}(s)$ is given by the Wigner surmise, then $$\label{IIAwig}
p^{(n)}(s)=\left(\frac{\pi}{2}\right)^{n+1}\int_{0<x_1<x_2\cdots<x_n<s} x_1 e^{-\frac{\pi}{4}x{_1}^2}\cdots(s-x_n) e^{-\frac{\pi}{4}(s-x_n)^2}dx_1\cdots dx_n.$$ We can calculate the behavior of these functions for arbitrary values of $n$ in this limit $s\rightarrow\infty$ as we show next. From Ref. [@gonzalez] we know that at least the first two spacing distribution functions decay like Gaussian functions, then, we assume that for arbitrary values of $n$ these functions have the form $p_{asy}^{(n)}(s)=M_n s^{\gamma_n} e^{-N_n s^2} $ in the limit $s\rightarrow\infty$. In order to eliminate the integrals in equation (\[IIAwig\]) we use the Laplace transformation $$\label{Laplace-Wigner}
\widetilde{p}^{(n)}(l)=\left(1-l e^{l^2/\pi} \mathrm{erfc}\left(\frac{l}{\sqrt{\pi}}\right)\,\right)^{n+1},$$ where ${\mathop{\text{erfc}}}(z)=(2/\sqrt{\pi})\int_{z}^{\infty} e^{-t^2}\,dt$ is the complementary Gaussian error function. In the same way we take the Laplace transform in $p_{asy}^{(n)}$. Additionally, we expand both transformations in Taylor series around $l=0$. Let be $Z_j$ the $j^{th}$ coefficient in the expansion of equation (\[Laplace-Wigner\]) and $Y_j$ is the one for the Laplace transform of $p_{asy}^{(n)}(s)$. We find that the coefficients of both expansions satisfy the relation $Y_{i}/Z_{i}=Y_{j}/Z_{j}$ in the limit $i,j\rightarrow\infty$. By using this method we can find $M_n$, $N_n$ and $\beta_n$. If fact we find that $N_n=\frac{\pi}{4 n}$ and $\gamma_n=n+1$. In general, if we know $p^{(0)}(s)$ we can calculate the asymptotic behavior of $p^{(n)}(s)$ under the assumption that the IIA is valid for $s\rightarrow\infty$, but, as we will see in next sections, this is not true always.
In the figure \[ida\] we compare the exact statistical behavior of IIA with the generalized Wigner surmise, i.e., with a fit developed by using the behavior of $p^{(n)}(s)$ in the limits $s\rightarrow 0$ and $s\rightarrow \infty$, because of that from now on we will call it local fit. Also, we compare the global fit which was developed by using equations (\[alfagen\]) to (\[bgen\]) and the complete behavior of $p^{(n)}(s)$ in the interval $[0,\infty]$. By using the values of $\alpha_n$ found in the global fit, we developed a new fit to determine the global behavior of $\alpha_n$, explicitly in this case we have $$\label{alfaidaglobal}
\alpha_n=1.8268 n+0.9954,$$ this result is close to the exact exponent (\[alfaida\]), even when we use wrong functions in the fit; for example, the exact result for $p^{(1)}(s)$ is, see Ref. [@gonzalez]
$$\label{IIAp1}
p^{\left(1\right)}(s)=\frac{\pi}{16}e^{-\frac{\pi s^2}{4}}\left(4 s +\sqrt{2}e^{\frac{\pi s^2}{8}}\left(-4+\pi s^2\right)\mathrm{erf}\left(\frac{1}{2}\sqrt{\frac{\pi}{2}}s\right)\right),$$
which is very different form our surmise, however, both functions (\[pgen\]) and (\[IIAp1\]) have the same type of behavior in the limits $s\rightarrow 0$ and $s\rightarrow \infty$. Equation (\[gcoeap\]) for the correlation function it is still valid in both cases, global and local fit, we only must use equation (\[alfaida\]) and (\[alfaidaglobal\]) respectively. The main problem in the global fit approximation it is the use of not integer exponents in the level repulsion. Figure \[ida1\] show the differences between the three cases for small values of $s$, naturally in this region the graph of the global fit is not parallel to graph of the exact result as it actually happens in the local fit approximation. In figure \[ida2\] we can see the linear behavior of $p^{(n)}(s)$ in limit $s\rightarrow\infty$, which implies that the distribution functions decay like a Gaussian function as it was to be expected.
\[!htp\]
![Comparison between the exact statistical behavior of IIA, the generalized Wigner surmise (local fit) and the global fit.[]{data-label="ida"}](ida.eps)
\[!htp\]
![Log-Log graphic for the spacing distribution functions for IIA.[]{data-label="ida1"}](ida1.eps)
\[!htp\]
![Asymptotic behavior for $s\rightarrow\infty$ of the spacing distribution functions.[]{data-label="ida2"}](ida2.eps)
Coalescing random walk (CRW)
----------------------------
In the coalescing random walk the particles describe independent random walks along a one dimensional lattice and they are subjected to the reaction $A+A\rightarrow A$. This system is well studied [@ben; @doering; @ben0; @ben1] and its analytical solution is well know, because of that is used as approximation to more complex systems. Let $q^{\left(n\right)}(s)$ be the conditional probability that given one particle its next neighbor is at a distance of $s$. From its definition $q^{\left(n\right)}(s)$ is given by
$$\label{pn}
q^{\left(n\right)}(s)=\int_{0<y_1,\cdots,<y_n<s}\omega^{\left(n+2\right)}(y_1,\cdots,y_n,s)dy_1,\cdots,dy_n,$$
with $$\label{omega}
\omega^{\left(n\right)}(x_1,\cdots,x_n)=-\left.\frac{\partial^n E^{\left(n-1\right)}(x_1,y_1,\cdots,x_{n-1},y_{n-1})}{\partial x_1\cdots \partial x_{n-1}\partial y_{n-1}}\right|_{y_1=x_2,\cdots,\,y_{n-1}=x_n},$$
$$\label{en}
E^{(n)}(x_1,y_1,\cdots,x_n,y_n,t)=\sum^{(2n-1)!!}_{p=1}\sigma_p E^{(1)}(z_{1,p},z_{2,p},t)\cdots E^{(1)}(z_{2n-1,p},z_{2n,p},t),$$
where $z_{1,p}, z_{2,p}, . . . , z_{2n,p}$ symbolize an ordered permutation, $p$, of the variables $x_1, y_1, . . . , x_n, y_n$, such that
$$z_{1,p} < z_{2,p}, z_{3,p} < z_{4,p}, \cdots , z_{2n-1,p} < z_{2n,p},$$
and
$$z_{1,p} < z_{3,p} < z_{5,p} \cdots < z_{2n-1,p}.$$
The function $E^{(1)}(x_1,y_1,t)$ is the probability that from $x_1$ to $y_1$ the lattice is empty at time $t$. Then it is possible generate the complete solution for the CRW from $E^{(1)}(x_1,y_1,t)$, which is given by the solution of the diffusion equation under the suitable boundary conditions (see Ref. [@ben]). In fact, the exact expression for this function is $$\label{e1}
E^{(1)}(x_1,y_1,t)=\mathrm{erfc}\left(\frac{y_1-x_1}{\sqrt{8 D t}}\right),$$ with $D$ the diffusion constant and $t$ the time, for additionally information see Ref. [@ben]. For practical purposes, the solution given by equations (\[pn\]) to (\[e1\]) is hard to evaluate for arbitrary values of $n$ but it can be evaluated in the limit $s\rightarrow 0$ using Taylor series. The case $n=0$ is trivial, the Taylor expansion for equation (\[e1\]) is $$\label{expane1}
E^{(1)}(x_1,y_1,t)=1-\frac{y_1-x_1}{\sqrt{2 \pi}(D t)^{1/2}}+\frac{(y_1-x_1)^3}{24\sqrt{2 \pi}(D t)^{3/2}}-\frac{(y_1-x_1)^5}{640\sqrt{2 \pi}(D t)^{5/2}}+O(x,y)^7,$$ then $$q^{\left(0\right)}(x_2,x_1)=\omega^{\left(2\right)}(x_1,x_2)=-\left.\frac{\partial^2}{\partial x_1\partial y_1}E^{(1)}(x_1,y_1,t)\right|_{y_1=x_2},$$ $$q^{\left(0\right)}(x_2,x_1)=\frac{x_2-x_1}{4\sqrt{2\pi}(D t)^{3/2}}-\frac{(x_2-x_1)^3}{32\sqrt{2\pi}(D t)^{5/2}}+O(x)^5.$$ Making the variable change $s=\frac{x_2-x_1}{\sqrt{2\pi D t}}$ and taking into account that $p^{\left(0\right)}(s)=2\pi D t\,q^{\left(0\right)}(x_2,x_1)$, the product $D t$ disappears (dynamical scaling) in the above equation. Then, to first order, we have $$p^{\left(0\right)}(s)=\frac{s \pi}{2}+O(s)^3.$$ For small values of $s$, $p^{\left(0\right)}(s)$ has a linear behavior, i.e., $\alpha_0=1$. The case $n=1$ is more complicated, in fact we have $$\omega^{\left(3\right)}(x_1,x_2,x_3)=-\left.\frac{\partial^3}{\partial x_1\partial x_2\partial y_2}E^{(2)}(x_1,y_1,x_2,y_2,t)\right|_{y_1=x_2,y_2=x_3},$$ where $$\begin{aligned}
E^{(2)}\left(x_{1},y_{1},x_{2},y_{2},t\right) & = & E\left(x_{1},y_{1},t\right)E\left(x_{2},y_{2},t\right)\nonumber\\
& + & E\left(x_{1},y_{2},t\right)E\left(y_{1},x_{2},t\right) \nonumber\\
& - & E\left(x_{1},x_{2},t\right)E\left(y_{1},y_{2},t\right),\nonumber\\\end{aligned}$$ then $$\omega^{\left(3\right)}(x_1,x_2,x_3)=\frac{(x_2-x_1)(x_3-x_1)(x_3-x_2)}{32\pi (Dt)^3}+O(x)^4,$$ in that way $q^{\left(1\right)}(x_1,x_3,t)$ is given by $$q^{\left(1\right)}(x_3,x_1,t)=\int^{x_3}_{x_1}\frac{(x_2-x_1)(x_3-x_1)(x_3-x_2)}{32\pi (Dt)^3} dx_2+O(x)^5.$$ Integrating $$q^{\left(1\right)}(x_3,x_1,t)=\frac{(x_3-x_1)^4}{192\pi (Dt)^3}+O(x)^5.$$ Using again the variable change, it is straigthfoward to find $$p^{\left(1\right)}(s)=\frac{\pi^2 s^4}{24}+O(x)^5,$$ we conclude that $\alpha_1=4$. In general for an arbitrary value of $n$, we find that the first term in the expansion is $$q^{\left(n\right)}(x_1,x_n,t)\propto \int^{x_n}_{x_1}\cdots \int^{x_3}_{x_1}\prod_{1\leq i<j\leq n} (x_j-x_i)dx_2\cdots dx_{n-1},$$ therefore, the above equation has $(n+1)(n+2)/2$ different factors which implies that the integrand is proportional to $x_{i}^{(n+1)(n+2)/2}$. Making the integral and the usual variable change, the final expression for small values of $s$ is proportional to $s^{(n+1)(n+2)/2+n}$, explicitly, we have $$\label{alfacrw0}
\alpha_n=n+\frac{(n+1)(n+2)}{2}.$$
This is the same result reported in Ref. [@abdul] for the GOE/COE case and coincides with the partial result presented in Ref. [@ben1] for the CRW. We made again both fits, global and local. The global fit was made with the data from our simulation where we use a lattice with $1000$ sites and $500$ particles in $t=0$. The data to build the histograms was taken at three different times $T=50$, $T=100$ and $T=200$ over $50000$ realizations. In this case the global fit is not as accurate as in the IIA case as we can see in figure \[crw\] but it still is a good approximation. We use again equations (\[alfagen\]) to (\[bgen\]); and additionally we supposed a Gaussian decay ($\beta=2$). The global fit gives
$$\label{alfacrw}
\alpha_n=2.8688n+0.8621.$$
The global fit gives an erroneous exponent which depend linearly with $n$, this result it does not coincide with the analytical result (\[alfacrw0\]), where, $\alpha_n$ is a quadratic function of $n$. The local fit it is very different from the simulation results and coincides with the statistical behavior of the COE/GOE ensembles.
\[!htp\]
![Comparison between the statistical behavior of CRW, the global fit and the local fit.[]{data-label="crw"}](crw.eps)
\[!htp\]
![Log-Log graphic for the spacing distribution functions of the CRW.[]{data-label="crw1"}](crw1.eps)
\[!htp\]
![Statistical behavior of CRW for small values of $s$.[]{data-label="crw1a"}](crw1a.eps)
Although the global and local fit models are approximate, we can use them as a good approximations in some cases. For example, in Ref. [@ben1] the authors find an exact relation for the nearest neighbor distribution $p^{(0)}_{ann}(s)$ in the annihilation random walk in terms of $p^{(n)}(s)$ of the coalescing random walk. Explicitly, they found
$$\label{modeloaniquilacion}
p^{(0)}_{ann}(s)=\sum_{n\geq 0}\frac{1}{2^n}p^{(n)}(2s),$$
In order to test the validity of our approximations, we implement a simulation for the annihilation random walk for a one dimensional lattice with $2000$ sites, $100$ particles at $t=0$ over $20000$ realizations, the histogram was build by using three times $T=1000$, $T=1500$ and $T=2000$. By using the global and the local fit for the distribution functions $p^{(n)}(s)$ of the CRW, with equation (\[modeloaniquilacion\]), we find two analytical models for the annihilation random walk. We can see in figure \[crw2\] that the global and local fits provides a good approximation for the nearest neighbor distribution of the annihilation random walk. Additionally, figure \[crw3\] compare global and local fit with the asymptotic result $p^{(0)}_{ann}(s)\approx1.8167 e^{-1.3062s}$ given in Ref. [@ben1].
\[!htp\]
![Approximation for $p^{(0)}_{ann}(s)$ by using global and local fits.[]{data-label="crw2"}](crw2.eps)
\[!htp\]
![Asymptotic behavior of $p^{(0)}_{ann}(s)$.[]{data-label="crw3"}](crw3.eps)
Spin System
-----------
This system was introduced in Ref. [@cornell], where the authors consider a chain of $L$ Ising spins with nearest neighbor ferromagnetic interaction $J$. The chain is subject to spin-exchange dynamics with a driving force $E$ that favors motion of up spins to the right over motion to the left. In this case we do not have an analytical solution for the spacing distribution functions, because of that, we must start exploring numerically the behavior of $p^{(n)}(s)$ for small and large values of $s$. In figure \[espines1\], we can see the linear behavior of the spacing distribution function for $s\rightarrow0$. Using values in this region we develop a fit which suggest that $\alpha_1=3$ and $\alpha_2=6$ approximately. Naturally $\alpha_0=1$, however it is very difficult to know using this method the next exponents because it is not possible develop a numerical simulation with enough precision.
\[!htp\]
![Log-Log graphic for the spacing distribution functions of spin system.[]{data-label="espines1"}](espines1.eps)
\[!htp\]
![Asymptotic behavior of the spin system.[]{data-label="espines2"}](espines2.eps)
Curiously, these exponents for $n=0,1,2$ are given by the equation $$\alpha_n=\frac{(n+1)(n+2)}{2},$$ which is very similar to its counterpart in COE and GOE cases. For $s\rightarrow\infty$, $p^{(n)}(s)$ decay like a Gaussian function as we can see in figure \[espines2\]. In this case the global fit gives
$$\label{alfaespines}
\alpha_n=1.270 n + 0.920.$$
In figure \[espines\], we show the results given by equations (\[alfagen\]) to (\[bgen\]) for the global fit in comparison with the simulation results which was made with a lattice with $1000$ sites, equal number of spins up and down taken at two times $t=34$ and $t=48$ to build the histograms. The result for $g(r)$ is very good with a maximum error of $2.5\%$. Unfortunately this approximation is not good enough for $p^{(n)}(s)$ but at least it reproduce qualitatively the behavior of the real functions for $s\rightarrow\infty$. The local fit gives terrible results as it happens in the CRW case.
\[!h\]
![Comparison between the statistical behavior of the spin system and the global fit.[]{data-label="espines"}](espines.eps)
Gas System
----------
This system was originally studied in [@mettetal]. There, the authors studied the biased diffusion of two species in a fully periodic $2\times L$ rectangular lattice half filled with two equal number of two types of particles (labeled by their charge $+$ or $-$). An infinite external field drives the two species in opposite directions along the $x$ axis (long axis). The only interaction between particles is an excluded volume constraint, i.e., each lattice site can be occupied at most by only one particle. As it happens in the spin system, we do not know an analytic solution for the spacing and pair correlation functions. We follow the same method used in the spin system. In figure \[gas\], we can see the linear behavior of $p^{(n)}(s)$ which, by fit, give us $\alpha_1=3$ and $\alpha_2=5$ approximately, and of course $\alpha_0=1$. This fact suggest a linear behavior for $\alpha_n$ given by $$\alpha_n=2n+1$$ but again we could not find the next exponents with enough precision in order to validate the above equation. For $s\rightarrow\infty$ we found that $p^{(0)}(s)$ decays like a Gaussian function $(\beta=2)$, but for $n>0$, we found that $\beta$ is an indeterminate function of $n$. For example in figure \[gastres\] we can see the asymptotic behavior for two consecutive spacing distribution functions, the figure suggest $\beta=2$ for $p^{(0)}(s)$ and $\beta\neq2$ for $p^{(1)}(s)$ as it happens in Ref. [@aarao]. Because it is difficult determine the exact value of $\beta$ from the graphics, we implement a linear regression to find which value of $\beta$ give us a better “straight” line. With this method we find for example, that $\beta=2.6$ for $n=1$, $\beta=3$ for $n=5$ and $\beta=3.2$ for $n=8$. In the linear regressions we took values between $5.5\geq s
\geq2.5$, $11.7\geq s \geq7$ and $15.5\geq s \geq10$ respectively. Because of that, for the gas system we propose a model where $\beta$ depends on $n$. In particular we choose $\beta=2.6+0.1(n-1)$. With this model, the global fit gives $$\alpha_n=1.016 n + 0.788.$$ The results of the global fit are show in figure \[gas2\], again we find good fit for $g(r)$ with a maximum error of $2\%$ approximately but the agreement for $p^{(n)}(s)$ is not so good. Additionally, we include the first spacing distribution obtained with the local fit and our model for $\beta_n$.
\[!htp\]
![Log-Log graphic for the spacing distribution functions of gas system.[]{data-label="gas"}](gas.eps)
$\begin{array}{cc}
\includegraphics[scale=0.5]{gas1b.eps} & \includegraphics[scale=0.5]{gas1a.eps}\\
(a) & (b) \\
\includegraphics[scale=0.5]{gas1c.eps} & \includegraphics[scale=0.5]{gas1.eps}\\
(c) & (d) \\
\end{array}$
\[!htp\]
![Comparison between the statistical behavior of the gas system and the global fit.[]{data-label="gas2"}](gas2.eps)
Conclusion
==========
In COE and GOE ensembles, the spacing distribution functions $p^{(n)}(s)$ can be well described by using their behavior for small and large values of $s$ (local fit) as it happens in IIA case, however, this is not true for more complex systems like CRW, spin and gas systems. This result was to be expected because in general the spacing distribution functions are characterized also by their inter medium behavior. In general, the global fit gives better results in comparison with the local fit but it fails to reproduce the level repulsion, in fact, gives non integer exponents. The level repulsion for the CRW has the same behavior that the circular and Gaussian orthogonal ensembles, i.e., both systems are equivalents for $s\rightarrow 0$. The numerical results suggest that the IIA and the gas system are also equivalents in that region. We find numerical evidence that the spacing distributions functions for gas system is described by a non universal function, in fact, they decay as $M_n
s^{\gamma_n} e^{-N_n~s^{\beta_n}}$ for $n>0$, with $\beta_n$ an indeterminate function of $n$. In general the global and local fit provides a first approximation for $p^{(n)}(s)$ and $g(r)$, which can be used as a good approximation as it happens in the annihilation random walk case. These approximations also serve to classify the spacing distribution functions according to their level of repulsion and their decay functional form.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by an ECOS Nord/COLCIENCIAS action of French and Colombian cooperation and by the Faculty of Sciences of Los Andes University.
M. Mehta, Random matrices $2^{ed}$, Academic press (1991). A. Y. Abdul-Magd and M. H. Simbel, Wigner surmise for high-order level spacing distribution of chaotic systems, Phys. Rev E. **60**:5371–5374 (1999). D. L. González and G. Téllez, Statistical behavior of domain systems, Phys. Rev. E. **76**:011126 (2007). D. ben-Avraham and S. Havlin. Diffusion and reactions in fractals and disordered systems, Cambridge University Press (2000). S. J. Cornell and A. J. Bray, Domain growth in a one-dimensional driven diffusive system, Phys. Rev. E **54**:1153–1160 (1996). J. Mettetal, B. Schmittmann and R. Zia, Coarsening dynamics of a quasi one-dimensional driven lattice gas, Europhysics Lett. **58**:653–659 (2002). Z. W. Salsburg, R. W. Zwanzig and J. G. Kirkwood, Molecular distribution functions in a one-dimensional fluid, J. Chem. Phys. **21**:1098–1107 (1953). Lett. **58**:653–659 (2002). P. A. Alemany and D. ben-Avraham, Inter-particle distribution functions for one-species diffusion-limited annihilation, $A+A\to0$, Phys. Lett. A. **206**:18–25 (1995). C. Doering. Physica A, Microscopic spatial correlations induced by external noise in a reaction-diffusion system, **188**:386-403 (1992). D. ben-Avraham, Complete exact solution of diffusion-limited coalescence, $A+A\rightarrow A$, Phys. Rev. Lett. **81**:4756–4759 (1998). D. ben-Avraham and É. Brunet, On the relation between one-species diffusion-limited coalescence and annihilation in one dimension. J. Phys. A: Math. Gen. **38**:3247–3252 (2005). F. D. A. Aarão and R. B. Stinchcombe, Non universal coarsening and universal distributions in far-from-equilibrium systems. Physical Rev. E **71**:026110 (2005).
[^1]: die-gon1@uniandes.edu.co
[^2]: gtellez@uniandes.edu.co
|
---
abstract: 'We present the main results on the energy spectrum and composition of the highest energy cosmic rays of energy exceeding 10$^{18}$ eV obtained by the High Resolution Fly’s Eye and the Southern Auger Observatory. The current results are somewhat contradictory and raise interesting questions about the origin and character of these particles.'
author:
- Todor Stanev
title: 'Ultra high energy cosmic rays: A review'
---
Introduction
============
There is not an exact definition which cosmic rays should be called of ultrahigh energy. Generally the term is applied to those cosmic rays that we think are not accelerated in our Galaxy, i.e. are of extragalactic origin. This definition is certainly model dependent, but it always includes cosmic rays of energy above 10$^{19}$ eV. Showers generated by such high energy cosmic rays cosmic rays were first detected by the Vulcano Ranch air shower array in New Mexico, U.S.A. [@JL1]. The first cosmic ray of energy 10$^{20}$ eV, and possibly higher, was also detected by the same array [@JL2]. In this article we will call all particles of energy above 10$^{18}$ eV ultrahigh energy cosmic rays, abbreviated to UHECR.
At the time (1963) this did not cause any surprize: all physicists expected that the cosmic ray spectrum will go forever to higher and higher energies and the only reason we do not see such particles is that their flux is very low. It was only after the discovery of the microwave background that Greisen [@Greisen] in U.S. and Zatsepin & Kuzmin [@ZK] in the Soviet Union predicted the end of the cosmi ray spectrum because of the cosmic rays interactions with the microwave background. This feature is now called the [*GZK*]{} feature or cutoff.
In order to give the reader an impression of how big these showers are we show in Fig. \[highest\] the shower profile (number of particles as a function of the atmospheric depth) of the highest energy cosmic ray shower detected by the Fly’s Eye experiment [@highest]. The energy estimate for this shower is 3$\times$10$^{20}$ eV and the number of shower electrons at the shower maximum development ($N_{max}$) is more than 2$\times$10$^{11}$. The energy estimate of such showers consistes of an integration on the shower profile times the average electron eneregy.
![ Shower profile of the highest energy shower (3$\times$10$^{20}$ eV) detected by the Fly’s Eye experiment.[]{data-label="highest"}](stanev_2010_01_fig01.ps){width="10truecm"}
The next figure shows the state of our knowledge of the energy spectrum of the UHECR before the current generation of experiments, HiRes and Auger. One of the data sets shown in Fig. \[old\_spectrum\], Agasa, attracts the attention since it does include more than 10 events of energy above 10$^{20}$ eV [@Agasa]. At the time Agasa was the biggest (100 sq. km.) air shower array. Since we do not expect UHECR of this energy to be able to propagate from their sources to us, this result called for UHECR production models different from acceleration at astrophysical objects. As a result tens of exotic models explaining the existence of these particles as a result of ultraheavy $X$-particles decay appeared in the literature. The UHECRs in shuch models are most likely to be either $\gamma$-rays or neutrinos because most of the decay products have to be mesons rather than nucleons.
![ UHECR spectrum as determined before the current generation of experiments, HiRes and Auger.[]{data-label="old_spectrum"}](stanev_2010_01_fig02.ps){width="10truecm"}
Whatever the origin of the UHECR, acceleration at astrophysical objects or $X$-particle decay it is important to note that the energy spectrum detected at Earth is not the same as the acceleration and production spectrum is a result of the UHECR energy loss in propagation from their sources to us. We can use the detected energy spectrum and cosmic rays propagation calculations to study what the production spectrum is. Although all interaction properties are well known, propagation calculations still have to make simplifying assumptions:\
$\bullet$ All sources have identical acceleration/production spectra, and\
$\bullet$ Sources are isotropically distributed in the Universe.
UHECR energy spectrum
=====================
In this section we will discuss the comtemporary measurements of the UHECR energy spectrum that were done by the High Resolution Fly’s Eye (HiRes) detector and the Southern Auger Onservatory. The HiRes experiment consists of two fluorescent telescopes, HiRes 1 and 2. Fluorescent light is emitted by the air Nitrogen atoms excited by the ionization of the shower particles. The ligh emission is isotropic and the yield at see level is 4 UV photons per one meter of electron track. This yeld depends on the air density and temperature but 4 photons per meter of track is good enough for a simple idea for the photon fluxes at the telescope. HiRes 1 looks at elevations between 3 and 17$^o$ above the horizon, while HiRes 2 doubles the field of view up to elevation of 31$^o$. The telescopes work both individually or in stereo mode. Stereoscopic detection makes the shower analysis much more accurate. The energy estimate couples the integral of the shower profile and the average electron energy.
The Southern Auger Observatory is a hybrid detector cosisting of a huge air shower array of total area 3,000 km$^2$ and 24 fluorescent telescopes that observe the air above it. The detectors of the surface array (SD) are water Cherenkov tanks that register the Cherenkov light of the shower particles that hit the tank. Each water Cherenkov tank has a surface area of 10 m$^2$ and a depth of 1.2 m and is viewed by 3 photomultiplier tubes. The fluorescent telescopes are organized in 4 stations that occasionally can observe the fluorescent light in stereo. The surface array is fully efficient above shower energy of 3$\times$10$^{18}$ eV. The energy estimate of the surface array is obtained by the correlation of the shower signal at 1000 m from the shower axis, $S_{1000}$ to the fluorescent signals in events detected by both detectors. The lower energy part of the spectrum is measured by hybrid events, where at least one of the surface detectors triggered in coincidence with the fluorescent telescope.
Fig. \[spectrum\] shows the cosmic ray spectrum as measured by HiRes and Auger. The independ analyses of HiRes 1 and 2 [@HiRes_prl] are shown with black and grey points while the stereo analysis [@HiRes_stereo] is shown with empty circles. The Auger surface array spectrum [@Auger_prl] is shown with black squares and the hybrid measurement [@Auger_hybrid] is shown with empty squares. Since the surface detector has much higher statistics the hybrid data are only important for energies below 10$^{19}$ eV.
![ UHECR spectrum as measured by HiRes and Auger.[]{data-label="spectrum"}](stanev_2010_01_fig03.ps){width="10truecm"}
The first conclusion from Fig. \[spectrum\] is that both experiments confirm the GZK cutoff. After energy of about 4$\times$10$^{19}$ eV the cosmic ray spectrums steepens and there are very few events above that energy. Note that the measured spectrum is multiplied by E$^3$ which makes the small differences in the energy calibration look significantly bigger. There are, though, some differences that steer the interpretations of the energy spectra towards different models. The Auger energy spectrum is fitted to E$^{-2.6}$ behaviour over an order of magnitude above 3$\times$10$^{18}$ eV, while the HiRes spectrum is somewhat steeper. The same is true for the exact position of the GZK cutoff, where the experimental statistics is low.
In spite of the similarity of the two spectra they allow quite different interpretations for the origine and type of UHECR. The HiRes spectrum is fully consistent with the model of Berezinsky et al [@beretal05] which assumes a steep cosmic ray acceleration spectrum (E$^{-2.7}$) and a pure proton composition. There is no need for a cosmological evolution of the cosmic ray sources.
The interpretation of the Auger energy spectrum is much more complicated as it allows several different models. The first one is not dissimilar to that of Berezinsky et al - pure proton composition, E$^{-2.55}$ acceleration spectrum and no cosmological distribution of the UHECR sources. The second proton composition model requires flatter E$^{-2.3}$ acceleration spectrum, a very strong (proportional to $(1 + z)^5$) cosmological evolution of the sources. The third model is that of mixed composition, i.e. the same nuclei that exist in the galactic cosmic rays are accelerated at the powerful extragalactic sources. The acceleration spectra are relatively flat and the exact parameters depend on the composition of the accelerated cosmic rays.
Cosmic ray composition
======================
The measurement of the cosmic ray composition at high energy depends on the parameters of the showers that different nuclei generate. At energy around 10$^{15}$-10$^{16}$ eV the main composition parameter is the ratio of muons to electrons in the shower. At higher energy such a measurement becomes difficult since the muon counters are much more expensive and the main composition measurement is the depth of the shower maximum development X$_{max}$. Using a very simple analytic shower model Matthews [@Matthews2005] showed that $$X^A_{max}\; = \; X^p_{max}\; - ; X_0 ln A$$ where $X_0$ is the radiation length. Showers induced by heavy nuclei develop significantly faster than those of protons. The contemporary fluorescent detectors can measure X$_{max}$ with an accuracy of about 20 g/cm$^2$ while the difference between the average X$_{max}$ in proton and iron showers is about 100 g/cm$^2$.
Showers initiated by $\gamma$-rays or neutrinos have also development characteristics that are different from those of showers initiated by nuclei. Both HiRes [@HR_neu] and Auger [@Auger_neu] were able to put limits on the fluxes of ultrahigh energy neutrinos. These limits were set on specific neutrino flavors but the limit the total neutrino flux since oscillations would led to $\nu_e$ : $\nu_\mu$ : $\nu_\tau$ ratio of approximately 1 : 1 : 1.
The Auger Observatory also put limits on the fraction of $\gamma$-rays in the total cosmic ray flux. The limits come from the fact that very high energy $\gamma$-ray showers develop significantly deeper in the atmosphere than proton showers do. Two different techniques were used again: hybrid showers between 10$^{18}$ and 10$^{19}$ eV [@Auger_g1] and surface detector [@Auger_g2] above that energy. The fraction of $\gamma$-rays in the cosmic ray flux above 2$\times$10$^{18}$ eV was limited to less than 4% and the fraction above 10$^{19}$ eV to 2%. These limits almost eliminate the exotic models for UHECR production although it is still possible that highest energy particles, where the statistics is too low to set limits, are still $\gamma$-rays.
An important parameter in the study of the shower depth of maximum is the elongation rate $D_{10}$ - the rate of change of X$_{max}$ per a decade of energy. $D_{10}$ is approximately (1 - B)X$_0$ln10, where B is a parameter that depends on the hadronic interaction model. For different models D$_{10}$ is between 50 and 60 g/cm$^2$ if the cosmic ray composition is constant. In case that the composition becomes lighter with energy D$_{10}$ grows. If it becomes heavier D$_{10}$ is lower. Fig. \[compos\] shows the results on X$_{max}$ energy dependence as measured by the HiRes and Auger together with the predictions of three different interaction models for proton and Fe showers.
![Depth of maximum as a function of the shower energy as measured by HiRes and Auger and the predictions of three different interaction models for proton and iron showers.[]{data-label="compos"}](stanev_2010_01_fig04.ps){width="10truecm"}
The points labeled HiRes 05 show the first result with relatively small statistics published by HiRes in 2005 [@HR2005]. These points imply that the shower depth of maximum increases faster than any of the interaction models predict. This suggests that the cosmic ray composition becomes lighter with energy, although it can be heavier than pure protons, especially if the EPOS 1.99 model is correct. The black squares show the X$_{max}$ results of the Auger Observatory [@Auger_comp]. The general behavior is shown with the gray line. The three lowest energy points suggest D$_{10}$ of 106 g/cm$^2$ with a large error bar. At higher energy the elongation rate becomes 24$\pm$3 g/cm$^2$ . The interpretation in terms of cosmic ray composition should be that up to 3$\times$10$^{18}$ eV the cosmic ray composition becomes lighter and at higher energy it becomes heavier. The highest energy point at about 4$\times$10$^{19}$ eV is closer to iron than it is to protons in any of the interaction models. The open squares show the HiRes data set taken in stereo [@HR10]. Some of these points are even higher than expected for Fe showers in the QGSJet II model. There is an obvious disagreement between the two experiments.
This disagreement carries over when the two experiments examined the width of the X$_{max}$ distributions in each of the energy bins shown in Fig. \[RMS\]. The width of these distributions also reflects the cosmoc ray composition. The predictions for proton showers are of order 60 g/cm$^2$ while for Fe showers they are about 20 g/cm$^2$. In the case of Auger the decrease of the rms values follow the average depth of maximum and both of them suggest a composition that becomes heavier with the increasing energy, In the case of HiRes the best fit is a straight line. Note that the definitions of [*width*]{} are not identical. Auger uses directly the rms value of the distribution while HiRes gives the Gaussian width after the long tale of the distribution is cut off.
![Width of the X$_{max}$ distributions measured by Auger and HiRes.[]{data-label="RMS"}](stanev_2010_01_fig05.ps){width="10truecm"}
One possibility for the disagreement is that HiRes and Auger select their event samples in different ways. The reason is that none of the fluorescent detectors observes the whole sky and only elevations up to 31$^o$ in the best case. The experiments want to be certain they do are not biased toward early or late developing showers and apply very different cuts to the data. Auger, for example only uses hybrid events which implies selecting showers relatively close to the fluorescent detector while HiRes does not use showers closer than 10 km to any of their telescopes.
Arrival directions of the highest energy cosmic rays
====================================================
The idea is that UHECR do not scatter much in the galactic and extragalactic magnetic fields so they will cluster arround their sources and thus reveal them. The Southern Auger Observatory is the biggest UHECR detector and has the highest chance to look for the sources of these particles. Early in the game Auger made a trial scan, identified the procedure to follow and in 2007 they published a paper [@Auger_science] on the correlation of their events of energy exceeding 57 EeV with the active galactic nuclei from the VCV [@VCV] catalog at redshifts smaller than 0.018.
Out of 27 highest energy events 19 events came from directions not exceeding 3.1$^o$ from an AGN. Most of the events that did not correlate passed less than 12$^o$ from the galactic plane where the galactic magnetic field is the strongest. Accounting for the scans in particle energy, AGN distance, and distance from the AGNs the significance of the correlation was not huge, but still exceeding 3$\sigma$. These events and the AGNs with $|b|$ > 12$^o$ are shown in Fig. \[corr\].
![Correlation of the arrival directions of the Auger and HiRes highest energy events with the VCV catalog.[]{data-label="corr"}](stanev_2010_01_fig06.ps){width="12.5truecm"}
HiRes looked for correlation of their highest energy 13 events with the AGNs from the same catalog and did not see any correlation [@HiRes_corr]. The fields of view of the two experiments do not coincide, but there is enough overlap to look for correlations of both data sets.
The Auger Observatory has more than doubled its statistics by the time of of International Cosmic Ray Conference in 2009. The level of correlation, however, decreased significantly to about 2$\sigma$ [@Auger_corr09]. Since the VCV catalog does not appear to be the best one to contain possible UHECR sources the collaboration attempted to correlate the arrival directions of their highest energy events with objects from different ones with similar results. Only about 30% of UHECR seem to correlate with the directions of powerful extragalactic objects while the rest seem to come from an isotropic distribution. The exception is the direction of the nearby (less than 4 Mpc) radio galaxy Centaurus A around which several events are clustered.
There is obviously an internal contradiction between the idea of correlation of the arrival directions with extragalactic objects and that of heavy cosmic ray composition. If the highest energy events are indeed heavy nuclei they may scatter a lot and appear to come to us from isotropic directions. The question of UHECR astronomy can most likely be solved only with vastly increased statistics. Auger is now proposing a new, much larger Northern Observatory.
Summary
=======
The main question raised by the results of the Agasa collaboration, the energy extent of the cosmic ray spectrum, is now solved. Both HiRes and Auger observe its end, the GZK cutoff. With a small change in the energy assignment, less than the systematic error of 20%, the two measured spectra will be fully consistent.
The results on the cosmic ray composition are much more contradictory. HiRes sees cosmic ray composition close to a purely proton one while Auger observes increasily heavier composition above 3$\times$10$^{18}$ eV. The reasons for this disagreement are not obvious and it will take the two groups lots of work and collaboration to understand them. There maybe some help from the new hybrid experiment Telescope Array which combines scintillator counters with fluorescent telescopes.
The question about the sources of these particles is not yet solved. Less than 1/3 of the highest Auger events correlate in arrival direction with the possible sources from different catalogs. The solution will most likely require significant increase of the experimental statistics.\
[**DISCUSSION**]{}\
[**DANIELE FARGION**]{} Why we do not see UHECR events in Auger towards Virgo and why are no events toward Norma?\
[**TODOR STANEV**]{} The answer of Auger is that after accounting for the exposure (1/3) and for the dostance (1/25) they expect 75 times less events from Virgo for equal luminosity. There are models, including yours, that can explain the lack of events.\
[**SERGIO PETRERA**]{} I have a comment on the HiRes spectrum - They show that their data are well fit by the model of Berezinsky et al. They never published fits with other models, but mixed composition models can successfully fit data as well because of the many handles they have.\
[**TODOR STANEV**]{} The HiRes group used an existing model to fit their spectrum. After it worked they did not present other fits.\
[**FRANCESCO VISSANI**]{} Greisen, Zatsepin and Kuzmin put together their proposal [*before*]{} the data were known. Many of the models you discussed were proposed [*after*]{} the data were available, for instance the superheavy decaying particle model. Don’t you believe that this consideration already puts these theories in a different position?\
[**TODOR STANEV**]{} You are correct. The GZK was the original model. Contemporary models are made to fit observations. It would be better to call them fits to data rather than models.
[99]{} J. Linsley, L. Scarsi & B. Rossi, Phys.Rev.Lett. [**6**]{}, 485 (1961). J. Linsley, Phys. Rev. Lett., [**10**]{}, 146 (1963) K. Greizen, Phys. Rev. lett., 16:748 (1966) G.T. Zatsepin & V.A. Kuzmin, JETP Lett 4:78 (1966) The HiRes Collaboration (D.J. Bird et al.), Ap.J. [**441**]{},144 (1995) The Agasa Collaboration (M. Takeda et al.), Phys. Rev. Lett., 81:1163 (1998) The HiRes Collaboration (R.U. Abbasi et al), Phys.Rev.Lett. [**100**]{}:101101 (2008) The HiRes Collaboration (R.U. Abbasi et al), Astropart.Phys. [**32**]{}, 53 (2009) The Auger Collaboration (J. Abraham et al), Phys.Rev.Lett. [**101**]{}:061101 (2008) The Auger Collaboration (J. Abraham et al), Phys.Lett.B[**685**]{}, 239 (2010) V. Berezinsky et al., Phys. Rev., D[**74**]{}:043005 (2006) J. Matthews, Astropart.Phys. [**22**]{}, 387 (2005) The HiRes Collaboration (R.U. Abbasi et al) Ap.J, [**684**]{}, 790 (2008) The Auger Collaboration (J. Abraham et al), Phys.Rev.D [**79**]{}:102001 (2009) The Auger Collaboration (J. Abraham et al), Astropart.Phys. [**31**]{}, 399 (2009) The Auger Collaboration (J. Abraham et al), Astropart.Phys. [**29**]{}, 243 (2008) The HiRes Collaboration (R.U. Abbasi et al), ApJ, [**622**]{}, 910 (2005) The Auger Collaboration (J. Abraham et al), Phys.Rev.Lett [**104**]{}:091101 (2010) The HiRes Collaboration (R.U. Abbasi et al), Phys.Rev.Lett. [**104**]{}:161101 (2010) The Auger Collaboration (J. Abraham et al), Science, [**318**]{}:938 (2007) M.-P. Véron-Cetty & P. Véron, A&A, [**455**]{}, 773 (2006) The HiRes Collaboration (R.U. Abbasi et al), Astropart. Phys., [**30**]{}, 175 (2008) J. Abraham et al. (Auger Collaboration) Proc. 31st Int. Cosmic Ray Conf. (Lodz, Poland, 2009), arXiv:0906.2347
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---
abstract: 'It is found that 1E 1207.4-5209 could be a low-mass bare strange star if its small radius or low altitude cyclotron formation can be identified. The age problems of five sources could be solved by a fossil-disk-assisted torque. The magnetic dipole radiation dominates the evolution of PSR B1757-24 at present, and the others are in propeller (or tracking) phases.'
author:
- 'Y. Shi, R. X. Xu'
title: 'Can the age discrepancies of neutron stars be circumvented by an accretion-assisted torque?'
---
Introduction
============
The age of neutron star is an essential parameter, which is relevant to the physics of supernova explosion and thereafter the evolution of stars. However, it is still a big problem now to determine generally an exact value of age (except the crab pulsar). It is a conventional and convenient way to obtain the age for rotation-powered neutron stars by equalizing the energy lose rate of spindown to that of magnetodipole radiation, assuming that the inclination angle between magnetic and rotational axes is $\alpha=90^{\rm o}$ (e.g., Manchester & Taylor 1977). The conclusion keeps quantitatively for any $\alpha$, as long as the braking torques due to magnetodipole radiation and the unipolar generator are combined (Xu & Qiao 2001). The resultant age, the so-called characteristic age, is $T_{\rm
c}=P/(2{\dot P})$ if the initial period $P_0$ is much smaller than the present period $P$. The age, $T_{\rm c}$, is generally considered as the true one for a neutron star with $P>100$ ms since most newborn neutron stars could rotate initially at $P_0\sim (20-30)$ ms (e.g., Xu et al. 2002).
It challenges the opinion above that the ages of a few supernova remnants are inconsistent with $T_{\rm c}$ of their related isolated stars (Table 1), which implies that some additional torque mechanisms do contribute to star braking. Among the five stars, three of them have $T_{\rm c} > 10 T_{\rm
SNR}$ (the age of supernova remnant), and other two $T_{\rm c} \la
2 T_{\rm SNR}$. In addition, electron cyclotron resonant lines are detected in two of the five neutron stars (1E 1207.4-5209: Bignami et al. 2003; 1E 2259+568: Iwasawa et al. 1992), but the inferred magnetic fields, $B_{\rm cyc}$, from which are significantly smaller than that in the magnetic dipole radiation model, $B_{\rm d}$. The most prominent one in this age discrepancy issue is 1E 1207.4-5209, which has $T_{\rm c} \ga 30 T_{\rm SNR}$ and $B_{\rm
cyc}\la B_{\rm d}/30$.
One probable and popularly-discussed way to solve the age problem is through an additional accretion torque (Marsden et al. 2001, Alpar et al. 2001, Menou et al. 2001). In this paper, whether an additional accretion torque can possibly solve the age discrepancy is investigated, including discussions about possible astrophysical implications.
The case of 1E 1207.4-5209
==========================
The key point in the age discrepancy of 1E 1207.4-5209 is how to spin down from $P_0\sim 20$ ms to 424 ms in a short time of $T_{\rm SNR}\sim 7$ kyrs if its true age is $T_{\rm SNR}$. Certainly the problem disappears if one assumes a long initial period $P_0\sim 400$ ms or a large braking index $n\sim 50$ (Pavlov et al. 2002); but this is not of Occam’s razor since it is generally believed that rotation-powered radio pulsars born with $\sim 20$ ms, and brake with index $\la 3$. May an additional accretion torque help the spindown? Actually, in an effort to reconcile $B_{\rm d}$ with $B_{\rm
cyc}$, an accretion model for 1E 1207.4-5209 was proposed (Xu et al. 2003). However, a very difficulty in the model is how to choose a time-dependent accretion rate ${\dot M}_{\rm d}(t)$, and to determine the propeller torque with the rate ${\dot M}_{\rm d}$.
Nevertheless, the propeller phase works in the centrifugal inhibition regime when $r_{\rm m}>r_{\rm c}$; for a star with mass $M$ and magnetic moment $\mu$, the corotation radius $r_{\rm
c}=[GM/(4\pi^2)]^{1/3}P^{2/3}$ and the magnetospheric radius $
r_{\rm m}= [\mu^2/({\dot M}_{\rm d}\sqrt{2GM})]^{2/7}$. To avoid the complex calculations of magnetohydrodynamics, the rotation energy loss due to propeller torque could be simply introduced as ${\dot E}_{\rm a}=-G{\dot M}_{\rm d}M/R_{\rm m}$, based on the energy conservation law. This is unphysical, but should be an limit for accretion braking. As $r_{\rm m}$ decreases ($\rightarrow r_{\rm c}^+$), ${\dot
M}_{\rm d}$ increases, and $|{\dot E}_{\rm a}|$ increases too. Therefore the most efficient spindown (MESD) takes place when $r_{\rm m}\rightarrow r_{\rm c}^+$.
In a model where the propeller and electromagnetic torques are combined, in the MESD case, one can derive the period evolution $$P<1.1 B_{12}^2R_6^4(M/M_\odot)^{-2}(t/{\rm yrs})+P_0~({\rm ms}),
%
\label{P}$$ where $B_{12}=B/(10^{12}$G) and $R_6=R/(10^6$cm). The right hand of Eq.(\[P\]) is an upper limit of $P$ because 1, a realistic accretion rate may not be as high as that of MESD; and 2, the corresponding braking torque is not so effective. If 1E 1207.4-5209 is a conventional neutron star with mass $\sim
M_\odot$ and radius $\sim 10^6$ cm, and the line features are related to cyclotron absorptions near the surface (Xu et al. 2003; the polar magnetic field is thus $6\times 10^{10}$ G), one has $P<3.8(t/{\rm kyrs})+P_0$ (ms).
Therefore, assuming 1E 1207.4-5209 has a true age $t\sim 7$ kyrs and an initial period $P_0\sim 20$ ms, the upper limit of the present period is $\sim 40$ ms ($\ll P=434$ ms), and the age discrepancy can then not be solved in the conventional neutron star model. However, if 1E 1207.4-5209 is a strange star with low mass, for instance $R=1$ km (and the mass is thus $\sim 10^{-3}M_\odot$, since low-mass strange stars have almost a homogenous density $\sim 4\times 10^{14}$g/cm$^3$; Alcock et al. 1986), the upper limit is then $P\simeq 110 B_{12}^2(t/{\rm yrs})+P_0$ (ms). In this case, 1E 1207.4-5209 could spin down to $\sim 2.8$ s during $\sim 7$ kyrs if its polar magnetic field $6\times 10^{10}$ G. In fact, the fitted radius of 1E 1207.4-5209 with a blackbody model is only $\sim 1$km (Mereghetti et al. 1996; Vasisht et al. 1997) although a lager radius is possible if a light-element atmosphere is applied (Zavlin, Pavlov & Trümper 1998). The best-fit tow blackbody model of [*XMM-Newton*]{} data indicates an emitting radius $\sim 3$ km for the soft component with temperature $\sim 200$ eV (Bignami et al. 2003). Combined with its non-atomic feature spectrum, we may suggest that 1E 1207.4-5209 is a low-mass strange star with bare quark surface (Xu 2002, Xu et al. 2003).
An alternative possibility is that 1E 1207.4-5209 is a conventional neutron star, but the cyclotron resonant absorption forms far away from the surface. The polar magnetic field is $\sim 6\times 10^{10}[(R+h)/R]^3$G if the resonant lines form at a height $h$. From Eq.(\[P\]), $424 < 1.1\times 7\times 10^3 B_{12}^2+20$, we estimate a low limit of the polar magnetic field to be $\sim 2.3
\times 10^{11}$ G. This implies that the resonant absorption region should be at a level of $>16$ km height from the surface. Certainly, in case of no propeller torque (i.e., the polar magnetic field is $(1.7-3.6)\times 10^{12}$G), the height of resonant absorption region is $(30-40)$km.
It is worth noting that 1E 1207.4-5209 could be a low-mass neutron star with polar magnetic field $B_{12}=0.06$. From Eq.(\[P\]) and the conditions of MESD, one has $R_6^2>4(M/M_\odot)$ for $P_0\sim 20$ ms. This implies a neutron star with radius $R>10$ km but mass $M<M_\odot$ (e.g., Shapiro & Teukolsky 1983). This result may have difficulties in explaining 1, a non-atomic spectrum (Xu et al. 2003), and 2, a possible small radius observed (Mereghetti et al. 1996; Vasisht et al. 1997; Bignami et al. 2003) and even the fitting result of a neutron star with 10 km and $1.4M_\odot$ (Zavlin et al. 1998).
Other sources
=============
If other sources list in Table 1 are neutron stars with $R_6=1$ and $M=M_\odot$, the low limits of polar magnetic fields are $6.1\times 10^{11}$ G for 1E 2259+586, $4.9\times 10^{10}$ G for PSR B1757-24, $1.6\times 10^{11}$ G for PSR J1811-1925, and $(2.5-5.6)\times 10^{11}$ G for PSR J1846-0258. Among these sources, only possible cyclotron absorption is found in 1E 2259+586, and the limit field is within the range of that inferred from cyclotron line. This suggests that the cyclotron resonant may take place just above the stellar surface.
An interesting question is: how much mass could accrete during the propeller phase in case of MESD? Certainly, only a very small part of this matter can accrete onto the stellar surface. When MESD works, one obtains the accretion rate ${\dot M}_{\rm d}\sim 2^{11/6}\pi^{7/3}\mu^2(GM)^{-5/3}P^{-7/3}$ from $r_{\rm m}\simeq r_{\rm c}$. If the quantities are re-scaled, $m=M_{\rm d}/M_\odot$, $\tau=t/{\rm yr}$, and $p=P/{\rm ms}$, one has ${\rm d}m/{\rm d}\tau\sim 0.24~
\mu_{30}^2(M/M_\odot)^{-5/3}p^{-7/3}$. Combining with Eq.(\[P\]), one comes to $$m<\int_0^\infty {\rm d}\tau =
0.16(M/M_\odot)^{-2/3}I_{45}p_0^{-4/3},
%
\label{m}$$ where $p_0=P_0/{\rm ms}$. Note that the upper limit of accretion mass, $M_{\rm d}$, in the right hand of Eq.(\[m\]) does not depend on the magnetic momentum. Typically for $p_0=20$, the upper limit of accretion mass is $2.95\times 10^{-3}M_\odot$, which is reasonable since the amount of the fall-back material after supernova explosion could be as high as $0.1M_\odot$ (Lin, Woosley & Bodenheimer 1991; Chevalier 1989). Due to $r$-mode instability, a nascent neutron star may loss rapidly its angular momentum though gravitational radiation if the initial period is less than $\sim 3-5$ ms (Andersson & Kokkotas 2001). The upper limit of accretion mass in case of MESD is $\sim
0.04M_\odot$ for $p_0=3$. These results indicate that the fall-back matter is enough to brake the center stars by propeller torque in MESD case.
A model with self-similar accretion rate
========================================
Though the study about MESD torque provides some useful information on the accretion model, including the appropriate magnetic field and the mass of fall-back disk around a neutron star, the accretion rate of MESD torque is questionable in realistic cases. After a dynamical time a fossil disk may form. For a viscosity-driven disk, the accretion could be in a self-similar way, with an accretion rate of (Cannizzo, Lee & Goodman 1990) $${\dot m}={\dot m_0},~0<t<T;~~
%
{\dot m}={\dot m_0}(t/T)^{-\alpha},~t\geq T,
%
\label{dotm}$$ where $T$ is of order the dynamical time, ${\dot m}={\rm d}m/{\rm
d}t$. Assuming $T\sim 1$ms, an initial disk mass $\sim 0.006M_\odot$, and an opacity dominated by electron scattering ($\alpha=7/6$), Chatterjee, Hernquist and Narayan (2000) develop a first detailed model of fossil disk accretion for anomalous X-ray pulsars (AXPs). However it is noted by Francischelli and Wijers (2002) that Kramers opacity may prevail in the fossil disk (i.e., $\alpha=1.25$). In the regime of conventional neutron stars, we will calculate the accretion torque through the realistic accretion rate of Eq.(\[dotm\]), assuming $\alpha=1.25$ and $T=1$ms, with the inclusion of magnetic dipole radiation. The spinup/spindown torque proposed by Menou et al. (1999), $${\dot J}=2{\dot M}_{\rm d}r_{\rm m}^2\Omega_{\rm k}(r_{\rm
m})[1-\Omega/\Omega_{\rm k}(r_{\rm m})],$$ is applied for the action of fossil disk in the model, where $\Omega=2\pi/P$, $\Omega_{\rm k}(r_{\rm m})$ is the Keplerian angular velocity at the magnetospheric boundary.
One may compute the accretion rate, $\dot m$, as well as the spin evolution $P(t)$. The total disk mass could be $m=\int_0^\infty{\dot m}{\rm d}t$. It is worth noting that the disk mass obtained in this way could be much larger than that in MESD case because of the inclusion of the high accretion in the initial period. We think that the accretion rate characterized by Eq.(\[dotm\]) is on average in a sense. The period derivative, $\dot P$, may be affected by dynamical instabilities or some stochastic processes, whereas the period, $P$, is of the integration over a very long time. We therefore calculate $P(T_{\rm SNR})$ for any disk mass, $m$, and polar magnetic field, $B_{12}$, of neutron stars. For 1E 1207.4-5209, the calculated contour of period relative error, $|P(T_{\rm SNR})-P|/P$ ($P=0.424$s for 1E 1207.4-5209), is shown in Fig.1. A reasonable parameter set (disk mass $m$ and polar magnetic field $B_{12}$) is chosen if the following criteria are met: 1, the period relative error is smaller; 2, $m<0.1$; 3, $B_{\rm
12,cyc}<B_{12}<B_{\rm 12,d}$ (Table 1). We then have $m=0.054$ and $B_{12}=3.55$. The parameter sets for other sources can also be obtained in this way, which are listed on Fig.2 (except for PSR J1811-1925). The period evolution curves, with these parameters, are drawn in Fig.2. Note these curves do not change significantly if the parameter sets shift reasonably.
The heights, $h$, of cyclotron resonant scattering regions can be obtained, based on the differences of parametric magnetic field, $B_{12}$, and the field inferred from absorption features, $B_{\rm
12,cyc}$. It is found that $h\sim 29$ km and $\sim (8.8-15)$ km for 1E 1207.4-5209 and 1E 2259+586, respectively, in the model.
Whether or not the disk will influence the spindown of the neutron star or suppress the radio emission will depend on the location of $r_{\rm m}$ relative to the light cylinder radius, $r_{\rm L}$, and the corotation radius $r_{\rm c}$ (Chatterjee et al. 2000). Magnetic dipole radiation dominates, and the disk and the star will effectively evolve independently if $r_{\rm m}>r_{\rm L}$; but in other cases, accretion onto the star will lead to accretion-induced X-ray emission with radio quiet. We see from Fig.2 that the condition of $r_{\rm m}>r_{\rm L}$ is satisfied only for PSR B1757-24 when it is older than $\sim 10^3$ years. We are therefore not surprise that PSR B1757-24 is now radio loud whereas the others (1E 1207.4-5209, 1E 2259+586, and PSR J1846-0258) are radio quiet. The AXP 1E 2259+586 is in a tracking phase, and we expect other two (1E 1207.4-5209 and PSR J1846-0258) will evolve to be AXPs when they are in tracking phases too.
PSR J1811-1925 is an interesting exception among the five sources, whose age is certain if it has physical association with the remnant of a supernova recorded in A.D. 386. In its calculated contour, we can only choose {$B_{12}=2.6, m=0.1$} or {$B_{12}=1.6, m=2.2$}; both the parameter sets are not reasonable (i.e., can not meet those 3 criteria). This may imply that the accretion of PSR J1811-1925 is not self-similar. Recalling that the low limit of polar field is only $1.6\times 10^{11}$G if in MESD case, we could suggest that PSR J1811-1925 has a field within $(1.6-17)\times 10^{11}$G, with an accretion stronger than that of Eq.(\[dotm\]) but weaker than that in MESD case. This result hints that PSR J1811-1925 is radio quiet (Crawford et al. 1998). In addition, the parametric field, $B_{12}=3.55$, chosen for 1E 1207.4-5209, which is close to $B_{\rm 12,d}=3.6$, would also indicate that the real accretion is not described by Eq.(\[dotm\]). In fact the accretion of Eq.(\[dotm\]) is for the capture of material by black holes where magnetic field is not important, which could differ from that for neutron stars with strong fields.
Conclusions and Discussions
===========================
The possibility of solving the age discrepancy by an accretion-assisted torque is discussed. We find that: 1, 1E 1207.4-5209 can not be a neutron star, but a low-mass bare strange star, if the cyclotron resonant region is near the polar cap with a magnetic field of $6\times 10^{10}$G; whereas it could be a conventional neutron star if the cyclotron lines form at a height of $(16-40)$km. An identification of smaller radius or low altitude cyclotron formation favors a low-mass bare strange star model for 1E 1207.4-5209. 2, Among the five sources with age problems, the magnetic dipole radiation dominates the evolution of PSR B1757-24 at present, and the others are in propeller (or tracking) phases. 3, The real accretion around these sources may differ from a self-similar one (Eq.(\[dotm\])), at least for PSR J1811-1925. 4, By a calculation with self-similar accretion, it is suggested that PSR J1846-0258 and 1E 1207.4-5209 (and probably PSR J1811-1925) would evolve to be anomalous X-ray pulsars in the future.
The debris disks formed following supernovae are currently conjectured also for interpreting other astrophysical phenomena, e.g, anomalous X-ray pulsars and soft $\gamma-$ray repeaters. Factually, these disks around the sources could be bright in a wide spectral range. Recent discoveries of possible optical and near-infrared emission from a few AXPs may be a hint of such kind fallback accretion disks (1E 2259+586: Hulleman et al. 2001; 1RXS J170849-400910: Israel et al. 2002; 1E 1048.1-5937: Wang & Chakrabarty 2002). Although a comparison of optical and near-infrared observations with theoretical predictions of spectra of disks around neutron stars (Perna et al. 2000) have helped rule out the presence of disks in some cases, more detailed studies in this aspect is still necessary, which may be an effective way to test the fossil-disk model for young neutron stars.
[*Acknowledgments*]{}: The helpful suggestions and discussions from an anonymous referee are sincerely acknowledged. This work is supported by National Nature Sciences Foundation of China (10273001) and the Special Funds for Major State Basic Research Projects of China (G2000077602).
Alcock, C., Farhi, E., Olinto, A. 1986, ApJ, 310, 261
Alpar, M. A., Ankay, A., Yazgan, E. 2001, ApJ, 557, L61
Andersson, N., Kokkotas, K.D. 2001, Int. J. Mod. Phys., D10, 381
Bignami, G. F., Caraveo, A., Luca, A. D., Mereghetti, S. 2003, Nature, 423, 725
Cannizzo, J.K., Lee, H.M., Goodman, J. 1990, ApJ, 351, 38
Chatterjee, P., Hernquist, L., Narayan, R. 2000, ApJ, 534, 373
Chevalier, R.A. 1989, ApJ, 346, 847
Crawford, F., et al. 1998, Mem. Soc. Astron. Italiana, 69, 951
Francischelli, G.J., Wijers, R.A.M.J. 2002, preprint (astro-ph/0205212)
Gavriil, F. P., Kaspi, V. M. 2002, ApJ, 567, 1067
Gotthelf, E. V., Vasisht, G., Boylan-Kolchin, M., Torii, K. 2000, ApJ, 542, L37
Hughes, V. A., Harten, R. H., van der Bergh, S. 1981, ApJ, 246, L127
Hulleman, F., et al. 2001, ApJ, 563, L49
Israel, G. L., et al. 2002, ApJ, 589, L93
Iwasawa, K., Koyama, K., Halpern, J. P. 1992, PASJ, 44, 9
Kaspi V. M., et al. 2003, ApJ, 588, L93
Lin, D.N.C., Woosley, S.E., Boddenheimer, P.H. 1991, Nature, 353, 827
Manchester, R. N. & Taylor, J. H.: 1977, [*Pulsars*]{}, (Freeman: San Francisco)
Marsden, D., Lingenfelter, R. E., Rothschild, R. E. 2001, ApJ, 547, L45
Menou, K., Esin, A.A., Narayan, R., Garcia, M.R., Lasota, J.-P., McClintock, J.E. 1999, ApJ, 520, 276
Menou, K., Perna, R., Hernquist, L. 2001, ApJ, 554, L63
Mereghetti, S., Bignami, G.F., Caraveo, P.A. 1996, ApJ, 464, 842
Pavlov, G. G., Zavlin, V. E., Sanwal, D., Trümper, J. 2002, ApJ, 569, L95
Perna R., Hernquist, L., Narayan, R. 2000, ApJ, 541, 344
Shapiro, S. L., Teukolsky, S. A.: 1983, [*Black holes, white dwarfs, and neutron stars: The physics of compact objects*]{} (Wiley-Interscience: New York)
Torii, K., et al. 1999, ApJ, 523, L69
Vasisht, G., Kulkarni, S. R., Anderson, S. B., Hamilton, T. T., Kawai, N. 1997, ApJ, 476, L43
Wang, Z. X., Chakrabarty, D. 2002, ApJ, 579, L33
Xu, R. X. 2002, ApJ, 570, L65
Xu, R. X., Qiao, G. J. 2001, ApJ, 561, L85
Xu R. X., Wang H. G., Qiao, G. 2002, Chin. Jour. Astro. Astrophys., 2, 533
Xu, R. X., Wang, H. G., Qiao, G. 2003, Chin. Phys. Lett., 20, 314
Zavlin, V. E., Pavlov, G. G., Trümper, J. 1998, A&A, 331, 821
[llccccccl]{} 1E 1207.4-5209 & PKS 1209-51/52 & 0.424 & 0.07-0.3 & 200-900 & $\sim 7$ & 1.7-3.6 & 0.06 & 1,2\
1E 2259+586$^{\dag}$ & CTB 109 & 6.98 & 4.84 & 228 & 17 & 59 & 0.4-0.9 & 3,4,5\
PSR B1757-24 & G5.4-1.2 & 0.125 & 1.28 & 16 & $>39$ & 4.0 & — & 6\
PSR J1811-1925 & G11.2-0.3 & 0.065 & 0.44 & 24 & 1.6 & 1.7 & — & 7\
PSR J1846-0258 & Kes 75 & 0.325 & 71 & 0.72 & 0.9-4.3 & 49 & — & 8\
|
---
abstract: |
Due to its hereditary nature, genomic data is not only linked to its owner but to that of close relatives as well. As a result, its sensitivity does not really degrade over time; in fact, the relevance of a genomic sequence is likely to be longer than the security provided by encryption. This prompts the need for specialized techniques providing *long-term* security for genomic data, yet the only available tool for this purpose is GenoGuard [@huang_genoguard:_2015]. By relying on [*Honey Encryption*]{}, GenoGuard is secure against an adversary that can brute force all possible keys; i.e., whenever an attacker tries to decrypt using an incorrect password, she will obtain an incorrect but plausible looking decoy sequence.
In this paper, we set to analyze the real-world security guarantees provided by GenoGuard; specifically, assess how much more information does access to a ciphertext encrypted using GenoGuard yield, compared to one that was not. Overall, we find that, if the adversary has access to side information in the form of partial information from the target sequence, the use of GenoGuard does appreciably increase her power in determining the rest of the sequence. We show that, in the case of a sequence encrypted using an easily guessable (low-entropy) password, the adversary is able to rule out most decoy sequences, and obtain the target sequence with just 2.5% of it available as side information. In the case of a harder-to-guess (high-entropy) password, we show that the adversary still obtains, on average, better accuracy in guessing the rest of the target sequences than using state-of-the-art genomic sequence inference methods, obtaining up to 15% improvement in accuracy.
author:
- Bristena Oprisanu
- Christophe Dessimoz
- Emiliano De Cristofaro
title: 'How Much Does GenoGuard Really “Guard”? An Empirical Analysis of Long-Term Security for Genomic Data'
---
Introduction {#sec:intro}
============
Over the past two decades, the cost of sequencing the human genome – i.e., determining a person’s complete DNA sequence – has plummeted from millions to thousands of dollars, and continues to drop [@genome2017org]. As a result, sequencing has not only become routine in biology and biomedics research, but is also increasingly used in clinical contexts, with treatments tailored to the patient’s genetic makeup [@ashley2016towards]. At the same time, the “direct-to-consumer” genetic testing market is booming [@adoption] with companies like 23andMe and AncestryDNA attracting millions of customers, and providing them with easy access to reports on their ancestry or genetic predisposition to health-related conditions. Progress and investments in genomics have also enabled public initiatives to gather genomic data for research purposes. For instance, in 2015, the US launched the “All of Us” program [@allofus2017], which aims to sequence one million people, while, in the UK, Genomics England is sequencing the genomes of 100,000 patients with rare diseases or cancer [@genomicsengland].
Alas, as more and more genomic data is generated, collected, and shared, serious privacy, security, and ethical concerns also become increasingly relevant. The genome contains very sensitive information related to, e.g., ethnic heritage, disease predispositions, and other phenotypic traits [@ayday2013chills]. Furthermore, even though most published genomes have been anonymized, previous work has shown that anonymization does not provide an effective safeguard for genomic data [@gymrek_identifying_2013]. While some individuals choose to donate their genome to science, or even publicly share it [@pgp], others might be concerned about their privacy, or fear discrimination by employers, government agencies, insurance providers, etc. [@burns_gop_nodate]. Worse yet, consequences of genomic data disclosure are not limited in time or to the data owner: due to its hereditary nature, access to one’s sequenced genome inherently implies access to many features that are relevant to their progeny and their close relatives. A case in point is the story of Henrietta Lacks, a patient who died of cancer in 1951. Some of her cancerous cells revealed to be useful for research because of their ability to keep on dividing. Unbeknownst to her family, the cells became the most commonly used “immortal cell line,” and their genome was eventually sequenced and published [@landry2013genomic]. This prompted serious privacy concerns among her family members, even 60 years later [@callaway2013hela].
Motivated by these challenges, the research community has produced a large body of work aiming to protect genomic privacy and enable privacy-preserving sharing and testing of human genomes [@sok]. Available solutions mostly rely on cryptographic tools, including encryption as well as Secure Computation, Homomorphic Encryption, Oblivious RAM, etc. [@aziz2017privacy]. However, modern encryption algorithms provide security guarantees only against computationally bounded adversary; essentially, their security is assumed to last for 30 to 50 years [@enisa]. While this timeframe is acceptable for most uses of encryption, it is not for genomic data.
To address the problem of “long-term security,” Huang et al. [@huang_genoguard:_2015] introduce GenoGuard, a tool based on Honey Encryption (HE) [@HE2] to provide confidentiality of genomic data even in the presence of an adversary who can brute force all possible encryption keys. GenoGuard uses a distribution transforming encoder (DTE) together with symmetric (password-based) encryption. In essence, whenever an attacker would try to decrypt a GenoGuard ciphertext using a wrong password, the decryption will give a wrong but plausible looking plaintext, which we denote as a [*honey sequence.*]{} HE schemes based on DTE-then-encrypt constructions (as is the case for GenoGuard) only provide security in the message recovery context. That is, having access to the ciphertext only gives an unbounded adversary a negligible advantage in guessing the correct plaintext. However, as first discussed by Jaeger et al. [@jaeger2016honey], ciphertexts obtained from DTE-then-encrypt HE might still leak a significant amount of information about the plaintexts.
[**Technical Roadmap.**]{} We evaluate GenoGuard security by analyzing ciphertexts obtained using easily guessable (low-entropy) passwords as well as hard (high-entropy) ones. In other words, in both cases, we decrypt a GenoGuard ciphertext using a corpus of passwords and analyze the resulting decryptions (honey sequences). In the low-entropy setting, we consider an adversary who aims to identify the correct sequence among a pool of honey sequences, whereas, in the high-entropy case, one that uses the GenoGuard ciphertext in order to obtain more information about the target sequence as opposed to inference methods for genomic data.
In our experimental evaluation, we show that, under a low-entropy password setting, an adversary who has access to side information about the target sequence can quickly eliminate the decoy sequences in order to have an increased advantage of guessing the correct sequence. This draws attention to the fact that if the attacker obtains a list of known passwords for a user (as passwords are often compromised and/or re-used), together with some side information about the user’s sequence, she can have a significant advantage in guessing the correct sequence.
In the high-entropy setting, not only we observe that access to the GenoGuard ciphertext improves an adversary’s accuracy in guessing SNVs from a target sequence when 10% or less of the target sequence is available to her as side information, but also draws attention to the fact that with enough side information, the adversary can predict a significant part of the target genome just by using state of the art inference methods for genomic sequences.
[**Contributions.**]{} In summary, our paper makes two main contributions. First, under a low-entropy password setting, we formally show that, if the adversary obtains side information about the target sequence, there is a significant lower bound in her advantage. This highlights that the system offers low security when the adversary has access to side information, as supported by empirical evidence. Second, in the high-entropy password setting, we quantify the privacy loss for a user as a result of using GenoGuard, compared to the best inference methods for genomic data; once again, showing that that it is non-negligible. [**Paper Organization.**]{} The rest of the paper is organized as follows. The next section reviews notions used throughout the paper, then, in Section \[sec:genoguard\], we introduce GenoGuard. Section \[sec:evaluation\] presents our evaluation methodology for low and high-entropy settings, while Section \[sec:results\] reports our experimental results. Finally, after reviewing related work in Section \[sec:related work\], the paper concludes in Section \[sec:conc\].
Preliminaries {#sec:preliminaries}
=============
This section provides some relevant background information used throughout the paper.
Genomics Primer {#sec:genomics}
---------------
[**Genome.**]{} In the nucleus of an organism’s cell, double stranded deoxyribonucleic acid (DNA) molecules are packaged into thread-like structures called chromosomes. DNA molecules consist of two long and complementary polymer chains of four units called nucleotides, described with the letters A, C, G, and T. All chromosomes together make up the [*genome*]{}, which represents the entirety of the organism’s hereditary information; in humans, the genome includes 3.2 billion nucleotides. A [*gene*]{} is a particular region of the genome that contain the information to produce functional molecules, in particular proteins. For instance, the BRCA2 [@yoshida_role_2004] is a human tumor suppressor gene (it encodes a protein responsible for repairing the DNA), and a mutation in that gene increases significantly the risk for breast cancer [@friedenson2007brca1]. [*Alleles*]{} are the different versions of genes, as organisms inherit two alleles for each gene, one from each parent. The set of genes is also called the [*genotype.*]{} Finally, the [*haplotype*]{} is a group of alleles in an organisms that are inherited together from a single parent [@clarke_disentangling_2005].
[**SNPs and SNVs.**]{} Humans share about 99.5% of the genome, while the rest differs due to genetic variations. The most common type of variants are Single Nucleotide Polymorphisms (SNPs) [@reference_what_nodate], which occur at a single position and in at least 1% of the population. More generally, variants at specific positions of a genome are referred to as Single-Nucleotide Variants (SNVs); they may be due to SNPs, to rare variants in the population, or to new mutations. Typically, SNPs and SNVs are encoded with a value in $\{0,1,2\}$, with $0$ denoting the most common variant (allele) in the population, and $1$ and $2$ denoting alternative alleles.
[**Allele Frequency (AF).**]{} The frequency of an allele at a certain position in a given population is known as Allele Frequency (AF). More specifically, it is the ratio of the number of times the allele appears in the population over the total number of copies of the gene. In a nutshell, it shows the genetic diversity of a species’ population.
[**Linkage Disequilibrium (LD).**]{} LD refers to the non-random association of alleles at two or more positions in the general population, defined as the difference between the frequency of a particular combination of alleles at different positions and the one expected by random association.
[**Recombination Rate (RR).**]{} The process of determining the frequency with which characteristics are inherited together is known as recombination. This is due to two chromosomes of similar composition coming together and performing a molecular crossover, thus, exchanging the genetic content. Because recombination can occur with small probability at any location along the chromosome, the frequency of recombination between two locations depends on the distance separating them. Therefore, for genes sufficiently distant on the same chromosome, the amount of crossover is high enough to destroy the correlation between alleles [@li_modeling_2003]. The recombination rate (RR), as defined in [@philips__nodate], is the probability that a transmitted haplotype constitutes a new combination of alleles different from that of either parental haplotype. An example of how a haplotype is created by copying parts from the other haplotypes is illustrated in Figure \[fig:recomb\].
![An example of a haplotype, $h_4$, built as an imperfect mosaic from $h_1, h_2, h_3$. $h_4$ is created by (imperfectly) “copying” parts from $h_1, h_2$, and $h_3$. Each column of circles represents a SNP locus, with the black and white circles denoting the two alleles – major and minor. (Adapted from [@li_modeling_2003]).[]{data-label="fig:recomb"}](figures/Recomb.png){width="0.735\columnwidth"}
Markov Chains
-------------
A Markov chain is a probabilistic model encoding a sequence of possible events: the probability of each one of them depends only on the state attained in the previous event [@MarCha].
In the context of genomes, a Markov chain can represent a series of SNVs ordered by their positions. In particular, a $k$-th order Markov chain, on genome sequences, can be used to encode a set of SNVs, where the value of each SNV$_i$ depends on the values of the $k$ preceding ones: $$\Pr(SNV_i) = \Pr(SNV_i|SNV_{i-1},\hdots,SNV_{i-k})$$
SNV Correlation Modeling {#sec:correlation}
------------------------
In order to model correlations between SNVs, and perform sequence inference (i.e. predicting the values of SNVs from a sequence), one can use a few different approaches (for more details on various SNV correlations, please refer to [@samani_quantifying_2015]). We choose three models; see next.
[**Most likely genotype.**]{} First, we use a model based on the 1st order Markov chain model from AF and LD. Given allele frequencies (AF) and linkage disequilibrium (LD), we predict each SNV using the highest conditional probability of the SNV occurring. For each SNV, the joint probability matrix is computed taking into consideration the LD with previous one and the AF. If a SNV is not in LD with the previous one, the probability is computed using only the allele frequency. When this model is used for inference, the highest value from the joint probability matrix or the highest probability given by the AF is chosen to predict the specific SNV.
[**Sampled genotype.**]{} The second model is built from the 1st order Markov chain model from AF and LD. For this model, the conditional probabilities are computed in a similar way as in the most likely genotype model. The main difference is in the choice of the value of the SNV, given the three computed probabilities for major homozygous $\Pr_0$, heterozygous $\Pr_1$, and minor homozygous allele $\Pr_2$. A seed $s$ is chosen uniformly at random from the interval $[0,1)$. If $s<\Pr_0$, then choose the SNV to be major homozygous; if $\Pr_0 \leq s<\Pr_1 +\Pr_0$, then the SNV is heterozygous; and minor homozygous otherwise.
[**RR Model.**]{} This is a high-order correlation model that relates LD patterns to the underlying recombination rate [@Li2213]. Given a set of $n$ sampled haplotypes, $\{h_1, h_2,...,h_n\}$, the model relates their distribution to the underlying recombination rate. Given the recombination parameter, $\rho$, we have: $$\begin{split}
&\Pr(h_1,...,h_n|\rho) =\\
&=\Pr(h_1|\rho)\cdot\Pr(h_2|h_1;\rho)\cdot\ldots\cdot\Pr(h_n|h_1,\hdots ,h_{n-1};\rho)
\end{split}$$ We use this model to determine the value of a SNP at a given position. At each SNP, $h_k$ is a possibly imperfect copy of one of $h_1,...,h_{k-1}$. Let $H_i$ denote which haplotype is copied at a position $i$. For instance, in the example presented in Figure \[fig:recomb\], for $h_4$, we have $(H_1,H_2,H_3, H_4) = (3,2,2,1)$. For a generic $h_k$, each $H_i$ can be modeled as a Markov chain on $\{1,\hdots, k-1\}$. Assuming that one part of $h_{k}$ comes from $h_i$, the next adjacent part can be copied from any of the $k-1$ haplotypes, and the probability depends on the recombination rates between these two parts. Overall, the probability of a particular haploid genotype $h_{k}$ can be computed as the sum over all possible event sequences of recombination and mutation that could lead to $h_{k}$. Let $h_{i,j+1}$ denote the allele found at position $j+1$ in haplotype $i$, and $h_{i,\leq j}$ denote the values of the first $j$ positions of haplotype $i$ (i.e. the prefix sequence of $h_{i,j+1}$). Then, we can compute the conditional probability of an allele $h_{k,j+1}$, given all preceding alleles as: $$\Pr(h_{k,j+1}| h_{k,j}, \hdots h_{k,1}) = \frac{\Pr(h_{k,\leq j+1})}{\Pr(h_{k,\leq j)}}$$
Honey Encryption {#sec:he}
----------------
Honey Encryption (HE) [@HE2] is a cryptographic primitive used to provide confidentiality guarantees in the presence of possible brute-force attacks. It is a variant of Password-based Encryption (PBE), in that it also uses an arbitrary string (password) to perform randomized encryption of a plaintext. Its main property is that all decryptions of a ciphertext will yield a plausible-looking plaintext, which is thus indistinguishable from the correct one.
The main building block of HE is the Distribution-Transforming Encoder (DTE). A DTE is a randomized encoding scheme [(encode,\
decode)]{} tailored on the target distribution. The [encode]{} algorithm takes as input a message $M$ from the message space $\mathcal{M}$, and outputs a value $S$ in a set $\mathcal{S}$, i.e., the seed space. Whereas, [decode]{} takes a seed $S \in \mathcal{S}$ and outputs a message $M \in \mathcal{M}$. A DTE scheme is [*correct*]{} if, for any $M \in \mathcal{M}$, $\Pr[$[decode$($encode$(M))$]{}$=M]=1$. The DTE-then-encrypt scheme presented in [@HE2] applies [encode]{} to a message, and then performs encryption using a secure symmetric encryption scheme (e.g., AES). Similarly, to decrypt a ciphertext, one first decrypts using the underlying cipher (e.g., AES), and then applies the [decode]{} algorithm.
[**Terminology.**]{} In the rest of the paper, to denote sequences decrypted from GenoGuard, we use the term *honey sequences*.
{width="90.00000%"}
GenoGuard {#sec:genoguard}
=========
In this section, we review GenoGuard [@huang_genoguard:_2015], along with a security analysis of the framework.
Construction {#subsec:ggdesc}
------------
GenoGuard is a framework providing long-term confidentiality for genomic data based on Honey Encryption [@HE2]. More specifically, it allows to encode genomic data, encrypt it using a secret password, and store in a database, in such a way that its confidentiality is preserved even against an attacker that can brute-force all possible passwords. In GenoGuard, genomes are represented as a sequence of single-nucleotide variants (SNVs), i.e., values in $\{0,1,2\}$.
[**Encoding.**]{} The construction uses a DTE scheme optimized for genome sequences. It assigns subspaces of seed space $\mathcal{S}$ to the prefixes of a sequence $M$, i.e., all the subsequences in the set $\{M_{1,i}| 1\leq i \leq n\}$, where $n$ is the length of the sequence. For example, the prefixes of the sequence $01102$ are $\{0, 01, 011, 0110, 01102\}$. The seed space $\mathcal{S}$ is the interval $[0,1)$, with each seed being a real number in this interval.
Let $\mathcal{M}$ be the set of all possible sequences (the plaintext space). To calculate the cumulative distribution function (CDF) of each sequence, a total order $\mathcal{O}$ is assigned to all sequences in $\mathcal{M}$. For any two different sequences $M$ and $M'$, we assume that they start to differ at SNV$_i$ and SNV$'_i$. If the value of SNV$_i$ is smaller than that of SNV’$_i$, then, $\mathcal{O}(M)< \mathcal{O}(M')$, and $\mathcal{O}(M)> \mathcal{O}(M')$ otherwise. The CDF of a sequence $M$ is then calculated as:
$CDF(M)=\sum_{M' \in \mathcal{M},\mathcal{O}(M')\leq \mathcal{O}(M)} \Pr_{SNV}(M')$
where $\Pr_{SNV}(M')$ is the probability of the sequence $M'$.
The encoding of a sequence can be performed using a perfect ternary tree, as depicted in Figure \[fig:tree\]. (Note that the plot was generated using code obtained from GenoGuard’s Github page.[^1]) Each node in the tree represents a prefix of a sequence, and each leaf a complete sequence. Nodes have an interval $[L_i^j, U_i^j)$, where $i$ is the depth of the node in the tree and $j$ its order at a given depth $i$. The first node has the interval $[L_0^0, U_0^0) = [0,1)$. Depending on the value of the SNV at position $i{+}1$, the encoding proceeds from the node that represents $M_{1,i}$ with order $j$ at depth $i$ to depth $i+1$ as follows:
- If SNV$_{i+1} = 0$, go to the left branch and attach an interval $[L_{i+1}^{3j},U_{i+1}^{3j}) = [L_i^j, L_i^j + (U_i^j-L_i^j)\times \Pr(SNV_{i+1}=0|M_{1,i}))$
- If SNV$_{i+1}=1$, go to the middle branch and attach an interval $[L_{i+1}^{3j+1},U_{i+1}^{3j+1}) = [L_i^j + (U_i^j-L_i^j)\times \Pr(SNV_{i+1}=0|M_{1,i}), L_i^j + (U_i^j-L_i^j)\times( \Pr(SNV_{i+1}=0|M_{1,i} + \Pr(SNV_{i+1}=1|M_{1,i})))$
- If SNV$_{i+1} = 2$, go to the right branch and attach an interval $[L_{i+1}^{3j+2},U_{i+1}^{3j+2}) = [L_i^j + (U_i^j-L_i^j)\times( \Pr(SNV_{i+1}=0|M_{1,i} + \Pr(SNV_{i+1}=1|M_{1,i})), U^j_i)$.
In order to compute the conditional probabilities, Huang et al. [@huang_genoguard:_2015] consider several models and compare their goodness of fit for real-world genome datasets. Specifically, they experiment with Linkage Disequilibrium (LD), Allele Frequencies (AF), building $k$-th order Markov chains on the dataset and recombination rates (RR), and find the latter to perform best. Finally, when a leaf is reached, a seed is picked uniformly from this range as the encoding of the corresponding sequence, and then fed into a Password-based Encryption (PBE) scheme to perform encryption, using a password chosen by the user. [**Decoding.**]{} To decode an encoded-then-encrypted sequence, the ciphertext is first decrypted (as per the PBE scheme) using the user-chosen password; this recovers the seed. Then, the decoding process proceeds similar to the encoding one. That is, given the seed $S \in [0,1)$, at each step, the algorithm computes three intervals for the three branches, chooses the interval in which the seed $S$ falls, and moves down the tree. Once a leaf node is reached, the path from the root to the leaf is outputted as the decoded sequence.
[**Finite Precision.**]{} Note that the Honey Encryption encoding model, as described in Section \[subsec:ggdesc\], requires the seed space $\mathcal{S}$ to be a real number domain with infinite precision. In the case of DNA sequences, this would yield a very long floating-point representation, and thus a high storage overhead. Therefore, GenoGuard uses a modification of the DTE scheme for finite precision. Specifically, for a sequence of length $n$, where each SNV takes three possible values, at least $n \cdot \log_23$ bits are needed for storing the sequence. Hence, a storage overhead parameter $h > \log_23$ is selected, and each sequence is encoded over $h\cdot n$ bits. The algorithm works as before, by selecting intervals according to the values of the respective SNVs based on conditional probabilities. The root interval is $[0,2^{hn}{-}1]$. At each branch at depth $i$, the algorithm will allocate a seed space of size $3^{n-i-1}$, and each following step will segment an input interval into three parts of equal size. Hence, any subinterval of the $j$-th node at depth $i$ will contain $3^{n-i-1}$ integers.
Security {#sec:security}
--------
Huang et al. [@huang_genoguard:_2015] evaluate the security of GenoGuard vis-à-vis the probability of an unbounded adversary recovering the encrypted sequence. That is, given the encryption of a message, what is the probability of the adversary recovering the correct message, even if she can brute-force all possible encryption keys for the underlying PBE scheme?
[**Upper Bound.**]{} More formally, they prove an upper bound to the probability an adversary recovers the correct message to be: $$\label{eq:secu}
Pr_{p_m,p_k} \leq w(1+\delta) + \frac{3^n +1/w}{2^{(h-\log_23)n}}$$ where $p_m$ is the original sequence distribution with maximum sequence probability $\gamma$, $p_k$ is a key (password) distribution with maximum weight $w$ (i.e., the most probable password has probability $w$), $n$ is the length of the sequence, $h$ the overhead parameter, and $\delta$ a parameter depending on $w$ and $\gamma$. Let $\Delta$ denote the fraction $\frac{3^n +1/w}{2^{(h-\log_23)n}}$ in Equation \[eq:secu\]. Note that $\Delta$ is a security loss term, since the upper bound on plaintext recovery probability should be $w$, as an adversary who trivially decrypts the ciphertext with the most probable key and outputs the result can recover the original message with probability $w$. $\Delta$ is essentially the security lost due to DTE imperfectness when moving to finite precision, i.e., given by the difference between the original message distribution and the DTE distribution. As shown in [@huang_genoguard:_2015], for $n = 20{,}000$, $h=4$, $w = \frac{1}{100}$, and $\gamma = 2.89\times 10^{-44}$, $\Delta$ is approximately $2^{-16600}$.
[**Empirical Evaluation.**]{} Huang et al. [@huang_genoguard:_2015] also present an [*empirical*]{} security analysis based on two experiments. In both, the chromosome 22 of a victim is encrypted using a password pool consisting of numbers from 1 to 1000, with “539” assumed to be the correct one. Then, in order to rule out wrong passwords, the interval size of each of the decrypted sequences is computed. In the first experiment, a genome is encoded by assuming a uniform distribution (i.e., each branch has weight $1/3$ at all depths), and a PBE scheme is used to encrypt the seed. In the second experiment, GenoGuard is used to encrypt the victim’s sequence. Hence, the size of the interval of a leaf in the ternary tree is proportional to the probability of the corresponding sequence. The results of their experiments, reported in Figure 10 in [@huang_genoguard:_2015], show that a simple classifier can distinguish the correct sequence in the first experiment, while, in the second one, it is “buried” among all the decrypted sequences.
Evaluation Methods {#sec:evaluation}
==================
We now describe our evaluation methods, for both low and high-entropy password settings. Before doing so, we introduce the notation used in the rest of the paper in Table \[tab:notations\].
Low-Entropy vs High-Entropy Password
------------------------------------
We use different approaches for evaluating GenoGuard under two different password types, namely low-entropy and high-entropy passwords. In other words, we encrypt a sequence with GenoGuard using either an easy to guess, low-entropy password ($\approx$7 bits), or using a harder password with a higher entropy ($\approx$72 bits).
The difference in the evaluation of the two approaches is given by the adversary’s goal. Specifically, in the low-entropy password case, the adversary attempts to use the side information in order to distinguish the original encrypted sequence among a pool of honey sequences. By contrast, in the high-entropy setting, the adversary uses both the honey sequences and the side information in order to predict the value of each SNV at each position in the target sequence.
[**Symbol**]{} [**Meaning**]{}
---------------- -------------------------------------------
MR Message recovery
SI Side information
HEnc Honey Encryption
HDec Honey Decryption
$\mathcal{K}$ Key space
$\mathcal{M}$ Message space
$p_k$ Key distribution
$p_m$ Message distribution
$\adv$ Adversary
$\adv^{SI}$ Adversary with access to side information
: Notation.[]{data-label="tab:notations"}
Threat Model {#subsec:threat-low}
------------
We use the same system and threat model presented in the GenoGuard paper [@huang_genoguard:_2015], i.e., we assume a genomic sequence of a user is to be stored, encrypted, at a third-party database, e.g., a biobank. We consider an adversary that has access to the encrypted data (for instance, she breaks into the biobank and gets access to the encrypted database, or the biobank itself is adversarial) and has access to public knowledge as well as to some side information (as discussed below).
[**Low-Entropy Password.**]{} The main adversarial goal in this case is to identify the target sequence among a pool of honey sequences, using the side information available, i.e. “message recovery” with side information (**MR-SI**).
[**High-Entropy Password.**]{} The main adversarial goal is to obtain as much information as possible about the sequence that was encrypted. Note that this adversarial goal is different from “message recovery,” according to which Huang et al. [@huang_genoguard:_2015] evaluate GenoGuard’s security (cf. Section \[sec:security\]). The main intuition is that, as also hypothesized by [@jaeger2016honey], using Honey Encryption might actually leak non-negligible information about the sequences encrypted using GenoGuard, even if the adversary cannot correctly recover the full plaintext with non-negligible probability.
Adversary’s Side Information {#subsec:adv}
----------------------------
As mentioned above, the adversary has access to the victim’s encrypted sequence as well as to public information such as, Linkage Disequilibrium, Allele Frequencies, Recombination Rate (see Section \[sec:genomics\]). In addition, we assume that the adversary may have some side information about the victim.
When referring to side information, note that we do [*not*]{} consider knowledge of common traits from phenotype-genotype associations, e.g., gender, ancestry, or other information about the victim that could be obtained, e.g., from social media. In fact, this is covered by GenoGuard’s guidelines, which state that the user should include as much side information about their genome as possible when performing the encoding. Whereas, even though assuming the user can knowingly enumerate all possible side information is quite a strong assumption, we actually consider the case where the victim undertakes some specific tests, and the adversary learns additional information about the victim from the outcome of those tests. Additionally, the victim might choose to re-encrypt their genome after obtaining the test results in order to incorporate them in the encoding, and the adversary could use the new ciphertext to extract information about the old ciphertext.
In the high-entropy password setting, we also evaluate the case where an adversary has no side information about the target sequence, in order to quantify the information leakage that might occur from using GenoGuard against baseline inference methods for genomic sequences. Overall, we consider different types of side information available to the adversary:
*No Side Information:* The adversary has access only to the encrypted sequence. (NB: this is only evaluated for the high-entropy password setting)
*Sparse SNVs:* The adversary has access to SNV values sparsely distributed in the target sequence.
*Consecutive SNVs:* The adversary has access to values from a cluster of consecutive SNVs in the target sequence.
Low-Entropy Password {#sec:low ent}
--------------------
We now formally provide a lower bound for the adversary’s advantage in the case where she obtains side information about the target sequence and encryption is done using a low-entropy password.
We present a lower bound on the adversary’s advantage when she has access to side information about the encrypted sequence and can exhaustively search the message space. We prove the bound formally, building on [@jaeger2016honey], which shows the impossibility of known-message attack (KMA) security with low-entropy passwords. However, instead of the adversary having access to message-ciphertext pairs, we assume that the adversary has access to (position, value) pairs from the encrypted sequence. The game defining message recovery security with side information is denoted as **MR-SI$^\adv_{HE,p_m, p_k}$** and illustrated in Figure \[fig:mrsi\].
Given a ciphertext $C^*$, an adversary $\adv^{SI}$, with access to side information, is allowed to guess the message by brute force. The adversary $\adv^{SI}$ wins the game if her output message is the same as the original message.
Our intuition is that the advantage of the adversary $^{SI}$ (Figure \[fig:a-si\]), for a number $q$ ($q\leq 2n$, where $n= [\log_2|\mathcal{K}|]$) of positions and values, from the original sequence, is equal to the probability that a randomly chosen key that decrypts correctly all values at the given positions, will also decrypt the rest of the sequence, i.e., = $\Pr[\mathrm{MR-SI}_{HE,p_m,p_k}^\adv]$. We denote by $K_q$ the number of keys consistent with the positions and values used as side information.
Hence, we use Lemma 4.2 from [@jaeger2016honey], as follows:
\[lemma\] If $s_0, s_1, ..., s_{n}$ are positive integer-valued random variables such that $s_0\leq2^{n}$ and $s_{q+1}\leq s_q$, for $q\in \mathbb{Z}_{n}$, then $\mathrm{max}_{q\in \mathbb{Z}_{n}} \expect{s_{q+1}/s_q}\geq \frac{1}{2n}$.
See [@jaeger2016honey].
Using Lemma \[lemma\], we can compute the adversary’s advantage as follows:
Let HE be an encryption scheme and $n = [\log_2|K|]$. Then, for any $p_m, p_k$, the adversary $^{SI}$ who obtains at most $n-1$ positions and values from the original sequence will have advantage:
\[(\^[SI]{})\] $\geq \frac{1}{2n^2}$
\[thm\]
The advantage $\advantage{MR-SI}{HE,p_m,p_k}[(\adv^{SI})]$, is equal to $\Pr[\mathrm{Game\ 1\ Retuns\ 1}]$ where Game 1 is defined in Figure \[fig:g1\]. This is due to the fact that Game 1 is MR-SI$^\adv_{HE,p_m, p_k}$ together with Adversary $\adv^{SI}$(C$^*$). By applying a few transformations to Game 1 and changing the final check, i.e. instead of checking if $M=M^*$ before returning 0 or 1, it checks if the key $K$ is in the subset that decrypts $C^*$ to $M^*$ we obtain an equivalent game, Game 2 (Figure \[fig:g2\]). Thus, $\Pr[\mathrm{Game\ 1} \mathrm{\ Returns\ } 1] = \Pr[\mathrm{Game\ 2} \mathrm{\ Returns\ } 1]$.
Since $K_{q+1}\subseteq K_q$, for fixed $q$, the probability that Game 2 will return 1 is $\expect{\frac{|K_{q+1}|}{|K_q|}}$. So we have $\Pr[\mathrm{Game\ 2} \mathrm{\ Returns\ } 1] = \sum^{n}_{q=0}\frac{1}{n}\expect{\frac{|K_{q+1}|}{|K_q|}}\vspace{0.1cm}$.
We then define Experiment 1 (Figure \[fig:e1\]), which shows that the distribution of $K_{q+1}$ and $K_q$ for $q \in \mathbb{Z}_n$ is the same as the distribution in Game 1. Let $s_q$ denote $|K_q|$ and $\epsilon = max_{q\in \mathbb{Z}_n}\expect{\frac{s_{q+1}}{s_q}}$, where the expectation is taken in Experiment 1. Since all $K_q$ contain at least the key $K^*$, they all are positive. Thus, by applying Lemma \[lemma\] we have $\epsilon \geq \frac{1}{2n}$. Then:
\[(\^[SI]{})\] = $\Pr[\mathrm{Game\ 2\ Returns\ 1} ]\newline = \sum^{n}_{q=0}\frac{1}{n}\expect{\frac{|K_{q+1}|}{|K_q|}} \geq \frac{1}{n}\cdot \epsilon \geq \frac{1}{2n^2}$
This shows that the security of the systems is weak even with a small number of pairs (position, value) from the target sequence available to the attacker, as opposed to having multiple known ciphertext-plaintext pairs.
High-Entropy Password {#sec:attacks-high}
---------------------
We now give an overview of our inference strategy using the GenoGuard ciphertext and discuss the baseline inference methods we evaluate our strategy against.
### Baseline Inferences {#sec:inference}
We compare the performance of our inference strategy to baselines for genomic sequence inference. For these baselines, we assume that the adversary has access only to side information, as discussed in Section \[subsec:adv\], but not the ciphertext resulting from GenoGuard’s encode-then-encrypt method.
As done by Samani et al. [@samani_quantifying_2015], we set to infer the value of an unknown SNV$_i$, given a probabilistic modeling of genome sequences. More specifically, we use the following models for SNV correlation:
*B1:* 1st order Markov chain model from AF and LD: most likely genotype.
*B2:* 1st order Markov chain model from AF and LD: sampled genotype.
*B3:* RR Model.
### GenoGuard Inference Methods {#subsec:attack-high}
Our method is based on exploiting the similarities between the honey sequences in order to obtain information about the target sequence. More specifically, we use two strategies:
1. *G1.* Side information-weighted SNVs: We assign a weight to each of the honey sequences according to the amount of side information contained. We then consider only the sequences with the highest weight and output the most common SNVs among them as our candidate SNVs for the target sequence. In the case of no side information, we consider the most common SNVs across all honey sequences.
2. *G2.* Interval and Side information-weighted SNVs: Similar to the previous method, however, we also adjust the weight of each sequence when considering the most common SNVs by the size of the interval that the seed of the respective sequence will fall into. In the case of no side information, we take the most common SNVs from all honey sequences, weighted by the previously mentioned interval size.
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Experimental Evaluation {#sec:results}
=======================
In this section, we present the datasets used for the experimental evaluation and the results obtained for both evaluation methods, i.e., low-entropy and high-entropy passwords.
Dataset {#subsec:data}
-------
We use the Phase III data from the HapMap dataset, i.e., the third release from the HapMap project.[^2] HapMap was an international project [@international2003international], run between 2002 and 2009, aimed at developing a haplotype map of the human genome, and describe the common patterns of human genetic variation. The HapMap data has been made publicly available and used for various research purposes, e.g., to research genetic variants affecting health, disease and responses to drugs and environmental factors, etc.
The Phase III release increased the number of DNA samples to 1,301 and included 11 different populations. In our experiments, we select data from three populations:
ASW (African ancestry in Southwest USA),
CEU (Utah residents with Northern and Western European ancestry from the CEPH collection),
CHB (Han Chinese in Beijing, China).
We sample 50 sequences at random from each of them, for a total of 150 sequences.
For all three populations presented above, we test the same SNVs positions.
Low-Entropy Password {#subsec:low-results}
--------------------
### Experiment Overview {#subsec:attack-low}
We use the following strategy for our evaluation:
Encrypt a sequence using GenoGuard’s DTE-then-encrypt method: for each of the 150 sequences, we select and encrypt 1,000 positions from chromosome 13, with a storage overhead $h = 4$ (the same as in the experimental evaluation of GenoGuard), using a low-entropy password.
Decrypt the ciphertext, using the top 10,000 most common passwords released by Daniel Miessler[^3] (with the encryption password in the set), to obtain plausible looking honey sequences;
Exclude the sequences which do not contain the side information.
Output the number of remaining sequences, given how many of the possible passwords match the side information.
### Adversary’s Advantage.
The performance of the adversary is calculated as the probability of the adversary guessing the target sequence within the remaining pool of honey sequences.
### Sparse SNVs from the Target Sequence
Figure \[fig:low\_rnd\_seqs\] illustrates how the log number of candidate sequences decreases with more side information available. With 1% side information (10 SNVs), the number of sequences that match the side information reduces to approximately 44 on average across the three populations. Figure \[fig:low\_rnd\] shows the increase of the adversary’s advantage, averaged over 1000 rounds, vis-à-vis the number of SNVs available to her. 2.5% side information (25 SNVs) gives the adversary an advantage of approximately 80% on average for the ASW and CEU populations and close to 90% for the CHB population. With more side information, the adversary’s advantage increases to over 90% for all populations.
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### Consecutive SNVs form the Target Sequence
When the adversary has access to side information as a cluster of consecutive SNVs, she needs more side information to achieve comparable results to the Sparse SNVs case. Figure \[fig:low\_consec\_seqs\] shows the decrease of the log number of candidate sequences with increasing side information available. We observe the fastest decrease in the number of sequences with increasing side information available is for the ASW population when less than 10% of the sequence available. Figure \[fig:low\_consec\] shows the increase of the adversary’s advantage, averaged over 1000 rounds, vis-à-vis the number of SNVs available to her. The increase in the adversary’s advantage is slower as well, with an average of 70% across the three populations for 20% of the sequence available to the attacker.
High-Entropy Password {#subsec:high-results}
---------------------
### Experiment Overview {#subsec:overview}
The brute-force experiment presented in GenoGuard indicates that, when decrypting the same ciphertext with multiple passwords, the correct sequence would be “buried” among the incorrect ones. Hence, there is some similarity between the original sequence and the honey sequences. As a result, we set to quantify the corresponding privacy loss, i.e. [***how much more information does an adversary obtain via access to ciphertext encrypted using GenoGuard obtains, compared to one that was not***]{}.
Overall, we use the following evaluation strategy:
Encrypt a sequence using GenoGuard’s DTE-then-encrypt method: for each of the 150 sequences, we select and encrypt 1,000 positions from chromosome 13, with a storage overhead $h = 4$, using a random, high-entropy password (approx. 72 bits).
Decrypt the ciphertext, using the top 10,000 most common passwords released by Daniel Miessler, to obtain plausible looking honey sequences;
Infer the victim’s sequence using the honey sequences.
### Accuracy
To measure the performance and assess the potential leakage that access to the GenoGuard ciphertext might yield, we measure the accuracy as the number of correctly guessed SNVs over the total number or SNVs guessed.
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### Sparse SNVs from the Target Sequence
Figure \[fig:sparse\_high\] shows the inference results in this case for the three population groups, averaged over 1,000 rounds. In the case where no side information is available to the attacker, for all populations, the attacker can infer approximately 2% more of the target sequence from the GenoGuard ciphertext than just by using baseline inferences based on the population. For the ASW population (Figure \[fig:ASW\_rnd\]), over 80% of the target SNVs are guessed correctly with 2.5% (25 SNVs) or more of the target sequence available to the attacker. For the CEU population (Figure \[fig:CEU\_rnd\]), approximately 79% of the target SNVs are guessed correctly with 2.5% of the original sequence available to the attacker and over 83% of the target SNVs are guessed correctly with 5% (50 SNVs) or more are available. In the case of the CHB population (Figure \[fig:CHB\_rnd\]), the accuracy is of the GenoGuard inference is the lowest among the three populations, with over 73% accuracy in the cases where 2.5% SNVs are available to the attacker. The accuracy surpasses 80% for the CHB population when 10% or more of the target SNVs are available to the attacker.
In Figure \[fig:delta random\], we illustrate the difference between the best performing inference method using the GenoGuard ciphertext and the best performing baseline inference method. On average, having access to the GenoGuard ciphertext improves the inference accuracy. The peak of the improvement in accuracy (approximately 15%) over the baseline models can be observed when the attacker has access to 5% sparse SNVs from the target sequence. After this, we can see a decline in this difference with increasing SNVs available for the attacker, as the baseline inference becomes more accurate with more information available. In fact, for the CHB population, the best performing baseline (B3) for the case when 20% of the target sequence is available to the attacker provides an accuracy comparable to the GenoGuard inferences ($\approx$83.8$\%$).
### Consecutive SNVs form the Target Sequence
In Figure \[fig:consec\_high\], we illustrate the accuracy of the inference methods across the three populations when the adversary obtains, as side information, a cluster of consecutive SNVs, averaged over 1,000 rounds. For the ASW population (Figure \[fig:ASW\_consec\]), the accuracy of inferred SNVs from the correct sequence using the GenoGuard ciphertext is over 73% for 2.5% or more of the target SNVs available as side information, and over 80% when 10% or more of the sequence is available to the attacker. The GenoGuard inference for the CEU population (Figure \[fig:CEU\_consec\]) is over 70% when 2.5% or more of the target sequence is available to the attacker. For the CHB population (Figure \[fig:CHB\_consec\]), the GenoGuard inferences have the lowest accuracy across the three populations, obtaining 70% or more accuracy only when 5% or more of the target sequence is available to the attacker.
Figure \[fig:delta\_consec\] shows the difference between the best performing GenoGuard inference method and the best performing baseline inference method. On average, the inference using the GenoGuard ciphertext gives better accuracy than the baseline methods, but overall less than the previous case where sparse SNVs are available as side information. In this case, the peak in the improvement of accuracy compared to the baseline methods is approximately 7%, on average, across the three populations, when 5% of the target SNVs are available to the attacker. For the CHB population, when 20% of the sequence is available as side information to the attacker, we observe, as in the case of sparse SNVs, that the best performing baseline inference method (B3) obtains a comparable accuracy to that of the GenoGuard inferences ($\approx$73$\%$).
Take-Aways
----------
Overall, our experimental evaluation shows that, when the adversary has access to some side information, access to a ciphertext encrypted using GenoGuard can help her recover a remarkably high percentage of the SNVs from the target sequence or significantly increase her advantage in recovering the correct sequence.
Therefore, users need to include as much side information as possible when encrypting their genomic sequence. However, this prompts a parallel problem, with respect to how much that user is willing to publicly share (as this information is saved together with the ciphertext), considering that even without access to the GenoGuard ciphertext, it can enable attackers to correctly predict most of the target genome.
Related Work {#sec:related work}
============
In this section, we review relevant prior work on genome privacy and honey encryption.
Genome Privacy
--------------
[**Re-identification.**]{} Genomic data is hard to anonymize, due to the genome’s uniqueness as well as correlations within different regions. For instance, Gymrek et al. [@gymrek_identifying_2013] demonstrate that surnames of genomic data donors can be inferred using data publicly available from recreational genealogy databases. They also discuss how, through deep genealogical ties, publishing even a few markers can lead to the identification of another person who might have no acquaintance with the one who released their genetic data. In follow-up work, Erlich et al. [@Erlich690] show that a genetic database which covers only 2% of the target population can be used to find a third-cousin of nearly any individual.
[**Membership inference.**]{} Homer et al. [@homer_resolving_2008] present a membership inference attack in which they infer the presence of an individual’s genotype within a complex genomic DNA mixture. Wang et al. [@wang2009learning] improve on the attack using correlation statistics of just a few hundreds SNPs, while Im et al. [@im2012sharing] rely on regression coefficients. Shringarpure and Bustamante [@beacon_SB] perform membership inference against the Beacon network.[^4] They use a likelihood-ratio test to predict whether an individual is present in the Beacon, detecting membership within a Beacon with 1,000 individuals using 5,000 queries. Also, Von Thenen et al. [@von_Thenen200147] reduce the number of queries to less than 0.5%. Their best performing attack uses a high-order Markov chain to model the SNP correlations, as described in [@samani_quantifying_2015]. Note that, as part of the attacks described in this paper, we use inference methods from [@samani_quantifying_2015] as our baseline inference methods.
[**Data sharing.**]{} Progress in genomics research is dependent on collaboration and data sharing among different institutions. Given the sensitive nature of the data, as well as regulatory and ethics constraints, this often proves to be a challenging task. Kamm et al. [@kamm_new_2013] propose the use of secret sharing to distribute data among several entities and, using secure multi-party computations, support privacy-friendly computations across multiple entities. Wang et al. [@Wang2015] present GENSETS, a genome-wide, privacy-preserving similar patients querying system using genomic edit distance approximation and private set difference protocols. Then, Chen et al. [@chen_princess:_2017] use Software Guard Extensions (SGX) to build a privacy-preserving international collaboration tool; this enables secure and distributed computations over encrypted data, thus supporting the analysis of rare disease genetic data across different continents. Finally, Oprisanu and De Cristofaro [@oprisanu2018anonimme] present a framework (“AnoniMME”) geared supporting anonymous queries within the Matchmaker Exchange platform, which allows researchers to perform queries for rare genetic disease discovery over multiple federated databases.
[**Privacy-friendly testing.**]{} Another line of work focuses on protecting privacy in the context of personal genomic testing, i.e., computational tests run on sequenced genomes to assess, e.g., genetic susceptibility to diseases, determining the best course of treatment, etc. Baldi et al. [@baldi2011countering] assume that each individual keeps a copy of their data and consents to tests done in such a way that only the outcome is disclosed. They present a few cryptographic protocols allowing researchers to privately search mutations in specific genes. Ayday et al. [@ayday_protecting_2013] rely on a semi-trusted party to store an encrypted copy of the individual’s genomic data: using additively homomorphic encryption and proxy re-encryption, they allow a Medical Center to privately perform disease susceptibility tests on patients’ SNPs. Naveed et al. [@naveed14] introduce a new cryptographic primitive called Controlled Functional Encryption (CFE), which allows users to learn only certain functions of the (encrypted) data, using keys obtained from an authority; however, the client is required to send a fresh key request to the authority every time they want to evaluate a function on a ciphertext. Overall, for an overview of privacy-enhancing technologies applied to genetic testing, we refer the reader to [@sok].
[**Long-term security.**]{} As the sensitivity of genomic data does not degrade over time, access to an individual’s genome poses a threat to her descendants, even years after she has deceased. To the best of our knowledge, GenoGuard [@huang_genoguard:_2015] is the only attempt to provide long-term security. GenoGuard, reviewed in Section \[sec:genoguard\], relies on Honey Encryption [@HE2], aiming to provide confidentiality in the presence of brute-force attacks; it only serves as a storage mechanism, i.e., it does not support selective retrieval or testing on encrypted data (as such, it is not “composable” with other techniques supporting privacy-preserving testing or data sharing). In this paper, we provide a security analysis of GenoGuard. In parallel to our work, Cheng et al. [@cheng] recently propose attacks against probability model transforming encoders, and also evaluate them on GenoGuard. Using machine learning, they train a classifier to distinguish between the real and the decoy sequences, and exclude all decoy data for approximately 48% of the individuals in the tested dataset.
Honey Encryption {#honey-encryption}
----------------
Juels and Ristenpart [@HE2] introduce Honey Encryption (HE) as a general approach to encrypt messages using low min-entropy keys such as passwords. HE, reviewed in Section \[sec:he\], is designed to yield plausible-looking ciphertexts, called honey messages, even when decrypted with a wrong password. In a nutshell, it uses a distribution-transforming-encoder (DTE) to encode a-priori knowledge of the message distribution, aiming to provide [*message recovery*]{} security against computationally unbounded adversaries. It was originally designed to encrypt credit card information, RSA secret keys, etc. [@tyagi2015honey].
Message recovery security can be defined as follows [@jaeger2016honey]: given a message encrypted under a key whose maximum probability of taking on any particular value is at most $1/2^\mu$, an unbounded adversary’s ability to guess the correct message, even given the ciphertext, is at most $1/2^\mu$ plus a negligible amount. However, Jaeger et al. [@jaeger2016honey] discuss deficiencies of message recovery security as per modern security goals. More specifically, not only they prove the impossibility of known-message attack security in the case of low-entropy keys, but also mention that schemes meeting message recovery security might actually leak a significant amount of information about the plaintexts, even if the adversary cannot correctly recover the full message with non-negligible probability. Although this serves as an inspiration to our work, note that the context of our evaluation is different, as in the low-entropy setting, we show that a lower bound also applies to the adversary’s advantage when partial information from the target sequence is available to the attacker, compared to having pairs of ciphertext and plaintext. Another work studying attacks against HE is that by Cheng et al. [@cheng], which we have reviewed above.
[**Honeywords.**]{} Before Honey Encryption [@HE2], Juels and Rivest [@juels2013honeywords] introduced the concept of “honeywords” to improve the security of password databases. They propose adding honeywords (false passwords) to a password database together with the actual password (hashed with salt) of each user. This way, an adversary who hacks into the password database and inverts the hash function cannot know whether she has found the password or a honeyword.
Wang et al. [@wang2018security] present an evaluation of the honeyword system [@juels2013honeywords], finding it to be vulnerable to a number of attacks. More specifically, an adversary that wants to distinguish between real and decoy passwords can do so with a success rate of 30% compared to an expected 5%. In the case of a targeted attack, when the adversary is assumed to know some personal information about the user, they show that the adversary’s success rate is further improved to about 60%. Our attacks differ from those in [@wang2018security], first, as they target the honeywords system [@juels2013honeywords], while we focus on Honey Encryption [@HE2], and in particular its application to GenoGuard [@huang_genoguard:_2015]. Moreover, their attack only aims to identify the correct password from a given password pool, while we also examine the case when the correct password is not found within the tried passwords.
Conclusion {#sec:conc}
==========
Motivated by the decreasing cost of genomic sequencing and the related arising privacy challenges, the research community has produced a large body of work on genomic privacy. Most of the techniques focus on cryptographic tools, but fail to address the need for long term confidentiality for genomic data. In fact, GenoGuard [@huang_genoguard:_2015] is the only tool available for ensuring the long term encryption needed for genomic data [@sok].
In this paper, we set to determine whether GenoGuard can be safely used as an encryption tool, quantifying the additional privacy leakage arising from using it. We analyzed GenoGuard under two scenarios, based on the encryption password, for an adversary which has access to side information about the target sequence in the form of some values of SNVs from the target sequence. First, we assumed that the user encrypts his genomic sequence using a low-entropy, easily guessable password. In this case, we found that the adversary can easily exclude decoy passwords from the pool of possible passwords, and can guess the correct sequence with high probability by having access to 2.5% sparse SNVs or 20% or more consecutive SNVs from the target sequence.
Second, we assumed that the user encrypts his sequence using a high-entropy password. In this case, since elimination of decoy passwords might not yield any sequence, we use the honey sequences to obtain as much information as possible from the target sequence, exploiting the similarity between the original sequence and the honey sequences [@huang_genoguard:_2015]. We then compared the sequence obtained from the honey sequences to state-of-the-art methods from genome sequence inference in order to observe the privacy leakage. Even with no side information available to the attacker, the sequence obtained from the honey sequences had a 2% improvement on average over all tested baseline methods. With side information in the form of sparse SNVs from the target sequence, the improvement in accuracy compared to the baseline inference models raises to up to 15% on average when 5% of the target sequence is available to the attacker, predicting more than 82% (on average) of the target sequence correctly. When the attacker obtains consecutive SNVs from the target sequence, the accuracy of the attacker decreases slightly from the previous case, yielding 73% accuracy when 5% of the target sequence is known, with an average improvement of 7% over the baseline methods.
In conclusion, we argue that the research community should invest more resources toward the design of long-term encryption tools for genomic data. Overall, GenoGuard could be a viable solution when the user incorporates [*all*]{} side information into the encryption. However, given the fact that all this information needs to be stored together with the ciphertext, it also prompts the question of how much is a user willing to disclose, considering that only the baseline methods can predict, with high accuracy, the correct sequence (e.g. with 20% sparse SNVs available to the attacker, her accuracy is, on average, over 82%). Users who have already used GenoGuard for long-term encryption purposes need to be aware that if further genomic information can be obtained by the attacker, it will severely diminish the security of the system. As part of future work, we plan to analyze the security of GenoGuard for side information arising from kinship associations. [**Acknowledgments.**]{} This work was supported by a Google Faculty Award on “Enabling Progress in Genomic Research via Privacy Preserving Data Sharing,” the European Union’s Horizon 2020 Research and Innovation program under the Marie Skłodowska-Curie “Privacy&Us” project (GA No. 675730), and the Swiss National Science Foundation (Grant 150654).
[^1]: <https://github.com/acs6610987/GenoGuard>
[^2]: <https://www.sanger.ac.uk/resources/downloads/human/hapmap3.html>
[^3]: <https://github.com/danielmiessler/SecLists/blob/master/Passwords/Common-Credentials/10k-most-common.txt>
[^4]: Beacons are web servers that answer questions e.g. “does your dataset include a genome that has a specific nucleotide at a specific genomic coordinate?” to which the Beacon responds yes or no, without referring to a specific individual; see: <https://github.com/ga4gh-beacon/specification>.
|
---
abstract: 'We demonstrate an implementation for an approximate rank-k SVD factorization, combining well-known randomized projection techniques with previously implemented map/reduce solutions in order to compute steps of the random projection based SVD procedure, such QR and SVD. We structure the problem in a way that it reduces to Cholesky and SVD factorizations on $k \times k$ matrices computed on a single machine, greatly easing the computability of the problem.'
author:
- Burak Bayramlý
date: 'October 15, 2013'
title: 'SVD Factorization for Tall-and-Fat Matrices on Map/Reduce Architectures'
---
Introduction {#intro}
============
[@gleich] presents many excellent techniques for utilizing map/reduce architectures to compute QR and SVD for the so-called tall-and-skinny matrices. QR factorization is turned into an $A^TA$ computation problem to be computed in parallel using map/reduce, and its key element the Cholesky decomposition, can be performed on a single machine. Let’s use $C = A^TA$ and, since
$$C = A^TA = (QR)^T(QR) = R^TQ^TQR = R^TR$$
and because Cholesky factorization of an $n \times n$ symmetric positive definite matrix is
$$C = LL^T$$
where $L$ is an $n \times n$ lower triangular matrix, and R is upper triangular, we can conclude if we factorize $C$ into $L$ and $L^T$, this implies $C = LL^T = RR^T$, we have a method of calculating $R$ of QR using Cholesky factorization on $A^TA$. The key observation here is $A^TA$ computation results an $n \times n$ matrix and if $A$ is “skinny” then $n$ is relatively small (in the thousands), then Cholesky decomposition can be executed on a small $n \times n$ matrix on a single computer utilizing an already available function in a scientific computing library. $Q$ is computed simply as $Q = AR^{-1}$. This again is relatively cheap because R is $n \times n$, the inverse is computed locallly, matrix multiplication with $A$ can be performed through map/reduce.
SVD is an additional step. SVD decomposition is
$$A = U \Sigma V^T$$
If we expand it with $A = QR$
$$QR = U \Sigma V^T$$
$$R = Q^T U \Sigma V^T$$
Let’s call $\tilde{U} = Q^T U$
$$R = \tilde{U} \Sigma V^T$$
This means if we run a local SVD on $R$ (we just calculated above with Cholesky) which is an $n \times n$ matrix, we will have calculated $\tilde{U}$, the real $\Sigma$, and real $V^T$.
Now we have a map/reduce way of calculating QR and SVD on $m \times n$ matrices where $n$ is small.
Approximate rank-k SVD
----------------------
Switching gears, we look at another method for calculating SVD. The motivation is while computing SVD, if $n$ is large, creating a “fat” matrix which might have columns in the billions would require reducing the dimensionality of the problem. According to [@halko], one way to achieve is through random projection. First we draw an $n \times k$ Gaussian random matrix $\Omega$. Then we calculate
$$Y = A \Omega$$
We perform QR decomposition on $Y$
$$Y = QR$$
Then form $k \times n$ matrix
$$B = Q^T A \label{bt}$$
Then we can calculate SVD on this small matrix
$$B = \hat{U} \Sigma V^T$$
Then form the matrix
$$U = Q \hat{U}$$
The main idea is based on
$$A = QQ^T A$$
if replace $Q$ which comes from random projection $Y$,
$$A \approx \tilde{Q}\tilde{Q}^T A$$
$Q$ and $R$ of the projection are close to that of $A$. In the multiplication above $R$ is called $B$ where $B = \tilde{Q}^T A $, and,
$$A \approx \tilde{Q}B$$
then, as in [@gleich], we can take SVD of $B$ and apply the same transition rules to obtain an approximate $U$ of $A$.
This approximation works because of the fact that projecting points to a random subspace preserves distances between points, or in detail, projecting the n-point subset onto a random subspace of $O(\log
n/\epsilon^2)$ dimensions only changes the interpoint distances by $(1 \pm
\epsilon)$ with positive probability [@gupta]. It is also said that $Y$ is a good representation of the span of $A$.
Combining Both Methods
----------------------
Our idea was using approximate k-rank SVD calculation steps where $k << n$, and using map/reduce based QR and SVD methods to implement those steps. By utilizing random projection, we would be able to work in a smaller dimension which would translate to local Cholesky, and SVD calls on $k
\times k$ matrices that can be performed in a speedy manner. Below we outline each map/reduce job.
${\mbox{\texttt{random\_projection\_map}}}(key, value)$\
[****']{}input $A$\
[****']{}returns $Y$\
[****']{}Tokenize $value$ and pick out id value pairs\
[****']{}result = ${\mbox{\texttt{zeros}}}$(1,$k$)\
[****']{}${\mbox{\texttt{for each}}}$ $j^{th}$ $token$ $\in value$\
[****']{} Initialize seed with $j$\
[****']{} $j$ = generate $k$ random numbers\
[****']{} $result = result + r \cdot token[j]$\
[****']{}${\mbox{\texttt{emit}}}$ key, result
First random projection job (whose reduce is a no-op). Each value of $A$ will arrive to the algorithm as a key and value pair. Key is line number or other identifier per row of $A$. Value is a collection of id value pairs where id is column id this time, and value is the value for that column. Sparsity is handled through this format, if an id for a column does not appear in a row of A, it is assumed to be zero. The resulting $Y$ matrix has dimensions $m \times k$.
$A^TA {\mbox{\texttt{cholesky\_job\_map(key k,value a)}}}$\
[****']{}${\mbox{\texttt{for }}}$ $i,row$ in ${\mbox{\texttt{enumerate}}} a^Ta$\
[****']{} ${\mbox{\texttt{emit }}}$ $i,row$
${\mbox{\texttt{cholesky\_job\_reduce}}}(key,value)$\
[****']{}${\mbox{\texttt{emit}}}$ $k,{\mbox{\texttt{sum}}}(v_j^k)$
${\mbox{\texttt{cholesky\_job\_final\_local\_reduce}}}(key,value)$\
[****']{}$result = {\mbox{\texttt{cholesky}}}(A_{sum})$\
[****']{}${\mbox{\texttt{emit }}}result$
The ${\mbox{\texttt{cholesky\_job\_final\_local\_reduce}}}$ step is a function provided in most map/reduce frameworks, it is a central point that collects the output of all reducers, naturally a single machine which makes it ideal to execute the final Cholesky call on by now a very small ($k \times k$) matrix. The output is $R$.
${\mbox{\texttt{Q\_job\_map}}}(key,value)$\
[****']{}During initialization, $R_{inv} = R^{-1}$, store it once for each mapper\
[****']{}${\mbox{\texttt{for }}}$ $row$ in $Y$\
[****']{} ${\mbox{\texttt{emit }}}key, row \cdot R_{inv}$
There is no reducer in the $Q{\mbox{\texttt{\_job}}}$, it is a very simple procedure, it merely computes multiplication between row of $Y$ and a local matrix $R$. Matrix $R$ is very small, $k \times k$, hence it can be kept locally in every node. The initialiation is used to store the inverse of $R$ locally, once the mapper is initialized, it will always use the same $R^{-1}$ for every multiplication.
$A^TQ{\mbox{\texttt{\_job\_map}}}(key,value)$\
[****']{}$left = row$ from $A$\
[****']{}$right = row$ from $Q$\
[****']{}${\mbox{\texttt{for }}}$ each non-zero $j^{th}$ cell in $left$\
[****']{} ${\mbox{\texttt{emit }}} j, left[j] \cdot right$
$A^TQ{\mbox{\texttt{\_job\_reduce}}}(key,value)$\
[****']{}returs $B^T$\
[****']{}$result = {\mbox{\texttt{zeros}}}(1,k)$\
[****']{}${\mbox{\texttt{for }}}row$ in $value$\
[****']{} $result = result + row$\
[****']{}${\mbox{\texttt{emit }}} key, result$
The job above takes an $AQ$ matrix which is assumed to be a join between $A$ and $Q$, per row, based on key. We split the row and deduce the $A$ part and the $Q$ part, then iterate cells of $A$ one by one, which is assumed to be sparse, and multiply the entire row of $Q$. Then for each $j^{th}$ non-zero cell of $A$, we multiply this value with the row from $Q$ and emit the multiplication result with key $j$.
The $Q^TA$ job’s formula can be seen at \[bt\]. For implementation purposes we changed this formula into
$$B^T = A^TQ$$
because as output we needed to have a $n \times k$ matrix instead of a $k
\times n$ one, which would allow us to use map/reduce SVD that translates into a local Cholesky and SVD on $k \times k$ matrices. Since we take SVD of $B^T$ instead of $B$, that changes the output as well,
$$B = U\Sigma V^T$$
becomes
$$B^T = V\Sigma U^T$$
In other words, in order to obtain $U$ of $B$, we need to take $(U_{BT}^T)^T$ from the SVD of $B^T$. This is how $A^TA$ Cholesky Job is called, this time with $B^T$ as its input data.
$Q\tilde{U}{\mbox{\texttt{\_job\_map}}}(key, value)$\
[****']{}input $Q,R$\
[****']{}returns $U$\
[****']{}initialization $\tilde{U}$ = ${\mbox{\texttt{svd}}}$ of $R$\
[****']{}${\mbox{\texttt{for }}}$ row in $Q$\
[****']{} ${\mbox{\texttt{emit }}} key, row \cdot \tilde{U}$
${\mbox{\texttt{map\_reduce\_svd}}}$\
[****']{}$Y$ = ${\mbox{\texttt{random\_projection\_map}}}(A)$\
[****']{}$R_Y$ = $A^TA{\mbox{\texttt{\_cholesky\_job}}}(Y)$\
[****']{}$Q_Y$ = $Q{\mbox{\texttt{\_job}}}$\
[****']{}$R_{BT}$ = $A^TA{\mbox{\texttt{\_cholesky\_job}}}(B^T)$\
[****']{}$U$ = $Q\tilde{U}{\mbox{\texttt{\_job}}}(R_{BT},Q)$
Discussion
----------
We performed our experiments on the Netflix dataset which has about 100 million from over 480,000 customers on 17770 movies. The implementation was programmed on Sasha distributed framework [@bayramli1], and SVD calculation on the full dataset with $k=7$ on two notebook computers, utilizing in total 6 cores took 20 minutes. Scipy SVD calculation on the same dataset is much faster, however, we need to stress our algorithms are prepared for cases where $N$ is very large, i.e. in the billions. As such, for example during projection we did not simply create and pre-store a $n
\times k$ random matrix and multiply multiple rows of $A$ with this matrix. This would certainly be possible for Netflix data where $n$ is relatively small, but would not work well in cases where $A$ is “fat”. All code relevant for this paper can be found here [@bayramli2].
There are only two passes necessary on the full dataset, and three passes on $m$ rows but with reduced $k$ dimensions this time. Perhaps predictably, the procedure spends most of its time at $A^TQ$ Job. This step performs not only a join between $A$ and $Q$, it also emits $k$ cells per non-zero value of $A$’s rows, then creates partial sums these $k$ vectors creating $n \times k$ result. If for simplicity we assume $k$ non-zero cells in each $A$ row, the complexity of this step would be $O(mk)$.
[1]{}
Gleich, Benson, Demmel, *Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures*, [arXiv:1301.1071 \[cs.DC\]]{}, 2013
N. Halko, *Randomized methods for computing low-rank approximations of matrices*, University of Colorado, Boulder, 2010
S. Dangupta, A. Gupta *An Elementary Proof of a Theorem of Johnson and Lindenstrauss*, Wiley Periodicals, 2002
M. Kurucz, A. A. Benczúr, K. Csalogány, *Methods for large scale SVD with missing values*, ACM, 2007
B. Bayramli, *Sasha Framework*, [git@github.com:burakbayramli/sasha.git](git@github.com:burakbayramli/sasha.git) Github, 2013
B. Bayramli, *Map/Reduce Code for Netflix SVD Analysis*, <http://github.com/burakbayramli/classnotes/tree/master/stat/stat_mr_rnd_svd/sasha>, Github, 2013
|
---
abstract: 'By combining new long-slit spectral data obtained with the Southern African Large Telescope (SALT) for 9 galaxies with previously published our observations for additional 12 galaxies we study the stellar and gaseous kinematics as well as radially resolved stellar population properties and ionized-gas metallicity and excitation for a sample of isolated lenticular galaxies. We have found that there is no particular time frame of formation for the isolated lenticular galaxies: the mean stellar ages of the bulges and disks are distributed between 1 and $>13$ Gyr, and the bulge and the disk in every galaxy formed synchronously demonstrating similar stellar ages and magnesium-to-iron ratios. Extended ionized-gas disks are found in the majority of the isolated lenticular galaxies, in 72%$\pm 11$%. The half of all extended gaseous disks demonstrate visible counterrotation with respect to their stellar counterparts. We argue that just such fraction of projected counterrotation is expected if all the gas in isolated lenticular galaxies is accreted from outside, under the assumption of isotropically distributed external sources. A very narrow range of the gas oxygen abundances found by us for the outer ionized-gas disks excited by young stars, \[O/H\] from 0.0 to $+0.2$ dex, gives evidence for the satellite merging as the most probable source of this accretion. At last we formulate a hypothesis that morphological type of a field disk galaxy is completely determined by the outer cold-gas accretion regime.'
author:
- 'Ivan Yu. Katkov'
- 'Alexei Yu. Kniazev'
- 'Olga K. Sil’chenko'
bibliography:
- 'ilg\_salt.bib'
title: Kinematics and stellar population in isolated lenticular galaxies
---
=1
Introduction
============
Understanding processes of galaxy formation and evolution is the greatest challenge for modern extragalactic astrophysics. A huge variety of physical processes are involved into galaxy shaping, and it is a key problem to select the dominant agents driving evolution of galaxies of different morphological types.
Lenticular galaxies were introduced by Edwin @hubble_1936 when proposing his famous morphological scheme, ‘Hubble’s fork’, as a hypothetical intermediate type between ellipticals (to the left) and spirals (to the right): the so called S0s placed by him in the centre of the ‘Hubble’s fork’ were to possess large-scale stellar disks as spiral galaxies, but lacked patchy patterns of HII-regions and spiral arms unlike them. Their smooth reddish appearance implied their resemblance to elliptical galaxies as concerning the stellar populations ages. It was also long been suggested that the intermediate position of the S0s between pure spheroids (elliptical galaxies) and disk-dominated late-type spiral galaxies obliged them to have very large bulges. However direct photometric decompositions of the digital images of a representative sample of lenticular galaxies has proved that the bulges of any size can be met in S0s, from very large to tiny ones [@laurikainen_2010]. So the early idea of Sidney @van_den_bergh_1976 that within the Hubble’s morphological scheme the S0 galaxies constitute in fact a morphological sequence parallel to the sequence of spiral galaxies, matching their bulge-to-disk ratios at every morphological subtype, becomes now more and more acceptable [@kormendy_bender_2012; @cappellari_atlas7_2011]. At first glance, this re-forming of the Hubble morphological sequence strengthens the general opinion that S0s may be formed by quenching star formation in the disks of spiral galaxies – this transformation step must be easier to do when the bulge-to-disk ratios are the same in the progenitor and the descendant. But at this point we would like to note that if the bulges of S0s and spirals are indeed similar, the possibility of obtaining a [*spiral*]{} galaxy from a [*S0 progenitor*]{} arises while this direction of evolution was excluded when the bulges of S0s were supposed to be systematically larger than the bulges of spirals.
A great variety of physical processes that can in principle quench star formation in a disk of a spiral galaxy, to transform it into a lenticular one, are currently discussed: a very incomplete list includes direct collisions [@spitzer_1951; @icke_1985], tides from a cluster/group dark halo potential [@byrd_1990], ‘harrasment’ – high-speed encounters between galaxies in dense environments [@moore_1996], ram pressure by the hot intergalactic medium [@gunn_gott_1972; @quilis_2000], starvation of star formation because of removing external gas reservoir [@larson_1980; @shaya_tully_1984]. All these processes are inevitably related to dense environments: it is suggested that only clusters and rich groups of galaxies, having massive dark haloes enclosing many individual galaxies, can provide necessary density of intergalactic medium for effective ram pressure and galaxy tight packing for effective gravitational tides. On one hand, indeed, S0 galaxies are known to be the dominant galaxy population of nearby clusters, their fraction in clusters reaching 60% [@dressler_1980]. On the other hand, there is an even larger number of S0 galaxies in the field: the galaxy content of the nearby Universe includes 15% of lenticulars [@naim_1995]. Some quite isolated lenticular galaxies even exist [@sulentic_2006]. What are the mechanisms of their formation? Can be they quite different from those acting in dense environments? This question has not even been considered.
Despite the obvious scarcity of possible galaxy transformation mechanisms beyond the clusters and rich groups, it would be erroneous to think that an isolated galaxy evolves as a ‘closed box’. Recently we have studied an isolated early-type spiral galaxy NGC 7217. By analysing a full complexity of its properties including disk structure, dynamical state, inner gas polar disk, and stellar population characteristics along the radius, we have shown that its present structure requires at least two satellite infalls (minor merging) for the last 5 Gyr [@sil_2011_n7217]. A noticeable gas presence in S0s is not rare, and in particular, off-cluster environments have appeared to favor an external (accretion) origin of this gas [@davis_2011]. Moreover, we have shown that in extremely sparse environments, namely, in the quite isolated S0s the warm extended gas is [*always*]{} accreted from outside [@ilg_gas]. So external gas acquisition, related to smooth cold-gas accretion and/or to merging small late-type gas-rich satellites, together with the inner cold-disk instabilities, remain the only possible drivers of isolated lenticular galaxy formation and evolution. By taking this idea in mind, we have undertaken a study of the kinematical and stellar population properties of [*isolated*]{} lenticular galaxies, by hoping to single out the evolutionary paths related just to the gas/satellite accretion regime. Here we must also note that the gas accretion and/or minor merging allowed for the field disk galaxies are able not only to quench star formation in a large-scale disk, but to feed and provoke it in the disks where it has not proceeded before (e.g. @birnboim07 [@sancisi_rev]).
To achieve this goal, we have compiled a list of strictly isolated nearby ($v_r<4000$ ) lenticular galaxies and have undertaken deep long-slit spectroscopy of a small representative sample of them to study the kinematics of the stars and of the gas and the ages and chemical compositions of the stellar populations as well as the ionization mechanisms and metallicity of the warm-gas component. With these results in hands, we hope to restore formation and evolutionary histories of the isolated lenticular galaxies. In this paper we show and analyze the data on the southern part of the sample: the galaxies having been observed at the Southern Africa Large Telescope (SALT) are presented. The paper is organized as follows: [Section]{} \[txt:Sample\] describes the sample, [Section]{} \[txt:Obs\_and\_Red\] gives the description of the observations and data reduction, in [Section]{} \[txt:results\] we present our results and discuss them in [Section]{} \[txt:discus\], the conclusions drawn from this study are summarized in [Section]{} \[txt:summ\].
Sample selection {#txt:Sample}
================
Our approach to compile a sample of strictly isolated lenticular galaxies is based on a set of methods developped in the Laboratory of Extragalactic Astrophysics and Cosmology of the Special Astrophysical Observatory of the Russian Academy of Sciences by Igor Karachentsev, Dmitry Makarov, and their coauthors. They proposed a new group-finding algorithm which was intended to be applied to their Nearby Galaxy Catalog [@Karachentsev_neigb_gal_2004; @Karachentsev_neigb_gal_update_2013]. By extending their study of the local large-scale structures, they have also used their algorithms to identify galaxy pairs [@karachentsev_pairs_2008], triplets [@makarov_triplets_2009], groups [@makarov_groups_2011], and isolated galaxies [@karachentsev_isol_2011] up to Hubble velocities of $v_r\le 4000$ . The updated HyperLEDA and NED databases extended by measurements coming from the surveys SDSS, 6dF, HIPASS, and ALFALFA, provided line-of-sight systemic velocities, apparent magnitudes, and morphological types of the galaxies under consideration. The profit of their group-finding approach is that the individual properties of galaxies, in particular an integrated luminosity in the $K$-band as a stellar mass proxy, are taken into account. They assumed that velocity difference and visible separation of galaxies belonging to a physical pair must both satisfy the condition of negative total energy, and the pair components must be enclosed within the sphere of ‘zero-velocity’ that means that the pair components are separated from the Hubble expansion flow. This algorithm for galaxy grouping is iterative: galaxy-galaxy physical pairs are identified during the first iteration, and after that at the subsequent iterations the galaxy pairs are tied into groups through the common members. The isolation index ($II$) characterizing isolation degree of any galaxy within the sphere of $v_r\le 4000$ is a by-product of all the galaxy grouping procedures. The $II$ value of an unbound galaxy pair is larger than one and indicates a factor by which the mass of one of the components should be increased in order to create a gravitationally bound pair. Correspondingly, the $II$ values of the galaxies belonging to multiple systems are less than one.
Dmitry Makarov has kindly provided us with the information about the isolation indices for all galaxies of the Local Supercluster and its surroundings. To define our sample of isolated lenticular galaxies, we have selected early morphological types, $-3 \le T \le 0$, with the isolation indices $II > 2.5$. Also we have taken some galaxies having faint companions with $1< II < 2.5$ but with the $K$-magnitude difference of 3 mag and larger having in mind that the low-mass satellites, with the mass of ten percent and less relative to their host, cannot gravitationally affect the evolution of their hosts (unless they merge). The whole sample of the isolated lenticular galaxies, both of the northern and southern skies, lists 281 objects. We have started spectral observations of a representative part of this sample. Firstly, we have carried out spectroscopic observations of 12 northern targets from the sample of isolated S0 galaxies at the 6-m Russian telescope by using universal SCORPIO spectrograph; the results are published in @ilg_gas [@katkov_ilg_stpop]. In this paper we present results of the long-slit spectroscopic study of 9 targets of the southern hemisphere undertaken at the Southern African Large Telescope (SALT). As a Discussion, some summary of the results for the unified sample of northern and southern isolated S0s is also given at the end of the paper.
Long-slit spectroscopy {#txt:Obs_and_Red}
======================
Observations and data reduction
-------------------------------
The observations were performed with the Robert Stobie Spectrograph [RSS; @Burgh03; @Kobul03] at the Southern African Large Telescope (SALT) [@Buck06; @Dono06]. The long-slit spectroscopy mode of the RSS was used with a 1.25 arcsec slit width for the most observations. The total time of one observational block with the SALT is limited by the track-time of about an hour for our targets. For this reason and because the SALT is a queue-scheduled telescope, most of our galaxies were observed more than once and all observations were done during different nights. All observational details are summarised in Table \[tbl\_logobs\]. The slit was oriented along the major axis for every galaxy except NGC 7693. The grating GR900 was used for our program to cover finally the spectral range of 3760$-$6860 Å with a final reciprocal dispersion of $\approx0.97$ Å pixel$^{-1}$ and FWHM spectral resolution of 5.5 Å. The seeing during observations was in the range 1.5$-$3.0 arcsec. The RSS pixel scale is 01267, and the effective field of view is 8 in diameter. We utilised a binning factor of 2 or 4 to give final spatial sampling of 0258 pixel$^{-1}$ and 0507 pixel$^{-1}$ respectively. Spectrum of an Ar comparison arc was obtained to calibrate the wavelength scale after each observation as well as spectral flats were observed regularly to correct for the pixel-to-pixel variations. Spectrophotometric standard stars were observed during twilights, after observations of objects, for the relative flux calibration.
[cclccccc]{}
& 07.11.2012 & 620x3 & 2$\times$4 & 1.25 & 350 & 2.0\
& 04.01.2013 & 620x3 & 2$\times$4 & 1.25 & 350 & 2.0\
IC4653 & 11.05.2012 & 1200x3 & 2$\times$2 & 1.25 & 52 & 2.0\
& 05.10.2011 & 820x3 & 2$\times$2 & 1.25 & 210 & 3.0\
& 22.11.2011 & 900x3 & 2$\times$2 & 1.25 & 210 & 3.0\
& 22.12.2011 & 1030x2,730 & 2$\times$2 & 1.25 & 210 & 3.0\
& 25.12.2011 & 1000x3 & 2$\times$2 & 1.25 & 210 & 3.0\
& 17.12.2012 & 900x2 & 2$\times$4 & 1.25 & 169 & 3.0\
& 06.01.2013 & 900x3 & 2$\times$4 & 1.25 & 169 & 3.0\
& 15.01.2013 & 900x3 & 2$\times$4 & 1.25 & 169 & 3.0\
& 15.02.2013 & 900x2,700 & 2$\times$4 & 1.25 & 169 & 3.0\
& 17.02.2012 & 850x2,472 & 2$\times$2 & 1.25 & 130 & 2.0\
& 23.02.2012 & 800x3 & 2$\times$2 & 1.25 & 130 & 2.0\
& 28.02.2012 & 800x3 & 2$\times$2 & 1.25 & 130 & 2.0\
& 14.01.2013 & 600x3 & 2$\times$4 & 1.25 & 283 & 2.0\
& 19.03.2013 & 600x3 & 2$\times$4 & 1.25 & 283 & 2.0\
NGC6010 & 05.04.2013 & 750x3 & 2$\times$4 & 1.00 & 105 & 2.0\
& 10.07.2012 & 650x3 & 2$\times$2 & 1.25 & 210 & 2.0\
& 04.09.2012 & 650x3 & 2$\times$2 & 1.25 & 30 & 2.0\
& 22.09.2012 & 650x3 & 2$\times$2 & 1.25 & 30 & 2.0\
& 10.06.2012 & 700x3 & 2$\times$2 & 1.25 & 175 & 2.0\
& 10.07.2012 & 650 & 2$\times$2 & 1.25 & 175 & 2.0\
Primary data reduction was done with the SALT science pipeline [@Cr2010]. After that, the bias and gain corrected and mosaicked long-slit data were reduced in the way described in @Kn08. The accuracy of the spectral linearisation was checked using the sky line \[Oi\] $\lambda$5577; the RMS scatter of its wavelength measured along the slit is 0.04 Å. The slit length is approximately 8, so sky spectra from the slit edges were used to estimate the background during the galaxy exposures.
The Lick index system at the SALT/RSS
-------------------------------------
To derive stellar population properties from the integrated absorption-line spectra of a stellar system, in particular of a galaxy or its part, one can use the equivalent widths (EWs) of the stellar absorption spectral lines. Lick indices [@Faber_1985; @licksystem1; @Worthey_Ottaviani_1997] is a uniform, strictly established system of set line parameters measured in part as EWs of strong absorption lines in the spectral range of 4000-6200 Å. The system is named after a 20-yr spectral survey of nearby galaxies and stars with the 3-m Lick telescope using a photon-counting detector IDS at the Cassegrain spectrograph. The line and continuum border definitions within the Lick system are tied to the spectral resolution of the Lick spectrograph, $\approx$8 Åbut slightly varying with wavelength. The aim was to include the spectral lines fully into the integrated spectral ranges. The necessity to apply the Lick definitions of the absorption-line EW measurements to galactic spectra was strengthened by the fact that many evolutionary synthesis models of simple stellar populations, starting from the work of @licksystem1, used calibrations of the [*stellar*]{} Lick indices on the stellar effective temperatures and metallicities as their input data. These calibration relations were obtained from observations of more than 460 nearby stars with exactly the same Lick spectral setup.
The Volume Phase Holographic (VPH) grating of 900 g/mm of the SALT/RSS has a spectral resolution of about 5.5 Å that differs from the standard Lick resolution, $\approx 8$ Å. We hence need to calibrate the instrumental absorption-line indices obtained from the RSS spectra by integrating the spectral fluxes in the bands prescribed by the Lick system to the standard Lick system. This was done by observing a sample of Lick standard stars visible in southern sky from the list of @licksystem1. In total, 10 giant and dwarf stars with spectral types in-between F4 and K4 were observed with the VPH900 grating and the slit width of 1.25 arcsec. All observations of these bright stars were done either within twilight time or during bright moon time with poor seeing and cloudy conditions. For all obtained spectra we calculated the instrumental-system Lick indices H$\beta$, Mgb, Fe5270, and Fe5335, by integrating fluxes within the prescribed wavelength intervals for the lines, as well as their blue and red continuum points, as recommended by @licksystem1.
![The calibration of the instrumental Lick indices of the RSS/SALT with grating VPH900 onto the standard Lick system. The straight solid lines are the best-fit relations while the dashed lines are the equality relations. []{data-label="indsystem"}](katkov_fig1){width="55.00000%"}
The instrumental-system Lick indices were then compared to the tabular values provided by @licksystem1. The linear dependencies between the two sets of data were recovered, and the linear regressions were calculated, and are also shown in Fig. \[indsystem\]:
$$\mbox{H} \beta (\mbox{Lick}) = (1.084\pm 0.060) \times \mbox{H} \beta
(\mbox{\small RSS}) - (0.146\pm 0.158),$$ the rms scatter of the points around the straight line is 0.20 Å.
$$\mbox{Mgb} (\mbox{Lick}) = (1.091\pm 0.069) \times \mbox{Mgb}
(\mbox{\small RSS}) - (0.077\pm 0.189),$$ the rms scatter of the points around the straight line is 0.34 Å.
$$\mbox{Fe5270} (\mbox{Lick}) = (0.974\pm 0.053) \times \mbox{Fe5270}
(\mbox{\small RSS}) + (0.004\pm 0.113),$$ the rms scatter of the points around the straight line is 0.20 Å.
$$\mbox{Fe5335} (\mbox{Lick}) = (1.080\pm 0.059) \times \mbox{Fe5335}
(\mbox{\small RSS}) - (0.115\pm 0.116),$$ without HD10700, with the remaining 9 stars, the rms scatter of the points around the straight line is 0.22 Å.
Comparing the derived rms scatter of the individual stars around the best-fit straight lines with the mean accuracy of the tabular Lick indices mentioned by @licksystem1, namely 0.22 Å for H$\beta$, 0.23 Åfor Mgb, 0.28 Å for Fe5270, and 0.26 Å for Fe5335, we conclude that the scatter of the points in the plots of Fig. \[indsystem\] is produced mostly by the errors of the tabular indices.
Full spectral fitting
---------------------
Besides the Lick index measurements, we have also applied full spectral fitting approach to our spectra; it is valuable when strong emission lines are present in spectra, and the age-sensitive index H$\beta$ is strongly contaminated by the Balmer emission.
In order to perform full spectral fitting of the synthetic spectra to the observed data, we have used an <span style="font-variant:small-caps;">IDL</span>-based package <span style="font-variant:small-caps;">NBursts</span> [@nbursts_a; @nbursts_b]. This package implements a pixel-space fitting algorithm, that involves the non-linear least-squares optimization using Levenberg-Marquardt algorithm. The observed spectrum is approximated by a stellar population model broadened by line-of-sight velocity distribution (LOSVD); the parameters of the stellar population model, metallicity and age, are determined during the same minimization loop as the internal kinematical parameters – line-of-sight velocity and stellar velocity dispersion.
In our study, we use intermediate spectral resolution (R=10000) simple stellar population (SSP) models generated by evolutionary synthesis code <span style="font-variant:small-caps;">pegase.hr</span> [@pegasehr] in a wavelength range 3900-6800 Å for the @SalpeterIMF initial mass function based on ELODIE3.1 stellar library [@elodie3.1]. The grid of synthetic SSP spectra was pre-convolved with spectral line spread function (LSF) of the RSS spectrograph, which was determined by fitting spectrum of one of Lick standard stars against the R=10000 spectrum for the same star taking from ELODIE3.1 library. During the main minimization loop the template spectrum is extracted from the grid of models by interpolation to the current age $T$ and metallicity \[Z/H\]. Then template is convolved with LOSVD, which is defined by Gauss-Hermite series of orthogonal functions with parameters $v$, $\sigma$, $h_3$, $h_4$ [@gausshermite]. The model includes a multiplicative continuum aimed to take into account flux calibration uncertainties both in observations and in the models as well as possible dust attenuation of galactic spectrum.
In order to avoid systematic errors in the solution we masked narrow 15 Å– wide regions around ionized-gas emission lines and around traces of the subtracted strong airglow lines. As shown by @nbursts_a and @chil09_virgo, excluding age-sensitive Balmer lines from the full spectral fit neither biases age estimates nor significantly degrades the quality of the age determination. To achieve the required signal-to-noise ratio of 20-30 per spatial bin, we performed adaptive binning of the long-slit spectra along the slit.
A number of similar approaches of the full spectral fitting techniques exist, for instance <span style="font-variant:small-caps;">ppxf</span> by @ppxf, <span style="font-variant:small-caps;">starlight</span> by @starlight, <span style="font-variant:small-caps;">steckmap</span> by @steckmap, and other packages. The main difference between the <span style="font-variant:small-caps;">NBursts</span> package and the current version of the popular <span style="font-variant:small-caps;">ppxf</span> code as well as <span style="font-variant:small-caps;">starlight</span> and <span style="font-variant:small-caps;">steckmap</span> is that the <span style="font-variant:small-caps;">NBursts</span> specifies template spectrum as a single SSP spectrum with age and metallicity as free parameters. Other packages construct template spectrum as a linear combination of SSPs with fixed ages and metallicities and SSP weights taken as free parameters. In these cases the star formation history can be in principle derived from the observed galaxy spectra, but that requires very high signal-to-noise spectra [@steckmap]. Indeed, insufficient signal-to-noise level of spectra leads to degeneracy between weights of the different SSPs and unreliable star formation history. Due to understanding this effect, the majority of studies where linear combination of SSPs are used provide only mass- and/or light-weighted SSP-equivalent parameters of stellar populations that correspond to <span style="font-variant:small-caps;">NBursts</span> fitting parameters by definition.
Final choice of the stellar population parameters
-------------------------------------------------
To derive the stellar population parameters, we have tried both approaches: we have fitted ‘pixel-by-pixel’ all our spectra along the slit and we have calculated the Lick indices H$\beta$, Mgb, Fe5270, and Fe5335. The full spectral fitting included the use of the evolutionary synthesis code <span style="font-variant:small-caps;">pegase.hr</span> [@pegasehr]. The measured Lick indices were confronted to the models by @Thomasstpop allowing to analyze magnesium-overabundant stellar populations. To obtain the stellar magnesium-to-iron ratios, we were restricted to the use of Lick indices only, because the full spectral fitting assumed solar abundance ratios. On the other hand, the full spectral fitting has great advantage in deriving the stellar population ages when we deal with the spectra containing strong Balmer emission lines: when the Lick index H$\beta$ is contaminated by the hydrogen emission line, the full spectral fitting is much more safe because it allows to exclude spectral ranges polluted by emission lines. By calculating the Lick index H$\beta$, we tried to correct it for the emission by applying our approach basing on the measurements of the H$\alpha$ emission-line equivalent width [@sil2006]; however when the emission is strong, and the ionized-gas excitation is uncertain, the correction cannot be perfect. Unfortunately, among our southern sample of the isolated S0s, almost all galaxies demonstrate rich emission-line spectra. So for the present sample in particular we shall analyze mostly the results on the ages and metallicities obtained through the full spectral fitting. However, we wonder to know if two approaches give consistent results, and for a few our galaxies with weak or absent emission lines we have compared the ages and metallicities of the bulges and disks obtained by full spectral fitting using the PEGASE stellar population models with those obtained through the Lick indices H$\beta$ and \[MgFe\] using the models by @Thomasstpop. To get a sufficient statistical level of the comparison, we have involved the results for our previous sample of the northern isolated S0s [@katkov_ilg_stpop], and the final comparison can be inspected in Fig. \[stpopcomparison\]. The ages are consistent within the accuracy of their determination, and the metallicities may have a systematic shift by some 0.1 dex, perhaps due to slightly non-solar magnesium-to-iron ratios of the stellar populations in the S0s studied by us.
![The comparison of the ages and metallicities derived by two different methods: by full spectral fitting using the PEGASE stellar population models (‘NBursts’) and through the Lick indices H$\beta$ and \[MgFe\] using the models by @Thomasstpop. The dotted line represents the equality locus. Filled circles correspond to measurements in the bulges, open squares - to disks. Green circles show bulges with presence of emissions in the spectra.[]{data-label="stpopcomparison"}](katkov_fig2a "fig:"){width="27.00000%"} ![The comparison of the ages and metallicities derived by two different methods: by full spectral fitting using the PEGASE stellar population models (‘NBursts’) and through the Lick indices H$\beta$ and \[MgFe\] using the models by @Thomasstpop. The dotted line represents the equality locus. Filled circles correspond to measurements in the bulges, open squares - to disks. Green circles show bulges with presence of emissions in the spectra.[]{data-label="stpopcomparison"}](katkov_fig2b "fig:"){width="27.00000%"}
Ionized gas
-----------
A warm-gas emission-line spectrum can be obtained by subtracting the stellar component contribution (i.e., the best-fitting stellar population model) from the observed spectrum at every spatial bin. The resulted pure emission-line spectra are uncontaminated by absorption lines of the stellar components that is especially important for the Balmer lines. Then we fitted emission lines with Gaussians pre-convolved with the instrumental LSF in order to determine the LOS velocities of the ionized gas and emission-line fluxes.
Results {#txt:results}
=======
By applying the above-mentioned techniques to every galaxy spectrum along the major axis, we have derived the radial profiles of the following characteristics: stellar rotation velocity, stellar velocity dispersion, SSP-equivalent stellar ages, metallicities, and magnesium-to-iron ratios, ionized-gas rotation velocity, warm-gas velocity dispersion, emission-line flux ratios. The latter characteristics can be plotted at the classical excitation-diagnostic diagrams, so called BPT (after Baldwin, Phillips, & Terlevich) [@BPTdiag], to identify the gas excitation mechanism. If we see that the gas is ionized by young stars, we can apply so called ‘strong-line calibrations’ to estimate the gas metallicity. We have explored the formulae from the paper by @PPcalib and have estimated oxygen abundance using emission-line indices $O3N2=\log_{10}\left(([O\iii]\lambda5007/H\beta)/([N\ii]\lambda6583/H\alpha)\right)$ and $N2=\log_{10}([N\ii]\lambda6583/H\alpha)$ which are calibrated by @PPcalib against $12+\log (O/H)$ by using the data on 137 H-regions with known electronic temperatures. In the case of very noisy and \[O\] emission lines we used another calibration from @PPcalib that involves only $N2$ index. Due to exploration of ratios of nearest emission lines the dust attenuation does no affect the oxygen abundance estimations.
Below we present the results for every galaxy in graphical way and give brief description of the individual properties of the galaxies.
[**IC 1608.**]{} It is one of the most luminous galaxies of our sample. The galaxy demonstrates fast stellar rotation; however the stellar disk is rather hot dynamically, $\sigma _* \ge 100$ . The galaxy is very gas-rich; we observe strong emission lines over the full slit extension. The nuclear emission-line spectrum is a typical LINER-like one; starting from the radius of about 10 arcsec, the ionized gas is excited mostly by young stars, and in the outer part, $R\approx 40$, a starforming ring is clearly seen. The gas subsystem looks cold only within the starforming ring; elsewhere in the disk the gas velocity dispersion matches that of stars. The gas rotation curve coincides exactly with the stellar one so we can be sure that the gas is confined to the main galaxy plane. The gas oxygen abundance at $R>30\arcsec$ is observed to be nearly solar or slightly higher while the stellar mean metallicity there is very poor, about $-0.6\div -0.9$ dex. The magnesium-to-iron ratio in the stellar component is homogeneous along the radius and close to $+0.1$ dex; the mean stellar age looks intermediate, 3–5 Gyr, over the whole galaxy too. We may so suspect that low-level starforming events like the current one have occured multiple times during the galaxy evolution for the last several billion years; perhaps they have been provoked by small gas-rich satellites merging.\
[**IC 4653.**]{} This dwarf galaxy is classified as SB0/a pec in the NED[^1] database. However despite this relatively late morphological type and the elongated isophote shape, our spectrograph slit aligned with the isophote major axis reveals very weak rotation and rather large stellar velocity dispersion. We would re-classified the galaxy as a dwarf elliptical one and exclude it from the further consideration of our sample of isolated lenticular galaxies. Interestingly, the strong emission lines excited by the current star formation are seen over the whole galaxy including its nucleus.
[**NGC 1211.**]{} This luminous, almost face-on galaxy has a bar and two rings – the inner reddish one at $R\approx 20\arcsec$ and the blue starforming outer one at $R\approx 60\arcsec$. Surveys in the 21 cm line reported earlier large neutral hydrogen content in this galaxy, 5.5 billion solar masses of HI [@Garcia-Appadoo_2009], so the strong emission lines were also expected in its optical spectrum. The discrepancy between the rotation velocities of the ionized gas and the stars at $R<10\arcsec$ can be explained both by asymmetric drift and bar influence; beyond the radius of 10 the ionized gas and stars rotate together, and we conclude that the gas is confined to the main galaxy plane. The gas is excited by the LINER-like nucleus and by shock waves (from the bar?) in the central part of the galaxy, at $R<10\arcsec$, and by young stars beyond this radius; the oxygen abundance of the gas in the outer starforming ring is the solar one. Meantime the low-surface brightness stellar disk at $R\approx 30\arcsec$ demonstrates very old age, $T\ge 10$ Gyr, and very low stellar metallicity, \[Z/H\]$\approx -1.5$ dex. The bulge and the lens which we relate to the bar ends, demonstrate intermediate stellar ages and strong metallicity gradient along the radius.
[**NGC 2917.**]{} This very luminous S0$^+$ galaxy is strongly inclined to the line of sight, but is not exactly edge-on, so we can distinguish a dust ring and no signs of bar in the galaxy. The bulge is so small that the galaxy has been included into the list of ‘flat’ late-type galaxies by @Mitronova_2004. The systemic velocity of NGC 2917 given in the NED, 3675 , is erroneous coming from a very weak spurious 21-cm signal detected by @Richter_1987. Our optical spectral observations give $v_{sys}=5377$ for this galaxy, so the galaxy is even more luminous than it has been thought before. Though we have traced the stellar component of NGC 2917 almost up to its optical border, $R_{25}=38\arcsec$ (RC3[^2]), we have only measured its lens; the outer stellar disk is very low-surface brightness one and could not be detected in our observations. While the bulge has intermediate stellar-population characteristics, the lens looks rather young, $T=2-3$ Gyr, that is consistent with the ionized gas excited by current star formation at $R\ge 10\arcsec$. The ionized-gas metallicity is high. The fall of the ionized-gas rotation velocity at the southern edge of the galaxy accompanied by the rise of the velocity dispersion of the gas clouds seems to be real. Are there any traces of interaction?
[**NGC 3375.**]{} Another galaxy which being previously classified as a lenticular one is in fact an elliptical: its stellar component does not rotate regularly, and the stellar velocity dispersion exceeds 150 everywhere through the galaxy. Emission lines are absent in the spectrum.
[**NGC 4240.**]{} The galaxy is classified in RC3 as between E and S0 ($T=-3.8\pm
0.5$). In the frame of the APM survey [@naim_1995] when 6 independent researchers classified it ‘by eye’, three voted for S0 and three voted for E (see ‘Detailed classification’ option in the NED). However our long-slit cross-section along the major axis reveals rather fast rotation of the stellar component; and in the photometric data we see an exponential disk at $R>15\arcsec$. So we consider NGC 4240 as a lenticular galaxy. The stellar population properties were spectrally studied by @Reda_2007 up to the distance about 10 from the center; a slightly subsolar metallicity and a rather old age were measured by the Lick index method. The kinematics was examined by @Hau_2006; however their slit was obviously off the dynamical center, and they did not report gas counterrotation for this galaxy which is striking. We have traced the stellar rotation and stellar population properties toward $R\approx 25\arcsec$ so measuring not only the bulge but also the large-scale stellar disk at $R>15\arcsec$. Both the bulge and the disk have an intermediate stellar age, about 5 Gyr, but the disk is very metal-poor, \[Z/H\]$\approx -1.0$, while the bulge has only \[Z/H\]$\approx -0.3$. The ionized gas counterrotates the stars in the bulge-dominated area; while the stellar velocity dispersions of the stars and gas clouds are comparable, the rotation velocities differ significantly, and we conclude that the ionized gas may rotate in the plane which does not coincide with the plane of the stellar disk. The gas metallicity in the outer starforming ring, at $R\approx 15\arcsec$, is close to the solar one.
[**NGC 6010.**]{} It is another small-bulge, edge-on S0 galaxy included into the catalogue of ‘flat galaxies’ by @Mitronova_2004. Also we must note that in our present sample it is the only S0-galaxy without strong emission lines in the spectra. We see only weak narrow emission lines with LINER-like excitation in the very circumnuclear region; some signs of the ionized-gas counterrotation are however detected. Meantime the neutral hydrogen is found in this galaxy by @Springob_2005, but no signs of recent star formation are present. The stellar ages of the nucleus ($T=9$ Gyr) and of the bulge ($T=8$ Gyr) are slightly older than in other galaxies of our sample. However, the characteristics of the disk in the radius range $R=20\arcsec - 40\arcsec$ are quite typical – $T=5$ Gyr and \[Z/H\]$=-0.4$.
[**NGC 7693.**]{} Due to instrumental problems, the galaxy was observed with the slit turned by some 40 degree to the major axis. However, even so, the observed stellar rotation is too slow, and the ionized-gas velocities are quite decoupled from the stellar ones. Consequently, no signs of current star formation is seen in this galaxy, though both the bulge and the disk look very young, 1–3 Gyr old. The magnesium-to-iron ratio over the whole galaxy is solar so it seems that continuous star formation ceased rather recently, due to perhaps just minor merger from an inclined orbit.
[**UGC 9980.**]{} The galaxy demonstrates fast regular rotation, looking quite similar in the stellar and ionized-gas components. The gas is spread over the whole galaxy, and starting outward from the radius $R > 10\arcsec$ it is excited by young stars. However, both the bulge and the large-scale stellar disk possess rather old stellar populations, $T=7$ Gyr in the former and $T=10$ Gyr in the latter, so the widespread star formation has evidently started quite recently: unless the case of NGC 7693, this time minor merging has stimulated star formation, not ceased it. The difference of metallicities – \[Z/H\]$=-1.0$ dex in the stellar disk and –0.2 dex in the gaseous disk – indicates also the external origin of the current fuel for the star formation. The inner stellar ring related to ansae at the ends of the bar, at $R\approx
10\arcsec$, is distinguished by slightly younger stellar age, $T\approx 5$ Gyr. However, this ancient ring-like star formation burst was probably related not to interaction but to the bar affecting gaseous disk of the galaxy which was perhaps more gas-rich at $z=0.5$ than it is at the present epoch.
After obtaining the full radial profiles of the stellar characteristics in the galaxies studied, which are presented in the Appendix, we have wished to extract mean characteristics for the large-scale galaxy components – bulges and disks. To identify radius ranges that correspond to the bulge and disk domination in the integrated light, we have undertaken photometric decomposition of the images of the galaxies. For this purpose we have used mostly the SDSS public database, Data Release 9; the $r$-band images as the images with the highest signal-to-noise ratio have been taken. For the southern galaxy NGC 4240 which was not observed in the frame of the SDSS we have decomposed the 2MASS composite, $J+H+K$, image. For one galaxy, IC 1608, very deep $gri$ photometric data obtained during test observations of the LCOGT project [@sil_lcogt]. For every galaxy, we have performed an isophotal analysis and have derived azimuthally-averaged surface-brightness radial profiles. By inspecting these profiles, we have found the outer radial segments where the surface-brightness radial profiles can be well approximated by exponential laws, and the isophote ellipticity stays constant. These outer parts of galaxies are identified by us as disk-dominated. To characterize the bulges which are mostly compact in the galaxies of our sample we fix the radial range of $4\arcsec -7\arcsec$ that is beyond the unresolved nucleus contamination under our seeing conditions. In some galaxies we have also distinguished the radial ranges where we see rings of enhanced stellar brightness or flat brightness profile segments betraying the presence of lenses. The corresponding segments for each component are shown by shaded gray lines in figures in the Appendix. The mean stellar ages, metallicities, and magnesium-to-iron ratios for the bulges, disks, rings, and lenses of the galaxies studied here are presented in the Table \[tbl\_stpop\].
[lrrrrr]{}
\
IC 1608 & 4 & $ 4.7^{\pm 0.3}$ & $ -0.21^{\pm 0.07}$ & $ 0.12^{\pm 0.09}$ & $ 149^{\pm 9}$\
NGC 1211 & 3 & $ 4.5^{\pm 0.4}$ & $ -0.16^{\pm 0.05}$ & $ 0.11^{\pm 0.07}$ & $ 156^{\pm 18}$\
NGC 2917 & 4 & $ 6.1^{\pm 1.3}$ & $ -0.21^{\pm 0.06}$ & $ 0.27^{\pm 0.08}$ & $ 191^{\pm 4}$\
NGC 4240 & 4 & $ 4.6^{\pm 0.3}$ & $ -0.32^{\pm 0.08}$ & $ 0.18^{\pm 0.09}$ & $ 108^{\pm 11}$\
NGC 6010 & 4 & $ 8.16^{\pm 0.45}$ & $ -0.19^{\pm 0.07}$ & $ 0.19^{\pm 0.06}$ & $ 154^{\pm 11}$\
NGC 7693 & 8 & $ 1.35^{\pm 0.18}$ & $ -0.38^{\pm 0.10}$ & $ -0.02^{\pm 0.02}$ & $ 82^{\pm 14}$\
UGC 9980 & 6 & $ 7.4^{\pm 1.4}$ & $ -0.30^{\pm 0.07}$ & $ 0.18^{\pm 0.11}$ & $ 138^{\pm 20}$\
\
IC 1608 & 8 & $ 3.5^{\pm 0.8}$ & $ -0.46^{\pm 0.14}$ & $ 0.18^{\pm 0.15}$ & $ 138^{\pm 32}$\
NGC 1211 & 2 & $ 10.5^{\pm 4.1}$ & $ -1.50^{\pm 0.14}$ & & $ 141^{\pm 12}$\
NGC 2917 & 0 & & & &\
NGC 4240 & 5 & $ 5.4^{\pm 2.1}$ & $ -1.02^{\pm 0.11}$ & $ 0.33^{\pm 0.13}$ & $ 112^{\pm 8}$\
NGC 6010 & 12 & $ 5.4^{\pm 2.4}$ & $ -0.36^{\pm 0.16}$ & $ 0.18^{\pm 0.04}$ & $ 113^{\pm 21}$\
NGC 7693 & 11 & $ 1.5^{\pm 0.9}$ & $ -0.67^{\pm 0.21}$ & $ 0.15^{\pm 0.13}$ & $ 106^{\pm 24}$\
UGC 9980 & 5 & $ 9.8^{\pm 2.8}$ & $ -0.99^{\pm 0.12}$ & $ 0.21^{\pm 0.20}$ & $ 83^{\pm 5}$\
\
IC 1608 & 10 & $ 4.6^{\pm 2.8}$ & $ -0.77^{\pm 0.20}$ & $ 0.24^{\pm 0.08}$ & $ 98^{\pm 5}$\
NGC 1211 & 12 & $ 5.6^{\pm 2.7}$ & $ -0.82^{\pm 0.23}$ & $ 0.20^{\pm 0.18}$ & $ 148^{\pm 38}$\
NGC 2917 & 10 & $ 2.6^{\pm 0.6}$ & $ -0.34^{\pm 0.08}$ & $ 0.24^{\pm 0.07}$ & $ 130^{\pm 21}$\
NGC 4240 & 0 & & & &\
NGC 6010 & 0 & & & &\
NGC 7693 & 0 & & & &\
UGC 9980 & 7 & $ 4.6^{\pm 2.2}$ & $ -0.48^{\pm 0.18}$ & $ 0.22^{\pm 0.19}$ & $ 114^{\pm 8}$\
Discussion {#txt:discus}
==========
In this paper we have described the results of spectral study for 7 isolated lenticular galaxies of the southern sky (another 2 galaxies observed so far have been re-classified here as ellipticals basing on their stellar kinematics and do not take part in the analysis below). Earlier we have already published the results of the similar study for 11 isolated lenticulars of the northern sky which were observed at the Russian 6m telescope using the SCORPIO and SCORPIO-2 spectrograph [@ilg_gas; @katkov_ilg_stpop]. With the totality of 18 isolated lenticular galaxies observed with the long-slit spectrographs of two large telescopes, we can now discuss some statistical properties concerning the kinematics, the stellar population parameters, and the ionized-gas features in isolated lenticular galaxies. The overall distributions of the parameters of the stellar component for the bulges, disks, and rings/lenses are presented for the total sample in Fig. \[allhist\]. The distribution of the absolute magnitudes in $B$- and $K$-band are shown in Fig. \[mabshist\].
Bulges vs. Disks
----------------
Fig. \[binneydia\] demonstrates first of all the dynamical status of the bulges and disks in our sample of isolated lenticular galaxies. The diagrams presented in the two left plots for the disks and for the bulges correspondingly, confront the ratio of the regular rotation velocity to the stellar velocity dispersion versus the visible ellipticity of the isophotes. It was proposed by @Illingworth_1977 and theoretically calculated by @Binney_1978_rotellip [@Binney_1978_prol] to check if the shape of a galaxy spheroid is supported by rotation. The main theoretical locus at this diagram signifies so called oblate spheroids – ones round in the equatorial plane, with isotropic velocity dispersion, whose smaller third axis is completely explained through flattening by rotation. Many true elliptical galaxies are found well below this locus because they rotate too slowly, and their shapes are supported by anisotropy of the velocity dispersion distributions. Jonh @Kormendy_1993 (see also @Kormendy_Kennicutt_2004 for the updated version of this diagram) used this diagram for the bulges of disk galaxies to separate so called ‘classical bulges’ which can be considered as analogues of elliptical galaxies, from the ‘pseudobulges’ which reveal the disk-like kinematics. If the observed characteristics place some bulges above the theoretical line for oblate spheroids, we would classify them as ‘pseudobulges’ formed from the disk material during secular dynamical evolution. Fig. \[binneydia\], middle plot, gives evidence for the roughly equal proportion of ‘classical bulges’ and ‘pseudobulges’ among the isolated lenticular galaxies: the points are oscillating around the theoretical locus for the oblate isotropic spheroids. When inspecting Fig. \[binneydia\] (right plot), we make sure again that in 8-10 bulges of 18 the stellar velocity dispersion is the same as in the surrounding disks so indeed these are ‘pseudobulges’.
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
Fig. \[bulgedisc\] gives comparison of the characteristics of the stellar populations in the bulges and in the disk structures: it is the comparison of age–age, metallicity–metallicity, Mg/Fe ratio–Mg/Fe ratio. The first and the third plots demonstrate correlations between the properties of the bulges and of the disks: covering all the range of possible ages, from 1 Gyr to 17 Gyr, the mean stellar ages of the bulges and disks tend to be similar in the galaxies studied, and the magnesium-to-iron ratios are strictly the same in the bulges and in the disk structures. It is an opposition to the properties of S0 galaxies in denser environments: @sil_s0 found for a sample of mostly group S0 members that the disks appear to be usually older than the bulges covering homogeneously the upper left corner of the diagram similar to the Fig. \[bulgedisc\] (left plot), and @virgo_gemini have found just the same effect for all the Virgo S0s studied by them. In both dense-environment samples the stellar disks appear to be much more magnesium-overabundant than the bulges. We can conclude that when placed beyond the outer gravitational and hot-medium influence, the bulges and the disks in S0 galaxies formed synchronously: star formation started simultaneously here and there and ceased at one moment. Interestingly, despite the synchronous star formation, the mean stellar metallicities of the disk structures are significantly lower than the metallicities of the bulges (Fig. \[bulgedisc\], middle plot). Does it mean that pristine outer gas was accreted by the outer disks and fueled star formation there, while the nearly simultaneous star formation in the bulges was fed by the gas pre-processed and enriched in the disks?
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
Let us inspect some scaling relations connecting evolutionary and dynamical characteristics of the stellar components which are commonly studied for the elliptical galaxies. Fig. \[scalrel\] (left plot) confronts the mean stellar ages of the different structural components with their magnesium-to-iron ratio which characterizes the duration of the main starforming episode, from a very brief, shorter than $10^9$ years (\[Mg/Fe\]$=+0.3$), to several Gigayears (\[Mg/Fe\]$=0.0$). We see a cloud of points limited at the down right by a linear law which marks probably the initial epoch of launching star formation in S0s at $z\approx 3$: star formation starting 12 Gyr ago and ceasing just immediately would give \[Mg/Fe\]$=+0.3$, and star formation starting 12 Gyr ago and lasting to 4 Gyr ago would give \[Mg/Fe\]$=0.0$. However there is a lot of points, relating both to the bulges and to the disks, which are expanding to the left of this limiting line. Obviously these are the stellar systems which have started their formation much later than at $z=2-3$: to get the mean stellar age of 3 Gyr and \[Mg/Fe\]$=+0.3$ signifying the duration of star formation less than 1 Gyr, the process had to be launched at $z=0.4$. From this plot, we conclude that main star formation events both in the bulges and in the disks of the isolated S0 galaxies have no a single fixed epoch, but are homogeneously spread from very high redshifts to rather recent ones.
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
Two other plots of the Fig. \[scalrel\] confront the chemical properties of the stellar populations to the dynamical parameter $(v^2+\sigma ^2)^{0.5}$, where $v$ and $\sigma$ are the rotation velocity and stellar velocity dispersion averaged over the disk-dominated or bulge-dominated area; this dynamical parameter characterizes, under the condition of virialization, the local gravitational potential. The correlation of the magnesium-to-iron ratio with the gravitational potential well, in particular, with the central stellar velocity dispersion, is well known for the elliptical galaxies (e.g. @trager). In our data, we see that the bulges (spheroids) and the disks behaves similarly as concerning the duration (the effectiveness?) of star formation in a particular gravitational potential well: the deeper the well the shorter star formation. However, the similarity of the bulges and disks disappears when we inspect not the Mg/Fe ratio, but the global metallicity versus the dynamical parameter (Fig. \[scalrel\], right plot): while the bulges follow the well-known mass–metallicity relation, the larger mass the higher metallicity, this correlation vanishes completely for the disks.
We can put our results on the stellar population properties in the bulges of isolated lenticular galaxies into a wider context by referring to the study of nearby lenticular galaxies with the integral-field spectrograph of the Russian 6-meter telescope, MPFS, by @sil2006 [@Silchenko2008_procIAU]. In this work [@Silchenko2008_procIAU] the data for the nuclei and bulges of more than 50 nearby S0s were presented; the sample included galaxies over a wide range of environments and was divided into two parts – ‘dense environments’ (Virgo cluster and central galaxies of rich groups) and ‘sparse environments’ (mostly peripheries of groups). In Fig. \[stpop\_sig\_comparison\] we reproduce the Fig. 2 from @Silchenko2008_procIAU where we overplot our present results on the bulges of completely isolated S0s. Besides the data on the bulges of nearby S0s, this figure contains also mean relations for the integrated stellar populations properties of elliptical galaxies in clusters [@nelan05], in the field [@howell], and in both types of environments [@thomas05]. We must note here that among these comparison samples only ellipticals were considered by @howell; @nelan05 and @thomas05 have mixed ellipticals and lenticulars. In Fig. \[stpop\_sig\_comparison\] we see that the relations connecting the ages and the \[Mg/Fe\] ratios with the stellar velocity dispersion found for spheroids in a wide range of environments are broadly consistent with our results on the bulges of isolated lenticular galaxies. However the stellar metallicities of the bulges are on average twice lower in the isolated S0s with respect to all other samples. We can speculate that this difference may be related to the possible difference in gas accretion sources in S0s galaxies in different environments, if the SSP-equivalent metallicity of the bulges is biased toward the metallicity of the last stellar generation born during some bulge rejuvenation event. \[stpop\_bulge\_comparison\]
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Rings and lenses
----------------
General structure of lenticular galaxies differs from that of other disk galaxies by often revealing such disk features as stellar rings (most frequent in S0/a, @deLapparent_2011) and lenses (most frequent in S0s, @Laurikainen_2009). It is a common view that the stellar lenses are very old and dynamically hot though this point of view is based on very rare observations of a few objects [@Kormendy_1984; @Laurikainen_2013]. We have succeeded to measure kinematical and stellar population characteristics for 9 lenses and rings, and our results contradict to this common view. The stellar velocity dispersions are generally the same in the lenses/rings and in the surrounding disks (Fig. \[stpop\_discs\_lens\], right bottom plot) so dynamically they are indistinguishable; perhaps, there are some hints that the rings and lenses can be found mostly in dynamically hot disks. Fig. \[stpop\_discs\_lens\] gives also evidence for identical chemical properties of the lenses/rings and their surrounding disks. But there is however one important distinction between the disks and the lenses/rings: while the mean stellar ages of the disks fill out the complete range of possible values, between 1 and 12 Gyr, the ages of the rings and lenses are predominantly concentrated in the narrow range between 2 and 6 Gyr (Fig. \[stpop\_discs\_lens\], left upper plot). An exception is the galaxy from SCORPIO sample - NGC 6615 which has largest ring among entire sample with the age of $\approx13$ Gyr. We can so state that the last starforming episodes took place in these substructures at $z<1$. Here we see an association with the fact that strong bars are predicted to appear in galactic disks only after $z=1$ [@Kraljic_2012]. Since lenses in S0s are commonly related with dissolved or weakened bars [@Buta_2010] and since star formation in rings is usual at the resonance radii of the bars [@Buta_Comb_1996], we would like to connect the epoch of the last starforming episodes in the rings and lenses and the epoch of the rapid bar arising in the stellar disks after $z<1$.
Another interesting finding can be seen in Fig. \[stpopcomp\]. While the bulges show the correlation between their ages and the stellar velocity dispersion, just as is known for elliptical galaxies, the disks and rings/lenses ages do not correlate with the observed stellar velocity dispersion. These findings are supported by the evaluation of Spearman correlation coefficient between the ages and velocity dispersion of the bulges, $r_s=0.58$, with the probability of the correlation to be insignificant $p=0.012$, while the correlation coefficients for the disks, $r_s=0.07$, $p=0.80$, and for the rings/lenses, $r_s=-0.17$, $p=0.61$, prove that here the dependencies are absent. In addition, we found slight anticorrelation between the stellar metallicity and velocity dispersion in the ring/lens structures ($r_s=-0.46$, $p=0.15$) while for the disks and bulges such correlation is insignificant ($r_s=-0.26$, $p=0.34$ for the disks and $r_s=0.12$, $p=0.62$ for the bulges, correspondingly). It is obvious that such anticorrelation, if exists, has an evolutionary census; but extension of the galaxy sample with reliable measurements of the stellar population properties in the disk substructures is needed to strengthen the relation and to propose a particular scenario to explain it.
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
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Ionized-gas characteristics
---------------------------
By analyzing the half of our sample observed at the Russian 6m telescope, we have noted that, firstly, the majority of isolated lenticular galaxies contain extended ionized-gas disks, and secondly, the rotation and orientation of the ionized-gas disks are often decoupled from the rotation and orientation of the stellar disks [@ilg_gas]. Now, with the complete sample in hands, we can refine the statistics of the ionized-gas content of the isolated lenticular galaxies. Among 18 galaxies studied, 13 galaxies demonstrate extended ionized-gas emission (72%$\pm 11$%); and among 13 galaxies with the extended gas emission, 7 galaxies (54%$\pm 14$%) demonstrate visible counterrotation of the ionized gas with respect to their stellar components. Our spectral observations are ‘one-dimensional’: the long slit aligned with the major axis of the continuum isophotes characterizing the line of nodes of the [*stellar*]{} disk cannot help to determine the orientation of the [*gas*]{} rotation plane. Following the logic proposed by @bertola92, if we suppose that the gas in S0s is accreted from external sources, and the orbital momentum of the accreted gas is oriented accidentally with respect to the angular momentum of the galaxy, we should see equal proportions of corotating and counterrotating gas by studying only the gas velocity projection onto the stellar disk lines of nodes. It is just what we have found from our observations of the isolated lenticular galaxies. So we may conclude that the statistics of the ionized-gas rotation in our sample of the isolated lenticular galaxies gives evidence for the [*all*]{} gas having been accreted from external sources isotropically distributed around the galaxies.
What can these sources be? Since our galaxies are [*isolated*]{} and do not have neighbouring large galaxies which may be donors of the gas, we can propose only two probable sources of the decoupled gas acquisition: minor merging of small gas-rich satellites [@Reshetnikov_Sotnikova_1997; @Bournaud_Combes_2003] or gas inflow from cosmological filaments of the Universe large-scale structure [@keres_flows; @dekel_flows; @Bournaud_Elmegreen_2009]. We suggest that metallicity of the gas can help to identify exactly the gas origin: cosmological filaments of the Universe large-scale structure must contain the pristine gas so it must be very metal-poor [@Agertz_2009]. By pursuing this aim, we have picked out in our galaxies the radial ranges along the slit where the ionized gas is excited by young stars, according to the BPT-diagram diagnostics, and then we have added the spectra over these ranges for every galaxy. To these rather high-S/N spectra, we have applied so called ‘strong-line calibrations’ allowing to estimate oxygen abundance in the HII-regions with only a few emission lines, namely, with the Balmer lines H$\alpha$ and H$\beta$, low-excitation \[NII\]$\lambda$6583, and high-excitation \[OIII\]$\lambda$5007. We have succeeded to estimate metallicities of the ionized gas in 8 isolated lenticular galaxies. The results are presented in the Table \[table\_abund\]. Despite the wide range of galaxy luminosities, the ionized-gas metallicities have appeared to be confined to a very narrow range of values near the solar metallicity or slightly higher. So we think that we can exclude cosmological filaments as the source of gas accretion in this particular case. Obviously, we see the consequences of gas-rich satellite merging.
[rcc]{}
\
& ( -51.4; -31.1) & 8.78 ( 0.09) $\pm 0.47$\
& ( 29.7; 43.4) & 8.80 ( 0.11) $\pm 0.26$\
& ( 9.4; 15.8) & 8.72 ( 0.03) $\pm 0.41$\
& ( 32.3; 37.3) & 8.73 ( 0.04) $\pm 0.41$\
& ( -15.7; -5.6) & 8.90 ( 0.21) $\pm 0.27$\
& ( 9.6; 19.8) & 8.82 ( 0.13) $\pm 0.26$\
& ( -11.6; -6.6) & 8.80 ( 0.11) $\pm 0.41$\
& ( 4.1; 12.2) & 8.78 ( 0.09) $\pm 0.41$\
& ( -13.1; -4.3) & 8.82 ( 0.13) $\pm 0.42$\
& ( 8.4; 20.8) & 8.71 ( 0.02) $\pm 0.42$\
\
NGC 2350 & ( -1.6; 2.0) & 8.68 ( -0.01) $\pm 0.25$\
& ( -34.1; -27.7) & 8.71 ( 0.02) $\pm 0.41$\
& ( 29.1; 36.6) & 8.73 ( 0.04) $\pm 0.41$\
NGC 7351 & ( -2.7; 3.8) & 8.64 ( -0.05) $\pm 0.25$\
Though formally we cannot determine the orientation of the ionized-gas rotation plane with the only long-slit spectroscopy, we can note some possible signatures of the gas confinement to the plane of the stellar disk: it may be consistency of the rotation velocity estimates for the stars and for the gas in the outer parts of the galaxies where the stellar velocity dispersion is low and does not affect strongly the line-of-sight velocity profiles through the asymmetric drift. Among our sample, such consistency is demonstrated by the galaxies with co-rotating ionized gas IC 1608, NGC 1211, NGC 2350, NGC 2917, UGC 9980, and by the galaxies with the [*counterrotating*]{} ionized gas NGC 4240 and NGC 6798; just these galaxies figure in the Table \[table\_abund\] with their outer ionized gas excited by young stars. Other galaxies where we can suspect the gas rotating off the main symmetry plane show mostly other types of excitation – by shock waves or by old post-AGB stars placing the emission-line flux ratios at the BPT-diagrams to the right from the dividing curve. We can here remind theoretical consideration by @wakamatsu93 who noted that inclined gaseous disks/rings must experience shocks developed because of the gas crossing gravitational potential well of a stellar disk. The shock waves must heat the gas and prevent its cooling necessary for a star formation burst. We suggest then that conditions for star formation starting in the accreted gas must include, besides the gas amount, also the favorable geometry of the gas accretion.
Origin of isolated S0s
----------------------
Just from this point we would like to start discussion about the origin of isolated lenticular galaxies. Indeed, all the mechanisms proposed so far to quench star formation in the disks of spiral galaxies and to transform them into lenticulars act only in dense environments – in clusters and massive rich groups. It remained so quite unclear how lenticular galaxies in the field might form. The situation changes if we accept the new paradigm for evolution of disk galaxies proposed by @sil_s0: all disk galaxies started their evolution as lenticulars at the redshifts of $z=2-3$, and only after $z<1$ most of them became spirals by undergoing persistent outer gas accretion onto their disks that resulted in dynamical cooling and subsequent spiral-arm development and star formation ignition. Then the key point for galaxy morphological shaping becomes outer-gas accretion regime. In clusters the outer cold gas accretion is almost impossible due to tide-induced starvation and hot intracluster medium ram pressure so in clusters the most disk galaxies remain lenticulars for all their lifes. In the field the conditions for outer cold gas accretion can be quite various. If we regard gas-rich satellite merging as a main outer gas source, then we can expect the following variety of satellite system properties for the isolated disk galaxies: How many satellites has the host galaxy? Are they distributed isotropically around it or are confined to some dedicated plane as it is observed in our Galaxy and in M 31? Is the satellite system dynamically cold or hot (related perhaps to the mass of the host dark matter halo)? Concerning the last point, there is a curious fact noticed by @kara11. They considered faint companions of isolated galaxies from their catalog 2MIG. They found that the companions of isolated [*early-type*]{} galaxies have in average larger line-of-sight velocity difference with their hosts than the late-type ones. By inspecting their Fig. 4 we have ascertained that there is no practically companions of isolated early-type galaxies with the line-of-sight velocity difference less than 50 while the isolated late-type hosts possess a lot of such companions. It is a natural suggestion that the accretion of companions with a large flyby velocity is more difficult than that of slow ones, so we come to a conclusion that perhaps just a ‘hot’ orbital population of companions defines an early morphological type of the host isolated galaxy. Perhaps, the orbital composition of a satellite system relates stochastically to initial conditions, or may be a present-day isolated lenticular have merged all its slow companions several Gyr ago and has no ones presently... In other cases, accretion of a gas-rich satellite from a highly inclined orbit may lead to gas heating and prevent star formation and spiral-arm development which requires cold, gravitationally unstable disk. By varying possible variants of satellite merging regime, we can easily get an isolated galaxy of any morphological type – in opposition to the tight accretion conditions in dense environments which provide a subsequent tight range of morphological types, mostly S0s.
Summary {#txt:summ}
=======
We have observed 9 galaxies from our sample of isolated lenticular galaxies at the 10m Southern African Large Telescope with the Robert Stobie Spectrograph in the long-slit mode. The radial variations of the kinematical characteristics, line-of-sight velocities and velocity dispersions, are studied both for the stellar component and for the ionized gas which is found in the most part of the sample. Also we have derived radial profiles of the mean stellar metallicity and ages, as well as the gas excitation characteristics and oxygen abundances far outward, into the disk-dominated regions of the galaxies. By joining two subsamples of isolated lenticular galaxies studied by us here and earlier, the northern and southern ones, with a totality of 18 isolated lenticular galaxies, we analyze the statistics of the stellar population properties and ionized-gas features for this morphological type of galaxies in extremely rarefied environments.
We have found that there is no particular time frame for shaping the isolated lenticular galaxies: the mean stellar ages of the bulges and disks are homogeneously distributed between 1 and $>13$ Gyr, and the bulges and disks tend to form synchronously having mostly similar ages and magnesium-to-iron ratios. In some galaxies we have found stellar disk substructures – rings and lenses; their mean stellar ages are confined to a rather narrow range, from 2 to 5 Gyr. We relate the appearance of these structures to strong bars arising in disk galaxies after $z<1$.
An ionized-gas extended emission is found in the majority of our galaxies, in 13 of 18 (72%$\pm 11$%). And the half of all extended gaseous disks demonstrate visible counterrotation with respect to their stellar counterparts. Just this proportion is expected if all the gas in isolated lenticular galaxies is accreted from isotropically distributed external sources. A very narrow range of the oxygen abundances, \[O/H\] from 0.0 to $+0.2$ dex estimated by us for the outer ionized-gas disks excited by young stars, gives evidences for the satellite merging as the most probable source of this accretion. At last we formulate a hypothesis that morphological type of a field disk galaxy is completely determined by the outer cold gas accretion regime.
Acknowledgments {#acknowledgments .unnumbered}
===============
The observations reported in this paper were obtained with the Southern African Large Telescope (SALT). AYK acknowledges the support from the National Research Foundation (NRF) of South Africa. The study of isolated lenticular galaxies is supported by the grant No. 13-02-00059a of the Russian Foundation for Basic Research. IYK is grateful to Dmitry Zimin’s non-profit Dynasty Foundation.
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
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![NGC 3375. The same as previous figure.[]{data-label="fig_n3375"}](app_fig5.pdf){width="\textwidth"}
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![NGC 6010. The same as previous figure.[]{data-label="fig_n6010"}](app_fig7.pdf){width="\textwidth"}
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\[lastpage\]
[^1]: NASA/IPAC Extragalactic Database
[^2]: Third Reference Catalogue of Bright Galaxies.
|
---
abstract: 'We propose an effective exponent ruling the algebraic decay of the average quantum return probability for discrete Schrödinger operators. We compute it for some non-periodic substitution potentials with different degrees of randomness, and do not find a complete qualitative agreement with the spectral type of the substitution sequences themselves, i.e., more random the sequence smaller such exponent.'
author:
- |
César R. de Oliveira[^1] [and]{} Giancarlo Q. Pellegrino[^2]\
*Departamento de Matemática – UFSCar, *São Carlos, SP, 13560-970 Brazil**
date:
title: Quantum Return Probability for Substitution Potentials
---
PACS numbers: 05.45.Pq, 02.30.-f, 71.30.+h, 71.55.Jv
Anomalous transport in non-periodic structures is due to intricate quantum interferences which may also lead to localization of wave functions. Another possibility is ballistic motion, mainly related to periodic structures. Here we consider transport properties in nearest-neighbours tight-binding models in ${\ensuremath{\sf{Z\!\!Z}}}$ whose general Hamiltonian $H$ is given by $$\label{hamiltonian} (H\psi)_n=\psi_{n+1}+\psi_{n-1}+\lambda V_n\psi_n,$$ with $\lambda>0$ and potentials $V=(V_n)_{n\in{\ensuremath{\sf{Z\!\!Z}}}}$ generated by some non-periodic substitution sequences.
Among the characterizations of (de)localization and transport we single out the (average) moments of the “position” operator $$\label{moments} m_\alpha(T)=\frac{1}{T}\int_0^T dt\sum_{n=-\infty}^\infty |n-n_0|^\alpha |\psi_n(t)|^2,\;\;
\alpha\neq 0,$$ and the (average) return probability $$\label{return} C(T)=\frac{1}{T}\int_0^T dt\; |\psi_{n_0}(t)|^2 .$$ In relations (\[moments\]) and (\[return\]) it is implicitly assumed that the initial condition is $\psi_n=\delta_{n,n_0}$. Both quantities $m_\alpha(T)$ and $C(T)$ have strong physical appeal and in some cases are attainable to theoretical and numerical investigations. We notice that the return probability was one of the first quantities considered in the seminal paper by Anderson on localization in disordered structures [@A].
It has been found that for large $T$ [@G1; @GM1; @KPG; @H; @BCM; @L] $$\label{exponents} m_\alpha (T)\sim T^{\alpha\beta(\alpha)}\;\; {\rm and }\;\; C(T)\sim T^{-\Delta}.$$ Localization should be characterized by vanishing exponents $\beta$ and $\Delta$, ballistic motion by $\beta(\alpha)=\Delta=1$, while anomalous transport by $0<\beta,\Delta<1$. Notice that $\beta(2)$ is related to the direct conductivity via the anomalous Drude formula [@SBB1; @SBB2].
In this work we consider potentials $V$ in (\[hamiltonian\]) generated by some substitution sequences and compute the exponent for the decay of the return probability as a function of the degree of randomness of those sequences. As it will be seen later, in most cases $\Delta$ can not be obtained directly from numerical integration of the time-dependent Schrödinger equation; namely, the standard fitting procedure given by equation (\[exponents\]) works only for the well-investigated case of Fibonacci potentials (to be defined below). We then propose an alternative approach, based on the energy spectral decomposition of the initial quantum state, which not only is able to retrieve the known Fibonacci results but permits us to exhibit an effective exponent also for other non-periodic substitution potentials such as Thue-Morse, Rudin-Shapiro, paperfolding and period doubling (see below for their precise descriptions). The main conclusion will be that there is no perfect correspondence of the exponent ruling the algebraic decay of the return probability and the degree of randomness of the own substitution sequences, as one expects based on general considerations (i.e., more random the sequence smaller such exponent). Now we proceed to the details of the points just outlined, including a justification for the choice of $\Delta$ as our exponent of interest.
The RAGE theorem and Wiener lemma give direct physical meaning to the standard types of spectra, i.e., point and continuous (absolutely and singular) in the sense that one of their corollaries is that for continuous spectra $\beta(\alpha)>0$ and $\;\Delta>0$, while for point spectra $\Delta=0$; it is left room for $\beta>0$ even for point spectra due to the tail of eigenfunctions and domain intricateness [@dRJLS1; @dRJLS2]. The determination of such exponents is a quantitative step from RAGE and Wiener lemma which, by its turn, is still related to deep spectral quantities, i.e., generalized dimensions of the (positive) spectral measures associated to the initial state $\psi$. It has been rigorously proven that $\beta(\alpha)$ is bounded from below by the information dimension $D_1$ [@G1; @GM1] (all dimensions are related to the corresponding spectral measure) and also conjectured that $\beta(2)\approx D_0$ ($D_0$ denotes the fractal dimension of the spectrum). In an interesting paper Guarneri and Mantica [@GM2] have presented examples of homogeneous fractal spectral measures, i.e., with generalized dimensions $D_q=D_0$ for any $q$, for which $\beta(\alpha)$ is not constant and no simple exact relation seems to hold between the thermodynamics of the spectrum and the exponent $\beta$, so that “multiscaling does not require multifractality” [@GM2]; this was called quantum intermitency in [@GM2; @M1]. See [@BSB] for some recent results on a particular class of systems and other references.
On the other hand, it was rigorously proven [@H; @BCM] that the exponent $\Delta$ ruling the algebraic decay of the return probability equals the correlation dimension $D_2$. We note that such relation supposes the limit defining the dimension $D_2$ does exist. See also [@KPG] where this relation was first proposed in the context of anomalous diffusion. Therefore we have selected the return probability, its corresponding exponent $\Delta$ and correlation dimension $D_2$, as the main tools for analyzing our systems.
Relevant examples of anomalous diffusion are generated by almost-periodic potentials $V$; an important class of such potentials is given by $V$ induced by non-periodic (primitive) substitution sequences [@Q; @AG]. These sequences form a convenient laboratory for the study of anomalous transport since in all rigorously analyzed cases they generate singular continuous spectra for the tight-binding model (\[hamiltonian\]), although the own spectral types of substitution sequences are not equal; for example, Fibonacci ([Fcc]{}), paperfolding ([PF]{}) and period doubling ([PD]{}) substitution sequences have point autocorrelation measures, Thue-Morse ([TM]{}) has singular continuous autocorrelation measure, and the autocorrelation measure of Rudin-Shapiro ([RS]{}) substitution sequence is absolutely continuous. Although all these sequences are almost periodic, their spectral properties characterize them qualitatively from “ordered to random” since periodic and quasiperiodic sequences have pure point autocorrelation measures (as [Fcc, PF]{} and [PD]{} do), whereas independent random sequences have absolutely continuous autocorrelation measures (as [RS]{} does). The [TM]{} sequence lies in an intermediate place. It is also worth noting that all these sequences give rise to strictly ergodic dynamical systems with zero topological and generalized entropy [@Q; @B].
Due to different degrees of randomness of the substitution sequences, it was expected differences in the spectral properties of the corresponding tight-binding Hamiltonians (\[hamiltonian\]), but as already commented above all rigorously studied cases have presented singular continuous spectrum [@BBG; @BG1; @BG2; @Su; @HKS; @HM] (the RS case is an important open problem [@BG1; @HKS; @Al]; the [PF]{} case is also open). Our main goal in this Letter is to investigate whether the spectral character of the sequence generating the potential is responsible for different physics through details of the return probability behaviour. To this end we consider potentials $V$ in (\[hamiltonian\]) generated by the five non-periodic sequences [Fcc, PF, PD, TM]{} and [RS]{}. $\Delta$ can be computed either from numerical integration of the time-dependent Schrödinger equation, and then fitting a straight line on $\log
C(t)\times\log t$, or directly computing $D_2$ from its definition (which also involves a straight line fitting—see below). However, as already anticipated, we have faced problems in linear fittings in both procedures (except for the well-investigated case of [Fcc]{} potentials) and we propose a pragmatic approach to get such exponents which is able to recover the [Fcc]{} known results.
Now we present the rules describing the sequences we use to generate $V$. [Fcc, PD]{} and [TM]{} sequences are constructed with an alphabet of two letters $\{a,b\}$ through the substitutions $$a\rightarrow ab, \;b\rightarrow a\; {\rm(Fcc), \;\;\;}\;\;\; a\rightarrow ab,
\;b\rightarrow ba \;\;\;{\rm(TM),}$$ $$a\rightarrow ab, \;b\rightarrow aa\; {\rm(PD).}$$ Beginning with $a$ and applying successively the substitution rules, non-periodic sequences are obtained; e.g., the Thue-Morse sequence is given by $$abbabaabbaababba\cdots$$ The [RS]{} and [PF]{} sequences can be obtained with an alphabet of four letters $\{a',b',c',d'\}$, the substitutions $$a'\rightarrow a'b', \;\;\;b'\rightarrow a'c',\;\;\; c'\to d'b',\;\;\; d'\to d'c' \;\;\;\;{\rm(RS)},$$ $$a'\rightarrow a'b', \;\;\;b'\rightarrow c'b',\;\;\; c'\to a'd',\;\;\; d'\to c'd' \;\;\;\;{\rm(PF)},$$ and then the identifications $a',b'\to a$ and $c',d'\to b$ in both cases; the first elements of the RS sequence are $$aaabaabaaaabbb\cdots$$
We then use these substitution sequences to define our potentials $V$; we take $V_n=0$ if the $n$-th letter of the sequence is $a$ and $V_n=1$ in case it is $b$. There are standard ways to extend the potential for negative values of $n$ [@BG2; @HKS], but we avoid such issue by taking a finite sample of $N$ sites, with $n\geq0$, and using the initial wavefunction $\psi_n=\delta_{N/2,n_0}$. In this way we construct the almost-periodic substitution potentials $\lambda V$ and investigate $\Delta$ as function of the degree of randomness of the potential and its intensity $\lambda$.
It is known that [Fcc, PD]{} and [TM]{} generate potentials whose spectra of (\[hamiltonian\]) are singular continuous for all $\lambda\neq0$. The case of [RS]{} has been numerically investigated in [@DJR1; @DJR2] indicating point spectrum for $\lambda> 2$ and mixed spectrum, i.e., point and singular continuous, for $0<\lambda\leq2$ (notice we use a scale for the potential values which is different from [@DJR1]). For Hamiltonian (\[hamiltonian\]) with [PF]{} potential it is only known that its spectrum has no absolutely continuous component, since it is primitive [@HKS]; from a rigorous point of view the lack/presence of eigenvalues in this case is also an open question.
The case of [Fcc]{} Hamiltonian has also been considered in [@KPG] and a good agreement between the value of $\Delta$ from numerical integration of Schrödinger equation and $D_2$ was found. Let’s recall the definition of $D_2$ associated to a spectral measure $\mu$ and how it is usually estimated [@KPG; @BCM]. For $\varepsilon>0$ let $B_\varepsilon (x)$ denote the open ball of centre $x$ and radius $\varepsilon/2$ and set $$\gamma(\varepsilon)=\int\mu(B_\varepsilon(x))d\mu(x);$$ the correlation dimension of $\mu$ is given by the limit $$\label{D2} D_2=\lim_{\varepsilon\to 0} \frac{\log\gamma(\varepsilon)}{\log\varepsilon}.$$ If this limit does not exist one defines $D_2^+$ and $D_2^-$ via $\limsup$ and $\liminf$, respectively. The latter remark is important here since we have found numerical indications that the limit in the definition (\[D2\]) does not exist for general substitution potentials. In numerical practice we have a finite basis approximation for (\[hamiltonian\]) whose spectrum is composed of eigenvalues $\chi_k$; then we divide the energy range into boxes $B_j$ of length $\varepsilon$, approximate $$\label{gamaTil}
\gamma(\varepsilon)\approx \gamma^*(\varepsilon) = \sum_j (\sum_{\chi_k\in B_j} |a_k|^2)^2$$ and get $D_2$ from the linear fitting of $\log \gamma^*(\varepsilon)\times\log\varepsilon$. $a_k$ is the projection of the initial wavefunction $\psi$ on the eigenvector with eigenvalue $\chi_k$. We have used these procedures to recover $D_2$ and $\Delta$ found in [@KPG] for the [Fcc]{} case as illustrated in figure 1. However such techniques do not work for substitution potentials distinct from [Fcc]{} (as far as we have checked), since no clear region with linear behavior is found in the plots $\log \gamma^*(\varepsilon)\times\log\varepsilon$ and $\log C(t)\times\log t$, as exemplified in figure 2 for the [PD]{} potential with $\lambda=1.6$. We suspect this behaviour is an indication that the limits defining the scale exponents $D_2$ and $\Delta$ are not well defined in such situations; then we propose a pragmatic approach to extract effective exponents $D_2$ by selecting a particular value $\varepsilon^*$ of $\varepsilon$. Before presenting our approach we stress we have also tried to get well-defined exponents by site averaging on samples beginning at locations $0,1\times10^4,2\times10^4,\cdots,5\times10^4$, but quite similar behaviours were found.
We begin our argument with the remark that if $\varepsilon$ is smaller than the least eigenvalue spacing (we just ignore the possibility of degenerate eigenvalues in this argument) then $$\gamma^*(\varepsilon)= \sum_k |a_k|^4$$ which resembles the so-called inverse participation ratio (which does not depend on $\varepsilon$); this also gives a physical interpretation for $D_2$. As a naive first guess for an effective exponent one could try to use $\sum_k |a_k|^4$ instead of $\gamma^*(\varepsilon)$, but the exact value of $\varepsilon$ to be used in an approximation to (\[D2\]) is not clear at all. The theoretical determination of $D_2$ involves the limit $\varepsilon\to 0$; finite basis approximations preclude this limit and also too small values of $\varepsilon$ are meaningless, despite the inverse participation ratio interpretation. For sufficient small values of $\varepsilon$ we have $$\label{divisao}
\frac{\gamma^*(\varepsilon)}{ \sum_k |a_k|^4}\approx 1;$$ we suggest to pick $\varepsilon^*$ as the smallest value of $\varepsilon$ such that the l.h.s. of (\[divisao\]) considerably deviates from $1$, so still keeping track of $\gamma^*(\varepsilon)$ and also the inverse participation ratio interpretation in operation. Then we estimate the effective $D_2$ as $D^*_2$ given by $$\label{epsStar}D^*_2= \frac{\log\gamma^*(\varepsilon^*)}{\log\varepsilon^*}.$$ Let’s be more precise on how we have picked up $\varepsilon^*$ in practice. By using double precision (16 digits) in our code, we adopted that after diagonalization we can numerically resolve the spectral quantities with 8 digits, i.e., half of the number of digits of the code precision, so that $\varepsilon^*$ is given by the smallest value of $\varepsilon$ such that $\gamma^*(\varepsilon)-\sum_k |a_k|^4\geq10^{-8}$ or, equivalently, the smallest $\varepsilon$ such that $$|\frac{\gamma^*(\varepsilon)}{ \sum_k |a_k|^4}-1|\geq\frac{10^{-8}}{\sum_k |a_k|^4}.$$ We remark that in most cases $\varepsilon^*$ can also be obtained directly from visual inspection, as in figure 2b, and the precise value $10^{-8}$ is not so relevant since in general $\gamma^*(\varepsilon)$ has a pronounced jump at $\varepsilon=\varepsilon^*$.
We have tested our approach in the [Fcc]{} case and have got very good agreement with the computed values of $D_2$ from our linear fittings and the values reported in [@KPG]. In figure 2b we show a typical curve used to estimate $\varepsilon^*$, and in figure 3 we compare the values of the exponents $\Delta$ and $D_2$ as calculated in figure 1 and also the matching values of $D^*_2$ from equation (\[epsStar\]) for the [Fcc]{} case. From now on we use this procedure to estimate the exponents $D^*_2$ for the other substitution Hamiltonians (\[hamiltonian\]) considered here.
Its now time to discuss our numerical results and details of their implementations. The return probability $C(T)$ was calculated by direct diagonalization of the Hamiltonian and eigenfunction expansion of the initial state; we used bases of size $N\approx
1\times10^3$ and checked some results with bases of size $N\approx 2\times10^3$. The initial condition was always concentrated on the centre of the basis $n_0$ and we have followed its time evolution until time $T_f$ for which the modulus of the amplitude at one of the border sites reaches $1\times10^{-6}$. For the calculation of $D_2$ we have considered subdivisions of the spectrum in subintervals of size $\varepsilon$ ranging from the least eigenvalues spacing (we disregarded multiple eigenvalues) up to $10^{-2}$. We could seldom conceive a linear behaviour in such $\log-\log$ plots in both $D_2$ and $\Delta$ cases in order to extract faithful exponents, so that we were left with the task of finding $\varepsilon^*$ and computing only $D_2^*$.
In figure 4 we present a summary of our main numerical results, i.e., the values of $D_2^*$ for some substitution potentials as function of the potential intensity $\lambda$. Since we are not aware of any complete rigorous spectral classification for Hamiltonian (\[hamiltonian\]) in the cases of [RS]{} and [PF]{} sequences, we have also used $T_f$ as indication of any possible (de)localization transition; this is the reason for the restriction of the [RS]{} case to $\lambda\leq1.7$; for all substitution sequences we have found $T_f\leq10^3$ for $\lambda\leq2$, but for [RS]{} $T_f$ jumps from $T_f\approx10^3$ for $\lambda=1.7$ to $T_f\approx10^5$ for $\lambda=1.8$, which characterizes absence (at least numerically) of extended states. If non-localized states are present their “amounts” suffer a drastic reduction at $\lambda\approx1.8$ so that we have not detected them. Recall that in [@DJR1; @DJR2] it is argued that all states of [RS]{} Hamiltonian should be localized for $\lambda>2.0$.
Since no such sharp transition in $T_f$ was found for the [PF]{} Hamiltonian, its values of $D_2^*$ are close to the corresponding values for [PD]{}, and both substitution sequences have point autocorrelation measures, we conjecture that the [PF]{} Hamiltonian (\[hamiltonian\]) has singular continuous spectrum for $0\neq|\lambda|\leq 2$ (maybe also for any $\lambda\neq 0$) as [PD]{} Hamiltonian does [@BBG].
Besides the above conjecture we see from figure 4 that for all sequences the exponent $D_2^*$ decreases as $\lambda$ increases (as physically expected). Since different exponents were found despite the proven singular continuous spectra of [Fcc, TM]{} and [PD]{} Hamiltonians, we see that $D_2^*$ is able to discern these operators. The values of $D_2^*$ for the [RS]{} are very close to the [TM]{} case, but in principle one would not expect this since the autocorrelation measure of the [RS]{} sequence is Lebesgue measure (the same for random sequences). Notice that the spectral classification of the underlying sequence generating the potential does not reflect exactly in $D_2^*$, since these values for [PF]{} and [PD]{} are below the corresponding ones for [TM]{} and [RS]{}, the latter sequences being considered “more random” than the former ones. Notice, however, that for the most popular sequences [Fcc, TM]{} and [RS]{} we have found agreements with the classification through the sequence spectral type, i.e., this order implying decreasing values of $D_2^*$.
The higher values of the exponents for [RS]{} compared to [PD]{} indicates that the presence of extended states mixed with numerically found localized states [@DJR1; @DJR2] does not necessarily imply lower exponents $D_2^*$.
Summing up, we have found that for substitution potentials in general the dynamical exponent $\Delta$ and the correlation dimension $D_2$ are difficult to be obtained from direct linear fittings (at least with the basis sizes we used; we suspect this is a consequence of the quantum intermitency and multiscaling in time of the dynamics [@GM2; @M1]) and we have proposed $D_2^*$ as an effective exponent, which has recovered $D_2$ in the cases it can be directly obtained. We then computed $D_2^*$ for some substitution potentials and have not found a complete qualitative agreement with the spectral type of the substitution sequences themselves, i.e., more random the sequence smaller $D_2$. Only for [RS]{} we have got indications of a spectral transition from extended (critical) to localized states, although its values of $D_2^*$ are higher than those for [PD]{} and [PF]{} cases.
### Acknowledgments {#acknowledgments .unnumbered}
[CRO was partially supported by CNPq (Brazil); discussions with U. Grimm at the Max Planck Institute for the Physics of Complex Systems (Dresden, Germany) are acknowledged. GQP thanks the support by FAPESP (Brazil).]{}
[99]{}
Anderson P W 1958 [*Phys. Rev.*]{} [**109**]{} 1492
Guarneri I 1993 [*Europhys. Lett.*]{} [**21**]{} 729
Guarneri I and Mantica G 1994 [*Ann. Inst. H. Poincaré*]{} [**A61**]{} 369
Ketzmerick R, Petschel G and Geisel T 1992 [*Phys. Rev. Lett.*]{} [**69**]{} 695
Holschneider M 1994 [*Commun. Math. Phys.*]{} [**160**]{} 457
Barbaroux J-M, Combes J-M and Montcho R 1997 [*J. Math. Anal. Appl.*]{} [**213**]{} 698
Last Y 1996 [*J. Funct. Anal.*]{} [**142**]{} 402
Schulz-Baldes H and Bellissard J 1998 [*Rev. Math. Phys.*]{} [**10**]{} 1
Schulz-Baldes H and Bellissard J 1998 [*J. Stat. Phys.*]{} [**91**]{} 991
del Rio R, Jitomirskaya S, Last Y and Simon B 1995 [*Phys. Rev. Lett.*]{} [**75**]{} 117
del Rio R, Jitomirskaya S, Last Y and Simon B 1996 [*J. Anal. Math.*]{} [**69**]{} 153
Guarneri I and Mantica G 1994 [*Phys. Rev. Lett.*]{} [**73**]{} 3379
Mantica G 1997 [*Physica*]{} [**D 109**]{} 113
Barbaroux J-M and Schultz-Baldes H 1999 [*Anomalous quantum transport in presence of self-similar spectra*]{} (Technishe Univ. Berlin, preprint)
Queffélec M 1987 [*Substitution Dynamical Systems—Spectral Analysis*]{}, Lecture Notes in Mathematics Vol. 1294 (Berlin: Springer)
Axel F and Gratias D (eds.) 1995 [*Beyond Quasicrystals*]{} (Berlin: Les Editions de Physique and Springer Verlag)
Berthé V 1994 [*J. Phys. A: Math. Gen.*]{} [**27**]{} 7993
Bellissard J, Bovier A and Ghez J-M 1991 [*Commun. Math. Phys.*]{} [**135**]{} 379
Bovier A and Ghez J-M 1993 [*Commun. Math. Phys.*]{} [**158**]{} 45
Bovier A and Ghez J-M 1993 [*Commun. Math. Phys.*]{} [**166**]{} 431
Sütő A 1989 [*J. Stat. Phys.*]{} [**56**]{} 525
Hof A, Knill O and Simon B 1995 [*Commun. Math. Phys.*]{} [**174**]{} 149
Hörnquist M and Magnus J 1995 [*J Phys A: Math. Gen.*]{} [**28**]{} 479
Allouche J-P 1997 [*J. Math. Phys.*]{} [**38**]{} 1843
Dulea M, Johansson M and Riklund R 1992 [*Phys. Rev.*]{} [**B 45**]{} 105
Dulea M, Johansson M and Riklund R 1993 [*Phys. Rev.*]{} [**B 47**]{} 8547
Figure Captions {#figure-captions .unnumbered}
===============
a\) Log-log (base 10) of the return probability (dashed line) $C(t)$ for the [Fcc]{} potential with $\lambda=1.0$ versus time. The slope of the straight line fitting (full line) corresponding to $\Delta$ is indicated.
\(b) Log-log (base 10) of $\gamma^*$ (dashed line) for the [Fcc]{} potential with $\lambda=1.0$ versus $\varepsilon$. The slope of the straight line fitting (full line) corresponding to $D_2$ is indicated.
[[**Figure 2**]{}]{}
Same as in figure 1 but for the [PD]{} potential with $\lambda=1.6$. The arrow in b) indicates $\varepsilon^*$; the first point at left in b) corresponds to the least eigenvalue spacing, for which $\gamma*=\sum_k |a_k|^4$. No linear fitting is shown.
[[**Figure 3**]{}]{}
Scaling exponents $D_2$, $D_2^*$ and $\Delta$ for various potential intensities $\lambda$ of the [Fcc]{} potential.
[[**Figure 4**]{}]{}
Effective scaling exponents $D_2^*$ as function of $\lambda$ for some substitution potentials.
[^1]: e-mail: oliveira@dm.ufscar.br
[^2]: e-mail: gian@dm.ufscar.br
|
---
abstract: 'In this work we consider the two-dimensional percolation model arising from the majority dynamics process at a given time $t\in{\mathbb{R}}_+$. We show the emergence of a sharp threshold phenomenon for the box crossing event at the critical probability parameter $p_c(t)$. We then use this result in order to obtain stretched-exponential bounds on the one-arm event probability in the subcritical phase. Our results are based on differential inequalities derived from the OSSS inequality [@osss], inspired by the recent developments [@abgm; @dcrt1; @dcrt2].'
address:
- 'Institute of Mathematics, University of Leipzig – Augustusplatz 10, 04109 Leipzig, Germany'
- 'Bar-Ilan University, 5290002, Ramat Gan, Israel'
author:
- Caio Alves
- Rangel Baldasso
bibliography:
- 'mybib.bib'
title: |
Sharp threshold for two-dimensional\
majority dynamics percolation
---
Introduction
============
In recent years, the study of sharp threshold phenomena in percolation has received great attention. This is mainly due to the development of new techniques that allow the treatment of dependent models [@dcrt1; @dcrt2]. Following this line, in this paper we prove that, for each fixed $t \geq 0$, percolation in two-dimensional majority dynamics undergoes a sharp phase transition in the density parameter.
In two-dimensional majority dynamics, each vertex $x \in {\mathbb{Z}}^{2}$ receives an initial opinion which can be either zero or one[^1]. With rate one, the vertex $x$ updates its opinion to match the majority of its neighbors. In the case of a tie, the original opinion is kept. Denote by ${\mathbb{P}}_{p,t}$ the distribution of the process at time $t$ when the initial density of ones is $p \in [0,1]$. Our interest lies in understanding the critical percolation function defined as $$\label{eq:perc_function}
p_{c}(t) = \inf \left\{ p \in [0,1]: {\mathbb{P}}_{p,t}\left[\begin{array}{c} \text{there exists an} \\ \text{infinite open path} \end{array} \right]>0\right\}.$$ For each $t>0$, we have $p_{c}(t) \in \left[\frac{1}{2}, p_{c}^{site}\right)$, where $p_{c}^{site}$ is the critical threshold for two-dimensional site percolation (see [@ab]).
Our main result here regards crossing events. For each $n \in {\mathbb{N}}$, let $R_{n} = [1,n]^{2}$, and consider the crossing event $$H(n,n) = \left\{\begin{array}{c} \text{there exists an open path contained in } R_{n} \\ \text{ that connencts } \{1\} \times [1,n] \text{ to } \{n\} \times [1,n] \end{array} \right\}.$$
\[t:sharp\_thresholds\] For each $t \geq 0$, there exists $\gamma=\gamma(t)>0$ such that $${\mathbb{P}}_{p_{c}(t)-n^{-\gamma},t}[H(n,n)] \to 0 \quad \text{and} \quad {\mathbb{P}}_{p_{c}(t)+n^{-\gamma},t}[H(n,n)] \to 1,$$ as $n$ grows.
In the theorem above, we consider crossings of the squares $R_{n}$. In fact, the proof applies to rectangles of any fixed aspect ratio: for each fixed $\lambda>0$, we can consider horizontal crossings of the rectangles $R_{n}^{\lambda}=[1,\lambda n] \times [1,n]$.
[\[c:decay\]]{}
As a consequence of Theorem \[t:sharp\_thresholds\], together with a general multiscale renormalisation argument, we obtain stretched-exponential decay of one-arm probabilities in the subcritical phase.
\[t:exp\_decay\] For any $t \geq 0$ and $p < p_{c}(t)$, there exists a positive constant ${c_{\textnormal{\tiny \ref{c:decay}}}}={c_{\textnormal{\tiny \ref{c:decay}}}}(p,t)>0$ such that $${\mathbb{P}}_{p,t}\left[\begin{array}{c} \text{there exists an open path connecting } 0 \\ \text{ to the boundary of the ball } B(0,n) \end{array} \right] \leq \exp\left\{-{c_{\textnormal{\tiny \ref{c:decay}}}}\frac{n}{\log n}\right\}.$$
We remark that the above result follows from a general statement inspired by [@pt] we prove here about dependent percolation with fast decay of correlations. Under general conditions (see Proposition \[p:1armdecaygen\]), this result together with Russo-Seymour-Welsh-type arguments imply stretched-exponential decay in the subcritical phase.
**Overview of the proofs.** The proof of Theorem \[t:sharp\_thresholds\] relies on exploiting the relation between Boolean functions and randomized algorithms obtained through OSSS inequality. Here it is possible to write the existence of a crossing at time $t$ as a random Boolean function of the initial configuration, with randomness coming from the evolution of majority dynamics. A first approach would then be to consider the quenched configuration, where the clocks of the Markov process are fixed, and try to use these tools directly on the space of initial configurations, for each possible realization of the Poisson clocks in the evolution. This idea fails, since quenched configurations lack the homogeneity needed for our arguments.
To circumvent this difficulty, we need to consider the randomness that comes from the evolution together with the one from the initial configuration. We then revisit the idea developed in [@abgm], and further explored in [@att] and [@ahlbergb], of using a two-stage construction of the process to obtain a discretization of it that still retains relevant properties of the annealed evolution. The central idea is to construct the process in a way that each vertex is associated to a Poisson point process of clocks of intensity $k \in {\mathbb{N}}$, with $k$ large. Whenever a clock in a given vertex rings, we keep this ringing with probability $\frac{1}{k}$ and, in this case, update the opinion of the vertex to agree with the majority of its neighbors.
This artificial increase of the density of clock ringings allows us now to consider quenched probabilities, as we condition on the denser Poisson process, and still retain good properties of the annealed configuration with large probability. Given a collection of clock ringings, we obtain a Boolean function by considering the initial opinions and the selection of the clock-ringings that are kept for the evolution.
We then proceed to analyze this quenched random Boolean function. First, we devise an algorithm that determines the outcome of the function and bound its revealment. This algorithm is a simple exploration process that discovers the open components that intersect a random line crossing the rectangle $R_{n}$ by querying the initial state of sites and which clock ringings are selected to compose the evolution. The bound on the revealment will follow from one-arm estimates in the quenched setting (see Proposition \[prop:one\_arm\]). These estimates in turn are derived from Russo-Seymour-Welsh-type results stated in [@ab] and inspired by [@tassion].
Since we are considering randomness that comes from the time evolution as well, when applying OSSS inequality it will be necessary to control the influence of clock ringings. We relate time-pivolality to spatial-pivotality, bounding the influence of a clock ringing by a combination of the influences of the initial positions (see Proposition \[prop:piv\]). This pivotality relation is the most original and sensitive part of our proof, and fails, for example, if one considers the contact process instead of majority dynamics as the rule for the time evolution of the opinions. Nevertheless, we can also prove a similar result for the voter model (see Section \[sec:further\_models\]). With this relation in hands, we are able to conclude the proof of Theorem \[t:sharp\_thresholds\].
Let us now turn our attention to the proof of Theorem \[t:exp\_decay\]. Here, we provide a general statement on the decay rate of the one-arm probability in percolation models with fast decaying correlations. We prove that, provided the annulus crossing probability goes to $0$ as the size of the annulus goes to infinity, the rate of decay of the one-arm probability is at least stretched exponential in the ball’s radius. Combining this with Theorem \[t:sharp\_thresholds\] yields Theorem \[t:exp\_decay\]. The proof of this statement relies on a multiscale renormalisation argument adapted from [@pt].
Our technique is somewhat general and might be applied to other dynamics. As an example, in Section \[sec:further\_models\], we explain how to adapt it to the case when the opinions follow the voter model. The greatest obstacle to a broader generalisation is the lemma relating time- and space-pivotality, whose proof is strongly model-dependent.
Camia, Newmann, and Sidoravicius in [@cns] prove that fixation of the opinions happens with stretched-exponential speed in a sub-interval of the supercritical phase. The idea of the proof is to observe that, if $p$ is larger than $p_{c}^{site}$ (the critical probability for Bernoulli site percolation in ${\mathbb{Z}}^2$), one can obtain a random partition of ${\mathbb{Z}}^{2}$ into finite subsets whose boundaries are circuits of constant initial opinion which are preserved by the dynamics, reducing the evolution to finite random subsets. This, together with the uniform bound on the number of changes in opinion each vertex can have (see Tamuz and Tessler [@tt]), allows one to conclude that the speed of convergence is stretched exponential. They further improve the proof by performing an enhancement on the initial configuration, and conclude that stretched exponential decay also holds for values of $p$ slightly smaller than $p_{c}^{site}$. We remark that the same idea can be applied together with Theorem \[t:exp\_decay\] to verify that streched-exponential decay of the non-fixation probability also holds for $p \in \left( \lim p_{c}(t), 1 \right]$. Symmetry considerations imply an analogous result for $p \in \left[0, 1- \lim p_{c}(t) \right)$.
**Related works.** Russo’s approximate 0-1 law [@russo] may be seen as a first result regarding sharp thresholds in independent percolation. It says that a sequence of monotone Boolean functions exhibits a sharp threshold, provided the supremum of the influences converges to zero. The use of randomized algorithms and OSSS inequality to understand threshold phenomena is much more recent and so far has proven to be a very powerful technique. Duminil-Copin, Raoufi and Tassion [@dcrt1; @dcrt2] use these techniques to study the subcritical phase of Voronoi percolation and threshold phenomena for the random-cluster and Potts models.
After these seminal works, other applications of such techniques were found. Muirhead and Vanneuville [@mv] use this approach to conclude that level-set percolation for a wide class of smooth Gaussian processes undergoes a sharp phase transition. Dereudre and Houdebert [@dh] conclude similar statements for the Widom-Rowlinson model.
The collection of upper invariant measures for the contact process was also studied. Van den Berg [@vdb] considers the two-dimensional case, and proves the existence of a sharp phase transition without relying on the OSSS inequality. More recently, Beekenkamp [@beekenkamp] generalized this result for any dimension $d \geq 2$. The proof relies on three central steps. First, estimates on the speed of convergence of the distributions are used in order to reduce the analysis to configurations in finite time. Second, for a given fixed time $t$, a discretization scheme is developed for the graphical construction. With this in hands, the machinery developed for discrete settings can be used to deduce the existence of a sharp threshold phenomena. Crucially, in this model the sharp threshold phenomenon is exhibited in the rate of infection $\lambda$ associated to the process. Therefore, the techniques employed cannot be easily translated into our context, since the object of our study is the density of the initial configuration.
The discretization we use here is more in line with the one considered in Ahlberg, Broman, Griffiths, and Morris [@abgm], where the authors prove noise sensitivity for the critical Boolean model. With a similar discretization, and relying on Talagrand’s inequality [@talagrand], Ahlberg, Tassion, and Teixeira [@att] deduce that Boolean percolation undergoes a sharp phase transition. Furthermore, Ahlberg, in collaboration with the second author [@ahlbergb], employs this technique to two-dimensional Voronoi percolation to study noise sensitivity and conclude, as a corollary, the existence of a sharp threshold with polynomial window.
Regarding percolation in majority dynamics, Amir and the second author [@ab] prove that the percolation function $p_{c}(t)$ is a continuous non-increasing function that is strictly decreasing on zero and that $p_{c}(t) \geq \frac{1}{2}$. Besides, they also obtain that there is no percolation at criticality for any time $t \geq 0$.
**Open problems.** Regarding the percolation function $p_{c}(t)$ (see Equation ), it is know that it is a continuous non-increasing function that is strictly decreasing at zero. Whether or not it is strictly decreasing in the whole non-negative real line it is still not known. We hope our new estimates on the connectivity decay of the subcritical phase might help.
Our techniques are reliant on RSW theory, and are therefore limited to two dimensions. We believe our results to be valid for any dimension and for a large class of particle system models, and that with future developments in the field such general problems will be tractable.
**Organization of the paper.** In Section \[sec:properties\], we state properties of the majority dynamics and some results that will be used throughout the text. Section \[sec:construction\] contains a graphical construction of majority dynamics that will be used in our results, while Section \[sec:influence\] discusses the concept of influences and pivotality in the quenched setting. We present a randomized algorithm and bound its revealment in Section \[sec:alg\], and use this algorithm to conclude the proof of Theorem \[t:sharp\_thresholds\] in Section \[sec:thresholds\]. In Section \[sec:one\_arm\], we provide quenched one-arm estimates for the model that were previously assumed in the proof of Theorem \[t:sharp\_thresholds\]. Theorem \[t:exp\_decay\] is proved in Section \[sec:decay\]. Finally, we discuss how to modify our result to the case when the dynamics follows the voter model in Section \[sec:further\_models\].
**Acknowledgments.** The authors thank Daniel Valesin for valuable discussions during the elaboration of this work. CA is supported by the DFG grant SA 3465/1-1. RB is supported by the Israel Science Foundation through grant 575/16 and by the German Israeli Foundation through grant I-1363-304.6/2016
Basic properties {#sec:properties}
================
We denote by $\eta\equiv\eta(p)=(\eta_t)_{t \in {\mathbb{R}}{+}}$ the two-dimensional majority dynamics with initial configuration $\eta_0\in\{0,1\}^{{\mathbb{Z}}^2}$, which assigns i.i.d. $\mathrm{Bernoulli}(p)$ random variables to each vertex of ${\mathbb{Z}}^2$. As mentioned in the Introduction, we denote by ${\mathbb{P}}_{p,t}$ the law of $\eta_t=\eta_t(p)$. We collect here facts about this collection of measures. A complete proof of these facts can be found in [@ab] and references therein.
Notice that, as a consequence of Harris [@harris] and a correlation decay estimate (see Equation \[eq:correlation\_decay\_radius\]), the measures ${\mathbb{P}}_{p,t}$ are positively associated. This is the same as stating that ${\mathbb{P}}_{p,t}$ satisfies the FKG inequality: for any two events $A$ and $B$ that are increasing with respect to the partial ordering[^2] of $\{0,1\}^{{\mathbb{Z}}^{2}}$, it holds that $${\mathbb{P}}_{p,t}[A \cap B] \geq {\mathbb{P}}_{p,t}[A] {\mathbb{P}}_{p,t}[B].$$
Given two disjoint subsets $A$ and $B$ of ${\mathbb{Z}}^{2}$ and $X \subset {\mathbb{Z}}^{2}$ such that $A \cup B \subset X$, we define the event $$\label{eq:open_connection}
\left[A \overset{X}{\longleftrightarrow} B\right]$$ as the existence of an open path contained in $X$ connecting a vertex in $A$ to a vertex in $B$. We omit $X$ in the notation above when $X={\mathbb{Z}}^{2}$. The event where percolation holds is defined as the existence of an infinite open path. Standard arguments yield that $${\mathbb{P}}_{p,t}[\eta \text{ percolates}]>0 \quad \text{if, and only if,} \quad \inf_{n}{\mathbb{P}}_{p,t}\left[\{0\} \leftrightarrow \partial B(0,n)\right]>0,$$ where $\partial B(0,n) = \{x \in {\mathbb{Z}}^{2}: {\left\|x\right\|_{\infty}}=n\} $ is the boundary of the ball $B(0,n)=[-n,n]^{2}$.
[\[c:correlation\]]{} [\[c:rsw\]]{}
Let us now list some properties of the probabilities ${\mathbb{P}}_{p,t}$ for a fixed $t$. First of all, we state correlation decay for these measures, which is a consequence of standard cone-of-light estimates. For each $t \geq 0$, there exists a constant ${c_{\textnormal{\tiny \ref{c:correlation}}}}={c_{\textnormal{\tiny \ref{c:correlation}}}}(t)$ such that, if $A$ is an event that depends on the configuration $\eta_{t}(x)$ only on sites inside $[-n,n]^{2}$ and $B$ is an event that depends on the configuration on sites outside $[-2n,2n]^{2}$, then, for every $p \in [0,1]$, $$\label{eq:correlation_decay_radius}
\Big|{\mathbb{P}}_{p,t}[A \cap B] - {\mathbb{P}}_{p,t}[A]{\mathbb{P}}_{p,t}[B] \Big| \leq {c_{\textnormal{\tiny \ref{c:correlation}}}}n^{2}e^{-\frac{n}{2}}.$$
Given $\lambda>0$, denote by $H(\lambda n, n)$ the crossing event $$H(\lambda n,n) = \left[ \{1\} \times [1,n] \overset{R_{n}}{\longleftrightarrow} \{\lfloor \lambda n \rfloor \} \times [1,n] \right],$$ where $R_{n}\equiv R_n(\lambda)=[1,\lambda n] \times [1,n]$, and let $H^{*}(\lambda n, n)$ denote the event of the existence of a closed horizontal $*$-crossing[^3] of the rectangle $R_{n}$. The main result regarding crossing events is the RSW theory, that we can obtain by adapting the proofs of Tassion [@tassion], since they rely on the invariance of the percolation measure under certain simmetries of ${\mathbb{Z}}^2$, decay of correlations, and the FKG inequality, properties that are also available to us.
\[prop:RSW\] For each fixed value of $t \geq 0$ and each $\lambda>0$, there exists a positive constant ${c_{\textnormal{\tiny \ref{c:rsw}}}}={c_{\textnormal{\tiny \ref{c:rsw}}}}(\lambda, t)>0$ such that $$\label{eq:rsw}
{c_{\textnormal{\tiny \ref{c:rsw}}}} \leq {\mathbb{P}}_{p_{c}(t),t}\left[H(\lambda n,n) \right] \leq 1-{c_{\textnormal{\tiny \ref{c:rsw}}}},$$ for all $n \in {\mathbb{N}}$.
Since $H(\lambda n, n)$ holds if, and only if, there is no closed vertical $*$-crossing of $R_{n}=[0,\lambda n] \times [0,n]$, one can easily deduce from the proposition above that an analogous result holds for the event $H^{*}(\lambda n, n)$. Furthermore, monotonicity considerations imply that, for all $p \geq p_{c}(t)$, $$\label{eq:rsw2}
\inf_{n} {\mathbb{P}}_{p,t}\left[H(\lambda n,n) \right] \geq {c_{\textnormal{\tiny \ref{c:rsw}}}}(\lambda, t),$$ and, for all $p \leq p_{c}(t)$, $$\label{eq:rsw3}
\inf_{n} {\mathbb{P}}_{p,t}\left[H^{*}(\lambda n,n) \right] \geq {c_{\textnormal{\tiny \ref{c:rsw}}}}(\lambda^{-1}, t).$$
**The OSSS inequality.** Let us quickly recall the version of the OSSS inequality we use here. Fix $f:\{0,1\}^{n} \to \{0,1\}$ a Boolean function and, for a vector $\bold{p} = (p_{1}, \dots p_{n})$, let ${\mathbb{P}}_{\bold{p}}$ denote the probability measure on $\{0,1\}^{n}$ where each entry is independent and the $i$-th entry has probability $p_{i}$ of being one. For each $i \in [n]$, we define the influence of the bit $i$ as $$\operatorname{Inf}_{\bold{p}}(f,i) = {\mathbb{P}}_{\bold{p}}[f(\omega) \neq f(\omega^{i})],$$ where $\omega^{i}$ is obtained from $\omega$ by changing the $i$-th entry of the vector.
A (randomized) algorithm $\mathcal{A}$ is a rule that outputs a value zero or one, by querying entries of the vector $\omega$, and whose choice of the next entry to be queried is allowed to depend on the previous observations. An algorithm can determine its output before querying all bits, in this case we say the algorithm *stops*. We say that the algorithm determines $f$ if its outcome coincides with $f(\omega)$, for every $\omega$. The revealment of the bit $i$ for an algorithm $\mathcal{A}$ is the quantity $$\delta(\mathcal{A},i) = {\mathbb{P}}_{\bold{p}}[\mathcal{A} \text{ queries } i \text{ before stopping}].$$
The OSSS inequality (see [@osss]) provides the bound $$\label{eq:osss}
\operatorname{Var}(f) \leq \sum_{i=1}^{n}\delta(\mathcal{A},i)\operatorname{Inf}_{\bold{p}}(f,i).$$ Poincaré’s inequality can be recovered from the above inequality by bounding all the revealments by one. We remark that the above version is not the original OSSS inequality, since we consider different parameters for each entry, but the same proof applies for this case.
The two-stage construction {#sec:construction}
==========================
In this section we present a graphical construction of the majority dynamics that will be used in the rest of the paper. We begin by presenting the usual Harris construction, since we will use a simple modification of it.
Consider a collection $\mathscr{P}=\big( \mathscr{P}_{x} \big)_{x \in {\mathbb{Z}}^2}$ of i.i.d. Poisson processes in the interval $[0,t]$ with rate one. For each $x \in {\mathbb{Z}}^2$, the clocks $\mathscr{P}_{x}$ will control the updates in that site: whenever the clock at $x$ rings, the opinion at $x$ is updated to match the majority of its neighbors. In case of a draw, the site keeps its original opinion. With this construction, we can fully determine the state of the system at any given time with the collection of clocks $\mathscr{P}$ and the initial configuration $\eta_{0}$.
\[remark:voter\_model\] It is possible to obtain the voter model with the same graphical construction, just by modifying the way sites are updated: instead of choosing the new opinion to be the majority of the neighboring opinions, the update is made by copying the opinion of a randomly selected neighbor.
The construction we will use is a slight modification of the one presented above. Instead of considering the collection of clocks $\mathscr{P}$, we start with a denser collection of clocks $\mathscr{P}^{k}=\big( \mathscr{P}_{x}^{k} \big)_{x \in {\mathbb{Z}}^2}$ distributed as i.i.d. Poisson processes on the interval $[0,t]$ with rate $k$, where $k$ is a fixed positive integer number that will be taken to be large. With this collection of clocks in hand, we need some additional randomness in order to define the process: whenever a clock rings, we perform the update at the respective site with probability $\frac{1}{k}$ (this can be realized by considering an independent $\operatorname{Bernoulli}\left(\frac{1}{k}\right)$ random variable for each clock ringing of $\mathscr{P}^{k}$). In this case, conditioned on the realization of the clocks $\mathscr{P}^{k}$, we can obtain the state of the system at any given time $t \geq 0$ by using the initial configuration $\eta_{0}$ and the collection of random variables that verify whether or not each update is performed. We will denote by ${\mathbb{P}}_{kt}$ the distribution of $\mathscr{P}^{k}$ and by $P_{p, \frac{1}{k}}$ the joint distribution of the initial condition and the additional randomness necessary in order to determine the process $(\eta_{s})_{s \geq 0}$.
The advantage of the last construction presented above lies in the fact that the model at time $t$ may be seen as a random Boolean function: for each realization of $\mathscr{P}^{k}$, we obtain a Booelan function whose entries select the initial configuration and which updates are performed. By choosing the value of $k$ large enough, we can ensure that these random functions are well-behaved, in a sense that we will make clear later.
We will work with the process conditioned on the realization $\mathscr{P}^{k}$. In this case, we may write the characteristic function of the crossing event $\left[ \eta_{t} \in H(\lambda n,n)\right]$ as a Boolean function $f_{n}: \{0,1\}^{\Lambda} \to \{0,1\}$, where $$\Lambda = {\mathbb{Z}}^2 \cup \{(x,s): x \in {\mathbb{Z}}^2, \, s \in \mathscr{P}^{k}_{x} \cap [0,t]\},$$ and such that each configuration describes the entries at time zero and whether each clock ringing before time $t$ is accepted or not. We will denote a configuration on $\{0,1\}^{\Lambda}$ by a pair $(\eta_{0}, \mathscr{P})$, where the first coordinate contains the initial opinions of each site and the second retains the information of which clock ringings are kept. Moreover, each entry of $\eta_{0}$ will be distributed as a $\operatorname{Bernoulli}(p)$ random variable, where $p \in [0,1]$ is the initial density of the process, and each entry of $\mathscr{P}$ will have distribution $\operatorname{Bernoulli}\left(\frac{1}{k}\right)$.
Since, almost surely (on $\mathscr{P}^{k}$), one only needs to observe only a finite amount of sites in order to verify if $H(\lambda n,n)$ holds or not, the domain of $f_{n}$ is almost surely finite and hence this is a well-defined Boolean function.
The main reason we consider this construction is the following lemma.
\[lemma:variance\_decay\] For every integer $k \geq 2$ and $p \in (0,1)$, we have $$\operatorname{Var}\Big({\mathbb{E}}\left[f_{n}(\eta_{0}, \mathscr{P})|\mathscr{P}^{k}\right]\Big) \leq \frac{1}{k}.$$
The proof follows simply by considering a particular construction of $\mathscr{P}^{k}$ and $\mathscr{P}$: Let $\mathscr{P}_{1},\mathscr{P}_{2},\ldots,\mathscr{P}_{k}$ be independent copies of $\mathscr{P}$, and let $\kappa$ be chosen uniformly in $[k]\equiv\{1,\dots,k\}$. Observe that $$\begin{aligned}
\operatorname{Var}\left({\mathbb{E}}\left[f_{n}(\eta_{0}, \mathscr{P})|\mathscr{P}^{k}\right]\right) \,& \leq\, \operatorname{Var}\Big({\mathbb{E}}\left[f_{n}(\eta_{0}, \mathscr{P}_{\kappa})|(\mathscr{P}_{i})_{i=1}^{k}\right]\Big)\\
& =\, \operatorname{Var}\bigg(\frac{1}{k}\sum_{i=1}^{k}{\mathbb{E}}_{\eta_0}[f_{n}(\eta_{0}, \mathscr{P}_{i})]\bigg),
\end{aligned}$$ where the last variance above is with respect to the collection $(\mathscr{P}_{i})_{i=1}^{k}$. The result then follows from the independence of the $\mathscr{P}_{i}$.
[\[c:cir\]]{}
We can use the above lemma together with RSW theory to bound quenched probabilities in good events. Let $$\label{eq:cir}
\operatorname{Circ}(m) = \left\{ \begin{array}{c} \text{there exists an open circuit} \\ \text{contained in } B\left(0, 3m \right) \setminus B\left(0, m \right) \end{array}\right\},$$ and write $\operatorname{Circ}^{*}(m)$ for the equivalent event, but asking for the existence of a closed $*$-circuit. Notice that Equations (\[eq:rsw2\]) and (\[eq:rsw3\]) and the FKG inequality imply that there exists a positive constant ${c_{\textnormal{\tiny \ref{c:cir}}}}={c_{\textnormal{\tiny \ref{c:cir}}}}(t)>0$ such that $$\label{eq:RSW_cir}
\inf_{n}{\mathbb{P}}_{p,t}\left[\operatorname{Circ}(n)\right] \geq {c_{\textnormal{\tiny \ref{c:cir}}}},$$ if $p \geq p_{c}(t)$, and $$\label{eq:RSW_cir2}
\inf_{n}{\mathbb{P}}_{p,t}\left[\operatorname{Circ}^{*}(n)\right] \geq {c_{\textnormal{\tiny \ref{c:cir}}}},$$ for $p \leq p_{c}(t)$.
\[lemma:quenched\_cir\] For any fixed $t \geq 0$ and $k \geq 2$, $${\mathbb{P}}_{kt} \Big[{\mathbb{P}}_{p , \frac{1}{k}}\left[ \operatorname{Circ}(n)| \mathscr{P}^{k}\right] \leq \frac{{c_{\textnormal{\tiny \ref{c:cir}}}}}{2} \Big] \leq \frac{4}{{c_{\textnormal{\tiny \ref{c:cir}}}}^{2}k},$$ for all $n \geq 1$ and $p \geq p_{c}(t)$. An analogous estimate holds for $\operatorname{Circ}^{*}(n)$, for $p \leq p_{c}(t)$.
The same proof of Lemma \[lemma:variance\_decay\] can be used to the characteristic function of $\operatorname{Circ}(n)$. Combining this with Chebyshev inequality and implies $$\begin{split}
{\mathbb{P}}_{kt} & \Big[{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Circ}(n)| \mathscr{P}^{k}\right] \leq \frac{{c_{\textnormal{\tiny \ref{c:cir}}}}}{2} \Big]\\
& \qquad\leq
{\mathbb{P}}_{kt} \left[\Big|{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Circ}(n)| \mathscr{P}^{k}\right]-{\mathbb{P}}_{p, t}[\operatorname{Circ}(n)]\Big| \geq \frac{{c_{\textnormal{\tiny \ref{c:cir}}}}}{2}\right] \\
& \qquad \leq \frac{4}{{c_{\textnormal{\tiny \ref{c:cir}}}}^{2} k},
\end{split}$$ for all $k \geq 2$. To conclude the statement for $\operatorname{Circ}^{*}(n)$, one proceeds in the same way, but with instead of .
The result above provides quenched estimates for the existence of circuits and can be applied to deduce quenched one-arm estimates. For each $n$, let $\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})$ denote the event that there exists an open path connecting the boundary of the ball $B\left(0, n^{\sfrac{1}{4}}\right)$ to the boundary of the ball $B\left(0, n^{\sfrac{1}{2}}\right)$. This path can be chosen to be entirely contained inside $B\left(0, n^{\sfrac{1}{2}}\right) \setminus B\left(0, n^{\sfrac{1}{4}}\right)$. Denote also by $\operatorname{Arm}^{*}_{n}(\eta_{0}, \mathscr{P})$ the corresponding event, but asking for a closed $*$-path with the same properties.
\[prop:one\_arm\] There exists $\nu>0$ such that, for all $\gamma>0$, there exists $k_{0} \geq 2$ such that, for any $k \geq k_{0}$ and $p \leq p_{c}(t)$, if $n \geq n_{0}=n_{0}(k)$, then $${\mathbb{P}}_{kt}\Big[{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})|\mathscr{P}^{k}\right] \geq n^{-\nu} \Big] \leq n^{-\gamma}.$$ An analogous result holds for $\operatorname{Arm}^{*}_{n}(\eta_{0}, \mathscr{P})$ instead of $\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})$ if we assume $p \geq p_{c}(t)$.
The proof of the above Proposition relies on observing that $\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})$ holds if, and only if, there is no closed $*$-circuit inside $B\left(0, n^{\sfrac{1}{2}}\right) \setminus B\left(0, n^{\sfrac{1}{4}}\right)$. Since it is possible to find a logarithmic amount of disjoint and distant annuli in this set, we can repeatedly apply Lemma \[lemma:quenched\_cir\] to obtain that the probability of not having such a circuit in any of the annuli is small. A complete proof requires additional care to control dependencies between the disjoint annuli, and we postpone it to Section \[sec:one\_arm\].
To conclude this section, we present cone-of-light estimates for the denser collection of clock ringings. Given $\mathscr{P}^{k}$ we define the (past) cone of light $C_{k,t}^{\leftarrow}(x)$ to be the collection of vertices one needs to observe in order to determine $\eta_{s}(x)$, for all $s \in [0,t]$, varying over every possible pair $(\eta_{0}, \mathscr{P})$ of initial configurations and clock-ringing selections. We also define the future cone of light $C_{k,t}^{\rightarrow}(x)$ as the set of vertices that can be influenced by $x$ up to time $t$, that is, $$C_{k,t}^{\rightarrow}(x):=\{y\in{\mathbb{Z}}^2; x\in C_{k,t}^{\leftarrow}(y) \}.$$
\[prop:col\] Given $k \in {\mathbb{N}}$ and $t \geq 0$, if $n$ is large enough, $$\begin{split}
{\mathbb{P}}_{kt}\Big[C_{k,t}^{\leftarrow}(x) \cap \partial B(x,n) \neq \emptyset \Big] &\leq e^{-\frac{1}{8}n \log n},
\\
{\mathbb{P}}_{kt}\Big[ C_{k,t}^{\rightarrow}(x) \cap \partial B(x,n) \neq \emptyset \Big] &\leq e^{-\frac{1}{8}n \log n}.
\end{split}$$
Without loss of generality, we consider $x=0$ and prove the bound for $C_{k,t}^{\leftarrow}(x)$, the other bound following by analogous reasoning. Notice that, in order for $C_{k,t}^{\leftarrow}(0)$ to intersect $\partial B(0, n)$, it is necessary that there exists a path of length at least $n$ whose vertices’ associated Poisson clocks ring in decreasing order. That is, there must exist a (not necessarily simple) path $0=x_0,x_1,\dots,x_m\in\partial B(0,n)$, $m\geq n$, and a sequence of times $$t_0>t_1>\dots>t_m \text{ such that }t_j \text{ is a mark in }\mathcal{P}^{k}_{x_j}.$$ Besides, it is necessary that this collection of ordered clocks all ring before time $t$. Combining the fact that these clocks are i.i.d with distribution $\operatorname{Exponential}(k)$, the relation between Poisson and Exponential distributions, and union bounds, we obtain $$\begin{split}
\lefteqn{{\mathbb{P}}_{kt}\Big[C_{k,t}^{\leftarrow}(x) \cap \partial B(x,n) \neq \emptyset \Big] }\phantom{********}
\\
&\leq \sum_{m\geq n}{\mathbb{P}}_{kt} \left[ \begin{array}{c} \text{there exists a path of size } m \\ \text{starting at $0$ such that all clocks} \\ \text{ring before time $t$ in decreasing order} \end{array} \right] \\
& \leq \sum_{m\geq n} 4^m {\mathbb{P}}[\operatorname{Poisson}(kt) \geq m] \\
& = e^{-kt}\sum_{m \geq n} 4^m \sum_{j \geq m} \frac{(kt)^{j}}{j!}
\\
&\leq e^{-kt}\sum_{m \geq n} e^{kt} \frac{(4tk)^{m}}{m!} \\
& \leq e^{4tk} \frac{(4tk)^{n}}{\left(\frac{n}{2}\right)^{\frac{n}{2}}}\\
& \leq e^{-\frac{1}{8} n \log n},
\end{split}$$ if $n \geq \left(16 kt \right)^{8}$. This concludes the proof.
Influence and space-pivotality {#sec:influence}
==============================
Given a realization of $\mathscr{P}^{k}$, the quenched influence of a bit $x \in {\mathbb{Z}}^2$ or $(x,s) \in \{x\} \times \mathscr{P}^{k}_{x}$ is defined respectively as $$\operatorname{Inf}_{x}(f_{n}, \mathscr{P}^{k}) = {\mathbb{P}}_{p, \frac{1}{k}}\left[ f_{n}(\eta_{0}, \mathscr{P}) \neq f_{n}(\eta_{0}^{x}, \mathscr{P})|\mathscr{P}^{k} \right],$$ and $$\operatorname{Inf}_{(x,s)}(f_{n}, \mathscr{P}^{k}) = {\mathbb{P}}_{p, \frac{1}{k}}\left[ f_{n}(\eta_{0}, \mathscr{P}) \neq f_{n}(\eta_{0}, \mathscr{P}^{(x,s)})|\mathscr{P}^{k} \right],$$ where $\eta_{0}^{x}$ and $\mathscr{P}^{(x,s)}$ are obtained from $\eta_{0}$ and $\mathscr{P}$ by exchanging the entries at $x$ and $(x,s)$, respectively.
The crossing functions $f_{n}$ are monotone non-decreasing in the space variables $\eta_{0}$. Furthermore, the set $\bigcup_{y\in R_n}C_{k,t}^{\leftarrow}(y)$ comprised of vertices whose opinions at time $0$ can influence the output of $f_n(\eta_0,\mathcal{P})$ is almost surely finite. Classical arguments then show that Russo’s Formula applies to the derivative with respect to $p$ and one obtains $$\label{eq:russo}
\frac{\partial}{\partial p}{\mathbb{E}}_{p,\frac{1}{k}}\left[f_n(\eta_{0},\mathscr{P})| \mathscr{P}^{k}\right] = \sum_{x \in {\mathbb{Z}}^2} \operatorname{Inf}_{x}(f_n, \mathscr{P}^{k}).$$ Since $$\left|\frac{\partial}{\partial p}{\mathbb{E}}_{p,\frac{1}{k}}\left[f_n(\eta_{0},\mathscr{P})| \mathscr{P}^{k}\right]\right| \leq \bigg| \bigcup_{y\in R_n}C_{k,t}^{\leftarrow}(y) \bigg|,$$ as a direct consequence of the bounded convergence Theorem and Proposition \[prop:col\], it is possible to conclude $$\begin{split}
\frac{\partial}{\partial p}{\mathbb{E}}_{p,t}\left[f_n(\eta_{0},\mathscr{P})\right]
&=
{\mathbb{E}}_{kt}\left[\frac{\partial}{\partial p}{\mathbb{E}}_{p,\frac{1}{k}}\left[f_n(\eta_{0},\mathscr{P})| \mathscr{P}^{k}\right]\right]
\\
&= {\mathbb{E}}_{kt}\bigg[\sum_{x \in {\mathbb{Z}}^2} \operatorname{Inf}_{x}(f_n, \mathscr{P}^{k})\bigg].
\end{split}$$
[\[c:piv\]]{}
Regarding pivotality of clock ringings, we present a proposition that allows us to relate it to space-pivotality, provided we are in the event where the collection $\mathscr{P}^{k}$ is well behaved. Recall that $R_{n}=[1,n]^{2}$ and that $C_{k,t}^{\rightarrow}(x)$ denotes the future cone of light of the vertex $x$ associated to the collection of clocks $\mathscr{P}^{k}$. For $\epsilon >0$, consider the event $$\label{eq:large_cone}
E(\epsilon) = \left\{\begin{array}{c} \text{there exists } x \in R_{n} \text{ such that} \\ C_{k,t}^{\rightarrow}(x) \cap \partial B(x,\epsilon \log n) \neq \emptyset \end{array} \right\}.$$ Our next proposition relates time-pivotality to space-pivotality, provided we are in the event $E(\epsilon)^{c}$.
\[prop:piv\] Given $k \geq 2$ and $p \in (0,1)$, there exists a positive constant ${c_{\textnormal{\tiny \ref{c:piv}}}}={c_{\textnormal{\tiny \ref{c:piv}}}}(k,p)>0$ such that the following holds: for any $\mu>0$, there exists $\epsilon>0$ such that, for any bit associated to $(x,s) \in \{x\} \times \mathscr{P}_{x}^{k}$, $$\operatorname{Inf}_{(x,s)}(f_n, \mathscr{P}^{k}) \mathbf{1}_{E(\epsilon)^{c}}(\mathscr{P}^{k}) \leq {c_{\textnormal{\tiny \ref{c:piv}}}} n^{\mu} \sum_{y \, \in \, \partial B(x, 3\epsilon \log n)} \operatorname{Inf}_{y}(f_n, \mathscr{P}^{k}).$$ Furthermore, if $p$ varies in a compact subset of $(0,1)$, the value of $\epsilon$ and ${c_{\textnormal{\tiny \ref{c:piv}}}}$ can be chosen to be uniformly positive and bounded.
Observe first that $|\partial B(x,3\epsilon \log n)| \leq 24 \epsilon \log n +8$.
Fix a configuration $\mathscr{P}^{k}$ in $E(\epsilon)^{c}$ and assume that the presence of the clock ringing $(x,s)$ is pivotal. This can happen in two ways: first, it might be that adding the clock ringing allows us to obtain a crossing, while, with the removal of such clock ringing, no open crossings exist. The second possibility is the opposite: the addition of the clock ringing prevents the existence of a crossing, while its removal implies on the presence of a crossing. We will consider only the first case, since the second can be treated similarly.
When the clock ringing is present in the configuration (which we can assure by paying a finite multiplicative factor of $k$ in the probabilities), all possible crossings of the square $R_{n}$ intersect the cone of light $C_{k,t}^{\rightarrow}(x)$. In particular, since the clock ringing is pivotal and we are in the event $E(\epsilon)^{c}$, these crossings necessarily intersect the box $B(x, 3\epsilon \log n)$. Hence, if we declare all vertices in $\partial B(x,3\epsilon \log n)$ as closed at time zero, no crossing can be found at time $t$. *This is because closed nearest-neighbor cycles are stable in the majority dynamics*. Every vertex in a “monochromatic” cycle is surrounded by at least two neighbors of the same opinion, and therefore its opinion remains forever unchanged.
We now proceed by successively changing each entry in $(\eta_0(y))_{y\in\partial B(x,3\epsilon \log n)}$ which is one to zero. After all changes are performed, we obtain a configuration that has no crossing at time $t$. In particular, at some step, one of the entries of $(\eta_0(y))_{y\in\partial B(x,3\epsilon \log n)}$ is space-pivotal for the configuration. Since in order to perform each of these changes we need to pay a multiplicative factor in the probabilities that is bounded from above by $\left(p \wedge (1-p) \right)^{-1}$, we can estimate $$\begin{gathered}
\operatorname{Inf}_{(x,s)}(f_n, \mathscr{P}^{k}) \mathbf{1}_{E(\epsilon)^{c}}(\mathscr{P}^{k}) \\
\leq k \left(p \wedge (1-p) \right)^{-(24 \epsilon \log n +8)} \sum_{y \, \in \, \partial B(x,3\epsilon \log n)} \operatorname{Inf}_{y}(f_n, \mathscr{P}^{k}).\end{gathered}$$ The proof is completed by choosing $\epsilon>0$ small enough.
Low-revealment algorithms {#sec:alg}
=========================
In order to apply the OSSS inequality to the crossing functions, we need to develop an algorithm that determines the existence of such crossings in the quenched case, when the realization of $\mathscr{P}^{k}$ is fixed, and bound its revealment. This is the goal of this section, where we define an algorithm with the desired properties and provide bounds on its revealment.
The algorithm
-------------
We begin by presenting the algorithm we will study. This algorithm will be a simple exploration process: we start with a random vertical line contained in the rectangle and query the opinion at time $t$ of all vertices that are in the given line. When we have this realisation, we start exploring the components of open vertices that intersect this line. The existence of a crossing is equivalent to the existence of an open component that intersects this line and connects both sides of the rectangle.
For the rest of this subsection, we fix a realization $\mathscr{P}^{k}$ of the denser collection of clock ringings. Since we are working with a fixed realization of $\mathscr{P}^{k}$, the sets $C_{k,t}^{\leftarrow}(x)$ are not random and depend only on the realization of $\mathscr{P}^{k}$. Of course, when we reveal the realization of $\mathscr{P}_{y}$, for all $y \in C_{k,t}^{\leftarrow}(x)$, together with $\eta_{0}(y)$, we can determine $\eta_{s}(x)$, for all $s \in [0,t]$. In view of this, whenever we *query* the state of a vertex $x \in {\mathbb{Z}}^2$, we observe the initial opinions and selection of clock ringings for all vertices $y \in C_{k,t}^{\leftarrow}(x)$.
We are now in position to present the algorithm we will consider. Recall that $R_{n}=[1,n]^2$ and the notation $\Lambda = {\mathbb{Z}}^2 \cup \{(x,s): x \in {\mathbb{Z}}^2, \, s \in \mathscr{P}^{k}_{x} \cap [0,t]\}$.
**Input:** $\mathscr{P}^{k}$ and $(\eta_{0}, \mathscr{P}) \in \{0,1\}^{\Lambda}$. If there exists $x \in R_n$ and $y \in C_{k,t}^{\leftarrow}(x)$ such that ${\left\|x-y\right\|_{1}} \geq \log n$, query all vertices of $R_n$. Choose an integer point $x_0$ uniformly in the set $\left[\frac{n}{3}, \frac{2n}{3}\right]\cap {\mathbb{Z}}$. Query all vertices of $R_{n}$ whose first space-coordinate is $x_{0}$, and declare these vertices as explored. Proceed to query all vertices that are neighbors to an open explored vertex, and declare all these vertices explored. Repeat Step 5 until all open connected components inside $R_n$ that intersect $\{x=x_0\}$ are discovered. If there exists a connected open component inside $R_n$ that connects $\{x=1\}$ to $\{x=n\}$, return 1. Otherwise, return 0.
Notice that the algorithm above clearly determines the existence of open crossings, since any open crossing intersects any vertical line $\{x=x_{0}\} \cap R_{n}$. Furthermore, one can define an analogous algorithm that determines the existence of a closed vertical $*$-crossing of the box. When analyzing the revealment of the algorithm, we will consider Algorithm \[alg:crossing\] for $p \leq p_{c}(t)$ and its alternative formulation in terms of closed vertical $*$-crossings for $p > p_{c}(t)$.
We now proceed to bound the revealment of the algorithm above (the bound on the alternative version is obtained analogously). Observe first that the revealment depends only on the sites $y \in {\mathbb{Z}}^2$, since we reveal all clock ringings of a given site $y$ at once, together with its initial opinion. We can therefore talk about the revealment of a site $y \in {\mathbb{Z}}^2$. Given a vertex $y \in {\mathbb{Z}}^{2}$, there are three different possibilities that might lead us to reveal it. The first case that comes from Step 2 in the algorithm is when, for some $x \in R_{n}$, $C^{\leftarrow}_{k,t}(x)$ is large. Second, it might be the case that $y \in C_{k,t}^{\leftarrow}(z)$, for some site $z$ in the vertical line segment $\{x=x_{0}\}\cap R_{n}$. Finally, there is the case when $y \in C_{k,t}^{\leftarrow}(z)$ and some vertex adjacent to $z$ is connected to the selected vertical line segment by an open path.
In order to bound the revealment, we consider each of the three cases separately. The first and second cases can be easily controlled. As for the third case, we need finer estimates given by the one-arm estimates provided by Proposition \[prop:one\_arm\].
\[prop:revealment\] Let $\mathcal{A}$ denote the Algorithm \[alg:crossing\], and let $\mathcal{A}^*$ denote the analogous algorithm that looks for vertical closed $*$-crossings. Consider the revealments $$\delta_\mathcal{A}(\mathscr{P}^k):=\sup_{x\in R_n} \delta(\mathcal{A},x);\quad
\delta_{\mathcal{A}^*}(\mathscr{P}^k):=\sup_{x\in R_n} \delta({\mathcal{A}^*},x)$$ There exist $\nu>0$ and $k_{0}>0$ such that, for all $k \geq k_{0}$, there exists $n_{0}=n_{0}(k)$ such that, if $n \geq n_{0}$ and $p \leq p_{c}(t)$, then $$\label{eq:revealmentA}
{\mathbb{P}}_{kt}\left[\delta_\mathcal{A}(\mathscr{P}^k) > n^{-\nu} \right] \leq n^{-50},$$ and if $p > p_{c}(t)$, then $$\label{eq:revealmentA*}
{\mathbb{P}}_{kt}\left[\delta_{\mathcal{A}^{*}}(\mathscr{P}^k) > n^{-\nu} \right] \leq n^{-50}.$$
We will prove Equation , following by the same reasoning. We examine separately the revealment of bits. First, we consider the case when $C_{k,t}^{\leftarrow}(x)$ is large, for some $x \in [1,n]^{2}$. Define the event $$A = \left\{\begin{array}{c} \text{there exists $x \in R_{n}$ such that} \\ C_{k,t}^{\leftarrow}(x) \cap B(x,\log n) \neq \emptyset\end{array} \right\},$$ and observe that Lemma \[prop:col\] implies $${\mathbb{P}}_{kt}\left[A\right] \leq n^{2}e^{-\frac{1}{8} \log n \log \log n}.$$ Second, consider the event $$B = \left\{\begin{array}{c} \text{there exists $x \in R_{n}$ such that} \\ {\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Arm}_{\sqrt n}(x, \eta_{0}, \mathscr{P})|\mathscr{P}^{k}\right] \geq n^{-\nu'} \end{array} \right\},$$ where $\nu'$ is obtained from Proposition \[prop:one\_arm\] by choosing $\gamma=100$, and observe that $${\mathbb{P}}_{kt}\left[B\right] \leq n^{2}n^{-100} = n^{-98}.$$
We now bound the revealment on the event $A^{c} \cap B^{c}$. In this case, we split the revealment in two cases. Either the distance from the site $x$ to the random selected line is smaller then $2\sqrt{n}$, which is unlikely due to the randomness in selecting the line, or $x \in C_{k,t}^{\leftarrow}(y)$, for some $y$ such that a neighbor of it is connected to the random line by an open path. Since we are in the event $A^{c}$, we may assume that $y$ is close to $x$ and hence that, in the last case, $\operatorname{Arm}_{\sqrt n}(x,\eta_{0},\mathscr{P})$ holds. This leads to the bound $$\begin{split}
\lefteqn{\delta_\mathcal{A}(\mathscr{P}^k) \mathbf{1}_{A^{c} \cap B^{c}}(\mathscr{P}^{k}) }\phantom{******}
\\
&\leq \left(\max_{x \in R_{n}} {\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Arm}_{\sqrt n}(x, \eta_{0}, \mathscr{P})|\mathscr{P}^{k}\right]\right)\mathbf{1}_{A^{c} \cap B^{c}}(\mathscr{P}^{k}) + \frac{4\sqrt{n}}{\frac{n}{3}} \\
& \leq n^{-\nu'} + \frac{12}{n^{\sfrac{1}{2}}} \leq n^{-\nu},
\end{split}$$ if $\nu$ is small enough and $n$ large enough. In particular, we obtain from Proposition \[prop:one\_arm\], by choosing $k$ and $n$ sufficiently large, $$\begin{split}
{\mathbb{P}}_{kt}\left[\delta_\mathcal{A}(\mathscr{P}^k) > n^{-\nu} \right]& \leq {\mathbb{P}}_{kt}\left[A \cup B\right] \\
& \leq n^{2}e^{-\frac{1}{8} \log n \log \log n}+n^{-98} \leq n^{-50},
\end{split}$$ for sufficiently large $n$, concluding the proof.
Sharp thresholds {#sec:thresholds}
================
In this section,we combine the results from the previous sections to conclude the proof of Theorem\[t:sharp\_thresholds\].
Given $\alpha>0$, consider the interval $$I_{\alpha}(n)=\left\{p \in \left[\frac{1}{10},\frac{9}{10}\right]: {\mathbb{P}}_{p,t}[H(n,n)] \in [\alpha, 1-\alpha]\right\}.$$ Our goal is to prove that the length of this interval is bounded by $cn^{-\gamma}$, for some positive constants $c$ and $\gamma$. This is enough to conclude the proof, once we know that $p_{c}(t) \in I_{\alpha}(n)$, for all $n \in {\mathbb{N}}$, provided $\alpha$ is small enough.
We begin by introducing the event where the process is well behaved inside the box $R_{n}$. Recall the definition of the event $E(\epsilon)$ in and consider $$A(\epsilon)=E(\epsilon) \cup \left\{|\mathscr{P}^{k}_{x}| \geq \log n, \text{ for some } x \in [-n,2n]^{2} \right\},$$ and notice that, as a consequence of Proposition \[prop:col\] and standard bounds on the tail of the Poisson distribution, we obtain $${\mathbb{P}}_{kt}[A(\epsilon)] \leq 10 n^{2}\exp\left\{-\frac{\epsilon}{8} \log n \log \left(\epsilon \log n\right)\right\}$$ if $n$ is large enough, depending on $k$ and $t$.
Given $p \in I_{\alpha}(n)$, consider the events $$B=\left\{ {\mathbb{P}}_{p,\frac{1}{k}}[f(\eta_{0}, \mathscr{P})=1|\mathscr{P}^{k}] \notin \left(\frac{\alpha}{2}, 1-\frac{\alpha}{2}\right)\right\}$$ and $$C = \left\{
\begin{array}{c}
\delta_\mathcal{A}(\mathscr{P}^k) \geq n^{-\nu} \text{ for all } p \in I_{\alpha}(n)\cap(0,p_c(t)];
\\
\delta_\mathcal{A^*}(\mathscr{P}^k) \geq n^{-\nu} \text{ for all } p \in I_{\alpha}(n)\cap(p_c(t),1).
\end{array}
\right\},$$ where $\mathcal{A}$ denotes Algorithm \[alg:crossing\], $\mathcal{A}^*$ denotes the analogous algorithm that looks for vertical closed $*$-crossings, and $\nu>0$ is given by Proposition \[prop:revealment\]. Here we observe that the revealment of our algorithm (or its analogue) is monotone in $p$, since it is related to connection probabilities. This can be used to bound the probability of the above event, by considering only the case $p=p_{c}(t)$. We claim that $${\mathbb{P}}_{kt}\left[B \cup C\right] \leq \frac{4}{\alpha^{2}k}+2n^{-50}.$$ The above bound follows partly from Proposition \[prop:revealment\] and partly from an analogous reasoning to the proof of Lemma \[lemma:quenched\_cir\]. If we take $k$ large enough, and $n$ large depending on $k$ and $t$, we have $$P_{kt}[A(\epsilon) \cup B \cup C] \leq \frac{1}{2}.$$
We now use OSSS inequality in the quenched setting. We assume that $p\leq p_c(t)$, the other case following analogously. If $$\mathscr{P}^{k} \in \left(A(\epsilon) \cup B \cup C \right)^{c},$$ we can use Proposition \[prop:piv\] and Russo’s Formula with $\mu=\frac{\nu}{2}$ to estimate $$\nonumber
\begin{split}
\lefteqn{\operatorname{Var}\Big( f_{n}(\eta_{0}, \mathscr{P}) \Big|\mathscr{P}^{k}\Big)}
\phantom{**}
\\
&\leq \sum_{x} \delta_\mathcal{A}(\mathscr{P}^k)\Bigg(\operatorname{Inf}_{x}\left(f_{n}, \mathscr{P}^{k}\right)+\sum_{s \in \mathscr{P}^{k}_{x}}\operatorname{Inf}_{(x,s)}\left(f_{n}, \mathscr{P}^{k}\right)\Bigg) \\
&\leq \sum_{x} \delta_\mathcal{A}(\mathscr{P}^k)\Bigg(\operatorname{Inf}_{x}\left(f_{n}, \mathscr{P}^{k}\right)+{c_{\textnormal{\tiny \ref{c:piv}}}} n^{\frac{\nu}{2}}|\mathscr{P}^{k}_{x}|\sum_{y \in\partial B(x, 3\epsilon \log n)}\operatorname{Inf}_{y}\left(f_{n}, \mathscr{P}^{k}\right)\Bigg) \\
&\leq n^{-\frac{\nu}{2}}\sum_{x} \operatorname{Inf}_{x}\left(f_{n}, \mathscr{P}^{k}\right) \Big(1+{c_{\textnormal{\tiny \ref{c:piv}}}} \log n \Big|\partial B(x, 3\epsilon \log n)\Big|\Big) \\
&\leq 25 {c_{\textnormal{\tiny \ref{c:piv}}}} n^{-\frac{\nu}{2}}\log^{2} n\sum_{x} \operatorname{Inf}_{x}\left(f_{n}, \mathscr{P}^{k}\right) \\
&\leq 25 {c_{\textnormal{\tiny \ref{c:piv}}}} n^{-\frac{\nu}{2}}\log^{2} n \frac{\partial}{\partial p} {\mathbb{P}}_{p, \frac{1}{k}}\left[f_{n}(\eta_{0}, \mathscr{P})=1|\mathscr{P}^{k}\right] \\
&\leq n^{-\frac{\nu}{3}}\frac{\partial}{\partial p} {\mathbb{P}}_{p, \frac{1}{k}}\left[f_{n}(\eta_{0}, \mathscr{P})=1|\mathscr{P}^{k}\right],
\end{split}$$ if $n$ is large enough.
In particular, for $p \in I_{\alpha}(n)$, using the fact that $f_{n}(\eta_{0}, \mathscr{P})$ is a $\operatorname{Bernoulli}$ variable, $$\begin{split}
\frac{\partial}{\partial p} {\mathbb{P}}_{p, t}\left[H(n,n)\right] & = \frac{\partial}{\partial p} {\mathbb{E}}_{kt} \left[ {\mathbb{P}}_{p, \frac{1}{k}}\left[f_{n}(\eta_{0}, \mathscr{P})=1|\mathscr{P}^{k}\right]\right] \\
& \geq {\mathbb{E}}_{kt}\left[ \frac{\partial}{\partial p} {\mathbb{P}}_{p, \frac{1}{k}}\left[f_{n}(\eta_{0}, \mathscr{P})=1|\mathscr{P}^{k}\right] \mathbf{1}_{\left(A(\epsilon) \cup B \cup C \right)^{c}}\left(\mathscr{P}^{k}\right) \right] \\
& \geq n^{\frac{\nu}{3}}{\mathbb{E}}_{kt}\left[ \operatorname{Var}\Big( f_{n}(\eta_{0}, \mathscr{P}) \Big|\mathscr{P}^{k}\Big) \mathbf{1}_{\left(A(\epsilon) \cup B \cup C \right)^{c}}\left(\mathscr{P}^{k}\right) \right] \\
& \geq n^{\frac{\nu}{3}}\frac{\alpha^{2}}{4} {\mathbb{P}}_{p,t}\left[\left(A(\epsilon) \cup B \cup C \right)^{c}\right] \geq n^{\frac{\nu}{3}}\frac{\alpha^{2}}{8}.
\end{split}$$
This implies $$1 \geq \int_{I_{\alpha}(n)}\frac{\partial}{\partial p} {\mathbb{P}}_{p, t}\left[H(n,n)\right] \, {{\rm d}}p \geq n^{\frac{\nu}{3}}\frac{\alpha^{2}}{8}|I_{\alpha}(n)|,$$ which gives the bound $$|I_{\alpha}(n)| \leq \frac{8}{\alpha^{2}}n^{-\frac{\nu}{3}},$$ and concludes the proof.
One-arm estimates {#sec:one_arm}
=================
The goal of this section is to conclude the proof of the quenched one-arm estimates stated as Proposition \[prop:one\_arm\].
We will work on the event where all cones of light are well behaved. For each $n$, define $$E_n:=\left\{C_{k,t}^{\leftarrow}(x) \cap \partial B\left(x, n^{\sfrac{1}{4}}\right) = \emptyset\text{ for every }x \in B\left(0, n^{\sfrac{1}{2}}\right) \right\}.$$ Proposition \[prop:col\] implies, for sufficiently large $n$, $$\label{eq:one_arm_1}
{\mathbb{P}}_{kt}[E_{n}^{c}] \leq 16n e^{-\frac{1}{32}n^{\sfrac{1}{4}} \log n}.$$
Consider the collection of indices $$J=\left\{j \in 2{\mathbb{N}}: n^{\frac{1}{4}} \leq 4^{j}n^{\frac{1}{4}} \leq n^{\frac{1}{2}}\right\},$$ and, for $j \in J$, denote by $A_{j}$ the set of vertices $$A_{j}=B\left(0, 2 \cdot 3^{j+1} n^{\sfrac{1}{4}}\right) \setminus B\left(0, 2\cdot 3^{j-1} n^{\sfrac{1}{4}}\right)$$ and recall the definition of $\operatorname{Circ}^{*}(m)$ immediately after . Notice that, on $E_{n}$, $\operatorname{Circ}^{*}\left(3^{j}n^{\sfrac{1}{4}}\right)$ depends on $(\eta_{0}(x), \mathscr{P}_{x})$ only for $x \in A_{j}$. In particular, on $E_{n}$, we can estimate $$\label{eq:one_arm_2}
\begin{split}
\lefteqn{{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})|\mathscr{P}^{k} \right]\mathbf{1}_{E_{n}}(\mathscr{P}^{k}) }\phantom{******}
\\
&\leq {\mathbb{P}}_{p, \frac{1}{k}}\left[\left.\bigcap_{j \in J}\operatorname{Circ}^{*}\left(3^{j}n^{\sfrac{1}{4}}\right)^{c}\right|\mathscr{P}^{k} \right]\mathbf{1}_{E_{n}}(\mathscr{P}^{k}) \\
& = \prod_{j \in J}{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Circ}^{*}\left(3^{j}n^{\sfrac{1}{4}}\right)^{c}|\mathscr{P}^{k} \right]\mathbf{1}_{E_{n}}(\mathscr{P}^{k}).
\end{split}$$ Consider now the event $$D_{j}=\left\{{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Circ}^{*}\left(3^{j}n^{\sfrac{1}{4}}\right)|\mathscr{P}^{k} \right] \geq \frac{{c_{\textnormal{\tiny \ref{c:cir}}}}}{2} \right\},$$ and denote by $D$ the event where $D_{j}$ holds for at least half of the indices $j \in J$.
From , we obtain $$\begin{split}
{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})|\mathscr{P}^{k} \right] & \mathbf{1}_{E_{n}\cap D}(\mathscr{P}^{k}) \\
& \leq \prod_{j \in J}{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Circ}^{*}\left(3^{j}n^{\sfrac{1}{4}}\right)^{c}|\mathscr{P}^{k} \right]\mathbf{1}_{E_{n} \cap D}(\mathscr{P}^{k}) \\
& \leq \left(1-\frac{{c_{\textnormal{\tiny \ref{c:cir}}}}}{2}\right)^{\frac{|J|}{2}} \leq n^{-\nu},
\end{split}$$ for some $\nu$ small enough, since $|J|$ is of order $\log n$. This implies that $$\label{eq:one_arm_2.5}
{\mathbb{P}}_{kt}\Big[{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})|\mathscr{P}^{k}\right] \geq n^{-\nu} \Big] \leq {\mathbb{P}}_{kt}\left[E_{n}^{c} \cup D^{c}\right],$$ so it remains to bound the right hand side probability above.
We begin by estimating $$\label{eq:one_arm_3}
{\mathbb{P}}_{kt}\left[E_{n}^{c} \cup D^{c}\right] \leq {\mathbb{P}}_{kt}\left[E_{n}^{c}\right]+ \sup_{I} {\mathbb{P}}_{kt}\left[E_{n} \cap \bigcap_{j \in I} D_{j}^{c}\right],$$ where the supremum is taken over all subsets of $J$ with at least $\frac{|J|}{2}$ indices. From Lemma \[lemma:quenched\_cir\], we obtain $${\mathbb{P}}_{kt}[D_{j}^{c}] \leq \frac{4}{{c_{\textnormal{\tiny \ref{c:cir}}}}^{2} k},$$ provided $k$ is taken large enough. The same type of independence argument used in can be used here to bound $$\label{eq:one_arm_4}
\begin{split}
{\mathbb{P}}_{kt}\left[E_{n} \cap \bigcap_{j \in I} D_{j}^{c}\right] & \leq \prod_{j \in I} \left({\mathbb{P}}_{kt}\left[D_{j}^{c}\right]+{\mathbb{P}}_{kt}\left[E_{n}^{c}\right] \right) \\
& \leq \prod_{j \in I} \left(\frac{4}{{c_{\textnormal{\tiny \ref{c:cir}}}}^{2} k}+16ne^{-\frac{1}{32}n^{\sfrac{1}{4}} \log n}\right) \\
& \leq \left(\frac{4}{{c_{\textnormal{\tiny \ref{c:cir}}}}^{2} k}+16ne^{-\frac{1}{32}n^{\sfrac{1}{4}} \log n}\right)^{\frac{|J|}{2}},
\end{split}$$ whenever $I \geq \frac{|J|}{2}$ and $k$ is large enough.
Combining Equations , , , and yields $${\mathbb{P}}_{kt}\Big[{\mathbb{P}}_{p, \frac{1}{k}}\left[\operatorname{Arm}_{\sqrt n}(\eta_{0}, \mathscr{P})|\mathscr{P}^{k}\right] \geq n^{-\nu} \Big] \leq n^{-\gamma},$$ for all $n$ large enough, by further increasing the value of $k$ if necessary. This concludes the proof of the result.
Stretched-exponential decay of the one-arm event probability {#sec:decay}
============================================================
[\[c:1armexpgen1\]]{} [\[c:1armexpgen2\]]{} In this section we prove will prove Theorem \[t:exp\_decay\] using the results so far obtained. We will in fact prove a more general result, based on the proof of Theorem $3.1$ of [@pt], which, together with a decoupling inequality and Theorem \[t:sharp\_thresholds\], will imply the desired rate of decay.
We first develop some notation needed before we state te result. Given $L\in{\mathbb{R}}_+$ and $x\in{\mathbb{Z}}^d$, we define the subsets $$\begin{split}
\label{eq:annulusdef}
C_x(L):=[0,L)^d + x,\quad\quad D_x(L):=[-L,2L)^d\cap{\mathbb{Z}}^d + x.
\end{split}$$ In accordance with , we denote by $\{A \longleftrightarrow B\}$ for the event where there exists a nearest-neighbor open path starting at $A$ and ending at $B$. For $x\in{\mathbb{Z}}^d$, $L\in{\mathbb{R}}_+$, we define the annulus-crossing event $$A_x(L) := \{C_x(L){\longleftrightarrow} {\mathbb{Z}}^d\setminus D_x(L)\}.$$
\[p:1armdecaygen\] Let $\tilde {\mathbb{P}}$ denote a probability distribution over $\{0,1\}^{{\mathbb{Z}}^d}$, invariant under translations of ${\mathbb{Z}}^d$. Assume that $$\label{eq:1armexpgen2}
\liminf_{L\to \infty} \tilde {\mathbb{P}}\left[A_0(L) \right]<\frac{1}{d^{2}\cdot 7^d},$$ and that there exists a positive constant ${c_{\textnormal{\tiny \ref{c:1armexpgen1}}}}>0$ such that, for every $L,R\in{\mathbb{R}}_+$ and every $x,y\in {\mathbb{Z}}^d$ with $\|x-y\|_\infty \geq 3L+R$, we have $$\label{eq:1armexpgen3}
\left| \tilde {\mathbb{P}}\left[ A_x(L) \cap A_y(L) \right] - \tilde{\mathbb{P}}\left[ A_x(L) \right] \tilde{\mathbb{P}}\left[ A_y(L) \right] \right| \leq L^{2d} \exp\left\{ -f(R) \right\},$$ where $f:{\mathbb{R}}_+\to{\mathbb{R}}_+$ is a non-decreasing function such that $$\begin{split}
\label{eq:1armexpgenf}
\liminf_{R\to \infty} \frac{f(R)}{R\log R}\geq {c_{\textnormal{\tiny \ref{c:1armexpgen1}}}}.
\end{split}$$ Then, there exists a positive constant ${c_{\textnormal{\tiny \ref{c:1armexpgen2}}}}>0$ such that, for $n\in{\mathbb{N}}$, $$\label{eq:1armexpgen4}
\tilde {\mathbb{P}}\left[ \{0\} \longleftrightarrow \partial B(0,n) \right] \leq {c_{\textnormal{\tiny \ref{c:1armexpgen2}}}}^{-1}\exp\left\{
-{c_{\textnormal{\tiny \ref{c:1armexpgen2}}}}\frac{n}{\log n}
\right\}.$$
The proof is based on the proof of Theorem $3.1$ of [@pt], specifically, the proof of Equation $(3.5)$. Since in our case no sprinkling argument is needed in order to obtain a decoupling inequality, the argument here will be simpler.
The proof consists in a multi-scale renormalization argument. We start by inductively defining the sequence of scales $(L_k)_{k\in{\mathbb{N}}}$. Given $L_1\in{\mathbb{R}}_+$, which will chosen to be large, we let, for $k\in{\mathbb{N}}$, $$\begin{split}
\label{eq:Lkdef}
L_{k+1}=2\left(1+\frac{1}{(k+5)^{3/2}}\right)L_{k}.
\end{split}$$ We then have $$\begin{split}
\label{eq:Lkbound1}
2L_k=L_{k+1}-\frac{2L_k}{(k+5)^{3/2}}\leq L_{k+1}-\frac{2^k L_1}{(k+5)^{3/2}}\leq \lfloor L_{k+1}\rfloor-\frac{2^{k-1} L_1}{(k+5)^{3/2}},
\end{split}$$ for $L_1$ sufficiently large. Furthermore, $$\begin{split}
\label{eq:Lkbound2}
\log L_{k}&=(k-1)\log 2 +\log L_1 + \sum_{j=1}^{k-1}\log\left(1+\frac{1}{(j+5)^{3/2}}\right)
\\
&\leq (k-1)\log 2 +\log L_1 + \sum_{j=1}^{k-1}\frac{1}{(j+5)^{3/2}},
\end{split}$$ so that, denoting by $\zeta(3/2)$ the Riemann zeta function evaluated at $3/2$, we have, $$\begin{split}
\label{eq:Lkbound3}
L_1 2^{k-1} \leq L_k \leq e^{\zeta(3/2)} L_1 2^{k-1} .
\end{split}$$
Our goal is to use an induction argument in order to bound the probability $$\begin{split}
\label{eq:pkdef}
p_k:=\tilde {\mathbb{P}}\left[A_0(L_k)\right].
\end{split}$$ Recalling the sets defined in , note that, for $k\geq 1$, there exist two collection of points $\{x_i^k\}_{i=1}^{3d}$ and $\{y_j^k\}_{j=1}^{2d\cdot 7^{d-1}}$ such that $$\begin{split}
\label{eq:xiyiproperty}
&C_0(L_{k+1})=\cup_{i=1}^{3d}C_{x_i}(L_{k}),
\\
&\left(\cup_{j=1}^{2d\cdot 7^{d-1}}C_{y_j}(L_{k})\right)\cap D_0(L_{k+1})=\varnothing,
\\
&\partial({\mathbb{Z}}^d\setminus D_0(L_{k+1}))\subset\cup_{j=1}^{2d\cdot 7^{d-1}}C_{y_j}(L_{k}).
\end{split}$$
The above follows from Equation and the fact that, for every $k\geq 1$, $$\begin{split}
\nonumber
2\left(1+\frac{1}{(k+5)^{3/2}}\right)< 3,\quad\text{ and }\quad 6\left(1+\frac{1}{(k+5)^{3/2}}\right)< 7.
\end{split}$$ Properties then imply $$\begin{split}
\label{eq:akinduc}
A_0(L_{k+1})\subset \bigcup_{\substack{i \leq 3 d \\ j \leq 2d\cdot 7^{d-1}}} A_{x_i^k}(L_{k})\cap A_{y_j^k}(L_{k}),
\end{split}$$ see Figure \[f:multiscale\]. Furthermore, Equation implies that the distance between $D_{x_i^k}(L_k)$ and $D_{y_j^k}(L_k)$ is greater than $2(k+5)^{-3/2}L_k$ uniformly in $i$ and $j$. Property then implies, together with the above equation and the translation invariance of $\tilde {\mathbb{P}}$, for $k\geq 1$, $$\begin{split}
\label{eq:pkinduc}
p_{k+1}\leq d^{2} \cdot 7^d \left(
p_k^2 + L_k^{2d}\exp\left\{ - f\left( \frac{2L_k}{(k+5)^{3/2}} \right) \right\}
\right).
\end{split}$$ We now use the above equation and an induction argument to finish the proof of the Proposition. Consider $h_1\in{\mathbb{R}}_+$ such that $$\label{eq:h_1def}
\liminf_{L\to \infty} \tilde {\mathbb{P}}\left[A_0(L) \right] <e^{-h_1} < \frac{1}{d^{2}\cdot 7^d},$$ and, using , choose $L_1$ large enough so that, for every $k\geq 1$, $$\begin{split}
\label{eq:pkinduc2}
\lefteqn{d^{2}\cdot 7^d(e^{\zeta(3/2)} L_1 2^{k-1} )^{2d}\exp\left\{ - f\left( \frac{2^kL_1}{(k+5)^{3/2}} \right) +h_1 +\frac{2^{k+1}}{k+1} \right\}}\phantom{**************************}
\\
&<1-d^{2}\cdot 7^d e^{-h_1}.
\end{split}$$ Now, possibly taking $L_1$ even larger, we use to find $h_2\in(0,1)$ sufficiently small so that $$\label{eq:h_2def}
p_1\leq \exp\left\{ -h_1 -2 h_2\right\}.$$ We will prove by induction that $$\begin{split}
\label{eq:pkinduc3}
p_{k}\leq
\exp\left\{ -h_1 -\frac{2^k}{k} h_2\right\}.
\end{split}$$ Observe that the case $k=1$ is already verified in . Assume now that the above holds for a given $k$. We must show that $$\begin{split}
\label{eq:pkinduc4}
\frac{p_{k+1}}{\exp\left\{ -h_1 -2^{k+1}{(k+1)}^{-1} h_2\right\}}\leq 1 .
\end{split}$$ But note that $(\ref{eq:Lkbound3},\ref{eq:h_1def},\ref{eq:pkinduc},\ref{eq:pkinduc2},\ref{eq:pkinduc3},\ref{eq:pkinduc4})$, together with the fact that $h_2<1$, imply $$\begin{split}
\label{eq:pkinduc5}
\lefteqn{\frac{p_{k+1}}{\exp\left\{ -h_1 -2^{k+1}{(k+1)}^{-1} h_2\right\}}}\phantom{**}
\\
&\leq
d^{2}\cdot 7^d \left(
p_k^2 + L_k^{2d}\exp\left\{ - f\left( \frac{2L_k}{(k+5)^{3/2}} \right)
\right\}
\right)
\\
&\phantom{***************}\times\exp\left\{ h_1 +2^{k+1}{(k+1)}^{-1} h_2\right\}
\\
&\leq 1-d^{2}\cdot 7^d e^{-h_1} + d^{2}\cdot 7^d e^{-h_1}
\\
&\leq 1,
\end{split}$$ proving for general $k\geq 1$. Note then that, for $n\in[2L_k,2L_{k+1}]$, we have $$\label{eq:Lkton}
[\{0\} \longleftrightarrow \partial B(0,n)] \subseteq A_0(L_k),$$ and therefore $$\label{eq:Lkton2}
\tilde {\mathbb{P}}[\{0\} \longleftrightarrow \partial B(0,n) ] \leq \tilde {\mathbb{P}}[A_0(L_k)] \leq \exp\left\{ -h_1 -\frac{2^k}{k} h_2\right\}.$$ Equation then implies the result, for a suitably chosen constant ${c_{\textnormal{\tiny \ref{c:1armexpgen2}}}}$.
[\[c:cor\_decay\]]{}
We need a stronger decoupling inequality in order to apply the above result to our context. Proposition \[prop:col\] allows us to obtain the following strengthening of the bound :
\[prop:decoup\] For every $t \geq 0$, there exists a positive constant ${c_{\textnormal{\tiny \ref{c:cor_decay}}}}={c_{\textnormal{\tiny \ref{c:cor_decay}}}}(t)>0$ such that the following holds: for any pair of events $A$ and $B$ with respective supports inside the balls $D_x(L)$ and $D_y(L)$, with ${\left\|x-y\right\|_{\infty}} \geq 3L+R$, and $p \in [0,1]$, $$\label{eq:strong_correlation_decay}
\Big|{\mathbb{P}}_{p,t}[A \cap B] - {\mathbb{P}}_{p,t}[A]{\mathbb{P}}_{p,t}[B] \Big| \leq {c_{\textnormal{\tiny \ref{c:cor_decay}}}}^{-1}L^{2}e^{-{c_{\textnormal{\tiny \ref{c:cor_decay}}}}R \log R}.$$
If $C_{1,t}^{\leftarrow}(y) \cap \partial B\left(y,\frac{R}{2}\right) = \emptyset$ for all $y \in D_x(L) \cup D_y(L)$, then the occurrence of $A$ and $B$ are determined by disjoint (and hence independent) parts of the graphical construction. In particular, we can bound the LHS of by $${\mathbb{P}}_{p,t}\left[\begin{array}{c} C_{1,t}^{\leftarrow}(y) \cap \partial B\left(y,\frac{R}{2}\right) \neq \emptyset \\ \text{for all } y \in D_x(L) \cup D_y(L) \end{array}\right] \leq 18 L^{2}e^{\frac{1}{16}R \log \frac{R}{2}},$$ where the last bound above is a consequence of Proposition \[prop:col\] for large values of $R$. Choosing the constant in to cover the remaining cases concludes the proof.
For $p<p_c(t)$, Theorem \[t:sharp\_thresholds\] implies $$\limsup_{n}{\mathbb{P}}_{p,t}\left[ \operatorname{Circ}(n)\right] \leq 4 \limsup_{n} {\mathbb{P}}_{p,t}[H(n,3n)] = 0.$$ Proposition \[prop:decoup\] and basic properties of the majority dynamic imply that ${\mathbb{P}}_{p,t}$ satisfies the hypotheses of Proposition \[p:1armdecaygen\], which then implies the desired result.
Further models {#sec:further_models}
==============
As we already mentioned, our technique can be applied to other particle systems as long as some basic properties can be verified. In particular, we require equivalent formulations of Lemmas \[lemma:variance\_decay\] and of Propositions \[prop:one\_arm\], \[prop:col\] and \[prop:piv\]. Here, we extend our results for the voter model in the two-dimensional lattice ${\mathbb{Z}}^{2}$.
The voter model is very similar to majority dynamics, in the sense that it differs just in the way each vertex selects its new opinion once its clock ringings. In this case, the new opinion is selected randomly among the neighbors’ opinions.
Once again, for each fixed time $t$, there exists a non-trivial critical parameter $p^{VM}_{c}(t) \in (0,1)$ for the existence of percolation at time $t$. We remark that the non-triviality of $p_{c}^{VM}(t)$ follows by a standard renormalisation argument, since, by applying Proposition \[prop:col\], we can derive decoupling estimates that are uniform in the value of $p \in [0,1]$.
The usual graphical construction of the voter model (see Remark \[remark:voter\_model\]) can be modified exactly as we did in Section \[sec:construction\], and Lemma \[lemma:variance\_decay\] and Proposition \[prop:one\_arm\] can be obtained from general results, as in the case of majority dynamics. Furthermore, we can apply the same proof to obtain Proposition \[prop:col\]. The most delicate part is in establishing a relation between time-pivotality and space-pivotality.
Let us now describe how one approaches Proposition \[prop:piv\] here. In this case, we use the fact that the opinion of each vertex at any time $s \geq 0$ is a copy of one of the initial opinions that are contained in the cone of light. Not only that, but changing this opinion at time zero implies that the opinion changes at time $s$. This last observation allows us to conclude that time-pivotality implies space-pivotality for some vertex in the cone of light. From this, we derive the bound $$\operatorname{Inf}_{(x,s)}(f_n, \mathscr{P}^{k}) \leq c \sum_{y \, \in \, C^{k}_{t}(x)} \operatorname{Inf}_{y}(f_n, \mathscr{P}^{k}),$$ for some positive constant $c>0$. This yields a version of Proposition \[prop:piv\] that can be used to conclude Theorem \[t:sharp\_thresholds\] for the voter model.
[^1]: Following the usual notation in percolation theory, we refer to sites with opinion zero as closed and to sites with opinion one as open.
[^2]: We say $\eta \preceq \xi$ if $\eta(x) \leq \xi(x)$, for all $x \in {\mathbb{Z}}^{2}$. An event $A$ is increasing with respect to this partial ordering if $\eta \in A$ and $\eta \preceq \xi$ imply $\xi \in A$.
[^3]: A $*$-path in ${\mathbb{Z}}^{2}$ is a path $x_{1}, x_{2}, \dots, x_{n}$ of vertices in ${\mathbb{Z}}^{2}$ such that ${\left\|x_{i+1}-x_{i}\right\|_{\infty}} = 1$, for all $i=1,2, \dots, n-1$. In other words, it is a path that is allowed to cross diagonals on the lattice ${\mathbb{Z}}^{2}$.
|
---
address: 'FZK, Institut für Kernphysik, Postfach 3640, 76021 Karlsruhe, GERMANY'
author:
- 'T. PIEROG (H. J. DRESCHER, F.M. LIU, S. OSTAPCHENKO AND K. WERNER)'
title: HIGH PT SUPPRESSION WITHOUT JET QUENCHING IN AU+AU COLLISIONS IN NEXUS
---
Introduction \[intro\]
======================
High [$P_{T}$ ]{}suppression in RHIC Au+Au data is one of the most exciting results of this experiment [@d'Enterria:2002bw; @Adcox:2002pe; @Adler:2002xw; @Adams:2003kv]. Together with back-to-back high [$P_{T}$ ]{}hadron correlations [@Adler:2002tq] and new results for d+Au collisions [@Adams:2003im], this suppression appears to be an effect of final state interactions. Indeed, jet quenching [@Wang:xy] in association with Cronin effect [@Cronin:zm] and nuclear shadowing [@Close:1989ca] , can describe the data reasonably well [@Adams:2003kv]. For a qualitative description it is enough, but for a precise quantitative description of high [$P_{T}$ ]{}and all the other observables in ultra-relativistic heavy ion collisions, we have to take care about other effects. Those can lead to significant difference in the energy-loss parameter for instance.
Then it is important to have a proper description of the initial state of this kind of interaction. The most sophisticated approach to high energy hadronic interactions is the so-called Gribov-Regge theory [@Gribov:fc]. This is an effective field theory, which allows multiple interactions to happen “in parallel”, with phenomenological objects called *Pomerons* representing elementary interactions [@Baker:cv].
We recently presented a new approach [@Werner:ze; @Drescher:1999js; @Drescher:2000ha; @Pierog:2002], for hadronic interactions and the initial stage of nuclear collisions, which is able to solve several of the problems of the Gribov-Regge theory, such as a consistent approach to include both soft and hard processes, and the energy conservation both for cross section and particle production calculations. In both cases, energy is properly shared between the different interactions happening in parallel. This is the most important new aspect of our approach, which we consider a first necessary step to construct a consistent model for high energy nuclear scattering. And this leads to interesting results.
[[neXus]{} ]{}\[nexus\]
=======================
We will discuss the basic features of the new approach in a qualitative fashion. It is an effective theory based on effective elementary interactions. Multiple interactions happen in parallel in proton-proton and nucleus-nucleus collisions. An elementary interaction is referred to as a Pomeron, and can be either elastic (uncut Pomeron) or inelastic (cut Pomeron). The spectators of each proton form remnants, see Fig. \[allin1\]a.
Since a Pomeron is finally identified with two strings, the Pomeron aspect (to obtain probabilities) and the string aspect (to obtain particles) are treated in a completely consistent way. *In both cases energy sharing is considered in a rigorous way, and in both cases all Pomerons are identical.* To share the energy of the nucleons, we made a strong and simple assumption that the partition function does not depend of the number of elementary interactions. We will discuss this important point in the followings.
This theory provides also a consistent treatment for hard and soft processes: each Pomeron can be expressed in terms of contributions of different types, soft, hard and semihard ones, cf. Fig. \[allin1\]b. A hard Pomeron stands for a hard interaction between valence quarks of initial hadrons. A semihard one stands for an interaction between sea quarks or gluons but in which a perturbative process involves in the middle. The high [$P_{T}$ ]{} particles come from this middle part of the semi-hard (or hard) Pomeron. No perturbative process occures at all in soft Pomerons.
A Pomeron is an elementary interaction. But those Pomerons may interact with each other at high energy [@Baker:cv; @kai86], then they give another type of interaction called *enhanced diagram*. There are many types of enhanced diagrams depending on the number of Pomerons for each vertices and on the number of vertices. In our model, effective first order of triple-Pomeron vertices (Y diagrams see Fig. \[allin1\]c) are enough to cure unitarity problem which occure at high energy without this kind of diagram [@Pierog:2002]. Indeed, Y-type diagrams are screening corrections which are negative contributions to the cross-section. The inelastic contributions (cut enhanced diagrams on Fig. \[allin1\]d) of this diagrams contribute to the increase of the fluctuations in particle production, and in case of nuclear collision, this type of diagramms correspond to a kind of shadowing (modification of the structure function of the nucleons inside a nucleus).
The model [[neXus]{} ]{}is designed to reproduce proton-proton interactions. The initial stage of nuclear interaction is obtained by a sophisticated extension of the formalism with some approximations for the numeric solution. As a consequence, there is neither Cronin Effect nor partonic or hadronic final state interaction as jet quenching, hydrodynamic or rescattering. Comparison with the data should then be done carrefully.
Results \[resu\]
================
Since [[neXus]{} ]{}results for AA are just an extrapolation of pp, but with a proper energy sharing scheme, we can compare the high [$P_{T}$ ]{}production of Au+Au collisions at 200 Gev in [[neXus]{} ]{}with the data, to see what is the effect of the energy-momentum conservation. As there is no hadronic final state interactions, the results for a $P_{T}<3-4$ GeV should not be regarded as a realistic one. To quantifythe medium effect , we use the nuclear modification factor $R_{AA}$ defined in eq. \[raa\] or the ratio $R_{CP}$ of the central yield to the peripheral yield defined in eq. \[rcp\], where in both cases $<N_{coll}>$ is the mean number of binary collisions for a given centrality region in the Glauber model (which does not take into account the energy-momentum conservation).
$$R_{AA}=\frac{\left(1/N_{AA}^{evt}\right)d^{2}N_{AA}/dy dp_{T}}{<N_{coll}>/\sigma _{pp}^{ine}\, \times \, d^{2}\sigma _{pp}/dy /dp_{T}}\label{raa}$$
$$R_{CP}=\frac{<N_{coll}^{peri}>\, \times \, d^{2}N_{cent}/dy dp_{T}}{<N_{coll}^{cent}>\, \times \, d^{2}N_{peri}/dy /dp_{T}}\label{rcp}$$
In fig. \[results\], the experimental ratio $R_{AA}$ for $\pi^0$ is compared to [[neXus]{} ]{}predictions for the 0-10% central events (top) and 70-80% peripheral events (bottom left-hand side), together with the $R_{CP}$ for charged hadrons ($(h^++h^-)/2$) where central means 0-5% central events and peripheral means 40-50% (top) or 60-80% peripheral events (bottom right-hand size).
We can see that in all cases, an energy-momentum conservation scheme at the level of the cross-section calculation (which fixes the number of binary collisions) can lead to a suppression of the high [$P_{T}$ ]{}produced particles which is compatible with the data.
Discussion
==========
Of course, we are not claiming that this kind of process explains the high [$P_{T}$ ]{}suppression. The recent d+Au data do not show any particular suppression for high [$P_{T}$ ]{}, while in [[neXus]{} ]{}a suppression appear. A Cronin effect would explain a part of the difference, but surely not all. In fact in our scheme, the suppression due to the energy conservation mechanism is maximal because of our simple choice for the partition function. It has to be seen as a maximum limit of this effect. A better understanding of this part of our formalism, which can be done partially with pp data, should lead to a weaker suppression. Together with a strong Cronin effect, it could appear like a weak Cronin effect in d+Au reaction. For heavy ion reactions, this can change the value of the needed energy-loss. Thus we want to emphasize that for a quantitative description of heavy ion collision data, and for a real comprehension of the complex processes involved in this kind of reaction, it is important to take care about energy-momentum conservation. It can play a non-negligeable role to fix the proportion of all the other processes like the Cronin effect or the jet quenching.
S. O. was supported by the German Ministry for Education and Research (BMBF, grant 05 CU1VK1/9) and F.M. L. is supported by the German Alexander von Humboldt Foundation.
References {#references .unnumbered}
==========
[99]{}
D. d’Enterria \[PHENIX Collaboration\], arXiv:hep-ex/0209051. K. Adcox [*et al.*]{} \[PHENIX Collaboration\], Phys. Lett. B [**561**]{} (2003) 82 \[arXiv:nucl-ex/0207009\]. C. Adler [*et al.*]{}, Phys. Rev. Lett. [**89**]{} (2002) 202301 \[arXiv:nucl-ex/0206011\]. J. Adams [*et al.*]{} \[STAR Collaboration\], arXiv:nucl-ex/0305015. C. Adler [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett. [**90**]{} (2003) 082302 \[arXiv:nucl-ex/0210033\]. J. Adams, arXiv:nucl-ex/0306024. X. N. Wang and M. Gyulassy, Phys. Rev. Lett. [**68**]{} (1992) 1480. J. W. Cronin, H. J. Frisch, M. J. Shochet, J. P. Boymond, R. Mermod, P. A. Piroue and R. L. Sumner, Phys. Rev. D [**11**]{} (1975) 3105. F. E. Close, J. w. Qiu and R. G. Roberts, Phys. Rev. D [**40**]{} (1989) 2820.
V. N. Gribov, Sov. Phys. JETP [**26**]{}, 414 (1968) \[Zh. Eksp. Teor. Fiz. [**53**]{}, 654 (1968)\]. M. Baker and K. A. Ter-Martirosian, Phys. Rept. [**28**]{}, 1 (1976).
K. Werner, H. J. Drescher, E. Furler, M. Hladik and S. Ostapchenko, [*Given at 5th International Conference on Relativistic Aspects of Nuclear Physics, Rio de Janeiro, Brazil, 27-29 Aug 1997*]{}.
H. J. Drescher, M. Hladik, S. Ostapchenko and K. Werner, Nucl. Phys. A [**661**]{}, 604 (1999) \[arXiv:hep-ph/9906428\]. H. J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog and K. Werner, Phys. Rept. [**350**]{}, 93 (2001) \[arXiv:hep-ph/0007198\]. T. Pierog, Thèse de doctorat de l’universitée de Nantes (2002).
A. B. Kaidalov, L. A. Pomomarev, and K. A. Ter-Martirosian, Sov. J. Nucl.Phys [**44**]{}, 468 (1986).
D. d’Enterria \[PHENIX Collaboration\], QM2002 Presentation.
J. Klay \[STAR Collaboration\], QM2002 Presentation.
|
---
author:
- 'Yu-Ting Chen[^1]'
title: '**Approximating Laplace transforms of meeting times for some symmetric Markov chains**'
---
\[mthm\][Corollary]{} \[section\] \[thm\][Theorem]{} \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Definition]{} \[thm\][Problem]{} \[thm\][Discussion]{} \[thm\][Key Observation/Assumption]{} \[thm\][Assumption]{} \[thm\][Research remark]{} \[thm\][Remark]{} \[thm\][Example]{} \[thm\][Quest]{} \[thm\][Comment]{} \[thm\][Notation]{}
Introduction {#sec:intro}
============
In this paper we investigate distributions of (first) meeting times by two independent continuous-time Markov chains. Meeting times arise as fundamental objects in the study of finite stochastic spatial models in mathematical biology, including, for example, voter models [@C:CRW; @CCC:WF], Kimura’s stepping-stone model and its generalizations [@K:SSM; @S:SSM; @CD:SSM; @CDZ:SSM2], and evolutionary games on graphs [@ATTN; @C:BC]. In contrast to the frequently visited subject in most of these studies on approximating meeting time distributions under large time scales by exponential variables (cf. Keilson [@K:AE] and Aldous and Fill [@AF:RMC Section 3.5.4] for the classical results), our main focus in this paper is on the detailed meeting time distributions *up to* large times through their Laplace transforms. This direction concerns in particular the biological application of approximating certain critical values for the emergence of cooperation under some evolutionary games. For this, a brief discussion of the background and pointers to the literature will be given in Section \[sec:EGT\].
Unless otherwise mentioned, we consider throughout this paper irreducible (conservative) Markov kernels $Q$ on finite state spaces $E$ with $\#E=N> 8$ and subject to the symmetry condition: $$\begin{aligned}
\label{ass:sym}
Q(x,y)=Q(y,x)\quad \forall\;x,y\in E\end{aligned}$$ as well as the zero-trace condition: $$\begin{aligned}
\label{ass:trace}
{{\sf tr}}(Q)=0.\end{aligned}$$ For such a Markov kernel $Q$, we denote by $M_{x,y}$ the (first) meeting time of two independent continuous-time rate-$1$ $Q$-chains started at $x$ and $y$, and hence, $M_{x,x}\equiv 0$. In addition, we write $H_{x,y}$ for the first hitting time of $y$ by a rate-$1$ $Q$-chain $X^x$ started at $x$.
One of the conventional treatments of meeting times is based on the validation of the following exact distributional reduction: $$\begin{aligned}
\label{eq:MT}
M_{x,y}\stackrel{(\rm d)}{=}H_{x,y}/2.\end{aligned}$$ Observe that (\[eq:MT\]) means a significant dimensionality reduction in invoking both the number of Markov chains and the size of sets to be hit, and so the driving Markov kernel can be made stand out through the associated Green functions for a detailed study of meeting time distributions by $$\begin{aligned}
\label{eq:HTGF}
\begin{split}
&{{\mathbb E}}\big[e^{-\lambda H_{x,y}/2}\big]\\
&\hspace{.5cm}=\left.{{\mathbb E}}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{y\}}(X^x_t)dt\right]\right/{{\mathbb E}}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{y\}}(X^y_t)dt\right].
\end{split}\end{aligned}$$ Although (\[eq:MT\]) can lead to this important relation and is immediate for some special Markov kernels (the classical examples are random walks on groups), obtaining the reduction (\[eq:MT\]) in general poses some problems. For example, while we restrict our attention to Markov kernels subject to the symmetry condition (\[ass:sym\]) in this paper, it is readily pointed out in Aldous [@A:MTMC p.188] by the counterexample of random walks on star graphs that (\[eq:MT\]) can fail when the symmetry condition is disobeyed. In addition, it seems that before the present paper, the reduction (\[eq:MT\]) can only be obtained by few arguments which require various strong links among all pointwise transition probabilities, and thus, are insufficient for a further study of whether some “non-global” properties of Markov chains similar to those allowing for the exact equality (\[eq:MT\]) can give an approximating reduction.
Our goal in this paper is to determine whether the distributional reduction (\[eq:MT\]) is approximately correct in terms of Laplace transforms, for a class of “weakly inhomogeneous” Markov kernels. (See also Section \[sec:EGT\] for more on our motivation.) Informally, we consider kernels $Q$ on large, but finite, state spaces $E$ under which pointwise return probabilities are almost identical up to some large numbers of steps, and the presence of few “wild” points which disagree significantly with the majority on such almost homogeneity of finite-range return probabilities is allowed (see our discussion after Theorem \[thmm:main-1\] for the example of large random regular graphs). To formalize this, we choose the largest sets of points which almost agree in $s$-step return probabilities under $Q$, for $s\in \Bbb Z_+$, from $$\begin{aligned}
\label{def:RQ}
\mathcal R_Q^\gamma(x,s)\triangleq \{y\in E;|Q^s(x,x)-Q^s(y,y)|\leq \gamma\},\quad x\in E,\;\gamma\in [0,1],\end{aligned}$$ and consider, for every $s_0\in \Bbb N$, the magnitude of $$\begin{aligned}
\label{context}
\frac{\min_{x\in E}[N-\# \mathcal R^{\gamma}_{Q}(x,s)]}{N}+\gamma\end{aligned}$$ for all $s\in \{0,\cdots, s_0\}$. We stress that *no* connections between such kernels and those which allow for the conventional distributional reduction of meeting times will be assumed a priori, and the novelty of this paper as we see it lies in the treatment of this missing link in order to understand how Laplace transforms of some meeting times and those of the associated hitting times of points (as in (\[eq:MT\])) can still be related. See Section \[sec:method\] for further discussions.
To facilitate the present purpose to study meeting times on large sets, let us introduce some meeting times started from pairs of random points independent of the underlying $Q$-Markov chains. The first natural candidate is the pair of random points $(U,V_\infty)$ whose coordinates are independent and uniformly distributed on the state space $E$. See, for example, Durrett [@D:Epi] for a discussion of the corresponding meeting time $M_{U,V_\infty}$, and Chen, Choi and Cox [@CCC:WF] for its role in diffusion approximation of voter models defined on large sets. On the other hand, if we consider $V_s$ for $s\in \Bbb N$ being distributed according to the $s$-step distribution $Q^s(U,\,\cdot\,)$ of the discrete-time $Q$-chain started at $U$ when we condition on $U$, then the meeting times $M_{U,V_s}$ have a very different nature. For example, if we consider random walk on a large graph with relatively smaller degree, then informally, $M_{U,V_s}$ for a small $s\in \Bbb N$ is the first meeting time of two Markov chains started with points close to each other (with respect to graph distance) such that the local structure of the underlying graph becomes an essential part of its distribution, whereas $M_{U,V_\infty}$ is for starting points far away from each other and should depend much less on the local structure of the graph (see Oliveira [@O:MFC] and Chen et al. [@CCC:WF Theorem 4.1] for some rigorous treatments).
For $s\in \Bbb N\cup \{\infty\}$, we will call $M_{U,V_s}$ the [**${\boldsymbol}s$-th order meeting time**]{}, but put emphasis on the special case $s=1$ throughout this paper. There are several reasons why we choose to make this restriction. First, the distributions of $M_{U,V_1}$ and $M_{U,V_\infty}$ always determine each other through an explicit integral equation (cf. Aldous and Fill [@AF:RMC Proposition 3.21] or Section 3 of Chen et al. [@CCC:WF]). More importantly, since the mass of $M_{U,V_\infty}$ always escapes to infinity in the limit of a large state space (see [@CCC:WF Corollary 3.4]) and our interest here is for meeting times under the natural time scale, the viable object of study need be the distribution of $M_{U,V_1}$. Besides, the tail distribution of $M_{U,V_{s_0}}$ for finite $s_0$ can be approximated by that of $M_{U,V_1}$ if it holds that (\[context\]) is small for $s $ up to $s_0$ (Section \[sec:MT\]). In accordance with this emphasis on the first-order meeting time and to save notation, we write $V$ for $V_1$ from now on.
The main result of this paper is given by Theorem \[thmm:main-1\] below. For the statement of the theorem, we define $$\begin{aligned}
\label{def:deltaQ}
\Delta_Q^{\gamma}\triangleq \min_{s\in \Bbb N}\left\{\frac{\min_{x\in E}[N-\#\mathcal R^\gamma_Q(x,s)]}{N}+\gamma,\;\;\frac{{{\sf tr}}(|Q|^s)}{N}\right\}\end{aligned}$$ (recall (\[context\])), where $f(Q)$ is defined by functional calculus of $Q$ for every complex function $f$ defined on an open set in $\Bbb C$ containing the line segment $ [-1,1]$ and ${{\sf tr}}(\,\cdot\,)$ denotes trace, so that ${{\sf tr}}\big(f(Q)\big)=\sum f(q)$ for $q$ ranging over all eigenvalues of $Q$.
\[thmm:main-1\] Let $Q$ be an irreducible finite Markov kernel subject to the symmetry condition (\[ass:sym\]) and the zero-trace condition (\[ass:trace\]). Suppose that $Q$ is defined on a finite set $E$ of size $N=\#E> 8$. Let ${{\varepsilon}}\in (0,1]$ and $\lambda\in ({{\varepsilon}},\infty)$ such that $(\lambda-{{\varepsilon}})N-\lambda{{\varepsilon}}>0$, and let $m\in \Bbb N$ and $\gamma\in [0,1]$ be auxiliary parameters left to be chosen. We have $$\begin{aligned}
\label{crit:main0}
&\left|{{\mathbb E}}[e^{-\lambda M_{U,V}}]-\left.{{\mathbb E}}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{U})dt\right]\right/{{\mathbb E}}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{V})dt\right]\right|\notag\\
\begin{split}
\leq &4\Big(1+\frac{\lambda}{N}\Big)^{-mN}+\left[\Big(1+\frac{\lambda}{N}\Big)\Big(1-\frac{{{\varepsilon}}}{N}\Big)\right]^{-(mN+1)}\times \frac{C_{{\varepsilon}}\lambda(N-{{\varepsilon}})}{(\lambda-{{\varepsilon}})N-\lambda{{\varepsilon}}}\\
&\hspace{.5cm}+80\Delta^\gamma_Q\times \left(1+\frac{6}{N}\right)^{mN},
\end{split}\end{aligned}$$ where $C_{{\varepsilon}}$ is a constant which diverges as ${{\varepsilon}}\searrow 0$ and can be chosen to be $$\begin{aligned}
C_{{\varepsilon}}&=\frac{1}{\pi}\left(2{{\varepsilon}}^{-1}+9+\pi+\frac{32\sqrt{2}}{(1-\cos 1)^{1/2}}\right)\times \frac{4+{{\varepsilon}}}{{{\varepsilon}}},\label{def:Cvep}\end{aligned}$$ and $\Delta_Q^{\gamma}$ is defined as in (\[def:deltaQ\]).
We will discuss the method of proof for Theorem \[thmm:main-1\] in Section \[sec:method\], and start its proof in Section \[sec:tpc\], with the conclusion set in Section \[sec:lap-2\].
Observe that the ratio of Green functions on the left-hand side of (\[crit:main0\]) bears a strong resemblance to the Laplace transform ${{\mathbb E}}[e^{-\lambda H_{U,V}/2}]$ if we recall (\[eq:HTGF\]). For this reason, we regard (\[crit:main0\]) as an approximation in a weak sense for the classical distributional reduction (\[eq:MT\]). In addition, this ratio can be written as $$\begin{aligned}
\begin{split}\label{ratio}
&\left.{{\mathbb E}}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{U})dt\right]\right/{{\mathbb E}}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{V})dt\right]\\
&\hspace{3.5cm}=\left.{{\sf tr}}\left(\frac{Q}{\lambda+2(1-Q)}\right)\right/{{\sf tr}}\left(\frac{1}{\lambda+2(1-Q)}\right),
\end{split}\end{aligned}$$ which depends only on the eigenvalue distribution of $Q$ (see the proof of Lemma \[lem:MUV-exp\] (iii)).
Let us discuss informally some contexts for which Theorem \[thmm:main-1\] may allow for “good” approximation of the Laplace transform of $M_{U,V}$ by the associated ratio of Green functions. The key, of course, is to identify kernels $Q$ such that the associated parameters $\Delta^\gamma_Q$ in (\[def:deltaQ\]) are small. We consider a kernel $Q$ such that the quantity in (\[context\]) is small for $s$ up to some $s_0$, so the task of bounding $\Delta^\gamma_Q$ falls upon controlling ${{\sf tr}}(|Q|^{s_0})/N$. In this direction, if the eigenvalues of $Q$ are sufficiently bounded away from $\pm 1$ except for a small fraction of them, then the normalized trace term ${{\sf tr}}(|Q|^{s_0})/N$ is small if the magnitude of $s_0$ can be chosen relatively larger. (See also the role of spectral gaps of Markov chains in obtaining almost exponentiality of hitting times in Aldous and Fill [@AF:RMC Section 3.5.4].)
In fact, if $Q$ is the random walk kernel on a (simple connected) $k$-regular graph on $N$ vertices with $k\geq 2$ and $s_0$ is even, we can alternatively bound ${{\sf tr}}(|Q|^{s_0})/N$ by means of partial geometry of the graph and the analogous spectral functional on the infinite $k$-regular tree. (Here and in what follows, see Biggs [@B:AGT] for terminology in algebraic graph theory.) Indeed, we have $$\frac{{{\sf tr}}(|Q|^{s_0})}{N}\leq\frac{N-N(s_0)}{N}+ \int_{-1}^1 q^{s_0}f^{(k)}(q)dq,$$ where $N(s_0)$ is the number of vertices $x$ in the regular graph such that the subgraph induced by the vertices at graph distance $\leq s_0/2$ from $x$ defines a tree (cf. McKay [@M:EED Lemma 2.2]), and $$\begin{aligned}
\label{KM:SM}
f^{(k)}(q)=\frac{\sqrt{4(k-1)-(kq)^2}}{2\pi (1-q^2)},\quad q\in \left[-\frac{2\sqrt{k-1}}{k},\frac{2\sqrt{k-1}}{k}\,\right],\end{aligned}$$ is the Kesten-McKay density [@K:SRG; @M:EED] for the spectral measure of the random walk kernel on the infinite $k$-regular tree and satisfies $\lim_{s\to\infty}\int q^{s}f^{(k)}(q)dq=0$. For approximation of the ratio of Green functions as in (\[crit:main0\]) by explicit values using the spectral representation (\[ratio\]) on regular graphs with large girth, see McKay [@M:EED Theorem 4.4].
\[thmm:main-1-0\] Let $\{Q^{(n)}\}$ be a sequence of irreducible finite Markov kernels subject to the symmetry condition (\[ass:sym\]) and the zero-trace condition (\[ass:trace\]). Assume that these kernels are defined on growing state spaces. If $$\begin{aligned}
\label{crit:main}
\inf_{\gamma\in [0,1]}\limsup_{n\to\infty}
\Delta_{Q^{(n)}}^{\gamma}=0,\end{aligned}$$ then $$\begin{aligned}
\begin{split}\label{conv:main}
&\lim_{n\to\infty}\Bigg\{{{\mathbb E}}^{(n)}[e^{-\lambda M_{U,V}}]-\left.{{\mathbb E}}^{(n)}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{V\}}(X^U_t)dt\right]\right/\\
&\hspace{2.5cm}{{\mathbb E}}^{(n)}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{V\}}(X^{V}_t)dt\right]\Bigg\}=0,\quad \forall\;\lambda\in (0,\infty),
\end{split}\end{aligned}$$ where ${{\mathbb E}}^{(n)}$ denotes expectation with respect to $Q^{(n)}$.
Fix $({{\varepsilon}},\lambda)$ such that ${{\varepsilon}}\in (0,1]$ and $0<{{\varepsilon}}<\lambda<\infty$. Under (\[crit:main\]), we pass limit for the analogues of (\[crit:main0\]) for $Q^{(n)}$ first along the state space size. For the resulting error bounds, we take infimum over $\gamma\in [0,1]$ and then pass limit along $m\to\infty$.
By Corollary \[thmm:main-1-0\], we obtain the following particular result for random regular graphs (see, e.g., Wormald [@W:MRG] for a survey of random regular graphs). A further discussion of this result will be given in Section \[sec:EGT\].
\[cor:main2\] Fix an integer $k\geq 2$. With respect to the sequence of random walk kernels $\{Q^{(n)}\}$ associated with an i.i.d. sequence of growing (uniform) random $k$-regular graphs, it holds that $$\begin{aligned}
\label{eq:RGconv}
\lim_{n\to\infty}{{\mathbb E}}^{(n)}[e^{-\lambda M_{U,V}}]={{\mathbb E}}^{(\infty)}[e^{-\lambda H_{x,y}/2};H_{x,y}<\infty]\quad \forall\;\lambda\in (0,\infty)\end{aligned}$$ almost surely with respect to the randomness that the graphs are chosen. Here $H_{x,y}$ under ${{\mathbb E}}^{(\infty)}$ denotes the first hitting time of $y$ by a rate-$1$ random walk on the (transient) infinite $k$-regular tree started at $x$ for any vertices $x$ and $y$ which are adjacent to each other.
The proof of Corollary \[cor:main2\] is relegated to Section \[sec:cor-main2\].
We recall that with probability tending to $1$ in the limit of a large graph size, random regular graphs with a fixed degree do not have homogeneity in return probabilities in all numbers of steps (cf. Wormald [@W:MRG Section 2.3]). Nonetheless (\[crit:main\]) holds almost surely with respect to the randomness that the graphs are chosen, as will be explained in Section \[sec:cor-main2\].
Application to evolutionary game theory {#sec:EGT}
---------------------------------------
The present study was in part motivated by Allen, Traulsen, Tarnita and Nowak [@ATTN], and let us discuss briefly the role of the meeting times under consideration in their studies. The paper [@ATTN] investigates the influence of mutation on forming cliques of strategy types under some evolutionary games. The underlying game players are arranged according to finite vertex-transitive graphs, and their strategy types are updated indefinitely in some Markovian manners. Analogues of the key quantities on some infinite vertex-transitive graphs are also considered in [@ATTN]. See Nowak [@N:ED] for an authoritative introduction to evolutionary games, Section 1 and 2 in the supplementary information of Nowak, Tarnita and Wilson [@NTW:EU] for an analysis of critical values under general evolutionary games, and the recent survey paper by Allen and Nowak [@AN:GG] for critical values under games closely related to those in [@ATTN] on weighted vertex-transitive graphs.
Allen et al. [@ATTN] obtains explicit results on certain critical values for the emergence of cooperation between game players, when special finite vertex-transitive graphs are taken into consideration. This is based on the result that on general finite vertex-transitive graphs, the so-called identity-by-descent probabilities for two game players occupying adjacent vertices [@ATTN Section 2.3] are the only quantities left to be solved for the critical values [@ATTN Appendix C]. These identity-by-descent probabilities are equivalent to the Laplace transforms ${{\mathbb E}}[e^{-\lambda M_{x,y}}]$ of the first meeting times $M_{x,y}$ by independent continuous-time (rate-$1$) random walks started at adjacent vertices $x$ and $y$ (cf. the algebraic equations in [@ATTN Appendix B]), where $\lambda$ are strictly positive parameters depending only on the underlying mutation rates. More generally, it is not difficult to check that on regular graphs, the Laplace transforms ${{\mathbb E}}[e^{-\lambda M_{U,V}}]$ determine the critical values for cooperation under the games in [@ATTN] (see [@ATTN Appendix A to C] and Corollary \[cor:Mtime1\]).
While the same evolutionary games considered in Allen et al. [@ATTN] in the absence of mutation allow for rather complete explicit results (see e.g. [@OHLN; @TDW; @CDP:VMP; @C:BC]), we are unable to expect so for the critical values in general when mutation enters because Laplace transforms of meeting times are now involved. Nonetheless, as one special case, Allen et al. [@ATTN Section 2.2 and 3.2] circumvented the difficulty of handling the identity-by-descent probabilities equivalent to ${{\mathbb E}}[e^{-\lambda M_{U,V}}]$ on general finite regular graphs, by considering large random regular graphs and turning to the analogous identity-by-descent probabilities on infinite regular trees instead (or equivalently, ${{\mathbb E}}^{(\infty)}[e^{-\lambda M_{x,y}}]$ by the notation of Corollary \[cor:main2\]). See also Szabó and Fáth [@SF:EGG] for related discussions on the application of random regular graphs for evolutionary games on graphs.
Corollary \[thmm:main-1-0\] gives a justification of the practice in Allen et al. [@ATTN] discussed above if we assume a fixed mutation rate and take limit of the identity-by-descent probabilities on growing random regular graphs with a constant degree. In fact, by formalizing the discussion on approximating the Laplace transforms of first-order meeting times after Theorem \[thmm:main-1\], a quantitative approximation of the critical values on finite regular graphs with bounded degree and large girth without passing to the limit is now also possible.
Method of proof {#sec:method}
---------------
We consider a linear analysis for first meeting time distributions. Our point of view starts with the duality between coalescing Markov chains and voter models, and we apply the linear coupling of voter models defined by products of i.i.d. random linear operators acting on initial configurations, which makes some computations natural and straightforward. This coupling of voter models and a study of similarly defined interacting particle systems can be found in Liggett [@L:IPS Chapter IX] (see e.g. [@FF:RAP; @AL:AP; @FY:LS] for recent studies).
Let us recall coalescing Markov chains, voter models and a particular result of their duality. A system of coalescing Markov chains consists of continuous-time rate-$1$ $Q$-Markov chains started at all points of the state space $E$. They move independently before meeting and together afterward. The associated voter model is a continuous-time rate-$N$ Markov chain $(\xi_t)$ taking values in the space of configurations of two possible “opinions”, say $1$ and $0$, at points of the state space so that, at each epoch time, the opinion of a randomly chosen point, say $U'$, is changed to the opinion of another chosen according to $Q(U',\,\cdot\,)$. We consider the following particular consequence of the duality between the voter model and the coalescing Markov chains: $$\begin{aligned}
\label{eq:MTVM}
&{{\mathbb P}}(M_{x,y}>t)=\frac{1}{u(1-u)}\Big(u-{{\mathbb E}}_{\beta_u}\left[\xi_t(x)\xi_t(y)\right]\Big),\quad u\in (0,1),\end{aligned}$$ for $x,y\in E$. Here, under ${{\mathbb E}}_{\xi}$, the voter model starts at configuration $\xi\in \{1,0\}^E$, and ${{\mathbb E}}_{\beta_u}$ means that the initial configuration is randomized according to the random configuration $\beta_u$ which is defined by placing i.i.d. Bernoulli random variables $\beta_u(x)$ with mean $u\in (0,1)$ at all points $x$. See Liggett [@L:IPS] for a general account of voter models and coalescing Markov chains.
Let us discuss how the duality method is applied for the study of meeting time distributions. It should be clear from the above description that the voter model can be identified as a Markov chain on configurations which is updated sequentially by a family of i.i.d. random linear operators $(T_n)$ independent of the initial configuration. We may assume that the voter model is obtained by time-changing a Markov sequence $(\xi_n)$ by an independent rate-$N$ Poisson process, where $(\xi_n)$ is defined recursively by $$\begin{aligned}
\label{coupling}
\xi_n=T_n\xi_{n-1},\quad n\in \Bbb N.\end{aligned}$$ Here, configurations are regarded as column vectors with coordinates indexed by points of $E$ (with respect to a fixed order). (A detailed description of the distribution of these random linear operators is given in the proof of Proposition \[prop:conj\].) Write $J$ for the square matrix with entries identically equal to $1/N$, and let us use the standard bra-ket notation (see Paratharasy [@P:QSC]) so that $\langle \beta_u|$ and $|\beta_u\rangle$ mean the row vector and the column vector corresponding to the Bernoulli random configuration $\beta_u$. Then by the independence of the underlying objects defining the voter model started at $\beta_u$, the discrete-time analogues of the two-point correlations ${{\mathbb E}}_{\beta_u}[\xi_t(U)\xi_t(V_\infty)]$, which determine the distribution of the $\infty$-order meeting time $M_{U,V_\infty}$ through the duality equation (\[eq:MTVM\]), can be written as: $$\begin{aligned}
\label{eq:coupling}
{{\mathbb E}}_{\beta_u}[\xi_n(U)\xi_n(V_\infty)]=&
\frac{1}{N}{{\mathbb E}}[\langle \beta_u|T_1^*\cdots T_n^*JT_n\cdots T_1|\beta_u\rangle]\notag\notag\\
=&\frac{1}{N}{{\mathbb E}}[\langle \beta_u|L^{n}(J)|\beta_u\rangle],\quad n\in \Bbb N,\end{aligned}$$ where $L$ is a (deterministic) linear operator defined by the expected matrix “congruence” $$\begin{aligned}
\label{eq:T*T}
L(C)\triangleq {{\mathbb E}}[T^*CT]\end{aligned}$$ on the space of square matrices indexed by points of the state space. Here in (\[eq:T\*T\]), $T$ has the same distribution as a random linear operator $T_n$ in the linear coupling (\[coupling\]) of the voter model, and $*$ denotes transpose. See also Liggett [@L:IPS Section IX.3] for characterizations of two-point correlations for general linear interacting particle systems by linear evolution equations.
It is true that in general, the key matrices $L^n(J)$ in (\[eq:coupling\]) are complicated, and hence, so are the associated discrete-time two-point correlations there. When, however, the return probabilities $Q^s(x,x)$ of the discrete-time $Q$-chain do not depend on points $x$ for every $s\in \Bbb Z_+$ or, simply, $Q$ is [**walk-regular**]{} (see Godsil and McKay [@GM:WRG]), we find a detailed description for $L^n(J)$: $$\begin{aligned}
\label{eq:Ln-intro}
L^n(J)=J+\sum_{s=0}^\infty \alpha(n,s)Q^s.\end{aligned}$$ Here, $\alpha$ is a two-parameter discrete function defined by an explicit partial recurrence equation, which in part takes the form of a discrete transport equation, has a finite-speed of propagation and is defined explicitly in terms of the traces ${{\sf tr}}(Q^s)$ of products of $Q$ (see Lemma \[lem:alpha\] for its definition). In particular, the resulting characterization of the discrete-time two-point correlations ${{\mathbb E}}_{\beta_u}[\xi_n(U)\xi_n(V_\infty)]$ is enough for an explicit expression of the Laplace transform of the *first-order* meeting time, which is essentially due to the connection mentioned above between the distributions of $M_{U,V}$ and $M_{U,V_\infty}$ (see Section \[sec:lap-1\] for the details). Then it holds that $$\begin{aligned}
\label{eq:HT}
\begin{split}
&{{\mathbb E}}[e^{-\lambda M_{U,V}}]=\\
&\hspace{.5cm}=\left.{{\mathbb E}}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{U})dt\right]\right/{{\mathbb E}}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{V})dt\right]
\end{split}\end{aligned}$$ (Corollary \[cor:MUV-wr\]), and so a comparison with the analogous representation (\[eq:HTGF\]) for ${{\mathbb E}}[e^{-\lambda H_{U,V}/2}]$ entails the classical reduction $$M_{U,V}\stackrel{(\rm d)}{=}H_{U,V}/2$$ *under* the assumption of walk-regularity of the underlying Markov kernel. We remark that the simple first-step recurrence argument is in fact enough to yield the equality (\[eq:HT\]) (cf. Lemma 12 in [@AN:GG]). Nonetheless, it appears less clear why the explicit forms of the discrete-time two-point correlations ${{\mathbb E}}_{\beta_u}[\xi_n(U)\xi_n(V_\infty)]$ of the voter model, implied by the infinite series (\[eq:Ln-intro\]) for $L^n(J)$ and (\[eq:coupling\]), can be as well obtained by inverting explicit forms for Laplace transforms of meeting times.
To establish an approximating equality for (\[eq:HT\]) when walk-regularity of $Q$ fails, we introduce the terms $$\begin{aligned}
\label{eq:pc}
\frac{1}{N}{{\mathbb E}}[\langle \beta_u|L_0^n(J)|\beta_u\rangle]\end{aligned}$$ as substitutes of the true correlations in (\[eq:coupling\]), where the linear operator $L_0$ is chosen to be an “approximating” version of $L$ and such that the infinite series expression in (\[eq:Ln-intro\]) remains valid with $L^n(J)$ replaced by the new products $L^n_0(J)$. (See (\[def:L0\]) for our choice of $L_0$.) For convenience, we will call the above terms [**approximating correlations**]{} in view of (\[eq:coupling\]), even though in general their probabilistic interpretation is not clear to us. The main technical issues as a result of adopting these approximation correlations arise essentially in the context of large state spaces. We have to understand the asymptotic behavior of $L_0^n(J)$ for large $n$ with respect to the size of the state space, and bound the errors from replacing $L$ with $L_0$ in formulating the approximating correlations so that they do not grow too fast up to moderately large times even on large state spaces. The details are given in Section \[sec:lap-2\].
#### **Organization of the paper.**
In Section \[sec:tpc\], we study basic properties of the operator $L$ defined by (\[eq:T\*T\]), which gives the two-point correlations (\[eq:coupling\]) of voter models, from its explicit expression to some related operator norms. Based on the explicit form, we define in Section \[sec:ptpc\] the operator $L_0$ in making the approximating correlations (\[eq:pc\]), and study generating functions of the matrices $L_0^n(J)$. We show a preliminary connection among the distribution of $M_{U,V}$, the true correlations (\[eq:coupling\]), and the approximating correlations (\[eq:pc\]) in Section \[sec:lap-1\], and then in Section \[sec:lap-2\], we prove the main approximation theorem (Theorem \[thmm:main-1\]) on the distributional reduction of the first-order meeting times. Details of the proof of Corollary \[cor:main2\] are given afterward in Section \[sec:cor-main2\]. We close this paper with some relations between meeting times of all finite orders in Section \[sec:MT\].
Correlations of voter models {#sec:tpc}
============================
Recall the linear operator $L$ defined in (\[eq:T\*T\]) by a matrix expectation. In this section, we derive its explicit representation and study related operator norms. From now on, we index entries of $N\times N$ matrices over $\Bbb C$ by points of $E$ (with a fixed order as before), and $\mathsf M_E$ stands for the linear space of $N\times N$ matrices subject to such convention. In addition, we will consistently use the bra-ket notation for matrix and vector multiplications.
To start with, let us set some matrix notation for the explicit representation of $L$. We define a linear operator ${{\sf diag}}$ on $\mathsf M_E$ by $$\begin{aligned}
{{\sf diag}}(C)\triangleq \sum_{x\in E}|x\rangle\langle x|C|x\rangle \langle x|,\label{def:diag}\end{aligned}$$ that is ${{\sf diag}}(C)$ is the diagonal matrix obtained from $C\in \mathsf M_E$ by setting all off-diagonal entries to zero and leaving all diagonal terms unchanged. Here in (\[def:diag\]), $|x{\rangle}$ and ${\langle}x|$ denote respectively the column vector and the row vector with $1$ at $x$-th coordinate and zero otherwise. In addition, we write $J$ for the probability matrix defined by $$\begin{aligned}
\forall\;x,y\in E,\quad \langle x|J|y\rangle\triangleq \frac{1}{N}.\label{def:J}\end{aligned}$$
\[prop:conj\] Fix a Markov kernel $Q$, and recall the associated linear operator $L$ which is defined by the matrix expectation (\[eq:T\*T\]). Then $L$ admits the explicit expression: $$\begin{aligned}
\begin{split}
L(C)=&\frac{N-2}{N}C+\frac{1}{N}(CQ+QC)-\frac{1}{N}[{{\sf diag}}(C)Q+Q{{\sf diag}}(C)]\\
&+\frac{1}{N}{{\sf diag}}(C )
+{{\sf diag}}\big(Q{{\sf diag}}(C )J\big).\label{eq:tst}
\end{split}\end{aligned}$$
Fix a constant matrix $C\in \mathsf M_E$ throughout this proof.
For the present purpose to compute the explicit expression of $L$, we may assume that the operator $T$ in its definition (\[eq:T\*T\]) is given by $$\begin{aligned}
T=I-|U{\rangle}{\langle}U|+|U{\rangle}{\langle}V|.\end{aligned}$$ Here, as before, ${{\mathbb P}}(U=x,V=y)\equiv \frac{1}{N}\langle x|Q|y\rangle$. Then we expand the product $T^*CT$ defining $L$ as $$\begin{aligned}
\begin{split}\label{eq:T*CT}
T^*CT=&\big(I-|U\rangle\langle U|+|V\rangle \langle U|\big)C\big(I-|U\rangle\langle U|+|U\rangle \langle V|\big)\\
=&C-C|U\rangle \langle U|+C|U\rangle \langle V|\\
&-|U\rangle \langle U|C+|U\rangle \langle U|C|U\rangle \langle U|-|U\rangle \langle U|C|U\rangle \langle V|\\
&+|V\rangle \langle U|C-|V\rangle \langle U|C|U\rangle \langle U|+|V\rangle \langle U|C|U\rangle \langle V|.
\end{split}\end{aligned}$$ Taking expectation for both sides of the foregoing equality, we obtain $$\begin{aligned}
L(C)=&\,{{\mathbb E}}\left[T^*CT\right]
=C-\sum_{x\in E}\frac{1}{N}C|x\rangle \langle x|+\sum_{x,y\in E}\frac{1}{N}\langle x|Q|y\rangle C |x\rangle \langle y|\\
&-\sum_{x\in E}\frac{1}{N}|x\rangle \langle x| C+\sum_{x\in E}\frac{1}{N}|x\rangle \langle x|C|x\rangle \langle x|-\sum_{x,y\in E}\frac{1}{N}\langle x|Q|y\rangle|x\rangle \langle x|C|x\rangle \langle y|\\
&+\sum_{x,y\in E}\frac{1}{N}\langle x|Q|y\rangle |y\rangle \langle x|C-\sum_{x,y\in E}\frac{1}{N}\langle x|Q|y\rangle|y\rangle \langle x|C|x\rangle\langle x|\\
&+\sum_{x,y\in E}\frac{1}{N}\langle x|Q|y\rangle|y\rangle \langle x|C|x\rangle\langle y|.\end{aligned}$$ To simplify the right-hand side of the foregoing equality, we use the assumed symmetry of the Markov kernel $Q$ and the equality $I=\sum_{x\in E}|x\rangle \langle x|$, and invoke the operators ${{\sf diag}}(\,\cdot\,)$ and $J$ defined above in (\[def:diag\]) and (\[def:J\]), respectively. It follows that $$\begin{aligned}
\begin{split}\label{eq:tst1}
L(C)=&C-\frac{1}{N}CI+\frac{1}{N}CQ\\
&-\frac{1}{N}IC+\frac{1}{N}{{\sf diag}}(C)-\frac{1}{N}{{\sf diag}}(C)Q\\
&+\frac{1}{N}QC-\frac{1}{N}Q{{\sf diag}}(C)+{{\sf diag}}\big(Q{{\sf diag}}(C)J\big).
\end{split}\end{aligned}$$ We obtain the required equation (\[eq:tst\]) from (\[eq:tst1\]) after rearrangement. The proof is complete.
We consider some particular cases of (\[eq:tst\]).
\[cor:tst\] With respect to the operator $L$ defined by (\[eq:T\*T\]), we have $$\begin{aligned}
L(J)&=J+\frac{2}{N^2}(I-Q),\label{eq:itPi}\\
\begin{split}
L(Q^s)&=\frac{N-2}{N}Q^s+\frac{2}{N}Q^{s+1}-\frac{1}{N}[{{\sf diag}}(Q^s)Q+\frac{1}{N}Q{{\sf diag}}(Q^s)]\\
&\hspace{.5cm}+\frac{1}{N}{{\sf diag}}(Q^s)+{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big),\quad \forall\;s\in \Bbb Z_+.\label{eq:itQ}
\end{split}\end{aligned}$$ In particular, if the $s$-step return probabilities $\langle x|Q^s|x\rangle $ do not depend on $x$ for $s\in\Bbb Z_+ $, then (\[eq:itQ\]) simplifies to $$\begin{aligned}
L(Q^s)=&\frac{N-2}{N}Q^s+\frac{2}{N}Q^{s+1}-\frac{2{{\sf tr}}(Q^s)}{N^2}Q+\frac{2{{\sf tr}}(Q^s)}{N^2}I.\label{eq:itQw}\end{aligned}$$
To see (\[eq:itPi\]), we apply to (\[eq:tst\]) the fact that $QJ=JQ=J$, and obtain the required equation: $$\begin{aligned}
L(J)&=\frac{N-2}{N}J+\frac{1}{N}2J-\frac{1}{N^2}2Q +\frac{1}{N^2}I+\frac{1}{N^2}I
=J+\frac{2}{N^2}(I-Q).\end{aligned}$$
Equation (\[eq:itQ\]) is a straightforward consequence of (\[eq:tst\]). For the particular case that $\langle x|Q^s|x\rangle $ do not depend on $x$, we note that $\langle x|Q^s|x\rangle$ is equivalent to the arithmetic mean of the diagonal terms of $Q^s$ and so ${{\sf diag}}(Q^s)=[{{\sf tr}}(Q^s)/N]I$. It follows that (\[eq:itQ\]) can be written as $$\begin{aligned}
L(Q^s)
&=\frac{N-2}{N}Q^s+\frac{2}{N}Q^{s+1}-\frac{2{{\sf tr}}(Q^s)}{N^2}Q+\frac{{{\sf tr}}(Q^s)}{N^2}I+\frac{{{\sf tr}}(Q^s)}{N^2}I\\
&=\frac{N-2}{N}Q^s+\frac{2}{N}Q^{s+1}-\frac{2{{\sf tr}}(Q^s)}{N^2}Q+\frac{2{{\sf tr}}(Q^s)}{N^2}I,\end{aligned}$$ which is (\[eq:itQw\]).
Let us turn to some operator norms related to $L$. We equip $\mathsf M_E$ with the $\ell_1$-norm: $$\begin{aligned}
\label{eq:matrix-norm}
\|C\|\triangleq \sum_{x,y\in E}|\langle x|C|y\rangle|\end{aligned}$$ and consider the (induced) operator norms $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert S
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}\triangleq \max\{\|S(C)\|;\|C\|=1,C\in \mathsf M_E\}$$ for linear operators $S:\mathsf M_E{\longrightarrow}\mathsf M_E$. The following proposition will be used in Section \[sec:lap-2\] for the proof of Theorem \[thmm:main-1\].
\[prop:norm\] We have the following bounds:
1. $\displaystyle {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert L
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}= 1$,
2. $\displaystyle {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert L-I
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}\leq \frac{4}{N}$.
We note that by (\[eq:tst\]), $$\begin{aligned}
L(C)&=\frac{N-1}{N}C+{{\sf diag}}(QCJ)\quad\mbox{if }C={{\sf diag}}(C),\label{eq:diagL}\\
L(C)&=\frac{N-2}{N}C+\frac{1}{N}(CQ+QC)\quad\mbox{if }{{\sf diag}}(C)=0.\label{eq:offdiagL}\end{aligned}$$ By (\[eq:diagL\]), we have $$\begin{aligned}
\|L(|x{\rangle}{\langle}x|)\|&=\frac{N-1}{N}+\sum_{y\in E}\sum_{z\in E}\langle y|Q|z\rangle \langle z|x{\rangle}{\langle}x|z\rangle\langle z|J|y\rangle
=1,
\label{eq:diag1}\end{aligned}$$ and by (\[eq:offdiagL\]), for $x\neq y$, $$\begin{aligned}
\label{Ixy}
\|L(|x{\rangle}{\langle}y|)\|=\frac{N-2}{N}+\frac{1}{N}\sum_{z\in E}\langle y|Q|z\rangle+\frac{1}{N}\sum_{z\in E}\langle z|Q|x\rangle=1.\end{aligned}$$ Since $\big\||x{\rangle}{\langle}y|\big\|= 1$ for all $x,y\in E$, we deduce (i) from (\[eq:diag1\]) and (\[Ixy\]).
By (\[eq:diagL\]) and (\[eq:offdiagL\]) again, we obtain, respectively, $$\begin{aligned}
\big\|L(|x{\rangle}{\langle}x|)-|x{\rangle}{\langle}x|\big\|=&\sum_{y\in E}\left|\frac{-1}{N}\langle y|x{\rangle}{\langle}x|y{\rangle}+\sum_{z\in E}{\langle}y|Q|z{\rangle}{\langle}z|x{\rangle}{\langle}x|z{\rangle}{\langle}z|J|y{\rangle}\right|\notag\\
=&\sum_{y\in E}\left|\frac{-1}{N}\langle y|x{\rangle}{\langle}x|y{\rangle}+\frac{1}{N}{\langle}x|Q|y{\rangle}\right|\notag\\
=&\frac{2}{N}(1-\langle x|Q|x\rangle)\leq \frac{2}{N}\label{L-IIxx}\end{aligned}$$ and, for $x\neq y$, $$\begin{aligned}
&\big\|L(|x{\rangle}{\langle}y|)-|x{\rangle}{\langle}y|\big\|\notag\\
=&\sum_{z,a\in E}\left|-\frac{2}{N}{\langle}z|x{\rangle}{\langle}y|a{\rangle}+\frac{1}{N}{\langle}z|x{\rangle}\langle y|Q|a\rangle+\frac{1}{N}{\langle}z|Q|x{\rangle}{\langle}y|a{\rangle}\right|\label{L-IIxy-0}\\
=&\frac{1}{N}\big(2-{\langle}y|Q|y{\rangle}-{\langle}x|Q|x{\rangle}\big)+\frac{1}{N}\sum_{a:a\neq y}\langle y|Q|a{\rangle}+\frac{1}{N}\sum_{z:z\neq x}{\langle}z|Q|x{\rangle}\label{L-IIxy-1}\\
= &\frac{2}{N}\big(2-{\langle}y|Q|y{\rangle}-{\langle}x|Q|x{\rangle}\big)\leq \frac{4}{N}, \label{L-IIxy}\end{aligned}$$ where (\[L-IIxy-1\]) follows by noting that the summands on the right-hand side of (\[L-IIxy-0\]) are nonzero only if $z=x$ or $a=y$. By (\[L-IIxx\]) and (\[L-IIxy\]), we deduce (ii).
Approximating correlations {#sec:ptpc}
==========================
The matrices $L^n(J)$ allow for simple expressions in the walk-regular case. This is pointed out in (\[eq:Ln-intro\]) in Section \[sec:intro\], and we will give the proof later on in Lemma \[lem:alpha\]. In general, we can still choose an operator $L_0$ which is similar to $L$ so that $L_0^n(J)$ gives the same expression (\[eq:Ln-intro\]) (with $L^n(J)$ replaced) even if the assumption of walk-regularity is not in force. One choice is to replace ${{\sf diag}}(\,\cdot\,)$ with $[{{\sf tr}}(\,\cdot\,)/N]I$ in the representation (\[eq:tst\]) of $L$, and consider accordingly $$\begin{aligned}
\label{def:L0}
L_0(C)&\triangleq \frac{N-2}{N}C+\frac{1}{N}(CQ+QC)-\frac{2{{\sf tr}}(C)}{N^2}Q
+\frac{2{{\sf tr}}(C)}{N^2}I\end{aligned}$$ for $C\in \mathsf M_E$. We will work with the matrices $L_0^n(J)$ throughout this section, and it will be helpful to keep in mind that they are meant to “approximate” $L^n(J)$, in a sense to be made precise in Section \[sec:lp\].
\[lem:alpha\] Consider a two-parameter function $\alpha:\Bbb Z_+\times \Bbb Z_+{\longrightarrow}{{\Bbb R}}$ defined recursively as follows. Set $\alpha(0,s)\equiv 0$ for all $s\in \Bbb Z_+$ and, for $n\in \Bbb Z_+$, $$\begin{aligned}
\alpha(n+1,0)&=\frac{2}{N^2}+\alpha(n,0)+\frac{2}{N^2}\sum_{s=1}^\infty \alpha(n,s){{\sf tr}}(Q^s),\label{rec1}\\
\alpha(n+1,1)&=\frac{-2}{N^2}+\frac{N-2}{N}\alpha(n,1)-\frac{2}{N^2}\sum_{s=1}^\infty \alpha(n,s){{\sf tr}}(Q^s),\label{rec2}\\
\alpha(n+1,s)&=\frac{N-2}{N}\alpha(n,s)+\frac{2}{N}\alpha(n,s-1),\quad s\geq 2.\label{rec3}\end{aligned}$$ Then the action of the $n$-iteration of the linear map $L_0$ defined by (\[def:L0\]) on $J$ satisfies $$\begin{aligned}
L_0^n(J)=J+\sum_{s=0}^\infty \alpha(n,s)Q^s,\quad \forall\;n\in \Bbb Z_+.\label{eq:L0n}\end{aligned}$$ In particular, if $Q$ is walk-regular, then $$\begin{aligned}
\label{eq:Ln}
L^n(J)=L_0^n(J)=J+\sum_{s=0}^\infty \alpha(n,s)Q^s,\quad \forall\;n\in \Bbb N.\end{aligned}$$
We note that $\alpha$ has a finite speed of propagation, or more precisely, $\alpha(n,s)=0$ whenever $s\geq n+1$, and (\[rec3\]) takes the form of a discrete transport equation.
We prove (\[eq:L0n\]) by an induction on $n\in\Bbb Z_+$. The case $n=0$ follows plainly by the assumed initial condition of $\alpha$. Suppose that (\[eq:L0n\]) is true for some $n\in \Bbb Z_+$. Then we have $$\begin{aligned}
L_0^{n+1}(J)=L_0\big(L^n_0(J)\big)=&L_0\left(J+\sum_{s=0}^\infty \alpha(n,s)Q^s\right)=L_0(J)+\sum_{s=0}^\infty \alpha(n,s)L_0(Q^s),\end{aligned}$$ where the infinite series are only finite sums since $\alpha$ has a finite speed of propagation. By the definition (\[def:L0\]) of $L_0$, we can express $L_0(J)$ and $L_0(Q^s)$ on the right-hand side of the foregoing equality by linear combinations of $J$ and $Q^s$ (cf. the proof of Corollary \[cor:tst\]), and get $$\begin{aligned}
L_0^{n+1}(J)
=&\left(J-\frac{2}{N^2}Q+\frac{2}{N^2}I\right)\\
&+\sum_{s=0}^\infty \alpha(n,s)\left(\frac{N-2}{N}Q^s+\frac{2}{N}Q^{s+1}-\frac{2{{\sf tr}}(Q^s)}{N^2}Q+\frac{2{{\sf tr}}(Q^s)}{N^2}I\right)\notag\\
\begin{split}
\hspace{-1cm}=&J+\left(\frac{2}{N^2}+\alpha(n,0)\frac{N-2}{N}+\sum_{s=0}^\infty \alpha(n,s)\frac{2{{\sf tr}}(Q^s)}{N^2}\right)I\\
&+\left(-\frac{2}{N^2}+\alpha(n,1)\frac{N-2}{N}+\alpha(n,0)\frac{2}{N}-\sum_{s=0}^\infty \alpha(n,s)\frac{2{{\sf tr}}(Q^s)}{N^2}\right)Q\\
&+\sum_{s=2}^\infty \left(\alpha(n,s)\frac{N-2}{N}+\alpha(n,s-1)\frac{2}{N}\right)Q^s.
\end{split}\end{aligned}$$ Since ${{\sf tr}}(I)=N$, (\[eq:L0n\]) with $n$ replaced by $n+1$ follows from the last equality and the definition of $\alpha(n+1,\,\cdot\,)$ (see (\[rec1\])–(\[rec3\])). By mathematical induction (\[eq:L0n\]) is true for all $n\in \Bbb Z_+$.
The proof of (\[eq:Ln\]) proceeds with an induction on $n\in \Bbb N$. We need two observations. Notice that when $Q$ is walk-regular, $$\begin{aligned}
\label{L=L0}
L(J)=L_0(J)\quad\mbox{ and }\quad L(Q^s)=L_0(Q^s)\quad\forall\; s\in \Bbb Z_+,\end{aligned}$$ by Corollary \[cor:tst\] and (\[def:L0\]). In addition, it follows from Hoffman’s theorem that $J=f(Q)$ for some polynomial $f$ over ${{\Bbb R}}$ (see [@B:AGT Proposition 3.2]). We see that $$\begin{aligned}
\label{LL0-rest}
\mbox{(\ref{L=L0}) holds}\quad \Longleftrightarrow \quad L=L_0\;\mbox{on}\;\mathcal A\triangleq \{g(Q)\}\end{aligned}$$ where $g$ ranges over all polynomials over $\Bbb C$, and $\mathcal A$ defines an algebra. In addition, it is plain from the definition (\[def:L0\]) of $L_0$ that $$\begin{aligned}
\label{L0-A}
L_0(\mathcal A)\subseteq \mathcal A.\end{aligned}$$
We are ready to give the proof of (\[eq:Ln\]). It holds for $n=1$ by (\[L=L0\]). If (\[eq:Ln\]) holds for some $n\in \Bbb N$, then we get $$L^{n+1}(J)=L\big(L_0^n(J)\big)=L_0\big(L_0^n(J)\big)=L_0^{n+1}(J),$$ where the second equality follows from (\[LL0-rest\]) and (\[L0-A\]). Hence, by mathematical induction, (\[eq:Ln\]) is true for all $n\in \Bbb N$. The proof is complete.
Next, we derive the generating function of the above discrete function $\alpha$, using functional calculus for the symmetric matrix $Q$. Recall that we assume the state space of $Q$ has size $N>8$.
\[thmm:GH\] Let $\alpha$ be the two-parameter function defined in Lemma \[lem:alpha\] for $L_0$. Write $\mathscr G\alpha(\zeta,q)$ for the two-parameter generating function of $\alpha$: $$\begin{aligned}
{\mathscr G}\alpha(\zeta,q)\triangleq \sum_{n=0}^\infty \zeta^n\sum_{s=0}^\infty q^s\alpha(n,s).
\label{def:A0}\end{aligned}$$ Then ${\mathscr G}\alpha(\zeta,q)$ converges absolutely for any $\zeta,q\in \Bbb C$ with $|\zeta|<1$ and $|q|\leq 1$. Moreover for such $\zeta,q$, $$\begin{aligned}
{\mathscr G}\alpha(\zeta,q)&=\frac{2\zeta(1-q)[N-\zeta( N-2+2 q)]^{-1}}{N(1-\zeta)^2{{\sf tr}}\left(\frac{1}{N-\zeta(N-2+2Q)}\right)}.\label{eq:A0}\end{aligned}$$
Apply the foregoing theorem to (\[eq:L0n\]) by the spectral representation of $f(Q)$ for complex functions $f$ which are analytic around $[-1,1]$. We obtain the following corollary immediately.
\[cor:L0R0-gen\] The generating function of the square matrices $L_0^n(J)$ with respect to exponent $n\in \Bbb Z_+$ is given by $$\begin{aligned}
\label{eq:genL}
\begin{split}
\mathscr G L_0 (J)(\zeta)&\triangleq \sum_{n=0}^\infty \zeta^n L_0^n(J)\\
&=\frac{1}{1-\zeta}J+\frac{2\zeta(I-Q)[N-\zeta( N-2+2 Q)]^{-1}}{N(1-\zeta)^2{{\sf tr}}\left(\frac{1}{N-\zeta(N-2+2Q)}\right)}.
\end{split}\end{aligned}$$ Here, the infinite series of matrices converges absolutely with respect to the $\ell_1$-norm on $\mathsf M_E$ (see (\[eq:matrix-norm\])) whenever $\zeta\in \Bbb C$ with $|\zeta|<1$.
The rest of this section is devoted to the proof of Theorem \[thmm:GH\]. The conclusion will be given at the end of this section.
We begin with an a priori estimate to ensure that the radius of convergence for ${\mathscr G}\alpha$ is not degenerate. See also Lemma \[lem:alpha-growth\] for different estimates.
\[lem:bdd-0\] The entries of $\alpha$ satisfy $$\begin{aligned}
\label{ineq:radii-a}
\max\left\{|\alpha(n,0)|,|\alpha(n,1)|,\sum_{s=2}^\infty|\alpha(n,s)|\right\}\leq \frac{1}{N}\left(1+\frac{6}{N}\right)^{n-1},\quad n\in \Bbb N.\end{aligned}$$
Note that $|{{\sf tr}}(Q^s)/N|\leq 1$ for all $s\in \Bbb Z_+$. Hence, (\[rec1\])–(\[rec3\]) imply $$\begin{aligned}
&|\alpha(n+1,0)|\leq \frac{2}{N^2}+|\alpha(n,0)|+\frac{2}{N}|\alpha(n,1)|+\frac{2}{N}\sum_{s=2}^\infty |\alpha(n,s)|,\\
&|\alpha(n+1,1)|\leq \frac{2}{N^2}+|\alpha(n,1)|+\frac{2}{N}\sum_{s=2}^\infty |\alpha(n,s)|,\\
&\sum_{s=2}^\infty |\alpha(n+1,s)|\leq \sum_{s=2}^\infty |\alpha(n,s)|+\frac{2}{N}|\alpha(n,1)|.\end{aligned}$$ Then (\[ineq:radii-a\]) can be checked by induction on $n$ (recall the initial condition of $\alpha$) and the foregoing inequalities. The proof is complete.
\[lem:solve\] Let $\zeta,q\in \Bbb C$ be such that $$\begin{aligned}
|\zeta|<\frac{N}{N+6}\quad\mbox{ and }\quad |q|\leq 1.\label{zetaq}\end{aligned}$$ Then the infinite series (\[def:A0\]) defining $\mathscr G\alpha(\zeta,q)$ converges absolutely, and we have $$\begin{aligned}
&\sum_{n=0}^\infty \zeta^n\alpha(n,0)=\frac{-[N-\zeta(N-2)]}{N(1-\zeta)}\times\sum_{n=0}^\infty \zeta^n\alpha(n,1),\label{alpha0-gen}\\
&\sum_{n=0}^\infty \zeta^n\alpha(n,1)=\frac{-2\zeta}{(1-\zeta)[N-\zeta(N-2)]^2{{\sf tr}}\left(\frac{1}{N-\zeta(N-2+2Q)}\right)},\label{alpha1-gen}\\
&\sum_{n=0}^\infty \zeta^n\sum_{s=2}^\infty q^s\alpha(n,s)=\frac{2\zeta q^2}{N-\zeta(N-2+2q)}\times
\sum_{n=0}^\infty \zeta^n\alpha(n,1).\label{alpha-gen}\end{aligned}$$ Hence, (\[eq:A0\]) holds for $\zeta,q\in \Bbb C$ satisfying (\[zetaq\]).
It follows immediately from Lemma \[lem:bdd-0\] that ${\mathscr G}\alpha(\zeta,q)$ converges absolutely for $\zeta,q\in \Bbb C$ satisfying (\[zetaq\]). We may assume throughout the proof that $\zeta\neq 0$.
First, we derive (\[alpha0-gen\]) and (\[alpha-gen\]) in order. We add up both sides of (\[rec1\]) and (\[rec2\]) and get $$\alpha(n+1,0)+\alpha(n+1,1)=\alpha(n,0)+\frac{N-2}{N}\alpha(n,1),\quad n\in \Bbb Z_+.$$ Using the foregoing equality and the fact that $\alpha(0,\cdot)\equiv 0$, we deduce that $$\begin{aligned}
\frac{1}{\zeta}\sum_{n=0}^\infty \zeta^n\alpha(n,0)+\frac{1}{\zeta}\sum_{n=0}^\infty \zeta^n\alpha(n,1)&=\sum_{n=0}^\infty \zeta^{n}\alpha(n+1,0)+\sum_{n=0}^\infty \zeta^{n}\alpha(n+1,1)
\\
&=\sum_{n=0}^\infty \zeta^n\alpha(n,0)+\frac{N-2}{N}\sum_{n=0}^\infty \zeta^n\alpha(n,1),\end{aligned}$$ which implies (\[alpha0-gen\]). Next, by (\[rec3\]) and the fact that $\alpha(0,\,\cdot\,)\equiv 0$, we get $$\begin{aligned}
&\frac{1}{\zeta}\sum_{n=0}^\infty \zeta^n\sum_{s=2}^\infty q^s \alpha(n,s)=
\sum_{n=0}^\infty \zeta^n\sum_{s=2}^\infty q^s\alpha(n+1,s)\\
&\hspace{.5cm}=\left(\frac{N-2}{N}+\frac{2q}{N}\right)\sum_{n=0}^\infty\zeta^n \sum_{s=2}^\infty q^s\alpha(n,s)+\frac{2q^2}{N}\sum_{n=0}^\infty \zeta^n\alpha(n,1),\end{aligned}$$ and the equality (\[alpha-gen\]) follows.
Let us solve for the series $\sum_{n=0}^\infty \zeta^n\alpha(n,1)$ which appears on both of the right-hand sides of (\[alpha0-gen\]) and (\[alpha-gen\]). By the initial condition $\alpha(0,\,\cdot\,)\equiv 0$ and (\[rec2\]), $$\begin{aligned}
\begin{split}
&\frac{1}{\zeta}\sum_{n=0}^\infty \zeta^n\alpha(n,1)
=\sum_{n=0}^\infty \zeta^n\alpha(n+1,1)\\
&\hspace{1cm}=\frac{-2}{N^2(1-\zeta)}
+\frac{N-2}{N}\sum_{n=0}^\infty \zeta^n\alpha(n,1)
-\frac{2{{\sf tr}}(Q )}{N^2}\sum_{n=0}^\infty \zeta^n\alpha(n,1)\\
&\hspace{1.5cm}-\frac{2}{N^2}{{\sf tr}}\left(\sum_{n=0}^\infty \zeta^n\sum_{s=2}^\infty Q^s\alpha(n,s)\right),
\end{split}\end{aligned}$$ where the absolute convergence of the infinite series is justified by Lemma \[lem:bdd-0\]. By the foregoing equality, we get $$\begin{aligned}
&\sum_{n=0}^\infty \zeta^n\alpha(n,1)\\
=&\frac{-2\zeta}{N^2(1-\zeta)}+\frac{\zeta (N-2)}{N}\sum_{n=0}^\infty \zeta^n\alpha(n,1)-\frac{2\zeta{{\sf tr}}(Q )}{N^2}\sum_{n=0}^\infty \zeta^n\alpha(n,1)\\
&-\frac{2\zeta}{N^2}{{\sf tr}}\left(\sum_{n=0}^\infty \zeta^n\sum_{s=2}^\infty Q^s\alpha(n,s)\right)\\
=&\frac{-2\zeta}{N^2(1-\zeta)}+\left[\frac{\zeta (N-2)}{N}- \frac{2\zeta{{\sf tr}}(Q )}{N^2}- \frac{4\zeta^2}{N^2}{{\sf tr}}\left( \frac{Q^2}{N-\zeta(N-2+2Q)}
\right)\right]\\
&\times \sum_{n=0}^\infty \zeta^n\alpha(n,1),\end{aligned}$$ where the second equality follows from (\[alpha-gen\]), and hence, $$\begin{aligned}
\label{alphan1}
\begin{split}
&\sum_{n=0}^\infty \zeta^n\alpha(n,1)\\
&\hspace{.5cm}=\frac{-2\zeta}{N(1-\zeta)\left[N-N\zeta+2\zeta+2\zeta\frac{{{\sf tr}}(Q)}{N}+\frac{4\zeta^2}{N} {{\sf tr}}\left( \frac{Q^2}{N-\zeta(N-2+2Q)}
\right)\right]}.
\end{split}\end{aligned}$$ By linearity of trace, we obtain $$\begin{aligned}
&N-N\zeta+2\zeta+2\zeta\frac{{{\sf tr}}(Q)}{N}+\frac{4\zeta^2}{N} {{\sf tr}}\left( \frac{Q^2}{N-\zeta(N-2+2Q)}
\right)\\
=&\frac{[N(1-\zeta)+2\zeta]^2}{N}{{\sf tr}}\left(\frac{1}{N-\zeta(N-2+2Q)}\right),\end{aligned}$$ and hence, (\[alpha1-gen\]) by the foregoing equality and (\[alphan1\]).
Finally, we recall (\[def:A0\]) and use (\[alpha0-gen\])–(\[alpha-gen\]) to obtain $$\begin{aligned}
{\mathscr G}\alpha(\zeta,q)&=\left\{\frac{-[N-\zeta(N-2)]}{N(1-\zeta)}+q+\frac{2\zeta q^2}{N-\zeta(N-2+2q)}\right\}
\\
&\hspace{1cm}\times\frac{-2\zeta}{(1-\zeta)[N-\zeta(N-2)]^2{{\sf tr}}\left(\frac{1}{N-\zeta(N-2+2Q)}\right)},\end{aligned}$$ from which we deduce the equality (\[eq:A0\]) for $ \zeta,q\in \Bbb C$ satisfying (\[zetaq\]). The proof is complete.
From now on, we write $C(0,R)$, $D(0,R)$ and $\overline{D}(0,R)$ for the circle, the open disc and the close disc, respectively, centered at $0$ with radius $R\in (0,\infty)$ in the complex plane. We study the denominator of the function on the right-hand side of (\[eq:A0\]).
\[lem:bdd-1\] For any ${{\varepsilon}}\in [0,N)$, $$\begin{aligned}
\begin{split}\label{ineq:Mobm}
&\min_{\zeta\in C(0,1-\frac{{{\varepsilon}}}{N})}
\left|\frac{1}{N}{{\sf tr}}\left(\frac{1-\zeta}{N-\zeta(N-2+2Q)}\right)\right|\\
&\hspace{4cm}=\frac{1}{N^2}{{\sf tr}}\Bigg(\frac{{{\varepsilon}}}{2-2Q+{{\varepsilon}}-\frac{{{\varepsilon}}}{N}(2-2Q)}\Bigg),\\
\end{split}\\
\begin{split}
&\max_{\zeta\in C(0,1-\frac{{{\varepsilon}}}{N})}
\left|\frac{1}{N}{{\sf tr}}\left(\frac{1-\zeta}{N-\zeta(N-2+2Q)}\right)\right|\\
&\hspace{4cm}= \frac{1}{N^2}{{\sf tr}}\Bigg(\frac{2-\frac{{{\varepsilon}}}{N}}{
1+\big(1-\frac{{{\varepsilon}}}{N}\big)\big(1-\frac{2}{N}+\frac{2Q}{N}\big)}\Bigg).\label{ineq:MobM}
\end{split}\end{aligned}$$ Here, meromorphic functions are defined at their removable singularities in the natural way, and the right-hand side of the equality in (\[ineq:Mobm\]) is read as $1/N^2$ if ${{\varepsilon}}=0$ (since $1$ is an eigenvalue of $Q$). Moreover, in the above display, the minimum and the maximum are attained at $1-\frac{{{\varepsilon}}}{N}$ and $-(1-\frac{{{\varepsilon}}}{N})$, respectively.
For every $q\in [-1,1]$, consider the Möbius transformation $$M_q(\zeta)\triangleq \frac{1-\zeta}{N-\zeta(N-2+2q)}.$$ Note that $M_1$ is just the constant map $1/N$.
Let us make some observations for $M_q$, when $q\in [-1,1)$. By a standard result of Möbius transformations, $M_q$ maps $C(0,1-\frac{{{\varepsilon}}}{N})$ to a nondegenerate circle, say $C_q$, in $\Bbb C$ because it is nonconstant and analytic in an open set containing $\overline{D}(0,1)$. The circle $C_q$ is symmetric about the real line because $M_q$ is defined by real coefficients, and plainly intersects the real line at $$\begin{aligned}
\label{+1}
\displaystyle M_q\left(1-\frac{{{\varepsilon}}}{N}\right)=\frac{{{\varepsilon}}/N}{2-2q+{{\varepsilon}}-\frac{{{\varepsilon}}}{N}(2-2q)}\end{aligned}$$ and $$\begin{aligned}
\label{-1}
M_q\left(-1+\frac{{{\varepsilon}}}{N}\right)=\frac{2-\frac{{{\varepsilon}}}{N}}{
N\big[1+\big(1-\frac{{{\varepsilon}}}{N}\big)\big(1-\frac{2}{N}+\frac{2q}{N}\big)\big]},\end{aligned}$$ which are distinct strictly positive real numbers. This means that $C_q$ is contained in the half plane $\{\zeta\in \Bbb C;\Re(\zeta)>0\}$. In addition, note that for each $q\in [-1,1)$, the value in (\[+1\]) is strictly less than the value in (\[-1\]), and $M_q(\sqrt{-1})$ has strictly negative imaginary part. We deduce that for $q\in [-1,1)$ and $\zeta=(1-\frac{{{\varepsilon}}}{N})e^{\sqrt{-1}\theta}$ with $\theta\in [0,2\pi]$, $$\begin{aligned}
\label{Mq:rotat}
M_q(\zeta)=a_q-b_qe^{\sqrt{-1}\theta}\end{aligned}$$ for some $a_q,b_q\in (0,\infty)$ independent of $\theta$. The foregoing equality holds trivially for $q=1$ if we set $a_1=1/N$ and $b_1=0$.
By the special form (\[Mq:rotat\]) of $M_q$ for all $q\in [-1,1]$, the optimization problems in (\[ineq:Mobm\]) and (\[ineq:MobM\]) take the forms of minimizing and maximizing $|A-Be^{\sqrt{-1}\theta}|$ subject to $\theta\in [0,2\pi]$, for fixed $A,B\in (0,\infty)$. For the latter two, solutions are given by $\theta=0$ and $\theta=\pi$, respectively. Hence, the equalities in (\[ineq:Mobm\]) and (\[ineq:MobM\]) follow upon using (\[+1\]) and (\[-1\]).
Note that [@P:ZPS Theorem 1] studies locations of zeros for complex functions taking the form $\zeta{\longmapsto}P(\zeta)+\zeta^k\int_{[0,1]}(1-\zeta q)^{-1}\mu(dq)$, which arises from the investigation of Riesz summability. Here, $P$ are polynomials of degree $k-1$ for $k\in \Bbb Z_+$ ($P(\zeta)\equiv 0$ if $k=0$) and $\mu$ is a finite measure on $[0,1]$. The context in [@P:ZPS Theorem 1] overlaps in part the context of Lemma \[lem:bdd-1\].
#### **Conclusion for the proof of Theorem \[thmm:GH\].**
We have seen that (\[eq:A0\]) holds when $\zeta,q\in \Bbb C$ satisfying (\[zetaq\]), by Lemma \[lem:solve\]. By a standard result of several complex variables (see, e.g., Hörmander [@H:SCV Theorem 2.2.1]) and Lemma \[lem:bdd-1\], the function in two complex variables on the right-hand side of (\[eq:A0\]) is analytic in the open polydisc $D=\{(\zeta,q)\in \Bbb C\times \Bbb C;|\zeta|<1,|q|<1\}$. Hence, by Cauchy’s inequalities for analytic functions in several complex variables [@H:SCV Theorem 2.2.7], ${\mathscr G}\alpha(\zeta,q)$ converges absolutely on the polydisc $D$. Next, if we fix $\zeta\in \Bbb C$ such that $|\zeta|<1$ and repeat the above argument with respect to the single complex variable $q$, then the assertion of Theorem \[thmm:GH\] can be extended up to the boundary case $|q|=1$. The proof is complete.
Laplace transforms of first-order meeting times {#sec:lp}
===============================================
Meeting time distributions, voter correlations and approximating correlations {#sec:lap-1}
-----------------------------------------------------------------------------
In this section, we derive an infinite series expression for the Laplace transform of $M_{U,V}$ in terms of the true correlations $$\frac{1}{N}{{\mathbb E}}[\langle \beta_u|L^n(J)|\beta_u\rangle]={{\mathbb E}}_{\beta_u}[\xi_n(U)\xi_n(V_\infty)]$$ (recall (\[eq:coupling\])), and compute the analogous series in terms of the approximating correlations $\frac{1}{N}{{\mathbb E}}[\langle \beta_u|L^n_0(J)|\beta_u\rangle]$. The results of the present section will be applied in Section \[sec:lap-2\] for the proof of Theorem \[thmm:main-1-0\].
From now on, we write $S_\xi(C)\equiv {\langle}\xi|C|\xi\rangle$ for all deterministic configurations $\xi$, and in addition, $$\begin{aligned}
\label{def:Sbetau}
S_{\beta_u}(C)\equiv {{\mathbb E}}[\langle \beta_u|C|\beta_u\rangle],\quad C\in\mathsf M_E,\end{aligned}$$ for the Bernoulli configurations $\beta_u=\sum_x \beta_u(x)|x{\rangle}$ (recall that $\beta_u(x)$ are i.i.d. Bernoulli with mean $u$).
\[lem:lap-ac-1\] For every $\lambda\in (0,\infty)$ and $\xi\in \{1,0\}^E$, $$\begin{aligned}
\label{BuLn-2}
\begin{split}
&\frac{\lambda}{N+\lambda}\sum_{n=0}^\infty\left(\frac{N}{N+\lambda}\right)^n \frac{1}{N}S_\xi L_0^n(J)
=p_1(\xi)^2\\
&\hspace{4cm}+\frac{1}{N}S_\xi\left(\frac{2(I-Q)(\lambda+2-2Q)^{-1}}{ \lambda{{\sf tr}}\left(\frac{1}{\lambda+2-2Q}\right)}\right),
\end{split}\end{aligned}$$ where the series on the left-hand side converges absolutely.
We apply $\frac{\lambda}{N+\lambda}\times \frac{1}{N}S_\xi$ to both sides of (\[eq:genL\]) with $\zeta$ set to be $N/(N+\lambda)$. Then notice that $$\begin{aligned}
\frac{1}{N}S_\xi(J)=\frac{\langle \xi|{\mathds 1}\rangle \langle {\mathds 1}|\xi\rangle}{N^2}=p_1(\xi)^2\end{aligned}$$ and the second term on the right-hand side of (\[eq:genL\]) with $\zeta$ set to be $N/(N+\lambda)$ becomes $$\begin{aligned}
\frac{2(N+\lambda) (I-Q)(\lambda+2-2Q)^{-1}}{\lambda^2{{\sf tr}}\left(\frac{1}{\lambda+2-2Q}\right)} .\end{aligned}$$ The equality (\[BuLn-2\]) now follows plainly.
Next, we relate the Laplace transform of $M_{U,V}$ to the generating function of $\frac{1}{N}S_{\beta_u}L_0^n(J)$ (in $n$) in the following lemma. We define the mean local density $p_{10}(\xi)$ of configuration $\xi$ by $$\begin{aligned}
p_{10}(\xi)\triangleq \sum_{x,y\in E}\frac{1}{N}\langle x|Q|y\rangle \xi(x)\xi(y)= \frac{1}{N}\langle \xi|Q|\xi\rangle.\label{def:p10}\end{aligned}$$
\[lem:MUV-exp\]
1. For every $\xi\in \{1,0\}^E$ and $n\in \Bbb Z_+$, $$\frac{1}{N}S_\xi L^{n+1}(J)-\frac{1}{N}S_\xi L^n(J)=\frac{2}{N^2}{{\mathbb E}}_\xi[p_{10}(\xi_n)].$$
2. For every $\lambda\in (0,\infty)$ and $u\in (0,1)$, $$\begin{aligned}
\begin{split}
&{{\mathbb E}}\big[e^{-\lambda M_{U,V}}\big]\\
&=1-\frac{\lambda N^2}{2u(1-u)(N+\lambda)}\sum_{n=0}^\infty \left(\frac{N}{N+\lambda}\right)^n\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J) \right),
\end{split}\end{aligned}$$ where the series on the right-hand side converges absolutely.
3. For every $\lambda\in (0,\infty)$ and $u\in (0,1)$, $$\begin{aligned}
\begin{split}
&\left.{{\mathbb E}}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{V\}}(X^U_t)dt\right]\right/\Bbb E\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{V\}}(X^V_t)dt\right]\\
&=1-\frac{\lambda N^2}{2u(1-u)(N+\lambda)}\sum_{n=0}^\infty \left(\frac{N}{N+\lambda}\right)^n\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right),
\end{split}\end{aligned}$$ where the series on the right-hand side converges absolutely.
The proof of (i) follows immediately from the fact: for every $\xi\in \{1,0\}^E$ and $n\in \Bbb Z_+$, $$\begin{aligned}
\label{p1:timeevolution}
\frac{1}{N}S_{\xi}L^n(J)=\frac{1}{N}S_\xi(J)+\frac{2}{N^2}\sum_{j=0}^{n-1}{{\mathbb E}}_\xi[p_{10}(\xi_j)],\end{aligned}$$ where $p_{10}(\xi)$ is defined by (\[def:p10\]). The above equation is implicit in the proof of [@CCC:WF Proposition 3.1 (iii)] since $\frac{1}{N}S_\xi L^n(J)\equiv {{\mathbb E}}_\xi[\xi_n(U)\xi_n(V_\infty)]$.
To prove (ii), we resort to the duality equation (\[eq:MTVM\]) and use the fact that $(\xi_t)$ is equal to $(\xi_n)$ time-changed by an independent Poisson process with rate $N$. We get $$\begin{aligned}
1-{{\mathbb E}}[e^{-\lambda M_{U,V}}]&=\int_0^\infty \lambda e^{-\lambda s}\frac{{{\mathbb E}}_{\beta_u}[p_{10}(\xi_s)]}{u(1-u)}ds\\
&\hspace{-2cm}=\frac{1}{u(1-u)}\frac{\lambda}{N+\lambda}\sum_{n=0}^\infty \left(\frac{N}{N+\lambda}\right)^n{{\mathbb E}}_{\beta_u}[p_{10}(\xi_n)]\\
&\hspace{-2cm}=\frac{\lambda N^2}{2u(1-u)(N+\lambda)}\sum_{n=0}^\infty \left(\frac{N}{N+\lambda}\right)^n \left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right),\end{aligned}$$ where the last equality follows from (i).
It remains to prove (iii). On one hand, we note that $$\begin{aligned}
&\left.{{\mathbb E}}\left[\int_0^\infty e^{-t\lambda/2}{\mathds 1}_{\{V\}}(X^{U}_t)dt\right]\right/ {{\mathbb E}}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{V\}}(X^{V}_t)dt\right]\\
&= \left.\frac{1}{N}\sum_{x,y\in E}\langle x|Q|y\rangle \left\langle x\left|\int_0^\infty e^{-t\lambda/2+t(Q-I)}dt\right|y\right\rangle\right/\frac{1}{N}\sum_{y\in E}\left\langle y\left|
\int_0^\infty e^{-t\lambda/2+t(I-Q)}dt
\right|y\right\rangle\\
&=\left.\frac{1}{N}{{\sf tr}}\Bigg(\frac{Q}{\frac{\lambda}{2}+(I-Q)}\Bigg)\right/\frac{1}{N}{{\sf tr}}\Bigg(\frac{1}{\frac{\lambda}{2}+(I-Q)}\Bigg)\\
&=\left.{{\sf tr}}\Bigg(\frac{Q}{\lambda+2(I-Q)}\Bigg)\right/{{\sf tr}}\Bigg(\frac{1}{\lambda+2(I-Q)}\Bigg),\end{aligned}$$ and so (iii) follows if we can show that $$\begin{aligned}
&\left.{{\sf tr}}\Bigg(\frac{Q}{\lambda+2(I-Q)}\Bigg)\right/{{\sf tr}}\Bigg(\frac{1}{\lambda+2(I-Q)}\Bigg)\\
=&1-\frac{\lambda N^2}{2u(1-u)(N+\lambda)}\sum_{n=0}^\infty \left(\frac{N}{N+\lambda}\right)^n\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right).\end{aligned}$$ The foregoing equality, however, is a consequence of (\[BuLn-2\]) if we apply a randomization by $\beta_u$ to both sides of (\[BuLn-2\]) and notice $$\begin{aligned}
\begin{split}\label{Sbetau-0}
&\frac{1}{N}S_{\beta_u}\Bigg(\frac{2(I-Q)(\lambda+2-2Q)^{-1}}{\lambda {{\sf tr}}\left(\frac{1}{\lambda+2-2Q}\right)}\Bigg)\\
&\hspace{1.8cm}=\left.\left[2(u-u^2){{\sf tr}}\left(\frac{1-Q}{\lambda+2-2Q}\right)\right]\right/\left[\lambda N {{\sf tr}}\left(\frac{1}{\lambda+2-2Q}\right)\right].
\end{split}\end{aligned}$$
To see (\[Sbetau-0\]), we first compute $$\begin{aligned}
S_{\beta_u}(Q^s)&={{\mathbb E}}\left[\sum_{x,y\in E}\beta(x)\langle x|Q^s|y\rangle\beta(y)\right]\notag\\
&=\sum_{\stackrel{\scriptstyle x,y\in E}{x\neq y}}\langle x|Q^s|y\rangle u^2+\sum_{x\in E}\langle x|Q^s|x\rangle u\notag\\
&=u^2N+(u-u^2){{\sf tr}}(Q^s),\quad \forall\;s\in \Bbb Z_+,\label{Sbetau}\end{aligned}$$ and similarly, $S_{\beta_u}(J)=(N-1)u^2+u$. Hence, we get $$\begin{aligned}
\label{SSSSS}
S_{\beta_u}\big(Q^s{\mathds 1}_{(-\infty,1)(Q)}\big)
&=S_{\beta_u}\left(Q^s-J\right)
=(u-u^2){{\sf tr}}\big(Q^s;Q<1\big).\end{aligned}$$ Here, we write $$\begin{aligned}
\label{eq:trace-set}
{{\sf tr}}\big(f(Q);Q\in A\big)\triangleq {{\sf tr}}\big(f(Q){\mathds 1}_{A}(Q)\big),\quad A\subseteq {{\Bbb R}}.\end{aligned}$$ Then by polynomial approximation, (\[SSSSS\]) implies $$\begin{aligned}
S_{\beta_u}\big((I-Q)(\lambda+2-2Q)^{-1}\big)&=S_{\beta_u}\big((I-Q)(\lambda+2-2Q)^{-1}{\mathds 1}_{(-\infty,1)}(Q)\big)\notag\\
&=(u-u^2)
{{\sf tr}}\left(\frac{1-Q}{\lambda+2-2Q};Q<1\right)\notag\\
&=(u-u^2)
{{\sf tr}}\left(\frac{1-Q}{\lambda+2-2Q}\right).\label{Subeta-2}\end{aligned}$$ The equality (\[Subeta-2\]) is enough to get (\[Sbetau-0\]), and the proof is complete.
\[cor:MUV-wr\] Suppose that $Q$ is walk-regular. Then $$\begin{aligned}
\begin{split}
&{{\mathbb E}}[e^{-\lambda M_{U,V}}]\\
&\hspace{.5cm}=\left.{{\mathbb E}}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{V\}}(X^U_t)dt\right]\right/
{{\mathbb E}}\left[\int_0^\infty e^{-\lambda t/2}{\mathds 1}_{\{V\}}(X^V_t)dt\right].\label{MUV-1}
\end{split}\end{aligned}$$
Equation (\[MUV-1\]) follows at once if we compare (ii) and (iii) of Lemma \[lem:MUV-exp\], since $$\frac{1}{N}S_{\beta_u}L^n(J)=\frac{1}{N}S_{\beta_u}L_0^n(J),\quad \forall\;n\in \Bbb Z_+,$$ by (\[eq:Ln\]) in Lemma \[lem:alpha\]. The proof is complete.
Approximation {#sec:lap-2}
-------------
This section is devoted to the proof of Theorem \[thmm:main-1-0\]. Recall the notation $S_\xi$ and $S_{\beta_u}$ defined at the beginning of Section \[sec:lap-1\]. To prove Theorem \[thmm:main-1\], it is enough to bound $$\begin{aligned}
\begin{split}\label{T}
&\frac{\lambda N^2}{2u(1-u)(N+\lambda)}
\sum_{n=0}^\infty \left(\frac{N}{N+\lambda}\right)^n\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)\\
&-\frac{\lambda N^2}{2u(1-u)(N+\lambda)}\sum_{n=0}^\infty \left(\frac{N}{N+\lambda}\right)^n\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right)
\end{split}\end{aligned}$$ by Lemma \[lem:MUV-exp\] (ii) and (iii). For fixed $m\in \Bbb N$, we will handle separately and in order the objects: (1) the partial sum of the first term in (\[T\]) for $n$ ranging from $mN+1$ to $\infty$, (2) the partial sum of the second term in (\[T\]) for $n$ ranging from $mN+1$ to $\infty$, and (3) the difference in (\[T\]) with the upper limits $\infty$ of the two series replaced with $mN$.
The first object described above is easy to deal with.
\[lem:T1\] For every $\lambda\in(0,\infty)$, $u\in (0,1)$ and $m\in \Bbb N$, $$\begin{aligned}
&\left|
\frac{\lambda N^2}{2u(1-u)(N+\lambda)}
\sum_{n=mN+1}^{\infty}\left(\frac{N}{N+\lambda}\right)^n\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)\right|\\
&\hspace{8cm}\leq \frac{1}{u(1-u)}\left(\frac{N}{N+\lambda}\right)^{mN}.\end{aligned}$$
By Lemma \[lem:MUV-exp\] (i), $$\frac{N^2}{2}\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)={{\mathbb E}}_{\beta_u}[p_{10}(\xi_n)]\in [0,1].$$ Hence, $$\begin{aligned}
&\left|\frac{\lambda N^2}{2u(1-u)(N+\lambda)}\sum_{n=mN}^\infty \left(\frac{N}{N+\lambda}\right)^n\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)\right|\\
&\hspace{.5cm}\leq \frac{\lambda}{u(1-u)(N+\lambda)}\sum_{n=mN}^\infty
\left(\frac{N}{N+\lambda}\right)^n=\frac{1}{u(1-u)}\left(\frac{N}{N+\lambda}\right)^{mN},\end{aligned}$$ as required.
To bound the second object described below (\[T\]), we turn to the asymptotics of the discrete function $\alpha$ defined in Lemma \[lem:alpha\].
\[lem:alpha-growth\] For every ${{\varepsilon}}\in (0,1]$ and $n\in \Bbb N$, we have $$\begin{aligned}
&|\alpha(n,0)|\leq \frac{({{\varepsilon}}^{-1}+4\sqrt{2})}{\pi}\times \frac{4+{{\varepsilon}}}{{{\varepsilon}}} \times\frac{1}{N}\left(\frac{N}{N-{{\varepsilon}}}\right)^{\max\{n-1,0\}}, \label{alpha0-exp-bdd}\\
&|\alpha(n,1)|\leq \frac{9}{\pi}\times \frac{4+{{\varepsilon}}}{{{\varepsilon}}} \times\frac{1}{N}\left(\frac{N}{N-{{\varepsilon}}}\right)^{\max\{n-1,0\}},\label{alpha1-exp-bdd}\\
\begin{split}
&\sup_{q\in [-1,1]}\left|\sum_{s=2}^\infty q^s\alpha(n,s)\right|\leq \frac{1}{\pi}\left({{\varepsilon}}^{-1}+\frac{16\sqrt{2}}{(1-\cos 1)^{1/2}}\right)\times \frac{4+{{\varepsilon}}}{{{\varepsilon}}}\\
&\hspace{6cm}\times\frac{1}{N}\left(\frac{N}{N-{{\varepsilon}}}\right)^{\max\{n-2,0\}}.\label{alpha-exp-bdd}
\end{split}\end{aligned}$$
Throughout this proof, we fix ${{\varepsilon}}\in (0,1]$. We start with the proof of (\[alpha0-exp-bdd\]), and fix $n\geq 1$. For $\zeta=(1-\frac{{{\varepsilon}}}{N})z$, it follows from (\[alpha0-gen\]), (\[alpha1-gen\]) and Cauchy’s integral formula that $$\begin{aligned}
\begin{split}\label{Gamma-goal}
&\alpha(n,0)=\frac{1}{2\pi \sqrt{-1}}\int_{C(0,1)}\frac{1}{z^{n+1}}\\
&\hspace{2cm}\times\frac{2z}{[N-N(1-\frac{{{\varepsilon}}}{N})z][N-(1-\frac{{{\varepsilon}}}{N})z(N-2)]}\\
&\hspace{2.5cm}\times \frac{1}{
\frac{1}{N}{{\sf tr}}\left(\frac{1-(1-\frac{{{\varepsilon}}}{N})z}{N-(1-\frac{{{\varepsilon}}}{N})z(N-2+2Q)}\right)}dz\times \frac{1}{N}\left(1-\frac{{{\varepsilon}}}{N}\right)^{-n+1}.
\end{split}\end{aligned}$$ In the following, we bound $$\begin{aligned}
\label{Gamma}
\begin{split}
&\Bigg|\frac{1}{2\pi \sqrt{-1}}\int_{\Gamma_j}\frac{1}{z^{n+1}}\times\frac{2z}{[N-N(1-\frac{{{\varepsilon}}}{N})z][N-(1-\frac{{{\varepsilon}}}{N})z(N-2)]}\\
&\hspace{5.5cm}\times \frac{1}{\frac{1}{N}
{{\sf tr}}\left(\frac{1-(1-\frac{{{\varepsilon}}}{N})z}{N-(1-\frac{{{\varepsilon}}}{N})z(N-2+2Q)}\right)}dz\Bigg|,
\end{split}\end{aligned}$$ for $j=1,2$, where the arcs $\Gamma_j$ are defined by $$\begin{aligned}
\Gamma_1&\triangleq \{z\in C(0,1)\setminus \{-1\};|\arg(z)|\leq 1/N\},\label{Gamma1}\\
\Gamma_2&\triangleq \{z\in C(0,1)\setminus \{-1\};|\arg(z)|\in (1/N,\pi)\}\cup \{-1\}\label{Gamma2}\end{aligned}$$ and their disjoint union is equal to $C(0,1)$. Here for any $z\in \Bbb C\setminus (-\infty,0]$, we write $\arg(z)=\theta$ for $z=|z|e^{i\theta}$ for $\theta\in (-\pi,\pi)$.
We handle (\[Gamma\]) for $j=1$ first. Using (\[ineq:Mobm\]), we see that $$\begin{aligned}
\label{Mobm-bdd}
\min_{\zeta\in C(0,1-\frac{{{\varepsilon}}}{N})}
\left|\frac{1}{N}{{\sf tr}}\left(\frac{1-\zeta}{N-\zeta(N-2+2Q)}\right)\right|\geq \left(\frac{{{\varepsilon}}}{4+{{\varepsilon}}}\right)\frac{1}{N}.\end{aligned}$$ By (\[Mobm-bdd\]) and the fact that $\min_{z\in C(0,1)}\big|1-az\big|=1-a$ for any $a\in (0,1)$, we have $$\begin{aligned}
&\Bigg|\frac{1}{2\pi \sqrt{-1}}\int_{\Gamma_1}\frac{1}{z^{n+1}}\times\frac{2z}{[N-N(1-\frac{{{\varepsilon}}}{N})z][N-(1-\frac{{{\varepsilon}}}{N})z(N-2)]}\notag\\
&\hspace{5.5cm}\times \frac{1}{
\frac{1}{N}{{\sf tr}}\left(\frac{1-(1-\frac{{{\varepsilon}}}{N})z}{N-(1-\frac{{{\varepsilon}}}{N})z(N-2+2Q)}\right)}dz\Bigg|\notag\\
&\hspace{.5cm}\leq \frac{1}{2\pi} \int_{-1/N}^{1/N}\frac{2}{{{\varepsilon}}\left(2+{{\varepsilon}}-\frac{2{{\varepsilon}}}{N}\right)\times {{\varepsilon}}/(4+{{\varepsilon}})\times (1/N)} d\theta\leq \frac{1}{{{\varepsilon}}\pi }\times \frac{4+{{\varepsilon}}}{{{\varepsilon}}}.\label{Gamma1-1}\end{aligned}$$
Next, we bound (\[Gamma\]) for $j=2$. Note that $$\begin{aligned}
\forall\; \delta\in (0,1)\;\mbox{and}\; \theta\in [0,2\pi],\quad |1-(1-\delta)e^{\sqrt{-1}\theta}|
\geq
&\sqrt{2(1-\delta)}\sqrt{1-\cos \theta}.\end{aligned}$$ Applying the foregoing inequality to $$\begin{aligned}
&\frac{1}{N}\left|N-N\left(1-\frac{{{\varepsilon}}}{N}\right)z\right|=\left|1-\left(1-\frac{{{\varepsilon}}}{N}\right)\right|
\quad\mbox{and}\\
&\frac{1}{N}\left|N-\left(1-\frac{{{\varepsilon}}}{N}\right)z(N-2)\right|=\left|1-\left(1-\frac{{{\varepsilon}}}{N}\right)\left(1-\frac{2}{N}\right)z\right|\quad \mbox{for }z\in C(0,1),\end{aligned}$$ and using (\[Mobm-bdd\]), we see that $$\begin{aligned}
\begin{split}\notag
&\Bigg|\frac{1}{2\pi \sqrt{-1}}\int_{\Gamma_2}\frac{1}{z^{n+1}}\times\frac{2z}{[N-N(1-\frac{{{\varepsilon}}}{N})z][N-(1-\frac{{{\varepsilon}}}{N})z(N-2)]}\notag\\
&\hspace{5.5cm}\times \frac{1}{\frac{1}{N}
{{\sf tr}}\left(\frac{1-(1-\frac{{{\varepsilon}}}{N})z}{N-(1-\frac{{{\varepsilon}}}{N})z(N-2+2Q)}\right)}dz\Bigg|\notag\\
\end{split}\\
&\hspace{.2cm}\leq \frac{1}{2\pi}\int_{\{\theta\in {{\Bbb R}}:1/N\leq |\theta|\leq \pi\}}\frac{2}{N\sqrt{2\big(1-\frac{{{\varepsilon}}}{N}\big)(1-\cos \theta)}} \notag\\
&\hspace{.5cm}\times\frac{1}{N\sqrt{2\left(1-\frac{{{\varepsilon}}}{N}\right)\left(1-\frac{2}{N}\right)(1-\cos\theta)}}\times \frac{1}{{{\varepsilon}}/(4+{{\varepsilon}})\times (1/N)}d\theta\notag\\
&\leq \frac{8^{1/2}}{\pi}\times\frac{4+{{\varepsilon}}}{{{\varepsilon}}}\times \frac{1}{N}\int_{1/N}^\pi \frac{1}{1-\cos \theta}d\theta\leq \frac{4\sqrt{2}}{\pi}\times \frac{4+{{\varepsilon}}}{{{\varepsilon}}}
,\label{B1:Gamma2}\end{aligned}$$ where the next to the last inequality follows since ${{\varepsilon}}/N,2/N\leq 1/2$ and $$\begin{aligned}
\label{ineq:cos}
\sup_{r\in (0,1]}r\int_r^\pi \frac{d\theta}{1-\cos \theta}d\theta=2.\end{aligned}$$ Applying (\[Gamma1-1\]) and (\[B1:Gamma2\]) to (\[Gamma-goal\]), we see that $|\alpha(n,0)|$ can be bounded as in (\[alpha0-exp-bdd\]).
The proofs of (\[alpha1-exp-bdd\]) and (\[alpha-exp-bdd\]) follow the same lines as the proof of (\[alpha0-exp-bdd\]), except for minor changes and the application of the inequality $$\sup_{r\in (0,1]}r^2\int_r^\pi \frac{d\theta}{(1-\cos \theta)^{3/2}}\leq \frac{2}{(1-\cos 1)^{1/2}}$$ instead of (\[ineq:cos\]) in proving (\[alpha-exp-bdd\]). The details are left to the readers.
\[lem:T2\] Fix ${{\varepsilon}}\in (0,1]$. Then for every $\lambda\in ({{\varepsilon}},\infty)$ such that $(\lambda-{{\varepsilon}})N>\lambda{{\varepsilon}}$, $u\in (0,1)$ and $m\in \Bbb N$, we have $$\begin{aligned}
\label{eq:A0nalytic2}
&\left|\frac{\lambda N^2 }{2u(1-u)(N+\lambda)}\sum_{n=mN+1}^\infty \left(\frac{N}{N+\lambda}\right)^n
\left(\frac{1}{N}S_{\beta_u} L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u} L_0^n(J)\right)\right|\notag\\
&\hspace{.5cm}\leq \left[\Big(1+\frac{\lambda}{N}\Big)\left(1-\frac{{{\varepsilon}}}{N}\right)\right]^{-(mN+1)}\times \frac{C_{{\varepsilon}}\lambda (N-{{\varepsilon}})}{(\lambda-{{\varepsilon}})N-\lambda{{\varepsilon}}},\end{aligned}$$ where the constant $C_{{\varepsilon}}$ is defined by (\[def:Cvep\]).
First we claim the following inequality: for all $n\in \Bbb Z_+$ $$\begin{aligned}
\begin{split}
&\left|\frac{1}{N}S_{\beta_u}L^{n+1}_0(J)-\frac{1}{N}S_{\beta_u}L^n_0(J)\right|\leq \frac{2C_{{\varepsilon}}u(1-u) }{N^2}
\left(\frac{N}{N-{{\varepsilon}}}\right)^n,\label{Budiff-growth}
\end{split}\end{aligned}$$ where the constant $C_{{\varepsilon}}$ is defined by (\[def:Cvep\]). By (\[eq:L0n\]) and (\[Sbetau\]), we have $$\begin{aligned}
&\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n_0(J)
=\sum_{s=0}^\infty [\alpha(n+1,s)-\alpha(n,s)]
\frac{u^2N+(u-u^2){{\sf tr}}(Q^s)}{N}.\end{aligned}$$ Note that $\sum_{s=0}^\infty \alpha(n,s)=0$, which can be seen by using the initial condition of $\alpha$ and adding up both sides of (\[rec1\])–(\[rec3\]). Hence, by the foregoing display and then the recursive equations (\[rec1\])–(\[rec3\]) for $\alpha$, we get $$\begin{aligned}
&\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)
=\frac{u(1-u)}{N}\sum_{s=0}^\infty [\alpha(n+1,s)-\alpha(n,s)]{{\sf tr}}(Q^s)\notag\\
=&\frac{u(1-u)}{N}\Bigg\{\left[\frac{2}{N^2}+\frac{2}{N^2}\sum_{s=1}^\infty \alpha(n,s){{\sf tr}}(Q^s)\right]{{\sf tr}}(I)\\
&\hspace{1cm}+\left[\frac{-2}{N^2}-\frac{2}{N}\alpha(n,1)-\frac{2}{N^2}\sum_{s=1}^\infty \alpha(n,s){{\sf tr}}(Q^s)\right]{{\sf tr}}(Q)\notag\\
&\hspace{1.5cm}-\frac{2}{N}\sum_{s=2}^\infty [\alpha(n,s)-\alpha(n,s-1)]{{\sf tr}}(Q^s)\Bigg\}\\
=&\frac{u(1-u)}{N}\Bigg\{\left[\frac{2}{N}-\frac{2{{\sf tr}}(Q)}{N^2}\right]+\left[\frac{2{{\sf tr}}(Q^2)}{N}-\frac{2{{\sf tr}}(Q)^2}{N^2}\right]\alpha(n,1)\\
&\hspace{1.5cm}-\frac{2{{\sf tr}}(Q)}{N^2}{{\sf tr}}\left(\sum_{s=2}^\infty Q^s\alpha(n,s)\right)+\frac{2}{N}{{\sf tr}}\left(Q\sum_{s=2}^\infty Q^{s}\alpha(n,s)\right)\Bigg\}.\end{aligned}$$ Our claim (\[Budiff-growth\]) follows upon applying Lemma \[lem:alpha-growth\] to the right-hand of the foregoing equality.
To finish the proof of (\[eq:A0nalytic2\]), we use (\[Budiff-growth\]) and get $$\begin{aligned}
&\Bigg|\frac{\lambda N^2 }{2u(1-u)(N+\lambda)}\sum_{n=mN+1}^\infty \left(\frac{N}{N+\lambda}\right)^n
\left(\frac{1}{N}S_{\beta_u}L^{n+1}_0(J)-\frac{1}{N}S_{\beta_u}L^{n}_0(J)\right)\Bigg|\\
\leq &\frac{\lambda N^2}{2u(1-u)(N+\lambda)}\times \frac{2C_{{\varepsilon}}u(1-u) }{N^2}\times \sum_{n=mN+1}^\infty \left(\frac{N}{N+\lambda}\right)^n \left(\frac{N}{N-{{\varepsilon}}}\right)^n\\
=& \left[\Big(1+\frac{\lambda}{N}\Big)\left(1-\frac{{{\varepsilon}}}{N}\right)\right]^{-(mN+1)}\times \frac{C_{{\varepsilon}}\lambda (N-{{\varepsilon}})}{(\lambda-{{\varepsilon}})N-\lambda{{\varepsilon}}},\end{aligned}$$ as stated in in the required inequality (\[eq:A0nalytic2\]). The proof is complete.
The final stage is on bounding the third object described below (\[T\]): $$\begin{aligned}
\begin{split}\label{T'}
&\frac{\lambda N^2}{2u(1-u)(N+\lambda)}
\sum_{n=0}^{mN} \left(\frac{N}{N+\lambda}\right)^n\Bigg[\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)\\
&\hspace{4.5cm}-\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right)\Bigg].
\end{split}\end{aligned}$$
\[lem:LL0-diff\] For all $n\in \Bbb N$, $$\begin{aligned}
\label{eq:LLn}
L^{n}(J)-L^{n}_0(J)=\sum_{j=0}^{n-1}L^j(\eta_{n-j}),\end{aligned}$$ where $\eta_n\triangleq L\big(L_0^{n-1}(J)\big)-L_0^n(J)$ satisfies $$\begin{aligned}
\label{eq:etan}
\begin{split}
\eta_n
=&\sum_{s=0}^\infty \Bigg\{\left[\frac{1}{N}{{\sf diag}}(Q^s)+{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)-2\frac{{{\sf tr}}(Q^s)}{N^2}\right]I\\
&\hspace{.5cm}+\left[-\frac{{{\sf diag}}(Q^s)}{N}Q-Q\frac{{{\sf diag}}(Q^s)}{N}+\frac{2{{\sf tr}}(Q^s)}{N^2}Q\right]\Bigg\}\alpha(n-1,s).
\end{split}\end{aligned}$$
Set ${{\varepsilon}}_n=L^n(J)-L^n_0(J)$. For any $n\in \Bbb N$, we have $$\begin{aligned}
L^{n}(J)=&L\big(L^{n-1}(J)\big)=L({{\varepsilon}}_{n-1})+L\big(L_0^{n-1}(J)\big)
=L({{\varepsilon}}_{n-1})+\eta_{n}+ L_0^{n}(J)\end{aligned}$$ by the definition of $\eta_{n}$, and it follows that $${{\varepsilon}}_{n}=L({{\varepsilon}}_{n-1})+\eta_{n}.$$ We obtain (\[eq:LLn\]) by iterating the above equality and using the equality ${{\varepsilon}}_1=\eta_1$ (recall Corollary \[cor:tst\] and (\[def:L0\])).
Next, we show the explicit form (\[eq:etan\]) of $\eta_n$. Fix $n\in \Bbb N$, and recall the definition (\[def:L0\]) of $L_0$. By (\[eq:L0n\]) and Corollary \[cor:tst\], we have $$\begin{aligned}
&L\big(L_0^{n-1}(J)\big)
=\left[J+\frac{2}{N^2}(I-Q)\right]+\sum_{s=0}^\infty \alpha(n-1,s)L(Q^s)\\
=&J+\frac{2}{N^2}(I-Q)+\sum_{s=0}^\infty \alpha(n-1,s)\Bigg\{\frac{N-2}{N}Q^s+\frac{2}{N}Q^{s+1}\\
&-\frac{1}{N}[{{\sf diag}}(Q^s)Q+Q{{\sf diag}}(Q^s)]+\frac{1}{N}{{\sf diag}}(Q^s)+{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)\Bigg\}\\
=&J+\Bigg[\frac{2}{N^2}+\frac{N-2}{N}\alpha(n-1,0)+\sum_{s=0}^\infty \alpha(n-1,s)\frac{1}{N}{{\sf diag}}(Q^s)\\
&\hspace{.5cm}+\sum_{s=0}^\infty \alpha(n-1,s){{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)\Bigg] I\\
&+\Bigg\{\left[-\frac{2}{N^2}+\frac{N-2}{N}\alpha(n-1,1)+\frac{2}{N}\alpha(n-1,0)-\sum_{s=0}^\infty \alpha(n-1,s)\frac{1}{N}{{\sf diag}}(Q^s)\right]Q\\
&\hspace{.5cm}-Q\left[\sum_{s=0}^\infty\alpha(n-1,s)\frac{1}{N}{{\sf diag}}(Q^s)\right]\Bigg\}\\
&+\sum_{s=2}^\infty \left[\frac{N-2}{N}\alpha(n-1,s)+\frac{2}{N}\alpha(n-1,s-1)\right]Q^s.\end{aligned}$$ On the other hand, if we express the coefficients $\alpha(n,\,\cdot\,)$ of $L^n_0(J)$ in (\[eq:L0n\]) by $\alpha(n-1,\,\cdot\,)$ using the partial recurrence equations (\[rec1\])–(\[rec3\]) obeyed by $\alpha$, then $$\begin{aligned}
L^n_0(J)
=&J+\left[\frac{2}{N^2}+\alpha(n-1,0)+\frac{2}{N^2}\sum_{s=1}^\infty \alpha(n-1,s){{\sf tr}}(Q^s)\right]I\\
&+\left[-\frac{2}{N^2}+\frac{N-2}{N}\alpha(n-1,1)-\frac{2}{N^2}\sum_{s=1}^\infty \alpha(n-1,s){{\sf tr}}(Q^s)\right]Q\\
&+\sum_{s=2}^\infty \left[\frac{N-2}{N}\alpha(n-1,s)+\frac{2}{N}\alpha(n-1,s-1)\right]Q^s\\
=&J+\left[\frac{2}{N^2}+\frac{N-2}{N}\alpha(n-1,0)+\sum_{s=0}^\infty \alpha(n-1,s)\frac{2{{\sf tr}}(Q^s)}{N^2}\right]I\\
&+\left[-\frac{2}{N^2}+\frac{N-2}{N}\alpha(n-1,1)+\frac{2}{N}\alpha(n-1,0)-\sum_{s=0}^\infty \alpha(n-1,s)\frac{2{{\sf tr}}(Q^s)}{N^2}\right]Q\\
&+\sum_{s=2}^\infty \left[\frac{N-2}{N}\alpha(n-1,s)+\frac{2}{N}\alpha(n-1,s-1)\right]Q^s.\end{aligned}$$ Comparing the last equalities in the foregoing two displays, we deduce that $\eta_n$ satisfies the required expression (\[eq:etan\]). The proof is complete.
The next lemma is on bounding the square matrices followed by $\alpha(n-1,s)$ in the expression (\[eq:etan\]). Recall the definition (\[def:RQ\]) of $\mathcal R_Q^\gamma(x,s)$ and the $\ell_1$-norm on $\mathsf M_E$ defined by (\[eq:matrix-norm\]).
\[lem:eigenbdd\] For all $s\in \Bbb Z_+$ and $\gamma\in [0,1]$, $$\begin{aligned}
\begin{split}
& \left\|{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|\leq \left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|\\
&\hspace{1.5cm} \leq \min\left\{\frac{4\min_{x\in E}[N-\#\mathcal R_Q^\gamma(x,s)]}{N}+\gamma,\frac{2{{\sf tr}}(|Q|^s;Q<1)}{N}\right\}
\label{delta-bdd}
\end{split}\end{aligned}$$ (for the last trace term, recall the notation (\[eq:trace-set\])).
Fix $s\in \Bbb Z_+$. We will prove (\[delta-bdd\]) in four steps.\
[**(Step 1).**]{} Note that $$\begin{aligned}
\left\|{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|&=\frac{1}{N}\sum_{y\in E}\left|\sum_{x\in E} {\langle}y|Q|x{\rangle}{\langle}x|Q^s|x{\rangle}-\frac{{{\sf tr}}(Q^s)}{N}\right|\\
&=\frac{1}{N}\sum_{y\in E}\left|\sum_{x\in E} {\langle}x|Q|y{\rangle}\left[{\langle}x|Q^s|x{\rangle}-\frac{{{\sf tr}}(Q^s)}{N}\right]\right|\\
&\leq \sum_{x\in E} \left|\frac{1}{N}{\langle}x|Q^s|x{\rangle}-\frac{{{\sf tr}}(Q^s)}{N^2}\right|\\
&=\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|.\end{aligned}$$ The first inequality in (\[delta-bdd\]) follows from the foregoing display.\
[**(Step 2).**]{} Before giving the proof of the second inequality in (\[delta-bdd\]), we claim that for any $x\in E$, $$\begin{aligned}
\left|\langle x|Q^s|x\rangle-\frac{{{\sf tr}}(Q^s)}{N}\right|\leq \frac{2[N-\#\mathcal R_Q^\gamma(x,s)]}{N}+\gamma.\label{eq:locbdd-1}\end{aligned}$$ To see this, consider $$\begin{aligned}
\left|\langle x|Q^s|x\rangle-\frac{{{\sf tr}}(Q^s)}{N}\right|&\leq \frac{1}{\#\mathcal R_Q^\gamma(x,s)}\sum_{y\in \mathcal R_Q^\gamma(x,s)}\left|\langle x|Q^s|x\rangle-\frac{\#\mathcal R_Q^\gamma(x,s)}{N}\langle y|Q^s|y\rangle\right|\\
&\hspace{.5cm}+\frac{N-\#\mathcal R_Q^\gamma(x,s)}{N}\\
&\leq \frac{N-\#\mathcal R_Q^\gamma(x,s)}{N}\left[\langle x|Q^s|x\rangle+1\right]+\gamma\\
&\leq \frac{2[N-\#\mathcal R^\gamma_Q(x,s)]}{N}+\gamma,\end{aligned}$$ which gives our claim (\[eq:locbdd-1\]).\
[**(Step 3).**]{} We claim that $$\begin{aligned}
\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|\leq \frac{4\min_{x\in E}[N-\#\mathcal R_Q^\gamma(x,s)]}{N}+\gamma.\label{delta-bdd-1-1}\end{aligned}$$ For any $x\in E$, $$\begin{aligned}
\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|=&\frac{1}{N}\sum_{y\in E} \left|\langle y|Q^s|y\rangle-\frac{{{\sf tr}}(Q^s)}{N}\right|\\
\leq &\frac{1}{ N}\sum_{y\in \mathcal R_Q^\gamma(x,s)}\left|\langle y|Q^s|y\rangle-\frac{{{\sf tr}}(Q^s)}{N}\right|+\frac{2[N-\#\mathcal R_Q^\gamma(x,s)]}{N}\\
\leq &\frac{\# \mathcal R_Q^\gamma(x,s)}{N}\left(\frac{2[N-\#\mathcal R^Q_\gamma(x,s)]}{N}+\gamma\right)+\frac{2[N-\#\mathcal R_Q^\gamma(x,s)]}{N}\\
\leq &\frac{4[N-\#\mathcal R_Q^\gamma(x,s)]}{N}+\gamma,\end{aligned}$$ where the second inequality follows from the inequality (\[eq:locbdd-1\]). Since the last inequality holds for arbitrary $x\in E$, (\[delta-bdd-1-1\]) follows.\
[**(Step 4).**]{} Finally, we claim that $$\begin{aligned}
\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|\leq \frac{2}{N}{{\sf tr}}\left(|Q|^s;Q<1\right).\label{delta-bdd-1-2}\end{aligned}$$ Let $\{\psi_q;q\in \sigma(Q)\} $ be a basis of $\Bbb C^E$ corresponding to the eigenvalues $q$ of $Q$ ($\sigma(Q)$ denotes the spectrum of $Q$). We may assume that the eigenfunctions $\psi_q$ are orthonormal with respect to the counting measure over $E$. Then we have $$\begin{aligned}
\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|&=\frac{1}{N^2}\sum_{x\in E}\left|\sum_{y\in E}\Big(\langle x|Q^s|x\rangle-\langle y|Q^s|y\rangle\Big)\right|\\
&=\frac{1}{N^2}\sum_{x\in E}\left|\sum_{y\in E}\sum_{q\in \sigma(Q)}\Big(\psi_q(x)^2-\psi_q(y)^2\Big) q^s\right|\\
&\leq \frac{2{{\sf tr}}(|Q|^s;Q<1)}{N},\end{aligned}$$ where the last equality follows since the unique eigenfunction corresponding to $1$ is a constant function. The last inequality gives (\[delta-bdd-1-2\]).
By (\[delta-bdd-1-1\]) and (\[delta-bdd-1-2\]), the second inequality in (\[delta-bdd\]) follows at once. The proof is complete.
\[lem:T3\] For any $\lambda\in (0,\infty)$, $u\in (0,1)$, $m\in \Bbb N$ and $\gamma\in [0,1]$, we have $$\begin{aligned}
\begin{split}\label{diff0}
&\Bigg|\frac{\lambda N^2}{2u(1-u)(N+\lambda)}
\sum_{n=0}^{mN} \left(\frac{N}{N+\lambda}\right)^n\Bigg[\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)\\
&\hspace{1cm}-\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right)\Bigg]\Bigg|\leq \frac{40\Delta^\gamma_Q}{(1-u)}\left(1+\frac{6}{N}\right)^{mN}
\end{split}\end{aligned}$$ where $\Delta_Q^{\gamma}$ is defined in (\[def:deltaQ\]).
We study the summands of the term in (\[T’\]) for $1\leq n\leq mN$ (the summand with $n=0$ is zero by Corollary \[cor:tst\] and (\[def:L0\])). By (\[eq:LLn\]), we have $$\begin{aligned}
&\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)-\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right)
\notag\\
=&\frac{S_{\beta_u}}{N}\left(\sum_{j=0}^nL^j\big(\eta_{n+1-j}\big)-\sum_{j=0}^{n-1}L^j\big(\eta_{n-j}\big)\right)\notag\\
\begin{split}\label{p1L0}
=&\frac{S_{\beta_u}}{N}
\left(\eta_{n+1}+\sum_{j=0}^{n-1}(L-I)L^j\big(\eta_{n-j}\big)\right).
\end{split}\end{aligned}$$ Since ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert S_{\beta_u}
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}=u$, (\[p1L0\]) implies $$\begin{aligned}
&\left|\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)-\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right)\right|\notag\\
\leq &\frac{u}{N}\Big(\|\eta_{n+1}\|+{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert L-I
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}\times\|\eta_n\|+\cdots +{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert L-I
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}\times {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert L
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}^{n-1}\times \|\eta_1\|\Big)\notag\\
\leq &\frac{u}{N}\left(\|\eta_{n+1}\|+\frac{4}{N}\|\eta_n\|+\cdots +\frac{4}{N}\|\eta_1\|\right),
\label{eq:LL0-diff01}\end{aligned}$$ where in the last equality we use ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert L
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}=1$ and ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert L-I
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}\leq 4/N$ by Lemma \[prop:norm\].
To handle the right-hand side of (\[eq:LL0-diff01\]), we consider the $\ell_1$-norms (defined as in (\[eq:matrix-norm\])) of the matrices followed by $\alpha(n-1,s)$ in the expression (\[eq:etan\]) for $\eta_n$. We have $$\begin{aligned}
&\Bigg\|\left[\frac{1}{N}{{\sf diag}}(Q^s)+{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)-2\frac{{{\sf tr}}(Q^s)}{N^2}\right]I\notag\\
&+\left[-\frac{1}{N}{{\sf diag}}(Q^s)Q-Q\frac{1}{N}{{\sf diag}}(Q^s)+\frac{2{{\sf tr}}(Q^s)}{N^2}Q\right]\Bigg\|\notag\\
\leq &\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|+\left\|{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|\notag\\
&+\left\|\left(-\frac{1}{N}{{\sf diag}}(Q^s)+\frac{{{\sf tr}}(Q^s)}{N^2}I\right)Q\right\|\notag\\
&+\left\|Q\left(-\frac{1}{N}{{\sf diag}}(Q^s)+\frac{{{\sf tr}}(Q^s)}{N^2}I\right)\right\|\notag\\
\leq &\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|+\left\|{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|\notag\\
&+\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|+\left\|\frac{1}{N}{{\sf diag}}(Q^s)-\frac{{{\sf tr}}(Q^s)}{N^2}I\right\|,\label{eq:LL0-diff2}\end{aligned}$$ where the next to the last equality follows from the fact that $Q$ is a symmetric probability matrix. Then apply Lemma \[lem:eigenbdd\] to (\[eq:LL0-diff2\]), and we obtain $$\begin{aligned}
&\Bigg\|\left[\frac{1}{N}{{\sf diag}}(Q^s)+{{\sf diag}}\big(Q{{\sf diag}}(Q^s)J\big)-2\frac{{{\sf tr}}(Q^s)}{N^2}\right]I\notag\\
&\hspace{1cm}+\left[-\frac{1}{N}{{\sf diag}}(Q^s)Q-\frac{1}{N}Q{{\sf diag}}(Q^s)+\frac{2{{\sf tr}}(Q^s)}{N^2}Q\right]\Bigg\|
\leq 16\Delta_Q^{\gamma}\end{aligned}$$ for $\Delta_Q^{\gamma}$ defined by (\[def:deltaQ\]). Applying (\[eq:LL0-diff2\]) to the right-hand side of (\[eq:etan\]), we find that for every $n\in \Bbb N$, $$\begin{aligned}
\|\eta_n\|\leq &16\Delta_Q^{\gamma}\times \sum_{s=0}^\infty |\alpha(n-1,s)|
\leq \frac{48\Delta_Q^{\gamma}}{N}\left(1+\frac{6}{N}\right)^{n-1},\end{aligned}$$ where the last equality follows from Lemma \[lem:bdd-0\]. By the foregoing inequality and (\[eq:LL0-diff01\]), we obtain that for all $1\leq n\leq mN$, $$\begin{aligned}
&\left|\left(\frac{1}{N}S_{\beta_u}L^{n+1}(J)-\frac{1}{N}S_{\beta_u}L^n(J)\right)-\left(\frac{1}{N}S_{\beta_u}L_0^{n+1}(J)-\frac{1}{N}S_{\beta_u}L_0^n(J)\right)\right|\notag\\
&\hspace{.5cm}\leq \frac{48 \Delta_Q^{\gamma}u}{N^2}\left\{\left(1+\frac{6}{N}\right)^{n}+\frac{4}{N}\times \left(1+\frac{6}{N}\right)^{n-1}+\cdots+\frac{4}{N}\times\left(1+\frac{6}{N}\right)+\frac{4}{N}
\right\}\\
&\hspace{.5cm}\leq \frac{80\Delta_Q^{\gamma}u}{N^2}\left(1+\frac{6}{N}\right)^n.\end{aligned}$$ The foregoing inequality is enough to obtain the required inequality (\[diff0\]) (recall that the summand with $n=0$ is zero).
#### **Conclusion for the proof of Theorem \[thmm:main-1\].**
The required inequality (\[crit:main0\]) of the theorem follows plainly by applying Lemma \[lem:T1\], Lemma \[lem:T2\] and Lemma \[lem:T3\], if we recall Lemma \[lem:MUV-exp\] (ii) and (iii) and take $u$ to be $1/2$.
Application to large random regular graphs {#sec:cor-main2}
------------------------------------------
We give the proof of Corollary \[cor:main2\] in this section, and fix $k\geq 2$. Assume that the state space on which $Q^{(n)}$ lives is given by $E_n$ and has size $N_n\nearrow \infty$. It follows from a standard result of random regular graphs that for every $s\in \Bbb Z_+$, $$\lim_{n\to\infty}\frac{\min_{x\in E_n}[N_n-\#\mathcal R^0_{Q^{(n)}}(x,s)]}{N_n}=0\quad\mbox{a.s.}$$ (cf. Wormald [@W:MRG Section 2] for the above convergence as well as the present assumption that the random regular graphs are simple and connected). We also know that a.s., the sequence of empirical eigenvalues distributions of $Q^{(n)}$, namely $$\frac{1}{N_n}\#\{r\in (-\infty,q];r\mbox{ is an eigenvalue of }Q^{(n)}\},\quad q\in {{\Bbb R}},$$ converges weakly to the (normalized) Kesten-McKay distribution with density given by (\[KM:SM\]), and hence, (\[crit:main\]) holds. See Kesten [@K:SRG] for the fact that the function in (\[KM:SM\]) is a density of the spectral measure of the random walk kernel on infinite $k$-regular tree, and McKay [@M:EED Theorem 4.3] for the above convergence of empirical eigenvalue distributions.
To use (\[conv:main\]) and find the limit of the ratios of Green functions there, we apply the spectral representation (\[ratio\]) and deduce that $$\begin{aligned}
&\lim_{n\to\infty}\left.{{\mathbb E}}^{(n)}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{U})dt\right]\right/
{{\mathbb E}}^{(n)}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{V\}}(X_t^{V})dt\right]\\
=&\left.{{\mathbb E}}^{(\infty)}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{y\}}(X_t^{x})dt\right]\right/
{{\mathbb E}}^{(\infty)}\left[\int_0^\infty e^{-t\lambda /2}{\mathds 1}_{\{y\}}(X_t^{y})dt\right]\\
=&\,{{\mathbb E}}^{(\infty)}\big[e^{-\lambda H_{x,y}/2}\big],\end{aligned}$$ almost surely with respect to the randomness that the graphs are chosen, where $x$ and $y$ are any adjacent vertices on the infinite $k$-regular tree. This proves Corollary \[cor:main2\].
Meeting times of higher orders {#sec:MT}
==============================
Let $\{U_n;n\in \Bbb Z_+\}$ and $\{V_n;n\in \Bbb Z_+\}$ be two sequences of $Q$-chains with $U_0=V_0$ so that $U_0$ has uniform distribution on $E$, and conditioned on $U_0$, the two chains are independent. We assume in addition that $\{U_n\}$ and $\{V_n\}$ are independent of a system of coalescing $Q$-chains which we may denote it by $\{X^x\}$. Below we consider meeting times of all orders from a slightly more general point of view, using the pairs $(U_m,V_n)$ as starting points.
\[prop:MT\] For all $\ell,m,n\in \Bbb Z_+$, $$\begin{aligned}
\label{idd-M}
M_{V_\ell,V_{\ell+m+n}} \stackrel{(\rm d)}{=} M_{U_m,V_n}\end{aligned}$$ and $$\begin{aligned}
\begin{split}\label{eq:shifttime0}
&\int_0^t 2e^{-2(t-v)}{{\mathbb P}}(M_{V_0,V_{m+n+1}}>v)dv\\
=&{{\mathbb P}}(M_{U_m,V_n}>t)-e^{-2t}\left(1-\frac{{{\sf tr}}(Q^{m+n})}{N}\right)\\
&+\int_0^t 2e^{-2(t-v)}\frac{1}{N}\sum_{x,y\in E}\langle x|Q^{m+n}|x\rangle\langle x|Q|y\rangle{{\mathbb P}}(M_{x,y}>v)dv.
\end{split}\end{aligned}$$
Recall the definition (\[def:RQ\]) of the sets $\mathcal R^\gamma_Q(x,s)$, and note that, for the last integral in (\[eq:shifttime0\]), $$\begin{aligned}
&\left|\frac{1}{N}\sum_{x,y\in E}\langle x|Q^{m+n}|x\rangle \langle x|Q|y\rangle {{\mathbb P}}(M_{x,y}>v)-\frac{{{\sf tr}}(Q^{m+n})}{N}{{\mathbb P}}(M_{U,V}>v)\right|\\
\leq &\left\|\frac{{{\sf diag}}(Q^{m+n})}{N}-\frac{{{\sf tr}}(Q^{m+n})}{N^2}I\right\|\\
\leq &\min\left\{\frac{4\min_{x\in E}[N-\#\mathcal R_Q^\gamma(x,m+n)]}{N}+\gamma,\frac{2{{\sf tr}}(|Q|^{m+n};Q<1)}{N}\right\}\end{aligned}$$ by Lemma \[lem:eigenbdd\], where the bound is independent of $v$.
By the mass-transport equation $$\begin{aligned}
\label{mass-transport}
{{\mathbb E}}[f(U_{n'+1},V_{n'})]={{\mathbb E}}[f(U_{n'},V_{n'+1})],\quad \forall\; f:E\times E{\longrightarrow}{{\Bbb R}},\;n'\in \Bbb Z_+,\end{aligned}$$ which follows from the reversibility of the $Q$-chains $\{U_{n'}\}$ and $\{V_{n'}\}$, it holds that $M_{U_m,V_n}$ has the same distribution as $M_{U_0,V_{m+n}}=M_{V_0,V_{m+n}}$. By this fact and the stationarity of the chain $\{V_{n'}\}$, (\[idd-M\]) follows.
To obtain (\[eq:shifttime0\]), we condition $(X^{U_n},X^{V_m})$ at the first epoch time and obtain $$\begin{aligned}
{{\mathbb P}}(M_{U_n,V_m}>t)=&e^{-2t}{{\mathbb P}}(U_n\neq V_m)\notag\\
&+\int_0^t 2e^{-2(t-v)}{{\mathbb P}}(U_n\neq V_{m},M_{U_n,V_{m+1}}>v)dv.\label{eq:st1}\end{aligned}$$ Here on the right-hand side of (\[eq:st1\]), the first term and the distribution ${{\mathbb P}}(U_n\neq V_m,U_n=x,V_{m+1}=y)$ can be determined respectively as ${{\mathbb P}}(U_n=V_m)={{\sf tr}}(Q^{m+n})/N$ and $$\begin{aligned}
&{{\mathbb P}}(U_n\neq V_m,U_n=x,V_{m+1}=y)\\
&\hspace{2.1cm}={{\mathbb P}}(U_n=x,V_{m+1}=y)-{{\mathbb P}}(U_n=V_m,U_n=x,V_{m+1}=y)\\
&\hspace{2.1cm}=\frac{1}{N}\langle x|Q^{n+m+1}|y\rangle-\frac{1}{N}\langle x|Q^{n+m}|x\rangle \langle x|Q|y\rangle.\end{aligned}$$ The proof is complete.
\[cor:Mtime1\] If, for some $n\in \Bbb Z_+$, the return probabilities $\langle x|Q^n|x\rangle $ do not depend on $x$, then $$\begin{aligned}
\begin{split}
&\int_0^t 2e^{-2(t-v)}{{\mathbb P}}(M_{V_0,V_{n+1}}>v)dv\\
=&{{\mathbb P}}(M_{V_0,V_n}>t)-e^{-2t}\left(1-\frac{{{\sf tr}}(Q^{n})}{N}\right)
+\frac{{{\sf tr}}(Q^{n})}{N}\int_0^t 2e^{-2(t-v)}{{\mathbb P}}(M_{U,V}>v)dv.
\end{split}\end{aligned}$$
Hence, in the case that $Q$ is walk regular, the distributions of $M_{U,V_n}$ for all $n\in \Bbb N$ can be completely determined by $M_{U,V}$ by iteration and differential calculus.
[9]{} \[A:EHT\] [Aldous, D. J.]{} (1982). Markov chains with almost exponential hitting times. *Stochastic Process. Appl.* [**13**]{} (1982) 305–310. \[A:MTMC\] [Aldous, D. J.]{} (1991). Meeting times for independent Markov chains. *Stochastic Process. Appl.* [**38**]{} 185–193. \[AF:RMC\] [Aldous, D. J.]{} and [Fill, J. A.]{} (2002). *Reversible Markov Chains and Random Walks on Graphs*. Unfinished monograph, recompiled 2014, available at [http://www.stat.berkeley.edu/\$\\sim\$aldous/RWG/book.html](http://www.stat.berkeley.edu/$\sim$aldous/RWG/book.html). \[AL:AP\] [Aldous, D. J.]{} and [Lanoue, D.]{} (2012). A lecture on the averaging process. *Probab. Surv.* [**9**]{} 90–102. \[AN:GG\] [Allen, B.]{} and [Nowak, M. A.]{} (2014). Games on graphs. *EMS Surv. Math. Sci.* [**1**]{} 113–151. \[ATTN\] [Allen, B.]{}, [Traulsen, A.]{}, [Tarnita, C. E.]{} and [Nowak, M. A.]{} (2012). How mutation affects evolutionary games on graphs. *J. Theoret. Biol.* [**299**]{} 97–105. \[B:AGT\] [Biggs, N.]{} (1993). *Algebraic Graph Theory*, 2nd ed. Cambridge University Press, Cambridge. \[C:BC\] [Chen, Y.-T.]{} (2013). Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs. *Ann. Appl. Probab.* [**23**]{} 637–664. \[CCC:WF\] [Chen, Y.-T.]{}, [Choi, C.]{} and [Cox, J. T.]{} (2013). On the convergence of densities of finite voter models to the Wright-Fisher diffusion. To appear in *Ann. Inst. Henri Poincaré Probab. Stat.*, available at <http://arxiv.org/abs/1311.5786>. \[C:CRW\] [Cox, J. T.]{} (1989). Coalescing random walks and voter model consensus times on the torus in $\Bbb Z^d$. *Ann. Probab.* [**17**]{} 1333–1366. \[C:SSM\] [Cox, J. T.]{} (2010). Intermediate range migration in the two-dimensional stepping stone model. *Ann. Appl. Probab.* [**20**]{} 785–805. \[CD:SSM\] [Cox, J. T.]{} and [Durrett, R.]{} (2002). The stepping stone model: new formulas expose old myths. *Ann. Appl. Probab.* [**12**]{} 1348–1377. \[CDP:VMP\] [Cox, J. T.]{}, [Durrett, R.]{} and [Perkins, E. A.]{} (2013). Voter model perturbations and reaction diffusion equations. *Astérisque* [**349**]{}. \[D:Epi\] [Durrett, R.]{} (2010). Some features of the spread of epidemics and information on a random graph. *Proc. Natl. Acad. Sci. U. S. A.* [**107**]{} 4491–4498. \[FF:RAP\] [Ferrari, P. A.]{} and [Fontes, L. R. G.]{} (1998). Fluctuations of a surface submitted to a random average process. *Electron. J. Probab.* [**3**]{} 1–34. and [Yoshida, N.]{} (2012). On exponential growth for a certain class of linear systems. *ALEA Lat. Am. J. Probab. Math. Stat.* [**9**]{} 323–336. \[GM:WRG\][Godsil, C. D. and McKay, B. D.]{} (1980). Feasibility conditions for the existence of walk-regular graphs. *Linear Algebra Appl.* [**30**]{} 51–61. \[H:SCV\] [Hörmander, L.]{} (1990). *An Introduction to Complex Analysis in Several Variables*, 3rd ed. North-Holland Publishing Co., Amsterdam. \[K:AE\] [Keilson, J.]{} (1979). *Markov Chain Models–Rarity and Exponentiality*. Springer-Verlag, New York-Berlin. \[K:SRG\] [Kesten, H.]{} (1959). Symmetric random walks on groups. *Trans. Amer. Math. Soc.* [**92**]{} 336–354. \[K:SSM\] [Kimura, M.]{} (1953). “Stepping-stone” model of population. *Annu. Rept. Natl. Inst. Genet. Jpn.* [**3**]{} 62–63. \[L:IPS\] [Liggett, T. M.]{} (2005). *Interacting Particle Systems*. Springer, Berlin. \[M:EED\] [McKay, B. D.]{} (1981). The expected eigenvalue distribution of a large regular graph. *Linear Algebra Appl.* [**40**]{} 203–216. \[N:ED\] [Nowak, M. A.]{}(2006). *Evolutionary Dynamics. Exploring the Equations of Life.* The Belknap Press of Harvard University Press, Cambridge, MA. \[NTW:EU\] [Nowak, M. A., Tarnita, C. E. and Wilson, E. O.]{} (2010). The evolution of eusociality. *Nature* [**466**]{} 1057-1062. \[OHLN\] [Ohtsuki, H.]{}, [Hauert, C.]{}, [Lieberman, E.]{} and [Nowak, M. A.]{} (2006). A simple rule for the evolution of cooperation on graphs and social networks. *Nature* [**441**]{} 502–505. \[O:MFC\] [Oliveira, R. I.]{} (2013). Mean field conditions for coalescing random walks. *Ann. Probab.* [**41**]{} 3420–3461. \[P:QSC\] [Parathasarathy, K. R.]{} (1992). *An Introduction to Quantum Stochastic calculus*. Birkhäuser/Springer Basel AG, Basel. \[P:ZPS\] [Peyerimhoff, A.]{} (1966). On the zeros of power series. *Michigan Math. J.* [**13**]{} 193–214. \[S:SSM\] [Sawyer, S.]{} (1976). Results for the stepping stone model for migration in population genetics. *Ann. Probab.* [**4**]{} 699–728. and [Fáth, G.]{} (2007). Evolutionary games on graphs. *Phys. Rep.* [**446**]{} 97–216. \[TDW\] [Taylor, P. D.]{}, [Day, T.]{} and [Wild, G.]{} (2007). Evolution of cooperation in a finite homogeneous graphs. *Nature* [**447**]{} 469–472. \[W:MRG\] [Wormald, N. C.]{} (1999). Models of random regular graphs. In *Surveys in Combinatorics*, 1999. London Math. Soc. Lecture Note Ser. [**276**]{} 239-298. \[CDZ:SSM2\] [Zähle, I.]{}, [Cox, J. T.]{} and [Durrett, R.]{} (2005). The stepping stone model. II. Genealogies and the infinite sites model. *Ann. Appl. Probab.* [**15**]{} 671–699.
[^1]: Center of Mathematical Sciences and Applications, Program for Evolutionary Dynamics, Harvard University
|
---
abstract: |
Let $\mathbf{T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property $$\textrm{dim} \; \textrm{ker} \; (\mathbf{T}-\boldsymbol\lambda) \ge \textrm{dim} \; \textrm{ker} \; (\mathbf{T} - {\boldsymbol\lambda})^*),$$ for every $\boldsymbol\lambda$ in the Taylor spectrum $\sigma(\mathbf{T})$ of $\mathbf{T}$. We prove that the Weyl spectrum of $\mathbf{T}$, $\omega(\mathbf{T})$, satisfies the identity $$\omega(\mathbf{T})=\sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T}),$$ where $\pi_{00}(\mathbf{T})$ denotes the set of isolated eigenvalues of finite multiplicity.
Our method of proof relies on a (strictly $2$-variable) fact about the topological boundary of the Taylor spectrum; as a result, our proof does not hold for $d$-tuples of commuting hyponormal operators with $d>2$.
address:
- |
Indian Institute of Technology Kanpur\
Kanpur- 208016, India
- |
University of Iowa\
Iowa City, Iowa 52242, USA
author:
- 'Sameer Chavan and Ra$\acute{\mbox{u}}$l Curto'
title: 'Weyl’s Theorem for Pairs of Commuting Hyponormal Operators'
---
[^1]
Weyl’s Theorem in Two Variables
===============================
The aim of this note is to present an analog of Weyl’s Theorem for commuting pairs of hyponormal operators. For the definitions and basic theory of various spectra including the Taylor and Harte spectra, the reader is referred to [@Cu] (see also [@Tay1]). For a commuting $d$-tuple $\mathbf{T}$, we reserve the symbols $\sigma(\mathbf{T})$, $\sigma_H(\mathbf{T})$, $\sigma_p(\mathbf{T})$ and $\sigma_e(\mathbf{T})$ for the Taylor spectrum, Harte spectrum, point spectrum and Taylor essential spectrum of $\mathbf{T}$, respectively. By a [*commuting $d$-tuple*]{} $\mathbf{T}$, we understand here and throughout this note a $d$-tuple of commuting bounded linear operators $T_1, \cdots, T_d$ on a complex, separable Hilbert space $\mathcal H$. For $d=1$, L. Coburn proved in [@C] Weyl’s Theorem for hyponormal operators; this led to a number of extensions to classes of operators containing the subnormal operators. For $d>1$, there are various notions of Weyl spectrum ([@Cho-1], [@H], [@P], [@L]). We recall in particular two notions of Weyl spectrum with which we will be primarily concerned. The [*joint Weyl spectrum*]{} $\omega(\mathbf{T})$ of a commuting $d$-tuple $\mathbf{T}$ is defined as $$\omega(\mathbf{T}):=\cap \{\sigma(\mathbf{T}+\mathbf{K}) : \mathbf{K} \in \mathcal K^{(d)}(\mathcal H)~\mbox{~such~that~} \mathbf{T}+\mathbf{K}~\mbox{~is~commuting}\},$$ where $\mathcal \mathbf{K}^{(d)}(\mathcal H)$ denotes the collection of $d$-tuples of compact operators on $\mathcal H$. The [*Taylor Weyl spectrum*]{} of $\mathbf{T}$ is defined as $$\sigma_W(\mathbf{T}):=\sigma_e(\mathbf{T}) \cup \{\boldsymbol\lambda \in \sigma(\mathbf{T}) \setminus \sigma_e(\mathbf{T}) : \mbox{ind}(\mathbf{T}-\boldsymbol\lambda) \neq 0\},$$ where $\boldsymbol\lambda:=(\lambda_1,\cdots,\lambda_d)$ and the [*Fredholm index*]{} $\mbox{ind}(\mathbf{S})$ of a $d$-tuple $\mathbf{S}$ of commuting operators is the Euler characteristic of the Koszul complex $K(\mathbf{S}, \mathcal{H})$ for $\mathbf{S}$, given by \[index\] ():= \_[k=0]{}\^d (-1)\^k H\^k(), with $H^k(\mathbf{S})$ denoting the $k$-th cohomology group in $K(\mathbf{S},\mathcal{H})$; observe that $H^0(\mathbf{S}) \cong \ker \; Q_{\mathbf{S}}(I)$ and $H^d(\mathbf{S}) \cong \ker \; Q_{\mathbf{S^*}}(I)$. By [*Weyl spectrum*]{} we understand [*any*]{} of the joint Weyl and Taylor Weyl spectra.
For future reference, we record the following elementary fact from [@H-K] about the relationship between the aforementioned two notions of Weyl spectra. For the sake of completeness, we provide an alternative verification of this result.
([@H-K Lemma 2]) \[inclu\] The joint Weyl spectrum and Taylor Weyl spectrum of a commuting $d$-tuple $\mathbf{T}$ satisfies the relation $\sigma_W(\mathbf{T}) \subseteq \omega(\mathbf{T})$.
Suppose that there exists a $d$-tuple $\mathbf{K}$ of compact operators such that $\mathbf{T}+\mathbf{K}$ is a commuting $d$-tuple and $\boldsymbol\lambda \notin \sigma(\mathbf{T}+\mathbf{K})$. Thus, $\mathbf{T}-\boldsymbol\lambda + \mathbf{K}$ is invertible, and hence Fredholm with Fredholm index equal to $0$. By the multivariable Atkinson Theorem [@Cu-0 Theorem 2], $\mathbf{T}-\boldsymbol\lambda$ is Fredholm with index equal to $0$, that is, $\boldsymbol\lambda \notin \sigma_W(\mathbf{T}).$
It is interesting to note that $\sigma_W(\mathbf{T}) = \omega(\mathbf{T})$ for all $\mathbf{T}$ if and only if $d=1$. This may be concluded from the discussion following [@R-2 Theorem], where it is shown that certain Fredholm $d$-tuples of index equal to $0$ cannot be perturbed by compact tuples to a Taylor invertible tuple (see also [@H-K]).
Before we present an analog of Weyl’s Theorem [@C Theorem 3.1] for commuting hyponormal tuples, recall that a bounded, linear operator $S$ on a Hilbert space $\mathcal H$ is [*hyponormal*]{} if its self-commutator $S^*S - SS^*$ is positive. Also, given a $d$-tuple $\mathbf{S} \equiv (S_1,\cdots,S_d)$ we let $$Q_{\mathbf{S}}(X):=\sum_{i=1}^d S^*_iXS_i \; \; (\textrm{for } X \textrm{ a bounded operator on } \mathcal{H}).$$
(cf. [@H-K]). A $d$-tuple $\mathbf{T}$ has the [*quasitriangular property*]{} if $\mathbf{T}$ satisfies \[qt\] Q\_[\_]{}(I) Q\_[\^\*\_]{}(I), for every $\boldsymbol\lambda \in \sigma(\mathbf{T})$, where $\mathbf{T}_{\boldsymbol\lambda}:=\mathbf{T}-\boldsymbol\lambda$.
We are now ready to state our main result.
\[Weyl\] Let $\mathbf{T}$ be a commuting $d$-tuple of hyponormal operators on a Hilbert space $\mathcal H$ and let $\pi_{00}(\mathbf{T})$ denote the set of isolated eigenvalues of $\mathbf{T}$ of finite multiplicity. The following statements are true. (i) $\omega(\mathbf{T}) \subseteq \sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T})$. (ii) Assume $d=2$ and that $\mathbf{T}$ satisfies (\[qt\]). Then $\sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T}) \subseteq \sigma_W(\mathbf{T})$.
Condition (\[qt\]) is equivalent to the statement that the dimension of the cohomology group at the first stage of the Koszul complex for $\mathbf{T}-\boldsymbol\lambda$ is greater than or equal to the dimension of the cohomology group at the last stage of the Koszul complex for $\mathbf{T}-\boldsymbol\lambda$. This property is closely related to the notion of quasitriangular operator (see the discussion following [@H-K Definition 3]). Finally, note that is satisfied by any $d$-tuple $\mathbf{T}$ such that $\sigma_p(T_i^*)=\emptyset$ for some $i=1,\cdots,d$.
The following is immediate from Theorem \[Weyl\] and Lemma \[inclu\].
\[Weyl-0\] Let $\mathbf{T}$ be a commuting pair of hyponormal operators on a Hilbert space $\mathcal H$ and let $\pi_{00}(\mathbf{T})$ denotes the set of isolated eigenvalues of $\mathbf{T}$ of finite multiplicity. If $\mathbf{T}$ satisfies the quasitriangular property then $$\label{Cor15}
\omega(\mathbf{T}) = \sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T}) = \sigma_W(\mathbf{T}).$$
By Theorem \[Weyl\](i), $$\omega(\mathbf{T}) \subseteq \sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T}),$$ and by Theorem \[Weyl\](ii), $$\sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T}) \subseteq \sigma_W(\mathbf{T}).$$ Since $\sigma_W(\mathbf{T}) \subseteq \omega(\mathbf{T})$ is always true (by Lemma \[inclu\]), (\[Cor15\]) follows.
(i) One cannot relax the condition . Indeed, let $\mathbf{T}=(U_+, 0)$, where $U_+$ denotes the unilateral shift on $\ell^2(\mathbb Z_+)$ (see the discussion following [@H-K Definition 5]). Indeed, for this commuting pair, $\sigma(\mathbf{T})=\bar{\mathbb D} \times {0}$, $\sigma_e(\mathbf{T})=\mathbb{T} \times {0}$, $\pi_{00}(\mathbf{T})=\emptyset$, $\omega(\mathbf{T})=\sigma(\mathbf{T})$ and $\sigma_W(\mathbf{T})=\sigma_e(\mathbf{T})$. (ii) Note further that the result above is not best possible. Indeed, it may happen that the conclusion of Weyl’s Theorem holds but is violated. For instance, let $\mathbf{T}$ be the Drury-Arveson $2$-variable weighted shift; then $T_1$ and $T_2$ are hyponormal operators such that $\pi_{00}(\mathbf{T})=\emptyset$ and $\omega(\mathbf{T}) = \sigma(\mathbf{T}) = \sigma_W(\mathbf{T})=\overline{\mathbb B}$ [@Ar3]. However, since $I - Q_\mathbf{T}(I) \leq 0$ and $I-Q_{\mathbf{T}^*}(I)$ is the orthogonal projection onto the scalars, $$0=\dim \ker Q_{\mathbf{T}}(I) \ngeq \dim \ker Q_{\mathbf{T}^*_{\boldsymbol\lambda}}(I)=1 \; \; \textrm{ for every } \boldsymbol\lambda \in \sigma(\mathbf{T}). \qed$$
The first part of Theorem \[Weyl\] generalizes [@Cho-1 Theorem 4], while Corollary \[Weyl-0\] generalizes [@H-K Theorem 6] when $d=2$ (see also [@Le Theorem 2.5.4]). All these were obtained under the additional assumption that $\mathbf{T}$ is [*doubly commuting*]{}, that is, a commuting $d$-tuple $\mathbf{T}$ such that $$T^*_iT_j=T_jT^*_i~\mbox{for~} 1 \leq i \neq j \leq d.$$ Although the conclusion of [@H-K Theorem 6] is stronger than that of Theorem \[Weyl\], our result does not assume double commutativity. On the other hand, our method of proof relies on a (consequence of a) strictly $2$-dimensional result about the topological boundary of the Taylor spectrum [@Cu Theorem 6.8], and hence does not extend to the case $d \geq 3.$
Proof of Theorem \[Weyl\]
=========================
The proof of Theorem \[Weyl\] presented below relies on a number of non-trivial results, including the Shilov Idempotent Theorem ([@Cu], [@Tay2]). We also need a result from [@Cu-1] pertaining to connections between Harte and Taylor spectra. We begin with a decomposition of tuples of commuting hyponormal operators; we believe this result is known, although we have not been able to find a concrete reference in the literature. In what follows, recall that $\boldsymbol\lambda := (\lambda_1, \cdots, \lambda_d)$.
\[Lem\] Let $\mathbf{T}$ be a $d$-tuple of commuting hyponormal operators $T_1, \cdots, T_d$. Then for any $\boldsymbol\lambda \equiv (\lambda_1,
\cdots, \lambda_d) \in \sigma(\mathbf{T}),$ $\mathcal M_1:=\ker
Q_{\mathbf{T}_{\boldsymbol\lambda}}(I)$ is a reducing subspace for $\mathbf{T}$, where $\mathbf{T}_{\boldsymbol\lambda}$ denotes the $d$-tuple $\mathbf{T} - \boldsymbol\lambda I=(T_1-\lambda_1 I,
\cdots, T_d - \lambda_d I)$, and $Q_{\mathbf{T}_{\boldsymbol\lambda}}(I):=\sum_{i=1}^d
(T_i-\lambda_i I)^*(T_i-\lambda_i I)$. Moreover, $\mathbf{T}$ decomposes into $(\lambda_1 I_{\mathcal M_1},\cdots, \newline \lambda_d I_{\mathcal M_1}) \oplus \mathbf{B}$ on the orthogonal direct sum $\mathcal H = \mathcal M_1 \oplus (\mathcal M_1)^{\perp},$ where $I_{\mathcal M_1}$ is the identity operator on $\mathcal M_1,$ and $\mathbf{B}$ is a $d$-tuple of commuting hyponormal operators such that $\ker
Q_{\mathbf{B}_{\boldsymbol\lambda}}(I_{\mathcal M^{\perp}_1})=\{0\}$.
Note that $\mathcal M_1=\ker Q_{\mathbf{T}_{\boldsymbol\lambda}}(I) = \cap_{i=1}^d \ker (T_i - \lambda_i I).$ Clearly, $T_i (\mathcal M_1) \subseteq \mathcal M_1$ for any $i=1, \cdots, d$. Since $T_i$ is hyponormal, $T_ix=\lambda_i x$ implies $T^*_ix = \bar{\lambda}_ix,$ and hence $T^*_i (\mathcal M_1)
\subseteq \mathcal M_1$ for any $i=1, \cdots, d$. Since $\mathbf{T}$ is an extension of $\mathbf{B}$, it follows immediately that $\mathbf{B}$ has the desired properties.
To state the next result, we recall that a pair $\mathbf{S}$ of commuting operators is said to be semi-Fredholm if all boundary maps in the Koszul complex $K(\mathbf{S},\mathcal{H})$ have closed range, and either $H^0(\mathbf{S})$ and $H^2(\mathbf{S})$ are finite dimensional or $H^1(\mathbf{S})$ is finite dimensional.
\[Cor610\] (cf. [@Cu-1 Corollary 3.6] and [@Cu Corollary 6.10]) Let $\mathbf{T}$ be a pair of commuting operators, and let $\boldsymbol\lambda$ be an isolated point of $\sigma_H(\mathbf{T})$. Assume that $\boldsymbol\lambda$ is in the semi-Fredholm domain of $\mathbf{T}$. Then $\boldsymbol\lambda$ is an isolated point of $\sigma(\mathbf{T})$ if and only if $\mbox{ind} (\mathbf{T}-\boldsymbol\lambda) = 0$.
\(i) To see the inclusion $ \omega(\mathbf{T}) \subseteq \sigma(\mathbf{T}) \setminus
\pi_{00}(\mathbf{T})$, it suffices to check that any isolated eigenvalue of $\mathbf{T}$ of finite multiplicity does not belong to the Taylor spectrum of some finite rank perturbation of $\mathbf{T}$. To see that, let $\boldsymbol\lambda=(\lambda_1, \cdots, \lambda_d)$ be an isolated point of the Taylor spectrum of $\mathbf{T}=(T_1, \cdots, T_d).$ Let $K_1:=\{\boldsymbol\lambda\}$ and let $K_2:=\sigma(\mathbf{T}) \setminus K_1$. By the Shilov Idempotent Theorem [@Cu Application 5.24], there exist invariant subspaces $\mathcal M_1, \mathcal M_2$ of $\mathbf{T}$ such that $\mathcal H = \mathcal M_1 \dotplus \mathcal M_2$ (Banach direct sum) and $\sigma(\mathbf{T}|_{M_i}) =K_i$ for $i=1, 2$. By the Spectral Mapping Property for the Taylor spectrum [@Cu Corollary 3.5], $\sigma(\mathbf{T}|_{\mathcal{M}_1} - \boldsymbol\lambda
I_{\mathcal M_1})=\{0\}$, where $I_{\mathcal M_1}$ is the identity operator on $\mathcal M_1$. Since $\mathbf{T}|_{\mathcal{M}_1} - \boldsymbol\lambda I_{\mathcal
M_1}$ is a commuting $d$-tuple of hyponormal operators, it follows from the Projection Property for the Taylor spectrum [@Cu Theorem 4.9] that $T_{i}|_{\mathcal{M}_1} = \lambda_i I_{\mathcal M_1}$ for $i=1, \cdots, d$. In particular, $\mathcal M_1 \subseteq \cap_{i=1}^d \ker (T_i -
\lambda_i I_{\mathcal M_1})$. Since $\boldsymbol\lambda \notin K_2,$ we must have $\mathcal M_1=\cap_{i=1}^d \ker (T_i - \lambda_i I_{\mathcal
M_1})$. By the preceding lemma, $\mathbf{T}$ decomposes into $\boldsymbol\lambda I \oplus \mathbf{B}$ on the orthogonal direct sum $\mathcal H = \mathcal M_1 \oplus \mathcal M^{\perp}_1,$ where $\mathbf{B}$ is a commuting $d$-tuple of hyponormal operators obtained by restricting $\mathbf{T}$ to $\mathcal M^{\perp}_1$.
Suppose now that $\boldsymbol \lambda \in \sigma(\mathbf{B})$. Since $\sigma(\mathbf{T}) =
\{\boldsymbol\lambda\} \cup \sigma(\mathbf{B})$ [@Cu Page 39], $\boldsymbol\lambda$ is an isolated point of $\sigma(\mathbf{B})$. Since $\mathbf{B}=(B_1, \cdots, B_d)$ is a $d$-tuple of hyponormal operators, by the argument of the preceding paragraph, isolated points of $\sigma(\mathbf{B})$ must be eigenvalues of $\mathbf{B}$, and hence there exists $0 \ne y \in \mathcal{M}_1^{\perp}$ such that $y \in \cap_{i=1}^d \ker (B_i - \lambda_i I_{\mathcal M_1^{\perp}})$. It follows that $y \in \cap_{i=1}^d \ker (T_i - \lambda_i I_{\mathcal{H}})=\mathcal M_1$, which is a contradiction. Thus, $\boldsymbol\lambda \notin \sigma(\mathbf{B})$.
We have proved that if $\boldsymbol\lambda$ is an isolated eigenvalue of $\mathbf{T}$ having finite multiplicity then $\mathbf{B}=\mathbf{T} - \boldsymbol\lambda
I_{\mathcal M_1}$ is a finite-rank perturbation of $\mathbf{T}$, and $\boldsymbol\lambda \notin \sigma(\mathbf{B})$, as desired.
\(ii) Assume now that $d=2$ and that $\mathbf{T}$ has the quasitriangular property . Since each $T_i$ is hyponormal, \[qtp\] Q\_[\_]{}(I) = Q\_[\^\*\_]{}(I) ().Let $\boldsymbol\lambda \in \sigma(\mathbf{T})$ be such that $\mathbf{T}-\boldsymbol\lambda$ is Fredholm with Fredholm index equal to $0$. By the definition of the Fredholm index (see ) and , \[dim-cohom\] 2 Q\_[\_]{}(I) = H\^1(\_) = 2Q\_[\^\*\_]{}(I), where $H^1(\mathbf{S})$ is the middle cohomology group appearing in the Koszul complex of the commuting pair $\mathbf{S}$. Since $\boldsymbol\lambda \in \sigma(\mathbf{T}) \setminus
\sigma_e(\mathbf{T})$, by we must necessarily have $0 < \dim \ker
Q_{\mathbf{T}_{\boldsymbol\lambda}}(I) < \infty,$ and hence $\boldsymbol\lambda$ is an eigenvalue of $\mathbf{T}$ of finite multiplicity.
To see that $\boldsymbol\lambda$ is an isolated point of $\sigma(\mathbf{T})$, in view of Lemma \[Cor610\], it suffices to check that $\boldsymbol\lambda$ is an isolated point of the Harte spectrum $\sigma_H(\mathbf{T})$ of $\mathbf{T}$. If $\mathbf{B}$ is as in Lemma \[Lem\], then $\mathbf{B}-\boldsymbol\lambda I_{\mathcal M^{\perp}_1}$ is also Fredholm with index equal to $0$. Since $\ker Q_{\mathbf{B}_{\boldsymbol\lambda}}(I_{\mathcal M^{\perp}_1})
=\{0\}$ (by Lemma \[Lem\]), using we must have $\ker
Q_{\mathbf{B}^*_{{\boldsymbol\lambda}}}(I_{\mathcal M^{\perp}_1}) =\{0\}$.
On the other hand, $Q_{\mathbf{B}_{\boldsymbol\lambda}}(I_{\mathcal
M^{\perp}_1})$ and $Q_{\mathbf{B}^*_{\boldsymbol\lambda}}(I_{\mathcal M^{\perp}_1})$ are Fredholm (by item (v) in the paragraph immediately following [@Cu Remark 6.7]). As a result, $Q_{\mathbf{B}_{\boldsymbol\lambda}}(I_{\mathcal M^{\perp}_1})$ and $Q_{\mathbf{B}^*_{\boldsymbol\lambda}}(I_{\mathcal M^{\perp}_1})$ are invertible. It follows that $\boldsymbol\lambda$ cannot be in the Harte spectrum $\sigma_H(\mathbf{B})$. Since $\boldsymbol\lambda \in \sigma_H(\mathbf{T}) = \{\boldsymbol\lambda\} \cup
\sigma_H(\mathbf{B})$, $\boldsymbol\lambda$ must be an isolated point of $\sigma_H(\mathbf{T})$, as desired.
\[rem1\] Assume that the commuting pair $\mathbf{T}$ satisfies . If $\mathbf{T}$ has no normal direct summand, then by Lemma \[Lem\], $\mathbf{T}$ has no eigenvalues, and hence the Weyl spectrum of $\mathbf{T}$ coincides with the Taylor spectrum.
Some Consequences of Theorem \[Weyl\]
=====================================
In this section we discuss a couple of interesting consequences of Theorem \[Weyl\] (cf. [@C Corollary 3.2]). First, we recall the notion of jointly hyponormality for $d$-tuples. A $d$-tuple $\mathbf{T}=(T_1, \cdots, T_d)$ of bounded linear operators on ${\mathcal H}$ is said to be [*jointly hyponormal*]{} if the $d
\times d$ matrix $([T^*_j, T_i])_{1 \leq i, j \leq d}$ is positive definite, where $[A, B]$ stands for the commutator $AB-BA$ of $A$ and $B$.
Let $\mathbf{T}=(T_1, T_2)$ be a jointly hyponormal commuting pair with the quasitriangular property . If $\mathbf{T}$ has no isolated eigenvalues of finite multiplicity, then for any pair $\mathbf{K}=(K_1, K_2)$ of compact operators $K_1, K_2$ such that $\mathbf{T}+\mathbf{K}$ is commuting, we have T\^\*\_1T\_1 + T\^\*\_2T\_2 (T\_1+K\_1)\^\*(T\_1 +K\_1) + (T\_2+K\_2)\^\*(T\_2 +K\_2).
Since $\mathbf{T}$ has no isolated eigenvalues of finite multiplicity, Theorem \[Weyl\] implies that $$\sigma(\mathbf{T}) \subseteq \sigma(\mathbf{T}+\mathbf{K})$$ for any pair $\mathbf{K}$ of compact operators such that $\mathbf{T}+\mathbf{K}$ is commuting. Now apply [@CS Lemma 3.10] and [@Ch-4 Lemma 2.1] to conclude that T\^\*\_1T\_1 + T\^\*\_2T\_2 &=& r()\^2 r(+)\^2\
&& (T\_1+K\_1)\^\*(T\_1 +K\_1) + (T\_2+K\_2)\^\*(T\_2 +K\_2), where $$r(\mathbf{S}):=\sup \big\{\sqrt{|z_1|^2 + \cdots + |z_d|^2} : (z_1,
\cdots, z_d) \in \sigma(S)\big\}$$ denotes the geometric spectral radius of the $d$-tuple $\mathbf{S}$ of bounded linear operators on $\mathcal H$.
We do not know whether the conclusion of the last corollary holds for commuting pairs of hyponormal operators satisfying the quasitriangular property. (Recall that there exist commuting pairs of subnormal operators which are not jointly hyponormal; the Drury-Arveson $2$-variable weighted shift is such an example.)
Next, we obtain an analog of the Riesz-Schauder Theorem for commuting pairs of hyponormal operators (cf. [@Le Corollary 2.5.6]). Recall that a commuting $d$-tuple is [*Browder invertible*]{} if $\mathbf{T}$ is Fredholm such that there exists a deleted neighborhood of $\mathbf{0}$ disjoint from the Taylor spectrum of $\mathbf{T}$. The [*Browder spectrum*]{} $\sigma_b(\mathbf{T})$ of $\mathbf{T}$ is the collection of those $\boldsymbol\lambda \in \mathbb C^d$ for which $\mathbf{T}-\boldsymbol\lambda$ is not Browder invertible. It is not hard to see that $\sigma_b(\mathbf{T})$ is the union of $\sigma_e(\mathbf{T})$ and the accumulation points of the Taylor spectrum $\sigma(\mathbf{T})$ (cf. [@CuD]). (For basic facts about the Browder spectrum in the case $d=1$, the reader is referred to [@BDW].)
Let $\mathbf{T}=(T_1, T_2)$ be a jointly hyponormal $2$-tuple with the quasitriangular property . Then $\sigma_b(\mathbf{T})=\omega(\mathbf{T}).$
The inclusion $\omega(\mathbf{T}) \subseteq \sigma_b(\mathbf{T})$ is always true [@Le Lemma 2.5.3]. To see the reverse inclusion, let $\boldsymbol\lambda \notin \omega(\mathbf{T})$. By Lemma \[inclu\], $\boldsymbol\lambda \notin \sigma_W(\mathbf{T})$. In particular, $\boldsymbol\lambda \notin \sigma_e(\mathbf{T}).$ By Theorem \[Weyl\](ii), if $\boldsymbol\lambda \notin \sigma_W(\mathbf{T})$ then $\boldsymbol\lambda$ is an isolated point of $\sigma(\mathbf{T})$. Since $\sigma_b(\mathbf{T})$ is the union of $\sigma_e(\mathbf{T})$ and the accumulation points of $\sigma(\mathbf{T})$, it follows that $\boldsymbol\lambda \notin \sigma_b(\mathbf{T})$. The desired inclusion is now clear.
[HD]{}
, [*Subalgebras of $C\sp *$-algebras. III. Multivariable operator theory*]{}, [Acta Math.]{} [**181**]{} (1998), 159-228.
J. Buoni, A. Dash and B. Wadhwa, [*Joint Browder spectrum*]{}, Pacific J. Math. [**94**]{} (1981), 259-263.
S. Chavan, [*Irreducible Tuples without Boundary Property*]{}, Canadian Mathematical Bulletin, [**58**]{} (2015), 9-18.
S. Chavan and V. M. Sholapurkar, [*Completely hyperexpansive tuples of finite order*]{}, preprint 2016.
M. Ch$\bar{\mbox{o}}$, [*On the joint Weyl spectrum III*]{}, Acta Sci. Math. (Szeged) [**56**]{} (1992), 365-368.
L. Coburn, [*Weyl’s theorem for nonnormal operators*]{}, Michigan Math. J. [**13**]{} (1966), 285-288.
R. Curto, [*Fredholm and invertible n-tuples of operators. The deformation problem*]{}, Trans. Amer. Math. Soc. [**266**]{} (1981), 129-159.
R. Curto, [*Connections between Harte and Taylor spectra*]{}, Rev. Roumaine Math. Pures Appl. [**31**]{} (1986), 203-215.
, [*Applications of several complex variables to multiparameter spectral theory. Surveys of some recent results in operator theory*]{}, Vol. II, 25-90, Pitman Res. Notes Math. Ser. 192, Longman Sci. Tech. Harlow, 1988.
R. Curto and A. Dash, [*Browder spectral systems*]{}, Proc. Amer. Math. Soc. [**103**]{} (1988), 407-412.
R. Gelca, [*Compact perturbations of Fredholm n-tuples. II*]{}, Integral Equations Operator Theory [**19**]{} (1994), 360-363.
Y. Han and A. Kim, [*Weyl’s theorem in several variables*]{}, J. Math. Anal. Appl. [**370**]{} (2010), 538-542.
R. Harte, [*Invertibility and Singularity for Bounded Linear Operators*]{}, Dekker, New York, 1988.
W. Lee, Lecture Notes on Operator Theory, Graduate Texts, Spring 2010, Seoul National University, available on line.
R. Levy, [*On the Fredholm and Weyl spectra of several commuting operators*]{}, Integral Equations Operator Theory, [**59**]{} (2007), 35-51.
M. Putinar, [*On Weyl spectrum in several variables*]{}, Math. Japon. [**50**]{} (1999), 355-357.
J. L. Taylor, A joint spectrum for several commuting operators, *J. Funct. Anal.* 6(1970), 172-191.
J. L. Taylor, The analytic functional calculus for several commuting operators, *Acta Math.* 125(1970), 1-48.
[^1]: The second named author was partially supported by NSF Grant DMS-1302666.
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abstract: 'Analytical method is applied for description of the small angle Bhabha scattering at LEP1. Inclusive event selection for asymmetrical wide-narrow circular detectors is considered. The QED correction to the Born cross-section is calculated with leading and next-to-leading accuracy in the second order of perturbation theory and with leading one – in the third order. All contributions in the second order due to photonic radiative corrections and pair production are calculated starting from essential Feynman diagrams. The third order correction is computed by means of electron structure function. Numerical results illustrate the analytical calculsations.'
author:
- 'N.P.Merenkov'
title: ' Small angle Bhabha scattering at LEP1. Wide-narrow angular acceptance.'
---
=-2.0cm=175.mm =0.0cm=0.0cm
= 24pt
[*National Science Centre[^1] “Kharkov Institute of Physics and Technology”\
*]{} [*Academicheskaya Str.1, 310108, Kharkov, Ukraine*]{}\
PACS 12.15.Lk, 12.20.-m, 12.20.Ds, 13.40.-f
Introduction
=============
The small angle Bhabha scattering (SABS) process is used to measure the luminosity of electron-positron colliders. At LEP1 an experimental accuracy on the luminosity of $\delta\sigma/\sigma <0.1\%$ has been reached \[1\]. However, to obtain the total accuracy, a systematic theoretical error must also be added. The accurate determination the SABS cross-section, therefore, directly affects some physical values measured at LEP1 experiments \[2,3\]. That is why in recent years a considerable attention has been devoted to the Bhabha process \[3-11\]. The reached accuracy is, however, still inadequate. According to these evaluations the theoretical estimates are still incomplete.
The theoretical calculation of SABS cross-section at LEP1 has to cope with two somewhat different problems. The first one is the description of an experimental restrictions used for event selection in terms of final particles phase space. The second concludes in the writing of matrix element squared with required accuracy. There are two approaches for theoretical investigation of SABS at LEP1: the approach based on Monte Carlo calculation \[3-5,7\] and semi–analytical one\[6,9-11\].
The advantage of Monte Carlo method is the possibility to model different types of detectors and event selection \[3\]. But at this approach one can not use in a strightforward way the exact matrix element squared based on essential Feynman diagrams because of infrared divergence. Therefore, some additinal procedures (YFS factor exponentiation \[12\], utilization of the electron structure functions \[13\]) apply to get rid this problem and to take into account leading contribution in the higher orders . It needs to be carefully at this point because of a possibility of the double counting. Any way, up to now the next-to-leading second order correction remains uncertain, and this is transparent defect of Monte Carlo approach.
The advantage of analytical method is the possibility to use the exact matrix element squared. The infrared problem in the frame of this approach is solved by usual way taking into account virtual, real soft and hard photon emission as well as pair production in every order of perturbation theory. Any questions with double counting do not arise at analytical calculations. The defect of this method is its low mobility relative the change of an experimental conditions for event selection. Nevertheless, the analytical calculations have a great importance because allow to check numerous Monte Carlo calculations for different “ideal” detectors.
Up to now analytical formulae for SABS cross-section at LEP1 are published for the case of inclusive event selection (IES) when circular symmetrical detectors record only final electron and positron energies \[10,11\]. These define the first and second order corrections to Born cross-section with leading (of the order $~(\alpha L)^n~$) and next-to-leading (of the order $~\alpha^nL^{n-1}~$) accuracy as well as third order one with leading only. Just these contributions will have to be computed to reach required per mille accuracy for SABS cross-section at LEP1. Note that such accuracy selects only collinear (like two-jets final-state configuration) and semicollinear (like three-jets one) kinematics.
In this paper I list full analytical calculation for IES with wide-narrow angular acceptance. The first and second order corrections are derived with next-to-leading accuracy starting from Feynman diagrams for two-loops elastic electron-positron scattering, one-loop single photon emission, two photon emission and pair production. The third order one is obtained with leading accuracy by the help of the electron structure function method. The results for leading second and third order corrections in the case of CES are given too.
The contents of this paper can be outlined as follows. In Section 2 the “observable” cross-section $~\sigma_{exp}~$ is introdcued with cuts on angles and energies taken into account, and the first order correction is obtained. In Section 3 the second order corrections are investigated. These include the contributions of the processes of pair (real and virtual) production considered in Subsection 3.1 and two photons (as well real and virtual) emission. In Subsection 3.2 the correction due to one-side two photon emission is considered and in Subsection 3.3 – due to opposite-side one. The expression for the second order photonic correction is given in leading approximation only, while the next-to-leading conribution to it is written in Appendix A for both symmetrical and wide–narrow detectors. The latter does not contain auxiliary infrared parameter. In Section 4 the full leading third order correction is derived using the expansion of electron structure functions. In Section 5 the numerical results suitable for IES are presented. The correspondence of obtained results with another semi–analytical ones is dicussed in Conclusion. In Appendix B some relations are given which have been used in the process of analytical calculations and which will be very useful for numerical ones.
First order correcion
=======================
Let us introduce dimentionless quantity $$\label{1}
\Sigma = \frac{1}{4\pi\alpha^2}Q_1^2\sigma_{exp} \ ,$$ where $Q_1^2 = \epsilon^2\theta_1^2$ ($\epsilon $ is the beam energy and $
\theta_1$ is the minimal angle of the wide detector). The “experimetally” measurable cross section $\sigma_{exp}$ is defined as follows $$\label{2}
\sigma_{exp} = \int dx_1dx_2\Theta d^2q_1^{\bot}d^2q_2^{\bot}\Theta_1^c
\Theta_2^c\frac{d\sigma (e^{+}+e^{-} \rightarrow e^{+}+e^{-}+X)}{dx_1dx_2
d^2q_1^{\bot}d^2q_2^{\bot}} \ ,$$ where X is undetected final particles, $x_1 \ (x_2),\ \ q_1^{\bot},\
(q_2^{\bot})$ are the energy fraction and the transverse component of the momentum of the electron (positron) in the final state. Functions $\Theta_{i}^c$ do take into account angular cuts while function $\Theta$ - cutoff on invariant mass of detected electron and positron: $$\Theta_1^c = \theta(\theta_3 - \theta_{-})\theta(\theta_{-} - \theta_1) \ , \
\Theta_2^c = \theta(\theta_4 - \theta_{+})\theta(\theta_{+} - \theta_2) \ ,\
\Theta = \theta (x_1x_2-x_c) \ ,$$ $$\label{3}
\theta_{-} = \frac{\mid\vec q_1^{\bot}\mid}{x_1\epsilon} \ ,\ \ \
\theta_{+} = \frac{\mid\vec q_2^{\bot}\mid}{x_2\epsilon} \ .$$
In the case of symmetrical angular acceptance $$\theta_2 = \theta_1\ , \ \theta_3 = \theta_4 \ ,\ \ \rho =\frac{\theta_3}
{\theta_1} > 1 \ ,$$ but for wide-narrow one $$\theta_3 > \theta_4 > \theta_2 > \theta_1 \ ,\ \ \rho_i =\frac{\theta_i}
{\theta_1} > 1 \ .$$ Fof numerical calculation ones usually take $$\theta_1 = 0.024 \ ,\ \theta_3 =0.058 \ ,\ \theta_2 = 0.024+\frac{0.017}{8}
\ ,\ \theta_4 = 0.058 - \frac{0.017}{8} \ .$$
The first order correction $\Sigma_1$ includes the contributions of virtual and real soft and hard photon emission processes $$\label{4}
\Sigma_1 = \Sigma_{V+S} + \Sigma^H + \Sigma_H \ .$$ The contribution due to virtual and real soft photon (with the energy less than $ \Delta\epsilon ,\ \Delta\ll1 $ ) may be written as follows ( in this case $ x_1 = x_2 = 1,\ \ \vec q_1^{\bot} + \vec q_2^{\bot} = 0 $) $$\label{5}
\Sigma_{V+S} = 2\frac{\alpha}{\pi}\int \limits_{\rho_2^2}^{\rho_4^2}
\frac{dz}{z^2}[2(L-1)\ln\Delta +\frac{3}{2}L -2] \ ,\ \ L =
\ln\frac{\epsilon^2 \theta_1^2z}{m^2} \ ,$$ where $ z=(\vec q_2^{\bot})^2/Q_1^2$ and $m$ is electron mass.
The second term in r.h.s. of Eq.(4) represents the correction due to hard photon emission by the electron. In this case $$\label{6}
X=\gamma(1-x_1, \vec k^{\bot})\ ,\ \ x_2 = 1\ ,\ \ \vec k^{\bot} + \vec q_1^
{\bot} + \vec q_2^{\bot} = 0 \ ,\ \ x_c < x_1 < 1-\Delta \ .$$ It can be derived by integration of the differential cross section of single photon emission over the region $$\label{7}
\rho_2^2 < z < \rho_4^2 \ ,\ \ x^2 < z_1 = \frac{\vec q_1^{\bot 2}}{Q_1^2} <
x^2\rho_3^2 \ ,\ \ -1 < cos\varphi < 1 \ ,$$ where $ \varphi $ is the angle between vectors $ \vec q_1^{\bot}$ and $\vec q_2
^{\bot} $, in the same way as it has been done in \[10\] for symmetrical angular acceptance. But at this passage I would like to indicate the principle moments of method used largely to obtain the results of the Section 3 and based on the separate calculation of the contributions due to collinear kinematics and semi-collinear one \[14\].
In collinear kinematics emitted photon moves inside the cone within polar angles $\theta_{\gamma} < \theta_0 \ll 1$ centred along electron momentum direction (initial: $\vec k \| \vec p_1 $ or final: $\vec k \| \vec q_1 $). In semicollinear region photon moves outside this cones. Because such distinction no longer has physical meaning, the dependence on auxiliary parameter $ \theta_0
$ disappeares in total contribution. This is valid for IES as well as for CES.
Inside collinear kinematics it needs to keep electron mass in differential cross section $$d\sigma = \frac{2\alpha^3s}{\pi^2q^2}\biggl[\frac{1+x^2}{s_1t_1} - \frac{2m^2
}{q^2}\biggl(\frac{1}{s_1^2} + \frac{x^2}{t_1^2}\biggr)\biggr]d\Gamma ,$$ $$\label{8}
d\Gamma = \frac{d^3q_1d^3q_2d^3k}{\epsilon_1 \omega 2\epsilon}\delta^{(4)}(p_1
+p_2 - k - q_1 - q_2) \ ,$$ where $ q=p_1-k-q_1,\ s_1=2(kq_1),\ t_1 = 2(kp_1),\ s=(2p_1p_2)$ and $p_1(p_2)$ is 4-momentum of initial electron (positron). If photon moves inside initial electron cone $$s_1 = x(1-x)\epsilon^2\theta_-^2 ,\ t_1 = -m^2(1-x)(1+\eta),\ q^2 = -x^2
\epsilon^2\theta_{-}^2 = - \epsilon^2\theta_{+}^2 \ ,$$ $$\label{9}
d\Gamma = \frac{m^2}{s}\epsilon^2\pi^2x(1-x)dxd\eta d\theta_{-}^2,\ \
0 < \eta =
\frac{\theta_{\gamma}^2\epsilon^2}{m^2} < \frac{\theta_0^2\epsilon^2}{m^2} \ ,$$ and one can derive after integration relative $\eta$ $$\label{10}
\sigma_{\vec k\|\vec p_1} = \frac{2\alpha^3}{Q_1^2}\int \limits_{\rho_2^2}^
{\rho_4^2}\frac{dz}{z^2}\int\limits_{
x_c}^{1-\Delta}dx\biggl[\frac{1+x^2}{1-x}\ln\frac{ \theta_0^2\epsilon^2}{m^2}
- \frac{2x}{1-x}\biggr]\theta(x^2\rho_3^2 - z) \ .$$ The r.h.s.of Eq.(10) corresponds to the contribution of narrow strip with the width $2\sqrt{z}\lambda(1-x)$ centred around line $z = z_1$ in $(z,z_1)$ plane, where $\lambda = \theta_0/\theta_1$. Really, the condition $\theta_{\gamma}
< \theta_0$ for initial electron cone may be formulated as follows $$\label{11}
\mid\sqrt{z} - \sqrt{z_1}\mid < \lambda(1-x) \ ,\ \ -1 < cos\varphi < -1 +
\frac{\lambda^2(1-x)^2-(\sqrt{z_1}-\sqrt{z})^2}{2\sqrt{z_1z}} \ .$$
If photon moves inside final electron cone $$s_1 = \frac{1-x}{x}m^2(1+\zeta) \ ,\ t_1 = -(1-x)\epsilon^2\theta_{-}^2 \ , \
q^2 = -\epsilon^2\theta_{-}^2 = -\epsilon^2\theta_{+}^2 \ ,$$ $$\label{12}
d\Gamma = \frac{m^2}{s}\epsilon^2\pi^2x(1-x)dxd\zeta\frac{d\theta_{-}^2}{x^2}
\ , \ \ 0 < \zeta < \frac{\theta_{0}^2\epsilon^2x^2}{m^2} \ ,$$ and the integration relative $\zeta$ leads to $$\label{13}
\sigma_{\vec k\|\vec q_1} = \frac{2\alpha^3}{Q_1^2}\int\limits_{\rho_2^{^2}}^
{\rho_4^2}\frac{dz}{z^2}\int\limits_{x_c}^{1-\Delta}dx\biggl[\frac{1+x^2}{1-x}
\ln\frac{\theta_0^2\epsilon^2x^2}{m^2} - \frac{2x}{1-x}\biggr] \ .$$ The r.h.s. of Eq.(13) corresponds to the contribution of the strip with the width $ 2\sqrt{z}x^2(1-x)\lambda $ around line $z_1 = x^2z$ in plane $(z_1,z)$. Really, the condition $\theta_{\gamma}< \theta_{0}$ for final electron cone may be formulated as $\mid\vec r\mid < \theta_{0}$, where $\vec r = \vec k/\omega - \vec q_1^{\bot}/\epsilon_1$, and the last reads as $$\label{14}
\mid\sqrt{z_1} - x\sqrt{z}\mid < x(1-x)\lambda , \ \ -1 < cos\varphi < -1 +
\frac{\lambda^2x^2(1-x)^2 - (\sqrt{z_1} - x\sqrt{z})^2}{2x\sqrt{zz_1}} \ .$$
Having contributions due to collinear regions now it needs to find the contribution due to semicollinear ones. Supposing $m = 0$ in r.h.s. of Eq.(8) the differential cross section suitable for this case may be written as follows $$\label{15} d\sigma = \frac{\alpha^3d\varphi
dzdz_1(1+x^2)}{\pi Q_1^2z(z_1-xz)}\biggl[
\frac{1}{z_1+z+2\sqrt{z_1z}cos\varphi} - \frac{x}{z_1+x^2z+2x\sqrt{z_1z}
cos\varphi}\biggr]dx \ .$$ When integrating the first term into the brackets in r.h.s. of Eq.(15) one must use the restriction $\theta_{\gamma} > \theta_{0}$ or $$\mid\sqrt{z_1} - \sqrt{z}\mid > (1-x)\lambda \ ,\ \ -1 < cos\varphi < 1 \
;$$ $$\label{16}
\mid\sqrt{z_1} - \sqrt{z}\mid < (1-x)\lambda \ ,\ \ 1 > cos\varphi > -1 +
\frac{\lambda^2(1-x)^2 - (\sqrt{z_1} - \sqrt{z})^2}{2\sqrt{zz_1}} \ ,$$ while for the integration the second one – the restriction $\mid\vec r\mid >
\theta_{0}$ or $$\mid\sqrt{z_1} -x\sqrt{z}\mid > x(1-x)\lambda \ ,\ \ -1 < cos\varphi < 1\ ;$$ $$\label{17}
\mid\sqrt{z_1} - x\sqrt{z}\mid < x(1-x)\lambda \ ,\ 1 > cos\varphi > -1 +
\frac{\lambda^2x^2(1-x)^2 - (\sqrt{z_1} - x\sqrt{z})^2}{2x\sqrt{zz_1}} \ .$$
The integration (15) over the region (16) leads to $$\label{18}
\sigma_{a} = \frac{2\alpha^3}{Q_1^2}\int\limits_{\rho_2^2}^
{\rho_4^2}\frac{dz}{z^2}\int\limits_{x_c}^{1-\Delta}\frac{1+x^2}{1-x}dx
\biggl[\biggl(\ln\frac{z}{\lambda^2} + L_2\biggr)\theta_{3}^{(x)} +
L_3\overline \theta_3^{(x)}\biggr].$$ Analogous, the integration of r.h.s. of Eq.(15) over the region (17) gives $$\label{19}
\sigma_{b} = \frac{2\alpha^3}{Q_1^2}\int\limits_{\rho_2^2}^
{\rho_4^2}\frac{dz}{z^2}\int\limits_{x_c}^{1-\Delta}\frac{1+x^2}{1-x}dx
\biggl(\ln\frac{z}{x^2\lambda^2} + L_1\biggr) \ .$$ The values $L_{i}$ which enter into Eqs.(18) and (19) are defined as follows $$L_1 = \ln\left|{x^2(z-1)(\rho_3^2-z)}\over{(x-z)(x\rho_3^2-z)}\right| ,\ \
L_2 = \ln\left|{(z-x^2)(x^2\rho_3^2-z)}\over{x^2(x-z)(x\rho_3^2-z)}\right| ,\
\ L_3 = \ln\left|{(z-x^2)(x\rho_3^2-z)}\over{(x-z)(x^2\rho_3^2-z)}\right| .$$ Beside this the following notations for $\theta $- functions are used $$\theta_3^{(x)} = \theta(x^2\rho_3^2-z)\ ,\ \ \overline\theta_3^{(x)} = 1 -
\theta_3^{(x)} = \theta(z-x^2\rho_3^2) \ .$$
Thus, the $\Sigma^{H}$ may be represented as the sum of (10), (13), (18) and (19) divided by factor $4\pi\alpha^2/Q_1^2$ or $$\label{20}
\Sigma^{H} = \frac{\alpha}{2\pi}\int\limits_{\rho_2^2}^
{\rho_4^2}\frac{dz}{z^2}\int\limits_{x_c}^{1-\Delta}\frac{1+x^2}{1-x}
[(1+\theta_3^{(x)})(L-1) + K(x,z;\rho_3,1)]dx \ ,$$ $$K(x,z;\rho_3,1) = \frac{(1-x)^2}{1+x^2}(1+\theta_3^{(x)}) + L_1 +
\theta_3^{(x)}L_2 + \overline\theta_3^{(x)}L_3 \ .$$ Further I will use the short notations for $\theta$-functions, namely $$\theta_{i}^{(x)} = \theta(x^2\rho_{i}^2-z) \ ,\ \theta_{i} = \theta(\rho_{i}
^2-z) \ ,\ \overline\theta{i}^{(x)} = 1 - \theta_{i}^{(x)} \ ,\ \ \overline
\theta_{i} = 1 - \theta_{i} \ .$$
One may easy to see that $\Sigma^{H}$ for wide-narrow detectors can be derived from $\Sigma^{H}$ for symmetrical ones (see\[10\]) by the change z-integrations limits $$\label{21}
\int\limits_{1}^{\rho^2}dz \rightarrow \int\limits_{\rho_2^2}^{\rho_4^2}dz$$ and the substitution $\rho_3$ instead of $\rho $ under integral sign.
The third term in r.h.s. of Eq.(4) describes photon emission by the positron. It may be derived by full analogy with $\Sigma^{H}$ except restrictions on variables $z$ and $z_1$, namely $$\label{22}
1 < z < \rho_3^2 \ ,\ \ \ \ x^2\rho_2^2 < z_1 < x^2\rho_4^2 \ .$$ The contribution of the collinear kinematics ($\vec k \| \vec p_2$ and $\vec k
\| \vec q_2 $) to single hard photon emission cross section corresponds to the integration over the regions inside strips with width $ 2\sqrt{z}(1-x)\lambda
$ and $ 2\sqrt{z}x^2(1-x)\lambda $, respectively. It may be written as follows $$\sigma_{\vec k \| \vec p_2, \vec k \| \vec q_2} = \frac{2\alpha^3}{Q_1^2}
\int\limits_{1}^{\rho_3^2}\frac{dz}{z^2}\int\limits_{x_c}^{1-\Delta}
\frac{1+x^2}{1-x}dx\biggl\{\biggl(\ln\frac{\epsilon^2\theta_0^2}{m^2} -
\frac{2x} {1-x}\biggr)\Delta_{42}^{(x)} +$$ $$\label{23}
\biggl(\ln\frac{\epsilon^2\theta_0^2x^2}{m^2} - \frac{2x}{1-x}\biggr)
\Delta_{42}\biggr\} \ ,$$ where $$\label{24}
\Delta_{42}^{(x)} = \theta_4^{(x)} - \theta_2^{(x)} \ ,\ \ \
\Delta_{42} + \theta_4 - \theta_2 \ .$$
The contribution of semi-collinear kinematics may be derived by integration (15), taking into account the restrictions (16), (17) and (22). The latters correspond to regions outside narrow strips near $z_1 = z$ and $z_1 = x^2z$, respectively. The result is $$\sigma_{a}+\sigma_{b} = \frac{2\alpha^3}{Q_1^2}\int\limits_{1}^{\rho_3^2}
\frac{dz}{z^2}\int\limits_{x_c}^{1-\Delta}\frac{1+x^2}{1-x}dx\biggl[\ln\frac{z}
{\lambda^2}(\Delta_{42}+\Delta_{42}^{(x)}) + \overline L_2\Delta_{42}^{(x)}
+(\overline L_1 - 2\ln x)\Delta_{42} +$$ $$\label{25}
\overline L_3(\overline\theta_4^{(x)} - \theta_2^{(x)}) + \overline L_4(\overline\theta_4 -
\theta_2)\biggr] \ ,$$ where $$\overline L_1 = \ln\left|{(z-\rho_2^2)(\rho_4^2-z)x^2} \over
{(x\rho_4^2-z) (x\rho_2^2-z)}\right|,\ \ \overline L_2
=\ln\left|{(z-x^2\rho_2^2)(x^2\rho_4^2-z)} \over
{x^2(x\rho_4^2-z)(x\rho_2^2-z)}\right| ,$$ $$\label{26}
\overline L_3 = \ln\left|{(z-x^2\rho_2^2)(x\rho_4^2-z)} \over {(x^2\rho_4^2-z)
(x\rho_2^2-z)}\right|,\ \ \ \overline L_4
=\ln\left|{(z-\rho_2^2)(x\rho_4^2-z)} \over
{(\rho_4^2-z)(x\rho_2^2-z)}\right| .$$ The $ \Sigma_{H} $ is the sum of (23) and (25) divided by $ 4\pi\alpha^2/Q_1^2$: $$\label{27}
\Sigma_{H} = \frac{\alpha}{2\pi}\int\limits_{1}^{\rho_3^2}\frac{dz}{z^2}
\int\limits_{x_c}^{1-\Delta}\frac{1+x^2}{1-x}dx\biggl[(L-1)(\Delta_{42}+
\Delta_{42}^{(x)}) + \widetilde{K}(x,z;\rho_4,\rho_2)\biggr] \ ,$$ $$\widetilde{K} = \frac{(1-x)^2}{1+x^2}(\Delta_{42}+\Delta_{42}^{(x)}) +
\Delta_{42}\overline L_1 + \Delta_{42}^{(x)}\overline L_2 + (\overline\theta_4^
{(x)} - \theta_2^{(x)})\overline L_3+ (\overline\theta_4 -
\theta_2)\overline L_4 \ .$$
As one can see the auxiliary parameter $ \theta_0$ disappears in expressions for $\Sigma^{H} $ and $\Sigma_{H}$, and large logarithm acquires the right appearence. Thus, the separate investigation of contributions due to collinear and semi-collinear kinematics simplifies the calculations and gives also the dipper understanding of underlying physics. The experience of this approach is very important for the study of CES when it needs to describe events which belong to electron cluster (or positron one) in a different way as compared with events do not.
The different parts in r.h.s. of Eq.(4) depend on auxiliary infrared paramerter $\Delta $ but the sum does not. It has the following form: $$\Sigma_1 = \frac{\alpha}{2\pi}\biggl\{\int\limits_{1}^{\rho_3^2}\frac{dz}
{z^2}\biggl[-\Delta_{42} + \int\limits_{x_c}^{1}\biggl((L-1)P_1(x)
(\Delta_{42}+\Delta_{42}^{(x)}) + \frac{1+x^2}{1-x}\widetilde
K\biggr)dx\biggr]$$ $$\label{28} + \int\limits_{
\rho_2^2}^{\rho_4^2}\frac{dz}{z^2}\biggl[-1 + \int\limits_{
x_c}^{1}\biggl((L-1)P_1(x)(1 + \theta_3^{(x)}) + \frac{1+x^2}
{1-x}K\biggr)dx\biggr]\biggr\} \ ,$$ where $$P_1(x) = \frac{1+x^2}{1-x}\theta(1-x-\Delta) + (2\ln\Delta + \frac{3}
{2})\delta(1-x) \ ,\ \ \Delta \rightarrow 0 \ .$$ In order to make the elimination of $\Delta$ -dependence more transparent one can use the following relations: $$\int\limits_{x_c}^{1}P_1(x)dx = - \int\limits_{0}^{x_c}\frac{1+x^2}{1-x}dx \
, \int\limits_{x_c}^{1}P_1(x)\overline\theta_3^{(x)}dx = \overline\theta_3^
{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_3}\frac{1+x^2}{1-x}dx \ ,$$ $$\label{29}
\int\limits_{x_c}^{1}P_1(x)\overline\Delta_{42}^{(x)}dx = \theta_4 \overline
\theta_4^{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_4}\frac{1+x^2}{1-x}dx - \theta
_2 \overline\theta_2^{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_2}\frac{1+x^2}
{1-x}dx \ ,$$ where $ \overline\Delta_{42}^{(x)} = \Delta_{42} - \Delta_{42}^{(x)} \ .$
The r.h.s. of Eq.(28) is the full first order QED correction to born SABS cross section at LEP1 for IES with switched off vacuum polarization. The latter can be taken into account by insertion the quantity $ [1-\Pi(-zQ_1^2)]^{-2}$ under sign of z-integration (for $\Pi $ see \[3\] and references therein).
Second order correction
=========================
The second order corection contains the contributions due to double photons (real and vrtual) emission and pair production. As in symmetrical case one needs to distinguish between the situations when additional photons attach only one fermion line (one-side emission) and two fermion lines (opposite-side emission) in corresponding Feynman’s diagrams.
The contribution of pair production
-------------------------------------
Consider at first the contribution of the process of electron-positron pair production $\Sigma^{pair}$ to the second order correction: $$\label{30}
\Sigma^{pair} = \Sigma^{e^+e^-} + \Sigma_{e^+e^-} \ .$$ In order to get rid of the writing some formulae which have the same structure for both symmetrical and wide-narrow angular acceptance I will often send the reader to work \[11\] in which the details of computation are given for symmetrical case.
The experience of Section 2 allows to write the expression for $\Sigma^{
e^+e^-}$ when created electron-positron pair press to electron momentum direction, using the result of \[11\] for $ \Sigma^{e^+e^-} $ suitable for wide–wide angular acceptance. It needs only to change z-integration limits; $(\rho^2, 1) \rightarrow (\rho_4^2, \rho_2^2)
$ and substitute $ \rho_3 $ instead of $ \rho $ everywhere under integral sign. The result may be written as follows: $$\Sigma^{e^+e^-} = \frac{\alpha^2}{4\pi^2}\int\limits_{\rho_2^2}^{\rho_4^2}
\frac{dz}{z^2}L\biggl\{L\biggl(1 + \frac{4}{3}\ln(1-x_c) - \frac{2}{3}\int
\limits_{x_c}^{1}\frac{dx}{1-x}\overline\theta_3^{(x)}\biggr) - \frac{17}{3} -
\frac{8}{3}\zeta_2 -$$ $$- \frac{40}{9}\ln(1-x_c) +
\frac{8}{3}\ln^2(1-x_c) + \int\limits_{\ \ \
x_c}^{1}\frac{dx}{1-x}\overline\theta_3^ {(x)}\biggl(\frac{20}{9} -
\frac{8}{3}\ln(1-x)\biggr) +$$ $$\label{31}
+ \int\limits_{x_c}^{1}\biggl[L\overline R(x)(1+\theta_3^{(x)}) +
\theta_3^{(x)}C_1(x,z;\rho_3) + C_2(x) + d_2(x,z;\rho_3)\biggr]dx\biggr\} \ ,$$ $$\overline R(x) = (1+x)(\ln x - \frac{1}{3}) + \frac{1-x}{6x}(4 + 7x + 4x^2)
\ ,$$ $$C_1(x,z;\rho_3) = - \frac{113}{9} + \frac{142}{9}x -
\frac{2}{3}x^2 - \frac {4}{3x} - \frac{4}{3}(1+x)\ln(1-x) +
\frac{2(1+x^2)}{3(1-x)}\biggl[2\ln\left| {x^2\rho_3^2 - z} \over {x\rho_3^2 -
z}\right| -$$ $$- 3L_{i2}(1-x)\biggr] +
(8x^2+3x-9-\frac{8}{x}-\frac{7}{1-x})\ln x + \frac{2(5x^2-6)}{1-x}\ln^2x +
R(x)\ln\frac{(x^2\rho_3^2-z)^2}{\rho_3^4} \ ,$$ $$C_2(x) = - \frac{122}{9} + \frac{133}{9}x + \frac{4}{3}x^2 + \frac
{2}{3x} - \frac{4}{3}(1+x)\ln(1-x) + \frac{2(1+x^2)}{(1-x)}L_{i2}(1-x) +$$ $$+\frac{1}{3}(-8x^2-32x-20+\frac{8}{x}+\frac{13}{1-x})\ln x + 3(1+x)\ln^2x,
\ \ R(x) = 2\overline R(x) + \frac{2}{3}(1+x) \ ,$$ $$\label{32}
d_2(x,z;\rho_3) = \frac{2(1+x^2)}{3(1-x)}\ln\left|{(z-x^2)(\rho_3^2-z)(z-1)}
\over{(z-x)^2(x^2\rho_3^2-z)}\right| +
+ R(x)\ln\left|{(z-x^2)(\rho_3^2-z)(z-1)}\over {x^2\rho_3^2-z}\right| \ .$$
The r.h.s. of Eq.(31) does not contain infrared auxiliary parameter because it includes the contributions due to real and virtual pair production. The contribution of hard pair takes into account both, collinear and semi-collinear kinematics, and this ensures the next-to-leading accuracy.
If created elctron-positron pair is emitted along of the positron momentum direction the corresponding expression requires more modifications. The source of such modifications is the semi-collinear kinematics as we saw in Section 2 for the single photon emission.
The strightforward calculation shows that for contribution of the semi-collinear region $\vec p_+ \| \vec p_- $ (I use here notation $ \vec p_{\pm} $ for 3 - momentum of created positron (electron)) one has to write into formula (28) of \[11\] $$(\Delta_{42} + \Delta_{42}^{(x)})\ln\frac{z}{\lambda^2} +
\Delta_{42}\ln\left |{(z-\rho_2^2)(\rho_4^2-z)} \over
{(z-x\rho_2^2)(x\rho_4^2-z)}\right| +
\Delta_{42}^{(x)}\ln\left|{(z-x^2\rho_2^2)(x^2\rho_4^2-z)}\over {x^2(z-x\rho_
2^2)(x\rho_4^2-z)}\right| +$$ $$\label{33}
(\overline\theta_4-\theta_2)\ln\left|{(z-\rho_2^2)(x\rho_4^2-z)} \over
{(z-x\rho_2^ 2)(\rho_4^2-z)}\right|
+(\overline\theta_4^{(x)}-\theta_2^{(x)})\ln\left|{(z-x^2
\rho_2^2)(x\rho_4^2-z)} \over {(z-x\rho_2^2)(z-x^2\rho_4^2)}\right|$$ instead of expression in curle brackets and change the upper limit of z-integration: $\rho \rightarrow \rho_3 \ . $
For the contribution of semi-collinear region $\vec p_+ \| \vec q_1$ the correspnding expression is (see Eq.(33) in \[11\]) $$\label{34}
\Delta_{42}\biggl(\ln\frac{z}{\lambda^2} + \ln\left|{(z-\rho_2^2)(\rho_4^2-z)}
\over {x_2^2\rho_2^2\rho_4^2}\right|\biggr) +
(\overline\theta_4-\theta_2)\ln\left| {\rho_4^2(z-\rho_2^2)} \over
{\rho_2^2(z-\rho_4^2)}\right| ,$$ and for semi-collinear region $ \vec p_- \| \vec p_1 $ (see Eq.(38) in \[11\]) $$\label{35}
\Delta_{42}^{(x)}\biggl(\ln\frac{z}{\lambda^2} + \ln\left|{(z-x^2\rho_2^2)
(x^2\rho_4^2-z)}\over {x_1^2x^4\rho_2^2\rho_4^2}\right|\biggr) +
(\overline\theta_4^{(x)}-\theta_2^{(x)})\ln\left|
{\rho_4^2(z-x^2\rho_2^2)} \over {\rho_2^2(z-x^2\rho_4^2)}\right| .$$ For the symmetrical wide–wide angular acceptance $\rho_3 = \ \rho_4 = \ \rho \
, \ \ \rho_2 = 1 \ ,$ and $$\label{36}
\Delta_{42} \rightarrow \theta(\rho^2-z)\theta(z-1)\ ,\ \Delta_{42}^{(x)}
\rightarrow \theta(x^2\rho^2-z)\ ,\ \overline\theta_4^{(x)} \rightarrow
\theta(z-x^2\rho^2)\ ,\ \overline\theta_4\ ,\ \theta_2,\ \theta_2^{(x)}
\rightarrow 0 \ ,$$ and (33), (34), (35) reduce to corresponding expressions derived in \[11\] .
The modification of the contributions due to virtual, real soft and hard collinear pair production includes the change of z-integral upper limit : $\rho \rightarrow \rho_3 $ and trivial change of $\theta -$functions under integral sign, namely: $ \theta(x^2\rho^2-z) \rightarrow \Delta_{42}^{(x)},
\ \ 1 \rightarrow \Delta_{42}.$ The sum of all contributions has the following form: $$\Sigma_{e^+e^-} = \frac{\alpha^2}{4\pi^2}\int\limits_{
1}^{\rho_3^2} \frac{dz}{z^2}L\biggl\{L\biggl[\Delta_{42}(1 +
\frac{4}{3}\ln(1-x_c)) - \frac{2}{3}\int\limits_{
x_c}^{1}\frac{dx}{1-x}\overline\Delta_{42}^{(x)}\biggr] + \Delta_{42}\biggl(-
\frac{17}{3} -\frac{8}{3}\zeta_2 -$$ $$- \frac{40}{9}\ln(1-x_c) + \frac{8}{3}\ln^2(1-x_c) \biggr)+ \int\limits_{
x_c}^{1}\frac{dx}{1-x}\overline \Delta_{42}^{(x)}\biggl(\frac{20}{9} -
\frac{8}{3}\ln(1-x)\biggr) + \int\limits_{x_c}^{1}\biggl[L\overline
R(x)(\Delta_{42}+\Delta_{42}^{(x)}) +$$ $$+\Delta_{42}^{(x)}C_1(x,z;\rho_2) + \Delta_{42}(C_2(x) + \overline
d_2(x,z;\rho_2) ) +
(\overline\theta_4^{(x)}-\theta_4^{(x)})\biggl(\frac{2(1+x^2)}{3(1-x)}\ln\left|{(x^2\rho_2
^2-z)(x\rho_4^2-z)} \over {(x^2\rho_4^2-z)(x\rho_2^2-z)}\right| +$$ $$+ R(x)\ln\left|{(x^2\rho_2^2-z)\rho_4^2} \over {(x^2\rho_4^2-z)\rho_2^2}
\right|\biggr)+ (\overline\theta_4 -
\theta_4)\biggl(\frac{2(1+x^2)}{3(1-x)}\ln \left|{(x\rho_4^2-z)(z-\rho_2^2)}
\over {(x\rho_2^2-z)(z-\rho_4^2)}\right| +$$ $$+ R(x)\ln\left|{(\rho_2^2-z)\rho_4^2} \over {(\rho_4^2-z)\rho_2^2}\right|
\biggr)\biggr]\biggr\} \ ,$$ $$\label{37}
\overline d_2(x,z;\rho_2) = \frac{2(1+x^2)}{3(1-x)}\ln\frac{(z-\rho_2^2)^2}
{(z-x\rho_2^2)^2} + 2R(x)\ln\frac{z-\rho_2^2}{\rho_2^2} \ .$$ By the help of (36) one can verify that r.h.s. of Eq.(36) goes over in corresponding expression for symmetrical angular acceptance.
The contribution of one-side double photon emission
-----------------------------------------------------
In this Section I give the analytical expressions for all contributions into the second order correction which appear due to one-side two photon (real and virtual) emission. The master formula which does not contain infrared auxiliary parameter $\Delta$ is written only for leading approximation, and next-to-leading contribution to it is given in Apendix A.
As before it needs to differ the radiation along electron and positron momentum directions $$\Sigma_2 = \Sigma^{\gamma\gamma} + \Sigma_{\gamma\gamma} \ ,\ \ \Sigma^{\gamma
\gamma} = \Sigma^{(S+V)^2} + \Sigma^{(S+V)H} + \Sigma^{HH},$$ $$\label{38}
\Sigma_{\gamma\gamma} = \Sigma_{(S+V)^2} + \Sigma_{(S+V)H} + \Sigma_{HH}\ .$$
The contribution of virtual and real soft photon is the same for both the electron and the positron emission $$\Sigma_{(S+V)^2} = \Sigma^{(S+V)^2} = \frac{\alpha^2}{\pi^2}\int\limits_
{\rho_2^2}^{\rho_4^2}\frac{dz}{z^2}L\biggl[L(2\ln^2\Delta +
3\ln\Delta +\frac{9} {8})-$$ $$\label{39}
4\ln^2\Delta - 7\ln\Delta + 3\zeta_3 - \frac{3}{2}\zeta_2 -
\frac{45}{16}\biggr] \ .$$
Virtual and real soft photon correction to single hard photon emission already differs for photon moving along the electron momentum direction and the positron one. In the first case corresponding contribution may be derived by the help of result for symmetrical detector (see\[10\], formula(50)) using the substitutions $(\rho_4^2,\ \rho_2^2)$ instead of $ (\rho^2,\ 1) $ for z-integration limits and $ \rho_3 $ instead of $\rho $ under integral sign. Therefore, $$\Sigma^{(S+V)H} = \frac{\alpha^2}{2\pi^2}\int\limits_{
\rho_2^2}^{\rho_4^2} \frac{dz}{z^2}L\int\limits_{
x_c}^{1-\Delta}\frac{1+x^2}{1-x}dx\biggl\{(2\ln \Delta - \ln x
+\frac{3}{2})\biggl[K(x,z;\rho_3,1) +$$ $$+ (L-1)(1+\theta_3^{(x)})\biggr] + \frac{1}{2}\ln^2x -
\frac{(1-x)^2}{2(1+x^2)} + (1+\theta_3^{(x)})(-2+\ln x-2\ln\Delta) +
\overline\theta_3^{(x)}\biggl[\frac{1}{2} L\ln x +$$ $$\label{40}
+ 2\ln\Delta \ln x - \ln x\ln(1-x) - \ln^2x - L_{i2}(1-x) -
\frac{x(1-x)+4x\ln x}{2(1+x^2)} \biggr]\biggr\} \ .$$ In order to obtain the expression for $\Sigma_{(S+V)H} $ it needs to change in r.h.s. of Eq.(39): $$\mbox{ i) limits of z-integration:}\; (\rho_4^2,\ \rho_2^2) \rightarrow
(\rho_3^2,\ 1) \ ,$$ $$\label{41} ii) \ K(x,z:\rho_3,1) \rightarrow
\widetilde{K}(x,z:\rho_4,\rho_2)\ ;\ \theta_3^{(x)} \rightarrow
\Delta_{42}^{(x)}\ ,\ \overline\theta_3^{(x)} \rightarrow \overline\Delta_{42}^{(x)},\ \
1 \rightarrow \Delta_{42}\ .$$ The contribution of two hard photons emitted along electron momentum directon may be obtained in the same way as $ \Sigma^{(S+V)H} $, using the known result for symmetrical detectors (see \[10\], Eq.(54)), namely: $$\label{42}
\Sigma^{HH} = \frac{\alpha^2}{4\pi^2}\int\limits_{\rho_2^2}^{\rho_4^2}\frac{dz}
{z^2}L\int\limits_{x_c}^{1-2\Delta}dx\int\limits_{\Delta}^{1-x-\Delta}dx_1
\frac{I^{HH}}{x_1(1-x-x_1)(1-x_1)^2} \ ,$$ $$I^{HH} = \overline A\theta_3^{(x)} + \overline B + \overline C\theta_3^{(1-x_1)} ,$$ $$\overline A = \gamma\beta\biggl(\frac{L}{2} +
\ln\frac{(x^2\rho_3^2-z)^2}{x^2 (x(1-x_1)\rho_3^2-z)^2}\biggr)+ \zeta
\ln\frac{(1-x_1)^2(1-x-x_1)}{xx_1} + \gamma_A \ ,$$ $$\overline B = \gamma\beta\biggl(\frac{L}{2} +
\ln\left|{x^2(z-1)(\rho_3^2-z)(z-x^2) (z-(1-x_1)^2)^2(\rho_3^2x(1-x_1)-z)^2}
\over {(\rho_3^2(1-x_1)^2-z)^2
(z-(1-x_1))^2(z-x(1-x_1))^2(\rho_3^2x^2-z)}\right|\biggr) +$$ $$+ \zeta \ln\frac{(1-x_1)^2x_1}{x(1-x-x_1)} + \delta_B \ ,$$ $$\label{43}
\overline C = \gamma\beta\biggl(L + 2\ln\left|{x(\rho_3^2(1-x_1)^2-z)^2} \over
{(1-x_1)^2(\rho_3^2x(1-x_1)-z)(\rho_3^2(1-x_1)-z)}\right|\biggr) -
2(1-x_1)\beta-2x(1-x_1)\gamma \ ,$$ where $$\gamma = 1 + (1-x_1)^2,\ \ \beta = x^2 + (1-x_1)^2, \ \ \zeta =
x^2 + (1-x_1)^4,$$ $$\gamma_A = xx_1(1-x-x_1)-x_1^2(1-x-x_1)^2-2(1-x_1)\beta,\ \
\delta_B = xx_1(1-x-x_1)-x_1^2(1-x-x_1)^2-2x(1-x_1)\gamma .$$ Unfortunately, it is impossible to give such simple prescription as (41) in order to obtain $\Sigma_{HH}$ from Eqs.(42) and (43). In the case of radiation two hard photons along the positron momentum direction an additional detailed consideration of semi-collinear kinematics is required. All essential moments of such consideration shown in Section 2, and reader can make all calculations by the help of formulae given in Appendix B of ref.\[10\]. Here I give final result $$\label{44} \Sigma_{HH} =
\frac{\alpha^2}{4\pi^2}\int\limits_{1}^{\rho_3^2}\frac{dz}
{z^2}L\int\limits_{x_c}^{1-2\Delta}dx\int\limits_{\Delta}^{1-x-\Delta}dx_1
\frac{I_{HH}}{x_1(1-x-x_1)(1-x_1)^2} \ ,$$ $$I_{HH} = \widetilde A\Delta_{42}^{(x)} + \widetilde C\Delta_{42}^{(1-x_1)} +
\widetilde B\Delta_{42} + (\overline\theta_4^{(x)}-\theta_2^{(x)}){\it a} +
(\overline\theta_4^{(1-x_1)}-\theta_2^{(1-x_1)}){\it c} + (\overline\theta_4-\theta_2)
{\it b} \ ,$$ $${\it a} = \gamma\beta \ln\left|{(\rho_4^2x(1-x_1)-z)(\rho_2^2x^2-z)} \over
{(\rho_2^2x(1-x_1)-z)(\rho_4^2x^2-z)}\right|\ ,\ \ \
{\it b} = \gamma\beta \ln\left|{(\rho_4^2(1-x_1)-z)(\rho_2^2-z)} \over
{(\rho_2^2(1-x_1)-z)(\rho_4^2-z)}\right| ,$$ $${\it c} = \gamma\beta \ln\left|{(\rho_4^2x(1-x_1)-z)(\rho_2^2(1-x_1)^2-z)^2
(\rho_4^2(1-x_1)-z)} \over {(\rho_2^2x(1-x_1)-z)(\rho_4^2(1-x_1)^2-z)^2
(\rho_2^2(1-x_1)-z)}\right| ,$$ $$\widetilde A = \gamma\beta\biggl(\frac{L}{2} + \ln\left|{(\rho_4^2x^2-z)
(\rho_2^2x^2-z)} \over {x^2(\rho_4^2x(1-x_1)-z)(\rho_2^2x(1-x_1)-z)}\right|
\biggr) + \zeta \ln\frac{(1-x_1)^2(1-x-x_1)}{xx_1} + \gamma_A \ ,$$ $$\widetilde B = \gamma\beta\biggl(\frac{L}{2} + \ln\left|{x^2(\rho_4^2-z)
(\rho_2^2-z)} \over {(\rho_4^2(1-x_1)-z)(\rho_2^2(1-x_1)-z)}\right|
\biggr) + \zeta \ln\frac{(1-x_1)^2x_1}{x(1-x-x_1)} + \delta_B \ ,$$ $$\widetilde C = \gamma\beta\biggl(L + \ln\left|{x^2(\rho_4^2(1-x_1)^2-z)^2
(\rho_2^2(1-x_1)^2-z)^2} \over {(1-x_1)^4(\rho_4^2x(1-x_1)-z)
(\rho_2^2x(1-x_1)-z)(\rho_4^2(1-x_1)-z)(\rho_2^2(1-x_1)-z)}\right|\biggr)$$ $$-2(1-x_1)(\beta + x\gamma) \ .$$
As one can see the separate contributions in r.h.s. of Eq.(38) depend on infrared auxiliary parameter $\Delta $ but $\Sigma^{\gamma\gamma}$ and $\Sigma_{\gamma\gamma} $ do not. In order to eliminate $\Delta $-dependence analytically it needs to apply a lot efforts. Below I give leading terms and for next-to-leading ones see Appendix A. $$\label{45}
\Sigma^{\gamma\gamma L} = \frac{\alpha^2}{4\pi^2}\int\limits_{\rho_2^2}^
{\rho_4^2}\frac{dz}{z^2}L^2\int\limits_{x_c}^{1}dx\biggl[\frac{1}{2}(1+\theta
_3^{(x)})P_2(x) + \int\limits_{x}^{1}\frac{dt}{t}P_1(t)P_1\biggl(\frac{x}{t}
\biggr)\theta_3^{(t)}\biggr] ,$$ $$\label{46}
\Sigma_{\gamma\gamma}^L = \frac{\alpha^2}{4\pi^2}\int\limits_{1}^{\rho_3^2}
\frac{dz}{z^2}L^2\int\limits_{x_c}^{1}dx\biggl[\frac{1}{2}(\Delta_{42}+
\Delta_{42}^{(x)})P_2(x) + \int\limits_{x}^{1}\frac{dt}{t}P_1(t)P_1
\biggl(\frac{x}{t}\biggr)\Delta_{42}^{(t)}\biggr] ,$$ where $$P_2(x) = P_1\otimes P_1 = \int\limits_{\ \ \ x}^{1}\frac{dt}{t}P_1(t)P_1
\biggl(\frac{x}{t}\biggr) = \lim_{\Delta \to 0} \biggl\{\biggl[(2\ln\Delta +
\frac{3}{2})^2 - 4\zeta_2\biggr]\delta(1-x) +$$ $$\label{47}
+ 2\biggl[\frac{1+x^2}{1-x}(2\ln(1-x)-\ln x+\frac{3}{2}) +
\frac{1}{2}(1+x)\ln x -1 +x\biggr]\theta(1-x-\Delta)\biggr\},$$ $$\int\limits_{\ \ \ 0}^{1}P_2(x)dx = 0 \ .$$ The expressions (45) and (46) are not convenient for numerical calculations. The suitable ones may be written as follows $$\label{48} \Sigma^{\gamma\gamma L} =
\frac{\alpha^2}{4\pi^2}\biggl\{-2\int\limits_{\rho_2^2}
^{\rho_4^2}\frac{dz}{z^2}L^2\int\limits_{0}^{x_c}P_2(x)dx - \int\limits_
{m_{23}}^{\rho_4^2}\frac{dz}{z^2}L^2\int\limits_{x_c}^
{\sqrt{z}/\rho_3}\biggl[P_1(x)g\biggl(\frac{x_c}{x}\biggr) + \frac{1}{2}P_2(x)
\biggr]dx\biggr\}\ ,$$ $$\label{49}
\Sigma_{\gamma\gamma}^L = \frac{\alpha^2}{4\pi^2}\biggl\{-2\int\limits_
{\rho_2^2}^{\rho_4^2}\frac{dz}{z^2}L^2\int\limits_{0}^{x_c}P_2(x)dx -
\int\limits_{m_{14}}^{\rho_4^2}\frac{dz}{z^2}L^2\int\limits_{x_c}^
{\sqrt{z}/\rho_4}\biggl[P_1(x)g\biggl(\frac{x_c}{x}\biggr) + \frac{1}{2}P_2(x)
\biggr]dx\biggr\} +$$ $$+ \int\limits_{m_{12}}^{\rho_2^2}\frac{dz}{z^2}L^2\int\limits_{x_c}^
{\sqrt{z}/\rho_2}\biggl[P_1(x)g\biggl(\frac{x_c}{x}\biggr) + \frac{1}{2}P_2(x)
\biggr]dx\biggr\} ,$$ where $$g(y) = y +\frac{y^2}{2} + 2\ln(1-y)\ ,\ \ m_{23} = max(\rho_2^2\ ,\
x_c^2\rho_3^2)\ ,$$ $$m_{14} = max(1,\ x_c^2\rho_4^2) \ ,\ \ m_{12} = max(1,\ x_c^2\rho_2^2)\ .$$ The last two formulae can be derived by means the relations given in Appendix B. The integration relative $x$-variable in Eqs.(45) and (46) may be performed by the help of the following formulae $$\label{50}
\int\limits_{\ }^{x}P_2(y)dy=F_2(x)\ ,\quad \int\limits_{\ }^{x}P_1(y)g\biggl(
\frac{x_c}{y}\biggr)dy = F_g(x)\ , \ \
\int\limits_{\ }^{x}P_1(y)dy = -g(x)\ , \ \ x < 1\ ,$$ $$\label{51}
F_2(x) = -2x - \frac{x^2}{4} + (x+\frac{x^2}{2})\ln\frac{x^3}{(1-x)^4} +
4\ln(1-x)\ln\frac{x}{1-x} + 4L_{i2}(x) \ ,$$ $$F_g(x) = -\frac{x_c^2}{2x} + (2x+x^2)\ln x +
(x_c+\frac{x_c^2}{2})\ln\frac{x} {(1-x)^2} +
(2x_c+\frac{x_c^2}{2}-2x-\frac{x^2}{2})\ln(x-x_c) +$$ $$\label{52}
+4L_{i2}(x) + 4L_{i2}\biggl(\frac{1-x}{1-x_c}\biggr)\ , \quad x_c < x < 1 \ .$$ Therefore, the second order leading contribution to SABS cross section at LEP1 can be expressed through integral relative z-variable only.
It is useful to note also that for CES the leading contributions in all orders of perturbation theory take into account the emission of photons in initial state only. Thus, the corresponding correction due to one-side two photon (real and virtual) emission will be read in this case as follows: $$\label{53}
\Sigma_{CES}^{\gamma\gamma L} = - \frac{1}{8}\biggl(\frac{\alpha}{\pi}\biggr)
^2\int\limits_{\ \ \ \rho_2^2}^{\rho_4^2}\frac{dz}{z^2}L^2\biggl\{F_2(x_c) +
\biggl[F_2\biggl(\frac{\sqrt{z}}{\rho_3}\biggr) - F_2(x_c)\biggr]\overline\theta_3
^{(x_c)}\biggr\},$$ $$\label{54}
\Sigma_{\gamma\gamma \ CES}^L = - \frac{1}{8}\biggl(\frac{\alpha}{\pi}\biggr)^2
\biggl\{\int\limits_{\ \ \ \rho_2^2}^{\rho_4^2}\frac{dz}{z^2}L^2F_2(x_c) +
\int\limits_{\ \ \ 1}^{\rho_4^2}\frac{dz}{z^2}L^2\biggl[F_2\biggl(\frac
{\sqrt{z}}{\rho_4}\biggr) - F_2(x_c)\biggr]\overline\theta_4^{(x_c)} -$$ $$- \int\limits_{\ \ \ 1}^{\rho_2^2}\frac{dz}{z^2}L^2\biggl[F_2\biggl(\frac
{\sqrt{z}}{\rho_2}\biggr) - F_2(x_c)\biggr]\overline\theta_2^{(x_c)} \biggr\} .$$
Second order correction due to opposite-side photon emission
--------------------------------------------------------------
In this Section I calculate analytically the expression for $$\label{55}
\Sigma_{\gamma}^{\gamma} = \Sigma_{S+V}^{S+V} + \Sigma_{S+V}^{H} +
\Sigma_{H}^{S+V} + \Sigma_{H}^{H} .$$ The quantity $ \Sigma_{\gamma}^{\gamma} $ does not depend on infrared auxiliary parameter $\Delta$ because it contains all contributions due to virtual, real soft and hard photon emission.
The first term in r.h.s. of Eq.(55) takes into account only “oposite-side” virtual and real soft photon corrections $$\label{56}
\Sigma_{S+V}^{S+V} = \frac{\alpha^2}{\pi^2}\int\limits_{\rho_2^2}^{\rho_4
^2}\frac{dz}{z^2}L\biggl[L(4\ln^2\Delta + 6\ln\Delta +\frac{9}{4}) - 6 - 14
\ln\Delta - 8\ln^2\Delta\biggr] \ .$$
The contribution of one-loop virtual and real soft photon corrections to hard single photon emission may be written as follows $$\label{57}
\Sigma_{S+V}^{H} = \frac{\alpha^2}{2\pi^2}\int\limits_{\rho_2^2}^
{\rho_4^2}\frac{dz}{z^2}\biggl[2(L-1)\ln\Delta + \frac{3}{2}L - 2\biggr]
\int\limits_ {x_c}^{1-\Delta}\frac{1+x^2}{1-x}\biggl[(1 + \theta_3^
{(x)})(L-1) +K(x,z;\rho_3,1)\biggr]\ ,$$ $$\label{58}
\Sigma_{H}^{S+V} = \frac{\alpha^2}{2\pi^2}\int\limits_{\ \ \ 1}^{\rho_3^2}
\frac{dz}{z^2}\biggl[2(L-1)\ln\Delta + \frac{3}{2}L - 2\biggr]\int\limits_
{\ \ \ x_c}^{1-\Delta}\frac{1+x^2}{1-x}\biggl[(\Delta_{42} + \Delta_{42}^{(x)})(L-1) +
\widetilde K(x,z;\rho_4,\rho_2)\biggr]dx.$$
In order to find the contribution of two opposite-side hard photon emission into $~\Sigma_{\gamma}^{\gamma}~$ it is convenient to use the factorization theorem for differential cross-sections of two-jets processes in QCD \[16\]. It reads as: $$\label{59}
\Sigma_{H}^{H} = \frac{\alpha^2}{4\pi^2}\int\limits_{0}^{\infty}
\frac{dz}{z^2}\int\limits_{x_c}^{1-\Delta}dx_1\int\limits_{
{x_c}/{x_1}}^{1-\Delta}dx_2\frac{1+x_1^2}{1-x_1}\frac{1+x_2^2}{1-x_2}
\Phi(x_1,z,;\rho_3,1)\Phi(x_2,z;\rho_4,\rho_2)\ ,$$ $$\Phi(x,z,;\rho_3,1) = (\Delta_{31} + \Delta_{31}^{(x)})(L-1) + \frac{(1-x)^2}
{1+x^2}(\Delta_{31} + \Delta_{31}^{(x)}) + \Delta_{31}L_1 + \Delta_{31}^{(x)}
L_2 \ ,$$ $$\label{60}
(\overline\theta_3^{(x)} - \theta_1^{(x)})L_3 + (\overline\theta_3 - \theta_1)
\ln\left|{(x\rho_3^2-z)(z-1)} \over {(z-x)(\rho_3^2-z)}\right| ,$$ $$\label{61}
\Phi(x,z,;\rho_4,\rho_2) = (\Delta_{42} + \Delta_{42}^{(x)})(L-1) +
\widetilde K(x,z;\rho_4,\rho_2)\ ,$$ $$\Delta_{31} = \theta_3 - \theta_1\ , \quad \Delta_{31}^{(x)} = \theta_3^{(x)}
- \theta_1^{(x)}, \quad \theta_1 = \theta(1-z)\ , \quad \theta_1^{(x)} =
\theta(x^2-z) \ .$$
The $\Delta$-dependence of separate terms in r.h.s. of Eq.(55) can be eliminated analytically in the whole sum. The leading contribution is expressed in terms of electron structure functions as follows $$\label{62}
\Sigma_{\gamma}^{\gamma L} = \frac{\alpha^2}{4\pi^2}\int\limits_{0}^
{\infty}\frac{dz}{z^2}L^2\int\limits_{x_c}^{1}dx_1\int\limits_
{{x_c}/{x_1}}^{1}dx_2P_1(x_1)P_1(x_2)(\Delta_{31} + \Delta_{31}^{(x_1)})
(\Delta_{42} + \Delta_{42}^{(x_2)}) \ .$$ The next-to-leading contribution to $\Sigma_{\gamma}^{\gamma}$ is given in Appendix A.
The form of $\Sigma_{\gamma}^{\gamma}$ suitable for numerical counting may be written in terms of functions $ F_2(x)$ and $F_g(x) $ in the same manner as it was done at the end of Subsection 3.2 $$\Sigma_{\gamma}^{\gamma L} = \frac{\alpha^2}{4\pi^2}\biggl\{ -\int\limits_
{\rho_2^2}^{\rho_4^2}\frac{dz}{z^2}L^2\biggl[4(1)F_2(x_c) + 2(1)\biggl(
F_g\biggl(\frac{\sqrt{z}}{\rho_3}\biggr) - F_g(x_c)\biggr)\overline\theta_3^
{(x_c)} -$$ $$-\int\limits_{1}^{\rho_4^2}\frac{dz}{z^2}L^2 2(1)\biggl(F_g\biggl(
\frac{\sqrt{z}}{\rho_4}\biggr) - F_g(x_c)\biggr)\overline\theta_4^{(x_c)} +
\int\limits_{1}^{\rho_2^2}\frac{dz}{z^2}L^2 2(1)\biggl(F_g\biggl(
\frac{\sqrt{z}}{\rho_2}\biggr) - F_g(x_c)\biggr)\overline\theta_2^{(x_c)} +$$ $$+ \int\limits_{x_c\rho_3\rho_4}^{\rho_4^2}\frac{dz}{z^2}L^2\biggl[
F_g\biggl(\frac{\sqrt{z}}{\rho_4}\biggr) - F_g\biggl(\frac{x_c\rho_3}
{\sqrt{z}}\biggr) + g\biggl(\frac{\sqrt{z}}{\rho_3}\biggr)\biggl(g\biggl
(\frac{\sqrt{z}}{\rho_4}\biggr) - g\biggl(\frac{x_c\rho_3}{\sqrt{z}}\biggr)
\biggr)\biggr] +$$ $$+ \int\limits_{x_c\rho_2}^{1}\frac{dz}{z^2}L^2\biggl[
F_g(\sqrt{z}) - F_g\biggl(\frac{x_c\rho_2}{\sqrt{z}}\biggr) +
g(\frac{\sqrt{z}}{\rho_2})\biggl(g(\sqrt{z}) - g\biggl(\frac{x_c\rho_2}
{\sqrt{z}}\biggr)\biggr)\biggr] -$$ $$- \int\limits_{x_c\rho_4}^{1}\frac{dz}{z^2}L^2\biggl[
F_g\biggl(\frac{\sqrt{z}}{\rho_4}\biggr) - F_g\biggl(\frac{x_c}
{\sqrt{z}}\biggr) + g(\sqrt{z})\biggl(g\biggl(\frac{\sqrt{z}}{\rho_4}\biggr)
- g\biggl(\frac{x_c}{\sqrt{z}}\biggr)\biggr)\biggr] -$$ $$\label{63}
- \int\limits_{x_c\rho_3\rho_2}^{\rho_2^2}\frac{dz}{z^2}L^2\biggl[
F_g\biggl(\frac{\sqrt{z}}{\rho_3}\biggr) - F_g\biggl(\frac{x_c\rho_2}
{\sqrt{z}}\biggr) + g\biggl(\frac{\sqrt{z}}{\rho_2}\biggr)\biggl(g\biggl
(\frac{\sqrt{z}}{\rho_3}\biggr) - g\biggl(\frac{x_c\rho_2}{\sqrt{z}}\biggr)
\biggr)\biggr]\biggr\} \ .$$ In the r.h.s. of Eq.(63) the figures into brackets are suitable for CES, when only initial state radiation it needs to take into account.
Third order correction
========================
Inside the required accuracy it needs to keep only leading contribution into the third order correction. The latter becomes more important than next-to leading one for LEP2 because of increase of the energy. In order to evalulate it one can use the iteration up to the third order of the master equation for the electron structure function \[13\] $$\label{64}
D(x,\alpha_{eff}) = D^{NS}(x,\alpha_{eff}) + D^{S}(x,\alpha_{eff}) \ .$$ The iterative form of non-singlet component of Eq.(64) reads $$D^{NS}(x,\alpha_{eff}) = \delta(1-x) + \sum_{k=1}^{\infty}\frac{1}{k!}
\biggl(\frac{\alpha_{eff}}{2\pi}\biggr)^kP_1(x)^{\otimes k} ,$$ $$\label{65}
\underbrace{ P_1(x)\otimes\cdots\otimes P_1(x)}_{k} = P_1(x)^{\otimes k},
\qquad P_1(x)\otimes P_1(x) = \int\limits_{\ \ \
x}^{1}P_1(t)P_1\biggl(\frac{x}{t} \biggr)\frac{dt}{t} \ .$$
Up to third order singlet component of Eq.(64) looks as follows \[13\] $$\label{66}
D^{S}(x,\alpha_{eff}) = \frac{1}{2!}\biggl(\frac{\alpha_{eff}}{2\pi}\biggr)^2
R(x) + \frac{1}{3!}\biggl(\frac{\alpha_{eff}}{2\pi}\biggr)^3\biggl[2P_1\otimes
R(x) - \frac{2}{3}R(x)\biggr] \ ,$$ where R(x) is defined by Eq.(31). Effective coupling $\alpha_{eff}$ in Eqs. (64) - (66) represents integral of running QED constant $$\label{67}
\frac{\alpha_{eff}}{2\pi} = \int\limits_{0}^{L}\frac{\alpha dt}{2\pi
(1-{\alpha t}/{3\pi})} = \frac{3}{2}\ln\biggl(1-\frac{\alpha L}
{3\pi}\biggr)^{-1}.$$
The nonsinglet structure function describes the photon emission and pair production without taking into account the identity of final fermions, while singlet one is responsible just on identity effects.
Up to third order the electron structure function has the following form $$D(x,L) = \delta(1-x) + \frac{\alpha L}{2\pi}P_1(x) + \frac{1}{2}\biggl(
\frac{\alpha L}{2\pi}\biggr)^2\biggl(P_2(x) + \frac{2}{3}P_1(x) + R(x)\biggr) +$$ $$\label{68}
\frac{1}{3}\biggl(\frac{\alpha L}{2\pi}\biggr)^3\biggl[\frac{1}{2}P_3(x) +
P_2(x) + \frac{4}{9}P_1(x) + \frac{2}{3}R(x) + R^{^p}(x)\biggr] \ , \quad
R^{^{^p}}(x) = P_1\oplus R(x) \ .$$ For functions $P_3(x)$ and $R^{^p}(x) $ see \[6,13 MS\].
The factorization form of the differential cross-section \[16\] leads to $$\label{69}
\Sigma^{L} = \int\limits_{0}^{\infty}\frac{dz}{z^2}\int\limits_{
x_c}^{1}dx_1\int\limits_{{x_c}/{x_1}}^{1}dx_2C(x_1,L)C(x_2,L) \ ,$$ $$C(x_1,L) = \int\limits_{x_1}^{1}\frac{dt}{t}D(t)D\biggl(\frac{x_1}{t}\biggl)
\Delta_{31}^{(t)}\ ,\ \
C(x_2,L) = \int\limits_{x_2}^{1}\frac{dt}{t}D(t)D\biggl(\frac{x_2}{t}\biggl)
\Delta_{42}^{(t)} \ .$$
The expansion of $C(x_1,L)$ reads $$C(x_1,L) = \delta(1-x_1)\Delta_{31}^{(x_1)} + \frac{\alpha L}{2\pi}P_1(x_1)
(\Delta_{31}^{(x_1)} + \Delta_{31}) +$$ $$+ \biggl(\frac{\alpha L}{2\pi}\biggr)^2\biggl[C_2(x_1)(\Delta_{31}^{(x_1)} +
\Delta_{31}) + \int\limits_{x_1}^{1}\frac{dt}{t}\Delta_{31}^{(t)}
\overline C_2(x_1,t)\biggr] +$$ $$\label{70}
+ \biggl(\frac{\alpha L}{2\pi}\biggr)^3\biggl[C_3(x_1)(\Delta_{31}^{(x_1)} +
\Delta_{31}) + \int\limits_{x_1}^{1}\frac{dt}{t}\Delta_{31}^{(t)}
\overline C_3(x_1,t)\biggr] \ ,$$ $$C_2(x) = \frac{1}{2}P_2(x) + \frac{1}{3}P_1(x) + \frac{1}{2}R(x), \qquad
\overline C_2(x,t) = P_1(t)P_1\biggl(\frac{x}{t}\biggr) \ ,$$ $$C_3(x) = \frac{1}{6}P_3(x) + \frac{1}{3}P_2(x) + \frac{4}{27}P_1(x) +
\frac {2}{9}R(x) + \frac{1}{3}R^{^p}(x)\ ,$$ $$\label{71}
\overline C_3(x,t) = P_1(t)C_2\biggl(\frac{x}{t}\biggr) + C_2(t)P_1\biggl(\frac{x}
{t}\biggr)\ ,$$ and the same for $C(x_2,L)$ with the substitution $x_2$ instead of $x_1$ and $\Delta_{42}^{(x_2)}\ (\Delta_{42})$ instead of $\Delta_{31}^{(x_1)}\
(\Delta_{31})\ .$
Because of $\theta$ -functions under integral sign one has to distinguish between $$\int\limits_{x}^{1}\frac{dt}{t}A(t)B\left(\frac{x}{t}\right)\Delta_{31}^{(t)}
\quad\mbox{and}\quad \int\limits_{x}^{1}\frac{dt}{t}B(t)A\left(\frac{x}{t}
\right)\Delta_{31}^{(t)}\ .$$
In the case of CES one has to acount the initial-state radiation only. Therefore instead of (70) it needs to write $$\label{72}
C_{CES}(x_1,L) = \Delta_{31}^{(x_1)}\biggl[\delta(1-x_1) + \frac{\alpha L}
{2\pi}P_1(x_1) + \biggl(\frac{\alpha L}{2\pi}\biggr)^2C_2(x_1) + \biggl(
\frac{\alpha L}{2\pi}\biggr)^3C_3(x_1)\biggr]\ ,$$ and analogous for $C(x_2,L).$
The last step is to write third order contribution in r.h.s. of Eq.(69): $$\label{73}
\Sigma_3^L = \biggl(\frac{\alpha}{2\pi}\biggr)^3\int\limits_{0}^{\infty}
\frac{dz}{z^2}L^3\int\limits_{x_c}^{1}dx\biggl(Z_1 + \int\limits_{
{x_c}/{x}}^{1}dx_1Z_2\biggr)\ ,$$ $$Z_1 = (2\Delta_{42}+\Delta_{42}^{(x)}\Delta_{31}+\Delta_{31}^{(x)}\Delta_
{42})C_3(x) + \int\limits_{x}^{1} \frac{dt}{t}(\Delta_{42}
^{(t)}\Delta_{31} + \Delta_{31}^{(t)}\Delta_{42})\overline C_3(x,t)\ ,$$ $$Z_2 = [(\Delta_{31}+\Delta_{31}^{(x)})(\Delta_{42}+\Delta_{42}^{(x_1)}) +
(\Delta_{31}+\Delta_{31}^{(x_1)})(\Delta_{42}+\Delta_{42}^{(x)})]P_1(x)C_2(x_1)
+$$ $$+ P_1(x)\int\limits_{\ \ \ \ x_1}^{1}[\Delta_{31}^{(t)}\Delta_{42} +
\Delta_{42}^{(t)}\Delta_{31} + \Delta_{31}^{(x)}\Delta_{42}^{(t)} +
\Delta_{42}^{(x)}\Delta_{31}^{(t)}]\frac{dt}{t}\overline C_2(x_1,t)\ .$$ When writing expressions for $Z_1$ and $Z_2$ it is taken into account that $\Delta_{31}\Delta_{42} = \Delta_{42} .$ In the case of CES the expressions for $Z_1$ and $Z_2$ may be written as follows: $$\label{74}
Z_1 = (\Delta_{42}^{(x)}\Delta_{31}+\Delta_{31}^{(x)}\Delta_{42})C_3(x)\ ,
\quad Z_2 = (\Delta_{42}^{(x)}\Delta_{31}^{(x_1)} + \Delta_{42}^{(x_1)}
\Delta_{31}^{(x)})P_1(x)C_2(x_1)\ .$$
Using the relations given in Appendix B the r.h.s. of Eq.(73) may be represented in the form suitable for numerical calculations as double integral relative $z$- and $x$-variables. It may be written as follows: $$\label{75}
\Sigma_3^L = \Sigma_3^0 + \Sigma_0^3 + \Sigma_2^1 + \Sigma_1^2 \ ,$$ where upper (down) index shows the number of additional particles (real and virtual) emitted by the electron (the positron). The one-side emission contribute to the r.h.s. of Eq.(75) as $$\Sigma_3^0 + \Sigma_0^3 = \left(\frac{\alpha}{2\pi}\right)^3\biggl\{\int
\limits_{\rho_2^{^2}}^{\rho_4^{^2}}\frac{dz}{z^2}L^{^3}\biggl[-2\int
\limits_{0}^{x_c}F_p(x)dx + 2\int\limits_{x_c}^{1}F_r(x)dx -$$ $$- \overline\theta_3^{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_3}F_{pr}
(x,x_c)dx\biggr] - \int\limits_{1}^{\rho_4^{^2}}\frac{dz}{z^2}
L^{^3}\overline\theta_4^{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_4}
F_{pr}(x,x_c)dx +$$ $$\label{76}
+ \int\limits_{1}^{\rho_2^{^2}}\frac{dz}{z^2}
L^{^3}\overline\theta_2^{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_2}
F_{pr}(x,x_c)dx\biggr\} \ ,$$ where $$F_p(x) = \frac{4}{3}P_3(x) + \frac{4}{3}P_2(x) + \frac{8}{27}P_1(x) , \ \ \
F_r(x) = \frac{4}{9}R(x) + \frac{5}{3}R^{^p}(x)\ ,$$ $$F_{pr}(x,x_c) = \frac{1}{6}P_3(x) + \frac{1}{2}P_2(x)[\frac{2}{3} + g(\frac
{x_c}{x})] + P_1(x)[\frac{4}{27} + \frac{1}{2}f(\frac{x_c}{x}) +$$ $$+ \frac{2}{3}g(\frac{x_c}{x}) + \frac{1}{2}r(\frac{x_c}{x};1)] +
R(x)[\frac{2}{9} + \frac {1}{2}g(\frac{x_c}{x})] + \frac{1}{3}R^{^p}(x)\ ,$$ $$r(z,1) = \int\limits_{z}^{1}R(x)dx = -\frac {22}{9} +
z + z^2 + \frac{4}{9}z^3 - \biggl(\frac{4}{3} + 2z +z^2\biggr)\ln z \ ,$$ $$f(z) = - F_2(z) \ .$$
In the case of CES the corresponding contribution may be derived by insertion of functions $F_p^{^c},\ F_r^{^c}$ and $F_{pr}^{^c} $ into the r.h.s of Eq.(76) instead of functions $F_p,\ F_r $ and $F_{pr}$, respectively, where $$F_{pr}^{^c}(x) = C_3(x),\ \ \ F_p^{^c}(x) = \frac{1}{6}P_3(x) + \frac{1}{3}
P_2(x) + \frac{4}{27}P_1(x)\ ,
F_r^{^c}(x) = \frac{2}{9}R(x) + \frac{1}{3}R^{^p}(x)\ .$$
The contribution due to opposite-side emission to r.h.s. of Eq.(75) reads $$\Sigma_2^1 + \Sigma_1^2 = \biggl(\frac{\alpha}{2\pi}\biggr)^3\biggl\{
\int\limits_{\rho_2^{^2}}^{\rho_4^{^2}}\frac{dz}{z^2}L^{^3}\biggl[
\int\limits_{0}^{x_c}\biggl(-8P_3(x) - \frac{8}{3}P_2(x)\biggr)dx +$$ $$+ 4\int\limits_{x_c}^{1}R^{^p}(x)dx -
\overline\theta_3^{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_3}\biggl(
H(x,x_c) + 2g(\frac{x_c}{x})h(x;\sqrt{z}/\rho_3)\biggr)dx\biggr] -$$ $$- \int\limits_{1}^{\rho_4^{^2}}\frac{dz}{z^2}L^{^3}\overline\theta_4^
{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_4}\biggl(H(x,x_c) + 2g(\frac
{x_c}{x})h(x;\sqrt{z}/\rho_4)\biggr)dx +$$ $$+ \int\limits_{1}^{\rho_2^{^2}}\frac{dz}{z^2}L^{^3}\overline\theta_2^
{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_2}\biggl(H(x,x_c) + 2g(\frac
{x_c}{x})h(x;\sqrt{z}/\rho_2)\biggr)dx +$$ $$+ \int\limits_{x_c\rho_3\rho_4}^{\rho_4^{^2}}\frac{dz}{z^2}L^{^3}
\biggl[\int\limits_{{x_c\rho_4}/{\sqrt{z}}}^{\sqrt{z}/\rho_3}\biggl(
P_1(x)G\biggl(\frac{x_c}{x};\frac{\sqrt{z}}{\rho_4}\biggr) + g\biggl(\frac
{x_c}{x};\frac{\sqrt{z}}{\rho_4}\biggr)h\biggl(x;\frac{\sqrt{z}}{\rho_3}\biggr)
\biggr)dx + (\rho_3 \leftrightarrow \ \rho_4)\ \biggr] +$$ $$+ \int\limits_{x_c\rho_2}^{1}\frac{dz}{z^2}L^{^3}
\biggl[\int\limits_{{x_c\rho_2}/{\sqrt{z}}}^{\sqrt{z}/1}\biggl(
P_1(x)G\biggl(\frac{x_c}{x};\frac{\sqrt{z}}{\rho_2}\biggr) + g\biggl(\frac
{x_c}{x};\frac{\sqrt{z}}{\rho_2}\biggr)h\biggl(x;\frac{\sqrt{z}}{1}\biggr)
\biggr)dx + (\rho_2 \leftrightarrow \ 1)\ \biggr] -$$ $$- \int\limits_{x_c\rho_3\rho_2}^{\rho_2^{^2}}\frac{dz}{z^2}L^{^3}
\biggl[\int\limits_{{x_c\rho_2}/{\sqrt{z}}}^{\sqrt{z}/\rho_3}\biggl(
P_1(x)G\biggl(\frac{x_c}{x};\frac{\sqrt{z}}{\rho_2}\biggr) + g\biggl(\frac
{x_c}{x};\frac{\sqrt{z}}{\rho_2}\biggr)h\biggl(x;\frac{\sqrt{z}}{\rho_3}\biggr)
\biggr)dx + (\rho_3 \leftrightarrow \ \rho_2)\ \biggr] -$$ $$\label{77}
- \int\limits_{x_c\rho_4}^{1}\frac{dz}{z^2}L^{^3}
\biggl[\int\limits_{{x_c\rho_4}/{\sqrt{z}}}^{\sqrt{z}/1}\biggl(
P_1(x)G\biggl(\frac{x_c}{x};\frac{\sqrt{z}}{\rho_4}\biggr) + g\biggl(\frac
{x_c}{x};\frac{\sqrt{z}}{\rho_4}\biggr)h\biggl(x;\frac{\sqrt{z}}{1}\biggr)
\biggr)dx + (\rho_4 \leftrightarrow \ 1)\ \biggr] ,$$ where $$g(a;b) = g(a) - g(b) ,\ \ G(a;b) = G(a) - G(b) ,\ \ G(z) = \frac{1}{2}f(z)
+ \frac{1}{3}g(z) + \frac{1}{2}r(z)\ ,$$ $$H(x,x_c) = P_1(x)[2f(\frac{x_c}{x}) + \frac{4}{3}g(\frac{x_c}{x}) + r(\frac
{x_c}{x};1)] + g(\frac{x_c}{x})[P_2(x) + R(x)] \ ,$$ $$h(x;\sqrt{z}/\rho) = \int\limits_{\ \ \ \ x}^{\sqrt{z}/\rho}\frac{dt}{t}P_1
(t)P_1\biggl(\frac{x}{t}\biggr) =$$ $$\frac{1+x^2}{1-x}\biggl(\frac{3}{2} + 2\ln\frac{(\sqrt{z}/\rho-x)(1-x)}{(1-
\sqrt{z}/\rho)x}\biggr) - 1 + x - \frac{\sqrt{z}}{\rho} + \frac{x\rho}{\sqrt{z}}
- (1+x)\ln\frac{\sqrt{z}}{x\rho}\ .$$ Note that substitutions inside square brackets concern both, limits of $x$–integration and expressions under $x$–integral sign.
In the case of CES the r.h.s. of Eq.(77) requires the following modifications: i) coefficient at $P_3(x)$ has to be reduced eight times, coefficients at $P_2(x)$ and $R^{^p}(x)$ – four times; $\;$ ii) it needs to suppouse $h$ = 0 and to substitute $H^{^c}(x,x_c)$ instead of $H(x,x_c)$, where $$H^{^c}(x,x_c) = P_1(x)\biggl[\frac{1}{2}f(\frac{x_c}{x}) + \frac{2}{3}g(
\frac{x_c}{x}) + \frac{1}{2}r(\frac{x_c}{x};1)\biggr] + \frac{1}{2}g(\frac{x_c}
{x})[P_2(x) + R(x)]\ .$$
The numerical results
=======================
The numerical calculations carried out for the beam energy $\epsilon = 46.15
GeV, $ and limited angles of circular detectors as given after Eq.(3). The Born cross-section $$\label{78}
\sigma_B =
\frac{4\pi\alpha^2}{Q_1^2}\int\limits_{\rho_2^2}^{\rho_4^2}
\frac{dz}{z^2}\left(1 - \frac{z\theta_1^2}{2}\right)$$ (in symmetrical wide-wide case the limits of integration are 1 and $\rho_3^2)
$ equals 175.922nb for [**ww**]{} angular acceptance and 139.971nb for [**nn**]{} and [**wn**]{} ones. Formula (78) takes into account the contributions of the scattered diagram as well as the interference of scattered and annihilation ones. The contribution of pure annihilation diagram is proportional to $\theta_1^4$ and is negligible even on the born level. Note, that one has to reduce twice the coefficient at $\theta_1^2$ under integral sign in the r.h.s. of Eq.(78) if he want restrict himself with the contribution of the scattered diagram only. When calculating the QED corrections to the cross–section (78) I systematically ignore the terms proportional $\theta_1^2,$ which have the double logarithmic asymptotic behavior \[17\] and equal parametrically to $~(\alpha|t|)\ln^2(|t|/s)
/(\pi s).~$ The last value is about 0.1 $per \ mille$ as compared with unit for LEP1 conditions.
The results of the numerical calculations of QED correction with the switched off vacuum polarization are shown in the [**Tables 1–3**]{} . For comparsion we give also the corresponding numbers derived by the help of Monte Carlo program [**BLUMI**]{} based on the YFS exponentiation \[3\].
As one can see from the [**Table1**]{} there is an approximately constant difference on the level of 0.3 $per\ \ mille$ between our analytical and MC results inside first order correction. Because [**BLUMI**]{} compute the first order correction exactly \[18\] it may be think that this distiguish is caused by omitted in the present calculation terms mentioned above.
[|ccccccccc|]{}\
$x_c$ & [**blumi ww**]{} & [**ww**]{} & [**nn**]{} & [**wn**]{} & [**blumi ww**]{} & [**ww**]{} & [**nn**]{} & [**wn**]{}\
0.1 & 166.046 & 166.008 & 130.813 & 134.504 & 166.892 & 166.958 & 131.674 & 134.808\
0.3 & 164.740 & 164.702 & 129.797 & 133.416 & 165.374 & 165.447& 130.524 & 133.583\
0.5 & 162.241 & 162.203 & 128.001 & 131.428 & 162.530 & 162.574 & 128.474& 131.127\
0.7 & 155.431 & 155.390 & 122.922 & 125.809 & 155.668 & 155.597 & 123.206 & 125.225\
0.9 & 134.390 & 134.334 & 106.478 & 107.945 &137.342 & 137.153 & 108.820 & 109.667\
In the [**Table2**]{} I give the absolute values of the second order correction to SABS cross-section taking into account both leading and next-to-leading contributions. The correction due to pair production is small in accordance with the results of the work \[6\]. The second order photonic correction is represented as a sum of leading contribution and next-to-leading one. As one can see the next-to-leading part is not negligible .
[|rcrrccc|]{}\
$x_c$ & [**ww**]{} & [**nn**]{} & [**wn**]{} & [**ww**]{} & [**nn**]{} & [**wn**]{}\
0.1 & 0.007 & – 0.004 & 0.015 & 0.742+0.208 & 0.679+0.182 & 0.249+0.091\
0.3 &– 0.033 & – 0.033 & – 0.020 & 0.546+0.199& 0.556+0.171& 0.069+0.098\
0.5 & – 0.058 & – 0.050 & – 0.041 & 0.140+0.231& 0.291+0.182& – 0.314+0.134\
0.7 & – 0.090 & – 0.074 & – 0.069 & – 0.027+0.234& 0.117+0.187& – 0.571+0.170\
0.9 & – 0.142 & – 0.115 & – 0.115 & 2.961–0.142& 2.458–0.116& 1.822–0.090\
In the [**Table3**]{} the absolute value of the leading third order correction and SABS cross-section with all corrections obtained in this work are shown. The third order one takes into account three photon emission and pair production accompanied by single photon radiation. At large values of $x_c$ this correction is comparable with second order next-to-leading one. This effect will increase in the conditions of LEP2.
[|crrrccc|]{}\
$x_c$ & [**ww**]{} & [**nn**]{} & [**wn**]{} & [**ww**]{} & [**nn**]{} & [**wn**]{}\
0.1 & – 0.055 & – 0.047 & – 0.006 & 166.910& 131.623& 134.817\
0.3 & – 0.065& – 0.053 & – 0.018& 165.349& 10.438& 133.545\
0.5 & – 0.036 & – 0.040 & 0.004& 162.472& 128.384& 131.090\
0.7 & 0.089 & 0.058 & 0.124& 155.596& 123.190& 125.310\
0.9 & 0.291 & 0.220 & 0.331& 137.307& 108.927& 109.893\
As concerns the second order correction it needs to have the analytical formulae based on exponentiated form of the electron structure function in order to be consequent in the comparison with the [**BLUMI**]{} results. On the other hand, the comparison of given here the second order photonic correction, which includes the leading and next-to-leading contributions, with the corresponding numbers for non-exponentiated [**BLUMI**]{} version \[3\] was done recently in \[22\], and the agreement is very impressive.
Conclusion
============
In this paper analytical calculation of QED correction to SABS cross section at LEP1 are given for the case of inclusive event selection and wide-narrow angular acceptance. These include leading and next-to-leading contributions in first and second orders of perturbation theory and leading one in the third order. The leading contributions in the case of calorimeter event selection are obtained too for any form of final electron and positron clusters. Results are represented in the form of manifold integrals with definite limits, and functions under integral sign have not any physical singularities. No problem arises with infrared divergence and double counting.
The selection of essential Feynman diagrams, utilization of natural for this problem Sudakov’s variables, impact factor representation of differential cross section due to t-channel photon exchange as well as electron structure function method and investigation of underlying kinematics were very useful along of the whole this work. It needs to emphasize separately the simplifications connected with impact factor representation which allows to represent the differential cross sections of two-jets processes in QED by factorized form. The latter allows to use cut-off $\theta$ functions for the final electron and positron independently on the level of the differential cross-section. The calculation does not require to go to c.m.s. of underlying subprocess (as in \[6\]) and escapes corresponding complications.
At this point I want to comment the analytical calculation of leading contribution due to photon emission and pair production carried out in \[6\]. Authors of these articles used as the master formula for description QED corrections to the SABS cross-section due to initial-state radiation the representation valid for cross sections of Drell-Yan process \[19\], electron-positron annihilation into muons (or hadrons) \[20\] and large angle Bhabha scattering \[21\]. But inside this set the SABS process has a very particular feature caused by the existence of two different scales. The first one is the momentum transfer squared $t$, and just this scale defines the value of the cross-section. The second scale is full c.m.s. energy squared $s =
4\epsilon^2$, and the quantity $\theta^2 \sim |t|/s << 1$ has status of a small correction.
The $t$-scale physics is very simple and defined by peripheral interaction of the electron and the positron due to one photon exchange, provided momentum transfer is pure perpendicular : $t = - \vec q^2$. The $s$-scale physics is more complicated. On the born level it exhibits as contribution of an annihilation diagram and beside this permits the energy and longitudinal momentum exchange for the contribution of scattering diagram. The first order QED correction for $s$-scale cross-section includes the contributions of box diagrams, large angle photon emission and up–down interference because both, the eikonal representation for the scattering amplitude and the factorization form of the differential cross–section, breaks down. In the second order large angle pair production and appear.
The structure function used in \[6\] controls $t$-scale cross-section only and has not any relation to $s$-scale one because physics of different scales evolute by its own laws.
On the other hand, only scattered diagram contributes in born cross-section used in \[6\]. But everytime when somebody neglects annihilation diagram as compared with scattering one he must automatically neglect $\theta^2$ as compared with unit everywhere including the born cross-section (see comments to Eq.(78)) and experimental cuts in order to be consequent. Taking into account these arguments the master formula in \[6\] must be necessary simplified by eliminating terms proportional $\xi \sim |t|/s << 1$ and $\xi^2$ in the numerator of Eq.(5) and in the cutoff restrictions. After this it becames adequate to one obtained in \[10\] and used in this work.
Numerical evaluations shows good agreement with Monte Carlo calculations inside first order but the achievement of an agreement for high order corrections will require an additional efforts, connected with writing the version, based on the exponentiated form of the electron structure function for present analytical calculation.
[**Acknowlegement\
**]{}
Author thanks E. Kuraev, L. Trentadue, S. Jadach, B.W.L. Ward, G. Montagna and B. Pietrzyk for fruitful discussions and critical remarks as well as A. Arbuzov and G. Gach for the help in the numerical calculation. This work supported by INTAS grant 93-1867.
Appendix A {#appendixa .unnumbered}
==========
Let us begin with the consideration of the next-to-leading second order $\Delta$-independent contribution due to one-side two photons emission. At first I will give analytical expression for symmetrical case, because it was not published up to now in relevant form. (I do not introduce special notation for next-to-leading contribution to $\Sigma$ keeping in mind that only such kind terms are considered along this Appendix) $$\label{ea1}
\Sigma^{\gamma\gamma} = \Sigma_{\gamma\gamma} =
\frac{1}{4}\left(\frac{\alpha}{\pi}\right)^2\int\limits_{1}^{\rho^2}
\frac{dz}{z^2}L\; Y,$$ $$Y = y + \int\limits_{x_c}^{1}dx\;\biggl\{A + \int\limits_{0}^
{1-x}dx_1\; \biggl[\frac{1}{x_1}\; 4\;\frac{1+x^2}{1-x} (\theta_{\rho}^{(x)}
l_1+l_2) + \biggl(-1-\frac{1+x}{1-x_1} -$$ $$-\frac{x}{(1-x_1)^2}\biggr)(l_4 + \theta_{\rho}^{(x)}l_3 + 2\theta_{\rho}^
{(1-x_1)}l_5) + \frac{2(1+x)}{1-x_1}\theta_{\rho}^{(1-x_1)}\biggr] -$$ $$-4\;\frac{1+x^2}{1-x}\;\overline\theta_{\rho}^{(x)} \biggl[\int\limits_
{1-\sqrt{z}/\rho}^{1-x}dx_1\; \biggl(\frac{1}{x_1}l_5 + \frac{2}{x_2}\ln
\frac{x}{1-x_1}\biggr) + \int\limits_{0}^{\sqrt{z}/\rho-x}\frac{dx_1}
{x_1}\; l_6\biggr] \biggr\}\ ,$$ $$y = 12\zeta_3 + 10\zeta_2 - \frac{45}{4} - 16\ln^2(1-x_c) - 28\ln(1-x_c)\ ,$$ $$A = (1+\theta_{\rho}^{(x)})\biggl[2(5+2x)+4(x+3)\ln(1-x) + 4\;\frac{1+x^2}
{1-x}\ln x\biggr] +$$ $$+ 2\;\frac{1+x^2}{1-x}\biggl[(\frac{3}{2}-
\ln x)K(x,z;\rho,1) - \frac{1}{2}\ln^2x - \frac{(1-x)^2}{2(1+x^2)} +$$ $$+ 2\ln(1-x)\biggl(\theta_{\rho}^{(x)}\ln\left|\frac{x^2\rho^2-z} {x\rho^2-z}
\right|+ \ln\left|\frac{(z-1)(z-x^2)(\rho^2-z)} {(z-x)^2(x\rho^2-z)}\right|
\biggr)\biggr] +$$ $$+ \overline{\theta}_{\rho}^{(x)}\biggl[\frac{16}
{1-x}\ln(1-x) + \frac{14}{1-x} - (1-x)\ln x +$$ $$+ 2\;\frac{1+x^2}
{1-x}\biggl( - \frac{3}{2}\ln^2x + 3\ln x\ln(1-x) - L_{i2}(1-x) - \frac{x(1-x)+
4x\ln x}{2(1+x^2)} +$$ $$+ \frac{(1+x)^2}{1+x^2}\ln\left|\frac
{(\sqrt{z}-x\rho)} {\rho-\sqrt{z}}\right|+ 2\ln\left|\frac{\sqrt{z}-x\rho}
{\rho}\right| \ln\left|\frac{x(x\rho^2-z)}{x^2\rho^2-z}\right|\biggr)\biggr],$$ $$l_1 = \ln\left|\frac{(x^2\rho^2-z)(x\rho^2-z)} {(x(1-x_1)\rho^2-z)(x(x+x_1)
\rho^2-z)}\right|,\ \ l_3 = \ln\left|\frac{(1-x_1)^2(1-x-x_1)(x^2\rho^2-z)^2}
{x^3x_1(x(1-x_1)\rho^2-z)^2}\right|,$$ $$l_2 = \ln\left|\frac{(z-x)^2(z-(1-x_1)^2)(z-(x+x_1)^2)} {(z-(1-x_1))
(z-x(1-x_1))((x+x_1)-z)(x(x+x_1)-z)}\right| +$$ $$\ln\left|\frac{((1-x_1)^2\rho^2-z)((x+x_1)^2\rho^2-z)(x\rho^2-z)} {((x+x_1)
\rho^2-z)((1-x_1)\rho^2-z)(x^2\rho^2-z)}\right|,$$ $$l_4 = \ln\left|\frac{(1-x_1)^2xx_1(z-1)(z-x^2)(z-(1-x_1)^2)^2} {x_2(z-(1-x_1))^
2(z-x(1-x_1))^2}\right| + \ln\left|\frac{(\rho^2-z)(x(1-x_1)
\rho^2-z)^2} {(x^2\rho^2-z)((1-x_1)^2\rho^2-z)^2}\right|,$$ $$l_5 = \ln\left|\frac{x((1-x_1)^2\rho^2-z)^2} {(1-x_1)^2(x(1-x_1)\rho^2-z)
((1-x_1)\rho^2-z)^2}\right|,$$ $$l_6 = \ln\left|\frac{(x\rho^2-z)((x+x_1)^2\rho^2-z)^2} {(x^2\rho^2-z)
(x(x+x_1)\rho^2-z)((x+x_1)\rho^2-z)}\right|.$$
For wide-narrow angular acceptance it needs to consider only the case of the positron emission $\Sigma_{\gamma\gamma}$, because the corresponding expression for the electron emission $\Sigma^{^{\gamma\gamma}}$ is just eq.(A1) with $(\rho_4^2,\rho_2^2)$ as the limits of $z$-integration and $\rho_3$ instead of $\rho$ under the integral sign.
The analytical expression for $\Sigma_{\gamma\gamma}$ has the following form: $$\label{ea2}
\Sigma_{\gamma\gamma} = \frac{1}{4}\left(\frac{\alpha}{\pi}\right)^2\int
\limits_{1}^{\rho_3^2}\frac{dz}{z^2}L\; A^{W}_{N}\, ,$$ $$A^{W}_{N} = y\Delta_{42} + \int\limits_{x_c}^{1}dx\Biggl\{ \Delta_{42}
\biggl[4(4+3x)+6(x+3)\ln(1-x)+\biggl(x-1 +$$ $$+ 4\;\frac{1+x^2}{1-x}
\biggr)\ln x\biggr] + \Delta_{42}^{(x)}\biggl[(1-x)(3+\ln x) + 2(x+3)\ln(1-x)
+ 4\;\frac{1+x^2}{1-x}\ln x \biggr] + \overline{\Delta}_{42}^
{(x)} \frac{2}{1-x}(4+$$ $$+ (1+x)^2)\ln(1-x) + 2\;\frac{(1+x)^2}{1-x}
\biggl(\theta_4\overline{\theta}_4^{(x)} \ln\left|\frac{\sqrt{z}-x\rho_4}
{\rho_4-\sqrt{z}}\right| - \theta_2\overline{\theta}_2^{(x)}\ln\left|\frac
{\sqrt{z}-x\rho_2} {\rho_2-\sqrt{z}}\right|\biggr) +$$ $$+ \frac{1+x^2}{1-x}B+ \int\limits_{0}^{1-x}dx_1\biggl[2\;\frac{1+x^2}
{(1-x)x_1} \biggl(\Delta_{42}^{(x)}l_{1+} + \Delta_{42}l_{2+} +
(\overline{\theta}_4^{(x)} - \theta_2^{(x)})l_{1-} +
(\overline{\theta}_4 -\theta_2)l_{2-} \biggr) +$$ $$+ \biggl(-1-\frac{1+x}{1-x_1}-
\frac{x}{(1-x_1)^2}\biggr)\Biggl(\Delta_{42}^{(x)}\biggl(\ln\frac{(1-x_1)
^2x_2}{x^3x_1} + l_{3+}\biggr) + \Delta_{42}\biggl(\ln\frac{(1-x_1)^2xx_1}
{x_2} + l_{4+}\biggr) +$$ $$+ \Delta_{42}^{(1-x_1)}\biggl(2\ln\frac{x}{(1-x_1)^2} + l_{5+}\biggr)
+ (\overline{\theta}_4^{(x)}-\theta_2^{(x)})l_{3-} + (\overline{\theta}_4-
\theta_2)l_{4-} +$$ $$+ (\overline{\theta}_4^{(1-x_1)}-\theta_2^
{(1-x_1)})l_{5-}\Biggr) + 2\;\frac{1+x}{1-x_1}\Delta_{42}^{(1-x_1)} \biggr]
+ 2\;\frac{1+x^2}{1-x}\theta_4\overline{\theta}_4^{(x)}
\biggl[ \int\limits_{1-\sqrt{z}/\rho_4}^{1-x}dx_1 \biggl(\frac{1}{x_1}
\overline{l}_6 -$$ $$- \frac{4}{x_2}\ln\frac{x}{1-x_1}\biggr) + \int\limits_{0}
^{\sqrt{z}/\rho_4-x}\frac{dx_1}{x_1}\overline{l}_7\biggr]
+ 2\;\frac{1+x^2}{1-x}\theta_2\overline{\theta}_2^{(x)}\biggl[ \int\limits_
{1-\sqrt{z}/\rho_2}^{1-x}dx_1 \biggl(\frac{1}{x_1}\tilde{l}_6 +$$ $$+ \frac{4}{x_2}
\ln\frac{x}{1-x_1}\biggr) + \int\limits_{0}^{\sqrt{z}/\rho_2-x}\frac
{dx_1}{x_1}\tilde{l}_7\biggr] \Biggr\},$$ $$B = \Delta_{42}\biggl(-2\ln^2x+2\ln(1-x) \ln\left|\frac{x^4(z-\rho_2^2)^2
(z-x^2\rho_2^2)(x^2\rho_4^2-z)(\rho_4^2-z)^2} {(z-x\rho_2^2)^3(x\rho_4^2-z)^3}
\right|\biggr) +$$ $$+ \Delta_{42}^{(x)}\biggl(\ln^2x+2\ln(1-x)
\ln\left|\frac{(z-x^2\rho_2^2)(x^2\rho_4^2-z)} {x^4(z-x\rho_2^2)(x\rho_4^2-z)}
\right|\biggr) + (3-2\ln x)\widetilde{K}(x,z;\rho_4,\rho_2) +$$ $$+ \overline{\Delta}_{42}^{(x)}\biggl(7-2\ln x\ln(1-x) - 2\ln^2x
- 2L_{i2}(1-x) - \frac{x(1-x)+4x\ln x}{1+x^2}\biggr) +$$ $$+ 2(\overline{\theta}_4-\theta_2)\ln(1-x)\ln\left|\frac{(x\rho_4^2-z)^3
(z-\rho_2^2)^2(z-x^2\rho_2^2)}{(\rho_4^2-z)^2(x^2\rho_4^2-z) (z-x\rho_2^2)^3}
\right| +$$ $$+ 2(\overline{\theta}_4^{(x)}-\theta_2^{(x)})\ln(1-x) \ln\left|\frac
{(z-x^2\rho_2^2)(x\rho_4^2-z)} {(x^2\rho_4^2-z)(x\rho_2^2-z)}\right| +$$ $$+ 4\theta_4\overline{\theta}_4^{(x)}\ln\left|\frac{x\rho_4-
\sqrt{z}} {\rho_4}\right|\ln\left|\frac{x(x\rho_4^2-z)}{x^2\rho_4^2-z}\right|
+4\theta_2\overline{\theta}_2^{(x)}\ln\left|\frac{\sqrt{z}-x\rho_2}
{\rho_2}\right|\ln\left|\frac{z-x^2\rho_2^2}{x(z-x\rho_2^2)}\right|,$$ $$l_{1\pm} = (1\pm \hat{c})\ln\left|\frac{(z-x^2\rho_2^2)(z-x\rho_2^2)}
{(z-x(1-x_1)\rho_2^2)(z-x(x+x_1)\rho_2^2)}\right|,$$ $$l_{2\pm}= (1\pm \hat{c})\Biggl[\ln\left|\frac{(z-x\rho_2^2)^3(z-(1-
x_1)^2\rho_2^2)^2(z-(x+x_1)^2\rho_2^2)^2} {(z-x^2\rho_2^2)(z-x(1-x_1)\rho_2^2)
(z-x(x+x_1)\rho_2^2)(z-(1-x_1)\rho_2^2)^2(z-(x+x_1)\rho_2^2)^2}\right|
\Biggr],$$ $$l_{3\pm} = (1\pm \hat{c})\ln\left|\frac{z-x^2\rho_2^2} {z-x(1-x_1)\rho_2^2}
\right|, \quad
l_{4\pm} = (1\pm \hat{c})\ln\left|\frac{z-\rho_2^2} {z-(1-x_1)\rho_2^2}
\right|,$$ $$l_{5\pm} = (1\pm \hat{c})\ln\left|\frac{(z-(1-x_1)^2
\rho_2^2)^2} {(z-x(1-x_1)\rho_2^2)(z-(1-x_1)\rho_2^2)}\right|,$$ $$\tilde{l}_{6} = \ln\left|\frac{x^2(z-(1-x_1)^2\rho_2^2)^4} {(1-x_1)^4
(z-x(1-x_1)\rho_2^2)^2(z-(1-x_1)\rho_2^2)^2}\right|,$$ $$\tilde{l}_{7} = \ln\left|\frac{(z-x\rho_2^2)^2(z-(x+x_1)^2\rho_2^2)^4}
{(z-x^2\rho_2^2)^2(z-x(x+x_1)\rho_2^2)^2(z-(x+x_1)\rho_2^2)^2}\right|\ ,\ \
\overline{l}_6 = - \hat{c}\tilde{l}_6\ , \ \overline{l}_7 = - \hat{c}
\tilde{l}_7 \ ,$$ where $x_2=1-x-x_1$, and $\hat{c}$ is the operator of the substitution $$\begin{aligned}
\hat{c}f(\rho_2)=f(\rho_4) \ .\end{aligned}$$ One can verify that in the symmetrical limit formula (\[ea2\]) coincides with (\[ea1\]) one.
For opposite-side emission the next-to-leading contribution to $\Sigma$ in the symmetrical case reads $$\begin{aligned}
\label{ea4}
\Sigma^{\gamma}_{\gamma} = \left(\frac{\alpha}{\pi}\right)^2 L
\int\limits_{0}^{\infty}\frac{dz}{z^2}\; T, \end{aligned}$$ $$\begin{aligned}
T &=& A\theta_{\rho}\overline\theta_1 - \int\limits_{x_c}^{1}dx\;
\biggl[\frac{1+x^2}{2(1-x)}N(x,z;\rho,1) + \Xi(x) + \frac{\overline{\Xi}(x)}
{1-x}\biggr] \\ \nonumber &\times& \int\limits_{x_c/x_1}^{1}dx_1\;
\biggl[(1+x_1)\Xi(x_1) + \frac{2\overline{\Xi}(x_1)}{1-x_1}\biggr], \end{aligned}$$ where $$\begin{aligned}
A &=& - 6 - 14\ln(1-x_c) - 8\ln^2(1-x_c) + \int\limits_{\ \ \ \ x_c}^{1}dx
\biggl\{ 7(1+x) + \\ \nonumber &+& \frac{1+x^2}{2(1-x)}[3K(x,z;\rho,1) +
7\overline{\theta}_{\rho}^{(x)}]
+ 2\ln\frac{x-x_c}{x}\biggl[(3+x)(1+\theta_{\rho}^{(x)}) + \\ \nonumber
&+& \frac{4}{1-x}\overline{\theta}_{\rho}^{(x)} + \frac{1+x^2}{1-x}
N(x,z;\rho,1)\biggr] + \frac{8}{1-x}\ln\frac{x(1-x_c)}{x-x_c}\biggr\} \ .\end{aligned}$$ We introduce the following reduced notation for $\theta$-functions: $$\Xi(x)=\theta_{\rho}\overline{\theta}_1+\theta_{\rho}^{(x)}
\overline{\theta}_1^{(x)}, \quad \overline{\Xi}(x)=\theta_{\rho}
\overline{\theta}_{\rho}^{(x)} - \theta_1\overline{\theta}_1^{(x)} $$. The quantity $K(x,z;\rho,1)$ entering into espression for $A$ is the $K$– factor for single photon emission, and the quantity $N(x,z;\rho,1)$ may be derived by the help of Eq.(10) in the following way: $$N(x,z;\rho,1)=\biggl(\widetilde{K}(x,z;\rho_4,\rho_2) - \frac{(1-x)^2}
{1+x^2}(\Delta_{42} + \Delta_{42}^{(x)})\biggr) \bigg|_{\rho_4=\rho,
\ \rho_2=1} \ .$$ Note that $N(1,z;\rho,1)=0 \ .$
In the wide-narrow angular acceptance the corresponding formula for $\Sigma_{\gamma}^{\gamma}$ may be written as follows: $$\Sigma_{\gamma}^{\gamma}=\frac{\alpha^2}{\pi^2}L \int\limits_{0}^{\infty}
\frac{dz}{z^2}T_{N}^{W},$$ where $$\begin{aligned}
T_{N}^{W} &=& \widetilde{A} - \frac{1}{2}\;\Biggl\{ \int\limits_{x_c}
^{1}dx\;\biggl[
\frac{1+x^2}{2(1-x)}N(x,z;\rho_3,1) + \Xi_{31}(x) + \frac{1}{1-x}\overline{
\Delta}_{31}^{(x)}\biggr] \\ \nonumber &\times& \int\limits_{x_c/x}^{1}dx_1\;
\biggl[(1+x_1)\Xi_{42}(x) + \frac{2}{1-x_1}\overline{\Delta}_{42}^{(x)}\biggr]
+ \\ \nonumber
&+& \int\limits_{x_c}^{1}dx\;\biggl[ \frac{1+x^2}{2(1-x)}
N(x,z;\rho_4,\rho_2) + \Xi_{42}(x) + \frac{1}{1-x}\overline{\Delta}_{42}^{(x)}
\biggr] \\ \nonumber &\times& \int\limits_{x_c/x}^{1}dx_1\;\biggl
[ (1+x_1)\Xi_{31}(x) + \frac{2}{1-x_1}\overline{\Delta}_{31}^{(x)}\biggr]
\Biggr\} \ , \end{aligned}$$ where $$\begin{aligned}
\widetilde{A} &=& ( - 6 - 14\ln(1-x_c) - 8\ln^2(1-x_c))\Delta_{42} + \\
\nonumber &+& \int\limits_{x_c}^{1}dx\Biggl\{
\Delta_{42}\biggl[7(1+x) + \frac{8}{1-x}\ln\frac{x(1-x_c)}{x-x_c}\biggr] + \\
\nonumber
&+& \frac{1+x^2}{2(1-x)}\biggl[\frac{3}{2}\Delta_{42} \widetilde{K}(x,z;
\rho_3,1)
+ \frac{3}{2}\Delta_{31}\widetilde{K}(x,z;\rho_4,\rho_2) + \\ \nonumber
&+& \frac{7}{2}(\Delta_{42}\overline{\Delta}_{31}^{(x)} + \Delta_{31}
\overline{\Delta}_{42}^{(x)})\biggr] + \ln\frac{x-x_c}{x}\biggl[(3+x)
(\Delta_{31}\Xi_{42}(x) + \Delta_{42}\Xi_{31}(x)) + \\ \nonumber
&+& \frac{4}{1-x}(\overline{\Delta}_{42}^{(x)}\Delta_{31} + \overline{\Delta}_
{31}^{(x)}\Delta_{42}) + \frac{1+x^2}{1-x}(\Delta_{42}N(x,z;\rho_3,1) + \\
\nonumber &+& \Delta_{31}N(x,z;\rho_4,\rho_2))\biggr]\Biggr\}, \end{aligned}$$ and $$\Xi_{42}(x) = \theta_4\overline{\theta}_2+\theta_4^{(x)}\overline{\theta}
_2^{(x)} = \Delta_{42}+\Delta_{42}^{(x)}\ ,$$ $$\Xi_{31}(x)=\Delta_{31}+\Delta_{31}^{(x)}, \quad \overline{\Delta}_{31}^
{(x)}=\Delta_{31}-\Delta_{31}^{(x)}\ .$$ It is obvious that in symmetrical limit formula (A.7) coinsides with (A.4) one.
Appendix B {#appendixb .unnumbered}
==========
Here I give some relations which were used in the process of analytical calculations and at numerical computations. For the case of emission along the electron momentum direction they reads $$\int\limits_{\rho_2^2}^{\rho_4^2}dz\; \int\limits_{x_c}^
{1}dx\;\overline{\theta}_3^{(x)} = \int\limits_{\rho_2^2}^{\rho_4^2}dz
\;\overline{\theta}_3^{(x_c)} \int\limits_{x_c}^{\sqrt{z}/\rho_3}dx \ ,$$ $$\int\limits_{\rho_2^2}^{\rho_4^2}dz\; \int\limits_
{x_c}^{1}dx\;\int\limits_{0}^{1-x}dx_1 \overline{\theta}_3^
{(1-x_1)} =\int\limits_{\rho_2^2}^{\rho_4^2}dz\;
\overline{\theta}_3^{(x_c)} \int\limits_{x_c}^{\sqrt{z}/\rho_3}dx\;
\int\limits_{1-\sqrt{z}/\rho_3}^{1-x}dx_1\, .$$
For the case of the emission along the positron momentum direction: $$\begin{aligned}
&& \int\limits_{1}^{\rho_3^2}dz\;\int\limits_{x_c}^{1}dx\;
[\overline{\theta}_4^{(x)} - \theta_2^{(x)}] = \int\limits_{1}^
{\rho_3^2}dz\;\int\limits_{x_c}^{1}dx\; [\overline{\theta}_4-\theta_2
+\theta_4\overline{\theta}_4^{(x)} + \theta_2\overline{\theta}_2^{(x)}]
\nonumber
\\ && \quad = \int\limits_{1}^{\rho_3^2}dz\;\biggl\{ (\overline{\theta}_
4-\theta_2)\int\limits_{x_c}^{1}dx\; + \theta_4\overline{\theta}_4^{(x_c)}
\int\limits_{x_c}^{\sqrt{z}/\rho_4}dx\; + \theta_2\overline{\theta}_2
^{(x_c)}\int\limits_{x_c}^{\sqrt{z}/\rho_2}dx \biggr\}, \\ \nonumber
&& \int\limits_{1}^{\rho_3^2}dz\;
\int\limits_{x_c}^{1}dx\;\int\limits_{0}^{1-x}dx_1\;
[\overline{\theta}_4^{(1-x_1)}-\theta_2^{(1-x_1)}] = \int\limits_{1}^
{\rho_3^2}dz\;\int\limits_{x_c}^{1}dx\;\biggl\{ (\overline{\theta}_4-
\theta_2) + \int\limits_{0}^{1-x}dx_1\; \\ \nonumber && \quad
+ \overline{\theta}_4^{(x_c)}\theta_4\int\limits_{x_c}^{\sqrt{z}/\rho_
4}dx \int\limits_{1-{\sqrt{z}}/{\rho_4}}^{1-x}dx_1+\overline{\theta}_2^
{(x_c)} \theta_2\int\limits_{x_c}^{\sqrt{z}/\rho_2}dx \int\limits_
{1-\sqrt{z}/\rho_2}^{1-x}dx_1\biggr\}.\end{aligned}$$ Some additional relations arise for the case of the opposite-side emission. Let us consider first the integration limits restrictions for the product of $\theta$-functions in the symmetrical case: $$\theta_3\overline{\theta}_3^{(x_1)}\overline{\theta}_3^{(x_2)}, \quad \theta_
1\overline{\theta}_3^{(x_1)}\overline{\theta}_1^{(x_2)}, \quad \theta_1
\overline{\theta}_1^{(x_1)}\overline{\theta}_1^{(x_2)}.$$ At first it needs to use the formulae (B.1) and get rid $\;\overline{\theta}_i^{(x_2)}$ using the following changes: i)$\;\overline{\theta}_i^{(x_2)} \to
\overline{\theta}_i^{({x_c}/{x_1})}$, ii)$\;$ the upper limit of $x_2$ integration in the case of $\;\overline{\theta}_3^{(x_2)}$ has to be replaced by $({\sqrt{z}}/{\rho_3})$ and in the case of $\;\overline{\theta}_1^{(x_2)}$ by $\sqrt{z}$.
Thus, there are three regions defined by following curves in $(z,x_1)$ plane: $$\begin{aligned}
&& \rho^2=z,\quad z=x_1^2\rho^2,\quad z=\frac{x_c^2\rho^2}{x_1^2}\, ,
\\ \nonumber && 1=z,\quad z=x_1^2\rho^2,\quad z=\frac{x_1}{x_c^2}\, ,
\\ \nonumber && 1=z,\quad z=x_1^2,\quad z=\frac{x_1^2\rho^2}{x_c^2}.\end{aligned}$$ It easy to see that the limits of integrations may be transformed as follows: $$\begin{aligned}
&& \int\theta_3\overline{\theta}_3^{(x_1)}\overline{\theta}_3^{(x_2)} \
\rightarrow \int\limits_{x_c\rho^2}^{\rho^2}dz\; \int\limits_{
x_c\rho/\sqrt{z}}^{\sqrt{z}/\rho}dx_1\; \int\limits_{x_c/x_1}^
{\sqrt{z}/\rho}dx_2\; , \\ \nonumber && \int\theta_3\overline{\theta}_1^
{(x_1)}\overline{\theta}_1^{(x_2)} \rightarrow \int\limits_{x_c\rho}^
{1}dz\; \int\limits_{x_c/\sqrt{z}}^{\sqrt{z}/\rho}dx_1\; \int\limits_
{x_c/x_1}^{\sqrt{z}}dx_2\; , \end{aligned}$$ and for $\int\theta_1\overline{\theta}_1^{(x_1)}\overline{\theta}_1^{(x_2)}$ the formulae may be obtained from the above ones by putting $\rho=1$. For the wide-narrow angular acceptance the prescription is similar: $$\int\theta_4\overline{\theta}_4^{(x_1)}\overline{\theta}_3^{(x_2)}
\rightarrow \int\limits_{x_c\rho_3}^{\rho_4^2}dz\;
\int\limits_{x_c\rho_3/\sqrt{z}}^{\sqrt{z}/\rho_4}dx_1\;
\int\limits_{x_c/x_1}^{\sqrt{z}/\rho_3}dx_2\, .$$ The another variants of restrictions in wide-narrow ansular acceptancee may be transformed as follows: $$\begin{aligned}
&& \int\theta_1\overline{\theta}_2^{(x_1)}\overline{\theta}_1^{(x_2)}
\rightarrow \int\limits_{x_c\rho_2}^{1}dz\; \int\limits_{
{x_c\rho_2}/{\sqrt{z}}}^{\sqrt{z}}dx_1\; \int\limits_{{x_c}/{x_1}}^{\sqrt{z}/
\rho_2}dx_2\, , \\ \nonumber && \int\theta_1\overline{\theta}_4^{(x_1)}
\overline{\theta}_1^{(x_2)} \rightarrow \int\limits_{x_c\rho_4}^{1}dz
\; \int\limits_{x_c\rho_4/\sqrt{z}}^{\sqrt{z}}dx_1\; \int\limits_{x_c/x_1}
^{\sqrt{z}/\rho_4}dx_2\, , \\ \nonumber && \int\theta_2\overline{\theta}_2^
{(x_1)}\overline{\theta}_3^{(x_2)} \rightarrow \int\limits_{\ \ x_c\rho_2
\rho_3}^{\rho_2^2}dz\; \int\limits_{x_c\rho_3/\sqrt{z}}^{\sqrt{z}/
\rho_2}dx_1\; \int\limits_{x_c/x_1}^{\sqrt{z}/\rho_3}dx_2\, .\end{aligned}$$
References {#references .unnumbered}
==========
1. The LEP Collaboration: ALEPH, DELPI, L3 and OPAL and the LEP Electroweak Working Group, CERN–PPE/95; B. Pietrzyk, preprint LAPP-Exp-94.18. Invited talk at the Conference on “Radiative Corrections: Status and Outlook”. Galtinburg, TN, USA, 1994; I.C. Brock et al., Preprint CERN-PPE/96-89’ CMU-HEP/96-04, 27 June 1996.
2. Neutrino Counting in Z Physics at LEP, G. Barbiellini et al., L. Trentadue (conv.); G. Altarelli, R. Kleiss and C. Verzegnassi eds. CERN Report 89-08.
3. Events Generator for Bhabha Scattering, H. Anlauf et al., Conveners: S. Jadach and O. Nicrosini. Yellow Report CERN 96-01, v.2 p.229 - 298.
4. S. Jadach, E. Richter-Was, B.F.L. Ward and Z. Was, Comput.Phys.Commun. [**70**]{} (1992) 305.
5. G. Montagna et al., Comput.Phys.Commun. [**76**]{} (1993) 328; M. Cacciori, G. Montagna F. Piccinini, Comput.Phys.Commun. [**90**]{} (1995) 301, CERN-TH/95-169; G. Montagna et al., Nucl.Phys.[**B 401**]{} (1993) 3.
6. S. Jadach, M. Skrzypek and B.F.L. Ward, Phys.Rev. [**D 47**]{} (1993) 3733; S. Jadach, E. Richter-Was, B.F.L. Ward and Z. Was, Phys.Lett. [B 260]{} (1991) 438.
7. S. Jadach, E. Richter-Was, B.F.L. Ward and Z. Was, Phys.Lett. [B 353]{} (1995) 349, 362; S. Jadach, M. Melles, B.F.L. Ward and S.A. Yost, Phys.Lett.[**B 377**]{} (1996) 168.
8. W. Beenakker, F.A. Berends and S.C. van der Marck, Nucl.Phys. [**B 355**]{} (1991) 281; W. Beenakker and B. Pietrzyk, Phys.Lett. [**B 304**]{} (1993) 366.
9. M. Gaffo, H. Czyz, E. Remiddi, Nuovo Cim. [**105 A**]{} (1992) 271; Int.J.Mod.Phys. [**4**]{} (1993) 591; Phys.Lett. [/bf B 327]{} (1994) 369; G. Montagna, O. Nicrosini and F. Piccinini, Preprint FNT/T–96/8.
10. A.B. Arbuzov et al., Yellow Report CERN 95-03, p.369; preprint CERN-TH/95-313, UPRF-95-438 (to be published in Nucl.Phys.).
11. A.B. Arbuzov, E. Kuraev, N.P. Merenkov and L. Trentadue, JETP [**81**]{} (1995) 638;Preprint CERN-TH/95-241, JINR-E2-95-110.
12. D.R. Yennie, S.C. Frautchi and H. Suura, Ann.Phys. [**13**]{} (1961) 379.
13. L.N.Lipatov, Sov.J.Nucl.Phys. [**20**]{} (1974) 94; G. Altarelli and G. Parisi, Nucl.Phys. [**B 126**]{} (1977) 298; M. Skrzypek, Acta Phys.Pol. [**B 23**]{} (1992) 135.
14. N.P. Merenkov, Sov.J.Nucl.Phys. [**48**]{} (1988) 1073; [**50**]{} (1989) 469.
15. T.D. Lee and M. Nauenberg, Phys.Rev. [**B 133**]{} (1964) 1549.
16. H. Cheng and T.T. Wu, Phys.Rev.Lett. [**23**]{} (1969) 670; V.G. Zima and N.P. Merenkov, Yad.Fis. [**25**]{} (1976) 998; V.N. Baier, V.S. Fadin, V. Khoze and E. Kuraev, Phys.Rep. [**78**]{} (1981) 294.
17. V.G. Gorshkov, Uspechi Fiz. Nauk [**110**]{} (1973) 45; F.A. Berends et al., Nucl. Phys. [**B 57**]{} (1973) 371; E.A. Kuraev and G.V. Meledin, Nucl. Phys.[**B 122**]{} (1977) 3582.
18. S. Jadach and B.W.L. Ward, Phys. Rev. [**D 40**]{} (1989) 3582.
19. S. Drell and T.M. Yan, Phys. Rev. Lett, [**25**]{} (1970) 316.
20. E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys. [**41**]{} (1985) 466.
21. W. Beenakker, F.A. Berends and S.C. van der Marck, Nucl. Phys. [**B 349**]{} (1991) 323.
22. A.B. Arbuzov et al., Phys. Lett. [**B**]{} (in press).
[^1]: E–mail: kfti@rocket.kharkov.ua
|
---
abstract: |
The steadily growing use of license-free frequency bands requires reliable coexistence management and therefore proper . In this work, we propose a approach based upon a deep which classifies multiple IEEE 802.15.1, IEEE 802.11 b/g and IEEE 802.15.4 interfering signals in the presence of a utilized signal. The generated multi-label dataset contains frequency- and time-limited sensing snapshots with the bandwidth of and duration of , respectively. Each snapshot combines one utilized signal with up to multiple interfering signals.
The approach shows promising results for with a classification accuracy of approximately for IEEE 802.15.1 and IEEE 802.15.4 signals. For IEEE 802.11 b/g signals the accuracy increases for with at least .
author:
-
bibliography:
- 'literatur.bib'
title: 'Multi-Label Wireless Interference Identification with Convolutional Neural Networks'
---
{width="0.99\linewidth"}
Acknowledgement
===============
Part of this research was founded by KoMe (IGF 18350 BG/3 over DFAM, Germany) and HiFlecs (16KIS0266 over BMBF, Germany).
\[WTs\][wireless technologies]{}
|
---
author:
- 'G. Gentile'
- 'B. Famaey'
- 'W. J. G. de Blok'
title: THINGS about MOND
---
Introduction
============
\[sec:intr\] The current dominant paradigm is that galaxies are embedded in halos of cold dark matter (CDM), made of non-baryonic weakly-interacting massive particles (e.g., Bertone et al. 2005). However, an alternative way to explain the observed rotation curves of galaxies is the postulate of Milgrom (1983) that for gravitational accelerations below a certain value $a_0$, the true gravitational attraction $g$ approaches $(g_N a_0)^{1/2}$ where $g_N$ is the usual Newtonian gravitational field (as calculated from the observed distribution of visible matter): this paradigm is known as modified Newtonian dynamics (MOND).
MOND explains successfully many phenomena in galaxies, among which the following non-exhaustive list: (i) it predicted the shape of rotation curves of low surface-brightness (LSB) galaxies before any of them had ever been measured (e.g. McGaugh & de Blok 1998); (ii) tidal dwarf galaxies (TDG), which should be devoid of collisionless dark matter, still exhibit a mass-discrepancy in Newtonian dynamics, which is perfectly explained by MOND (Gentile et al. 2007); (iii) the baryonic Tully-Fisher relation (e.g., McGaugh 2005), one of the tightest observed relations in astrophysics, is a natural consequence of MOND, both for its slope and its zero-point; (iv) the first realistic simulations of galaxy merging in MOND were recently carried out, notably reproducing the morphology of the Antennae galaxies (Tiret & Combes 2008); (v) it naturally explains the universality of “dark" and baryonic surface densities within one core radius in galaxies (Donato et al. 2009, Gentile et al. 2009).
Recent theoretical developments have also added plausibility to the case for MOND through the work of, e.g., Bekenstein (2004), Sanders (2005), Zlosnik, Ferreira & Starkman (2007), Halle, Zhao & Li (2008), and Blanchet & Le Tiec (2008), who have all presented Lorentz-covariant theories yielding a MOND behavior in the weak field limit. Although still fine-tuned and far from a fundamental theory explaining the MOND paradigm, these effective theories remarkably allow for new predictions regarding cosmology (e.g., Skordis et al. 2006) and gravitational lensing (e.g., Angus et al. 2007, Shan et al. 2008). For reviews of MOND’s successes and weaknesses, both at the observational and theoretical level, as well as comparisons with dark matter results, see McGaugh & de Blok (1998), de Blok & McGaugh (1998), Sanders & McGaugh (2002), Bruneton & Esposito-Farèse (2008), Milgrom (2008), Skordis (2009), Famaey & Bruneton (2009), Ferreira & Starkman (2009).
One thing the MOND paradigm does not directly predict, though, is the shape of the interpolation between the MONDian regime where $g \ll a_0$ and the Newtonian regime where $g \gg a_0$, as well as the actual value of the acceleration constant $a_0$. The latter is in principle a free parameter, but once its value has been determined by some means, it must be identical for every astronomical object. Large variations of $a_0$ would invalidate MOND as a fundamental paradigm underpinned by new physics. Let us note that, as shown in Begeman, Broeils & Sanders (1991) fits with variable $a_0$ and fixed distance $D$ are essentially identical to fits with fixed $a_0$ and variable $D$ because the observed total gravitational acceleration is proportional to $1/D$. Ideally, the fitted distance should however generally conform to the independently determined one (e.g., Cepheids-based or RGB tip-based). Finally, a consequence of the absence of galactic dark matter within the MOND context is that the dynamical mass-to-light ratio that is derived from a rotation curve fit should agree with the true stellar mass-to-light ratio of the stellar disk (and sometimes bulge), as inferred from e.g. observed colours and stellar population synthesis models.
Here, we use results from The HI Nearby Galaxy Survey (THINGS; Walter et al 2008), which consists of high-resolution HI observations of a sample of 34 nearby galaxies, in order to constrain the transition function of MOND. In particular, we show that some individual galaxies that had been claimed to be potentially problematic for MOND such as NGC 2841 (Begeman et al. 1991) can yield good fits with the “simple" interpolating function.
We use a subset of the THINGS galaxies for which rotation curves could be derived in de Blok et al. (2008), restricting ourselves to galaxies which are not (obviously) dominated by non-circular motions. In Sect. 2, we summarize the popular choices that have been proposed in the literature for the transition between the MONDian and Newtonian regimes, in order to confront these different transitions with THINGS rotation curves. In Sect. 3, we explain how we selected the subsample of galaxies that we model in the context of MOND. Sect. 4.1. then presents the results for the value of the acceleration constant $a_0$, while Sect. 4.2. and 4.3. present the comparison of the rotation curve fits for the different transitions, especially the best-fit mass-to-light ratios and distances. Finally, in Sect. 4.4., we discuss NGC 3198, the only cases where the MOND fits perform significantly worse than dark matter fits in the context of Newtonian dynamics. Conclusions are drawn in Sect. 5.
MOND and its interpolating function
===================================
\[sec:mu\] The MOND paradigm stipulates that the Newtonian acceleration $\vec{g}_N$ produced by the visible matter is linked to the true gravitational acceleration $\vec{g}$ by means of an interpolating function $\mu$: $$\mu\left(\frac{g}{a_{0}}\right)\vec{g} = \vec{g}_{N},
\label{eq:A}$$ where $\mu(x) \sim x$ for $x \ll 1$ and $\mu(x) \sim 1$ for $x \gg 1$ (and $g=|\vec{g}|$). However, this expression cannot be exact for all orbits and all geometries, since it does not respect usual conservation laws. Such a modification of Newtonian dynamics could come at the classical (non-covariant) level from a modification of either the kinetic part or the gravitational part of the Newtonian action (with usual notations; $\phi_N$ being the Newtonian gravitational potential): $$S=\int \frac{1}{2}\rho v^2 d^3x \, dt \, - \int \left(\rho \phi_N + \frac{|\nabla \phi_N|^2}{8 \pi G}\right) d^3x \, dt,$$ where modifying the first term is referred to as [*modified inertia*]{} and modifying the second term as [*modified gravity*]{}. Milgrom (1994) has shown that within the modified inertia framework, Eq. 1 was exact [*only*]{} for circular orbits (for other orbits, predictions are difficult to make since the theory is non-local). On the other hand, Bekenstein & Milgrom (1984) have shown that within a modified gravity framework where $|\nabla \phi_N|^2$ is replaced by $a_0^2 F(|\nabla \phi|^2/a_0^2)$ in Eq. 2 ($\phi$ being the MONDian potential and $F'=\mu$), the right-hand side of Eq. 1 had to be replaced by $\vec{g}_N + \vec{s}$ where $\vec{s}$ is a solenoidal vector field determined by the condition that $\vec{g}$ can be expressed as the gradient of a MONDian potential. Milgrom (2010) has proposed another modified gravity formulation in which $|\nabla \phi_N|^2$ is replaced by $2 \nabla \phi \cdot \nabla \phi_N - a_0^2 Q(|\nabla \phi_N|^2/a_0^2)$ ($\phi$ being the MOND potential, $\phi_N$ remaining the Newtonian one, and $1/Q'=\mu$): in this case, the solenoidal field to be added to the right-hand-side of Eq. 1 is different from the one in the Bekenstein & Milgrom formulation (see also Zhao & Famaey 2010).
Although Brada & Milgrom (1995), Famaey et al. (2007) and Zhao & Famaey (2010) have shown that the expected differences in the predictions of the various formulations for rotation curves are not very large, they can be of the same order of magnitude as the differences produced by different choices for the $\mu$-function. In order to constrain $\mu$ within the modified gravity framework, one should calculate predictions of the modified Poisson formulations of Bekenstein & Milgrom (1984) or Milgrom (2010) numerically for each galaxy model, and for each choice of parameters. This is left for further works and we choose here to concentrate on the modified inertia formulation for circular orbits given by Eq. 1.
It is worth noting that other interesting constraints on MOND and its $\mu$-function could come from studies of the effect of the galactic gravitational field on the dynamics of the inner Solar System (Milgrom 2009), or from studies of the dynamics perpendicular to the galactic disk at the solar position (Bienaymé et al. 2009).
Various choices for the shape of the $\mu$-function have been proposed in the literature (see especially Milgrom & Sanders 2008 and McGaugh 2008), but we rather concentrate here on the two most popular choices that have been studied so far. The “standard" $\mu$-function (Milgrom 1983) yields a relatively sharp transition from the MONDian ($x \ll 1$, where $x=g/a_0$ and $g$ is the gravitational acceleration) to the Newtonian ($x \gg 1$) regime: $$\mu(x)= \frac{x}{\sqrt{1+x^2}},
\label{eqstandard}$$ while the “simple" $\mu$-function (Famaey & Binney 2005; Zhao & Famaey 2006) yields a more gradual transition: $$\mu(x) =\frac{x}{1+x}.
\label{eqsimple}$$
Fig. \[fig1\] displays those two $\mu$-functions as a function of $x$. Let us note that the simple function predicts that a constant acceleration equal to $a_0$ has to be added to the Newtonian gravitational acceleration for $g \gg a_0$. This is, for the values of $a_0$ compatible with galaxy rotation curves (Sect. 4.1), in strong disagreement with orbits of planets in the inner Solar System, and especially with measures of the perihelion precession of Mercury. A solution is to use an “improved simple" $\mu$-function that rapidly interpolates between the simple and standard ones for values of the gravitational acceleration $g \gtrsim 10 a_0$ (i.e. a higher value than those $g$ which are probed by galaxy rotation curves). Such an improved simple function is shown as an example on Fig. \[fig1\]. Nevertheless, we use the standard and simple $\mu$-functions hereafter (keeping in mind that the latter should be modified in the strong gravity regime) of Eqs. 3 and 4 in order to perform our MOND fits to THINGS galaxy rotation curves.
![The interpolating $\mu$-functions of Eq. 3 (standard, dashed line) and Eq. 4 (simple, dotted line) are displayed as a function of $x=g/a_0$. An improved simple function (solid line) interpolating between simple and standard for $x \gtrsim 10$ is also presented in order to show that a transition behaviour governed by the simple $\mu$-function in galaxies (where $x<10$) can a priori be in accordance with the Solar System constraints (where $x>>10$). []{data-label="fig1"}](3mus.pdf)
![Best-fit $a_0$ values (using the simple interpolating function) vs. central surface brightness in the 3.6 $\mu$m band. []{data-label="a0free"}](a0free.pdf)
The sample
==========
We use a subset of galaxies in the THINGS survey for which rotation curves were derived in de Blok et al. (2008). We restrict ourselves to galaxies which are not (obviously) dominated by non-circular motions. This means we omit the bright disk galaxies NGC 3031 and NGC 4736. While not necessarily dominated by non-circular motions, we also omit NGC 2366, IC 2574 and NGC 925. These galaxies have a neutral gas distribution that is dominated by holes and shells, the signature of which remains visible in the radial profile of the neutral gas. In these dwarf galaxies, the neutral gas profile dominates the total radial baryonic mass distribution, and as the MOND prediction is derived from the observed radial surface density distribution these remaining signatures of the holes and shells could possibly lead to erroneous results. The analysis of these specific rotation curves, interesting as they might be, is left for a forthcoming paper.
This leaves us with a total of 12 galaxies: some of these have already been discussed in Bottema et al. (2002), but with the higher resolution data available, and constrained stellar mass-to-light ratios as observed in the Spitzer IRAC $3.6 \mu {\rm m}$ band, we are able to perform a slightly more stringent test for MOND. In fact, the 12 rotation curves that we use here are the highest quality rotation curves currently available (in terms of spatial/spectral resolution and extent) for a sizeable sample of galaxies spanning a wide range of luminosities, and they therefore represent [an important]{} test for MOND or any theory that aims at fitting galaxy kinematics. In de Blok et al. (2008) some differences between their rotation curves and those of previous publications were highlighted. These differences could be caused by the different approach taken by de Blok et al. (2008) to derive the velocity field: they fit the velocity profiles using third-order Gauss-Hermite polynomials, instead of the more conventional intensity-weighted mean.
The two baryonic contributions to the rotation curve, necessary to compute the MOND mass model, were derived by de Blok et al. (2008) as follows. First, the shape (but not the amplitude) of $V_{\rm stars}$, the contribution of the stars to the rotation curve, was derived from the observed 3.6 $\mu$m surface brightness profile, and slightly modified to account for the observed $(J-K)$ colour gradients as a function of radius (which are an indication of a radially varying stellar mass-to-light $M/L$ ratio). Although there might be some contamination due to young stars and hot dust, this contamination is thought to be a negligible contribution to the flux at 3.6$\mu$m, see e.g. Pahre et al. 2004, Li et al. 2007, hence the $3.6 \mu {\rm m}$ emission is considered as good a tracer of stellar mass as the more commonly used K-band (Zhu et al. 2010). For the vertical distribution of the stellar disk, de Blok et al. (2008) assumed a sech$^2$ distribution with a scale height of $z_0 = h/5$, where $h$ is the radial exponential scale length. The amplitude of $V_{\rm stars}$ is scaled according to the [global]{} stellar mass-to-light $M/L$ ratio, which is left as a free parameter and then compared to the predictions of stellar population synthesis models (e.g. Bell & de Jong 2001). The contribution of the gaseous disk to the rotation curve, $V_{\rm gas}$, was derived from the observed HI surface density profiles, and then corrected for primordial He.
The galaxy distances are determined by various methods (Cepheids, tip of the Red Giant Branch, Hubble flow, brightest stars), and their quoted uncertainties are often close to 10%. They are listed in Table \[tab-dist\].
Name distance(Mpc) Method Ref
---------- ----------------- ----------------- -----
NGC 2403 $3.47\pm 0.29$ SN, Cepheids 1
NGC 2841 $14.1\pm 1.5 $ Cepheids 2
NGC 2903 $8.9\pm 2.2 $ brightest stars 3
NGC 2976 $3.56\pm 0.36 $ tip of the RGB 4
NGC 3198 $13.8\pm 1.5 $ Cepheids 5
NGC 3521 $10.7\pm 3.2 $ Hubble flow 6
NGC 3621 $6.64\pm 0.70 $ Cepheids 7
DDO 154 $4.30\pm 1.07 $ brightest stars 8
NGC 5055 $10.1\pm 3.0 $ Hubble flow 6
NGC 6946 $5.9\pm 1.5 $ brightest stars 8
NGC 7331 $14.72\pm1.29 $ Cepheids 9
NGC 7793 $3.91\pm 0.39 $ tip of the RGB 8
: Galaxy distances and the methods used to determine them. References are: 1: Vinkó et al. (2006). 2: Macri et al. (2001). 3: Drozdovsky & Karachentsev (2000). 4: Karachentsev et al. (2002). 5: Kelson et al. (1999), Freedman et al. (2001). 6: Walter et al. (2008). 7: Rawson et al. (1997). 8: Karachentsev et al. (2004). 9: Hughes et al. (1998).[]{data-label="tab-dist"}
Results
=======
The acceleration constant $a_0$ {#secta}
-------------------------------
The acceleration constant $a_0$ of MOND, though unknown from first principles, must be the same for all galaxies, therefore it has to be determined empirically, e.g. by fitting rotation curves. Begeman, Broeils & Sanders (1991) determined the value of the acceleration constant $a_0$ to be 1.21 $\times$ 10$^{-8}$ cm s$^{-2}$ from mass modelling of a number of nearby galaxies with the standard $\mu$-function of Eq. 3. This value was confirmed by Sanders & Verheijen (1998) using a sample of rotation curves of galaxies belonging to the Ursa Major galaxy group. However, Bottema et al. (2002) noted that using an updated value of the distance to the Ursa Major group would bring the value of $a_0$ down to 0.9 $\times$ 10$^{-8}$ cm s$^{-2}$.
The first fits that we performed were those with $a_0$ as a free parameter (the stellar $M/L$ ratio being the other free parameter). The distance in these fits was fixed at the values given in Table 1, the most accurate for each galaxy to date, to the best of our knowledge. We remind the reader that a fit with $a_0$ free and the distance fixed is equivalent to a fit with the distance free and $a_0$ fixed (Begeman et al. 1991), because the observed total gravitational acceleration is proportional to $1/D$, where $D$ is the distance.
Swaters, Sanders & McGaugh (2010) find a weak correlation between the R-band central surface brightness and the best-fit value of $a_0$ as a result of making MOND mass models of 27 dwarf and low surface brightness galaxies. They find that lower surface brightness galaxies have a tendency to have lower $a_0$. In Fig. \[a0free\] we look for a similar relation using our best-fit values of $a_0$ from the MOND fits using the simple $\mu$-function. We do not find the same correlation: indeed, the best-fit values of $a_0$ are scattered around the median value without any obvious correlation with central surface brightness. A thorough interpretation of the correlation (or lack thereof) between best-fit $a_0$ and central surface brightness goes beyond the aim of this paper, but our finding might not invalidate the interpretation by Swaters et al. (2010) that low surface brightness galaxies could be biased towards lower values of $a_0$ because of the external field effect (e.g. Milgrom 1983). In our sample, apart from DDO 154, the galaxies have a relatively high surface brightness.
The result is that the median values are (1.27 $\pm$ 0.30) $\times$ 10$^{-8}$ cm s$^{-2}$ for the standard $\mu$ function (eq. \[eqstandard\]) and (1.22 $\pm$ 0.33) $\times$ 10$^{-8}$ cm s$^{-2}$ for the simple $\mu$ function (eq. \[eqsimple\]), values that are remarkably similar to the estimates made in previous studies (the uncertainties are calculated following Müller 2000). However, for consistency we will now use these new values in the remainder of the paper. We note that our estimates of $a_0$ lie between $cH_0/(2 \pi) \approx 1.1 \times 10^{-8}$ cm s$^{-2}$ (where $H_0$ is the Hubble constant) and $c \sqrt{\Lambda}/(2 \pi) \approx 1.5 \times 10^{-8}$ cm s$^{-2}$ (where $\Lambda$ is the cosmological constant). However, the estimate of $a_0$ given by Bottema et al. (2002), 0.9 $\times$ 10$^{-8}$ cm s$^{-2}$, cannot be excluded by the present data, see Section \[sec\_distances\].
Mass-to-light ratios
--------------------
![Stellar $M/L$ ratio in the $3.6 \mu {\rm m}$ band vs. $(J-K)$ colour. The full circles are the results of the MOND fits (using the simple $\mu$-function of Eq. 4 and $a_0$ = 1.22 $\times$ 10$^{-8}$ cm s$^{-2}$) with the distance constrained within the uncertainties of its independently determined value, whereas the solid line represents the predictions of stellar population synthesis models (see text for details). []{data-label="colours"}](colours_simple_equiv.pdf)
![Stellar $M/L$ ratio in the B band vs. $(B-V)$ colour. See Fig. \[colours\] for the explanation of line and symbols. []{data-label="coloursb"}](colours_b_simple_equiv.pdf)
Starting from fixed distances to the 12 galaxies, we found in Section \[secta\] a common median value of $a_0$ corresponding to each interpolating $\mu$-function. Using this we perform 6 different types of fits to each galaxy rotation curve. For each of the two $\mu$-functions, we make fits with a fixed value of $a_0$ and (i) a fixed distance, (ii) a distance constrained to lie within the error bars from its independent determination, and (iii) a free distance with no constraints. In all cases, the stellar mass-to-light ratio of the disk (and bulge if present) is left as a free parameter. All the results are listed in Table 2.
From the $\chi^2$ values [^1], the lack of systematic deviations and the small number of highly discrepant data points, one can conclude that the fits are generally good, with only a few exceptions (cf. next Section). However, one has to check that, when a bulge is present, the mass-to-light ratio of the disk is smaller than that of the bulge, and that the stellar $M/L$ ratios are realistic.
There are five galaxies with a bulge. Using the standard $\mu$ function, two galaxies have the best-fit stellar $M/L$ of the disk larger than the one of the bulge. Of the remaining three cases, one is undetermined (NGC 2903: its best-fit $M/L$ of the bulge is zero but values larger than the best-fit $M/L$ of the disk give almost equally good fits; note also the likely presence of a bar, Leroy et al. 2009), and two are realistic (the $M/L$ of the disk is smaller than $M/L$ of the bulge). On the other hand, when using the simple $\mu$ function no such problems arise, and the $M/L$ of the bulge is always realistic. We therefore conclude, in line with Sanders & Noordermeer (2007), that the simple $\mu$ function gives superior fits. For the rest of the paper we will thus only use the simple $\mu$ function. This justifies our proposal in Fig. \[fig1\] of a $\mu$ function that resembles the simple one at typical galactic gravitational accelerations (and the standard one for higher accelerations representative of, e.g, the Solar System).
In order to check how realistic the fitted stellar $M/L$ ratios are, we compared them with the results of stellar population synthesis models. In Fig. \[colours\] we plot the best-fit “global” $M/L$ ratio in the $3.6 \mu {\rm m}$ band vs. $(J-K)$: $$M/L=\frac{(M/L)_{\rm disk} L_{\rm disk} + (M/L)_{\rm bulge} L_{\rm bulge}}{L_{\rm disk} + L_{\rm bulge}}$$ The solid line represents the population synthesis models prediction (from Bell & de Jong 2001, using also eq. 4 of de Blok et al. 2008) with a “diet-Salpeter" IMF. Although with some scatter, the points lie close to the prediction, and the $M/L$ ratios in the $3.6 \mu {\rm m}$ band vary very gradually with colour, staying constant around 0.5-1. It is also interesting to compare our results in a band where the predicted stellar $M/L$ varies more rapidly with colour. To achieve this, we converted our $M/L$ ratios to B-band (using the B-band luminosity given in de Blok et al. 2008), and we made use of the corrected $(B-V)$ colours given in the HyperLeda database (Paturel et al. 2003). For NGC 7793, the corrected $(B-V)$ colour was not available and we used the effective one. The results are shown in Fig. \[coloursb\]. The best-fit stellar $M/L$ of MOND closely follows the predictions of Bell & de Jong (2001), in that redder galaxies are best fitted with a higher stellar $M/L$ ratio.
The only three galaxies where the best-fit disk $M/L$ ratio differs from the population synthesis one by more than a factor of two are NGC 2903, NGC 2976, and NGC 7331 (see Fig. \[colours\] and Table 2). In NGC 2903 the MOND fit significantly overpredicts the $M/L$, a phenomenon that was also observed in Newtonian mass models with dark matter (de Blok et al. 2008). Let us note that, if one would follow population synthesis predictions, the predicted stellar disk would be surprisingly very sub-maximum for a massive galaxy with a rapidly increasing rotation curve with maximum velocity $\sim$ 215 km s$^{-1}$. In addition, in the central parts of NGC 2903 there is evidence for a bar (e.g. Leroy et al. 2009): the non-circular motions associated to it further increase the uncertainties on the mass modelling results (see also Sellwood & Zánmar Sánchez 2010). For NGC 2976, on the other hand, the MOND fits underpredict the stellar $M/L$ ratio; but again, this is also observed in the Newtonian mass models with dark matter. In addition, having a stellar disk that strongly dominates the kinematics over most of the extent of the gaseous disk in a galaxy with maximum velocity $\sim$ 85 km s$^{-1}$ would also be surprising. Mass model degeneracies in this galaxy (in particular the MOND mass models with distance free and distance constrained) are complicated by the very similar shapes of $V_{\rm stars}$ and $V_{\rm gas}$. In NGC 2976 too, Leroy et al. (2009) find an indication for a weak bar. Also in NGC 7331 the MOND fits give a lower $M/L$ compared to the expectations from the colours; this is the case also in the dark matter fits (see de Blok et al. 2008 where a strong dust ring is suggested as a possible explanation for the inflated stellar $M/L$ ratios predicted from the colours).
Distances {#sec_distances}
---------
One then also has to check that the fits with the distances constrained to lie within the error bars from their independent determinations are of good quality. When this is not the case, it means that MOND would predict another distance than what has been measured to date. NGC 2841 is, e.g., the most well-known and most persistently problematic galaxy for MOND. Begeman et al. (1991) pointed out that a good MOND fit could only be obtained if the galaxy was a factor $\sim$ 2 further away than the Hubble distance of $\sim$ 9.5 Mpc. This large discrepancy was alleviated somewhat when HST Cepheids measurements suggested a distance of 14.1 Mpc (Macri et al. 2001), but the discrepancy remained. However, these fits were performed with the standard $\mu$-function (Eq. 3) and not the simple one (Eq. 4). Our fits here show that the problem of the distance is solved when using the simple $\mu$-function (see the reduced $\chi^2$ in Table 2), and that the stellar mass-to-light ratio is also in accordance with population synthesis models.
All the fits of the 12 high-quality rotation curves, using the simple $\mu$-function of Eq. 4, $a_0=1.22 \times 10^{-8}$ cm s$^{-2}$, and a distance lying within the error bars coming from an independent distance determination, are shown in Fig. \[fits\]. The fits are clearly very good for 9 galaxies (including NGC 3521, whose high reduced $\chi^2$ value in Table 2 is dominated by the innermost two points, which have highly uncertain position angle and inclination, see de Blok et al. 2008). Among the galaxies with the 3 least good fits (NGC 3198, NGC 7793, and NGC 2976), we do not discuss further NGC 7793 and NGC 2976, since the MOND fits present the same failures as the dark matter fits (in Newtonian dynamics), therefore we do not consider them as evidence against MOND. We just briefly note that in NGC 7793 the value of the inclination angle fitted by de Blok et al. (2008) is low and presents large variations in adjacent radii, which results in a poorly constrained rotation curve; in NGC 2976 the amplitude of the non-circular motions (Trachternach et al. 2008) is correlated with the amplitude of the fit residuals.
Before proceeding with a detailed analysis of possible problems with the rotation curve of NGC 3198 in Sect. \[sec\_3198\], we finally consider the possibility that the true value of $a_0$ for all galaxies is actually at the lower end of our best-fit interval of Sect. \[secta\], i.e. a value compatible with the one determined by Bottema et al. (2002). As a matter of fact, a good reason for this is that the Ursa Major (UMa) galaxy group (e.g., Sanders & Verheijen 1998, Gentile, Zhao & Famaey 2008) is nowadays thought to be at a distance of 18.6 Mpc (Tully & Pierce 2000), implying a best-fit value of $a_0$ close to the one of Bottema et al. (2002), see e.g. Gentile et al. (2008). To get as good fits to the rotation curves of UMa galaxies as those obtained with $a_0=0.9 \times 10^{-8}$ cm s$^{-2}$ with a higher value of the order of $a_0=1.2 \times 10^{-8}$ cm s$^{-2}$, the distance of the group should be smaller, of the order of 15 Mpc (as originally assumed by Sanders & Verheijen 1998). For this reason, we plot in Fig. \[fits\_0p9\] the fits of the 12 rotation curves using $a_0=0.9 \times 10^{-8}$ cm s$^{-2}$ (and still the simple $\mu$-function and the distance constrained to lie within the error bars of Table \[tab-dist\]). As can be seen, the fits remain of approximately the same quality, apart for 3 galaxies: NGC 2841 and NGC 2403 have worse fits [^2]
![Rotation curve fit of NGC 3198 with $a_0=1.2 \times 10^{-8}$ cm s$^{-2}$ the distance as a free parameter ($\mu$ simple $d$ free in Table 2). The distance is 8.6 Mpc and the stellar mass-to-light ratio in the 3.6$\mu$m band is 1.01. The lines are described in Fig. \[fits\]. []{data-label="3198"}](rcfit_3198_simple.pdf)
![Rotation curve fit of NGC 3198 with $a_0=0.9 \times 10^{-8}$ cm s$^{-2}$, a distance of 11.2 Mpc (see Section \[sec\_3198\]), a best-fit stellar M/L ratio in the 3.6$\mu$m band of 0.76, and the simple $\mu$ function. The lines are described in Fig. \[fits\]. []{data-label="3198_11p2"}](rcfit_3198_simple_0p9_11p2.pdf)
NGC 3198 {#sec_3198}
--------
An excellent fit can be found with $a_0=1.2 \times 10^{-8}$ cm s$^{-2}$ by leaving the distance unconstrained. As already noted in previous studies (e.g. Bottema et al. 2002), MOND prefers a smaller distance. Fig. \[3198\] shows a very good fit with a distance of 8.6 Mpc, which is significantly lower than the Cepheids-based one of 13.8 Mpc. The distance that one would get by fixing the stellar $M/L$ ratio to the population synthesis value is also lower than the Cepheids value (9.6 Mpc). We note that other methods (the Tully-Fisher distance or the Hubble flow distance, both less accurate than the Cepheids) also yield lower values (see van Albada et al. 1985 and Bottema et al. 2002) than the Cepheids. It is however also interesting to note that the regions where the fit with the distance constrained is most discrepant (roughly, the first and last thirds of the rotation curve) are also those where the amplitude of the non-circular motions is higher, taken from Trachternach et al. (2008) and they are about of the same order of magnitude. The increase of the non-circular motions from the middle part of the rotation curve to the outer parts at $\sim10$ km s$^{-1}$ is also noted in Sellwood & Zánmar Sánchez (2010). The interpretation is further complicated by the fact that the IRAC 3.6 $\mu$m image shows what seems to be an end-on bar in the very inner part of NGC 3198. The main spiral arms of NGC 3198 originate here, and it is not clear how these affect the dynamics. The use of two different disks for the stellar contribution (see de Blok et al. 2008) does not change significanly the results. We thus conclude that a full MOND modelling of the two-dimensional velocity field of this galaxy, taking into account the full modified Poisson equation(s) of Bekenstein & Milgrom (1984) or Milgrom (2010), as well as the non-axisymmetry of the galactic potential, would lead to a benchmark test for the viability of MOND as a modification of gravity. We note, however, that a lower value of $a_0$would improve the fit of NGC 3198 although the fits of NGC 2841 and NGC 2403 would get worse. With $a_0=0.9 \times 10^{-8}$ cm s$^{-2}$, a fit equivalent to the one of Fig. \[3198\] would be obtained for a distance of 10.3 Mpc instead of 8.6 Mpc, closer to the Cepheid-based distance of 13.8 Mpc. Actually, it has been mentioned in the literature that there might be a reddening problem in determining the distance. In Macri et al. (2001), the H-band Cepheids distance of NGC 3198 is 11.2 Mpc. So, with $a_0=0.9 \times 10^{-8}$ cm s$^{-2}$, there might be no problem with the rotation curve of NGC 3198 (see Fig. \[3198\_11p2\]).
Scaling relations
-----------------
An interesting way to interpret the above MOND results is to phrase them in terms of the usual dark matter framework, considering MOND as a phenomenological, empirical, law encompassing the behavior of dark matter in galaxies. The additional gravity generated by MOND, compared to the Newtonian case, can indeed be attributed to what one would call dark matter in the Newtonian context, and this effective matter is called “phantom dark matter". For the gravity generated by baryons, we hereafter use the mass-to-light ratios from the fits made with the simple interpolation function and the distance constrained.
In Fig. \[mda\_things\_gn\], we display the scaling relation known as the Mass Discrepancy-Acceleration relation (McGaugh 2004), showing that the ratio of enclosed total dynamical mass (in Newtonian gravity) w.r.t. enclosed baryonic mass at any radius is a function of the gravity generated by the baryons at this radius. This relation precisely traces the $\mu$-function of MOND, and the small scatter around the line indicate the deviation from the MOND behavior, mostly consistent with observational errors.
Then, following Walker et al. (2010), we plot the gravity of (phantom) dark matter as a function of radius for the 12 galaxies of the sample. This is plotted on Fig. \[acc\_vs\_r\]. We remarkably find that the additional gravity predicted by MOND is in accordance with the mean and scatter in Fig. 1 of Walker et al. (2010). However, if one plots (Fig. \[phantom\_gravity\_gn\]) this additional gravity as a function of the baryonic [*gravitational acceleration*]{}, the scatter is much lower and samples the $\mu$-function. Since for the considered range of such gravitational accelerations, the range of “phantom dark matter" gravities is not very large, it gives the illusion of a dark matter gravity which is more or less constant with radius.
Conclusion
==========
We re-analysed the ability of the Modified Newtonian Dynamics (MOND; Milgrom 1983) paradigm to fit galaxy rotation curves, using the most up-to-date high-resolution HI data for nearby ($d<15$ Mpc) galaxies from the recent THINGS survey (Walter et al. 2008). We selected a subset of 12 galaxies not obviously dominated by non-circular motions, and yielding the most reliable mass models.
First, we redetermined the value of the acceleration parameter in MOND ($a_0$), which is unknown a priori but has to be the same for all galaxies. This was done for both commonly used interpolating functions $\mu$ of MOND. We find a median value of $a_0=(1.27$ $\pm$ 0.30) $\times$ 10$^{-8}$ cm s$^{-2}$ for the “standard" $\mu$ function (Eq. 3), and $a_0=(1.22$ $\pm$ 0.33) $\times$ 10$^{-8}$ cm s$^{-2}$ for the “simple" $\mu$ function (Eq. 4), very close to the value that had been determined in previous studies (e.g. Begeman et al. 1991).
Then, fixing these values for $a_0$, we performed three fits for each $\mu$ function: with the distance fixed at the value determined in an independent way, then by leaving the distance free but constrained within the uncertainties of this distance determination, and then with the distance as a free parameter with no constraints (Table 2). We find that the MOND fits with the distance “constrained" are of very good quality (Fig. \[fits\]), with three exceptions: two of these are galaxies that cannot give good fits using Newtonian dynamics plus dark matter (NGC 2976 and NGC 7793) either, see de Blok et al. (2008). For the remaining galaxy (NGC 3198) there is indeed some tension between observations and the MOND fit, that might be explained by the presence of non-circular motions, a small distance (see Fig. \[3198\]), or a value of $a_0$ at the lower end of our best-fit interval (see Fig. \[3198\_11p2\]). In any case, further observations (constraining the distance) and modelling of NGC 3198 in the MOND context should thus lead to a benchmark test for MOND as a modification of gravity. But we also show that MOND, as an empirical law encompassing the behavior of the gravitational field on galaxy scales, whatever its cause, is still very successful and summarizes old and new scaling relations with a remarkable consistency (Sect. 4.5).
We also conclude that, both from arguments of best-fit stellar mass-to-light ratios (Sect 4.2) and best-fit distances (Sect. 4.3), the simple $\mu$-function is preferred over the standard one. As noted by Famaey & Binney (2005) and McGaugh (2008), this is also the case when fitting the terminal velocity curve of our own Milky Way galaxy. Angus, Famaey & Diaferio (2010) also reached the same conclusion from using temperature profiles of the X-ray emitting gas of a sample of clusters, and from assuming that dark matter in MONDian galaxy clusters is made of 11eV fermionic particles[^3].
Let us however note that, in order to constrain $\mu$ from galaxy rotation curves within the modified gravity framework of MOND (see Sect. 2), one should actually calculate predictions of the modified Poisson formulations of Bekenstein & Milgrom (1984) or Milgrom (2010) numerically for each galaxy model, and for each choice of parameters. Our present conclusion for THINGS galaxy rotation curves does hold only for the modified inertia formulation for circular orbits given here by Eq. 1.
![Mass discrepancy-acceleration relation using the rotation curve data of our sample. The gravitational acceleration generated by baryons ($g_{\rm N}$) is measured in km$^{2}$ s$^{-2}$ kpc$^{-1}$ and they result from the fits made with the simple interpolation function and the distance constrained in Table 2. Black (open) circles represent the data points with an uncertainty larger than 5%. The data points with an uncertainty smaller than 5% are shown as red (full) circles. []{data-label="mda_things_gn"}](mda_things_gn.pdf)
![Gravitational acceleration generated by phantom dark matter ($g_{\rm phantomDM}$, measured in km$^{2}$ s$^{-2}$ pc$^{-1}$) versus radius (in pc). The values of $g_{\rm phantomDM}$ result from the fits made with the simple interpolation function and the distance constrained in Table 2. []{data-label="acc_vs_r"}](acc_vs_r.pdf)
![Gravitational acceleration generated by phantom dark matter ($g_{\rm phantomDM}$) versus gravitational acceleration generated by baryons ($g_{\rm N}$). The units are km$^{2}$ s$^{-2}$ pc$^{-1}$. The values of $g_{\rm phantomDM}$ and $g_{\rm N}$ result from the fits made with the simple interpolation function and the distance constrained in Table 2. The solid line represents the simple interpolation function of MOND (eq. \[eqsimple\]) []{data-label="phantom_gravity_gn"}](phantom_gravity_gn.pdf)
Acknowledgements {#acknowledgements .unnumbered}
================
GG is a postdoctoral researcher of the FWO-Vlaanderen (Belgium). BF is a CNRS Senior Resarch Associate (France) and acknowledges the support of the AvH foundation (Germany). The work of WJGdB is based upon research supported by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr). We are greatful to the whole THINGS (The HI Nearby Galaxy Survey) team, and in particular to Fabian Walter and Elias Brinks for valuable suggestions. We also wish to thank Garry Angus for very useful comments on the paper. Finally, we thank the referee, Bob Sanders, for valuable comments that improved the quality and scientific content of this paper.
Angus, G. W., Shan, H. Y., Zhao, H. S., & Famaey, B. 2007, ApJ, 654, L13 Angus, G. W., Famaey, B., & Diaferio, A. 2010, MNRAS, 402, 395 Begeman, K. G., Broeils, A. H., Sanders, R. H., 1991, MNRAS, 249, 523 Bekenstein, J., & Milgrom, M. 1984, ApJ, 286, 7 Bekenstein, J. D., 2004, Phys. Rev. D., 70, 083509 Bell, E. F., & de Jong, R. S. 2001, ApJ, 550, 212 Bertone, G., Hooper, D., & Silk, J. 2005, Physics Reports, 405, 279 Bienaym[é]{}, O., Famaey, B., Wu, X., Zhao, H. S., & Aubert, D. 2009, A&A, 500, 801 Blanchet, L., & Le Tiec, A. 2008, Phys. Rev. D., 78, 024031 Bottema, R., Pesta[ñ]{}a, J. L. G., Rothberg, B., & Sanders, R. H. 2002, A&A, 393, 453 Brada, R., & Milgrom, M. 1995, MNRAS, 276, 453 Bruneton, J.-P., & Esposito-Far[è]{}se, G. 2007, Phys. Rev. D., 76, 124012 de Blok, W. J. G., & McGaugh, S. S. 1998, ApJ, 508, 132 de Blok, W. J. G., Walter, F., Brinks, E., Trachternach, C., Oh, S.-H., & Kennicutt, R. C. 2008, AJ, 136, 2648 Donato, F., et al. 2009, MNRAS, 397, 1169 Drozdovsky, I. O., & Karachentsev, I. D. 2000, A&AS, 142, 425 Famaey, B., Binney, J., 2005, MNRAS, 363, 603 Famaey, B., Gentile, G., Bruneton, J.-P., & Zhao, H. 2007, Phys. Rev. D., 75, 063002 Famaey, B., & Bruneton, J.-P. 2009, arXiv:0906.0578 Ferreira P., & Starkman, G. 2009, Science, 326, 812 Ferreras, I., Mavromatos, N., Sakellariadou, M., Yusaf, M. 2009, Phys. Rev. D, 80, 103506 Fraternali, F., van Moorsel, G., Sancisi, R., & Oosterloo, T. 2002, AJ, 123, 3124 Freedman, W. L., et al. 2001, ApJ, 553, 47 Gentile, G., Famaey, B., Combes, F., Kroupa, P., Zhao, H. S., Tiret, O. 2007, A&A, 472, L25 Gentile, G., Zhao, H. S., & Famaey, B. 2008, MNRAS, 385, L68 Gentile, G., Famaey, B., Zhao, H., & Salucci, P. 2009, Nature, 461, 627 Halle, A., Zhao, H., & Li, B. 2008, ApJS, 177, 1 Hughes, S. M. G., et al. 1998, ApJ, 501, 32 Karachentsev, I. D., et al. 2002, A&A, 383, 125 Karachentsev, I. D., Karachentseva, V. E., Huchtmeier, W. K., & Makarov, D. I. 2004, AJ, 127, 2031 Kent, S. M. 1987, AJ, 93, 816 Kelson, D. D., et al. 1999, ApJ, 514, 614 Leroy, A. K., et al. 2009, AJ, 137, 4670 Li, H.-N., Wu, H., Cao, C., & Zhu, Y.-N. 2007, AJ, 134, 1315 Macri, L. M., Stetson, P. B., Bothun, G. D., Freedman, W. L., Garnavich, P. M., Jha, S., Madore, B. F., & Richmond, M. W. 2001, ApJ, 559, 243 McGaugh, S. S., & de Blok, W. J. G. 1998, ApJ, 499, 41 McGaugh, S. S., 2004, ApJ, 609, 652 McGaugh, S. S., 2005, ApJ, 632, 859 McGaugh, S. S. 2008, ApJ, 683, 137 Milgrom, M., 1983, ApJ, 270, 365 Milgrom, M. 1994, Annals of Physics, 229, 384 Milgrom, M. 2008, arXiv:0801.3133 Milgrom, M., & Sanders, R. H. 2008, ApJ, 678, 131 Milgrom, M. 2009, MNRAS, 399, 474 Milgrom, M. 2010, MNRAS, 403, 886 Müller, J. W. 2000, J. Res. Natl. Inst. Stand. Technol. 105, 551 Pahre, M. A., Ashby, M. L. N., Fazio, G. G., & Willner, S. P. 2004, ApJS, 154, 229 Paturel G., Petit C., Prugniel P., Theureau G., Rousseau J., Brouty M., Dubois P., Cambr[é]{}sy L., 2003, A&A, 412, 45 Rawson, D. M., et al. 1997, ApJ, 490, 517 Sanders, R. H., & Verheijen, M. A. W. 1998, ApJ, 503, 97 Sanders, R. H., McGaugh, S. S., 2002, ARA&A, 40, 263 Sanders, R. H. 2005, MNRAS, 363, 459 Sanders, R. H., & Noordermeer, E. 2007, MNRAS, 379, 702 Sellwood, J., & Zanmar Sanchez, R. 2010, MNRAS, 404, 1733 Shan, H. Y., Feix, M., Famaey, B., & Zhao, H. 2008, MNRAS, 387, 1303 Skordis, C., Mota, D. F., Ferreira, P. G., & B[œ]{}hm, C. 2006, Physical Review Letters, 96, 011301 Skordis, C. 2009, Classical and Quantum Gravity, 26, 143001 Swaters, R. A., Sanders, R. H., & McGaugh, S. S. 2010, arXiv:1005.5456 Tian, L., Hoekstra, H., & Zhao, H.S. 2009, MNRAS, 393, 885 Tiret, O., & Combes, F. 2008, Astronomical Society of the Pacific Conference Series, 396, 259 Trachternach, C., de Blok, W. J. G., Walter, F., Brinks, E., & Kennicutt, R. C. 2008, AJ, 136, 2720 Tully, R. B., & Pierce, M. J. 2000, ApJ, 533, 744 van Albada, T. S., Bahcall, J. N., Begeman, K., & Sancisi, R. 1985, ApJ, 295, 305 Vink[ó]{}, J., et al. 2006, MNRAS, 369, 1780 Walker, M., McGaugh, S., Mateo, M., Olszewski, E., & Kuzio de Naray, R. 2010, arXiv:1004.5228 Walter, F., Brinks, E., de Blok, W. J. G., Bigiel, F., Kennicutt, R. C., Thornley, M. D., & Leroy, A. 2008, AJ, 136, 2563 Zhao, H. S., & Famaey, B. 2006, ApJ, 638, L9 Zhao, H. S., & Famaey, B. 2010, Phys. Rev. D., 81, 087304 Zhu, Y.-N., Wu, H., Li, H.-N., & Cao, C. 2010, Research in Astronomy and Astrophysics, 10, 329 Zlosnik, T. G., Ferreira, P. G., & Starkman, G. D. 2007, Phys. Rev. D., 75, 044017
[^1]: \[footnotechi\]We note that in de Blok et al. (2008) the velocity difference between the approaching and receding side was considered in the error budget of each point of the rotation curve. In many cases this results in errorbars that are larger than the point-to-point scatter, which implies that the $\chi^2$ values cannot be used as probability indicators.
[^2]: In this case the less well fitted galaxy would be NGC 2403. Possible effects could be the fact that the $r$, J, H and K photometric profiles have a different shape from the $3.6 \mu {\rm m}$ band (see Kent 1987, Fraternali et al. 2002, de Blok et al. 2008). In addition, Fig. 7 of de Blok et al. (2008) shows that the outer parts of the rotation curve are quite uncertain. In that paper, the stellar component is also modelled with two separate components: the use of two different disks for the stellar contribution does not change significanly the results., but the quality of the fit of NGC 3198 improves.
[^3]: Let us note that such a hot dark matter component could also play a role in the strong and weak gravitational lensing of elliptical galaxies (Ferreras et al. 2009, Tian et al. 2009)
|
---
abstract: 'The description of the intersections of components of a Springer fiber is a very complex problem. Up to now only two cases have been described completely. The complete picture for the hook case has been obtained by N. Spaltenstein and J.A. Vargas, and for two-row case by F.Y.C. Fung. They have shown in particular that the intersection of a pair of components of a Springer fiber is either irreducible or empty. In both cases all the components are non-singular and the irreducibility of the intersections is strongly related to the non-singularity. As it has been shown in [@M-P] a bijection between orbital varieties and components of the corresponding Springer fiber in ${\rm GL}_n$ extends to a bijection between the irreducible components of the intersections of orbital varieties and the irreducible components of the intersections of components of Springer fiber preserving their codimensions. Here we use this bijection to compute the intersections of the irreducible components of Springer fibers for two-column case. In this case the components are in general singular. As we show the intersection of two components is non-empty. The main result of the paper is a necessary and sufficient condition for the intersection of two components of the Springer fiber to be irreducible in two-column case. The condition is purely combinatorial. As an application of this characterization, we give first examples of pairs of components with a reducible intersection having components of different dimensions.'
address:
- 'Department of Mathematics, University of Haifa, Haifa 31905, Israel.'
- ' Universität Duisburg-Essen Fachbereich Mathematik, Campus Essen 45117 Essen, Germany .'
author:
- 'A. Melnikov'
- 'N.G.J. Pagnon'
title: Reducibility of the intersections of components of a Springer Fiber
---
[^1]
Introduction
============
{#1.1}
Let $G$ denote the complex linear algebraic group ${\rm GL}_n$ with Lie algebra ${\mathfrak{g}}=\mathfrak{gl}_n$ on which $G$ acts by the adjoint action. For $g\in G$ and $u\in{\mathfrak{g}}$ we denote this action by $g.u:=gug^{-1}.$
We fix the standard triangular decomposition ${\mathfrak{g}}={\mathfrak{n}}\oplus{\mathfrak{h}}\oplus{\mathfrak{n}}^-$ where ${\mathfrak{n}}$ is the subalgebra of strictly upper-triangular matrices, ${\mathfrak{n}}^-$ is the subalgebra of strictly lower triangular matrices and ${\mathfrak{h}}$ is the subalgebra of diagonal matrices of ${\mathfrak{g}}.$ Let ${\mathfrak{b}}:={\mathfrak{h}}\oplus{\mathfrak{n}}$ be the standard Borel subalgebra so that ${\mathfrak{n}}$ is its nilpotent radical. Let $B$ be the (Borel) subgroup of invertible upper-triangular matrices in $G$ so that ${\mathfrak{b}}={\rm
Lie}(B)$. The associated Weyl group $W=\langle s_i\rangle_{i=1}^{n-1}$ where $s_i$ is a reflection w.r.t. a simple root $\alpha_i$ is identified with the symmetric group ${\mathbf{S}}_n$ by taking $s_i$ to be an elementary permutation interchanging $i$ and $i+1.$
Let ${\mathcal{F}}:=G/B$ denote the flag manifold. Let $G{\times}^{B}{\mathfrak{n}}$ be the space obtained as the quotient of $G\times {{\mathfrak{n}}}$ by the right action of $B$ given by $(g,x).b:=(gb,b^{-1}.x)$ with $g\in G$, $x\in {{\mathfrak{n}}}$ and $b\in B$. By the Killing form we identify the space $G{\times}^{B}{{\mathfrak{n}}}$ with the cotangent bundle of the flag manifold $ T^{*}(G/B)$. Let $g*x$ denote the class of $(g,x)$. The map $G{\times}^{B}{{\mathfrak{n}}}\rightarrow
{\mathcal{F}}\times {\mathfrak{g}}, g\ast x\mapsto(gB,g.x)$ is an embedding which identifies $G{\times}^{B}{{\mathfrak{n}}}$ with the following closed subvariety of ${\mathcal{F}}\times{\mathfrak{g}}$ (see. [@Slo1 p. 19]): $$\mathcal{Y}:=\{(gB,x)\ |\ x\in g.{{\mathfrak{n}}}\}$$ The map $f:G{\times}^{B}{{\mathfrak{n}}}\rightarrow {\mathfrak{g}}, g*x\mapsto g.x$ is called the [**Springer resolution**]{}. It embeds into the following commutative diagram:
(100,70) (-5,50)[$G{\times}^{B}{{\mathfrak{n}}}$]{}(90,50)[${\mathcal{F}}\times {\mathfrak{g}}$]{} (45,5)[${\mathfrak{g}}$]{}(30,55)[(1,0)[55]{}]{}(50,58)[$i$]{} (8,45)[(1,-1)[35]{}]{}(15,25)[$\scriptstyle f$]{} (85,45)[(-1,-1)[35]{}]{}(70,25)[$\scriptstyle pr_2$]{}
where ${pr}_{2}:{\mathcal{F}}\times {\mathfrak{g}}\rightarrow {\mathfrak{g}}, (gB,x)\mapsto x$ is the natural projection. Since ${\mathcal{F}}$ is complete and $i$ is closed embedding $f$ is proper (because $G/B$ is complete) and its image is exactly $G.{{\mathfrak{n}}}=\mathcal{N}$, the [**nilpotent variety**]{} of ${\mathfrak{g}}$ (cf. [@Spr1]).
Let $x$ be a nilpotent element in ${{\mathfrak{n}}}$. By the diagram above we have :\[description\] $${{\mathcal{F}}}_{x}:=f^{-1}(x)=\{gB\in {\mathcal{F}}\ |\ x\in g.{{\mathfrak{n}}}\ \}= \{gB\in {\mathcal{F}}\
|\ g^{-1}.x\in {{\mathfrak{n}}}\ \}.\eqno{(*)}$$ The variety ${{\mathcal{F}}}_{x}$ is called the [**Springer fiber**]{} above $x$. It has been studied by many authors. Springer fibers arise as fibers of Springer’s resolution of singularities of the nilpotent cone in [@Slo1; @Spr1]. In the course of these investigations, Springer defined ${\mathcal{W}}$-module structures on the rational homology groups $H_{*}({\mathcal{F}}_{x},\mathbb{Q})$ on which also the finite group $A(x)=Z_G(x)/{Z_G^o(x)}$ (where $Z_G(x)$ is a stabilizer of $x$ and $Z_G^o(x)$ is its identity component) acts compatibly. Recall that $A(x)$ is trivial for $G={\rm GL}_n.$ For $d={\rm dim}({\mathcal{F}}_{x})$ the $A(x)$-fixed subspace $H_{2d}({\mathcal{F}}_{x},\mathbb{Q})^{A(x)}$ of the top homology is known to be irreducible as a $W$-module[@Spr3].
In [@Kaz], D. Kazhdan and G. Lusztig tried to understand Springer’s work connecting nilpotent classes and representations of Weyl groups. Among problems posed there, Conjecture 6.3 in [@Kaz] has stimulated the research of the relation between the Kazhdan-Lusztig basis and Springer fibers.
Let $x\in{{\mathfrak{n}}}$ be a nilpotent element and let ${\mathcal{O}}_x=G.x$ be its orbit. Consider ${\mathcal{O}}_x\cap{{\mathfrak{n}}}.$ Its irreducible components are called [**orbital varieties**]{} associated to ${\mathcal{O}}_x.$ By Spaltenstein’s construction [@Spa] ${\mathcal{O}}_x\cap{{\mathfrak{n}}}$ is a translation of ${\mathcal{F}}_x$ (see Section \[2.1\]).
{#1.2}
For $x\in{\mathfrak{n}}$ its Jordan form is completely defined by $\lambda=(\lambda_1,\ldots \lambda_k)$ a partition of $n$ where $\lambda_i$ is the length of $i-$th Jordan block. Arrange the numbers in a partition $\lambda=(\lambda_1,\ldots \lambda_k)$ in the decreasing order (that is $\lambda_1\geqslant\lambda_2\geqslant\cdots\geqslant
\lambda_k\geqslant 1$) and write $J(x)=\lambda.$ Note that the nilpotent orbit ${\mathcal{O}}_x$ is completely defined by $J(x).$ We set ${\mathcal{O}}_{J(x)}:={\mathcal{O}}_x$ and ${{\rm sh\,}}({\mathcal{O}}_x):=J(x).$
In turn an ordered partition can be presented as a Young diagram $D_{\lambda}$ – an array of $k$ rows of boxes starting on the left with the $i$-th row containing $\lambda_i$ boxes. In such a way there is a bijection between Springer fibers (resp. nilpotent orbits) and Young diagrams.
Fill the boxes of Young diagram $D_\lambda$ with $n$ distinct positive integers. If the entries increase in rows from left to right and in columns from top to bottom we call such an array a Young tableau or simply a tableau of shape $\lambda.$ Let ${\mathbf{Tab}}_{\lambda}$ be the set of all Young tableaux of shape $\lambda.$ For $T\in{\mathbf{Tab}}_{\lambda}$ we put ${{\rm sh\,}}(T):=\lambda.$
By Spaltenstein ([@Spa0]) and Steinberg ([@Ste2]) for $x\in{\mathfrak{n}}$ such that $J(x)=\lambda$ there is a bijection between the set of irreducible components of ${\mathcal{F}}_x$ (resp. orbital varieties associated to ${\mathcal{O}}_\lambda$) and ${\mathbf{Tab}}_\lambda$ (cf. Section \[2.3\]). For $T\in {\mathbf{Tab}}_{\lambda}$, set ${\mathcal{F}}_T$ to be the corresponding component of ${\mathcal{F}}_x.$ Respectively set ${\mathcal{V}}_T$ to be the corresponding orbital variety associated to ${\mathcal{O}}_\lambda.$ Moreover, as it has been established in [@M-P] (cf. Section \[2.1\]) for $T,T'\in{\mathbf{Tab}}_\lambda$ the number of irreducible components and their codimensions in ${\mathcal{F}}_T\cap{\mathcal{F}}_{T{^{\prime}}}$ is equal to the number of irreducible components and their codimensions in ${\mathcal{V}}_T\cap {\mathcal{V}}_{T{^{\prime}}}.$ Thus, the study of intersections of irreducible components of ${\mathcal{F}}_x$ can be reduced to the study of the intersections of orbital varieties of ${\mathcal{O}}_x\cap{\mathfrak{n}}.$
The conjecture of Kazhdan and Lusztig mentioned above is equivalent to the irreducibility of certain characteristic varieties [@Bor Conjecture 4]. They have been shown to be reducible in general by Kashiwara and Saito [@Kas]. Nevertheless, the description of pairwise intersections of the irreducible components of the Springer fibers is still open.
The complete picture of the intersections of the components have been described by J.A. Vargas for hook case in [@Var] and by F.Y.C. Fung for two-row case in [@Fu]. Both in hook and two-row cases, all the components are non-singular, all the intersections are irreducible or empty.
In this paper we study the components of the intersection of a pair of components for two-column case (that is $\lambda=(2,2,\ldots)$ ). The two-column case and the hook case are two extreme cases in the following sense: For all nilpotent orbits of the given rank $k$ the orbit $\lambda=(k,1,1\ldots)$ is the most nondegenerate and the orbit $\lambda=(2,2,\ldots)$ (with dual partition $\lambda^*=(n-k,k)$) is the most degenerate, in the following sense $\overline{\mathcal{O}}_{(k,1,\ldots)}\supset
\overline{\mathcal{O}}_{\mu}\supset{\mathcal{O}}_{(2,\ldots,2,1,\cdots)}$ for any $\mu$ such that for $x\in{\mathcal{O}}_\mu$ one has ${{\rm Rank\,}}x=k.$ However, it seems that the general picture must be more close to the two-column case than to the hook case, which is too simple and beautiful.
{#section}
In general we have only Steinberg’s construction for orbital varieties. Via this construction orbital varieties in ${\mathcal{O}}_x\cap{\mathfrak{n}}$ are as complex from geometric point of view as irreducible components of ${\mathcal{F}}_x$. There is, however a nice exception: the case of orbital varieties in $\mathfrak{gl}_n$ associated to two-column Young diagrams. In this case each orbital variety is a union of a finite number of $B$-orbits and we can apply [@Mel1] to get the full picture of intersections of orbital varieties. In [@Mel1] the special so called rank matrix is attached to a $B$-orbit of $x\in{\mathfrak{n}}.$ In the case of $x$ of nilpotent order 2 it defines the corresponding $B$-orbit completely. Here we use the technique of these matrices to determine the intersection of two orbital varieties of nilpotent order two. In particular we show that the intersection of two orbital varieties associated to an orbit of nilpotent order 2 is not empty (see Proposition \[prop2\]). We give the purely combinatorial and easy to compute necessary and sufficient condition for the irreducibility of the intersection of two orbital varieties of nilpotent order 2 and provide some examples showing that in general such intersections are reducible and not necessary equidimensional (see examples in Section \[4.5\]).
In the subsequent paper (cf. [@M-P1]), we show that the intersections of codimension 1 in two-column case are irreducible. This together with computations in low rank cases permits us to conjecture
Given $S,T\in{\mathbf{Tab}}_\lambda$. If ${{\rm codim\,}}_{{\mathcal{F}}_S}{\mathcal{F}}_T\cap{\mathcal{F}}_S=1$ then ${\mathcal{F}}_T\cap{\mathcal{F}}_S$ is irreducible.
Let us now give a brief outline of the contents of the paper.\
$\bullet$ To make the paper as self contained as possible we present in Section 2 Spaltenstein’s and Steinberg’s constructions and quote the connected results essential in further analysis.
$\bullet$ In Section 3 we provide the main result of this paper, namely, a purely combinatorial necessary and sufficient condition for the intersection of two components of the Springer fiber to be irreducible in two-column case; as an application of this characterization, we give the first examples for which the intersections of two components of the Springer fiber are reducible and are not of pure dimension. This is the most technical part of the paper.
$\bullet$ In Section 4 we give some other counter-examples concerning the possible simplification of the construction of orbital varieties and of their intersections in codimension one.
General Construction
====================
{#2.1}
Given $x\in {\mathfrak{n}}$ denote $G_x=\{g\in G\ | \ g^{-1}xg\in{\mathfrak{n}}\}$. Set $f_1:G_x\rightarrow {\mathcal{O}}_x\cap{\mathfrak{n}}$ by $f_1(g)=g.x$ and $f_2:G_x\rightarrow {\mathcal{F}}_x$ by $f_2(g)=gB$. Define $\pi:{\mathcal{F}}_x\rightarrow {\mathcal{O}}_x\cap{\mathfrak{n}},\
gB\mapsto\pi(gB):=f_1(f_2^{-1}(gB))$. By Spaltenstein $\pi$ induces a surjection $\hat{\pi}$ from the set of irreducible components of ${\mathcal{F}}_x$ onto the set of irreducible components of ${\mathcal{O}}_x\cap{\mathfrak{n}}$, moreover the fiber of this surjective map is exactly an orbit under the action of the component group $A(x):=Z_G(x)/Z_G^o(x)$ (cf. [@Spa]). He showed also that ${\mathcal{F}}_x$ and ${\mathcal{O}}_x\cap{\mathfrak{n}}$ are equidimensional and got the following relations:
$$\begin{aligned}
\dim({\mathcal{O}}_x\cap{\mathfrak{n}})+\dim(Z_G(x))&=&\dim({\mathcal{F}}_x)+\dim(B)\\
\dim({\mathcal{O}}_x\cap{\mathfrak{n}})+\dim({\mathcal{F}}_x)&=&\dim({\mathfrak{n}})\\
\dim({\mathcal{O}}_x\cap{\mathfrak{n}})&=&\frac{1}{2}\dim({\mathcal{O}}_x).\end{aligned}$$
In our setting, for the case $G={\rm GL}_n$, the component is always trivial, so $\hat{\pi}$ is actually a bijection. As an extension of his work, we established in [@M-P] the following result
Let $x\in {\mathfrak{n}}$ and let ${\mathcal{F}}_1,{\mathcal{F}}_2$ be two irreducible components of ${\mathcal{F}}_x$ and ${\mathcal{V}}_1=\pi({\mathcal{F}}_1),\ {\mathcal{V}}_2=\pi({\mathcal{F}}_2)$ the corresponding orbital varieties. Let $\{{\mathcal{E}}_l\}_{l=1}^t$ be the set of irreducible components of ${\mathcal{F}}_1\cap{\mathcal{F}}_2.$ Then $\{\pi({\mathcal{E}}_l)\}_{l=1}^t$ is exactly the set of irreducible components of ${\mathcal{V}}_1\cap{\mathcal{V}}_2$ and ${{\rm codim\,}}_{{\mathcal{F}}_1}({\mathcal{E}}_l)={{\rm codim\,}}_{{\mathcal{V}}_1}(\pi({\mathcal{E}}_l))$.
This simple proposition shows that in the case of ${\rm GL}_n$, orbital varieties associated to ${\mathcal{O}}_x$ are equivalent to the irreducible components of ${\mathcal{F}}_x.$
{#2.3}
The parametrization of the irreducible components of ${\mathcal{F}}_x$ in ${\rm
GL}_n$ by standard Young tableaux is as follows.
In this case ${\mathcal{F}}$ is identified with the set of complete flags $\xi=(V_{1}\subset\cdots V_{n}=\mathbb{C}^{n})$ and ${\mathcal{F}}_{x}\cong
\{\xi=(V_{i})\in {\mathcal{F}}\ |\ x(V_{i})\subset V_{i-1}\}$.
Given $x\in{\mathfrak{n}}$ let $J(x)=\lambda.$ By a slight abuse of notation we will not distinguish between the partition $\lambda$ and its Young diagram. By R. Steinberg [@Ste3] and N. Spaltenstein [@Spa0] we have a parametrization of the irreducible components of ${{\mathcal{F}}}_{x}$ by the set ${\mathbf{Tab}}_{\lambda}$: Let $\xi=(V_{i})\in
{\mathcal{F}}_{x}$, then we get a sutured chain in the poset of Young diagrams $${\rm St}
(\xi):=(J(x)\supset J(x_{|V_{n-1}})\supset\cdots\supset
J(x_{|V_{2}})\supset J(x_{|V_{1}}))$$ where $x_{|V_{i}}$ is the nilpotent endomorphism induced by $x$ by restriction to the subspace $V_{i}$ and $J(x_{|V_{i+1}})$ differs from $J(x_{|V_i})$ by one corner box. It is easy to see that the data of such a sutured chain is equivalent to give a standard Young tableau. So we get a map ${\rm St}:{\mathcal{F}}_{x}\rightarrow {\mathbf{Tab}}_{\lambda}$. Then the collection $ \{{\rm St}^{-1}(T)\}_{T\in {\mathbf{Tab}}_{\lambda}}$ is a partition of ${\mathcal{F}}_{x}$ into smooth irreducible subvarieties of the same dimension and $\{\overline{{\rm St}^{-1}(T)}\}_{T\in
{\mathbf{Tab}}_{\lambda}}$ are the set of the irreducible components of ${{\mathcal{F}}}_{x}$ which will be denoted by ${\mathcal{F}}_T:=\overline{{\rm
St}^{-1}(T)}$ where $T\in{\mathbf{Tab}}_\lambda.$
On the level of orbital varieties the construction is as follows. For $1\leqslant i<j\leqslant n$ consider the canonical projections $\pi_{i,j}:{\mathfrak{n}}_n\rightarrow {\mathfrak{n}}_{j-i+1}$ acting on a matrix by deleting the first $i-1$ columns and rows and the last $n-j$ columns and rows. For any $u\in {\mathcal{O}}_\lambda\cap {\mathfrak{n}}$ set $J_n(u):=J(u)=\lambda$ and $J_{n-i}(u):=J(\pi_{1,n-i}(u))$ for any $i\ :\ 1\leqslant i\leqslant n-1.$ Exactly as in the previous construction we get a standard Young tableau corresponding to the sutured chain $(J_n(u)\supset\ldots\supset J_1(u)),$ therefore we get a map ${\rm St}_1:{\mathcal{O}}_\lambda\cap{\mathfrak{n}}\rightarrow {\mathbf{Tab}}_{\lambda}.$ Again the collection $\{{\rm St}_1^{-1}(T)\}_{T\in{\mathbf{Tab}}_\lambda}$ is a partition of ${\mathcal{O}}_\lambda\cap{\mathfrak{n}}$ into smooth irreducible subvarieties of the same dimensions and $\{\overline{{\rm
St}_1^{-1}(T)}\cap {\mathcal{O}}_\lambda\}_{T\in {\mathbf{Tab}}_{\lambda}}$ are orbital varieties associated to ${\mathcal{O}}_\lambda.$ Put ${\mathcal
V}_T:=\overline{{\rm St}_1^{-1}(T)}\cap {\mathcal{O}}_\lambda$ where $T\in{\mathbf{Tab}}_\lambda$; in particular, $\coprod\limits_{\lambda \vdash n}{\mathbf{Tab}}_\lambda$ parameterizes the set of orbital varieties contained in ${\mathfrak{n}}$.
{#1.4}
A general construction for orbital varieties by R. Steinberg (cf. [@Ste2]) is as follows. For $w\in {\mathbf{S}}_n$ consider the subspace $${\mathfrak{n}}\cap {^w {\mathfrak{n}}}:=\bigoplus\limits_{\alpha \in {{\mathcal{R}}}^{+}\cap
{^w{{\mathcal{R}}}^{+}}}{\mathfrak{g}}_{\alpha}$$ contained in ${\mathfrak{n}}$. Then $G.({\mathfrak{n}}\cap
{^w {\mathfrak{n}}})$ is an irreducible locally closed subvariety of the nilpotent variety ${\mathcal{N}}$. Since ${\mathcal{N}}$ is a finite union of nilpotent orbits, it follows that there is a unique nilpotent orbit ${{\mathcal{O}}}$ such that $\overline{G.({\mathfrak{n}}\cap {^w
{\mathfrak{n}}})}=\overline{{{\mathcal{O}}}}$. Moreover $\overline{B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})}\cap {{\mathcal{O}}}$ is an orbital variety associated to ${\mathcal{O}}$ and the fundamental result in Steinberg’s work is that every orbital variety can be obtained in this way [@Ste2]; in particular there is a surjective map $\varphi: {\mathbf{S}}_n\rightarrow \coprod\limits_{\lambda \vdash n}{\mathbf{Tab}}_\lambda$. The preimages of this map $\mathcal{C}_{T}:=\varphi^{-1}(T)$ are called the [**geometric**]{} (or [**left**]{}) [**cells**]{} of ${\mathbf{S}}_n$. The geometric cells are given by Robinson-Schensted correspondence, namely for $T\in {\mathbf{Tab}}_\lambda$, one has $\mathcal{C}_{T}=\{ {\rm RS}(T,S)\ :\ S\in{\mathbf{Tab}}_\lambda\}$, where RS represents the Robinson-Schensted correspondence.
Two-column case
===============
{#4.1}
In this section we use intensively the results of [@Mel1] and we adopt its notation.
Set ${\mathcal{X}}_2:=\{x\in{\mathfrak{n}}\ | \
x^{\scriptscriptstyle 2}=0\}$ to be the variety of nilpotent upper-triangular matrices of nilpotent order 2. Denote ${\mathbf{S}}_n^2:=\{\sigma\in{\mathbf{S}}_n\ | \
\sigma^{\scriptscriptstyle 2}=id\}$ the set of involutions of ${\mathbf{S}}_n$. For every $\sigma\in{\mathbf{S}}_n^2$, set $N_\sigma$ to be the “strictly upper-triangular part" of its corresponding permutation matrix, that is $$\begin{aligned}
(N_\sigma)_{i,j}:=\left\{\begin{array}{ll} 1&{\rm if}\ i<j\
{\rm and}\
\sigma(i)=j;\\
0&{\rm otherwise.}\\
\end{array}\right.\end{aligned}$$
Let ${\mathbf{Tab}}_n^2$ be the set of all Young tableaux of size $n$ with two columns. For $T\in{\mathbf{Tab}}_n^2$, write it as $T=(T_1,T_2),$ where $T_1=\left(\begin{array}{c}t_{1,1}\cr\vdots\cr
t_{n-k,1}\cr\end{array}\right)$ is the first column of $T$ and $T_2=\left(\begin{array}{c}t_{1,2}\cr\vdots\cr
t_{k,2}\cr\end{array}\right)$ is the second column of $T.$ And define the following involution
$$\begin{aligned}
\label{ecriture}
\sigma_{\scriptscriptstyle T}:=(i_1,j_1)\ldots (i_k,j_k)\end{aligned}$$
where $j_s:=t_{s,2};\ i_1:=t_{1,2}-1,$ and $i_s:=\max\{d\in T_1-\{i_1,\ldots,i_{s-1}\}\ | \ d<j_s\}$ for any $s>1.$ For example, take $$T=\begin{tabular}{|c|c|}
\hline
1 & 4 \\
\hline
2 & 5 \\
\hline
3 & 7 \\
\hline
6 & 8 \\
\hline
\end{tabular}$$ Then $\sigma_{\scriptscriptstyle T}=(3,4)(2,5)(6,7)(1,8).$
To define $T\in{\mathbf{Tab}}_n$ it is enough to know columns $T_i$ as sets (we denote them by $\langle T_i\rangle$), or equivalently the different column positions $c_T(i)$ of integers $i\ :\ 1\leqslant i\leqslant n$ since the entries increase from up to down in the columns. Thus given $\sigma_T$ we can reconstruct $T$. Indeed, $\langle
T_2\rangle=\{j_1,\ldots,j_k\}$ and $\langle
T_1\rangle=\{i\}_{i=1}^n\setminus\langle T_2\rangle$.
One has
\[4.1spec\]([@Mx2 2.2], [@Msmith 4.13])
- The variety ${\mathcal{X}}_2$ is a finite union of $B$-orbits, namely $${\mathcal{X}}_2=\coprod\limits_{\sigma\in{\mathbf{S}}_n^2}B.N_{\sigma}.$$
- For any $T\in {\mathbf{Tab}}_n^2$, one has $\overline{\mathcal{V}}_T=\overline{B.N}_{\sigma_T}$.
The finiteness property is particular for ${\mathcal{X}}_2.$ The fact that each orbital variety has a dense $B$-orbit is also particular for very few types of nilpotent orbits including orbits of nilpotent order 2 (cf. [@Msmith]). The first property permits us to compute the intersections of any two $B$-orbit closures in ${\mathcal{X}}_2$. The second one permits us to apply the results to the intersections of orbital varieties of nilpotent order 2.\
We begin with the general theory of the intersections of $\overline{B.N}_\sigma$ for $\sigma\in {\mathbf{S}}_n^2.$
{#4.2}
In this section we prefer to use the dual partition $\lambda^*$ instead of $\lambda$ since it will be more convenient to write it down for nilpotent orbits of nilpotent order 2. Indeed, for $x\in {\mathcal{X}}_2$ one has $J^*(x)=(n-k,k)$ where $k$ is number of Jordan blocks of length two in $J(x).$
For every element $x\in {\mathcal{X}}_2$, the integer ${\rm rk}(x)$ is exactly the number of blocks of length 2 in $J(x)$, so it defines the ${\rm GL}_n$-orbit of $x$.
Any element $\sigma\in {\mathbf{S}}_n^2$ can be written as a product of disjoint cycles of length 2. Order elements in increasing order inside the cycle and order cycles in increasing order according to the first entries. In that way we get a unique writing of every involution. Thus, $\sigma=(i_1,j_1)(i_2,j_2)\ldots(i_k,j_k)$ where $i_s<j_s$ for any $1\leqslant s\leqslant k$ and $i_s<i_{s+1}$ for any $1\leqslant s<k.$ Set $L(\sigma):=k$ \[do not confuse this notation with the length function\], and denote by ${\mathcal{O}}_\sigma$ the ${\rm GL}_n$-orbit of $N_\sigma$. By definition we have $L(\sigma)={\rm rk}(N_\sigma).$
Let us define the following number
$$\begin{aligned}
r_s(\sigma):={\rm card}\{i_p<i_s\ | \ j_p<j_s\}+ {\rm
card}\{j_p\ |\ j_p<i_s\}.\end{aligned}$$
Note that the definition of $r_s(\sigma)$ is independent of ordering cycles in increasing order according to the first entries. However if it is ordered then $r_1(\sigma)=0$ and to compute $r_s(\sigma)$ it is enough to check only the pairs $(i_p,j_p)$ where $p<s.$ For example, take $\sigma=(1,6)(3,4)(5,7).$ Then $L(\sigma)=3$ and $r_1(\sigma)=0,\
r_2(\sigma)=0,\ r_3(\sigma)=2+1=3.$
By [@Mx2 3.1] one has
For $\sigma=(i_1,j_1)(i_2,j_2)\ldots(i_k,j_k)\in {\mathbf{S}}_n^2$ one has $$\dim(B.N_\sigma)=kn-\sum\limits_{s=1}^k(j_s-i_s)-\sum\limits_{s=2}^k r_s(\sigma).$$
\[rem3\] By Theorem \[4.1spec\] (ii),the orbits $B.N_{\sigma_{T}}$ (where $({{\rm sh\,}}(T))^*=(n-k,k)$) are the only $B$-orbits of maximal dimension inside the variety ${\mathcal{O}}_{(n-k,k)^*}\cap{\mathfrak{n}}$ and $\dim(B.N_{\sigma_T})=k(n-k)$: Indeed any orbit $B.N_\sigma$ is irreducible and therefore lies inside an orbital variety ${\mathcal{V}}_T$, in particular it lies in $\overline{\mathcal{V}}_T$, so if $\dim
B.N_\sigma=\dim{\mathcal{V}}_T$ we get that $\overline {
B.N}_\sigma=\overline{\mathcal{V}}_T$ thus by Theorem \[4.1spec\] (ii) $\overline{B.N}_\sigma=\overline{B.N}_{\sigma_T}$ which provides $\sigma=\sigma_T.$
In particular if $\sigma=(i_1,j_1)\ldots(i_k,j_k)$ is such that $\dim(B.N_\sigma)=k(n-k)$, then $\sigma=\sigma_T$ where $T$ is the tableau obtained by $$c_T(s)=\left\{\begin{array}{ll} 2&{\rm if\ } s=j_p\ {\rm for\ some\ }p: \ 1\leqslant p\leqslant k\\
1&{\rm otherwise}\\
\end{array}\right.$$
{#4.3}
In [@Mel1] the combinatorial description of $\overline{B.N}_\sigma$ (with respect to Zariski topology) for $\sigma\in{\mathbf{S}}_n^2$ is provided. Let us formulate this result.
Recall from Section \[2.3\] the notion $\pi_{i,j}:{\mathfrak{n}}_n\rightarrow {\mathfrak{n}}_{j-i+1}$ and define the rank matrix $R_x$ of $x\in{\mathfrak{n}}$ to be
$$\begin{aligned}
(R_x)_{i,j}:=\left\{ \begin{array}{ll}
0&{\rm if}\ i\geqslant j;\\
{\rm rk} \,(\pi_{i,j}(x))&{\rm otherwise}.\\
\end{array}\right.\end{aligned}$$
Note that for any element $b\in B$, $\pi_{i,j}(b)$ is an invertible upper-triangular matrix in ${\rm GL}_{j-i+1}$. Therefore we can define an action of $B$ on ${\mathfrak{n}}_{j-i+1}$ by: $b.y:=\pi_{i,j}(b).y$ for $y\in{\mathfrak{n}}_{j-i+1}$ and $b\in B$.
Let us first establish a result
1. If $x,y\in {\mathfrak{n}}$ are in the same $B$-orbit, then they have the same rank matrix.
2. The morphism $\pi_{i,j}$ is $B$-invariant.
Note that for any two upper-triangular matrices $a,b$ and for any $i,j\ :\ 1\leqslant i<j\leqslant n$ one has $\pi_{i,j}(ab)=\pi_{i,j}(a)\pi_{i,j}(b).$ In particular, if $a\in B$ then $\pi_{i,j}(a^{-1})=(\pi_{i,j}(a))^{-1}.$ Applying this to $x\in
{\mathfrak{n}}$ and $y$ in its $B$ orbit (that is $y=b.x$ for some $b\in B$) we get $\pi_{i,j}(y)=\pi_{i,j}(b).\pi_{i,j}(x)$ so that the morphism $\pi_{i,j}$ is $B$-invariant and in particular ${\rm
rk}(\pi_{i,j}(y))={\rm rk}(\pi_{i,j}(x)).$ Hence $R_x=R_y.$
By this lemma we can define $R_\sigma:=R_{N_\sigma}$ as the rank matrix associated to orbit $B.N_\sigma$.
\[ones\] Note that computation of $(R_{N_\sigma})_{i,j}$ is trivial – this is exactly the number of non-zero entries in submatrix of $1,\dots,j$ columns and $i,\ldots,n$ rows of $N_\sigma$ or in other words the number of ones in $N_\sigma$ to the left-below of position $(i,j)$ (including position $(i,j)$).
Let ${\mathbb Z}^+$ be the set of non-negative integers. Put ${\mathbf R}^2_n:=\{R_\sigma\ |\ \sigma\in {\mathbf{S}}_n^2\}.$ By [@Mel1 3.1, 3.3] one has
\[ens\] $R=(R_{i,j})\in M_{n\times n}({\mathbb Z}^+)$ belongs to ${\mathbf R}^2_n$ if and only if it satisfies
- $R_{i,j}=0$ if $i\geqslant j;$
- For $i<j$ one has $R_{i+1,j}\leqslant R_{i,j}\leqslant R_{i+1,j}+1$ and $R_{i,j-1}\leqslant R_{i,j}\leqslant R_{i,j-1}+1;$
- If $R_{i,j}=R_{i+1,j}+1=R_{i,j-1}+1=R_{i+1,j-1}+1$ then
- $R_{i,k}=R_{i+1,k}$ for any $k<j$ and $R_{i,k}=R_{i+1,k}+1$ for any $k\geqslant j;$
- $R_{k,j}=R_{k,j-1}$ for any $k>i$ and $R_{k,j}=R_{k,j-1}+1$ for any $k\leqslant i;$
- $R_{j,k}=R_{j+1,k}$ and $R_{k,i}=R_{k,i-1}$ for any $k\ :\ 1\leqslant k\leqslant n.$
Fix $\sigma \in{\mathbf R}^2_n$, then the conditions (i) and (ii) are obvious from Remark \[ones\], and the conditions (iii) appears exactly for the coordinates $(i,j)$ in the matrix when $j=\sigma(i)$, with $i<j$; we draw the following picture (see Figure \[Fig4\] below) to help the reader to visualize the constraints (a), (b), (c) of (iii), with the following rule: the integers which are inside a same white polygon, are equal, and the integers in a same gray rectangle differ by one.
[cc]{}
(-6,-9)(7,6)
(0.05,-0.05)(1,-0.05)(1,-1)(1.95,-1)(1.95,-1.95)(0.05,-1.95) \[ur\](0.7,-1)[$+1$]{}
(-1.95,-0.05)(-1.05,-0.05)(-1.05,-1.95)(-1.95,-1.95) (-0.95,-0.05)(-0.05,-0.05)(-0.05,-1.95)(-0.95,-1.95) (-2.5,-2)(-2.75,-1.5)(-2.75,-0.5)(-2.5,0) \[ur\](-4,-1.5)[$=$]{} (-3.90,-0.5)(-2.15,-0.5) (-3.90,-1.5)(-2.15,-1.5)
(-5.95,-0.05)(-4.05,-0.05)(-4.05,-1.95)(-5.95,-1.95)
(-5.95,-2.05)(-4.05,-2.05)(-4.05,-2.95)(-5.95,-2.95) (-5.95,0.05)(-4.05,0.05)(-4.05,0.95)(-5.95,0.95)
(-6,-0.5)(-7,-0.5) (-6,-1.5)(-7,-1.5)
(-5.5,1.1)(-5.5,4.95) (-4.5,1.1)(-4.5,4.95)
(-5.5,-3.1)(-5.5,-5) (-4.5,-3.1)(-4.5,-5)
(0.05,-2.05)(1.95,-2.05)(1.95,-2.95)(0.05,-2.95) (0.05,-3.05)(1.95,-3.05)(1.95,-3.95)(0.05,-3.95) (0,-4.5)(0.5,-4.75)(1.5,-4.75)(2,-4.5) \[ur\](0.4,-5.5)[$=$]{} (0.5,-4.1)(0.5,-6) (1.5,-4.1)(1.5,-6)
(0.05,-6.05)(1.95,-6.05)(1.95,-7.95)(0.05,-7.95)
(0.5,-8.1)(0.5,-9) (1.5,-8.1)(1.5,-9)
(2.05,-6.05)(2.95,-6.05)(2.95,-7.95)(2.05,-7.95) (-0.05,-6.05)(-0.95,-6.05)(-0.95,-7.95)(-0.05,-7.95)
(3.1,-6.5)(6.95,-6.5) (3.1,-7.5)(6.95,-7.5)
(-1.1,-6.5)(-3,-6.5) (-1.1,-7.5)(-3,-7.5)
(2.05,0)(2.95,0)(2.95,-1.95)(2.05,-1.95) (3.05,0)(3.95,0)(3.95,-1.95)(3.05,-1.95) (2.05,-1)(2.2,-1)(2.45,-1)(2.60,-1) (2.8,-1)(2.95,-1) (3.05,-1)(3.2,-1)(3.45,-1)(3.60,-1) (3.8,-1)(3.95,-1) (4.1,-0.5)(6,-0.5) (4.1,-1.5)(6,-1.5) (4.5,-2)(4.75,-1.5)(4.75,-0.5)(4.5,0) \[ur\](4.5,-1.5) (6.05,0)(6.95,0)(6.95,-1.95)(6.05,-1.95) (6.05,-1)(6.2,-1)(6.45,-1)(6.60,-1) (6.8,-1)(6.95,-1)
\[ur\](7,-1) \[ur\](7,-7) \[ur\](7,-8)
(0.05,0.05)(1.95,0.05)(1.95,0.95)(0.05,0.95) (0.05,1.05)(1.95,1.05)(1.95,1.95)(0.05,1.95) (1,0.05)(1,0.30) (1,0.45)(1,0.60) (1,0.8)(1,0.95) (1,1.05)(1,1.30) (1,1.45)(1,1.60) (1,1.8)(1,1.95)
(0.5,2.25)(0.5,4.95) (1.5,2.25)(1.5,4.95) (0,2.5)(0.5,2.75)(1.5,2.75)(2,2.5) \[ur\](0.25,2.75) (0.05,4.05)(1.95,4.05)(1.95,4.95)(0.05,4.95) (1,4.05)(1,4.30)(1,4.45)(1,4.60) (1,4.8)(1,4.95)
\[ur\](1,5) \[ur\](-4.9,5) \[ur\](-6.5,5)
(-7,4.95)(6.95,4.95)(6.95,-8.95)
The first part of (c) can be explained in the following: since the integer $j$ appears already in the second entry of the cycle $(i,j)$, so it can not appear again in any other cycle; therefore in the matrix $N_{\sigma}$, the integers of the $j^{th}$ row are all 0, and that explains why we should have $(R_\sigma)_{j,k}=(R_\sigma)_{j+1,k}$ for $1\leqslant k\leqslant n$; the same explanation can also be done for the second part of (c).\
When the constrain (iii) appears, let us call the couple $(i,j)$ a [**position**]{} of constrain (iii).
\[rem5\] If two horizontal (resp. vertical) consecutive boxes of a matrix in ${\mathbf R}_n^2$ differ by one, then it is also the same for any consecutive horizontal (resp. vertical) boxes above (resp. on the right).
As an immediate corollary of Proposition \[ens\] we get
\[lemme1\] Let $\sigma,\sigma_1$ and $\sigma_2$ be involutions such that $\sigma=\sigma_1.\sigma_2$ and $L(\sigma)=L(\sigma_1)+L(\sigma_2)$, then $R_\sigma=R_{\sigma_1}+R_{\sigma_2}$; in particular we have $\sigma_1,\sigma_2 \preceq \sigma$.
The hypothesis $L(\sigma)=L(\sigma_1)+L(\sigma_2)$ means exactly that any integer appearing a cycle of $\sigma_1$ does not appear in any cycle of $\sigma_2$ and conversely \[note that it is also equivalent to say $\sigma_1.\sigma_2=\sigma_2.\sigma_1=\sigma$\]; this means in particular that when the coefficient 1 appears in the matrix $R_{\sigma_1}$ for the coordinate $(i,\sigma_1(i))$, then it can not appear in the $i^{th}$ line and in the $\sigma_1(i)^{th}$ column of $R_{\sigma_2}$ and conversely; therefore we get $N_\sigma=N_{\sigma_1}+N_{\sigma_2}$ and the result follows.
{#4.3a}
Define the following partial order on $M_{n\times n}({\mathbb Z}^+).$ For $A,B\in M_{n\times n}({\mathbb Z}^+)$ put $A\preceq B$ if for any $i,j\ :\ 1\leqslant i,j\leqslant n$ one has $A_{i,j}\leqslant B_{i,j}.$\
The restriction of this order to ${\mathbf R}_n^2$ induces a partial order on ${\mathbf{S}}_n^2$ by setting $\sigma'\preceq \sigma$ if $R_{\sigma'}\preceq R_{\sigma}$ for $\sigma,\sigma'\in {\mathbf{S}}_n^2.$ By [@Mel1 3.5] this partial order describes the closures of $B.N_\sigma$ for $\sigma\in{\mathbf{S}}_n^2$. Combining [@Mel1 3.5] with Remark \[rem3\] we get
\[thm1\] For any $\sigma\in {\mathbf{S}}_n^2$, one has $$\overline{B.N}_\sigma=\coprod\limits_{\sigma{^{\prime}}\preceq\sigma}B.N_{\sigma{^{\prime}}}.$$ In particular, for $T\in{\mathbf{Tab}}_n^2$ $${\mathcal{V}}_T=\coprod\limits_{\substack{\sigma{^{\prime}}\preceq\sigma_T \\L(\sigma')=L(\sigma_T)}}B.N_{\sigma{^{\prime}}}.$$
{#4.3b}
Let $\pi_{i,j}:{\mathfrak{n}}_{n}\rightarrow{\mathfrak{n}}_{j-i+1}$. If we denote by $\hat{\pi}_{s,t}:{\mathfrak{n}}_{j-i+1}\rightarrow{\mathfrak{n}}_{t-s+1}$ the same projection, but with the starting-space ${\mathfrak{n}}_{j-i+1}$, then we can easily check the following relation:
$$\begin{aligned}
\label{relation}
\hat{\pi}_{s,t}\circ\pi_{i,j}=\pi_{s+i-1,t+i-1}.\end{aligned}$$
Now if $R\in {\mathbf R}_n^2$, it is obvious by Remark \[ones\] that $\pi_{i,j}(R)$ fulfills the constraints (i), (ii) and (iii) of Proposition \[ens\]. Thus, we get
If $R\in {\mathbf R}_n^2$, then $\pi_{i,j}(R)\in {\mathbf R}_{j-i+1}^2$ for $1\leqslant i\leqslant j\leqslant n$.
Obviously, the converse is not true, as one can check for the matrix $\tiny{\left(
\begin{array}{ccc}
0 & 1 & 2 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{array}
\right)}.$\
By this lemma, for any $R_\sigma\in {\mathbf R}_n^2$, we have $\pi_{i,j}(R_\sigma)\in {\mathbf R}_{j-i+1}^2$; therefore $\pi_{i,j}$ induces a natural map from ${\mathbf{S}}_n^2$ onto ${\mathbf{S}}_{\langle
i,j\rangle}^2\cong{\mathbf{S}}_{j-i+1}^2$, symmetric group of the set $\{i,\cdots,j\}$. This projection will be also denoted by $\pi_{i,j}$. Moreover by (\[relation\]), and Remark \[ones\] one gets immediately:
$$\begin{aligned}
\label{relation1}
\pi_{i,j}(N_{\sigma})=N_{\pi_{i,j}(\sigma)}\ {\rm and}\
\pi_{i,j}(R_{\sigma})=R_{\pi_{i,j}(\sigma)}.\end{aligned}$$
Note that the resulting element $\pi_{i,j}(\sigma)$ is obtained from $\sigma$ by deleting all the cycles in which at least one entry does not belong to $\{i,\ldots, j\}.$ For every $\delta\in
{\mathbf{S}}_{\langle i,j\rangle}^2$, any element $\sigma\in \pi_{i,j}^{-1}(\delta)$ will be called a [**lifting**]{} of $\delta$. In the same way we will call the matrix $R_\sigma$ a lifting of $R_\delta$.
\[rem4\]
1. We will consider sometimes $\sigma\in {\mathbf{S}}_{\langle i,j\rangle}^2$ as an element of ${\mathbf{S}}_n^2$ (cf. proofs of Proposition \[prop2\], Lemma \[lemme2\] and Theorem \[intersection\]); in particular, with the description above we have $\sigma=\pi_{i,j}(\sigma)$
2. By note (i) and Lemma \[lemme1\] for any $\delta\in{\mathbf{S}}_{\langle i,j\rangle}^2$ and any $\sigma$ its lifting in ${\mathbf{S}}_n^2$ one has $\delta\preceq \sigma.$
3. By the relations (\[relation1\]), the projection $\pi_{i,j}$ respect the order $\preceq$: If $\sigma_1\preceq \sigma_2$, then $\pi_{i,j}(\sigma_1)\preceq \pi_{i,j}(\sigma_2)$.
{#4.3c}
Put ${\mathbf{S}}_n^2(k):=\{\sigma\in {\mathbf{S}}_n^2\ |\
L(\sigma)=k\}$ and respectively ${\mathbf{Tab}}_n^2(k):=\{T\in{\mathbf{Tab}}_n^2\ |\
{{\rm sh\,}}(T)=(n-k,k)^*\}$. As a corollary of partial order $\preceq$ on ${\mathbf{S}}_n^2$ we get
\[prop2\] $\sigma_o(k):=(1,n-k+1)(2,n-k+2)\ldots(k,n)$ is the unique minimal involution in ${\mathbf{S}}_n^2(k)$ and for any $\sigma\in {\mathbf{S}}_n^2(k)$ one has $\sigma_o(k)\preceq \sigma.$ In particular, for any $S,T\in{\mathbf{Tab}}_n^2(k)$ one has ${\mathcal{V}}_T\cap{\mathcal{V}}_S\ne\emptyset.$
Note that $N_{\sigma_o(k)}$ and respectively $R_{\sigma_o(k)}$ are $$\begin{aligned}
\centering
\psset{unit=0.5cm}
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\rput{U}(0,1){\tiny$\underbrace{n-k+1}$} \rput{U}(4,-3){\{\tiny$k$}
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\psline[linestyle=dotted,dotsep=3pt]{-}(1.5,-0.5)(2.5,-1.5)
\rput{U}(3,0){$0$}
\psline[linestyle=dotted,dotsep=3pt]{-}(1.5,0)(2.5,0)
\psline[linestyle=dotted,dotsep=3pt]{-}(3,-0.5)(3,-1.5)
\psline[linestyle=dotted,dotsep=3pt]{-}(3,-4.5)(3,-5.5)
\rput{U}(3,-6){$0$}
\rput{U}(-3,0){$0$}
\psline[linestyle=dotted,dotsep=3pt]{-}(-2.5,0)(-1.5,0)
\psline[linestyle=dotted,dotsep=3pt]{-}(-3,-0.5)(-3,-5.5)
\rput{U}(-3,-6){$0$}
\psline[linestyle=dotted,dotsep=3pt]{-}(-2.5,-6)(2.5,-6)
\rput{U}(-5,-3){$N_{\sigma_o(k)}=$}
\rput{U}(12,0){$1$} \rput{U}(13,-1){$1$} \rput{U}(15,-3){$1$}
\psline[linestyle=dotted,dotsep=3pt]{-}(13.5,-1.5)(14.5,-2.5)
\rput{U}(11,0){$0$} \rput{U}(12,-1){$0$} \rput{U}(15,-4){$0$}
\psline[linestyle=dotted,dotsep=3pt]{-}(12.5,-1.5)(14.5,-3.5)
\rput{U}(13,0){$2$} \rput{U}(15,-2){$2$}
\psline[linestyle=dotted,dotsep=3pt]{-}(13.5,-0.5)(14.5,-1.5)
\rput{U}(15,0){$k$}
\psline[linestyle=dotted,dotsep=3pt]{-}(13.5,0)(14.5,0)
\psline[linestyle=dotted,dotsep=3pt]{-}(15,-0.5)(15,-1.5)
\psline[linestyle=dotted,dotsep=3pt]{-}(15,-4.5)(15,-5.5)
\rput{U}(15,-6){$0$}
\rput{U}(9,0){$0$}
\psline[linestyle=dotted,dotsep=3pt]{-}(9.5,0)(10.5,0)
\psline[linestyle=dotted,dotsep=3pt]{-}(9,-0.5)(9,-5.5)
\rput{U}(9,-6){$0$}
\psline[linestyle=dotted,dotsep=3pt]{-}(9.5,-6)(14.5,-6)
\rput{U}(7,-3){$R_{\sigma_o(k)}=$} \rput{U}(5,-3){$,$}
\end{pspicture}\end{aligned}$$ so that $$(R_{\sigma_o(k)})_{i,j}=\left\{
\begin{array}{ll}
j-i+1-(n-k) &{\rm if}\ j-i>n-k-1\\
0 &{\rm otherwise}\end{array}\right.$$ On the other hand by Proposition \[ens\] (ii) for any $\sigma\in
{\mathbf{S}}_n^2$ one has $(R_\sigma)_{i,j}\geqslant
(R_\sigma)_{i-1,j}-1\geqslant (R_\sigma)_{i-2,j}-2\cdots\geqslant
(R_\sigma)_{1,j}-(i-1).$ In turn $(R_\sigma)_{1, j}\geqslant
(R_\sigma)_{1,j+1}-1\geqslant\ldots\geqslant (R_\sigma)_{1,n}-(n-j)$ so that $(R_\sigma)_{i,j}\geqslant (R_\sigma)_{1,n}-(n-j+i-1).$ Thus, for any $\sigma\in{\mathbf{S}}_n^2(k)$ one has $(R_\sigma)_{i,j}\geqslant j-i+1-(n-k).$ As well one has $(R_\sigma)_{i,j}\geqslant 0$ so that $(R_\sigma)_{i,j}\geqslant
\max\{0, j-i+1-(n-k)\}=(R_{\sigma_o(k)})_{i, j}.$ Thus, $\sigma\succeq \sigma_o(k).$
The second part is now a corollary of this result and Theorem \[thm1\].
{#4.4}
Given $\sigma,\sigma{^{\prime}}\in {\mathbf{S}}_n^2$ we define $R_{\sigma,\sigma{^{\prime}}}$ by $$\begin{aligned}
(R_{\sigma,\sigma{^{\prime}}})_{i,j}:=\min\{(R_\sigma)_{i,j},(R_{\sigma{^{\prime}}})_{i,j}\}.\end{aligned}$$ One has
\[intersection\] For any $\sigma,\sigma{^{\prime}}\in {\mathbf{S}}_n^2$ one has $$\overline
{B.N}_\sigma\cap\overline{B.N}_{\sigma{^{\prime}}}=\coprod\limits_{R_\varsigma\preceq
R_{\sigma,\sigma'}}B.N_\varsigma.$$ This intersection is irreducible if and only if $R_{\sigma,\sigma{^{\prime}}}\in {\mathbf R}_n^2.$
To establish this equivalence we need only to prove the “only if" part and to do this we need some preliminary result.
\[lemme2\] Suppose that $\overline
{B.N}_\sigma\cap\overline{B.N}_{\sigma{^{\prime}}}$ is irreducible. Denote $B'$ the Borel subgroup in ${\rm GL}_{j-i+1}$. Then $\overline
{B'.N}_{\pi_{i,j}(\sigma)}\cap\overline{B'.N}_{\pi_{i,j}(\sigma{^{\prime}})}$ is irreducible.
Let $\alpha,\beta$ be two maximal involutions in ${\mathbf{S}}_{\langle i,j\rangle}^2$ such $\alpha,\beta\preceq
\pi_{i,j}(\sigma),\pi_{i,j}(\sigma')$. By Remark \[rem4\] (ii), we have also $\alpha,\beta \preceq \sigma, \sigma'$. By hypothesis we have $\overline
{B.N}_\sigma\cap\overline{B.N}_{\sigma{^{\prime}}}=\overline{B.N}_{\delta}$ for an element $\delta\in {\mathbf{S}}_n^2$. In particular we get $\alpha,\beta \preceq \delta$. By Remarks \[rem4\] (i) and (iii) we get $\alpha=\pi_{i,j}(\alpha), \beta=\pi_{i,j}(\beta)\preceq
\pi_{i,j}(\delta)\preceq \pi_{i,j}(\sigma),\pi_{i,j}(\sigma')$. Since $\alpha$ and $\beta$ are maximal, we get $\alpha=\beta=\pi_{i,j}(\delta)$.
We prove the theorem by induction on $n.$ For $n=3$ all the intersections are irreducible so that the claim is trivially true.
Let now $n$ be minimal such that $\overline
{B.N}_\sigma\cap\overline{B.N}_{\sigma{^{\prime}}}$ is irreducible and $R_{\sigma,\sigma{^{\prime}}}\notin{\mathbf R}_n^2$. Note that constrains (i) and (ii) of Proposition \[ens\] are satisfied by any $R_{\sigma,\sigma'}.$ If $R_{\sigma,\sigma{^{\prime}}}\notin{\mathbf R}_n^2$ then at least one of the conditions (a), (b) and (c) of the constrain (iii) of Proposition \[ens\] is not fulfilled. By symmetry around the anti diagonal it is enough to check only Condition (a) and the first part of Condition (c).
As for the first relation in (\[relation1\]), we can easily check that
$$\begin{aligned}
\label{eq6}
R_{\pi_{i,j}(\sigma),\pi_{i,j}(\sigma')}=\pi_{i,j}(R_{\sigma,\sigma'}).\end{aligned}$$
Let $B'$ be the Borel subgroup of ${\rm GL}_{n-1}.$ By Lemma \[lemme2\] and Relation (\[eq6\]), we get that the varieties $\overline
{B'.N}_{\pi_{1,n-1}(\sigma)}\cap\overline{B'.N}_{\pi_{1,n-1}(\sigma{^{\prime}})}$, $\overline
{B'.N}_{\pi_{2,n}(\sigma)}\cap\overline{B'.N}_{\pi_{2,n}(\sigma{^{\prime}})}$ are irreducible. Thus by induction hypothesis
$$\begin{aligned}
\label{eq7}
\pi_{1,n-1}(R_{\sigma,\sigma'}),\ \pi_{2,n}(R_{\sigma,\sigma'})\in
{\mathbf R}_{n-1}^2,\end{aligned}$$
Put $\zeta\in{\mathbf{S}}_{n-1}^2$ to be such that $R_\zeta=\pi_{1,n-1}(R_{\sigma,\sigma'})$ and $\eta\in {\mathbf{S}}_{\langle
2,n\rangle}^2$ be such that $R_\eta=\pi_{2,n}(R_{\sigma,\sigma'}).$
Suppose that $R_{\sigma,\sigma{^{\prime}}}\notin{\mathbf R}_n^2$, denote $(i_o,j_o)$ the position of a constrain (iii) $\tiny{
\begin{tabular}{|c|c|} \hline
k & k+1 \\
\hline
k & k \\
\hline
\end{tabular}}$ which is not satisfied by the matrix $R_{\sigma,\sigma{^{\prime}}}$.
(a): If the first part of Condition (a) is not satisfied, it means that we can find two horizontal consecutive boxes below of the two boxes
k k
--- ---
which differ by one; but these two boxes and
k k
--- ---
will lies in $\pi_{2,n}(R_{\sigma,\sigma{^{\prime}}})\in{\mathbf R}_{n-1}^2$, which is impossible by Remark \[rem5\].
Now if the second part of Condition (a) is not satisfied, it means that we can find two equal vertical consecutive boxes
m
---
m
on the right of the boxes
k+1
-----
k
. By Relation (\[eq7\]), these four last boxes can not lie inside $\pi_{1,n-1}(R_{\sigma,\sigma'}),\
\pi_{2,n}(R_{\sigma,\sigma'})$; we deduce in particular that $i_o=1$ and that the boxes
m
---
m
belong to the last column. Since $R_{\sigma,\sigma'}$ satisfies Condition (ii) of Proposition \[ens\], the “North-East" corner of $R_{\sigma,\sigma'}$ must be
m m
----- ---
m-1 m
. Now if we look at $\zeta$ (resp. $\
\eta$) as its own lifting in ${\mathbf{S}}_n^2$, then its configuration in the “North-East" corner will be of the following
m m
----- -----
m-1 m-1
(resp.
m-1 m
----- ---
m-1 m
). Since the intersection is irreducible, we should find $\delta\in{\mathbf{S}}_n^2$ such that $\delta\succeq \zeta,\eta$ and $R_\delta\preceq R_{\sigma,\sigma'}.$ Since $(R_{\zeta})_{2,n-1}=(R_{\sigma,\sigma'})_{2,n-1}=m-1$ we get that also $(R_\delta)_{2,n-1}=m-1.$ Since $(R_{\zeta})_{1,n-1}=(R_{\sigma,\sigma'})_{1,n-1}=m$ we get that also $(R_\delta)_{1,n-1}=m.$ Since $(R_{\eta})_{2,n}=(R_{\sigma,\sigma'})_{2,n}=m$ we get that also $(R_\delta)_{2,n}=m.$ But then by Remark \[rem5\] the “North-East" corner of $R_\delta$ should be of the following configuration
m m+1
----- -----
m-1 m
, this is impossible since $(R_\delta)_{1,n}\leqslant
(R_{\sigma,\sigma'})_{1,n}=m$.
(c): Suppose that the first part of Condition (c) is not satisfied, it means that we can find two vertical consecutive boxes
m+1
-----
m
lying in the $j_o^{th}$ and $(j_o+1)^{th}$ lines. As above this problem can not appear inside the matrices $\pi_{1,n-1}(R_{\sigma,\sigma'})$ and $\
\pi_{2,n}(R_{\sigma,\sigma'})$; we deduce in particular that $i_o=1$ and that the boxes
m+1
-----
m
lie on the last column. Since $R_{\sigma,\sigma'}$ satisfies Condition (ii) of Proposition \[ens\], on the right side of the $j_o^{th}$ and $(j_o+1)^{th}$ lines of $R_{\sigma,\sigma'}$ we should find
m m+1
--- -----
m m
. Let us draw its configuration
$$\begin{aligned}
\label{eq8}
\centering \psset{unit=0.5cm}
\begin{pspicture}(-5,-4)(8,0)
\rput{U}(0,0){\tiny{
\begin{tabular}{|c|c|} \hline
k & k+1 \\
\hline
k & k \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-0.6,-0.7)(-0.6,-4)
\psline[linestyle=dotted,dotsep=5pt]{-}(0.6,-0.7)(0.6,-4)
\rput{U}(9,-3.1){{\tiny
\begin{tabular}{|c|c|}
\hline
m & m+1 \\
\hline
m & m \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-2.9)(7.5,-2.9)
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-3.5)(7.5,-3.5)
\rput{U}(-5,-2.6){$R_{\sigma,\sigma'}$=}
\end{pspicture}\end{aligned}$$
In the same way if we look at $R_\zeta$ and $R_\eta$ as elements of ${\mathbf R}_n^2$, then their configurations will be of the following
$$\begin{aligned}
\label{eq9}
\centering \psset{unit=0.5cm}
\begin{pspicture}(-5,-4)(8,0)
\rput{U}(0,0){\tiny{
\begin{tabular}{|c|c|} \hline
k & k+1 \\
\hline
k & k \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-0.6,-0.7)(-0.6,-4)
\psline[linestyle=dotted,dotsep=5pt]{-}(0.6,-0.7)(0.6,-4)
\rput{U}(9,-3.1){{\tiny
\begin{tabular}{|c|c|}
\hline
m & m \\
\hline
m & m \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-2.9)(7.5,-2.9)
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-3.5)(7.5,-3.5)
\rput{U}(-5,-2.6){$R_\zeta$=}
\end{pspicture}\end{aligned}$$
and
$$\begin{aligned}
\label{eq10}
\centering \psset{unit=0.5cm}
\begin{pspicture}(-5,-4)(8,0)
\rput{U}(0,0){\tiny{
\begin{tabular}{|c|c|} \hline
k & k \\
\hline
k & k \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-0.6,-0.7)(-0.6,-4)
\psline[linestyle=dotted,dotsep=5pt]{-}(0.6,-0.7)(0.6,-4)
\rput{U}(9,-3.1){{\tiny
\begin{tabular}{|c|c|}
\hline
m & m+1 \\
\hline
m & m \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-2.9)(7.5,-2.9)
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-3.5)(7.5,-3.5)
\rput{U}(-5,-2.6){$R_\eta$=}
\end{pspicture}\end{aligned}$$
Since $\delta\succeq \zeta,\eta$ and $R_\delta\preceq
R_{\sigma,\sigma'}$ combining the pictures in (\[eq8\]), (\[eq9\]) and (\[eq10\]), we get
$$\begin{aligned}
\centering \psset{unit=0.5cm}
\begin{pspicture}(-5,-4)(8,0)
\rput{U}(0,0){\tiny{
\begin{tabular}{|c|c|} \hline
k & k+1 \\
\hline
k & k \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-0.6,-0.7)(-0.6,-4)
\psline[linestyle=dotted,dotsep=5pt]{-}(0.6,-0.7)(0.6,-4)
\rput{U}(9,-3.1){{\tiny
\begin{tabular}{|c|c|}
\hline
m & m+1 \\
\hline
m & m \\
\hline
\end{tabular}}}
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-2.9)(7.5,-2.9)
\psline[linestyle=dotted,dotsep=5pt]{-}(-1,-3.5)(7.5,-3.5)
\rput{U}(-5,-2.6){$R_\delta$=}
\end{pspicture}\end{aligned}$$
which is impossible, because it does not satisfy Condition (iii) (c).
{#4.5}
Let us apply the previous subsection to the elements of the form $\sigma_T$ to show that in general the intersection ${\mathcal{V}}_T\cap{\mathcal{V}}_{T{^{\prime}}}$ is reducible and not equidimensional.
\[ex\]
1. For $n\leqslant 4$ all the intersections of $B$-orbit closures of nilpotent order 2 are irreducible. The first examples of reducible intersections of $B$ orbit closures occur in $n=5.$ In particular there is the unique example of the reducible intersection of orbital varieties and it is
$$T=\begin{tabular}{|c|c|}
\hline
1 & 2 \\
\hline
3 & 4 \\
\hline
5 \\
\cline{1-1}
\end{tabular}
,\ R_{\sigma_T}=\left(\tiny{\begin{array}{ccccc}
0&1&1&2&2\\
0&0&0&1&1\\
0&0&0&1&1\\
0&0&0&0&0\\
0&0&0&0&0\\\end{array}}\right)\ {\rm and } \
T'=\begin{tabular}{|c|c|}
\hline
1 & 3 \\
\hline
2 & 5 \\
\hline
4 \\
\cline{1-1}
\end{tabular}
,\ R_{\sigma_{T'}}=\left(\tiny{\begin{array}{ccccc}
0&0&1&1&2\\
0&0&1&1&2\\
0&0&0&0&1\\
0&0&0&0&1\\
0&0&0&0&0\\\end{array}}\right)$$ so that $$R_{\sigma_{T},\sigma_{T'}}=\left(\tiny{\begin{array}{ccccc}
0&0&1&1&2\\
0&0&0&1&1\\
0&0&0&0&1\\
0&0&0&0&0\\
0&0&0&0&0\\\end{array}}\right)$$ Since $(R_{\sigma_{T},\sigma_{T'}})_{1,3}=(R_{\sigma_{T},\sigma_{T'}})_{1,2}+1=(R_{\sigma_{T},\sigma_{T'}})_{2,2}+1=(R_{\sigma_{T},\sigma_{T'}})_{2,3}+1$ and $(R_{\sigma_{T},\sigma_{T'}})_{3,5}=(R_{\sigma_{T},\sigma_{T'}})_{4,5}+1$ we get that $R_{\sigma_{T},\sigma_{T'}}$ does not satisfy condition (iii)-(c) of Proposition \[ens\], therefore $R_{\sigma_{T},\sigma_{T'}}\not\in{\mathbf R}_5^2$. As well $(R_{\sigma_T,\sigma_{T'}})_{1,4},\
(R_{\sigma_T,\sigma_{T'}})_{2,5}$ do not satisfy Remark \[rem5\]. Accordingly we find three maximal elements $R,R',R{^{\prime\prime}}\in {\mathbf R}_5^2$ for which $R,R',R{^{\prime\prime}}\prec R_{\sigma_{T},\sigma_{T'}}$
$$R=R_{(1,3)(2,5)}=\left(\tiny{\begin{array}{ccccc}
0&0&1&1&2\\
0&0&0&0&1\\
0&0&0&0&0\\
0&0&0&0&0\\
0&0&0&0&0\\\end{array}}\right),\quad
R'=R_{(1,4)(3,5)}=\left(\tiny{\begin{array}{ccccc}
0&0&0&1&2\\
0&0&0&0&1\\
0&0&0&0&1\\
0&0&0&0&0\\
0&0&0&0&0\\\end{array}}\right),$$ $$R{^{\prime\prime}}=R_{(1,5)(2,4)}=\left(\tiny{\begin{array}{ccccc}
0&0&0&1&2\\
0&0&0&1&1\\
0&0&0&0&0\\
0&0&0&0&0\\
0&0&0&0&0\\\end{array}}\right)$$ Note that $\dim(B.N_{(1,3)(2,5)})=\dim(B.N_{(1,4)(3,5)})=\dim(B.N_{(1,5)(2,4)})=4$ so that ${\mathcal{V}}_T\cap{\mathcal{V}}_{T'}$ contains three components of codimension 2.
2. The first example of non-equidimensional intersection of orbital varieties occurs in $n=6$ and it is $$T=\begin{tabular}{|c|c|}
\hline
1 & 3 \\
\hline
2 & 6 \\
\hline
4 \\
\cline{1-1}
5 \\
\cline{1-1}
\end{tabular}
\ ,\quad R_{\sigma_{T}}=\left(\tiny{\begin{array}{cccccc}
0&0&1&1&1&2\\
0&0&1&1&1&2\\
0&0&0&0&0&1\\
0&0&0&0&0&1\\
0&0&0&0&0&1\\
0&0&0&0&0&0\\
\end{array}}\right)$$ and
$$T'=\begin{tabular}{|c|c|}
\hline
1 & 2 \\
\hline
3 & 5 \\
\hline
4 \\
\cline{1-1}
6 \\
\cline{1-1}
\end{tabular}
\ ,\quad R_{\sigma_{T'}}=\left(\tiny{\begin{array}{cccccc}
0&0&1&1&2&2\\
0&0&0&0&1&1\\
0&0&0&0&1&1\\
0&0&0&0&1&1\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
\end{array}}\right)$$ so that $$R_{\sigma_{T},\sigma_{T'}}=\left(\tiny{\begin{array}{cccccc}
0&0&1&1&1&2\\
0&0&0&0&1&1\\
0&0&0&0&0&1\\
0&0&0&0&0&1\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\\end{array}}\right)$$
Since $(R_{\sigma_{T},\sigma_{T'}})_{1,3}=(R_{\sigma_{T},\sigma_{T'}})_{1,2}+1=(R_{\sigma_{T},\sigma_{T'}})_{2,2}+1=(R_{\sigma_{T},\sigma_{T'}})_{2,3}+1$ and $(R_{\sigma_{T},\sigma_{T'}})_{1,5}=(R_{\sigma_{T},\sigma_{T'}})_{2,5}$ we get that $R_{\sigma_{T},\sigma_{T'}}$ does not satisfy condition (iii) (a) of Proposition \[ens\] and $(R_{\sigma_T,\sigma_{T'}})_{1,5},\
(R_{\sigma_T,\sigma_{T'}})_{2,6}$ do not satisfy Remark \[rem5\] so that $R_{\sigma_{T},\sigma_{T'}}\not\in{\mathbf R}_6^2$ and the maximal elements $R,R'\in {\mathbf R}_6^2$ for which $R,R'\prec R_{\sigma_{T},\sigma_{T'}}$ are $$R=R_{(1,3)(4,6)}=\left(\tiny{\begin{array}{cccccc}
0&0&1&1&1&2\\
0&0&0&0&0&1\\
0&0&0&0&0&1\\
0&0&0&0&0&1\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\\end{array}}\right)$$ and
$$R'=R_{(1,6)(2,5)}=\left(\tiny{\begin{array}{cccccc}
0&0&0&0&1&2\\
0&0&0&0&1&1\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
\end{array}}\right).$$ Note that $\dim(B.N_{(1,3)(4,6)})=6$ and $\dim(B.N_{(1,6)(2,5)})=4$ so that ${\mathcal{V}}_T\cap{\mathcal{V}}_{T'}$ contains one component of codimension 2 and another component of codimension 4.
Some other counter-examples
===========================
Cell graphs {#4.6}
------------
Let $T\in {\mathbf{Tab}}_{\lambda} $ be a standard tableau and $\mathcal{C}_T$ its corresponding left cell (cf. Section \[1.4\]). Steinberg’s construction provides the way to construct $\mathcal{V}_T$ with the help of elements of $\mathcal{C}_T$. In [@M-P], we got another geometric interpretation of $C_T$:
\[thm2\]([@M-P]) - Let $T\in {\mathbf{Tab}}_{\lambda}$ and let $w={\rm RS}(T,T{^{\prime}})\in \mathcal{C}_T$. Then for a $x\in\mathcal{V}_{T}\cap B.({\mathfrak{n}}\cap^w{\mathfrak{n}})$ in general position, the unique Schubert cell whose intersection with the irreducible component ${\mathcal{F}}_{T{^{\prime}}}$ of the Springer fiber is open and dense in ${\mathcal{F}}_{T{^{\prime}}}$ is indexed by $w$.
The cell $\mathcal{C}_T$ in ${\mathbf{S}}_n$ can be visualized as a [**cell graph**]{} $\Gamma_T$ where the vertices are labeled by ${\mathbf{Tab}}_{\lambda}$, and two vertices $T'$ and $T''$ are joined by an edge labeled by $k$ if $s_k{\rm RS}(T,T{^{\prime}})={\rm RS}(T,T'')$. One can easily see ( cf. [@M-P], for example) that if $T'$ and $T''$ are joined in $\Gamma_T$, then ${{\rm codim\,}}_{{\mathcal{F}}_{T'}}{\mathcal{F}}_{T{^{\prime}}}\cap{\mathcal{F}}_{T''}=1.$
Note that $T'$ and $T''$ can be joined by an edge in $\Gamma_T$ and not joined by an edge in $\Gamma_S$ for some $S,T\in{\mathbf{Tab}}_\lambda$. Is it true that ${{\rm codim\,}}_{{\mathcal{F}}_{T'}}{\mathcal{F}}_{T'}\cap{\mathcal{F}}_{T''}=1$ if and only if there exists $T\in{\mathbf{Tab}}_\lambda$ such that $T'$ and $T''$ are joined by an edge in $\Gamma_T$?
The answer is negative as we show by the example below.
As we show in [@M-P1] if $k\leqslant 2$ then ${{\rm codim\,}}_{{\mathcal{V}}_T}({\mathcal{V}}_T\cap{\mathcal{V}}_S)=1$ if and only if there exists $P\in{\mathbf{Tab}}_{(n-k,k)^*}$ such that $T$ and $S$ are joined by an edge in $\Gamma_P$ so that the first example occurs in $n=6$ for ${\mathbf{Tab}}_{(3,3)^*}.$ In that case $(3,3)^*=(2,2,2)$ and the corresponding orbital varieties are $9$-dimensional. Let us put $$T_{1}=\begin{tabular}{|c|c|}
\hline
1 & 4 \\
\hline
2 & 5 \\
\hline
3 & 6 \\
\hline
\end{tabular}
,\quad
T_{2}=\begin{tabular}{|c|c|}
\hline
1 & 3 \\
\hline
2 & 5 \\
\hline
4 & 6 \\
\hline
\end{tabular}\ ,\quad T_{3}=\begin{tabular}{|c|c|}
\hline
1 & 2 \\
\hline
3 & 5 \\
\hline
4 & 6 \\
\hline
\end{tabular}\ ,\quad
T_{4}=\begin{tabular}{|c|c|}
\hline
1 & 3 \\
\hline
2 & 4 \\
\hline
5 & 6 \\
\hline
\end{tabular}\ ,\quad T_{5}=\begin{tabular}{|c|c|}
\hline
1 & 2 \\
\hline
3 & 4 \\
\hline
5 & 6 \\
\hline
\end{tabular}\ .$$ One can check that all the cell graphs are the same this graph is
(70,120) (15,50)[$T_{4}$]{} (50,20)[$T_{5}$]{} (50,110)[$T_{1}$]{} (50,80)[$T_{2}$]{} (80,50)[$T_{3}$]{} (52,90)[(0,1)[17]{}]{}(55,97)[$\scriptstyle 3$]{} (28,55)[(1,1)[22]{}]{}(30,65)[$\scriptstyle 4$]{} (76,55)[(-1,1)[22]{}]{}(74,65)[$\scriptstyle 2$]{} (48,30)[(-1,1)[19]{}]{}(30,38)[$\scriptstyle 2$]{} (55,30)[(1,1)[20]{}]{}(71,38)[$\scriptstyle 4$]{}
On the other hand one has
$$R_{\sigma_{T_1},\sigma_{T_5}}=\left(\tiny{\begin{array}{cccccc}
0&0&0&1&2&3\\
0&0&0&1&1&2\\
0&0&0&1&1&1\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
\end{array}}\right)=R_{(1,5)(2,6)(3,4)}$$
and $\dim(B.N_{(1,5)(2,6)(3,4)})=8$, so that ${{\rm codim\,}}_{{\mathcal{V}}_{T_1}}({\mathcal{V}}_{T_1}\cap{\mathcal{V}}_{T_5})=1.$ As well the straight computations show that $\dim({\mathcal{V}}_{T_1}\cap{\mathcal{V}}_{T_4})=\dim({\mathcal{V}}_{T_1}\cap{\mathcal{V}}_{T_3})=
\dim({\mathcal{V}}_{T_2}\cap{\mathcal{V}}_{T_5})=\dim({\mathcal{V}}_{T_3}\cap{\mathcal{V}}_{T_4})=7$ so that all these intersections are of codimension 2. Further, ${\mathcal{V}}_{T_1}\cap{\mathcal{V}}_{T_4},\ {\mathcal{V}}_{T_1}\cap{\mathcal{V}}_{T_3}$ and ${\mathcal{V}}_{T_3}\cap{\mathcal{V}}_{T_4}$ are irreducible. ${\mathcal{V}}_{T_2}\cap{\mathcal{V}}_{T_5}$ has three components with the following dense $B$-orbits: $B.N_{(1,3)(2,5)(4,6)},\ B.N_{(1,5)(2,4)(3,6)},$ and $B.N_{(1,4)(2,6)(3,5)}.$ Below we draw the graph where two vertices are joined if the corresponding intersection is of codimension 1.
[cc]{}
(100,90)
(0,10)[$T_{4}$]{} (90,10)[$T_{5}$]{} (0,50)[$T_{2}$]{} (90,50)[$T_{3}$]{} (39,85)[$T_{1}$]{}
(2,20)[(0,1)[25]{}]{}(10,12)[(1,0)[75]{}]{}(92,20)[(0,1)[25]{}]{}(10,52)[(1,0)[75]{}]{}(9,58)[(5,4)[28]{}]{} (90,20)[(-2,3)[40]{}]{}
Orbital variety’s construction
------------------------------
Let us go back to Steinberg’s construction of an orbital variety (see Section \[1.4\]). Given $T\in{\mathbf{Tab}}_\lambda$ one has $\mathcal{V}_{T}=\overline{B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})}\cap {{\mathcal{O}}}_{\lambda}$ for any $w \in \mathcal{C}_T$. Obviously, $${\rm dim}(\overline{B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})}\cap {{\mathcal{O}}}_{\lambda})={\rm dim}(B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})\cap {{\mathcal{O}}}_{\lambda}),$$ so that ${\rm dim}(B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})\cap {{\mathcal{O}}}_{\lambda})={\rm dim}({\mathcal{O}}_{\lambda}\cap {\mathfrak{n}})$, therefore $B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})\cap {{\mathcal{O}}}_{\lambda}$ is also irreducible in ${\mathcal{O}}_{\lambda}\cap {\mathfrak{n}}$; in particular $B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})\cap {{\mathcal{O}}}_{\lambda}$ is an orbital variety if and only if $B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})\cap {{\mathcal{O}}}_{\lambda}$ is closed in ${\mathcal{O}}_{\lambda}\cap {\mathfrak{n}}$. The natural questions connected to the construction are\
[**Q1**]{}: May be one can always find $w\in{\mathcal C}_T$ such that $\mathcal{V}_T=B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})\cap {{\mathcal{O}}}_{\lambda}$?\
[**Q2**]{}: Or may be $\mathcal{V}_T=\bigcup\limits_{y\in \mathcal{C}_T}B.({\mathfrak{n}}\cap {^y {\mathfrak{n}}})\cap {{\mathcal{O}}}_{\lambda}$?\
The answers to both these questions are negative as we show by the following counter-example.
Let $T=\begin{tabular}{|c|c|}
\hline
1 & 3 \\
\hline
2 \\
\cline{1-1}
4 \\
\cline{1-1}
\end{tabular}$. The corresponding left cell is given by $$\mathcal{C}_T=\{w_1=\tiny{\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
4 & 2 & 3 & 1 \\
\end{array}
\right)},\ w_2=\tiny{\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
2 & 4 & 3 & 1 \\
\end{array}
\right)},\ w_3=\tiny{\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
4 & 2 & 1 & 3 \\
\end{array}
\right)}\}.$$
We draw here in green the corresponding space ${\mathfrak{n}}\cap {^w {\mathfrak{n}}}$:
{width="10cm" height="2.3cm"}
On the other hand by Theorem \[thm1\], $\mathcal{V}_T=B.N_{(2,3)}\cup B.N_{(2,4)}\cup B.N_{(1,3)}\cup B.N_{(1,4)}$. As one can see from the picture $N_{(1,4)}\not\in B.({\mathfrak{n}}\cap {^w {\mathfrak{n}}})$ for $w\in \{w_1,w_2,w_3\}$.
0.2 cm [**Acknowledgements.**]{} The second author would like to express his gratitude to A. Joseph, Lê Dũng Tráng, H. Esnault and E. Viehweg for the invitation to the Weizmann institute of Science, the Abdus Salam International Centre for Theoretical Physics and the University of Duisburg-Essen where this work was done. He would also like to thank these institutions for their kind hospitality and support.
INDEX OF NOTATION
Symbols appearing frequently are given below in order of appearance.
\[1.1\] ${\mathfrak{n}}, {\mathfrak{g}}_\alpha,\ \alpha_i,\ \Pi,\ \alpha_{i,j},\ B,\
{\mathbf{S}}_n,\ s_i,\ g.u,\ {\mathcal{F}}_x,\ {\mathcal{O}}_x;$
\[1.2\] $J(x),\ {\mathcal{O}}_\lambda,\ {{\rm sh\,}}({\mathcal{O}}),\ {{\rm sh\,}}(T),\ {\mathbf{Tab}}_\lambda,\
{\mathcal{F}}_T,\ {\mathcal{V}}_T;$
\[2.3\] $\pi_{i,j}:{\mathfrak{n}}_n{\rightarrow}{\mathfrak{n}}_{j+1-i}$
\[4.1\] ${\mathcal{X}}_2,\ {\mathbf{S}}_n^2,\ N_\sigma,\ {\mathbf{Tab}}_n^2,\ \sigma_T;$
\[4.2\] $L(\sigma),\ {\mathcal{O}}_\sigma;$
\[4.3\] $R_\sigma,\ {\mathbf R}_n^2;$
\[4.3b\] ${\mathbf{S}}_{\langle i,j\rangle}^2,\
\pi_{i,j}:{\mathbf{S}}_n^2\rightarrow{\mathbf{S}}_{\langle i,j\rangle}^2$
\[4.3c\] ${\mathbf{S}}_n^2(k),\ {\mathbf{Tab}}_n^2(k)$
[m]{}
W. Borho and J.L. Brylinski, Differential operators on homogeneous spaces III, Invent. Math. 80 (1985), pp. 1-68.
F.Y.C. Fung, On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory, Advances in Math. 178 (2003), pp.244-276.
M. Kashiwara, Y. Saito, Geometric construction of Crystal bases, Duke Math. J. 89 (1) (1997), pp. 9-36.
D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), pp. 165-184.
A.Melnikov, Orbital varieties in ${\mathfrak{s}\mathfrak{l}}_n$ and the Smith conjecture, J. of Algebra 200 (1998), pp. 1-31.
A.Melnikov, B-orbits in solutions to the equation $X^2=2$ in triangular matrices, J. of Algebra 223 (2000), pp. 101-108.
A. Melnikov, Description of B-orbit closures of order 2 in upper-triangular matrices, Transform. Groups 11 (2006), no. 2, pp. 217–247.
A. Melnikov, N.G.J. Pagnon, On the intersections of orbital varieties and components of Springer fiber, J. Algebra 298 (2006), pp. 1-14.
A. Melnikov, N.G.J. Pagnon, Intersections of components of a Springer fiber of codimension one for the two-column case, arXiv:math/0701178.
P. Slodowy, Four lectures on simple groups and singularities, Commun. Math. Inst., Rijksuniv. Utr. 11, 1980.
N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold, Indag. Math. 38 (1976), pp. 452-456.
N. Spaltenstein, On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology 16 (1977), pp. 203-204.
T.A. Springer, The unipotent variety of a semisimple group, Proc. of the Bombay Colloqu. in Algebraic Geometry, ed. S. Abhyankar, London, Oxford Univ. Press (1969), pp. 373-391.
T.A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), pp. 279-293.
T.A. Springer, Linear Algebraic Groups, Progress in Mathematics. Birkäuser Boston, 2nd edition 1998.
R. Steinberg, On the Desingularisation of the Unipotent Variety, Invent. Math. 36 (1976), pp. 209-224.
R. Steinberg, An Occurrence of the Robinson-Schensted Correspondence, J. Algebra 113 No. 2 (1988), pp. 523-528.
J.A. Vargas, Fixed points under the action of unipotent elements of ${\rm {SL}_{n}}$ in the flag variety, Bol. Soc. Mat. Mexicana 24 (1979), pp. 1-14.
[^1]: This work has been supported by the SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties” and by the Marie Curie Research Training Networks “Liegrits".
|
---
abstract: 'In the present paper we prove the existence outer derivations finite-dimensional solvable Lie algebras with nilradical of maximal rank and complementary subspace to nilardical of dimension less than rank of the nilradical.'
address:
- '\[K. K. Abdurasulov\] V.I.Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences, Tashkent 100170, Uzbekistan.'
- '\[R. Q. Gaybullayev\] National University of Uzbekistan, Tashkent, Uzbekistan.'
author:
- 'K.K. Abdurasulov, R.Q. Gaybullayev.'
title: Outer derivations of complex solvable Lie algebras with nilradical of maximal rank
---
Derivations play an important role in the theory of Lie algebras. They are two kinds of derivations (inner and outer derivations). It is know that every nilpotent Lie algebra over a field of arbitrary characteristic has an outer derivation, while any derivation of semisimple Lie algebras is inner [@jaci]. In the case of solvable Lie algebras the situation is more complicated. In the case of solvable Lie algebras we can not affirm innerness or outerness of derivations definitely.
In the paper [@xosiyat] study of solvable Lie algebras for which the dimension of the complementary space is equal to the number of generators of the nilradical. It is easy to see that all derivetions of the algebra are inner. In the present note we prove that it is a unique case when solvable Lie algebras with nilradical of maximal rank have no outer derivations. In other words, we prove that any solvable Lie algebra with nilradical of maximal rank and complementary subspace to nilradical of dimension less than rank admits an outer derivation.
In order to begin our study we recall the description of solvable Lie algebras whose nilradical has maximal rank (that is, maximal torus of the nilradical has dimension equal to the number of generator basis elements of nilradical) and dimension of complementary subspace to nilradical is equal to the rank of nilradical.
\[thm1\][@xosiyat] Let $R=N\oplus Q$ be a solvable Lie algebra such that $dimQ=dim N/N^2=k.$ Then $R$ admits a basis $\{e_1, e_2, \dots, e_k, e_{k+1}, \dots,
e_n, x_1, x_2, \dots, x_k\}$ such that the table of multiplications of $R$ in the basis has the following form: $$\left\{\begin{array}{ll}
[e_i,e_j]=\sum\limits_{t=k+1}^{n}\gamma_{i,j}^te_t,& 1\leq i, j\leq n,\\[1mm]
[e_i,x_i]=e_i,& 1\leq i\leq k,\\[1mm]
[e_i,x_j]=\alpha_{i,j}e_i,& k+1\leq i\leq n,\ \ 1\leq j\leq k,\\[1mm]
\end{array}\right.$$ where $\alpha_{i,j}$ is the number of entries of a generator basis element $e_j$ involved in forming non generator basis element $e_i$.
Structure constants $\alpha_{i,j}$ are roots in the decomposition of $N$ with respect to its maximal torus.
Further we shall also use the following result.
\[prop1\] [@leger] Let $L$ be a Lie algebra over a field of characteristic $0$ such that $Der(L)=Inder(L).$ If the center of the Lie algebra $L$ is non trivial, then $L$ is not solvable and the radical of $L$ is nilpotent.
Let $R$ be a solvable Lie algebra with nilradical $N$ and complementary subspace to nilradical $Q$, then $R=N\oplus Q$ as a direct sum of vector spaces.
We set $$N_{max}=\{ N \ | \ \ \mbox{there exists solvable } \ R \ \mbox{such that } dimQ=dimN/N^2\}.$$
Let $R=N_{max}\oplus Q$ is solvable Lie algebras such that $dimQ < dimN/N^2$. Then $R$ admits an outer derivation.
Set $dimQ=s$ and $dimN/ N^2=k$ and consider two possible cases.
***Case 1.*** Let $Center(R)=\{0\}$. Since $ad(R)$ is a solvable Lie algebra, $ad_x$ for $x \in Q$ has upper triangle form and we can write as follows: $$ad_{x_i}=d_i+d_{n_i},\ \quad 1\leq i\leq s,$$ where $d_i: R\rightarrow R$ is a diagonal derivation and $d_{n_i}:R\rightarrow R$ is nilpotent derivation whose matrix realization has is strictly upper triangle form.
If there exists $i_0$ such that $d_{n_{i_0}}\notin InDer(N)$. Then $d_{i_0}$ is outer derivation.
Suppose now that $d_{n_{i}}\in InDer(N)$ for any $1\leq i\leq s,$ that is, there are $z_i \in N$ such that $ad_{z_i}=d_{n_{i}}$. Consequently, $ad_{x_i}-ad_{z_i}=ad_{x_i-z_i}$ lie in a maximal torus of $N$ (denoted by $Tor_{max}$). Since $dim Tor_{max}=k>s$, then there exists $d'\in Tor_{max}\setminus Span\{ad_{x_1-z_1},ad_{x_2-z_2},\ldots, ad_{x_s-z_s}\}.$
Taking the change of basis $Q$ as follows $x_i'=x_i-z_i, \ 1\leq i \leq s.$ Then $d'(Q)=0$.
From the equality $$[z,[x_i',x_j']]=[[z,x_i'],x_j']-[[z,x_j'],x_i']=(ad_{x_j'}\circ ad_{x_i'}-ad_{x_i'}\circ ad_{x_j'})(z)=0$$ for any any $z\in R, \ 1\leq i,j\leq s$ we conclude that $[x_i',x_j']\in Center(R)=\{0\}$, hence $[Q,Q]=0$.
From the equalities $$d'([n,x])-[n,d'(x)]-[d'(n),x]=d'([n,x])-[d'(n),x]=-d'(ad_{x}(n))+ad_{x}(d'(n))=[ad_{x},d'](n)=0$$ for any $n\in N, x\in Q$ and $$d'([x_i',x_j'])-[x_i',d'(x_{j}')]-[d'(x_i'),x_j']=0,$$ we derive $d'\in Der(R)$.
Thus, we obtain that $d'$ is outer derivation(otherwise we get a contradiction with condition $dim Q=s$).
***Case b.*** Let $Center(R)\neq \{0\}$. Suppose that $Der(R)=Inner(R)$, then by Proposition \[prop1\] we conclude that algebra $R$ is nilpotent, that is a contradiction. Therefore, $Inner(R)\subsetneqq Der(R)$.
**Acknowledgement.** The authors are grateful to Professor B.A. Omirov for pointing out the property of considered algebras.
[99]{}
Jacobson N., Lie algebras, Interscience Publishers, Wiley, New York, 1962. Leger G., Derivations of Lie algebras III, Duke Mathematical Journal, Volume 30, Number 4 (1963), pp. 637-645.
Khalkulova Kh.A., Abdurasulov K.K., Solvable Lie algebras with maximal dimension of complementary space to nilradical, Uzbek Mathematical Journal, No 1, 2018, pp. 90-98.
|
---
abstract: 'We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.'
author:
- |
Cristian Lenart\
Department of Mathematics and Statistics, State University of New York at Albany,\
Albany, NY 12222, U.S.A. (e-mail: [clenart@albany.edu]{})\
Satoshi Naito\
Department of Mathematics, Tokyo Institute of Technology,\
2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan (e-mail: [naito@math.titech.ac.jp]{})\
Daisuke Sagaki\
Institute of Mathematics, University of Tsukuba,\
Tsukuba, Ibaraki 305-8571, Japan (e-mail: [sagaki@math.tsukuba.ac.jp]{})\
Anne Schilling\
Department of Mathematics, University of California,\
One Shields Avenue, Davis, CA 95616-8633, U.S.A. (e-mail: [anne@math.ucdavis.edu]{})\
Mark Shimozono\
Department of Mathematics, MC 0151, 460 McBryde Hall, Virginia Tech,\
225 Stanger St., Blacksburg, VA 24061, U.S.A. (e-mail: [mshimo@vt.edu]{})
title: |
**Quantum Lakshmibai-Seshadri paths\
and root operators**
---
Introduction. {#sec:intro}
=============
In our previous papers [@NS-IMRN], [@NS-Adv], [@NS-Tensor], we gave a combinatorial realization of the crystal bases of level-zero fundamental representations $W({\varpi}_{i})$, $i \in I_{0}$, and their tensor products $\bigotimes_{i \in I_{0}} W({\varpi}_{i})^{\otimes m_{i}}$, $m_{i} \in {\mathbb{Z}}_{\geq 0}$, over quantum affine algebras $U_{q}'({\mathfrak{g}})$, by using projected level-zero Lakshmibai-Seshadri (LS for short) paths. Here, for a level-zero dominant integral weight $\lambda = \sum_{i \in I_{0}} m_{i} {\varpi}_{i}$, with ${\varpi}_{i}$ the $i$-th level-zero fundamental weight, the set of projected level-zero LS paths of shape $\lambda$, which is a “simple” crystal denoted by ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, is obtained from the set ${\mathbb{B}}(\lambda)$ of LS paths of shape $\lambda$ (in the sense of [@Lit-A]) by factoring out the null root $\delta$ of an affine Lie algebra $\mathfrak{g}$. However, from the nature of the above definition of projected level-zero LS paths, our description of these objects in [@NS-IMRN], [@NS-Adv], [@NS-Tensor] was not as explicit as the one (given in [@Lit-I]) of usual LS paths, the shape of which is a dominant integral weight.
Recently, in [@LNSSS1], [@LNSSS2], we proved that a projected level-zero LS path is identical to a certain “rational path”, which we call a quantum LS path. A quantum LS path is described in terms of the (parabolic) quantum Bruhat graph (QBG for short), which was introduced by [@BFP] (and by [@LS] in the parabolic case) in the study of the quantum cohomology ring of the (partial) flag variety; see §\[subsec:def-QBG\] for the definition of the (parabolic) QBG. It is noteworthy that the description of a quantum LS path as a rational path is very similar to the one of a usual LS path given in [@Lit-I], in which we replace the Hasse diagram of the (parabolic) Bruhat graph by the (parabolic) QBG. Also, remark that the vertices of the (parabolic) QBG are the minimal-length representatives for the cosets of a parabolic subgroup $W_{0,\,J}$ of the finite Weyl group $W_{0}$, though we consider finite-dimensional representations $W({\varpi}_{i})$, $i \in I_{0}$, of quantum affine algebras $U_{q}'({\mathfrak{g}})$.
The purpose of this paper is to give an explicit description, in terms of rational paths, of the image of a quantum LS path ($=$ projected level-zero LS path) under root operators in a way similar to the one given in [@Lit-I]; see Theorem \[thm:main\] for details. This explicit description, together with the Diamond Lemmas [@LNSSS1 Lemma 5.14], for the parabolic QBG, provides us with a proof of the fact that the set of quantum LS paths (the shape of which is a level-zero dominant integral weight $\lambda$) is stable under the action of the root operators.
As a by-product of the stability property above, we obtain another (but somewhat roundabout) proof of the fact that a projected level-zero LS path is just a quantum LS path; see [@LNSSS1], [@LNSSS2] for a more direct proof. This new proof is accomplished by making use of a characterization (Theorem \[thm:charls\]) of the set ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ of projected level-zero LS paths of shape $\lambda$ in terms of root operators, which is based upon the connectedness of the (crystal graph for the) tensor product crystal $\bigotimes_{i \in I_{0}}
{\mathbb{B}}({\varpi}_{i})_{{\mathop{\rm cl}\nolimits}}^{\otimes m_{i}}
\simeq {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$; recall from [@NS-IMRN], [@NS-Adv], [@NS-Tensor] that for a level-zero dominant integral weight $\lambda = \sum_{i \in I_{0}} m_{i} {\varpi}_{i}$, the crystal ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ decomposes into the tensor product $\bigotimes_{i \in I_{0}}
{\mathbb{B}}({\varpi}_{i})^{\otimes m_{i}}_{{\mathop{\rm cl}\nolimits}}$ of crystals, and that ${\mathbb{B}}({\varpi}_{i})_{{\mathop{\rm cl}\nolimits}}$ for each $i \in I_{0}$ is isomorphic to the crystal basis of the level-zero fundamental representation $W({\varpi}_{i})$.
This paper is organized as follows. In §\[sec:LS\], we fix our fundamental notation, and recall some basic facts about (level-zero) LS path crystals. Also, we give a characterization (Theorem \[thm:charls\]) of projected level-zero LS paths, which is needed to obtain our main result (Theorem \[thm:main\]). In §\[sec:QLS\], we recall the notion of the (parabolic) quantum Bruhat graph, and then give the definition of quantum LS paths. In §\[sec:main\], we first state our main result. Then, after preparing several technical lemmas, we finally obtain an explicit description (Proposition \[prop:stable\]) of the image of a quantum LS path as a rational path under the action of root operators. Our main result follows immediately from this description, together with the characterization above of projected level-zero LS paths.
#### Acknowledgments.
C.L. was partially supported by the NSF grant DMS–1101264. S.N. was supported by Grant-in-Aid for Scientific Research (C), No.24540010, Japan. D.S. was supported by Grant-in-Aid for Young Scientists (B) No.23740003, Japan. A.S. was partially supported by the NSF grants DMS–1001256, OCI–1147247, and a grant from the Simons Foundation (\#226108 to Anne Schilling). M.S. was partially supported by the NSF grant DMS–1200804.
Lakshmibai-Seshadri paths. {#sec:LS}
==========================
Basic notation. {#subsec:notation}
---------------
Let ${\mathfrak{g}}$ be an untwisted affine Lie algebra over ${\mathbb{C}}$ with Cartan matrix $A=(a_{ij})_{i,\,j \in I}$; throughout this paper, the elements of the index set $I$ are numbered as in [@Kac §4.8, Table Aff 1]. Take a distinguished vertex $0 \in I$ as in [@Kac], and set $I_{0}:=I \setminus \{0\}$. Let ${\mathfrak{h}}=\bigl(\bigoplus_{j \in I} {\mathbb{C}}\alpha_{j}^{\vee}\bigr) \oplus {\mathbb{C}}d$ denote the Cartan subalgebra of ${\mathfrak{g}}$, where $\Pi^{\vee}:=\bigl\{\alpha_{j}^{\vee}\bigr\}_{j \in I} \subset {\mathfrak{h}}$ is the set of simple coroots, and $d \in {\mathfrak{h}}$ is the scaling element (or degree operator). Also, we denote by $\Pi:=\bigl\{\alpha_{j}\bigr\}_{j \in I} \subset
{\mathfrak{h}}^{\ast}:={\mathop{\rm Hom}\nolimits}_{{\mathbb{C}}}({\mathfrak{h}},{\mathbb{C}})$ the set of simple roots, and by $\Lambda_{j} \in {\mathfrak{h}}^{\ast}$, $j \in I$, the fundamental weights; note that $\alpha_{j}(d)=\delta_{j,0}$ and $\Lambda_{j}(d)=0$ for $j \in I$. Let $\delta=\sum_{j \in I} a_{j}\alpha_{j} \in {\mathfrak{h}}^{\ast}$ and $c=\sum_{j \in I} a^{\vee}_{j} \alpha_{j}^{\vee} \in {\mathfrak{h}}$ denote the null root and the canonical central element of ${\mathfrak{g}}$, respectively. The Weyl group $W$ of ${\mathfrak{g}}$ is defined by $W:=\langle r_{j} \mid j \in I\rangle \subset {\mathop{\rm GL}\nolimits}({\mathfrak{h}}^{\ast})$, where $r_{j} \in {\mathop{\rm GL}\nolimits}({\mathfrak{h}}^{\ast})$ denotes the simple reflection associated to $\alpha_{j}$ for $j \in I$, with $\ell:W \rightarrow {\mathbb{Z}}_{\ge 0}$ the length function on $W$. Denote by ${\Delta_{\mathrm{re}}}$ the set of real roots, i.e., ${\Delta_{\mathrm{re}}}:=W\Pi$, and by ${\Delta_{\mathrm{re}}^{+}}\subset {\Delta_{\mathrm{re}}}$ the set of positive real roots; for $\beta \in {\Delta_{\mathrm{re}}}$, we denote by $\beta^{\vee}$ the dual root of $\beta$, and by $r_{\beta} \in W$ the reflection with respect to $\beta$. We take a dual weight lattice $P^{\vee}$ and a weight lattice $P$ as follows: $$\label{eq:lattices}
P^{\vee}=
\left(\bigoplus_{j \in I} {\mathbb{Z}}\alpha_{j}^{\vee}\right) \oplus {\mathbb{Z}}d \,
\subset {\mathfrak{h}}\quad \text{and} \quad
P=
\left(\bigoplus_{j \in I} {\mathbb{Z}}\Lambda_{j}\right) \oplus
{\mathbb{Z}}\delta \subset {\mathfrak{h}}^{\ast}.$$ It is clear that $P$ contains $Q:=\bigoplus_{j \in I} {\mathbb{Z}}\alpha_{j}$, and that $P \cong {\mathop{\rm Hom}\nolimits}_{{\mathbb{Z}}}(P^{\vee},{\mathbb{Z}})$.
Let $W_{0}$ be the subgroup of $W$ generated by $r_{j}$, $j \in I_{0}$, and set $\Delta_{0}:=
{\Delta_{\mathrm{re}}}\cap \bigoplus_{j \in I_{0}}{\mathbb{Z}}\alpha_{j}$, $\Delta_{0}^{+}:=
{\Delta_{\mathrm{re}}}\cap \bigoplus_{j \in I_{0}}{\mathbb{Z}}_{\ge 0}\alpha_{j}$, and $\Delta_{0}^{-}:=-\Delta_{0}^{+}$. Note that $W_{0}$ (resp., $\Delta_{0}$, $\Delta_{0}^{+}$, $\Delta_{0}^{-}$) can be thought of as the (finite) Weyl group (resp., the set of roots, the set of positive roots, the set of negative roots) of the finite-dimensional simple Lie algebra corresponding to $I_{0}$. Denote by $\theta \in \Delta_{0}^{+}$ the highest root for the (finite) root system $\Delta_{0}$; note that $\alpha_{0}=-\theta+\delta$ and $\alpha_{0}^{\vee}=-\theta^{\vee}+c$.
\[dfn:lv0\]
An integral weight $\lambda \in P$ is said to be of level zero if ${\langle \lambda,\,c \rangle}=0$.
An integral weight $\lambda \in P$ is said to be level-zero dominant if ${\langle \lambda,\,c \rangle}=0$, and ${\langle \lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$ for all $j \in I_{0}=I \setminus \{0\}$.
\[rem:theta\] If $\lambda \in P$ is of level zero, then ${\langle \lambda,\,\alpha_{0}^{\vee} \rangle}=-{\langle \lambda,\,\theta^{\vee} \rangle}$.
For each $i \in I_{0}$, we define a level-zero fundamental weight ${\varpi}_{i} \in P$ by $${\varpi}_{i}:=\Lambda_{i}-a_{i}^{\vee}\Lambda_{0}.$$ The ${\varpi}_{i}$ for $i \in I_{0}$ is actually a level-zero dominant integral weight; indeed, ${\langle {\varpi}_{i},\,c \rangle}=0$ and ${\langle {\varpi}_{i},\,\alpha_{j}^{\vee} \rangle}=\delta_{i,j}$ for $j \in I_{0}$.
Let ${\mathop{\rm cl}\nolimits}:{\mathfrak{h}}^{\ast} \twoheadrightarrow {\mathfrak{h}}^{\ast}/{\mathbb{C}}\delta$ be the canonical projection from ${\mathfrak{h}}^{\ast}$ onto ${\mathfrak{h}}^{\ast}/{\mathbb{C}}\delta$, and define $P_{{\mathop{\rm cl}\nolimits}}$ and $P_{{\mathop{\rm cl}\nolimits}}^{\vee}$ by $$\label{eq:lat-cl}
P_{{\mathop{\rm cl}\nolimits}} := {\mathop{\rm cl}\nolimits}(P) =
\bigoplus_{j \in I} {\mathbb{Z}}{\mathop{\rm cl}\nolimits}(\Lambda_{j})
\quad \text{and} \quad
P_{{\mathop{\rm cl}\nolimits}}^{\vee} :=
\bigoplus_{j \in I} {\mathbb{Z}}\alpha_{j}^{\vee}
\subset P^{\vee}.$$ We see that $P_{{\mathop{\rm cl}\nolimits}} \cong P/{\mathbb{Z}}\delta$, and that $P_{{\mathop{\rm cl}\nolimits}}$ can be identified with ${\mathop{\rm Hom}\nolimits}_{{\mathbb{Z}}}(P_{{\mathop{\rm cl}\nolimits}}^{\vee},{\mathbb{Z}})$ as a ${\mathbb{Z}}$-module by $$\label{eq:pair}
{\langle {\mathop{\rm cl}\nolimits}(\lambda),\,h \rangle}={\langle \lambda,\,h \rangle} \quad
\text{for $\lambda \in P$ and $h \in P_{{\mathop{\rm cl}\nolimits}}^{\vee}$}.$$ Also, there exists a natural action of the Weyl group $W$ on ${\mathfrak{h}}^{\ast}/{\mathbb{C}}\delta$ induced by the one on ${\mathfrak{h}}^{\ast}$, since $W\delta=\delta$; it is obvious that $w \circ {\mathop{\rm cl}\nolimits}= {\mathop{\rm cl}\nolimits}\circ w$ for all $w \in W$.
\[rem:orbcl\] Let $\lambda \in P$ be a level-zero integral weight. It is easy to check that ${\mathop{\rm cl}\nolimits}(W\lambda)=W_{0}{\mathop{\rm cl}\nolimits}(\lambda)$ (see the proof of [@NS-LMS Lemma 2.3.3]). In particular, we have ${\mathop{\rm cl}\nolimits}(r_{0}\lambda)={\mathop{\rm cl}\nolimits}(r_{\theta}\lambda)$ since $\alpha_{0}=-\theta+\delta$ and $\alpha_{0}^{\vee}=-\theta^{\vee}+c$.
For simplicity of notation, we often write $\beta$ instead of ${\mathop{\rm cl}\nolimits}(\beta) \in P_{{\mathop{\rm cl}\nolimits}}$ for $\beta \in Q=\bigoplus_{j \in I} {\mathbb{Z}}\alpha_{j}$; note that $\alpha_{0}=-\theta$ in $P_{{\mathop{\rm cl}\nolimits}}$ since $\alpha_{0}=-\theta+\delta$ in $P$.
Paths and root operators. {#subsec:path}
-------------------------
A path with weight in $P_{{\mathop{\rm cl}\nolimits}}={\mathop{\rm cl}\nolimits}(P)$ is, by definition, a piecewise-linear, continuous map $\pi:[0,1] \rightarrow {\mathbb{R}}\otimes_{{\mathbb{Z}}} P_{{\mathop{\rm cl}\nolimits}}$ such that $\pi(0)=0$ and $\pi(1) \in P_{{\mathop{\rm cl}\nolimits}}$. We denote by ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ the set of all paths with weight in $P_{{\mathop{\rm cl}\nolimits}}$, and define ${\mathop{\rm wt}\nolimits}:{\mathbb{P}}_{{\mathop{\rm cl}\nolimits}} \rightarrow P_{{\mathop{\rm cl}\nolimits}}$ by $$\label{eq:wt}
{\mathop{\rm wt}\nolimits}(\eta):=\eta(1) \quad
\text{\rm for $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$}.$$ For $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ and $j \in I$, we set $$\label{eq:Hm}
\begin{array}{l}
H^{\eta}_{j}(t):={\langle \eta(t),\,\alpha_{j}^{\vee} \rangle}
\quad \text{for \,} t \in [0,1], \\[3mm]
m^{\eta}_{j}
:=\min\bigl\{H^{\eta}_{j}(t) \mid t \in [0,1]\bigr\}.
\end{array}
$$ For each $j \in I$, let ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$ denote the subset of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ consisting of all paths $\eta$ for which all local minima of the function $H^{\eta}_{j}(t)$ are integers; note that if $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, then $m^{\eta}_{j} \in {\mathbb{Z}}_{\le 0}$ and $H^{\eta}_{j}(1)-m^{\eta}_{j} \in {\mathbb{Z}}_{\ge 0}$. We set $${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}:=\bigcap_{j \in I} {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)};$$ see also [@NS-Tensor §2.3]. Here we should warn the reader that the set ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ itself is not necessarily stable under the action of the root operators $e_{j}$ and $f_{j}$ for $j \in I$, defined below.
Now, for $j \in I$ and $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, we define $e_{j}\eta$ as follows. If $m^{\eta}_{j} = 0$, then $e_{j}\eta:={{\bf 0}}$, where ${{\bf 0}}$ is an additional element not contained in ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$. If $m^{\eta}_{j} \le -1$, then we define $e_{j}\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ by $$\label{eq:ro_e}
(e_{j}\eta)(t):=
\begin{cases}
\eta(t) & \text{if \,} 0 \le t \le t_{0}, \\[2mm]
\eta(t_{0})+r_{j}(\eta(t)-\eta(t_{0}))
& \text{if \,} t_{0} \le t \le t_{1}, \\[2mm]
\eta(t)+\alpha_{j} & \text{if \,} t_{1} \le t \le 1,
\end{cases}
$$ where we set $$\label{eq:t1}
\begin{array}{l}
t_{1}:=\min\bigl\{t \in [0,1] \mid
H^{\eta}_{j}(t)=m^{\eta}_{j} \bigr\}, \\[2mm]
t_{0}:=\max\bigl\{t \in [0,t_{1}] \mid
H^{\eta}_{j}(t) = m^{\eta}_{j}+1 \bigr\};
\end{array}
$$ note that the function $H^{\eta}_{j}(t)$ is strictly decreasing on $[t_{0},t_{1}]$ since $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$. Because $$H^{e_{j}\eta}_{j}(t)=
\begin{cases}
H^{\eta}_{j}(t) & \text{if \,} 0 \le t \le t_{0}, \\[2mm]
2(m^{\eta}_{j}+1)-H^{\eta}_{j}(t)
& \text{if \,} t_{0} \le t \le t_{1}, \\[2mm]
H^{\eta}_{j}(t)+2 & \text{if \,} t_{1} \le t \le 1,
\end{cases}
$$ it is easily seen that $e_{j}\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, and $m^{e_{j}\eta}_{j}=m^{\eta}_{j}+1$. Therefore, if we set $$\label{eq:ve}
{\varepsilon}_{j}(\eta):=
\max\bigl\{n \ge 0 \mid e_{j}^{n}\eta \ne {{\bf 0}}\bigr\}$$ for $j \in I$ and $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, then ${\varepsilon}_{j}(\eta)=-m^{\eta}_{j}$ (see also [@Lit-A Lemma 2.1c)]). By convention, we set $e_{j}{{\bf 0}}:={{\bf 0}}$ for all $j \in I$.
Assume that $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(0)}$ satisfies the condition that $m^{\eta}_{0} \le -1$ and ${\langle \eta(t),\,c \rangle}=0$ for all $t \in [0,1]$. Then we have $$\label{eq:ro_e0}
(e_{0}\eta)(t)=
\begin{cases}
\eta(t) & \text{if \,} 0 \le t \le t_{0}, \\[2mm]
\eta(t_{0})+r_{\theta}(\eta(t)-\eta(t_{0}))
& \text{if \,} t_{0} \le t \le t_{1}, \\[2mm]
\eta(t)-\theta & \text{if \,} t_{1} \le t \le 1,
\end{cases}
$$ where $t_{0}$ and $t_{1}$ are defined by for $j=0$.
Similarly, for $j \in I$ and $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, we define $f_{j}\eta$ as follows. If $H^{\eta}_{j}(1)-m^{\eta}_{j}=0$, then $f_{j}\eta:={{\bf 0}}$. If $H^{\eta}_{j}(1)-m^{\eta}_{j} \ge 1$, then we define $f_{j}\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ by $$\label{eq:ro_f}
(f_{j}\eta)(t):=
\begin{cases}
\eta(t) & \text{if \,} 0 \le t \le t_{0}, \\[2mm]
\eta(t_{0})+r_{j}(\eta(t)-\eta(t_{0}))
& \text{if \,} t_{0} \le t \le t_{1}, \\[2mm]
\eta(t)-\alpha_{j} & \text{if \,} t_{1} \le t \le 1,
\end{cases}
$$ where we set $$\label{eq:t2}
\begin{array}{l}
t_{0}:=\max\bigl\{t \in [0,1] \mid
H^{\eta}_{j}(t)=m^{\eta}_{j}\bigr\}, \\[2mm]
t_{1}:=\min\bigl\{t \in [t_{0},1] \mid
H^{\eta}_{j}(t)=m^{\eta}_{j}+1\bigr\};
\end{array}
$$ note that the function $H^{\eta}_{j}(t)$ is strictly increasing on $[t_{0},t_{1}]$ since $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$. Because $$H^{f_{j}\eta}_{j}(t)=
\begin{cases}
H^{\eta}_{j}(t) & \text{if \,} 0 \le t \le t_{0}, \\[2mm]
2m^{\eta}_{j}-H^{\eta}_{j}(t)
& \text{if \,} t_{0} \le t \le t_{1}, \\[2mm]
H^{\eta}_{j}(t)-2 & \text{if \,} t_{1} \le t \le 1,
\end{cases}
$$ it is easily seen that $f_{j}\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, and $m^{f_{j}\eta}_{j}=m^{\eta}_{j}-1$. Therefore, if we set $$\label{eq:vp}
{\varphi}_{j}(\eta):=
\max\bigl\{n \ge 0 \mid f_{j}^{n}\eta \ne {{\bf 0}}\bigr\}$$ for $j \in I$ and $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, then ${\varphi}_{j}(\eta)=H^{\eta}_{j}(1)-m^{\eta}_{j}$ (see also [@Lit-A Lemma 2.1c)]). By convention, we set $f_{j}{{\bf 0}}:={{\bf 0}}$ for all $j \in I$.
\[rem:ro\_f0\] Assume that $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(0)}$ satisfies the condition that $H^{\eta}_{0}(1)-m^{\eta}_{0} \ge 1$ and ${\langle \eta(t),\,c \rangle}=0$ for all $t \in [0,1]$. Then we have $$\label{eq:ro_f0}
(f_{0}\eta)(t)=
\begin{cases}
\eta(t) & \text{if \,} 0 \le t \le t_{0}, \\[2mm]
\eta(t_{0})+r_{\theta}(\eta(t)-\eta(t_{0}))
& \text{if \,} t_{0} \le t \le t_{1}, \\[2mm]
\eta(t)+\theta & \text{if \,} t_{1} \le t \le 1,
\end{cases}
$$ where $t_{0}$ and $t_{1}$ are defined by for $j=0$.
We know the following theorem from [@Lit-A §2] (see also [@NS-Tensor Theorem 2.4]); for the definition of crystals, see [@Kas-OnC §7.2] or [@HK §4.5] for example.
\[thm:pc01\]
Let $j \in I$, and $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$. If $e_{j}\eta \ne {{\bf 0}}$, then $f_{j}e_{j}\eta=\eta$. Also, if $f_{j}\eta \ne {{\bf 0}}$, then $e_{j}f_{j}\eta=\eta$.
Let ${\mathbb{B}}$ be a subset of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ such that the set ${\mathbb{B}}\cup \{{{\bf 0}}\}$ is stable under the action of the root operators $e_{j}$ and $f_{j}$ for all $j \in I$. The set ${\mathbb{B}}$, equipped with the root operators $e_{j}$, $f_{j}$ for $j \in I$ and the maps , , , is a crystal with weights in $P_{{\mathop{\rm cl}\nolimits}}$.
In §\[subsec:LS\], we wiil give a typical example of a subset ${\mathbb{B}}$ of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ such that ${\mathbb{B}}\cup \{{{\bf 0}}\}$ is stable under the action of root operators.
For each path $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ and $N \in {\mathbb{Z}}_{\ge 1}$, we define a path $N\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ by: $(N\eta)(t)=N\eta(t)$ for $t \in [0,1]$; by convention, we set $N{{\bf 0}}:={{\bf 0}}$ for all $N \in {\mathbb{Z}}_{\ge 1}$. It is easily verified that if $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$ for some $j \in I$, then $N\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$ for all $N \in {\mathbb{Z}}_{\ge 1}$.
\[lem:N\] Let $j \in I$. For every $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$ and $N \in {\mathbb{Z}}_{\ge 1}$, we have $$\begin{aligned}
& {\varepsilon}_{j}(N\eta)=N{\varepsilon}_{j}(\eta) \quad \text{\rm and} \quad
{\varphi}_{j}(N\eta)=N{\varphi}_{j}(\eta), \\
& N(e_{j}\eta)=e_{j}^{N}(N\eta) \quad \text{\rm and} \quad
N(f_{j}\eta)=f_{j}^{N}(N\eta).\end{aligned}$$
For $j \in I$ and $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$, we define $e_{j}^{\max}\eta:=e_{j}^{{\varepsilon}_{j}(\eta)}\eta
\in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$ and $f_{j}^{\max}\eta:=f_{j}^{{\varphi}_{j}(\eta)}\eta
\in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$. The next lemma follows immediately from Lemma \[lem:N\].
\[lem:Ne\] Let $j \in I$. For every $\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}^{(j)}$ and $N \in {\mathbb{Z}}_{\ge 1}$, we have $e_{j}^{\max}(N\eta)=N(e_{j}^{\max}\eta)$ and $f_{j}^{\max}(N\eta)=N(f_{j}^{\max}\eta)$.
Now, for $\eta_{1},\,\eta_{2},\,\dots,\,\eta_{n} \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$, define the concatenation $\eta_{1} \ast \eta_{2} \ast \cdots \ast
\eta_{n} \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ by $$\begin{aligned}
\label{eq:cat}
& (\eta_{1} \ast \eta_{2} \ast \cdots \ast \eta_{n})(t):=
\sum_{l=1}^{k-1}
\eta_{l}(1)+ \eta_{k}(nt-k+1) \nonumber \\[-1.5mm]
& \hspace{60mm} \text{for \, } \frac{k-1}{n} \le t \le
\frac{k}{n} \text{\, and \,} 1 \le k \le n. \end{aligned}$$ For a subset ${\mathbb{B}}$ of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits}}$ and $n \in {\mathbb{Z}}_{\ge 1}$, we set ${\mathbb{B}}^{\ast n}:=\bigl\{
\eta_{1} \ast \eta_{2} \ast \cdots \ast \eta_{n} \mid
\text{$\eta_{k} \in {\mathbb{B}}$ for $1 \le k \le n$}
\bigr\}$.
\[prop:cat\] Let ${\mathbb{B}}$ be a subset of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ such that the set ${\mathbb{B}}\cup \{{{\bf 0}}\}$ is stable under the action of the root operators $e_{j}$ and $f_{j}$ for all $j \in I$; note that ${\mathbb{B}}$ is a crystal with weights in $P_{{\mathop{\rm cl}\nolimits}}$ by Theorem \[thm:pc01\].
For every $n \in {\mathbb{Z}}_{\ge 1}$, the set ${\mathbb{B}}^{\ast n} \cup \{{{\bf 0}}\}$ is stable under the root operators $e_{j}$ and $f_{j}$ for all $j \in I$. Therefore, ${\mathbb{B}}^{\ast n}$ is a crystal with weights in $P_{{\mathop{\rm cl}\nolimits}}$ by Theorem \[thm:pc01\].
For every $n \in {\mathbb{Z}}_{\ge 1}$, the crystal ${\mathbb{B}}^{\ast n}$ is isomorphic as a crystal to the tensor product ${\mathbb{B}}^{\otimes n}:={\mathbb{B}}\otimes \cdots \otimes {\mathbb{B}}$ ($n$ times), where the isomorphism is given by[:]{} $\eta_{1} \ast \eta_{2} \ast \cdots \ast \eta_{n} \mapsto
\eta_{1} \otimes \eta_{2} \otimes \cdots \otimes \eta_{n}$ for $\eta_{1} \ast \eta_{2} \ast \cdots \ast \eta_{n} \in {\mathbb{B}}^{\ast n}$.
Lakshmibai-Seshadri paths. {#subsec:LS}
--------------------------
Let us recall the definition of Lakshmibai-Seshadri (LS for short) paths from [@Lit-A §4]. In this subsection, we fix an integral weight $\lambda \in P$ which is not necessarily dominant.
\[dfn:Bruhat\] For $\mu,\,\nu \in W\lambda$, let us write $\mu \ge \nu$ if there exists a sequence $\mu=\mu_{0},\,\mu_{1},\,\dots,\,\mu_{n}=\nu$ of elements in $W\lambda$ and a sequence $\beta_{1},\,\dots,\,\beta_{n} \in {\Delta_{\mathrm{re}}^{+}}$ of positive real roots such that $\mu_{k}=r_{\beta_{k}}(\mu_{k-1})$ and ${\langle \mu_{k-1},\,\beta^{\vee}_{k} \rangle} < 0$ for $k=1,\,2,\,\dots,\,n$. If $\mu \ge \nu$, then we define ${\mathop{\rm dist}\nolimits}(\mu,\nu)$ to be the maximal length $n$ of all possible such sequences $\mu_{0},\,\mu_{1},\,\dots,\,\mu_{n}$ for $(\mu,\nu)$.
\[dfn:achain\] For $\mu,\,\nu \in W\lambda$ with $\mu > \nu$ and a rational number $0 < \sigma < 1$, a $\sigma$-chain for $(\mu,\nu)$ is, by definition, a sequence $\mu=\mu_{0} > \mu_{1} >
\dots > \mu_{n}=\nu$ of elements in $W\lambda$ such that ${\mathop{\rm dist}\nolimits}(\mu_{k-1},\mu_{k})=1$ and $\sigma{\langle \mu_{k-1},\,\beta_{k}^{\vee} \rangle} \in {\mathbb{Z}}_{< 0}$ for all $k=1,\,2,\,\dots,\,n$, where $\beta_{k}$ is the positive real root such that $r_{\beta_{k}}\mu_{k-1}=\mu_{k}$.
\[dfn:LS\] An LS path of shape $\lambda \in P$ is, by definition, a pair $({\underline{\nu}}\,;\,{\underline{\sigma}})$ of a sequence ${\underline{\nu}}:\nu_{1} > \nu_{2} > \cdots > \nu_{s}$ of elements in $W\lambda$ and a sequence ${\underline{\sigma}}:0=\sigma_{0} < \sigma_{1} < \cdots < \sigma_{s}=1$ of rational numbers satisfying the condition that there exists a $\sigma_{k}$-chain for $(\nu_{k},\,\nu_{k+1})$ for each $k=1,\,2,\,\dots,\,s-1$. We denote by ${\mathbb{B}}(\lambda)$ the set of all LS paths of shape $\lambda$.
Let $\pi=(\nu_{1},\,\nu_{2},\,\dots,\,\nu_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})$ be a pair of a sequence $\nu_{1},\,\nu_{2},\,\dots,\,\nu_{s}$ of integral weights with $\nu_{k} \ne \nu_{k+1}$ for $1 \le k \le s-1$ and a sequence $0=\sigma_{0} < \sigma_{1} < \cdots < \sigma_{s}=1$ of rational numbers. We identify $\pi$ with the following piecewise-linear, continuous map $\pi:[0,1] \rightarrow {\mathbb{R}}\otimes_{{\mathbb{Z}}} P$: $$\label{eq:path}
\pi(t)=\sum_{l=1}^{k-1}
(\sigma_{l}-\sigma_{l-1})\nu_{l}+
(t-\sigma_{k-1})\nu_{k} \quad
\text{for $\sigma_{k-1} \le t \le \sigma_{k}$, $1 \le k \le s$}.$$
\[rem:str01\] It is obvious from the definition that for each $\nu \in W\lambda$, $\pi_{\nu}:=(\nu\,;\,0,1)$ is an LS path of shape $\lambda$, which corresponds (under ) to the straight line $\pi_{\nu}(t)=t\nu$, $t \in [0,1]$, connecting $0$ to $\nu$.
For each $\pi \in {\mathbb{B}}(\lambda)$, we define ${\mathop{\rm cl}\nolimits}(\pi):[0,1]
\rightarrow {\mathbb{R}}\otimes_{{\mathbb{Z}}} P_{{\mathop{\rm cl}\nolimits}}$ by: $({\mathop{\rm cl}\nolimits}(\pi))(t)={\mathop{\rm cl}\nolimits}(\pi(t))$ for $t \in [0,1]$. We set $${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}:=
\bigl\{{\mathop{\rm cl}\nolimits}(\pi) \mid \pi \in {\mathbb{B}}(\lambda)\bigr\}.$$ We know from [@NS-Tensor §3.1] that ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ is a subset of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ such that ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \cup \{{{\bf 0}}\}$ is stable under the action of the root operators $e_{j}$ and $f_{j}$ for all $j \in I$. In particular, ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ is a crystal with weights in $P_{{\mathop{\rm cl}\nolimits}}$ by Theorem \[thm:pc01\].
Here we recall the notion of simple crystals. A crystal $B$ with weights in $P_{{\mathop{\rm cl}\nolimits}}$ is said to be regular if for every proper subset $J \subsetneq I$, $B$ is isomorphic as a crystal for $U_{q}({\mathfrak{g}}_{J})$ to the crystal basis of a finite-dimensional $U_{q}({\mathfrak{g}}_{J})$-module, where ${\mathfrak{g}}_{J}$ is the (finite-dimensional) Levi subalgebra of ${\mathfrak{g}}$ corresponding to $J$ (see [@Kas-OnL §2.2]). A regular crystal $B$ with weights $P_{{\mathop{\rm cl}\nolimits}}$ is said to be simple if the set of extremal elements in $B$ coincides with a $W$-orbit in $B$ through an (extremal) element in $B$ (cf. [@Kas-OnL Definition 4.9]).
\[rem:simple\]
The crystal graph of a simple crystal is connected (see [@Kas-OnL Lemma 4.10]).
A tensor product of simple crystals is also a simple crystal (see [@Kas-OnL Lemma 4.11]).
We know the following theorem from [@NS-IMRN Proposition 5.8], [@NS-Adv Theorem 2.1.1 and Proposition 3.4.2], and [@NS-Tensor Theorem 3.2].
\[thm:LScl\]
For each $i \in I_{0}$, the crystal ${\mathbb{B}}({\varpi}_{i})_{{\mathop{\rm cl}\nolimits}}$ is isomorphic, as a crystal with weights in $P_{{\mathop{\rm cl}\nolimits}}$, to the crystal basis of the level-zero fundamental representation $W({\varpi}_{i})$, introduced in [@Kas-OnL Theorem 5.17], of the quantum affine algebra $U_{q}'({\mathfrak{g}})$. In particular, ${\mathbb{B}}({\varpi}_{i})_{{\mathop{\rm cl}\nolimits}}$ is a simple crystal.
Let $i_{1},\,i_{2},\,\dots,\,i_{p}$ be an arbitrary sequence of elements of $I_{0}$ (with repetitions allowed), and set $\lambda:={\varpi}_{i_{1}}+{\varpi}_{i_{2}}+ \cdots + {\varpi}_{i_{p}}$. The crystal ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ is isomorphic, as a crystal with weights in $P_{{\mathop{\rm cl}\nolimits}}$, to the tensor product ${\mathbb{B}}({\varpi}_{i_{1}})_{{\mathop{\rm cl}\nolimits}} \otimes {\mathbb{B}}({\varpi}_{i_{2}})_{{\mathop{\rm cl}\nolimits}} \otimes
\cdots \otimes {\mathbb{B}}({\varpi}_{i_{p}})_{{\mathop{\rm cl}\nolimits}}$. In particular, ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ is also a simple crystal by Remark \[rem:simple\](2).
\[rem:extremal\] Let $\lambda \in \sum_{i \in I_{0}}{\mathbb{Z}}_{\ge 0}{\varpi}_{i}$ be a level-zero dominant integral weight.
It can be easily seen from Remark \[rem:str01\] that $\eta_{\mu}(t):=t\mu$ is contained in ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ for all $\mu \in {\mathop{\rm cl}\nolimits}(W\lambda)=W_{0}{\mathop{\rm cl}\nolimits}(\lambda)$.
We know from [@NS-Tensor Lemma 3.19] that $\eta_{{\mathop{\rm cl}\nolimits}(\lambda)} \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ is an extremal element in the sense of [@Kas-OnL §3.1]. Therefore, it follows from [@AK Lemma 1.5] and the definition of simple crystals that for each $\eta \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, there exist $j_{1},\,j_{2},\,\dots,\,j_{p} \in I$ such that $$e_{j_{p}}^{\max} \cdots e_{j_{2}}^{\max}e_{j_{1}}^{\max}\eta=
\eta_{{\mathop{\rm cl}\nolimits}(\lambda)}.$$ Also, by the same argument as for [@AK Lemma 1.5], we can show that for each $\eta \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, there exist $k_{1},\,k_{2},\,\dots,\,k_{q} \in I$ such that $$f_{k_{q}}^{\max} \cdots f_{k_{2}}^{\max}f_{k_{1}}^{\max}\eta=
\eta_{{\mathop{\rm cl}\nolimits}(\lambda)}.$$
\[lem:Nlambda\] Let $\lambda \in \sum_{i \in I_{0}}{\mathbb{Z}}_{\ge 0}{\varpi}_{i}$ be a level-zero dominant integral weight, and let $n \in {\mathbb{Z}}_{\ge 1}$. Then, the set ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n}$ is identical to ${\mathbb{B}}(n\lambda)_{{\mathop{\rm cl}\nolimits}}$.
First, let us show the inclusion ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n} \supset {\mathbb{B}}(n\lambda)_{{\mathop{\rm cl}\nolimits}}$. It is easily seen that the element $\eta_{{\mathop{\rm cl}\nolimits}(\lambda)} \ast \cdots \ast
\eta_{{\mathop{\rm cl}\nolimits}(\lambda)} \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n}$ is identical to $\eta_{{\mathop{\rm cl}\nolimits}(n\lambda)}$. Hence it follows that the crystal ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n}$ contains the connected component containing $\eta_{{\mathop{\rm cl}\nolimits}(n\lambda)} \in {\mathbb{B}}(n\lambda)_{{\mathop{\rm cl}\nolimits}}$. Here we recall that the crystal ${\mathbb{B}}(n\lambda)_{{\mathop{\rm cl}\nolimits}}$ is simple (see Theorem \[thm:LScl\]), and hence connected (see Remark \[rem:simple\](1)). Therefore, the connected component above is identical to ${\mathbb{B}}(n\lambda)_{{\mathop{\rm cl}\nolimits}}$. Thus, we have shown the inclusion ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n} \supset {\mathbb{B}}(n\lambda)_{{\mathop{\rm cl}\nolimits}}$.
Now, it follows from Proposition \[prop:cat\] that ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n}$ is isomorphic as a crystal to the tensor product ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\otimes n}$. Therefore, ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n} \cong {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\otimes n}$ is a simple crystal by Theorem \[thm:LScl\](2) and Remark \[rem:simple\](2), and hence connected by Remark \[rem:simple\](1). From this, we conclude that ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast n}=
{\mathbb{B}}(n\lambda)_{{\mathop{\rm cl}\nolimits}}$, as desired.
Characterization of the set ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. {#subsec:charls}
---------------------------------------------------------------------------------
\[thm:charls\] Let $\lambda \in \sum_{i \in I_{0}}{\mathbb{Z}}_{\ge 0}{\varpi}_{i}$ be a level-zero dominant integral weight. If a subset ${\mathbb{B}}$ of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ satisfies the following two conditions, then the set ${\mathbb{B}}$ is identical to ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$.
1. The set ${\mathbb{B}}\cup \{{{\bf 0}}\}$ is stable under the action of the root operators $f_{j}$ for all $j \in I$.
2. For each $\eta \in {\mathbb{B}}$, there exist a sequence $\mu_{1},\,\mu_{2},\,\dots,\,\mu_{s}$ of elements in ${\mathop{\rm cl}\nolimits}(W\lambda)=W_{0}{\mathop{\rm cl}\nolimits}(\lambda)$ and a sequence $0=\sigma_{0} < \sigma_{1} < \cdots < \sigma_{s}=1$ of rational numbers such that $$\label{eq:charls}
\eta(t)=\sum_{l=1}^{k-1}
(\sigma_{l}-\sigma_{l-1})\mu_{l}+
(t-\sigma_{k-1})\mu_{k} \quad
\text{\rm for $\sigma_{k-1} \le t \le \sigma_{k}$, $1 \le k \le s$}.$$
The equality ${\mathbb{B}}={\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ also holds when we replace the root operators $f_{j}$ for $j \in I$ by $e_{j}$ for $j \in I$ in the theorem above; for its proof, simply replace $f_{j}$’s by $e_{j}$’s in the proof below.
First, let us show the inclusion ${\mathbb{B}}\subset {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Fix an element $\eta \in {\mathbb{B}}$ arbitrarily, and assume that $\eta$ is of the form . Take $N \in {\mathbb{Z}}_{\ge 1}$ such that $N\sigma_{u} \in {\mathbb{Z}}$ for all $0 \le u \le s$. Then, the element $N\eta \in {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ is of the form: $$N\eta=
\underbrace{\eta_{\mu_{1}} \ast \cdots \ast \eta_{\mu_{1}}}_{ \text{$N(\sigma_{1}-\sigma_{0})$-times}} \ast
\underbrace{\eta_{\mu_{2}} \ast \cdots \ast \eta_{\mu_{2}}}_{ \text{$N(\sigma_{2}-\sigma_{1})$-times}} \ast \cdots \ast
\underbrace{\eta_{\mu_{s}} \ast \cdots \ast \eta_{\mu_{s}}}_{ \text{$N(\sigma_{s}-\sigma_{s-1})$-times}}.$$ Since $\eta_{\mu} \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ for every $\mu \in {\mathop{\rm cl}\nolimits}(W\lambda)$ (see Remark \[rem:extremal\](1)), we have $N\eta \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}^{\ast N}$, and hence $N\eta \in {\mathbb{B}}(N\lambda)_{{\mathop{\rm cl}\nolimits}}$ by Lemma \[lem:Nlambda\]. By Remark \[rem:extremal\], there exists $k_{1},\,k_{2},\,\dots,\,k_{q} \in I$ such that $$f_{k_{q}}^{\max} \cdots
f_{k_{2}}^{\max}f_{k_{1}}^{\max}(N\eta)=
\eta_{{\mathop{\rm cl}\nolimits}(N\lambda)}.$$ Using Lemma \[lem:Ne\] and condition (a) repeatedly, we deduce that $$f_{k_{q}}^{\max} \cdots
f_{k_{2}}^{\max}f_{k_{1}}^{\max}(N\eta) =
N(f_{k_{q}}^{\max} \cdots
f_{k_{2}}^{\max}f_{k_{1}}^{\max}\eta).$$ Combining these equalities, we obtain $N(f_{k_{q}}^{\max} \cdots
f_{k_{2}}^{\max}f_{k_{1}}^{\max}\eta) = \eta_{{\mathop{\rm cl}\nolimits}(N\lambda)}$. Since $\eta_{{\mathop{\rm cl}\nolimits}(N\lambda)}=N\eta_{{\mathop{\rm cl}\nolimits}(\lambda)}$, we get $$\label{eq:charls01}
f_{k_{q}}^{\max} \cdots
f_{k_{2}}^{\max}f_{k_{1}}^{\max}
\eta=\eta_{{\mathop{\rm cl}\nolimits}(\lambda)} \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}.$$ Therefore, by Theorem \[thm:pc01\](1), $\eta=
e_{k_{1}}^{c_{1}}e_{k_{2}}^{c_{2}} \cdots
e_{k_{q}}^{c_{q}}\eta_{{\mathop{\rm cl}\nolimits}(\lambda)} \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ for some $c_{1},\,c_{2},\,\dots,\,c_{q} \in {\mathbb{Z}}_{\ge 0}$. Thus we have shown the inclusion ${\mathbb{B}}\subset {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Also, we should remark that $\eta_{{\mathop{\rm cl}\nolimits}(\lambda)} \in {\mathbb{B}}$ by and condition (a).
Next, let us show the opposite inclusion ${\mathbb{B}}\supset {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Fix an element $\eta' \in {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ arbitrarily. By Remark \[rem:extremal\], there exists $j_{1},\,j_{2},\,\dots,\,j_{p} \in I$ such that $$e_{j_{p}}^{\max} \cdots
e_{j_{2}}^{\max}e_{j_{1}}^{\max}\eta'=
\eta_{{\mathop{\rm cl}\nolimits}(\lambda)}.$$ Therefore, by Theorem \[thm:pc01\](1), $\eta'=
f_{j_{1}}^{d_{1}}f_{j_{2}}^{d_{2}} \cdots
f_{j_{p}}^{d_{p}}\eta_{{\mathop{\rm cl}\nolimits}(\lambda)}$ for some $d_{1},\,d_{2},\,\dots,\,d_{p} \in {\mathbb{Z}}_{\ge 0}$. Since $\eta_{{\mathop{\rm cl}\nolimits}(\lambda)} \in {\mathbb{B}}$ as shown above, it follows from condition (a) that $\eta' \in {\mathbb{B}}$. Thus we have shown the inclusion ${\mathbb{B}}\supset {\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, thereby completing the proof of the theorem.
Quantum Lakshmibai-Seshadri paths. {#sec:QLS}
==================================
Quantum Bruhat graph. {#subsec:def-QBG}
---------------------
In this subsection, we fix a subset $J$ of $I_{0}$. Set $$W_{J}:=\langle r_{j} \mid j \in J \rangle \subset W_{0}.$$ It is well-known that each coset in $W_{0}/W_{J}$ has a unique element of minimal length, called the minimal coset representative for the coset; we denote by $W_{0}^{J} \subset W_{0}$ the set of minimal coset representatives for the cosets in $W_{0}/W_{J}$, and by ${\lfloor \,\cdot\, \rfloor}={\lfloor \,\cdot\, \rfloor}_{J}:
W_{0} \twoheadrightarrow W_{0}^{J} \cong W_{0}/W_{J}$ the canonical projection. Also, we set $\Delta_{J}:=\Delta_{0} \cap
\bigl(\bigoplus_{j \in J} {\mathbb{Z}}\alpha_{j}\bigr)$, $\Delta_{J}^{\pm}:=
\Delta_{0}^{\pm} \cap \bigl(\bigoplus_{j \in J} {\mathbb{Z}}\alpha_{j}\bigr)$, and $\rho:=(1/2)\sum_{\alpha \in \Delta_{0}^{+}}\alpha$, $\rho_{J}:=(1/2)\sum_{\alpha \in \Delta_{J}^{+}} \alpha$.
\[dfn:QBG\] The (parabolic) quantum Bruhat graph is a $(\Delta_{0}^{+} \setminus \Delta_{J}^{+})$-labeled, directed graph with vertex set $W_{0}^{J}$ and $(\Delta_{0}^{+} \setminus \Delta_{J}^{+})$-labeled, directed edges of the following form: ${\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} w$ for $w \in W_{0}^{J}$ and $\beta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$ such that either
1. $\ell({\lfloor wr_{\beta} \rfloor})=\ell(w)+1$, or
2. $\ell({\lfloor wr_{\beta} \rfloor})=\ell(w)-2{\langle \rho-\rho_{J},\,\beta^{\vee} \rangle}+1$;
if (i) holds (resp., (ii) holds), then the edge is called a Bruhat edge (resp., a quantum edge).
\[rem:QBG1\] If $w \in W_{0}^{J}$ and $\beta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$ satisfy the condition that $\ell({\lfloor wr_{\beta} \rfloor})=\ell(w)+1$, then $wr_{\beta} \in W_{0}^{J}$. Indeed, since $\ell(wr_{\beta}) \ge \ell({\lfloor wr_{\beta} \rfloor})=\ell(w)+1$, it follows that $wr_{\beta}$ is greater than $w$ in the ordinary Bruhat order. Therefore, by [@BB Proposition 2.5.1], ${\lfloor wr_{\beta} \rfloor}$ is greater than or equal to ${\lfloor w \rfloor}=w$ in the ordinary Bruhat order. Since $\ell({\lfloor wr_{\beta} \rfloor})=\ell(w)+1$ by the assumption, there exists $\gamma \in \Delta_{0}^{+}$ such that ${\lfloor wr_{\beta} \rfloor}=wr_{\gamma}$. Now, we take a dominant integral weight $\Lambda \in P_{{\mathop{\rm cl}\nolimits}}$ with respect to the finite root system $\Delta_{0}$ such that $\bigl\{j \in I_{0} \mid {\langle \Lambda,\,\alpha_{j}^{\vee} \rangle}=0\bigr\}=J$; note that ${\langle \Lambda,\,\beta^{\vee} \rangle} > 0$ since $\beta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$. Then we have $wr_{\beta}\Lambda={\lfloor wr_{\beta} \rfloor}\Lambda=wr_{\gamma}\Lambda$, and hence $r_{\beta}\Lambda=r_{\gamma}\Lambda$. It follows that ${\langle \Lambda,\,\beta^{\vee} \rangle}\beta=
{\langle \Lambda,\,\gamma^{\vee} \rangle}\gamma$. Since $\beta$ and $\gamma$ are both contained in $\Delta_{0}^{+}$, and since ${\langle \Lambda,\,\beta^{\vee} \rangle} > 0$, we deduce that $\beta=\gamma$. Thus, we obtain ${\lfloor wr_{\beta} \rfloor}=wr_{\gamma}=wr_{\beta}$, which implies that $wr_{\beta} \in W_{0}^{J}$.
\[rem:QBG\] We know from [@LS Lemma 10.18] that the condition (ii) above is equivalent to the following condition :
1. $\ell({\lfloor wr_{\beta} \rfloor})=
\ell(w)-2{\langle \rho-\rho_{J},\,\beta^{\vee} \rangle}+1$ and $\ell(wr_{\beta})=\ell(w)-2{\langle \rho,\,\beta^{\vee} \rangle}+1$.
Let $x,\,y \in W_{0}^{J}$. A directed path ${{\bf d}}$ from $y$ to $x$ in the parabolic quantum Bruhat graph is, by definition, a pair of a sequence $w_{0},\,w_{1},\,\dots,\,w_{n}$ of elements in $W_{0}^{J}$ and a sequence $\beta_{1},\,\beta_{2},\,\dots,\,\beta_{n}$ of elements in $\Delta_{0}^{+} \setminus \Delta_{J}^{+}$ such that $$\label{eq:dp}
{{\bf d}}:
x=
w_{0} \stackrel{\beta_{1}}{\leftarrow}
w_{1} \stackrel{\beta_{2}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow}
w_{n}=y.$$ A directed path ${{\bf d}}$ from $y$ to $x$ said to be shortest if its length $n$ is minimal among all possible directed paths from $y$ to $x$. Denote by ${\ell(y,\,x)}$ the length of a shortest directed path from $y$ to $x$ in the parabolic quantum Bruhat graph.
Definition of quantum Lakshmibai-Seshadri paths. {#subsec:QLS}
------------------------------------------------
In this subsection, we fix a level-zero dominant integral weight $\lambda \in \sum_{i \in I_{0}} {\mathbb{Z}}_{\ge 0} {\varpi}_{i}$, and set $\Lambda:={\mathop{\rm cl}\nolimits}(\lambda)$ for simplicity of notation. Also, we set $$J:=\bigl\{j \in I_{0} \mid {\langle \Lambda,\,\alpha_{j}^{\vee} \rangle}=0 \bigr\}
\subset I_{0}.$$
\[dfn:QBG-achain\] Let $x,\,y \in W_{0}^{J}$, and let $\sigma \in {\mathbb{Q}}$ be such that $0 < \sigma < 1$. A directed $\sigma$-path from $y$ to $x$ is, by definition, a directed path $$x=w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1}
\stackrel{\beta_{2}}{\leftarrow} w_{2}
\stackrel{\beta_{3}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=y$$ from $y$ to $x$ in the parabolic quantum Bruhat graph satisfying the condition that $$\sigma {\langle \Lambda,\,\beta_{k}^{\vee} \rangle} \in {\mathbb{Z}}\quad \text{for all $1 \le k \le n$}.$$
\[dfn:qLS\] Denote by ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ (resp., ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$) the set of all pairs $\eta=({\underline{x}}\,;\,{\underline{\sigma}})$ of a sequence ${\underline{x}}\,:\,x_{1},\,x_{2},\,\dots,\,x_{s}$ of elements in $W_{0}^{J}$, with $x_{k} \ne x_{k+1}$ for $1 \le k \le s-1$, and a sequence ${\underline{\sigma}}\,:\,
0=\sigma_{0} < \sigma_{1} < \cdots < \sigma_{s}=1$ of rational numbers satisfying the condition that there exists a directed $\sigma_{k}$-path (resp., a directed $\sigma_{k}$-path of length ${\ell(x_{k+1},\,x_{k})}$) from $x_{k+1}$ to $x_{k}$ for each $1 \le k \le s-1$; observe that ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \subset
{\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. We call an element of ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ a quantum Lakshmibai-Seshadri path of shape $\lambda$.
Let $\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})$ be a rational path, that is, a pair of a sequence $x_{1},\,x_{2},\,\dots,\,x_{s}$ of elements in $W_{0}^{J}$, with $x_{k} \ne x_{k+1}$ for $1 \le k \le s-1$, and a sequence $0=\sigma_{0} < \sigma_{1} < \cdots < \sigma_{s}=1$ of rational numbers. We identify $\eta$ with the following piecewise-linear, continuous map $\eta:[0,1] \rightarrow {\mathbb{R}}\otimes_{{\mathbb{Z}}} P_{{\mathop{\rm cl}\nolimits}}$ (cf. ): $$\label{eq:QBG-path}
\eta(t)=\sum_{l=1}^{k-1}
(\sigma_{l}-\sigma_{l-1})x_{l}\Lambda+
(t-\sigma_{k-1})x_{k}\Lambda \quad
\text{for $\sigma_{k-1} \le t \le \sigma_{k}$, $1 \le k \le s$};$$ note that the map $W_{0}^{J} \rightarrow W_{0}\Lambda$, $w \mapsto w\Lambda$, is bijective. We will prove that under this identification, both ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ and ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ can be regarded as a subset of ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ (see Proposition \[prop:ip\]). Furthermore, we will prove that both of the sets ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \cup \{{{\bf 0}}\}$ and ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \cup \{{{\bf 0}}\}$ are stable under the action of root operators (see Proposition \[prop:stable\]).
Main result. {#sec:main}
============
Statement and some technical lemmas. {#subsec:prf-main1}
------------------------------------
Throughout this section, we fix a level-zero dominant integral weight $\lambda \in \sum_{i \in I_{0}} {\mathbb{Z}}_{\ge 0} {\varpi}_{i}$. Set $\Lambda:={\mathop{\rm cl}\nolimits}(\lambda)$, and $$J:=\bigl\{j \in I_{0} \mid {\langle \Lambda,\,\alpha_{j}^{\vee} \rangle}=0 \bigr\}
\subset I_{0}.$$ The following theorem is the main result of this paper; it is obtained as a by-product of an explicit description, given in §\[subsec:ro\], of the image of a quantum LS path as a rational path under the action of root operators on quantum LS paths.
\[thm:main\] With the notation and setting above, we have $${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}=
{\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}={\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}.$$
In view of Theorem \[thm:charls\], in order to prove Theorem \[thm:main\], it suffices to prove that both ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ and ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ are contained in ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ (see Proposition \[prop:ip\] below), and that both of the sets ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \cup \{{{\bf 0}}\}$ and ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \cup \{{{\bf 0}}\}$ are stable under the action of the root operators $f_{j}$ for all $j \in I$ (see Proposition \[prop:stable\] below). To prove these, we need some lemmas.
\[lem:theta\] Let $w \in W_{0}^{J}$. If $w^{-1}\theta \in \Delta_{0}^{-}$, then there exists a quantum edge ${\lfloor r_{\theta}w \rfloor} \stackrel{-w^{-1}\theta}{\longleftarrow}
w$ from $w$ to ${\lfloor r_{\theta}w \rfloor}$ in the parabolic quantum Bruhat graph.
\[lem:mcr\] Let $w \in W_{0}^{J}$ and $j \in I_{0}$. If $w^{-1}\alpha_{j} \in \Delta_{0} \setminus \Delta_{J}$, then $r_{j}w \in W_{0}^{J}$.
\[lem:dist1\] Let $w \in W_{0}^{J}$ and $\beta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$ be such that ${\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} w$. Let $j \in I_{0}$.
If ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ and $w\beta \ne \pm \alpha_{j}$, then ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$. Also, both $r_{j}{\lfloor wr_{\beta} \rfloor}$ and $r_{j}w$ are contained in $W_{0}^{J}$, and $r_{j}{\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} r_{j}w$.
If ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ and $w\beta \ne \pm \alpha_{j}$, then ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$. Also, both $r_{j}{\lfloor wr_{\beta} \rfloor}$ and $r_{j}w$ are contained in $W_{0}^{J}$, and $r_{j}{\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} r_{j}w$.
If ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ and ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$, then $w\beta = \pm \alpha_{j}$.
If ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} \le 0$ and ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$, then $w\beta = \pm \alpha_{j}$.
\(1) Since ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$, we see that $w^{-1}\alpha_{j} \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$. By [@LNSSS1 Proposition 5.10(3)], there exists a Bruhat edge $r_{j}w \stackrel{w^{-1}\alpha_{j}}{\longleftarrow} w$ in the parabolic quantum Bruhat graph, with $r_{j}w \in W_{0}^{J}$. If the edge ${\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} w$ is a Bruhat (resp., quantum) edge, then it follows from the left diagram of (5.3) (resp., (5.4)) in part (1) (resp., part (2)) of [@LNSSS1 Lemma 5.14] that $r_{j}{\lfloor wr_{\beta} \rfloor}={\lfloor r_{j}wr_{\beta} \rfloor} \in W_{0}^{J}$, and there exists a Bruhat (resp., quantum) edge $r_{j}{\lfloor wr_{\beta} \rfloor}
\stackrel{\beta}{\longleftarrow} r_{j}w$ and a Bruhat edge $r_{j}{\lfloor wr_{\beta} \rfloor}
\stackrel{{\lfloor wr_{\beta} \rfloor}^{-1}\alpha_{j}}{\longleftarrow}
{\lfloor wr_{\beta} \rfloor}$ in the parabolic quantum Bruhat graph. In particular, we have ${\lfloor wr_{\beta} \rfloor}^{-1}\alpha_{j} \in
\Delta_{0}^{+} \setminus \Delta_{J}^{+}$, which implies that ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$. This proves part (1).
\(2) Since ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$, we see that ${\lfloor wr_{\beta} \rfloor}^{-1}\alpha_{j} \in
\Delta_{0}^{-} \setminus \Delta_{J}^{-}$. By [@LNSSS1 Proposition 5.10(1)], there exists a Bruhat edge ${\lfloor wr_{\beta} \rfloor} \stackrel{-{\lfloor wr_{\beta} \rfloor}^{-1}\alpha_{j}}{\longleftarrow}
r_{j}{\lfloor wr_{\beta} \rfloor}$ in the parabolic quantum Bruhat graph, with $r_{j}{\lfloor wr_{\beta} \rfloor} \in W_{0}^{J}$. If the edge ${\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} w$ is a Bruhat (resp., quantum) edge, then it follows from the right diagram of (5.3) (resp., (5.4)) in part (1) (resp., part (2)) of [@LNSSS1 Lemma 5.14] that $r_{j}w \in W_{0}^{J}$, and there exists a Bruhat (resp., quantum) edge $r_{j}{\lfloor wr_{\beta} \rfloor}
\stackrel{\beta}{\longleftarrow} r_{j}w$ and a Bruhat edge $w \stackrel{-w^{-1}\alpha_{j}}{\longleftarrow} r_{j}w$ in the parabolic quantum Bruhat graph. In particular, we have $w^{-1}\alpha_{j} \in
\Delta_{0}^{-} \setminus \Delta_{J}^{-}$, which implies that ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$. This proves part (2).
\(3) (resp., (4)) Assume that ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ and ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$ (resp., ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} \le 0$ and ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$). Suppose that $w\beta \ne \pm \alpha_{j}$. Then it follows from part (2) (resp., (1)) that ${\langle w\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ (resp., ${\langle wr_{\beta}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$), which is a contradiction. Thus we get $w\beta = \pm \alpha_{j}$. This completes the proof of Lemma \[lem:dist1\].
\[lem:dist2\] Let $w \in W_{0}^{J}$ and $\beta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$ be such that ${\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} w$. Let $z \in W_{J}$ be such that $r_{\theta}w={\lfloor r_{\theta}w \rfloor}z$; note that $z\beta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$.
If ${\langle w\Lambda,\,\alpha_{0}^{\vee} \rangle} > 0$ and $w\beta \ne \pm \theta$, then ${\langle wr_{\beta}\Lambda,\,\alpha_{0}^{\vee} \rangle} > 0$ and ${\lfloor r_{\theta}wr_{\beta} \rfloor}
\stackrel{z\beta}{\leftarrow} {\lfloor r_{\theta}w \rfloor}$.
If ${\langle wr_{\beta}\Lambda,\,\alpha_{0}^{\vee} \rangle} < 0$ and $w\beta \ne \pm \theta$, then ${\langle w\Lambda,\,\alpha_{0}^{\vee} \rangle} < 0$ and ${\lfloor r_{\theta}wr_{\beta} \rfloor}
\stackrel{z\beta}{\leftarrow} {\lfloor r_{\theta}w \rfloor}$.
If ${\langle wr_{\beta}\Lambda,\,\alpha_{0}^{\vee} \rangle} < 0$ and ${\langle w\Lambda,\,\alpha_{0}^{\vee} \rangle} \ge 0$, then $w\beta = \pm \theta$.
If ${\langle wr_{\beta}\Lambda,\,\alpha_{0}^{\vee} \rangle} \le 0$ and ${\langle w\Lambda,\,\alpha_{0}^{\vee} \rangle} > 0$, then $w\beta = \pm \theta$.
\(1) Since ${\langle w\Lambda,\,\alpha_{0}^{\vee} \rangle} > 0$, we see that $w^{-1}\theta \in \Delta_{0}^{-} \setminus \Delta_{J}^{-}$. By [@LNSSS1 Proposition 5.11(1)], there exists a quantum edge ${\lfloor r_{\theta}w \rfloor} \stackrel{-w^{-1}\theta}{\longleftarrow} w$ in the parabolic quantum Bruhat graph. If the edge ${\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} w$ is a Bruhat (resp., quantum) edge, then it follows from the left diagram of (5.5) or (5.6) (resp., (5.7) or (5.8)) in part (3) (resp., part (4)) of [@LNSSS1 Lemma 5.14] that there exists an edge ${\lfloor r_{\theta}wr_{\beta} \rfloor}
\stackrel{z\beta}{\leftarrow} {\lfloor r_{\theta}w \rfloor}$ and a quantum edge ${\lfloor r_{\theta}wr_{\beta} \rfloor}
\stackrel{-{\lfloor wr_{\beta} \rfloor}^{-1}\theta}{\longleftarrow}
{\lfloor wr_{\beta} \rfloor}$ in the parabolic quantum Bruhat graph. In particular, we have ${\lfloor wr_{\beta} \rfloor}^{-1}\theta \in \Delta_{0}^{-} \setminus \Delta_{J}^{-}$, which implies that ${\langle wr_{\beta}\Lambda,\,\alpha_{0}^{\vee} \rangle} > 0$. This proves part (1).
\(2) Since ${\langle wr_{\beta}\Lambda,\,\alpha_{0}^{\vee} \rangle} < 0$, we see that ${\lfloor wr_{\beta} \rfloor}^{-1}\theta \in
\Delta_{0}^{+} \setminus \Delta_{J}^{+}$. By [@LNSSS1 Proposition 5.11(3)], there exists a quantum edge ${\lfloor wr_{\beta} \rfloor} \stackrel{z'{\lfloor wr_{\beta} \rfloor}^{-1}\theta}{\longleftarrow}
{\lfloor r_{\theta}wr_{\beta} \rfloor}$ in the parabolic quantum Bruhat graph, where $z' \in W_{J}$ is defined by: $r_{\theta}{\lfloor wr_{\beta} \rfloor}={\lfloor r_{\theta}wr_{\beta} \rfloor}z'$. If the edge ${\lfloor wr_{\beta} \rfloor} \stackrel{\beta}{\leftarrow} w$ is a Bruhat (resp., quantum) edge, then it follows from the right diagram of (5.5) or (5.6) (resp., (5.7) or (5.8)) in part (3) (resp., part (4)) of [@LNSSS1 Lemma 5.14] that there exists an edge ${\lfloor r_{\theta}wr_{\beta} \rfloor}
\stackrel{z\beta}{\leftarrow} {\lfloor r_{\theta}w \rfloor}$ and a quantum edge $w \stackrel{zw^{-1}\theta}{\longleftarrow} {\lfloor r_{\theta}w \rfloor}$ in the parabolic quantum Bruhat graph. In particular, we have $w^{-1}\theta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$, which implies that ${\langle w\Lambda,\,\alpha_{0}^{\vee} \rangle} < 0$. This proves part (2).
Parts (3) and (4) can be shown by using parts (1) and (2) in the same way as parts (3) and (4) of Lemma \[lem:dist1\]. This completes the proof of Lemma \[lem:dist2\].
\[lem:sigma0\] Let $\lambda$, $\Lambda$, and $J$ be as above. Let $x,\,y \in W_{0}^{J}$, and let $\sigma \in {\mathbb{Q}}$ be such that $0 < \sigma < 1$. Assume that there exists a directed $\sigma$-path from $y$ to $x$ as follows: $$x=w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1}
\stackrel{\beta_{2}}{\leftarrow} w_{2}
\stackrel{\beta_{3}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=y.$$ Then, $\sigma(x\Lambda-y\Lambda)$ is contained in $Q_{0}:=\bigoplus_{j \in I_{0}}{\mathbb{Z}}\alpha_{j}$.
We have $$\begin{aligned}
\sigma (x\Lambda-y\Lambda)
& = \sum_{k=1}^{n} \sigma (w_{k-1}\Lambda-w_{k}\Lambda)
= \sum_{k=1}^{n} \sigma (w_{k}r_{\beta_{k}}\Lambda-w_{k}\Lambda) \\[1.5mm]
& = -\sum_{k=1}^{n} \sigma {\langle \Lambda,\,\beta_{k}^{\vee} \rangle}\,w_{k}\beta_{k}.\end{aligned}$$ It follows from the definition of a directed $\sigma$-path that $\sigma {\langle \Lambda,\,\beta_{k}^{\vee} \rangle} \in {\mathbb{Z}}$ for all $1 \le k \le n$. Also, it is obvious that $w_{k}\beta_{k} \in Q_{0}$ for all $1 \le k \le n$. Therefore, we conclude that $\sigma (x\Lambda-y\Lambda) \in Q_{0}$. This proves the lemma.
\[lem:weight\] Let $\lambda$, $\Lambda$, and $J$ be as above. If $\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then $\eta(1)$ is contained in $\Lambda+Q_{0}$, and hence in $P_{{\mathop{\rm cl}\nolimits}}$.
Let $\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Then we have (see ) $$\eta(1)=x_{s}\Lambda+
\sum_{k=1}^{s-1}\sigma_{k}(x_{k}\Lambda-x_{k+1}\Lambda).$$ It is obvious that $x_{s}\Lambda \in \Lambda+Q_{0}$. Also, it follows from Lemma \[lem:sigma0\] that $\sigma_{k}(x_{k}\Lambda-x_{k+1}\Lambda) \in Q_{0}$ for each $1 \le k \le s-1$. Therefore, we conclude that $\eta(1) \in \Lambda+Q_{0}$. This proves the lemma.
In what follows, we set $s_{j}:=r_{j}$ for $j \in I_{0}$, and $s_{0}:=r_{\theta} \in W_{0}$, in order to state our results and write their proofs in a way independent of whether $j=0$ or not.
\[lem:sigma1\] Let $\lambda$, $\Lambda$, and $J$ be as above. Let $x,\,y \in W_{0}^{J}$, and assume that there exists a directed path $$\label{eq:sigma1-0}
x=w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1}
\stackrel{\beta_{2}}{\leftarrow} w_{2}
\stackrel{\beta_{3}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=y.$$ from $y$ to $x$. Let $j \in I$.
If there exists $1 \le p \le n$ such that ${\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ for all $0 \le k \le p-1$ and ${\langle w_{p}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$, then ${\lfloor s_{j}w_{p-1} \rfloor}=w_{p}$, and there exists a directed path from $y$ to ${\lfloor s_{j}x \rfloor}$ of the form: $$\label{eq:sigma1-1}
{\lfloor s_{j}x \rfloor}=
{\lfloor s_{j}w_{0} \rfloor} \stackrel{z_{1}\beta_{1}}{\leftarrow} \cdots
\stackrel{z_{p-1}\beta_{p-1}}{\leftarrow}
{\lfloor s_{j}w_{p-1} \rfloor}=w_{p} \stackrel{\beta_{p+1}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=y.$$ Here, if $j \in I_{0}$, then we define $z_{k} \in W_{J}$ to be the identity element for all $1 \le k \le p-1$; if $j=0$, then we define $z_{k} \in W_{J}$ by $r_{\theta}w_{k}={\lfloor r_{\theta}w_{k} \rfloor}z_{k}$ for each $1 \le k \le p-1$.
If the directed path from $y$ to $x$ is shortest, i.e., ${\ell(y,\,x)}=n$, then the directed path from $y$ to ${\lfloor s_{j}x \rfloor}$ is also shortest, i.e., ${\ell(y,\,{\lfloor s_{j}x \rfloor})}=n-1$.
If the directed path is a directed $\sigma$-path from $y$ to $x$ for some rational number $0 < \sigma < 1$, then the directed path is a directed $\sigma$-path from $y$ to ${\lfloor s_{j}x \rfloor}$.
\(1) We give a proof only for the case $j \in I_{0}$. The proof for the case $j=0$ is similar; replace $\alpha_{j}$ and $\alpha_{j}^{\vee}$ by $-\theta$ and $-\theta^{\vee}$, respectively, and use Lemma \[lem:dist2\] instead of Lemma \[lem:dist1\]. First, let us check that $w_{k}\beta_{k} \ne \pm \alpha_{j}$ for any $1 \le k \le p-1$. Suppose, contrary to our claim, that $w_{k}\beta_{k} = \pm \alpha_{j}$ for some $1 \le k \le p-1$. Then, $$w_{k-1}\Lambda=w_{k}r_{\beta_{k}}\Lambda=
r_{w_{k}\beta_{k}}w_{k}\Lambda=s_{j}w_{k}\Lambda,$$ and hence ${\langle w_{k-1}\Lambda,\,\alpha_{j}^{\vee} \rangle}=
{\langle s_{j}w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle}=
-{\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$, which contradicts our assumption. Thus, $w_{k}\beta_{k} \ne \pm \alpha_{j}$ for any $1 \le k \le p-1$. It follows from Lemma \[lem:dist1\](2) and our assumption that ${\lfloor s_{j}w_{k-1} \rfloor} \stackrel{\beta_{k}}{\leftarrow} {\lfloor s_{j}w_{k} \rfloor}$ for all $1 \le k \le p-1$. Also, since ${\langle w_{p-1}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ and ${\langle w_{p}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$, it follows from Lemma \[lem:dist1\](3) that $w_{p}\beta_{p}=\pm \alpha_{j}$, and hence $$s_{j}w_{p-1}\Lambda=s_{j}w_{p}r_{\beta_{p}}\Lambda=
s_{j}r_{w_{p}\beta_{p}}w_{p}\Lambda=
s_{j}s_{j}w_{p}\Lambda=w_{p}\Lambda.$$ Thus, we obtain a directed path of the form from $y$ to ${\lfloor s_{j}x \rfloor}$. This proves part (1).
\(2) Assume that ${\ell(y,\,x)}=n$. By the argument above, we have ${\ell(y,\,{\lfloor s_{j}x \rfloor})} \le n-1$. Suppose, for a contradiction, that ${\ell(y,\,{\lfloor s_{j}x \rfloor})} < n-1$, and take a directed path $${\lfloor s_{j}x \rfloor}=z_{0} \stackrel{\gamma_{1}}{\leftarrow} z_{1}
\stackrel{\gamma_{2}}{\leftarrow} z_{2}
\stackrel{\gamma_{3}}{\leftarrow} \cdots
\stackrel{\gamma_{l}}{\leftarrow} z_{l}=y$$ from $y$ to ${\lfloor s_{j}x \rfloor}$ whose length $l$ is less than $n-1$. Let us show that $x \stackrel{\gamma}{\leftarrow} {\lfloor s_{j}x \rfloor}$ for some $\gamma \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$. Assume first that $j \in I_{0}$. Since ${\langle x\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ by the assumption, we have $x^{-1}\alpha_{j} \in \Delta_{0}^{-} \setminus \Delta_{J}^{-}$, and hence $\ell(x)=\ell(s_{j}x)+1$. Also, since $x \in W_{0}^{J}$, it follows from Lemma \[lem:mcr\] that $s_{j}x \in W_{0}^{J}$. Therefore, if we set $\gamma:=
x^{-1}s_{j}\alpha_{j}=-x^{-1}\alpha_{j}
\in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$, then we obtain $x \stackrel{\gamma}{\leftarrow} s_{j}x={\lfloor s_{j}x \rfloor}$. Assume next that $j=0$. Since ${\langle x\Lambda,\,-\theta^{\vee} \rangle}=
{\langle x\Lambda,\,\alpha_{0}^{\vee} \rangle} < 0$ by the assumption, we have $x^{-1}\theta \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$. Define an element $v \in W_{J}$ by $r_{\theta}x={\lfloor r_{\theta}x \rfloor}v$. Then we see that $\gamma:=vx^{-1}\theta$ is contained in $\Delta_{0}^{+} \setminus \Delta_{J}^{+}$, and that $${\lfloor {\lfloor s_0x \rfloor}r_{\gamma} \rfloor}=
{\lfloor {\lfloor r_{\theta}x \rfloor}r_{\gamma} \rfloor}=
{\lfloor r_{\theta}xv^{-1}r_{vx^{-1}\theta} \rfloor}=
{\lfloor r_{\theta}xv^{-1}vx^{-1}r_{\theta}xv^{-1} \rfloor}=
{\lfloor xv^{-1} \rfloor}=x$$ since $x \in W_{0}^{J}$ and $v \in W_{J}$. Also, note that ${\lfloor s_0x \rfloor}^{-1}\theta=
{\lfloor r_{\theta}x \rfloor}^{-1}\theta=
vx^{-1}r_{\theta}\theta=-\gamma \in
\Delta_{0}^{-} \setminus \Delta_{J}^{-}$. Therefore, we deduce from Lemma \[lem:theta\] that $$x={\lfloor {\lfloor s_0 x \rfloor}r_{\gamma} \rfloor}
\stackrel{\gamma}{\leftarrow}
{\lfloor r_{\theta}x \rfloor}={\lfloor s_{0}x \rfloor}.$$ Thus, we obtain a directed path $$x \stackrel{\gamma}{\leftarrow}
{\lfloor s_{j}x \rfloor}=z_{0} \stackrel{\gamma_{1}}{\leftarrow} z_{1}
\stackrel{\gamma_{2}}{\leftarrow} z_{2}
\stackrel{\gamma_{3}}{\leftarrow} \cdots
\stackrel{\gamma_{l}}{\leftarrow} z_{l}=y$$ from $y$ to $x$ whose length is $l+1 < n={\ell(y,\,x)}$. This contradicts the definition of ${\ell(y,\,x)}$. This proves part (2).
\(3) We should remark that ${\langle \Lambda,\,z_{k}\beta_{k}^{\vee} \rangle}={\langle \Lambda,\,\beta_{k}^{\vee} \rangle}$ for each $1 \le k \le p-1$, since $z_{k} \in W_{J}$. Hence the assertion of part (3) follows immediately from the definition of a directed $\sigma$-path. This completes the proof of Lemma \[lem:sigma1\].
The following lemma can be shown in the same way as Lemma \[lem:sigma1\]. If $j \in I_{0}$, then use Lemma \[lem:dist1\](1) and (4) instead of Lemma \[lem:dist1\](2) and (3), respectively; if $j=0$, then use Lemma \[lem:dist2\](1) and (4) instead of Lemma \[lem:dist2\](2) and (3), respectively.
\[lem:sigma2\] Keep the notation and setting in Lemma \[lem:sigma1\].
If there exists $1 \le p \le n$ such that ${\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ for all $p \le k \le n$ and ${\langle w_{p-1}\Lambda,\,\alpha_{j}^{\vee} \rangle} \le 0$, then $w_{p-1}={\lfloor s_{j}w_{p} \rfloor}$, and there exists a directed path from ${\lfloor s_{j}y \rfloor}$ to $x$ of the form: $$\label{eq:sigma2-1}
x=w_{0} \stackrel{\beta_{1}}{\leftarrow} \cdots
\stackrel{\beta_{p-1}}{\leftarrow}
w_{p-1}={\lfloor s_{j}w_{p} \rfloor}
\stackrel{z_{p+1}\beta_{p+1}}{\leftarrow} \cdots
\stackrel{z_{n}\beta_{n}}{\leftarrow}
{\lfloor s_{j}w_{n} \rfloor}={\lfloor s_{j}y \rfloor}.$$ Here, if $j \in I_{0}$, then we define $z_{k} \in W_{J}$ to be the identity element for all $p+1 \le k \le n$; if $j=0$, then we define $z_{k} \in W_{J}$ by $r_{\theta}w_{k}={\lfloor r_{\theta}w_{k} \rfloor}z_{k}$ for each $p+1 \le k \le n$.
If the directed path from $y$ to $x$ is shortest, i.e., ${\ell(y,\,x)}=n$, then the directed path from ${\lfloor s_{j}y \rfloor}$ to $x$ is also shortest, i.e., ${\ell({\lfloor s_{j}y \rfloor},\,x)}=n-1$.
If the directed path is a directed $\sigma$-path from $y$ to $x$ for some rational number $0 < \sigma < 1$, then the directed path is a directed $\sigma$-path from ${\lfloor s_{j}y \rfloor}$ to $x$.
\[lem:stable1\] Let $\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Let $j \in I$ and $1 \le u \le s-1$ be such that ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$. Let $$x_{u}=w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1}
\stackrel{\beta_{2}}{\leftarrow} w_{2}
\stackrel{\beta_{3}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=x_{u+1}$$ be a directed $\sigma_{u}$-path from $x_{u+1}$ to $x_{u}$. If there exists $0 \le k < n$ such that ${\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} \le 0$, then $H^{\eta}_{j}(\sigma_{u}) \in {\mathbb{Z}}$. In particular, if ${\langle x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle} \le 0$, then $H^{\eta}_{j}(\sigma_{u}) \in {\mathbb{Z}}$.
We see from the definition that $\eta':=(x_{1},\,x_{2},\,\dots,\,x_{u},\,x_{u+1}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{u},\,\sigma_{s})$ is an element of ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Also, observe that $\eta'(t)=\eta(t)$ for $0 \le t \le \sigma_{u+1}$, and hence $H^{\eta'}_{j}(t)=H^{\eta}_{j}(t)$ for $0 \le t \le \sigma_{u+1}$. It follows that $$H^{\eta}_{j}(\sigma_{u})=H^{\eta'}_{j}(\sigma_{u})=
H^{\eta'}_{j}(1)-(1-\sigma_{u}){\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle}.$$ Since $\eta'(1) \in P_{{\mathop{\rm cl}\nolimits}}$ (and hence $H^{\eta'}_{j}(1) \in {\mathbb{Z}}$) by Lemma \[lem:weight\], it suffices to show that $(1-\sigma_{u}){\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} \in {\mathbb{Z}}$.
We deduce from Lemma \[lem:sigma2\] that there exists a directed $\sigma_{u}$-path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to $x_{u}$. Therefore, $\eta''=
(x_{1},\,x_{2},\,\dots,\,x_{u},\,{\lfloor s_{j}x_{u+1} \rfloor}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{u},\,\sigma_{s})$ is also an element of ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Since both $\eta'(1)$ and $\eta''(1)$ are contained in $\Lambda+Q_{0}$ by Lemma \[lem:weight\], we have $\eta'(1)-\eta''(1) \in Q_{0}$. Also, we have $$\begin{aligned}
(Q_{0} \ni) \
\eta'(1)-\eta''(1) & =
(1-\sigma_{u})x_{u+1}\Lambda-(1-\sigma_{u})s_{j}x_{u+1}\Lambda \\[3mm]
& =
\begin{cases}
(1-\sigma_{u}){\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle}\alpha_{j}
& \text{if $j \in I_{0}$}, \\[1.5mm]
(1-\sigma_{u}){\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle}(-\theta)
& \text{if $j=0$}.
\end{cases}
$$ Here we remark that $\theta=\delta-\alpha_{0}=
\sum_{j \in I_{0}}a_{j}\alpha_{j}$, and the greatest common divisor of the $a_{j}$, $j \in I_{0}$, is equal to $1$. From these, we conclude that $(1-\sigma_{u}){\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} \in {\mathbb{Z}}$, thereby completing the proof of the proposition.
The following lemma can be shown in the same way as Lemma \[lem:stable1\]; noting that $\pi':=(x_{u},\,x_{u+1}\,\dots,\,x_{s};\,
\sigma_{0},\,\sigma_{u},\,\sigma_{u+1},\,\dots,\,\sigma_{s})$ is an element of ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, use $\pi'$ instead of $\eta'$ and the fact that $H^{\pi'}_{j}(1)-H^{\pi'}_{j}(1-t)=
H^{\eta}_{j}(1)-H^{\eta}_{j}(1-t)$ for $0 \le t \le 1-\sigma_{u-1}$.
\[lem:stable1a\] Let $\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Let $j \in I$ and $1 \le u \le s-1$ be such that ${\langle x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$. Let $$x_{u}=w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1}
\stackrel{\beta_{2}}{\leftarrow} w_{2}
\stackrel{\beta_{3}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=x_{u+1}$$ be a directed $\sigma_{u}$-path from $x_{u+1}$ to $x_{u}$. If there exists $0 < k \le n$ such that ${\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$, then $H^{\eta}_{j}(\sigma_{u}) \in {\mathbb{Z}}$. In particular, if ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$, then $H^{\eta}_{j}(\sigma_{u}) \in {\mathbb{Z}}$.
\[prop:ip\] Let $\lambda \in \sum_{i \in I_{0}} {\mathbb{Z}}_{\ge 0} {\varpi}_{i}$ be as above. Both ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ and ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ are contained in ${\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$ under the identification of a rational path with a piecewise-linear, continuous map.
Since ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \subset {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ by the definitions, it suffices to show that ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}} \subset {\mathbb{P}}_{{\mathop{\rm cl}\nolimits},\,{\mathop{\rm int}\nolimits}}$. Let $\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. We have shown that $\eta(1) \in P_{{\mathop{\rm cl}\nolimits}}$ for every $\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ (see Lemma \[lem:weight\]). It remains to show that for every $j \in I$, all local minima of the function $H_{j}^{\eta}(t)$ are integers. Fix $j \in I$, and assume that the function $H_{j}^{\eta}(t)$ attains a local minimum at $t' \in [0,1]$; we may assume that $t'=\sigma_{u}$ for some $0 \le u \le s$. If $u=0$ (resp., $u=s$), then $H_{j}^{\eta}(t')=H_{j}^{\eta}(0)=0 \in {\mathbb{Z}}$ (resp., $H_{j}^{\eta}(t')=H_{j}^{\eta}(1) \in {\mathbb{Z}}$) since $\eta(0)=0$ (resp., $\eta(1) \in P_{{\mathop{\rm cl}\nolimits}}$). If $0 < u < s$, then we have either ${\langle x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle} \le 0$ and ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$, or ${\langle x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ and ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$. Therefore, it follows from Lemma \[lem:stable1\] or \[lem:stable1a\] that $H^{\eta}_{j}(\sigma_{u}) \in {\mathbb{Z}}$. Thus, we have proved the proposition.
\[lem:stable2\] Let $\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s})
\in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$. Let $j \in I$ and $1 \le u \le s-1$ be such that ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ and $H^{\eta}_{j}(\sigma_{u}) \not\in {\mathbb{Z}}$. Let $$\label{eq:stable2-0}
x_{u}=w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1}
\stackrel{\beta_{2}}{\leftarrow} w_{2}
\stackrel{\beta_{3}}{\leftarrow} \cdots
\stackrel{\beta_{n}}{\leftarrow} w_{n}=x_{u+1}$$ be a directed $\sigma_{u}$-path from $x_{u+1}$ to $x_{u}$. Then, ${\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ for all $0 \le k \le n$, and there exists a directed $\sigma_{u}$-path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to ${\lfloor s_{j}x_{u} \rfloor}$ of the form: $$\label{eq:stable2-1}
{\lfloor s_{j}x_{u} \rfloor}={\lfloor s_{j}w_{0} \rfloor}
\stackrel{z_{1}\beta_{1}}{\leftarrow} {\lfloor s_{j}w_{1} \rfloor}
\stackrel{z_{2}\beta_{2}}{\leftarrow} \cdots
\stackrel{z_{n}\beta_{n}}{\leftarrow} {\lfloor s_{j}w_{n} \rfloor}=
{\lfloor s_{j}x_{u+1} \rfloor}.$$ Here, if $j \in I_{0}$, then we define $z_{k} \in W_{J}$ to be the identity element for all $1 \le k \le n$; if $j=0$, then we define $z_{k} \in W_{J}$ by $r_{\theta}w_{k}={\lfloor r_{\theta}w_{k} \rfloor}z_{k}$ for each $1 \le k \le n$. Moreover, if is a shortest directed path from $x_{u+1}$ to $x_{u}$, i.e., ${\ell(x_{u+1},\,x_{u})}=n$, then is a shortest directed path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to ${\lfloor s_{j}x_{u} \rfloor}$, i.e., ${\ell({\lfloor s_{j}x_{u+1} \rfloor},\,{\lfloor s_{j}x_{u} \rfloor})}=n$.
It follows from Lemma \[lem:stable1\] that if $H^{\eta}_{j}(\sigma_{u}) \not\in {\mathbb{Z}}$, then ${\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ for all $0 \le k \le n$ (in particular, ${\langle x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$). Assume that $j \in I_{0}$ (resp., $j=0$), and suppose, for a contradiction, that $w_{k}\beta_{k} = \pm \alpha_{j}$ (resp., $= \pm \theta$) for some $1 \le k \le n$. Then, $w_{k-1}\Lambda=w_{k}r_{\beta_{k}}\Lambda=
r_{w_{k}\beta_{k}}w_{k}\Lambda=s_{j}w_{k}\Lambda$, and hence ${\langle w_{k-1}\Lambda,\,\alpha_{j}^{\vee} \rangle}=
{\langle s_{j}w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle}=-{\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle}$, which contradicts the fact that ${\langle w_{k-1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ and ${\langle w_{k}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$. Thus, we conclude that $w_{k}\beta_{k} \ne \pm \alpha_{j}$ (resp., $\ne \pm \theta$) for any $1 \le k \le n$. Therefore, we deduce from Lemma \[lem:dist1\](1) (resp., Lemma \[lem:dist2\](1)) that there exists a directed path of the form from ${\lfloor s_{j}x_{u+1} \rfloor}$ to ${\lfloor s_{j}x_{u} \rfloor}$. Because the directed path is a directed $\sigma_{u}$-path, we have $\sigma_{u}{\langle \Lambda,\,\beta_{k}^{\vee} \rangle} \in {\mathbb{Z}}$. Also, it follows immediately that $\sigma_{u}{\langle \Lambda,\,z\beta_{k}^{\vee} \rangle}=
\sigma_{u}{\langle \Lambda,\,\beta_{k}^{\vee} \rangle} \in {\mathbb{Z}}$ since $z \in W_{J}$. Thus, the directed path is a directed $\sigma_{u}$-path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to ${\lfloor s_{j}x_{u} \rfloor}$.
Now, we assume that ${\ell(x_{u+1},\,x_{u})}=n$, and suppose, for a contradiction, that there exists a directed path $$\label{eq:dp11}
{\lfloor s_{j}x_{u} \rfloor}=z_{0}
\stackrel{\gamma_{1}}{\leftarrow} z_{1}
\stackrel{\gamma_{2}}{\leftarrow} z_{2}
\stackrel{\gamma_{3}}{\leftarrow} \cdots
\stackrel{\gamma_{l}}{\leftarrow} z_{l}={\lfloor s_{j}x_{u+1} \rfloor}$$ from ${\lfloor s_{j}x_{u+1} \rfloor}$ to ${\lfloor s_{j}x_{u} \rfloor}$ whose length $l$ is less than $n$. Let us show that ${\lfloor s_{j}x_{u+1} \rfloor} \stackrel{\gamma}{\leftarrow} x_{u+1}$ for some $\gamma \in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$. Assume first that $j \in I_{0}$. Since ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$, we have $\gamma:=x_{u+1}^{-1}\alpha_{j}
\in \Delta_{0}^{+} \setminus \Delta_{J}^{+}$, and hence $\ell(s_{j}x_{u+1})=\ell(x_{u+1})+1$. Also, by Lemma \[lem:mcr\], $s_{j}x_{u+1} \in W_{0}^{J}$. Since $s_{j}x_{u+1}=x_{u+1}r_{\gamma}$, we obtain ${\lfloor s_{j}x_{u+1} \rfloor}=s_{j}x_{u+1} \stackrel{\gamma}{\leftarrow} x_{u+1}$. Assume next that $j=0$. Since ${\langle x_{u+1}\Lambda,\,\theta^{\vee} \rangle}=-{\langle x_{u+1}\Lambda,\,\alpha_{0}^{\vee} \rangle} < 0$ by the assumption, it follows that $x_{u+1}^{-1}\theta \in \Delta_{0}^{-} \setminus \Delta_{J}^{-}$. Therefore, if we set $\gamma:=-x_{u+1}^{-1}\theta \in
\Delta_{0}^{+} \setminus \Delta_{J}^{+}$, then $s_{0}x_{u+1}=r_{\theta}x_{u+1}=x_{u+1}r_{\gamma}$, and we obtain ${\lfloor s_{0}x_{u+1} \rfloor} \stackrel{\gamma}{\leftarrow} x_{u+1}$ by Lemma \[lem:theta\]. By concatenating the directed path and ${\lfloor s_{j}x_{u+1} \rfloor} \stackrel{\gamma}{\leftarrow} x_{u+1}$, we obtain a directed path from $x_{u+1}$ to ${\lfloor s_{j}x_{u} \rfloor}$ whose length is $l+1$. Since ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ and ${\langle s_{j}x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle}=
-{\langle x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$, we deduce from Lemma \[lem:sigma1\](1) that there exists a directed path from $x_{u+1}$ to ${\lfloor s_{j}{\lfloor s_{j}x_{u} \rfloor} \rfloor}=x_{u}$ whose length is $(l+1)-1=l$. However, this contradicts the fact that $n={\ell(x_{u+1},\,x_{u})}$ since $l < n$. This proves the lemma.
Explicit description of the image of a quantum LS path under the action of root operators. {#subsec:ro}
------------------------------------------------------------------------------------------
In the course of the proof of the following proposition, we obtain an explicit description of the image of a quantum LS path as a rational path under the action of root operators; this description is similar to the one given in [@Lit-I].
\[prop:stable\] Both of the sets ${\widetilde{{\mathbb{B}}}}(\lambda) \cup \{{{\bf 0}}\}$ and ${\widehat{{\mathbb{B}}}}(\lambda) \cup \{{{\bf 0}}\}$ are stable under the action of the root operators $f_{j}$ for all $j \in I$.
Fix $j \in I$. Let $\eta=(x_{1},\,x_{2},\,\dots,\,x_{s}\,;\,
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{s}) \in
{\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, and assume that $f_{j}\eta \ne {{\bf 0}}$. It follows that the point $t_{0}=\max\bigl\{t \in [0,1] \mid
H^{\eta}_{j}(t)=m^{\eta}_{j}\bigr\}$ is equal to $\sigma_{u}$ for some $0 \le u < s$. Let $u \le m < s$ be such that $\sigma_{m} < t_{1} \le \sigma_{m+1}$; recall that $t_{1}=\min\bigl\{t \in [t_{0},1] \mid
H^{\eta}_{j}(t)=m^{\eta}_{j}+1\bigr\}$. Note that the function $H^{\eta}_{j}(t)$ is strictly increasing on $[t_{0},\,t_{1}]$, which implies that ${\langle x_{p}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ for all $u+1 \le p \le m+1$.
#### Case 1.
Assume that $x_{u} \ne {\lfloor s_{j}x_{u+1} \rfloor}$ or $u=0$, and that $\sigma_{m} < t_{1} < \sigma_{m+1}$. Then we deduce from the definition of the root operator $f_{j}$ (for the case $j=0$, see also Remark \[rem:ro\_f0\]; cf. [@Lit-A Proposition 4.7a)]) that $$\begin{aligned}
& f_{j}\eta=
(x_{1},\,x_{2},\,\dots,\,x_{u},\,{\lfloor s_{j}x_{u+1} \rfloor},\,\dots,\,
{\lfloor s_{j}x_{m} \rfloor},\,{\lfloor s_{j}x_{m+1} \rfloor},\,x_{m+1},\,x_{m+2},\,\dots,\,x_{s}\,;\, \\
& \hspace*{70mm}
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{u},\,\dots,\,
\sigma_{m},\,t_{1},\,\sigma_{m+1},\,\dots,\,\sigma_{s});\end{aligned}$$ note that ${\lfloor s_{j}x_{p} \rfloor} \ne {\lfloor s_{j}x_{p+1} \rfloor}$ for all $u+1 \le p \le m$, and that ${\lfloor s_{j}x_{m+1} \rfloor} \ne x_{m+1}$ since ${\langle x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$ as mentioned above. In order to prove that $f_{j}\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, we need to verify that
1. there exists a directed $\sigma_{u}$-path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to $x_{u}$ (when $u > 0$);
2. there exists a directed $\sigma_{p}$-path from ${\lfloor s_{j}x_{p+1} \rfloor}$ to ${\lfloor s_{j}x_{p} \rfloor}$ for each $u+1 \le p \le m$;
3. there exists a directed $t_{1}$-path from $x_{m+1}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$.
Also, we will show that if $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then the directed paths in (i)–(iii) above can be chosen from the shortest ones, which implies that $f_{j}\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$.
\(i) We deduce from the definition of $t_{0}=\sigma_{u}$ that ${\langle x_{u}\Lambda,\,\alpha_{j}^{\vee} \rangle} \le 0$ and ${\langle x_{u+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$. Since $\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, there exists a directed $\sigma_{u}$-path from $x_{u+1}$ to $x_{u}$. Hence it follows from Lemma \[lem:sigma2\](1),(3) that there exists a directed $\sigma_{u}$-path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to $x_{u}$. Furthermore, we see from the definition of ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ and Lemma \[lem:sigma2\](2) that if $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then there exists a directed $\sigma_{u}$-path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to $x_{u}$ whose length is equal to ${\ell({\lfloor s_{j}x_{u+1} \rfloor},\,x_{u})}$.
\(ii) Recall that $H^{\eta}_{j}(t)$ is strictly increasing on $[t_{0},t_{1}]$, and that $H^{\eta}_{j}(t_{0})=m^{\eta}_{j}$ and $H^{\eta}_{j}(t_{1})=m^{\eta}_{j}+1$. Hence it follows that $H^{\eta}_{j}(\sigma_{p}) \notin {\mathbb{Z}}$ for all $u+1 \le p \le m$. Therefore, we deduce from Lemma \[lem:stable2\] that there exists a directed $\sigma_{p}$-path from ${\lfloor s_{j}x_{p+1} \rfloor}$ to ${\lfloor s_{j}x_{p} \rfloor}$ for each $u+1 \le p \le m$. Furthermore, we see from the definition of ${\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ and Lemma \[lem:stable2\] that if $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then for each $u+1 \le p \le m$, there exists a directed $\sigma_{p}$-path from ${\lfloor s_{j}x_{p+1} \rfloor}$ to ${\lfloor s_{j}x_{p} \rfloor}$ whose length is equal to ${\ell({\lfloor s_{j}x_{p+1} \rfloor},\,{\lfloor s_{j}x_{p} \rfloor})}$.
\(iii) Since ${\langle x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$, by the same argument as in the second paragraph of the proof of Lemma \[lem:stable2\], we obtain ${\lfloor s_{j}x_{m+1} \rfloor} \stackrel{\gamma}{\leftarrow} x_{m+1}$, with $$\gamma:=
\begin{cases}
x_{m+1}^{-1}\alpha_{j} & \text{if $j \in I_{0}$}, \\[1.5mm]
x_{m+1}^{-1}(-\theta) & \text{if $j=0$};
\end{cases}$$ note that the directed path ${\lfloor s_{j}x_{m+1} \rfloor} \stackrel{\gamma}{\leftarrow} x_{m+1}$ is obviously shortest since its length is equal to $1$. Let us show that $t_{1}{\langle \Lambda,\,\gamma^{\vee} \rangle} \in {\mathbb{Z}}$. It is easily checked that ${\langle \Lambda,\,\gamma^{\vee} \rangle}=
{\langle x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle}$. Also, we have $\eta(t_{1})=t_{1}x_{m+1}\Lambda+
\sum_{k=1}^{m}\sigma_{k}(x_{k}\Lambda-x_{k+1}\Lambda)$, and hence $${\mathbb{Z}}\ni m^{\eta}_{j}+1=
H^{\eta}_{j}(t_{1})=t_{1}{\langle x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle}+
\sum_{k=1}^{m}{\langle \sigma_{k}(x_{k}\Lambda-x_{k+1}\Lambda),\,\alpha_{j}^{\vee} \rangle}.$$ Since $\sigma_{k}(x_{k}\Lambda-x_{k+1}\Lambda) \in Q_{0}$ for each $1 \le k \le m$ by Lemma \[lem:sigma0\], it follows from the equation above that $t_{1}{\langle x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} \in {\mathbb{Z}}$, and hence $t_{1}{\langle \Lambda,\,\gamma^{\vee} \rangle} \in {\mathbb{Z}}$. Thus, we have verified that there exists a directed $t_{1}$-path from $x_{m+1}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$ whose length is equal to ${\ell(x_{m+1},\,{\lfloor s_{j}x_{m+1} \rfloor})}=1$.
Combining these, we conclude that $f_{j}\eta$ is an element of ${\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, and that if $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then $f_{j}\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$.
#### Case 2.
Assume that $x_{u} \ne {\lfloor s_{j}x_{u+1} \rfloor}$ or $u=0$, and that $t_{1}=\sigma_{m+1}$. Then we deduce from the definition of the root operator $f_{j}$ (for the case $j=0$, see also Remark \[rem:ro\_f0\]; cf. [@Lit-A Proposition 4.7a) and Remark 4.8]) that $$\begin{aligned}
& f_{j}\eta=
(x_{1},\,x_{2},\,\dots,\,x_{u},\,{\lfloor s_{j}x_{u+1} \rfloor},\,\dots,\,
{\lfloor s_{j}x_{m} \rfloor},\,{\lfloor s_{j}x_{m+1} \rfloor},\,x_{m+2},\,\dots,\,x_{s}\,;\, \\
& \hspace*{70mm}
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{u},\,\dots,\,
\sigma_{m},\,t_{1}=\sigma_{m+1},\,\dots,\,\sigma_{s}).\end{aligned}$$ First, we observe that ${\langle x_{m+2}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$. Indeed, suppose, contrary to our claim, that ${\langle x_{m+2}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$. Since $H^{\eta}_{j}(\sigma_{m+1})=H^{\eta}_{j}(t_{1})=m^{\eta}_{j}+1$, it follows immediately that $H^{\eta}_{j}(\sigma_{m+1}+\epsilon) <
m^{\eta}_{j}+1$ for sufficiently small $\epsilon > 0$, and hence the minimum $M$ of the function $H^{\eta}_{j}(t)$ on $[t_{1},\,1]$ is (strictly) less than $m^{\eta}_{j}+1$. Here we recall from Proposition \[prop:ip\] that all local minima of the function $H^{\eta}_{j}(t)$ are integers. Hence we deduce that $M=m^{\eta}_{j}$, which contradicts the definition of $t_{0}$. Thus, we obtain ${\langle x_{m+2}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$. Since ${\langle x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$, and hence ${\langle s_{j}x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$, it follows that ${\lfloor s_{j}x_{m+1} \rfloor} \ne x_{m+2}$.
Now, in order to prove that $f_{j}\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, we need to verify that
1. there exists a directed $\sigma_{u}$-path from ${\lfloor s_{j}x_{u+1} \rfloor}$ to $x_{u}$ (when $u > 0$);
2. there exists a directed $\sigma_{p}$-path from ${\lfloor s_{j}x_{p+1} \rfloor}$ to ${\lfloor s_{j}x_{p} \rfloor}$ for each $u+1 \le p \le m$;
3. there exists a directed $\sigma_{m+1}$-path from $x_{m+2}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$ (when $m+1 < s$).
We can verify (i) and (ii) by the same argument as for (i) and (ii) in Case 1, respectively. Hence it remains to show (iv). Also, in order to prove that $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ implies $f_{j}\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, it suffices to check that the directed paths in (i), (ii), and (iv) above can be chosen from the shortest ones. We can show this claim for (i) and (ii) in the same way as for (i) and (ii) in Case 1, respectively. So, it remains to show it for (iv).
\(iv) As in the proof of (iii) in Case 1, it can be shown that there exists a directed $t_{1}$-path (and hence directed $\sigma_{m+1}$-path since $t_{1}=\sigma_{m+1}$ by the assumption) from $x_{m+1}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$ whose length is equal to $1$. Also, it follows from the definition that there exists a directed $\sigma_{m+1}$-path from $x_{m+2}$ to $x_{m+1}$. Concatenating these directed $\sigma_{m+1}$-paths, we obtain a directed $\sigma_{m+1}$-path from $x_{m+2}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$. Thus, we have proved that $f_{j}\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$.
Assume now that $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, and set $n:={\ell(x_{m+2},\,x_{m+1})}$. We see from the argument above that there exists a directed $\sigma_{m+1}$-path from $x_{m+2}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$ whose length is equal to $n+1$. Suppose, for a contradiction, that there exists a directed path from $x_{m+2}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$ whose length $l$ is less than $n+1$. Since ${\langle s_{j}x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ and ${\langle x_{m+2}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$ as seen above, we deduce from Lemma \[lem:sigma1\] that there exists a directed path from $x_{m+2}$ to ${\lfloor s_{j}{\lfloor s_{j}x_{m+1} \rfloor} \rfloor}={\lfloor x_{m+1} \rfloor}=x_{m+1}$ whose length is equal to $l-1 < n$, which contradicts $n={\ell(x_{m+2},\,x_{m+1})}$. Thus, we have proved that if $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then $f_{j}\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$.
#### Case 3.
Assume that $x_{u} = {\lfloor s_{j}x_{u+1} \rfloor}$ and $\sigma_{m} < t_{1} < \sigma_{m+1}$. Then we deduce from the definition of the root operator $f_{j}$ (for the case $j=0$, see also Remark \[rem:ro\_f0\]; cf. [@Lit-A Proposition 4.7a) and Remark 4.8]) that $$\begin{aligned}
& f_{j}\eta=
(x_{1},\,x_{2},\,\dots,\,x_{u}={\lfloor s_{j}x_{u+1} \rfloor},\,
{\lfloor s_{j}x_{u+2} \rfloor},\,\dots,\, \\
& \hspace*{50mm}
{\lfloor s_{j}x_{m} \rfloor},\,{\lfloor s_{j}x_{m+1} \rfloor},\,x_{m+1},\,x_{m+2},\,\dots,\,x_{s}\,;\, \\
& \hspace*{50mm}
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{u-1},\,\sigma_{u+1},\,\dots,\,
\sigma_{m},\,t_{1},\,\sigma_{m+1},\,\dots,\,\sigma_{s});\end{aligned}$$ note that ${\lfloor s_{j}x_{m+1} \rfloor} \ne x_{m+1}$ since ${\langle x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} > 0$. In order to prove that $f_{j}\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, we need to verify that
1. there exists a directed $\sigma_{p}$-path from ${\lfloor s_{j}x_{p+1} \rfloor}$ to ${\lfloor s_{j}x_{p} \rfloor}$ for each $u+1 \le p \le m$;
2. there exists a directed $t_{1}$-path from $x_{m+1}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$.
We can verify (ii) and (iii) by the same argument as for (ii) and (iii) in Case 1, respectively. Also, in the same way as in the proofs of (ii) and (iii) in Case 1, respectively, we can check that if $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then the directed paths in (ii) and (iii) above can be chosen from the shortest ones. Thus we have proved that $f_{j}\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, and that $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ implies $f_{j}\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$.
#### Case 4.
Assume that $x_{u} = {\lfloor s_{j}x_{u+1} \rfloor}$ and $t_{1}=\sigma_{m+1}$. Then we deduce from the definition of the root operator $f_{j}$ (for the case $j=0$, see also Remark \[rem:ro\_f0\]; cf. [@Lit-A Proposition 4.7a) and Remark 4.8]) that $$\begin{aligned}
& f_{j}\eta=
(x_{1},\,x_{2},\,\dots,\,x_{u}={\lfloor s_{j}x_{u+1} \rfloor},\,
{\lfloor s_{j}x_{u+2} \rfloor},\,\dots,\, \\
& \hspace*{50mm}
{\lfloor s_{j}x_{m} \rfloor},\,{\lfloor s_{j}x_{m+1} \rfloor},\,x_{m+2},\,\dots,\,x_{s}\,;\, \\
& \hspace*{50mm}
\sigma_{0},\,\sigma_{1},\,\dots,\,\sigma_{u-1},\,\sigma_{u+1},\,\dots,\,
\sigma_{m},\,t_{1}=\sigma_{m+1},\,\dots,\,\sigma_{s});\end{aligned}$$ note that ${\lfloor s_{j}x_{m+1} \rfloor} \ne x_{m+2}$ since ${\langle s_{j}x_{m+1}\Lambda,\,\alpha_{j}^{\vee} \rangle} < 0$ and ${\langle x_{m+2}\Lambda,\,\alpha_{j}^{\vee} \rangle} \ge 0$ (see Case 2 above). In order to prove that $f_{j}\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, we need to verify that
1. there exists a directed $\sigma_{p}$-path from ${\lfloor s_{j}x_{p+1} \rfloor}$ to ${\lfloor s_{j}x_{p} \rfloor}$ for each $u+1 \le p \le m$;
2. there exists a directed $\sigma_{m+1}$-path from $x_{m+2}$ to ${\lfloor s_{j}x_{m+1} \rfloor}$ (when $m+1 < s$).
We can verify (ii) and (iv) by the same argument as for (ii) in Case 1 and (iv) in Case 2, respectively. Also, as in the proofs of (ii) in Case 1 and (iv) in Case 2, we can check that if $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, then the directed paths in (ii) and (iv) above can be chosen from the shortest ones. Thus we have proved that $f_{j}\eta \in {\widetilde{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$, and that $\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ implies $f_{j}\eta \in {\widehat{{\mathbb{B}}}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$.
This completes the proof of Proposition \[prop:stable\].
Combining Theorem \[thm:charls\] with Propositions \[prop:ip\] and \[prop:stable\], we obtain Theorem \[thm:main\].
[XXXXXX]{}
T. Akasaka and M. Kashiwara, Finite-dimensional representations of quantum affine algebras, [*Publ. Res. Inst. Math. Sci.*]{} [**33**]{} (1997), 839–867.
F. Brenti, S. Fomin, A. Postnikov, Mixed Bruhat operators and Yang-Baxter equations for Weyl groups, [*Int. Math. Res. Not.*]{} [**1999**]{} (1999), no. 8, 419–441.
A. Björner and F. Brenti, “Combinatorics of Coxeter Groups”, Graduate Texts in Mathematics Vol. 231, Springer, New York, 2005.
J. Hong and S.-J. Kang, “Introduction to quantum groups and crystal bases”, Graduate Studies in Mathematics Vol. 42, Amer. Math. Soc., Providence, RI, 2002.
V. G. Kac, “Infinite Dimensional Lie Algebras”, 3rd Edition, Cambridge University Press, Cambridge, UK, 1990.
M. Kashiwara, On crystal bases, [*in*]{} “Representations of Groups” (B.N. Allison and G.H. Cliff, Eds.), CMS Conf. Proc. Vol. 16, pp. 155–197, Amer. Math. Soc., Providence, RI, 1995.
M. Kashiwara, On level-zero representations of quantized affine algebras, [*Duke Math. J.*]{} [**112**]{} (2002), 117–175.
T. Lam and M. Shimozono, Quantum cohomology of $G/P$ and homology of affine Grassmannian, [*Acta Math.*]{} [**204**]{} (2010), 49–90.
C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph, to appear in [*Int. Math. Res. Not.*]{}, doi:10.1093/imrn/rnt263, arXiv:1211.2042.
C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals II: Alcove model, path model, and $P=X$, preprint, arXiv:1402.2203.
P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, [*Invent. Math.*]{} [**116**]{} (1994), 329–346.
P. Littelmann, Paths and root operators in representation theory, [*Ann. of Math.*]{} (2) [**142**]{} (1995), 499–525.
S. Naito and D. Sagaki, Path model for a level-zero extremal weight module over a quantum affine algebra, [*Int. Math. Res. Not.*]{} [**2003**]{} (2003), no. 32, 1731–1754.
S. Naito and D. Sagaki, Crystal of Lakshmibai-Seshadri paths associated to an integral weight of level zero for an affine Lie algebra, [*Int. Math. Res. Not.*]{} [**2005**]{} (2005), no. 14, 815–840.
S. Naito and D. Sagaki, Path model for a level-zero extremal weight module over a quantum affine algebra. II, [*Adv. Math.*]{} [**200**]{} (2006), 102–124.
S. Naito and D. Sagaki, Crystal structure on the set of Lakshmibai-Seshadri paths of an arbitrary level-zero shape, [*Proc. Lond. Math. Soc.*]{} (3) [**96**]{} (2008), 582–622.
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---
abstract: 'We analyse the dynamics of the multiple planet system HD 181433. This consists of two gas giant planets (bodies $c$ and $d$) with $m \sin{i}=0.65$ M$_{\rm Jup}$ and 0.53 M$_{\rm Jup}$ orbiting with periods 975 and 2468 days, respectively. The two planets appear to be in a 5:2 mean motion resonance, as this is required for the system to be dynamically stable. A third planet with mass $m_b \sin{i} = 0.023$ M$_{\rm Jup}$ orbits close to the star with orbital period 9.37 days. Each planet orbit is significantly eccentric, with current values estimated to be $e_b=0.39$, $e_c=0.27$ and $e_d=0.47$. In this paper we assess different scenarios that may explain the origin of these eccentric orbits, with particular focus on the innermost body, noting that the large eccentricity of planet $b$ cannot be explained through secular interaction with the outer pair. We consider a scenario in which the system previously contained an additional giant planet that was ejected during a period of dynamical instability among the planets. N-body simulations are presented that demonstrate that during scattering and ejection among the outer planets a close encounter between a giant and the inner body can raise $e_b$ to its observed value. Such an outcome occurs with a frequency of a few percent. We also demonstrate, however, that obtaining the required value of $e_b$ and having the two surviving outer planets land in 5:2 resonance is a rare outcome, leading us to consider alternative scenarios involving secular resonances. We consider the possibility that an undetected planet in the system increases the secular forcing of planet [*b*]{} by the exterior giant planets, but we find that the resulting eccentricity is not large enough to agree with the observed one. We also consider a scenario in which the spin-down of the central star causes the system to pass through secular resonance. In its simplest form this latter scenario fails to produce the system observed today, with the mode of failure depending sensitively on the rate of stellar spin-down. For spin-down rates above a critical value, planet $b$ passes through the resonance too quickly, and the forced eccentricity only reaches maximum values $e_b \simeq 0.25$. Spin-down rates below the critical value lead to long-term capture of planet $b$ in secular resonance, driving the eccentricity toward unity. If additional short-period low mass planets are present in the system, however, we find that mutual scattering can release planet $b$ from the secular resonance, leading to a system with orbital parameters similar to those observed today.'
author:
- |
Giammarco Campanella$^{1}$[^1], Richard P. Nelson$^1$ and Craig B. Agnor$^1$\
$^{1}$Astronomy Unit, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, U.K.
date: 'Accepted 2013 May 31. Received 2013 May 30; in original form 2012 December 22'
title: Possible scenarios for eccentricity evolution in the extrasolar planetary system HD 181433
---
\[firstpage\]
planetary systems, dynamical evolution and stability, celestial mechanics, stars: individual: HD 181433 – methods: N-body simulations, numerical.
Introduction {#Sec:Intro}
============
Discovery of the first extrasolar planet around a sun-like star, 51 Peg b [@1995Natur.378..355M], and subsequent detection of more than 850 exoplanets raises many questions about the formation and dynamics of planetary systems. Now that we know of more than 120 multi-planet systems[^2], a broad range of planetary system architectures have been sampled. This variegate ensemble offers a wide range of features that are not displayed by our Solar System: giant planets in short period orbits (“hot Jupiters”), the wide eccentricity distribution of exoplanets, the relatively common detection of mean motion resonances (MMRs) in multiple planet systems offer interesting examples against which theories of planetary formation and evolution need to be tested [e.g. @b1].
Determining the physical processes that lead to the observed diversity of planetary systems remains an area of active research and debate. The discovery of dynamically quiescent, short-period multiplanet systems by the Kepler mission, such as Kepler 11 [@b39], provides clear evidence for disc-driven type I migration [@1997Icar..126..261W] of bodies that probably formed beyond the ice line in their protoplanetary discs. Similarly, giant planet systems in 2:1 mean-motion resonance such as GJ 876 [@2001ApJ...556..296M] provide evidence for disc-driven type II migration of gap-forming planets because of the slow convergent migration rates required for resonant capture , as do giant planets on short to intermediate period orbits that are too remote from their central stars to have undergone significant tidal evolution.
The relatively large eccentricities observed for the extrasolar planet population as a whole, however, indicate that disc-planet interactions do not tell the whole story. A plausible explanation for many of these systems is that they formed on near-circular orbits in the disc, but after disc dispersal dynamical instability led to planet-planet gravitational scattering [@b7]. Evidence is provided by the fact that numerical planet scattering experiments can reproduce the observed eccentricity distribution successfully [@b5; @2008ApJ...686..603J], and the observation that a number of short-period transiting planets have orbit planes inclined significantly with respect to the stellar equatorial plane [@2010ApJ...718L.145W]. Further ideas for explaining eccentric orbits include eccentricity driving through disc-driven migration of a resonant pair of planets [@b29], and sweeping secular resonances caused by various combinations of protoplanetary disc dispersal, spin-down of an initially rapidly spinning oblate central star, and general relativistic precession [@b31; @b8].
The three-planet extrasolar system HD 181433 has an architecture that makes it an interesting test-bed for exploring dynamical processes during and after planetary system formation. [@b11] report that the system consists of two outer gas giant planets (‘$c$’ and ‘$d$’) on eccentric orbits with periods $962 \pm 15$ and $2172 \pm 158$ days, and an inner super-Earth on a $9.3743 \pm 0.0019$ day orbit with eccentricity $e_b = 0.396 \pm 0.062$. Dynamical analysis of this system indicates that the parameters reported by [@b11] lead to global instability on very short time scales [@b3]. Long-term stability requires the two outer planets to be in 5:2 mean motion resonance. Parameters that lead to a minimum $\chi^2_{red}$ value of 4.96 for the orbital fits to the data are given in table 1, and these are the parameters of the system that we adopt in this paper. It is noteworthy that planet $b$ is dynamically isolated from the outer planets such that gravitational interaction over secular time scales is unable to excite the observed eccentricity of this body.
The best fit parameters presented in @b3 and reported in table 1 lead to the 5:2 resonant angles librating with large amplitude. Disc driven migration scenarios for resonant capture of giant planets normally result in capture in low-order resonances with relatively small libration amplitudes [@2002MNRAS.333L..26N]. Planet-planet scattering, however, may also result in resonant capture into higher order resonances with large libration amplitudes [@b2], and this would appear to be a more probable explanation for the currently inferred orbital parameters for the two outer planets $c$ and $d$.
In this paper, we present a case study of the evolution of the planetary system around HD 181433, and explore plausible scenarios of how the observed large eccentricities may have been generated after depletion of the protoplanetary disk. In particular we focus on scenarios that can lead to the current observed orbital configuration of planet *b*. In Section \[Properties\], we discuss the properties of the HD 181433 system, and study the locations of secular resonances and regions of stability. In Section \[tides\], we consider how tides may have influenced the evolution of the semi-major axis and eccentricity of planet *b* over the life-time of the system. In Section \[Scattering\], we explore the planet-planet scattering mechanism as a means of generating large eccentricities. We simulate the evolution in the presence of an additional giant planet whose role is to destabilise the system. We estimate the probability for planets *c* and *d* to be captured into the 5:2 MMR after scattering, and thus consider the joint probability for both resonant capture into 5:2 and excitation of the eccentricity of planet $b$. In Section \[spin\], we examine an alternative scenario in which sweeping of secular resonances induced by stellar spin-down force the eccentricity of HD 181433*b*. We study a range of initial stellar rotation periods and spin-down rates, and examine the effect of including additional terrestrial planets – whose presence we find to be necessary for the model to be successful. In Section \[concl\] we briefly discuss and summarize our findings.
Properties of the HD 181433 system {#Properties}
==================================
**Parameter** **HD 181433 b** **HD 181433 c** **HD 181433 d**
--------------------------- ----------------- ----------------- -----------------
P (days) 9.37 975 2468
$T_{peri}$ (BJD-2450000) 2788.92 2255.6 1844
e 0.39 0.27 0.47
$\varpi$ ($^{\circ}$) 202.04 22 319
$m\sin i$ ($M_{\rm Jup}$) 0.023 0.65 0.53
$m\sin i$ ($M_\oplus$) 7.4 208 167
a (AU) 0.080 1.77 3.29
\[param\]
The mass of the host star is reported to be $M_{*} = (0.781 \pm 0.10) M_\odot $ [@b21], its inferred radius is $R_{*} = (1.01 \pm 0.07) R_\odot $ [@b22], while an age of $(6.7 \pm 1.8)$ Gyr has been reported [@b23]. HD 181433 is believed to be of spectral type K3IV, and has a spin period of 54 days [@b11]. The orbital elements of the HD 181433 planets adopted in this paper are shown in Table \[param\]. These values are quoted from the best-fit orbital solution that is dynamically stable obtained by @b3.
The orbital period of HD 181433*b* is 9.4 days with an eccentricity of 0.39. The longitude of HD 181433*b*’s periastron advances mostly due to the relativistic correction of the Newtonian potential. Being a close-in planet in eccentric orbit, HD 181433*b* is undergoing tidal circularization. Given the age of the system and the measured $m \sin i$ value, this suggests that planet *b* has a tidal $Q$ factor substantially larger than inferred for terrestrial planets in the Solar System, for which $Q \sim 100$ (e.g. Murray & Dermott 1999). This point is discussed further in Section \[tides\].
The orbital parameter values quoted in Table \[param\] are specific to a particular moment in time. In fact, resonant and secular interactions cause $e_c$ and $e_d$ to reach values as large as 0.52 and 0.50, respectively [@b3]. In this paper, when discussing the eccentricities acquired by planets during dynamical evolution we will report the maximum values acquired during numerical computations once final stable systems have been established.
Formation and evolution scenarios {#formation}
---------------------------------
Before embarking on a detailed examination of the HD 181433 system, it is useful to present a qualitative description of the possible formation and evolution history. We envisage that the planets formed within a protoplanetary disc exterior to the snow-line. Planet *b* appears to have characteristics of a Neptune-like planet based on analysis of its tidal evolution (see later in the paper). It seems likely that this body formed largely from material beyond the snow-line and migrated in by type I migration, arriving at its observed location on a near-circular orbit. It is possible that it accreted during migration, possibly through giant impacts with other large bodies. If these occurred after migration was essentially complete then the body would be close to the star and the impactors would need to be at least as massive as the inferred minimum mass of *b* to generate significant eccentricity through scattering during the giant impacts phase. Interactions and giant impacts with bodies in the Earth-mass regime at semi-major axes $\sim 0.1$ AU would lead to $e_b < 0.1$. Therefore, we anticipate that planet *b* was formed with a small free eccentricity.
We anticipate that the outer giant planets *c* and *d* formed somewhat later than planet *b*, but were able to grow larger cores that enabled rapid gas accretion and formation of gas giant planets. Although disc driven migration may have brought the planets into close proximity, it is unlikely that it was responsible for establishing the 5:2 resonance. Disc-driven resonant locking tends to cause the resonant angles to librate with small amplitudes because the planets are pushed deep into the resonance, but planets *c* and *d* appear to be weakly embedded in the resonance [@b3]. Furthermore, [@2002MNRAS.333L..26N] undertook a study of disc-driven resonant capture and only obtained a 5:2 resonance when one of the two planets already had a high eccentricity equal to 0.3. A more plausible explanation for the resonance is capture as a result of planet-planet scattering, possibly involving an additional planet that was ejected during the dynamical instability [@b2]. Given the stabilisation provided by the eccentricity damping influence of the gas disc, this instability most likely occurred during the final stages of the disc life time or after it was dispersed altogether.
The rest of this paper largely concerns the question of how planet *b*’s eccentricity was excited, and when addressing this we consider the following possibilities: scattering by a giant planet during the previously mentioned dynamical instability that created the 5:2 resonance; by invoking the presence of an additional undetected planet in the system that enhances the forced eccentricity of planet *b*; passing planet *b* through a secular resonance through the process of stellar spin-down from an initially rapidly rotating state.
Secular evolution
-----------------
For the purpose of undertaking a simplified analysis of the dynamical evolution of the HD 18433 planetary system, we have implemented a Laplace-Lagrange secular model based on the discussion in chapter 7 of @b24, assuming that all planets are coplanar. Given the standard nature of this secular theory, we refer the interested reader to @b24 for description of the mathematical details. The disturbing function includes secular terms for the mutual gravitational interaction between all three planets that are second order in the eccentricities and first order in the masses. Precession due to general relativity (GR) is also included [e.g. see @b8], but our model does not take account of the 5:2 MMR occupied by the two outer planets. Although techniques have been developed to include first-order MMRs in secular treatments of planetary system evolution , this becomes significantly more involved technically for higher order MMRs because of the requirement to include terms that are higher order in the eccentricities. Our purpose here is to use the secular theory as a rough guide to the dynamics of the system, and not to accurately predict its long term evolution, so this omission is not crucial here. This point is illustrated later in the paper where we demonstrate, by means of direct numerical simulations, that the eccentricity evolution of the two outer planets maintains highly regular and periodic behaviour, reminiscent of secular evolution (see Figure \[ecc\] for example). It appears that the effect of the 5:2 MMR is rather weak, with the main influence being an alteration of the precession frequencies. The secular model, therefore, allows us to identify the existence and approximate locations of secular resonances, in addition to estimating the magnitudes of forced eccentricities experienced by additional low mass planets that may be present in the system today, or which were present in the past. All detailed analyses of the HD 181433 system dynamics undertaken in this paper use direct numerical simulations for accurate modelling of the mutual interactions, and all firm conclusions drawn about the past and future evolution are based on those simulations.
The secular system we examine is based on the osculating orbital elements listed in Table \[param\]. The left-hand panel of Figure \[analytic\] shows the variation of the free precession frequency $A$ experienced by a test particle located between 0 to 9 AU. On the same diagram the three eccentricity-pericentre eigenfrequencies of the system are denoted by solid horizontal lines. They are (in descending order) $g_2 = 0.0304 ^{\circ}$yr$^{-1}$, $g_3 = 0.0065 ^{\circ}$yr$^{-1}$ and $g_1 = 0.0049 ^{\circ}$yr$^{-1}$. Note that in the absence of relativistic precession $g_1 = 0.0009 ^{\circ}$yr$^{-1}$, with the other eigenfrequencies that are largely determined by the outer planets being essentially unaffected. The intersections of the lines with the curve show where the eigenfrequencies $g_i$ equal $A$, and identify the semi-major axes where large forced eccentricities can be expected. In the central panel of Figure \[analytic\], the values of the maximum forced eccentricity are shown as a function of semi-major axis. The singularities close to 0.18, 0.7, 4.7 and 6.3 AU represent locations where the value of $A$ is equal to one of the $g_i$ eigenfrequencies of the secular system, as anticipated above. This plot indicates that the system is more-or-less dynamically packed out to distances of $\sim 7$ AU, in the sense that additional planets located in the system will experience large forced eccentricities that may be destabilising. Even in the region between 0.2–0.6 AU (focus of the right-hand panel of Figure \[analytic\]), where a clear minimum exists in the forced eccentricity, a value of $e_{\rm forced} \ge 0.2$ is predicted. It is noteworthy that the habitable zone for this K3IV star is centred at $\sim 0.55$ AU [@b3].
{width="33.00000%"}{width="33.00000%"}{width="33.00000%"}
Full integration of the system {#sec:fullintegr}
------------------------------
We use the symplectic N-body code Mercury-6 [@b25], augmented to include the effects of GR [@b26], to perform a direct integration of the system using initial conditions from Table \[param\]. Unless stated otherwise, all simulations in this paper adopted the hybrid integrator option which utilises the second-order mixed-variable symplectic (MVS) algorithm for systems that are not strongly interacting, but switches to the Bulirsch-Stoer (BS) integrator when the minimum separation between objects $\le 4$ Hill radii. A time step of 1/20 of the orbital period of the innermost planets is adopted for the MVS integrator, and the Bulirsch-Stoer integrator is employed with an accuracy parameter of $10^{-11}$.
Figure \[GR\] shows two integrations of the system, one including the relativistic correction and one without it, and underlines how the relativistic correction for planet *b* is not relevant with regard to the amplitude of $e_b$, even if it is important with respect to the precession period. This differs from the $\upsilon Andromed{\ae}$ system, where the forced eccentricity of the inner planet depends sensitively on whether or not the post-Newtonian force is accounted for in secular evolution [@b10] because of the proximity of a secular resonance.
While the loose residence of planets c and d in the 5:2 mean motion resonances introduces some short-term oscillations, the eccentricity evolution of the system over longer periods appears regular and well-described by a simple three frequency secular model. Fourier analysis of the system’s apsidal behaviour indicates that the main effect of the 5:2 resonance is to simply modify the values of the principal precession frequencies that describe the giant planets (i.e., $g_2 = 0.0304^{\circ}\mbox{
/yr}\rightarrow 0.02305^{\circ}\mbox{
/yr}$ and $g_3 = 0.00652 ^{\circ}\mbox{
/yr}\rightarrow -0.04354^{\circ}\mbox{
/yr}$ making it inaccessible for resonant interaction with $g_1$). Figure \[ecc\] shows the evolution of the eccentricities of the HD 181433 planets retrieved from the Laplace-Lagrange secular model (left panel) and the numerical integration (right panel). The amplitudes of the eccentricity variation are very similar, while the periodicity of the eccentricity of *c* and *d* is approximately reduced of two thirds to $\sim 5000$ years as result of the 5:2 MMR. This illustrates the accuracy with which the secular model estimates the locations of the secular apsidal resonance, in addition to the strong regularity displayed by the full integration even though the outer two planets are in the 5:2 MMR. It is clear that the effects of the 5:2 MMR in this system are rather weak and make only a modest change to the secular behaviour, namely through shifting the precession frequencies of the outermost planet *d*.
Stability of an additional low mass planet {#stabadd}
------------------------------------------
To examine the consequences for an additional low mass planet in the system arising from the large forced eccentricities predicted in Figure \[analytic\], we have performed N-body simulations that include an additional planet. We place an Earth-mass object (which would be undetectable in current radial velocity surveys) at different distances between planets *b* and *c* to check for the stability of that region. The sampling used was of 0.05 AU for semi-major axes in the range $0.15-0.6$ AU and 0.1 AU for semi-major axes in the range $0.7-1.7$ AU.
We find that in full N-body simulations, the secular resonance at 0.18 AU moves inward leaving the region between 0.1– 0.35 AU stable for a single body, but unstable outside of this region. In particular, for semi-major axes $a > 0.26$ AU the forced eccentricity attains values $e_{\rm forced} \gtrsim 0.2$. Later in the paper we consider the influence of additional low mass planets, and we find that inserting two or three extra planets with at least one of them between 0.26 and 0.35 AU makes the system unstable (see Section \[addpl\]).
![Eccentricity evolution of HD 181433 *b*. Solid line (black) not considering GR, dashed line (red) including GR.[]{data-label="GR"}](Fig2.eps){width="\columnwidth"}
{width="50.00000%"}{width="50.00000%"}
Tidal evolution {#tides}
===============
The close proximity of planet *b* to its host star suggest it should be undergoing tidal evolution. Tidal effects circularize planetary orbits on a time scale given by [@b36] $$\tau_{e}=\frac{4}{63}Q\left(\frac{a^3}{GM_*}\right)^{1/2}\frac{m_p}{M_*}\left(\frac{a}{R_p}\right)^5.$$ The value of $Q$ calculated for rocky objects in the Solar System is of the order of $10^2$ [@b24]. Such a value would imply a circularization time scale of the order of $10^7$ years for planet *b*, much shorter than any reasonable estimate for the age of the system. If we reject the hypothesis that a recent event has generated the observed eccentricity, then the tidal factor $Q$ must be considerably greater than expected for a terrestrial body of mass $\lesssim 10$ M$_{\oplus}$. If we assume a $Q$ value and density typical of an ice giant such as Neptune ($Q=10^4$, $\rho_b = \rho_{Nep} = 1.638$ g cm$^{-3}$), then the circularization time scale becomes $\tau_{e} = 8.2$ Gyr, comfortably longer than the estimated system age. This would make HD 181433*b* a member of the class of low-mass, low-density, close-in planets exemplified by Kepler 11c and Kepler 11e [@b39; @2013arXiv1303.0227L]. To uncover the history of planet *b* and make an estimate for the orbital parameters that existed prior to significant tidal evolution, it is necessary to examine the influence of tides on past orbital evolution and estimate their influence of future evolution.
To study the orbital evolution of planet *b* we integrate the coupled tidal evolution equations for changes in $a$ and $e$ (see @b17 and references therein) backward and forward in time: $$\begin{aligned}
\frac{1}{a} \frac{da}{dt} &=& -\left[ \frac{63 \sqrt{G M_{*}^3} R_p^5}{2 Q_{\mathrm{p}} m_p} e^2 \right.\nonumber \\
&&+ \left.\frac{9 \sqrt{G/M_{*}} R_{*}^5 m_p}{2 Q_{*}} \left(1 + \frac{57}{4} e^2\right)\right] a^{-13/2} \\
\frac{1}{e} \frac{de}{dt} &=& -\left[ \frac{63 \sqrt{G M_{*}^3} R_{\mathrm{p}}^5}{4 Q_{\mathrm{p}} m_p} +
\frac{225 \sqrt{G/M_{*}} R_{*}^5 m_p}{16 Q_{*}} \right] a^{-13/2}.\textrm{~~~~}
\label{eq:tides}\end{aligned}$$ The effects of the tide raised on the star $Q_*$ as well as on the planet $Q_{\rm p}$ are both included. The stellar $Q$ value is typically estimated through the observed circularization of binary star orbits. The rate of tidal effects may be a very strong function of orbital frequency [@b38]. If this is the case, the planet may spend a lot of time at certain states where tidal effects are slow and rapidly pass through states where tidal effects are faster. This model assumes that the star is rotating slowly relative to the orbit of the planet (so when the star has already evolved toward the Main Sequence) and is second order in eccentricity. It describes tidal evolution associated with orbit circularization, as this operates most rapidly, and ignores other sources of tidal interaction which operate on longer time scales. Our intention is to use this simple tidal theory to develop an approximate picture for the likely orbital history for HD 181433*b*, and its possible future evolution.
We computed the orbital evolution for various pairs of values of $Q_{\rm p}$ (ranging between $10^3$ and $10^{6.5}$) and $Q_*$ (between $10^4$ and $10^{7}$). The chosen $Q$ values span the range that are plausible. The rate of tidal damping may depend on the interior structure of the planet and is likely to be different for different planets. We integrate the equations for the age of the star backward in time (6.7 Gyr) and for 10 Gyr into the future. Results show that only when $Q_*$ falls below $10^4$ does the behaviour change in a noticeable manner. We conduct our analysis using $Q_*=10^{5.5}$, which corresponds to the value for which @b13 have found the distribution of initial *e* values of close-in planets to match that of the general population. The top panel of Figure \[migration\] shows the migration of HD 181433*b* for different values of $Q_p$, while in the bottom panel the evolution in $a$-$e$ space is presented (obtained by assuming orbital angular momentum conservation). Since both $de/dt$ and $da/dt$ scale $\propto a^{-13/2}$, the rate of evolution is slower for larger initial values of $a$, but speeds up dramatically as $a$ decreases.
A smaller $Q_{\rm p}$ means a larger range of values is spanned during the evolution for a fixed age of the system. For example, $Q_{\rm p}=10^3$ implies that during the early main-sequence stage planet *b* would have had a semi-major axis of 0.15 AU and an eccentricity almost equal to $0.9$. However, assuming a current age of 6.7 Gyr, complete circularization of the orbit will take much less than one more gigayear to be achieved (with the migration terminating at about 0.07 AU). Without further detailed information about the system allowing us to constrain its physical nature, this evolutionary track and associated $Q_p$ value is equally probable as any others. Our purpose here is to define a plausible initial state of the system prior to significant tidal evolution, and in doing so we make the assumption that we are observing the system close to the midpoint of its tidal evolution. A value of $Q_{\rm p}=10^4$ means that planet *b* would have had initial values $a_b \simeq 0.1$ AU and $e_b \simeq 0.6$, and circularization would be completed within the next 10 Gyr. As noted earlier, $Q_{\rm p}=10^4$ is similar to the values inferred for Uranus and Neptune [@b24]. We find that a value $Q_p=10^5$ leads to tidal effects that are almost negligible such that the observed values of $a_b$ and $e_b$ would be very similar to those 6.7 Gyr ago. When considering mechanisms for exciting the eccentricity of planet *b*, we aim to achieve values between $0.4 \le e_b \le 0.6$, implying modest or essentially no tidal evolution has taken place since the excitation occurred.
![Tidal evolution for HD 181433*b* for the case $Q_*=10^{5.5}$. Top panel: Orbital migration for a range of $Q_{\rm p}$. Initially, a greater semi-major axis is expected in the case $Q_{\rm p}=10^{3}$, followed by the values $10^{4}$, $10^{5}$, $10^{6}$ and $10^{6.5}$. The point of intersection between all the lines represents the present state. Bottom: Evolution in $a-e$ space for the cases $Q_{\rm p}$ equal to $10^{3}$, $10^{4}$, $10^{5}$, $10^{6}$ and $10^{6.5}$. A smaller value of $Q_{\rm p}$ implies greater fraction of the $a-e$ space has been spanned. The evolution moves from right to left. For clarity, evolutionary trajectories for $Q_{\rm p}$ equal to $10^{4}$, $10^{5}$, $10^{6}$ and $10^{6.5}$ have been off-set by $\Delta e = + 0.04$ from each previous case.[]{data-label="migration"}](Fig12T.eps "fig:"){width="\columnwidth"} ![Tidal evolution for HD 181433*b* for the case $Q_*=10^{5.5}$. Top panel: Orbital migration for a range of $Q_{\rm p}$. Initially, a greater semi-major axis is expected in the case $Q_{\rm p}=10^{3}$, followed by the values $10^{4}$, $10^{5}$, $10^{6}$ and $10^{6.5}$. The point of intersection between all the lines represents the present state. Bottom: Evolution in $a-e$ space for the cases $Q_{\rm p}$ equal to $10^{3}$, $10^{4}$, $10^{5}$, $10^{6}$ and $10^{6.5}$. A smaller value of $Q_{\rm p}$ implies greater fraction of the $a-e$ space has been spanned. The evolution moves from right to left. For clarity, evolutionary trajectories for $Q_{\rm p}$ equal to $10^{4}$, $10^{5}$, $10^{6}$ and $10^{6.5}$ have been off-set by $\Delta e = + 0.04$ from each previous case.[]{data-label="migration"}](evsaQs5.5.eps "fig:"){width="\columnwidth"}
For completeness, we have also investigated how the tidal evolution may depend on the actual mass of *b*. We span a range for $\sin i$ between 0.1 and 1 and we assume a density $\rho_{Nep}$ in all cases. A smaller $\sin i$ implies a more massive planet: when $\sin i = 0.5$ planet [*b*]{} is about the mass of Uranus/Neptune. This is purely a qualitative analysis as a small $\sin i$ would imply a larger mass for planets *c* and *d* which could destabilize the system. As expected from equation \[eq:tides\], more massive planets experience significantly more rapid migration than less massive ones. Figure \[sini\] displays the case for the instance $Q_{\rm p}=10^4$. In the extreme case where $\sin i = 0.1$, planet *b* should have been found initially at around $0.13$ AU with an eccentricity of about 0.8; the circularization process speeds up significantly with time and would be fully accomplished in the next billion years for this particular example.
![Tidal evolution of HD 181433*b* for the range of $\sin i$ between 0.1 and 1 when $Q_*=10^{5.5}$ and $Q_{\rm p}=10^4$. The case $\sin i = 1$ can also be found in Figure \[migration\]. Top panel: Orbital migration. A smaller $\sin i$ corresponds to a greater initial semi-major axis and a quicker circularization. The point of intersection between all the lines represents the present state. Bottom: Evolution in $a-e$ space. A smaller value of $\sin i$ corresponds a greater fraction of spanned space. The evolution goes from right to left. For clarity, evolutionary trajectories for values of $\sin i $ from 0.2 to 1 have been off-set by $\Delta e = + 0.04$ from each previous case.[]{data-label="sini"}](Fig13T.eps "fig:"){width="\columnwidth"} ![Tidal evolution of HD 181433*b* for the range of $\sin i$ between 0.1 and 1 when $Q_*=10^{5.5}$ and $Q_{\rm p}=10^4$. The case $\sin i = 1$ can also be found in Figure \[migration\]. Top panel: Orbital migration. A smaller $\sin i$ corresponds to a greater initial semi-major axis and a quicker circularization. The point of intersection between all the lines represents the present state. Bottom: Evolution in $a-e$ space. A smaller value of $\sin i$ corresponds a greater fraction of spanned space. The evolution goes from right to left. For clarity, evolutionary trajectories for values of $\sin i $ from 0.2 to 1 have been off-set by $\Delta e = + 0.04$ from each previous case.[]{data-label="sini"}](evsasini.eps "fig:"){width="\columnwidth"}
We end our discussion of tidal evolution by noting that substantial tidal heating of a gaseous planet can in principle cause it to undergo Roche lobe overflow and orbital expansion [@b15; @b16; @b17]. As yet calculations have not been performed with parameter sets that would allow us to determine whether or not such a scenario is compatible with the current orbital configuration of HD 181433 *b*.
Planet-planet scattering {#Scattering}
========================
We now consider the origin of the eccentricities of the planets in the HD 181433 system, with our primary focus being the eccentricity of the inner-most body. Recent work has demonstrated that planet-planet scattering can explain the eccentricity distribution of the extrasolar planet population as a whole [e.g. @b5; @2008ApJ...686..603J]. Here we explore a scenario in which an additional giant planet was present in the system originally, orbiting close to the two existing outer giant planets, but shortly after dispersal of the protoplanetary disc planet-planet scattering caused this extra planet to be ejected from the system, leaving behind a three-planet system with eccentricities and semi-major axes similar to those observed today. Given the non-linear dynamics involved, the likelihood of producing a system with parameters very close to those of the observed HD 181433 system is exceedingly small. We therefore define two requirements that must be met for a simulation outcome to be deemed a success: the inner-most planet has a close encounter with one of the giant planets during the period of dynamical instability leading to significant growth of its eccentricity; two giant planets remain at the end of the simulation in 5:2 resonance. We are able to estimate the probability of each of these separate outcomes from our simulations, and hence the joint probability of both requirements being satisfied.
We use the Mercury-6 N-body code to study this problem, and employ the hybrid integrator with the characteristics described already in Section \[sec:fullintegr\]. The physical size for the bodies is determined by their mean densities. For low mass planets we adopt a value equal to 3 g cm$^{-3}$, and for giant planets we adopt the Jovian value (1.326 g cm$^{-3}$).
Scattering between three giant planets {#sec:3p-scattering}
--------------------------------------
In order for one of the giant planets to undergo a close encounter with planet *b* and scatter it onto an eccentric orbit with $0.4 \le e_b \le 0.6$, as required, the perturbing body needs to have a mass large enough to generate the required velocity perturbation. The perturbing body could be the additional planet ‘*x*’ or either of the planets *c* or *d*. We approximate the eccentricity as $e \simeq v_r/v_{\rm orb}$, where the $v_r$ is the perturbed radial velocity and $v_{\rm orb}=\sqrt{GM_*/a_b}=88$ km/s is the Keplerian orbital velocity of *b*. Assuming that $v_r$ due to an encounter is of the same order as the escape velocity from the perturbing planet gives $v_r \simeq \sqrt{2Gm_{\rm p}/R_{\rm p}}$. The mass required to generate eccentricity $e$ can then be written as $$m_{\rm p} \ge e^3 v_{\rm orb}^3 \left(\frac{3}{32 \pi G^3 \rho_{\rm p}} \right)^{\frac{1}{2}}$$ where $\rho_{\rm p}$ is the mean density of the perturber. Assuming a Jovian mean density gives a required mass for planet *x* in the range $0.19 \le m_x \le 0.64$ M$_{\rm Jup}$ if it plays the role of planet *b*’s perturber. We have performed simulations with $m_x=0.3$ M$_{\rm Jup}$ which is large enough to produce values of $e_b > 0.4$. If planets *c* or *d* act as the perturbers then larger values of $e_b$ are possible given their larger masses.
In addition to requiring that the perturbing planet can excite a large enough value of $e_b$, we also require that the initial conditions of our simulations at least in principle allow the observed configuration of HD 181433 to be attained once the dynamical stability has caused ejection of the additional planet. The combined specific energy of the two outer planets is given by $E_{\rm tot} =-G M_*/(2 a_c) - G M_*/(2 a_d)$, and we note that ejection of planet *x* requires a loss of specific energy from the system equal to $E_x=G M_*/(2 a_x)$. We therefore ensure that our initial conditions are such that if energy $E_x$ is lost from the system of outer giant planets the remaining energy equals $E_{\rm tot}$. We consider a number of basic initial configurations for our simulations, and each simulation set corresponds to a particular stellocentric ordering of the outer giant planets: *cdx*, *cxd*, *xcd*, *dcx* and *xdc*. The first and last letters in the labels correspond to the closest and further planets from the star. For each set, planet *x* is initially placed randomly between 2 and 5 mutual Hill radii ($R_{H,m} = 0.5(a_{b} + a_{x})[(m_{b} + m_{x})/3M_{*}]^{1/3}$) from its neighbour (for the set *cxd* planet *x* is initially closest to planet *d*), and the initial mean longitudes are also set randomly. The planets are initially on circular orbits with mutual inclinations $i\le 1$ degree. We note that with the ice line defined by $a_{\rm ice} = 2.7 \sqrt{L_*/L_\odot}$ AU gives $a_{\rm ice} = 1.50$ AU for HD 181433. Our initial set-up therefore concurs with the general expectation that giant planets emerge from the disk at locations beyond the ice line.
Throughout the integrations, close encounters and collisions between any two bodies were logged, as well as ejections and collisions with the parent star. We ran the simulations for 250 Myr but instabilities usually arise over much shorter time-scales. If dynamical instability resulted in ejection of one or more planet we calculated the orbital elements of the remaining planets. In particular, we are interested in the possibility that the two outer planets are in 5:2 resonance after the scattering, and the value attained for the eccentricity of planet *b* when the system has stabilised. In all simulations energy was conserved to a level better than 1 part in $10^4$, which is adequate for testing stability (e.g. Barnes & Quinn 2004). 300 simulations were performed overall, 50 for each of the sets described above except for set *cdx* where we ran 100 simulations.
The simulation results show that planet *x* is ejected in almost 50% of cases, but in none of our simulations do we find that planets *c* and *d* are in 5:2 mean motion resonance. The dynamical instability often leaves planets *c* and [d]{} more highly separated than in the observed configuration, with planet *d* in particular orbiting with significantly larger semi-major axis. Figure \[scatt\] shows the outcome from sets (*cdx*, *xcd*) that have undergone strong scattering, and we see that in each case the currently observed values of $e_b$, $e_c$ and $e_d$ are at, or close to, the upper limits of the eccentricities generated in the simulations. More importantly, these simulations also demonstrate the eccentricity of the short-period planet *b* can also be excited to the required value. Inclinations typically remain small in accordance to what was found by @b7.
![Comparison of our synthetic final planetary systems with the HD 181433 system showing maximum eccentricity versus semi-major axis. The values for the real planets are presented with filled black squares. Top panel: Cases from set *cdx*. Bottom: Cases from set *xcd*.[]{data-label="scatt"}](x22J.eps "fig:"){width="\columnwidth"} ![Comparison of our synthetic final planetary systems with the HD 181433 system showing maximum eccentricity versus semi-major axis. The values for the real planets are presented with filled black squares. Top panel: Cases from set *cdx*. Bottom: Cases from set *xcd*.[]{data-label="scatt"}](moreoutgiannew.eps "fig:"){width="\columnwidth"}
The left-hand panels of Figure \[highecc\] show a case from set *cdx* where planet *x* is ejected after close encounters, and planets *c* and *d* land close to their present locations (but are not in resonance). Planets *b*, *c* and *d* achieve maximum eccentricities of 0.09, 0.29 and 0.28, respectively. A case from set *xcd* is presented in the right panels of Figure \[highecc\]. Following close encounters, planet *x* is ejected with planets *c* and *d* landing at 1.6 and 4.4 AU, respectively. Planets *b*, *c* and *d* acquire maximum eccentricities of 0.19, 0.55 and 0.47, and maximum inclinations of 12$^{\circ}$, 11$^{\circ}$ and 9$^{\circ}$, respectively. Therefore, in this case planets *c* and *d* reach values for the eccentricities that are similar to the observed one, but planet *d* orbits at a greater distance.
{width="50.00000%"}{width="50.00000%"} {width="50.00000%"}{width="50.00000%"}
A case in which $e_b$ grows to the required value is displayed in Figure \[higheb\] where after many close encounters planet *x* is ejected and planet *b* gains a maximum eccentricity of $e_b=0.65$, which is a value that becomes relevant when considering tidal effects (see Section \[tides\]). The final semi-major axis for planet *d*, however, is equal to 10.3 AU so the overall final architecture of the system differs considerably from the observed system. This model is also characterized by orbital inclination growth: $i_b$ moves in the interval 25$^{\circ}$-83$^{\circ}$, $i_c$ in the range 12$^{\circ}$-24$^{\circ}$, while $i_d$ stays in the range 3$^{\circ}$-13$^{\circ}$. In this and similar cases we have checked that the large eccentricity arises because of scattering rather than the Kozai effect which can potentially become active when mutual inclinations exceed a value $\simeq 40$ degrees [@1962AJ.....67..591K]. Summing over all simulations, we find that 11%, 6% and 2% of the runs generate eccentricities for planet *b* of at least 0.2, 0.4 and 0.6, respectively. These simulations therefore demonstrate the feasibility of dynamical instability of the outer planet system causing all planets in the system to develop large eccentricities.
![Evolution of some orbital elements for a configuration from set cdx. Top panel: Time evolutions of the semi-major axes. Initially, in order of increasing distance planets *b*, *c*, *d* and *x* are located. Bottom: Time evolutions of the eccentricities. The solid (black) line indicates planet *b*, the dashed line (red) represents planet *c*, the dotted line (blue) denotes planet *d* and the dashed-dotted line (dark cyan) is planet *x*. When *x* is ejected around 4.5 My, the eccentricities oscillate stably on a secular time-scale. $e_b$ moves in the interval 0.20-0.65, $e_c$ moves in the interval 0.12-0.34, while $e_d$ in the range 0.23-0.30.[]{data-label="higheb"}](Fig7T.eps "fig:"){width="\columnwidth"} ![Evolution of some orbital elements for a configuration from set cdx. Top panel: Time evolutions of the semi-major axes. Initially, in order of increasing distance planets *b*, *c*, *d* and *x* are located. Bottom: Time evolutions of the eccentricities. The solid (black) line indicates planet *b*, the dashed line (red) represents planet *c*, the dotted line (blue) denotes planet *d* and the dashed-dotted line (dark cyan) is planet *x*. When *x* is ejected around 4.5 My, the eccentricities oscillate stably on a secular time-scale. $e_b$ moves in the interval 0.20-0.65, $e_c$ moves in the interval 0.12-0.34, while $e_d$ in the range 0.23-0.30.[]{data-label="higheb"}](Fig7B.eps "fig:"){width="\columnwidth"}
### Probability of resonant capture
Defining a successful outcome for the planet-planet scattering experiments is not straightforward. The nonlinear nature of the process clearly means we cannot reasonably expect that a relatively small number of N-body simulations will result in systems that are close analogues to the currently-observed HD 181433 system. Instead we use a more restricted definition of success in which planet *b* experiences an increase in its eccentricity and planets *c* and [d]{} end up in 5:2 resonance. As discussed above, our simulations have demonstrated that the eccentricity of planet *b* can be raised to the required value, and we also have outcomes in which planets *c* and *d* have period ratios that are quite close to 5:2. None of the simulations produce a system in 5:2 resonance, however, so we now examine the probability of capture in resonance by considering the width of the 5:2 resonance and the relative mean longitudes and longitudes of pericentre required for the planets to orbit stably in resonance.
@b2 show that planet-planet scattering may result in pairs of planets landing in high order MMRs. The resulting systems tend to have quite high eccentricities and resonant angles that librate with large amplitudes, characteristics that are displayed by the HD 181433 planets as described below and in @b3. The simulations presented by @b2 produced a few 5:2 MMRs for every one thousand simulations. Here we examine the probability of two planets being scattered into the 5:2 MMR by determining the width of the resonance using N-body simulations that explore the dynamics of two bodies in resonance. Our procedure follows that adopted by Soja et al. (2011) in their study of asteroids in resonance. We consider the presently inferred orbital elements of planets *c* and *d* and we study the width of the region inside of which libration occurs by varying the semi-major axis of *d* in steps of 0.0005 AU. For each case, we compute the maximum amplitude of the oscillations in semi-major axis by taking the difference between the maximum and minimum values over 50000 years of integration, normalized by the initial semi-major axis. The top panel of Figure \[reswidth\] shows how the amplitude of oscillations in semi-major axis varies in the resonant region. Planet *d* survives only in the range $3.2560 \le a \le 3.2995$ AU, and the system is disrupted when *d* is placed just outside this zone. The resonant argument for the 5:2 MMR of the planetary system HD 181433 is [@b3] $$\label{resarg}
\psi = 5\lambda_d-2\lambda_c-3\varpi_d$$ where $\lambda$ is the mean longitude and $\varpi$ is the longitude of pericentre. The bottom panel of Figure \[reswidth\] shows the libration amplitude of this resonant angle as a function of the semi-major axis of planet *d*. The libration amplitude at the location of *d* that corresponds to the best-fit stable orbital solution presented by @b3 agrees well with the value quoted in that paper. We find the width of the resonance to be $\Delta_a = 0.0435$ AU centred at $a=3.2775$ AU.
![The width of the 5:2 MMR. Top panel: The variation in size of resonant semi-major axis oscillations for different locations of planet *d*. Bottom: The variation in the resonant argument. The system is unstable outside the resonance.[]{data-label="reswidth"}](Fig18T.eps "fig:"){width="\columnwidth"} ![The width of the 5:2 MMR. Top panel: The variation in size of resonant semi-major axis oscillations for different locations of planet *d*. Bottom: The variation in the resonant argument. The system is unstable outside the resonance.[]{data-label="reswidth"}](Fig18B.eps "fig:"){width="\columnwidth"}
A simple estimate for the probability of the planets landing within the resonance after scattering is $$\label{Pdeltaa}
P(\Delta_a)=\frac{\Delta_a}{\overline{a_d-a_c}}$$ where ${\overline{a_d - a_c}}$ is the median value of $a_d - a_c$ at the end of all simulations for which strong plant-planet interactions occurred (we include only those runs for which final eccentricity of at least one of the planets $e > 0.1$). We obtain $P(\Delta_a)=0.0075$. Having planets land within the required $\Delta_a$ after scattering, however, does not guarantee that they will be in resonance. It also depends on the angles that define the mutual orientation of their eccentric orbits (the difference between their longitudes of pericentre $\varpi_c-\varpi_d$), and also the values of their mean anomalies at the beginning of their interaction once they land within the resonance width. To quantify this aspect of the problem we ran a set of simulations where we take the semi-major axis values that lie at the centre of the resonance, and the eccentricity values for the stable best fit solution. We vary $\varpi_c-\varpi_d$ in steps of $90^\circ$ and the mean anomalies $M_c$ and $M_d$ in steps of $45^\circ$, for a total of 256 simulations. We ran the integrations for 30 Myr. 34 pairs of planets survive, all in resonance and in anti-aligned mode. The resonant argument (equation \[resarg\]) librates with amplitudes that vary from a few degrees up to about $240^\circ$, as outlined by the lower panel of Figure \[reswidth\]. Finally, we are able to estimate the probability of resonance capture to be $P_{5:2} \sim 0.0075 \times 34/256 \simeq 10^{-3}$, in decent agreement with the larger sample of numerical simulations presented by @b2.
### Probability of scattering generating HD 181433 systems {#sec:probscat}
We have determined that planetary systems with global structure similar to HD 181433, but which originally had an additional gas giant planet orbiting close to the two outer giant planets, can lead to the excitation of the eccentricity of the inner super-Earth up to values $e_b \simeq 0.4$ during approximately 6 % of the time when the system experiences a global dynamical instability. This eccentricity excitation occurs because one or more of the outer planets has a close encounter (or a series of close encounters) with the inner planet during the chaotic phase of evolution. Treating the perturbation of the interior planet *b* onto an eccentric orbit, and the landing of the two outer planets in the 5:2 mean motion resonance, as being independent processes, the joint probability of eccentricity excitation and resonant capture becomes $P \simeq 6 \times 10^{-5}$. Taken at face value, this result suggests that systems with characteristics similar to HD 181433 occur through planet-planet scattering rather rarely.
Sweeping secular resonances due to stellar spin-down {#spin}
====================================================
Having determined that the planet-planet scattering hypothesis is a plausible scenario for excitation of the eccentricity of all planets in the HD 181433 system, but that the excitation of $e_b$ combined with resonant capture of planets *c* and *d* is likely to be a rare event, we now consider alternative scenarios for exciting planet *b*’s eccentricity. We consider the hypothesis that the eccentricities and resonant structure for the orbits of the outer planets were established after a period of dynamical instability once gas disc dispersal had occurred, and the eccentricity of planet *b* was established through the sweeping of a secular resonance with the outer planets caused by the spin-down of the central star from an initial state of very rapid rotation.
The idea that a short-period planet may experience excitation of its eccentricity and inclination because of stellar spin-down originates with @b30. They showed that if the primordial gravitational field of the Sun had a larger second-degree harmonic \[i.e. $J_2 \gtrsim {\cal O}(10^{-3})$\], equivalent to a rotation period $\lesssim 5.7$ hours, then subsequent solar spin-down would drive Mercury’s orbit through secular resonances capable of generating its large mean eccentricity and inclination. Secular resonances are generated when the orbits of two bodies precess synchronously. A small body in secular resonance with a large planet will have its eccentricity and inclination modified over relatively short time periods. As the mass distribution of a planetary system evolves, for example as a result of gas disc dispersal, orbital migration or stellar spin-down, the locations of secular resonances move. Planets located in regions through which the resonance sweeps are perturbed as the resonance drives the eccentricity and inclination. Given that we assume the current orbits of the outer planets were established shortly after gas disc dispersal, stellar spin-down provides the resonant sweeping in our model, and the rate of eccentricity/inclination forcing scales with the square root of the stellar spin-down time [@b31]. In principle it should therefore be possible to tune the spin-down time scale to obtain the desired eccentricity for planet *b*. There are other planetary systems with architectures similar to HD 181433 (i.e., a factor of $>10$ in orbit period between the inner and outer planets, and an eccentric inner planet) such as HD125612 and $\mu$ Arae. The same arguments used to constrain the orbital history of the HD 181433 system may apply to these systems too. We are investigating the more general implications of these evolutionary tuning processes and their application elsewhere [@AgnorLin2012b].
Stars are generally believed to lose their primordial angular momentum through the magnetic braking action of the stellar wind with a mass-loss rate orders of magnitude greater than that on the main sequence (e.g. Skumanich 1972). Pre-main sequence stars with masses over $0.25 M_\odot$ exhibit a bimodal period distribution with observed values clustered around 6–8 days and 2 days. The transition between the two peaks is fairly abrupt. T Tauri stars can have spin periods of the order of hours during their evolution, with a significant fraction of them showing this characteristic . In addition to providing a method of exciting the eccentricity of an interior planet through resonant sweeping [@b30], stellar spin-down from an initial rapid state of rotation has also been invoked to explain the low eccentricity of an interior planet. Using early system parameters for the then three planet $\upsilon Andromed{\ae}$ system, @b10 showed that the rotation period of the parent star had to be shorter than 2 days during dispersal of the gas disc so that the passage of the sweeping secular resonance near the orbit of the short-period planet *b* could have been avoided leaving its eccentricity at a low level. Our model differs in that we assume the eccentricities of the outer planets to have been established after gas disc removal rather than before or during its occurrence given its role in damping large planetary eccentricities (e.g. Papaloizou et al. 2001).
Secular model including stellar spin-down
-----------------------------------------
We begin our analysis by modifying the Laplace-Lagrange secular model described in Section \[Properties\] to account for stellar spin-down through inclusion of the $J_2$ contribution to the eigenfrequencies of the system. As previously remarked, the secular model provides only an approximate estimate of the locations of secular resonance and can therefore be used to quickly evaluate the hypothesis that a secular resonance may have swept the present-day semi-major axis of planet *b* at 0.080 AU during stellar spin-down. We adopt the method described in @b24 for including the $J_2$ terms, and we use the relation between the stellar spin period $P_*$ and $J_2$ provided by @b30. The mass of planet *b* is much smaller than the outer two giants so we treat it as a test particle in the secular model. The left panel of Figure \[sweep\] presents the free precession period of a test particle induced by planets *c* and *d*. It is important to point out the role of GR-induced precession in promoting secular resonances close to the star: neglecting GR, the precession rate would fade to zero very near to the star making it difficult to match any eigenfrequencies of the outer planet secular system (see Section \[Properties\] for how precession rates change due to GR). The central and right plots in Figure \[sweep\] show the sweeping of two secular resonances as the parent star spins down from a rotation period of 2 days to 30 days (for which the $J_2$ effects become insignificant). According to this simplified model, when the rotation period of HD 181433 was $P_* \approx 2.1$ days (equivalent to $J_2 \approx 2.2 \times 10^{-5}$), the free precession rate at the present location of planet *b* matched the one of the eigenfrequencies of the system. Later, the secular resonances move inward toward their present-day locations. Using the argument that the eccentricity excitation due to the passage of a secular resonance is inversely proportional to the square root of the changing rate of the resonant frequency [@b31], we can estimate the spin-down time $\tau= \Omega/\dot{\Omega}$, where $\Omega=2\pi/P_*$ is the rotational angular velocity of the star required to excite the present eccentricity. We have: $$\label{rateprec}
e \simeq \left(\frac{2\pi}{|\dot{A}|}\right)^{1/2} \mu_d$$ where, from @b24, $$\label{mud}
\mu_d = \sum_{j=1}^2 A_j e_{jd}$$ with $\mu_d$ modelling the resonance of interest, $e_{jd}$ being the components of the scaled eigenvector for planet *d*, $A_j$ defined by equation 7.142 in @b24, and $\dot{A}$ is the rate of change of the resonant precession frequency $A=\dot{\varpi}$ at resonance. Stellar rotation enhances the star’s oblateness. The effect of rotation on stellar oblateness can be encapsulated in the resulting $J_2$ coefficient which allows for simple inclusion into secular models and N-body simulations. From @b30, we have: $$\label{dotj2}
J_2 = \frac{2}{3}k\frac{\Omega^2 R_*^3}{GM_*}$$ where for the apsidal constant $k$ we take the value $8.16 \cdot 10^{-3}$ calculated for the Sun [@b30] Combining Eq.\[dotj2\] with secular theory for an oblate primary [@b24] gives the following relation between the spin down rate of the star $\dot{\Omega}$ and the rate of change in the precession frequency $\dot{A}$ $$\label{precMurray}
\dot{A} \simeq 2\left(\frac{R_*}{a} \right)^2
\left( \frac{\dot{\Omega}}{\Omega} \right) k\left(\frac{\Omega^2 R_*^3}{GM_*}\right)n.$$ Finally, for HD 181433 b this model suggests that a spin down timescale of $\tau \approx 1.9 \times 10^{7}$ years may be capable of accounting for its large observed free eccentricity of 0.39. This time scale is consistent with estimated mass loss rates from stars in the T-Tauri stage [@b30], suggesting that excitation of planet *b*’s eccentricity through sweeping secular resonances is a realistic hypothesis worthy of further more detailed exploration.
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N-body simulations of secular resonance sweeping
------------------------------------------------
As discussed in Section \[Properties\], the precession period for planet *d* is shorter than suggested by the secular model because of MMR effects. Therefore, we implement a stellar spin-down model in the BS algorithm of the integrator MERCURY-6 to account for the time dependent rotational flattening of the host star. Our model now includes effects due to GR and stellar oblateness, where we adopt the expression for the acceleration due to the oblate star given by @b10, in additional to the gravitational interaction between the three planets. We have run checks to ensure that the precession periods that arise from the simulations agree with analytical expression from secular theory. The stellar spin-down is modelled as a magnetic braking torque based on the empirical Skumanich law (Skumanich 1972): $$\label{down}
\frac{d{\bf \Omega}}{dt}=-\alpha \Omega^2 {\bf \Omega}$$ where ${\bf \Omega}$ is the stellar spin vector and $\alpha = 1.5
\times 10^{-14}$ years for a G or K dwarf (e.g., Barker & Ogilvie 2009). This parameter measures the speed of removal of angular momentum from a rotating star. Given the importance of angular momentum in determining the eccentricity evolution of the system we have confirmed that it is conserved in the simulations at a level better than 1 part in $10^6$.
### Evolution during resonant sweeping {#reson}
The secular theory suggests that a stellar spin period of between 11.1–16.7 hours would force planet *b* to precess with a frequency similar to one of the system’s eigenfrequencies, such that a resonance may be established. For comparison, the minimum rotation period for the star is $P_{cr}=2\pi\sqrt{R_{*}^3/GM_*} \approx 3.2$ hours.
We initiate N-body simulations with planets *c* and *d* in their inferred present day configuration, and with planet *b* on a circular orbit close to the star (later on we consider scenarios with differing implications for the long-term tidal evolution of the system discussed in Section \[tides\], and so place planet *b* at different semi-major axes). Here we consider evolution that implies very little tidal evolution of the system has occurred over its life-time, consistent with an adopted value of $Q_{\rm p} \gtrsim 10^5$ as discussed in Section \[tides\]. We therefore place planet *b* with its currently observed semi-major axis $a_b=0.08$ AU. We initiate simulations with a stellar spin period of 14 hours, and vary the value of the spin-down parameter $\alpha$, beginning with its nominal value given above.
Results are shown in the top panels of Figure \[alpha\] for the nominal value of $\alpha=1.5 \times 10^{-14}$ years. In the right panel it is possible to observe how the relative longitudes of pericentre $\varpi_b-\varpi_c$ evolve during the process: the initial growth of eccentricity begins when the precession rates of planets *b* and *c* match ($P_* \approx 16$ hours). The passage of the resonance is anticipated by the orbit of *b* precessing faster initially and then being overtaken by the precession rate of *c*. $e_b$ peaks at 0.16 and stabilizes later at a value of $e_b \simeq 0.13$. Setting the initial stellar rotation period to 5 hours instead of 14 hours produces the same result, with the eccentricity peaking when the stellar spin period is 17 hours.
Excitation of orbital eccentricity depends on the mass and eccentricity of the perturber. @b11 do not report an uncertainty on the mass of planet $c$, while the quoted errors on $e_c$ are relatively small with $\sigma_{e_c} = 0.02$. Such a small change in the value of $e_c$ would lead to only small changes in our results. We know the minimum mass of $c$. However, @b3 notes that the stable best–fit is found in a dynamically active region of phase space, and a value for $\sin i$ noticeably different from 1 would generate instabilities in the system. A slightly increased mass for planet $c$ would lead to only a slightly modified secular resonance.
As discussed above, the expected level of eccentricity excitation depends on the rate of resonant sweeping. A larger value of $e_b$ requires the spin-down rate to be slower, so we have examined how the evolution changes with smaller values of $\alpha$. Interestingly, we find that the system behaviour can be divided into two distinct modes that depend critically on the value of $\alpha$. For spin-down rates that exceed the critical value ($\alpha_{\rm crit}$ is 5.75 % smaller than the nominal value) the evolution is similar to that described above and illustrated in the upper panels of Figure \[alpha\]: temporary capture in the resonance and excitation of $e_b$ to values $e_b \lesssim 0.25$. Spin-down parameters equal to or smaller than $\alpha_{\rm crit}$ lead to long-term capture in the secular resonance (apparently indefinite capture) and growth of $e_b$ toward unity. This mode of evolution is shown in the lower panels of Figure \[alpha\] for a run with $\alpha=\alpha_{\rm crit}$, where over a run time of 18 Myr $e_b$ reaches a value of 0.7 and $\varpi_b-\varpi_c$ librates around zero with a semi-amplitude of $\sim 45$ degrees. Apparently a planet caught within this mode of evolution is driven to $e_b=1$ and collision with the central star unless tides are able to intervene for cases where $Q_{\rm p}$ is small enough to drive sufficiently rapid tidal damping of $e_b$.
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The reason for the existence of these two regimes can be sought in the expressions for the precession rate of the longitude of periastron due to GR and $J_2$, $\dot{\varpi}_{GR}$ (see e.g. Misner et al. 1973 for a derivation) and $\dot{\varpi}_{J_2}$ [@b34], respectively: $$\label{precrate}
{\dot \varpi}_{\rm GR}=\frac{3GM_*}{ac^2(1-e^2)}n\\
{\dot \varpi}_{J_2}=\frac{3}{2}\frac{J_2}{(1-e^2)^2}\left(\frac{R_*}{a}\right)^2n$$ where *n* is the mean motion, and both ${ \dot \varpi}_{GR}$ and ${ \dot \varpi}_{J_2}$ depend on the eccentricity such that an increase in $e$ leads to an increase in ${\dot \varpi}_{GR}$ and ${\dot \varpi}_{J_2}$. The condition for the resonance to be maintained during spin-down is given by $$\frac{\partial {\dot \varpi}_{J_2}}{\partial J_2} \frac{d J_2}{dt}
= - \left( \frac{\partial {\dot \varpi}_{J_2}}{\partial e} +
\frac{\partial {\dot \varpi}_{\rm GR}}{\partial e} \right) \frac{de}{dt}.
\label{eq:res-condition}$$ In other words, the reduction in precession rate due to stellar spin-down needs to be compensated by the increase in precession rate that occurs as eccentricity grows. We predict from equation \[eq:res-condition\] that removing the effects of GR will still allow long term secular resonant locking, but for slower values of the spin-down parameter $\alpha$. We have performed simulations to examine this by omitting the GR term in the equations of motion, and find that a rotation period of $\simeq 15$ hours is required to enter secular resonance and the spin-down parameter needs to be more than 20 % smaller than the nominal value to maintain long-term resonant capture, in agreement with our expectation. The plots in Figure \[spindownevst\] show the growth of $e_b$ for different values of $\alpha$ in cases in which GR effects are included (left panel) and neglected (right panel), demonstrating long-term resonant capture for $\alpha$ below a threshold value in each case.
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### Resonant sweeping with additional exterior planets {#addpl}
The simulations presented in the previous section indicate two modes of behaviour, but neither of them are able to explain the observed eccentricity of planet *b*. One results in an eccentricity that is too small, and the other apparently results in either an eccentricity which is too high or collision with the central star. One possibility that we explore here is that there may have been additional planets in the system orbiting relatively close to planet *b* during sweeping of the secular resonance. If the spin-down is below the critical value required for long-term capture then interactions with the additional planets when the eccentricity becomes large may release planet *b* from the secular resonance, resulting in a final eccentricity of the required magnitude.
We begin by exploring the evolution with one additional Earth-mass planet (so-called planet *x*) in the system located outside of the orbit of planet *b* on a circular orbit. We ran a suite of 12 simulations where planet *x* is located within 2-8 mutual Hill radii from the apocentre of planet *b* calculated when $e_b$ is in the range 0.3-0.7. The idea here is to induce planet-planet scattering when eccentricity growth is already underway; it is equivalent to placing planet *x* in the range 0.10-0.16 AU. Including planet *x* modifies the resonance condition, so for each simulation we have calculated the new spin period required to produce the necessary precession rate for *b*. We find that for an additional Earth-mass planet placed between 0.10-0.16 AU, that a stellar spin period from 0.75 days up to 5 days will enable resonant interaction with the exterior giants during stellar spin down.
The simulations yield the result that both planets *b* and *x* become trapped in the secular resonance with *c*, and each of them experiences eccentricity growth without limit. Figure \[1extra\] reports one example with planet *x* at 0.14 AU, a stellar spin period $\simeq 20$ hours necessary to generate the resonance, and spin-down parameter $\alpha$ that is 6% smaller than the nominal value. The left panel illustrates how the orbits of *b* and *x* cross as their eccentricities grow continuously (central plot). Collisions are avoided, however, because planet *x* is trapped in resonance with planet *b*, as demonstrated by the right panel. Planets *b* and *c* are in a secular apsidal resonance, with $\varpi_b-\varpi_c$ librating around 0$^{\circ}$ with a semi-amplitude of $\sim 45^{\circ}$. This behaviour is a feature of all runs for which we included one additional Earth-mass planet, and we note that in the absence of stellar spin-down all of the configurations that we considered were dynamically stable over $10^6$ years.
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For completeness we consider a model with three additional terrestrial planets to see whether this promotes the sought-after instability when planet *b* has reached the desired eccentricity $e_b \sim 0.5$. We construct systems consisting of three additional 0.5–1 Earth-mass planets. The inner-most additional body is placed 2–5 mutual Hill radii from the apocentre of the planet, calculated when its eccentricity is in the range 0.3–0.7. The second additional body is placed 2–5 mutual Hill radii from the apocentre of the first additional body, calculated when its eccentricity is in the range 0.3–0.7. The third additional planet is placed 2–5 mutual Hill radii from the apocentre of the second additional body, calculated when its eccentricity is in the range 0.2–0.7. We note that these three bodies are all placed interior 0.26 AU which is a stable zone. In fact, a planet with semi-major axis in the range 0.26 – 0.35 AU obtains a forced eccentricity $e_{\rm forced} \gtrsim 0.2$ (see Section \[stabadd\]), which will destabilize any planets in the range 0.1 – 0.26 AU before the secular resonance is entered. We consider a spin-down parameter 5.75 % smaller than the nominal $\alpha$ and we calculate the necessary spin period for each case to enter resonance. We set a density of 3 g/cm$^{3}$ for the additional planets and a Neptune density (1.638 g/cm$^{3}$) for *b*. We ran more than 100 simulations, varying the planetary mutual separations.
From the results of the N-body simulations we observe that all four inner planets become involved in the resonant trapping, with the eccentricity growing for all of them before instability occurs and strong mutual interactions take place. The outcomes of these simulations include mutual close encounters, collisions between the planets, and collisions with the central star. Occasionally the inner planets can disturb the fragile resonance between the two outer giants causing catastrophic ejections from the system. Out of 100 models we find two models that replicate the present configuration with a value for $e_b \gtrsim 0.4$. When a terrestrial planet survives in the process, the model is still compatible with the detected system because, for example using the Systemic Console [@b40], a 1 Earth-mass planet at 0.19 AU would be at the noise level of the radial velocity data with an F-test value of $\approx 40\%$. Figure \[3ext\] illustrates a successful model for which only planet *b* survives the instabilities and is able to achieve the required eccentricity. The necessary stellar rotation period to produce the resonance is $\approx 22.2$ hours. The right panel represents how all the inner planets are increasing their eccentricity during the passage of the resonance, leading eventually to close encounters and collisions which are displayed in the left panel. Orbital inclinations remain small in the system. We have tested the evolution of the systems neglecting the effects of stellar spin-down and found that the eccentricities remain small and the system is stable over runs times of $3.2$ Myr.
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### The resonance with an interior planet {#1int}
To test the generality of the results presented in the previous section, we evaluate how a hypothetical interior terrestrial planet may have influenced the sweeping of the resonance. We consider a single Earth-mass planet in circular orbit with semi-major axes in the range 0.03-0.055 AU. We want to assess if the additional planet will be trapped long-term into the resonance with planet *b* (as in Sect. \[addpl\]), or if its presence will release *b* from the resonance with the required eccentricity. We prepare a set of 100 runs with planet *b* initially located at 0.9-0.11 AU so that close encounters may start when a high eccentricity has already been reached. Each sub-set has its own stellar spin period for resonance capture. The particular behaviour depends on the individual run, but in general we observe that planets *b* and *x* are not quite trapped into a mutual resonance but instead experience differential precession, with the precession periods differing by approximately 2000 years. We find that collisions do occur, but because of the slow differential precession it takes on the order of $10^7$ years for them to happen. Figure \[int\] presents a successful case: the eccentricities of the two inner planets grow because of the passage of the resonance, the orbits cross each other, after 11 Myr instability arises leading to a collision, leaving planet *b* with the required eccentricity. Although planet *b* is initiated with a semi-major axis of 0.1, the inelastic collision causes the composite planet to effectively migrate to 0.08 AU, which is the observed value. Orbital inclinations stay small in the system. To generate the resonance this simulation required a stellar rotation period $\lesssim 13.4$ hours. A set of simulations with $a_b=0.105$ AU demands a stellar spin period of 11.3 hours. In these configurations the eccentricity stops growing and stabilizes around 0.45 as the secular resonance is disrupted.
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### Resonance passage considering stronger tidal evolution
In the previous section we showed that sweeping secular resonances due to stellar spin-down, combined with planet-planet scattering, can cause eccentricity excitation of planet *b* up to $e_b \simeq 0.4$, consistent with present day observations. The HD 181433 system, however, is unusual because of the large eccentricity of the inner planet in spite of the $\sim 6.7$ Gyr age of the star. The implication is that tidal dissipation inside the planet is weak compared with that measured for terrestrial bodies in the Solar System. If the value $e_b=0.4$ is primordial then it implies a value of $Q_{\rm p} \ge 10^5$.
A value of $Q_{\rm p} \sim 10^4$ is inferred for Neptune and Uranus [@b24], and we showed in Section \[tides\] that if such a value is adopted for HD 181433*b* then the planet is approximately half-way through the process of circularising from an initial eccentricity of $e_b \simeq 0.6$ and semi-major axis $a_b=0.1$. This raises the question of whether or not the sweeping secular resonance model with additional terrestrial-mass planets can also achieve an eccentricity this large, and we have addressed this issue below using a further batch of N-body simulations.
Here the planet *b* is placed further out so a shorter stellar spin period is required to generate the resonance. We infer that the star had to rotate in $\lesssim 11.0$ hours in order for the resonance to sweep the location of *b*. Using the nominal value of $\alpha$ (see equation \[down\]) in a simulation consisting of planets *b*, *c* and *d* only, $e_b$ peaks at 0.28 and stabilizes later around a value of 0.25. If we reduce the stellar spin-down rate to be equal to or less than the nominal value by just 5.75 %, $e_b$ grows indefinitely as described above.
Our original aim was to test the possibility of breaking the secular resonance for planet *b* by including three additional terrestrial planets orbiting beyond *b*, but the larger initial semi-major axis and larger final eccentricity required of $e_b=0.6$ mean that a stable system of three additional planets cannot be set-up because the outer one orbits outside the stable zone located within 0.26 AU (see Section \[Properties\] and \[addpl\]). For this reason we ran a suite of simulations with two Earth-mass planets orbiting outside of planet *b*. The qualitative evolution is similar to that described in Section \[addpl\]. An example of a successful simulation is displayed in Figure \[2ext\], showing the growth of eccentricity through secular resonance, planet-planet scattering and collision, and two surviving inner planets with planet *b* orbiting with $e_b \sim 0.6$ and $a_b \sim 0.1$. The necessary stellar rotation period to generate the resonance is 11.6 hours. The rocky planet which survives at 0.16 AU would be in the noise level of the measured radial velocities (F-test value of 84%). Its orbital inclination increases up to 12$^{\circ}$ for the smaller survivor while it remains much smaller for planet *b*. Taking account of tides with $Q_{\rm p} =10^4$, once the initially chaotic phase of evolution has finished the system will evolve to exhibit characteristics very similar to those observed today on a time scale of $\sim 6$ Gyr.
Note that we have tested the stability of the system without the resonance and find that it remains stable over a time-scale of 1 Myr.
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Finally, we report here also an example of evolution when an Earth-mass body is placed on an orbit interior to planet *b*. This scenario has already been discussed in Section \[1int\]. Figure \[inttide\] shows the increase of the eccentricities for the two inner planets (*b* and *x*). Instabilities arise after 18 My, only planet *b* survives and it is left with an eccentricity $e_b=0.56$, close to the required value. To generate the resonance a stellar rotation period $\lesssim 13.4$ hours is required.
![The passage of the resonance in the inner part of the HD 181433 system when an Earth-mass planet at 0.04 AU is included. The evolution of the eccentricities is show with $e_b$ growing faster and achieving a final value of about 0.56. The inner terrestrial planet is destroyed after 18 My.[]{data-label="inttide"}](Fig16.eps){width="\columnwidth"}
Large forced $e_b$ by an additional planet
------------------------------------------
The spin-down of the central star provides a means of enhancing the eccentricity of planet *b*. In the absence of a rapidly rotating star an additional undetected planet in the system can also enhance the forced eccentricity of planet *b* by strengthening coupling with the outer giant planets. By carefully choosing of the mass and location of an additional planet the forced eccentricity of planet *b* may be enhanced. We focus on the effect of a low mass planet near planet *b* below.
### Exterior low mass planet
@2004ApJ...614..955M, for example, have investigated the hypothesis that eccentric short-period planets have eccentricities that are excited by undetected outer companions. We have already discussed in Section \[stabadd\] that an additional planet can be stable in the region 0.1 – 0.35 AU. We use the secular model to estimate the planetary mass required in this range in order for the precession period of $b$ to match that of $c$ or of the hypothetical planet. We then refine the solution by means of numerical simulations, setting the eccentricities of the extra planet and *b* to small non-zero values and examining the evolution of 75 configurations for 10 Myr.
The maximum forced eccentricity obtained for these runs was $e_b=0.28$ for a 12 Earth mass companion orbiting at 0.35 AU or for a 15 Earth mass companion orbiting at 0.3 AU. All other simulations have resulted in values of $e_b < 0.28$. For example, a 11 Earth mass companion orbiting at 0.25 AU resulted in a peak value of $e_b=0.27$ (see left panel of Figure \[eccexcit\]).
### Interior low mass planet
We now consider the evolution when an extra planet is orbiting interior to $b$. We use the secular theory to estimate the planetary masses required between 0.008 – 0.062 AU for planet *b* to experience a significantly enhanced forced eccentricity. Results from this secular analysis indicate that an additional planet located at $a=0.008$ AU must have a mass equal to 45 $m_{\oplus}$. An additional planet located at $a=0.062$ AU must have a mass equal to $0.13 m_{\oplus}$. Numerical integration of 40 initial configurations, covering the range of semi-major axes and $m_p$ discussed above, show again that $e_b$ is not forced sufficiently to explain the currently observed value of $e_b=0.39$. The maximum forced eccentricity obtained was $e_b=0.21$, this happened in two cases: a 1.5 Earth mass companion orbiting at 0.041 AU (right panel of Figure \[eccexcit\]) and a 0.75 Earth mass companion orbiting at 0.056 AU.
### Detectability
We assess the detectability of these hypothetical planets assuming a radial velocity precision of 1 m/s [@b11]. In the range 0.1 – 0.35 AU planets with mass as small as 4–6 $M_\oplus$ would be detectable. This excludes the possibility of exciting a sufficiently large $e_b$ with an unseen planet in this region. In the inner zone, the detection limit is about 2 Earth-masses. This cannot exclude the generation of a forced eccentricity of 0.21.
If planet *b* had been formed with a free eccentricity of $\simeq 0.18$ then in principle the observed value of $e_b=0.39$ could have been obtained through excitation of a forced eccentricity equal to 0.21. This value for the free eccentricity, however, is rather large for either of the two most plausible scenarios for the arrival of planet *b* at its current location. Gas-disc driven orbital migration of a fully formed planet *b* to its observed location is likely to have left the planet in an essentially circular orbit. Formation [*in situ*]{} through a series of giant impacts may have left the planet with a remnant free eccentricity, but a value of $e_b=0.18$ requires the scattering bodies during the giant impacts phase to be approximately 6 Earth masses, which is only slightly smaller than the measured minimum mass of planet *b*.
{width="50.00000%"}{width="50.00000%"}
Discussion and conclusions {#concl}
==========================
We have investigated the dynamical evolution of the three-planet system orbiting the main sequence K-dwarf star HD 181433 [@b11]. The system consists of a close-in super-Earth with orbital period 9.37 days, and two sub-Jovian giant planets that orbit at much larger distance from the host star with orbital periods 975 and 2468 days. In order to be dynamically stable these outer giants need to be in a 5:2 mean motion resonance [@b3]. The corresponding semi-major axes and eccentricities are $a_b=0.08$, $a_c=1.77$ and $a_d=3.29$ AU. Of particular interest for our study are the large eccentricities displayed by the system: $e_b=0.39$, $e_c=0.27$, $e_d=0.47$. The value for the short-period super-Earth in particular raises interesting questions that we have addressed in this paper: given the large separation between the inner super-Earth and outer giants, what are the plausible mechanisms that can lead to excitation of $e_b$? Given the age of the system ($\sim 6$ Gyr), what are the implications of the observed value of $e_b$ for tidal evolution in the system and the $Q$-value for the inner body?
We assume that the HD 181433 planetary system attained its currently observed configuration shortly after dispersal of the protoplanetary disc. Analysis of the secular dynamics indicates that the eccentricity of the inner body cannot be explained through present day interactions between it and the outer giants. This analysis, however, does point to the existence of a nearby secular resonance that could have caused eccentricity growth during earlier evolution.
The inferred mass of HD 181433*b* is $m_b \sin{i}=7.4$ M$_{\oplus}$. If $\sin{i} \sim 1$ then this planet is a super-Earth. We have analysed the tidal evolution and conclude that if the tidal dissipation factor $Q_{\rm p} \le 10^3$ then tidal circularisation should be completed easily within the 6.7 Gyr age of the system. This suggests that HD 181433*b* is not a massive terrestrial-like planet. A value of $Q_{\rm p} \ge 10^5$ leads to very little tidal evolution, implying that the system observed now is similar to its primordial state if this $Q$-value is appropriate. A value of $Q_{\rm p} \simeq 10^4$, characteristic of Neptune and Uranus [@b24], would indicate that the system started out with $a_b \simeq 0.1$ AU and $e_b \simeq 0.6$ such that the system is essentially half-way through the tidal circularisation process. This might also indicate that HD 181433*b* should be considered as a hot-Neptune rather than a short-period super-Earth.
Given that all bodies in the system have quite large eccentricities, we began our study by analysing a scenario in which an additional outer giant planet was present originally, leading to global dynamical instability and ejection of this additional body. This is a natural starting point given the recent work showing that planet-planet scattering can explain the observed eccentricity distribution of the extrasolar planet population [@b5; @2008ApJ...686..603J]. The chaotic dynamics involved in such a scenario preclude us from obtaining a close-analogue to the HD 181433 system through N-body simulations, so we restrict our analysis to addressing the following two issues as a means of estimating the likelihood that the scenario may have operated: can the short-period super-Earth be perturbed onto an eccentric orbit with the required $e_b \ge 0.4$ during the dynamical instability, given that it orbits close to the central star?; what is the likelihood of the two outer planets *c* and *d* landing in the 5:2 resonance after scattering? From a suite of 300 N-body simulations we find that $e_b$ reaches $e_b \simeq 0.4$ in 6% of the runs, and reaches $e_b=0.6$ in 2% of them, indicating that we can obtain values of $e_b$ that are appropriate for a range of tidal histories. Further analysis of the width of the 5:2 resonance suggests a probability for landing in resonance of $\sim 10^{-3}$, in good agreement with the more extensive set of N-body simulations of planet-planet scattering reported by @b2. A naive estimate of the joint probability of eccentricity excitation for planet *b* and formation of the 5:2 resonance for planets *c* and *d* suggests a value $\sim 6 \times 10^{-5}$, indicating that the hypothetical planet-planet scattering scenario is plausible but only occurs as a rather rare event in planetary system evolution.
Given this conclusion regarding the planet-planet scattering scenario, we also considered the possibility that secular resonance caused excitation of planet *b*’s eccentricity. We examined how an additional planet in the system may have increase the secular forcing of planet *b*’s eccentricity, but this idea is discounted because the eccentricity obtained for masses below the detectability threshold was too small. During pre-main-sequence evolution most host stars are rapid rotators [e.g. @b35]. Rotational flattening introduces a $J_2$ component in the moments of the stellar gravitational potential. Precession caused by this effect can be important in determining the evolutionary fate of short-period planets [e.g. @b30; @b10]. The semi-major axis of planet *b* around HD 181433 is $\sim$ 0.08 AU, close to a secular resonance. Using customised N-body simulations that incorporate precession due to GR and stellar oblateness, and evolution of the stellar spin through magnetic braking, we have tested the hypothesis that the resonance swept past the location of *b*, generating the large eccentricity. We identify two distinct modes of evolution: for nominal values of the stellar spin-down parameter planet *b* is trapped in the resonance temporarily, leading to a maximum growth of eccentricity to $e_b=0.25$; for values of the stellar spin-down parameter that are marginally smaller than the nominal value (i.e only 5.75% slower), planet *b* becomes trapped in the resonance indefinitely with its eccentricity being driven toward unity. The final fate of the system in this latter case appears to be collision with the central star. Neither of these two outcomes leads to a system that looks like HD 181433*b*, but we find that the inclusion of additional short-period low mass planets in the system, orbiting in the vicinity of planet *b*, can perturb it out of long-term resonant capture when its eccentricity has reached large values. Such a scenario can account successfully for the observed orbital properties of planet *b* for a range of tidal histories, but in most cases we require a stellar spin period $\lesssim 20$ hours for resonant capture to occur. Given that we assume the resonant interaction occurs shortly after protoplanetary disc removal, this scenario requires that HD 181433*a* was a member of the young stellar population that displays rotation periods of less than 2 days. While these stars are in the minority, they are by no means rare [@b35], suggesting that this scenario provides a plausible explanation for the high eccentricity observed for HD 181433*b*.
Acknowledgements {#acknowledgements .unnumbered}
================
G.C. acknowledges the support of a PhD studentship from Queen Mary University of London. We are grateful to the referee Makiko Nagasawa for valuable comments that improved this paper.
[99]{} Adams F. & Laughlin G., 2006, ApJ, 649, 1004 Agnor, C. B., & Lin, D. N. C.2012, , 745, 143 Agnor C. B., Lin D. N. C., 2012, EPSC Conf., 946 Barnes R. & Quinn T., 2004, ApJ, 611, 494 Barker A. & Ogilvie G., 2009, MNRAS, 395, 2268 Baxter E., Corrales L., Yamada R., Esin A., 2008, ApJ, 689, 308 Bouchy F. et al., 2009, A&A, 496, 527 Butler R. et al., 2006, ApJ, 646, 505 Campanella G., 2011, MNRAS, 418, 1028 Chambers J., 1999, MNRAS, 304, 793 Chatterjee S., Ford E., Matsumura S., Rasio F., 2008, ApJ, 686, 580 Christou A. A., Murray C. D., 1999, MNRAS, 303, 806 Crida, A., S[á]{}ndor, Z., & Kley, W. 2008, , 483, 325 Ford E. & Rasio F., 2008, ApJ, 686, 621 Goldreich P. & Soter S., 1966, Icarus, 5, 375 Gu P., Lin D., Bodenheimer P., ApJ, 2003, 588, 509 Herbst, W., Bailer-Jones, C. A. L., Mundt, R., Meisenheimer, K., & Wackermann, R. 2002, , 396, 513 Ibgui L. & Burrows A., 2009, ApJ, 700, 1921 Iorio L., 2005, A&A, 433, 385 Jackson B., Greenberg R., Barnes R., ApJ, 2008, 678, 1396 Juri[ć]{}, M., & Tremaine, S. 2008, , 686, 603 Kozai, Y. 1962, , 67, 591 Lee, M. H., & Peale, S. J. 2002, , 567, 596 Lissauer J. et al., 2011, Nature, 470, 53 Lissauer, J. J., Jontof-Hutter, D., Rowe, J. F., et al. 2013, arXiv:1303.0227 Malhotra, R., Fox, K., Murray, C. D., & Nicholson, P. D. 1989, , 221, 348 Mardling, R. A., & Lin, D. N. C. 2004, , 614, 955 Marcy, G. W., Butler, R. P., Fischer, D., et al. 2001, , 556, 296 Mayor, M., & Queloz, D. 1995, Nature, 378, 355 Meschiari S., Wolf A., Rivera E., Laughlin G., Vogt S., Butler P., PASP, 2009, 121, 1016 Miller N., Fortney J., Jackson B., ApJ, 2009, 702, 1413 Misner C., Thorne K., Wheeler J., 1973, Gravitation, W. H. Freeman and Company Moorhead A. & Adams F., 2005, Icarus, 178, 517 Murray C. & Dermott S., 1999, Solar System Dynamics, Cambridge University Press Nagasawa M. & Lin D., 2005, ApJ, 632, 1140 Nagasawa M., Lin D., Thommes E., 2005, ApJ, 635, 578 Nelson, R. P., & Papaloizou, J. C. B. 2002, , 333, L26 Ogilvie G. & Lin D., 2004, ApJ, 610, 477 Papaloizou, J. C. B., Nelson, R. P., & Masset, F. 2001, , 366, 263 Quinn T., Tremaine S., Duncan M., 1991, AsJ, 101, 2287 Raymond S., Barnes R., Armitage P., Gorelick N., 2008, ApJ, 687, L107 Skumanich, A. 1972, , 171, 565 Snellgrove, M. D., Papaloizou, J. C. B., & Nelson, R. P. 2001, , 374, 1092 Sousa S. et al., 2008, A&A, 487, 373 Soja R., Baggaley W., Brown P., Hamilton D., 2011, MNRAS, 414, 1059 Torres G., Andersen J., Giménez A., 2010, A&ARv, 18, 67 Trevisan M., Barbuy B., Eriksson K., Gustafsson B., Grenon M., Pompéia L., 2011, A&A, 535, A42 Ward W., Colombo G., Franklin F., 1976, Icarus, 28, 441 Ward, W. R. 1997, , 126, 261 Winn, J. N., Fabrycky, D., Albrecht, S., & Johnson, J. A. 2010, , 718, L145
\[lastpage\]
[^1]: E-mail: g.campanella@qmul.ac.uk
[^2]: “The Extrasolar Planets Encyclopaedia” <http://www.exoplanet.eu>
|
---
abstract: 'We present our studies on jet–induced modifications of the characteristic of the bulk nuclear matter. To describe such a matter, we use efficient relativistic hydrodynamic simulations in (3+1)dimensions employing the Graphics Processing Unit (GPU) in the parallel programming framework. We use Cartesian coordinates in the calculations to ensure a high spatial resolution that is constant throughout the evolution of the system. We show our results on how jets modify the hydrodynamics fields and discuss the implications.'
address:
- '$^1$ Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, PL'
- '$^2$ Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, PL'
- '$^3$ Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, PL'
author:
- 'P Marcinkowski$^1$, M Słodkowski$^1$, D Kikoła$^1$, J Sikorski$^3$, J Porter-Sobieraj$^2$, P Gawryszewski$^1$ and B Zygmunt$^2$'
title: 'Jet–induced modifications of the characteristic of the bulk nuclear matter'
---
Simulation setup
================
The dynamics of the bulk matter is simulated using the ideal relativistic hydrodynamic equations with a source term [@hirano] $$\label{eqn:hydro}
\partial_{\mu} T^{\mu\nu}=
\partial_{\mu}\left((e+p)u^{\mu}u^{\nu}-pg^{\mu\nu} \right)=
S^{\nu}\\$$ where $u=u(x)$ represents a conserved variable, which include energy, momentum and charge densities. The source term represents energy deposited by jets in the system and it is defined as $$S^{\nu}(\vec{x})
=\sum_{i=1}^{n_{jet}}\left(-\frac{\mathrm{d} E}{\mathrm{d} t}, -\frac{\mathrm{d} \vec{M}}{\mathrm{d} t}\right)\delta ^{(3)}(\vec{x}-\vec{x_i}(t))$$ where $n_{jet}$ is the number of jets. We implemented several algorithms for solving the problem, including WENO [@shu1] ($5^{th}$ and $7^{th}$ order) and Musta-Force [@toro]. Currently we use $7^{th}$ order WENO, which has proven high performance and good quality in test problems with steep gradients. The numerical scheme is integrated over time using third order Runge-Kutta algorithm.\
The energy loss of jets propagating through the bulk nuclear matter is described by using two mechanisms: radiation of gluons and collisions of partons in dense medium and is given by $$\label{eqn:energy}
\left(-\frac{\mathrm{d} E}{\mathrm{d} x}\right )=
\kappa_{rad}\frac{C_R}{C_F}T^{3}x+\kappa_{coll}\frac{C_R}{C_F}T^{2}$$ where $T$ is local temperature and $\kappa_{rad},\kappa_{coll},C_R,C_F$ coefficients depend on jet flavor (quark or gluon) and its energy [@solana]. In the simulations, we assumed $\kappa_{rad}=4,\kappa_{coll}=2.5,C_R/C_F=1$. We assume that $\frac{\mathrm{d} E}{\mathrm{d} x}$ is small compared to the jet energy and the jet is not modified by the medium.\
The code is written using NVIDIA CUDA programming framework purposed for parallel computing on graphical processing units (GPU). Our implementation [@pub1; @pub2; @pub3; @pub4] speeds up more than 2 orders of magnitude in comparison to the equivalent CPU execution.
Results
=======
We simulated propagation and energy loss $\left(\frac{\mathrm{d} E}{\mathrm{d} x}\right)$ of a high-energy parton through the medium and analyzed the evolution of the system. The initial condidtions were based on ellipsoidal flow [@ellip] with initial simulation time $t_0=1$ fm/c.
![\[velocity\]Velocity profile in plane $xy$ at $z=0$ after $t=2.4$ fm/c.](plots/jetenergy.png){width="14pc"}
![\[velocity\]Velocity profile in plane $xy$ at $z=0$ after $t=2.4$ fm/c.](plots/jetvelocity.png){width="14pc"}
![\[evo\]Evolution of the system: energy density 1D sections at $y=z=0$ at different points of time (from 1.2 fm/c to 2 fm/c after simulation start).\
Simulation parameters: d$x=0.1$ fm, d$t=0.02$ fm/c, grid size $256\times 256\times 256$, EOS $p=e/3$.](plots/evoslim.png){width="16pc"}
In figures \[energy\] and \[velocity\], there is a clean signal of jet propagation in the medium: a Mach cone is visible both in energy and velocity distributions. Figure \[evo\] shows the evolution of energy density with time. There is a clear indication that jet energy loss has a significant impact on the system. The area where the jet interacted with system has much larger energy density compared to the surrounding matter and the difference increases with time.\
In the future we plan to employ more realistic jet energy loss algorithm and add freeze–out to study the effect on the experimental observables: elliptic flow and higher flow harmonics.
[ll]{} Parameter&Value\
d$x$ & 0.1 fm\
d$t$ & 0.02 fm/c\
grid size & $256\times 256\times 256$\
EOS & $p=e/3$\
References {#references .unnumbered}
==========
[11]{}
Tachibana Y and Hirano T 2014 [*Phys. Rev.*]{} C [**90**]{} 021902
Zhang X and Shu C W 2012 [*J. Comput. Phys.*]{} [**231**]{}(5) 2245-58
Toro E F 2006 [*Appl. Numer. Math.*]{} [**56**]{}(10-11) 1464-79
Casalderrey-Solana J, Gulhan D C, Milhano J G, Pablos D and Rajagopal K 2014 [*J. High Energy Phys.*]{} JHEP10(2014)019
Porter-Sobieraj J, Cygert S, Kikoła D, Sikorski J and Słodkowski M 2015 [*Concurrency and Computation: Practice and Experience*]{} [**27**]{}(6) 1591-602
Cygert S, Kikoła D, Porter-Sobieraj J, Sikorski J and Słodkowski M 2014 [*Lect. Notes Comput. Sc.*]{} [**8384**]{} 500-9
Sikorski J, Cygert S, Porter-Sobieraj J, Słodkowski M, Krzyżanowski P, Książek N and Duda P 2014 [*. J. Phys. Conf. Ser.*]{} [**509**]{} 012059
Cygert S, Porter-Sobieraj J, Kikoła D, Sikorski J and Słodkowski M 2013 [*Fed. Conf. on Computer Science and Information Systems (Kraków)*]{} pp 441-6
Sinyukov Y M and Karpenko I A 2005 [*Preprint*]{} nucl-th/0505041
|
---
abstract: 'We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Gröbner bases, factorization or sub-resultant computations.'
address:
- |
Mathematisches Institut\
Universität Tübingen\
Auf der Morgenstelle 10\
72076 Tübingen\
Germany
- |
Mathematisches Institut\
Universität Tübingen\
Auf der Morgenstelle 10\
72076 Tübingen\
Germany
author:
- Simon Keicher and Thomas Kremer
title: A test for monomial containment
---
[^1]
Introduction
============
Let $\KK$ be a field. Given an ideal $I\subseteq \KT{r}$, the [*monomial containment problem*]{} is to decide whether $I$ contains a monomial. Equivalently, one is interested in whether the intersection $V(I)\cap \TT^r$ of the zero set $V(I)\subseteq \b\KK^r$ with the algebraic torus $\TT^r := (\b\KK^*)^r$ is empty. The monomial containment problem occurs frequently when determining tropical varieties [@BoJeSpeStu] or when determining GIT-fans [@Ke]. The usual approach is via Gröbner bases: $I$ contains a monomial if and only if the saturation $I:(T_1\cdots T_r)^\infty$ contains $1\in \KT{r}$. This can also be decided by a radical membership test: $I$ contains a monomial if and only if $T_1\cdots T_r\in \sqrt{I}$.
In the present paper, we provide a direct approach involving neither Gröbner basis computations nor (sub-)resultants or factorization of polynomials. We consider more generally the following problem: given a polynomial $g\in \KT{r}$, prove or disprove the existence of an element $x\in \b\KK^r$ such that $$\begin{aligned}
\label{eq:solution}
f(x)\ =\ 0\quad
\text{for all }
f\in I,\qquad
g(x) \ne\ 0.\end{aligned}$$ Clearly, setting $g:= T_1\cdots T_r\in \KT{r}$ in , the existence of such $x$ is equivalent to the monomial containment problem. Our algorithm, Algorithm \[algo:containsmonomial\], proceeds in three steps:\[list:intro\]
1. Compute finite subsets $S_1,\ldots,S_m\subseteq \KT{r}$ that are in [*triangular shape*]{} and polynomials $g_1,\ldots,g_m$ such that the solutions of are preserved, i.e., the zero sets satisfy $$V(I)\setminus V(g)
\ \ =\ \
\bigcup V(S_i)\setminus V(g_i)
\ \ \subseteq\ \ \b\KK^r.$$
2. Making certain variables $T_j$ invertible, we obtain a function field $\LL$ and an embedding $\iota\colon \KT{r}\to \LL[T_{k_1},\ldots,T_{k_s}]$ such that the embedded equations $\iota(S_i)$ are [*dense*]{}, i.e., each variable $T_{k_j}$ corresponds to an equation.
3. Then an element $x\in \b\KK^r$ satisfying exists if and only if the minimal polynomial of the class $\b{\iota(g_i)} \in \LL[T_{k_1},\ldots,T_{k_s}]/\< S_i \>$ is not a monomial for some $i$.
Experiments with our implementation of Algorithm \[algo:containsmonomial\] suggest that it is competitive for certain classes of input; for instance, it usually beats the Gröbner basis approach when a solution exists, i.e., the ideal is monomial-free.
Note that the idea behind step (i) of the algorithm is quite common and similar concepts have been used by several authors for a more explicit study or even the explicit computation of solutions. See, e.g., [@AuLaMoMa; @AuMoma; @Chen; @Ritt; @Wu] for a series of papers with Gröbner basis-free algorithms for systems of equations. The methods of Wang [@Wa], Thomas [@thomas; @thomas2] as well as Bächler, Gerdt, Lange-Hegermann and Robertz [@BaeGeLaRo] can also deal with systems of equations and inequalities. They determine the solutions of such systems by means of certain triangular sets called [*simple systems*]{}; their computation involves sub-resultant computations. All algorithms, including ours in step (i), share the concept of [*triangular sets*]{}, certain finite subsets $S_i\subseteq \KT{r}$ such that $V(I)= \bigcup V(S_i)$ holds. The $S_i$ then give insight into the structure of the solution set $V(I)\subseteq \b\KK^r$. As we are only interested in solvability of , we will only need triangular sets with weaker properties but which can be computed more efficiently.
The structure of this paper is as follows. In Section \[sec:algotriag\], we show how to decompose the given ideal into a list of triangular sets with sufficient properties for our solvability test; this is step (i) in the previous list. Section \[sec:algomonomial\] is devoted to steps (ii) and (iii), i.e., we show how to reduce the problem to a dense system over a function field and how to determine the solvability of such a system by means of minimal polynomial computations. Explicit algorithms are given in each section. In Section \[sec:implem\], we present our algorithm for the monomial containment problem. We compare the experimental running time of the `perl` implementation [@monomtest:implem] of the algorithm to the Gröbner basis approach as well as to the methods of [@BaeGeLaRo; @Wa].
This paper builds on [@kremer]. We would like to thank Jürgen Hausen for helpful discussions.
Triangular shape {#sec:algotriag}
================
In this section, we treat item (i) of the list on page , i.e., we decompose a system as in with an ideal $I\subseteq \KT{r}$ and a polynomial $g\in \KT{r}$ into a list of finite sets of polynomials that are in [*triangular shape*]{}. We show how to compute this decomposition by iteratively applying a set of operations that do not change the solvability of .
We first define the notion of [*triangular shape*]{}. In the literature, they are also called [*triangular sets*]{} [@AuLaMoMa; @AuMoma; @DeLo; @GrePfi].
\[def:triag\] Fix the lexicographical ordering $T_1>\ldots>T_r$ on $\KT{r}$. We call polynomials $f_1,\ldots,f_s\in \KT{r}$ of [*triangular shape*]{} if for each $f_j$, there is $1\leq k(f_j)\leq r$ such that
1. we have $k(f_1)<\ldots <k(f_s)$,
2. $f_j\in \KK[T_{k(f_j)},\ldots, T_r]\setminus\KK[T_{k(f_j)+1},\ldots,T_r]$ holds for each $1\leq j\leq s$.
We denote by $\deg_{T_i}(f)$ the [*($T_i$-)degree*]{} of a polynomial $f\in \KT{r}$ considered as an element of the univariate polynomial ring $\KK[T_j;\ j\ne i][T_i]$. Moreover, we write $$\LC_{k(f_i)}(f_i)
\ \ \in\ \
R_{<k(f_i)}
\ :=\
\KK\left[T_{k(f_i)+1},\ldots,T_r\right].$$ for the leading coefficient of the polynomial $f_i$ considered in the ring $R_{<k(f_i)}[T_{k(f_i)}]$.
We now introduce the concept of [*(semi-) triangular systems*]{}. Assume $I$ is generated by polynomials $f_1,\ldots,f_s\in \KT{r}$. We sort them into two sets (and keep track of the inequality $g$): polynomials that are already in triangular shape $\FFt$ and remaining polynomials $\FFs$.
A [*semi-triangular system*]{} (of equations) is a tuple $(\mathcal F_\square, \FFt,k,g)$ consisting of finite subsets $\mathcal F_\square$, $\FFt\subseteq \KT{r}$, an integer $0\leq k\leq r$ and a polynomial $g\in \KT{r}$ such that
1. $\FFt$ is of triangular shape,
2. we have $\LC_{k(f)}(f)\mid g$ for all $f\in \FFt$,
3. the set $\{1,\ldots,k\}$ contains $\{k(f);\ f\in \FFt\}$,
4. for all $f\in \mathcal F_\square$ and each $1\leq i\leq k$ we have $\deg_{T_i}(f)=0$.
Moreover, we call a semi-triangular system $(\mathcal F_\square, \FFt,k,g)$ a [*triangular system*]{} if $\mathcal F_\square \subseteq \KK$ holds.
\[ex:triag\] Define in $\KK[T_1,T_2,T_{3}]$ the subsets $\mathcal F_\square := \emptyset$ and $\FFt := \{f_1,f_2,f_3\}$ where the $f_i$ and $k(f_i)$ are $$\begin{array}{rcrrcl}
f_1
&:=&
T_1^2 - (T_2 + T_3)T_1,
&
\qquad
k(f_1)
&=&
1,
\\
f_2
&:=&
T_2^2- T_3\qquad\ ,
&
k(f_2)
&=&
2,
\\
f_3
&:=&
T_3^2 - T_3,
&
k(f_3)
&=&
3.
\end{array}$$ Then $\FFt$ is of triangular shape and $(\mathcal F_\square, \FFt, 3, T_1T_2T_3)$ is a triangular system.
A list $\SSS$ of semi-triangular systems is called a [*triangle mush*]{}. Two triangle mushes $\SSS$ and $\SSS'$ are [*equivalent*]{} if we have $V(\SSS) = V(\SSS')$ with the [*solutions*]{} $$V(\SSS)
\ :=\!
\bigcup_{(\mathcal F_\square, \FFt,k,g)\in\SSS}
\!
V\left(
\mathcal F_\square \cup \FFt
\right)
\setminus
V(g)
\ \ \subseteq\ \
\b\KK^r.$$ For the case of a single element $\SSS = \{S\}$, we will use the same notions for $S$ instead of $\SSS$.
\[ex:mush\] Consider the triangle mush $\SSS := \{(\FFs, \emptyset, 0, g)\}$ in $\KT{4}$ where $g:=T_1T_2T_3$ and $\FFs$ consists of the two polynomials $$f_1
\ :=\
(T_3-T_1)(T_3-T_2)T_2
,
\qquad
f_2
\ :=\
(T_1+T_2-T_3)T_4.$$ Going through the different cases, one directly verifies that $V(\SSS)\subseteq \b\KK^4$ consists of all points $(x_1,x_2,x_1,0)$ and $(x_1,x_2,x_2,0)\in \b\KK^4$ where $x_i\in \b\KK^*$. We will continue this example in \[ex:containsmon\].
Given a triangle mush $\SSS$, we are interested in operations that transform $\SSS$ into an equivalent triangle mush $\SSS'$ that consists of triangular systems.
\[con:mushops\] Let $\SSS := \{(\mathcal F_\square, \FFt,k,g)\}$ consist of a semi-triangular system. Each of the following operations produces an equivalent triangle mush $\SSS'$.
1. [*Case-by-case analysis*]{}: If $f\in \KK[T_{k+1},\ldots,T_r]$ and $h\in \KT{r}$ are such that $g\mid h$ and $h\mid fg$, then one may choose $$\SSS'
\ :=\
\left\{
(\mathcal F_\square\cup\{f\}, \FFt,k,g),\,
(\mathcal F_\square, \FFt,k,h)
\right\}.$$
2. [*Polynomial division*]{}: Consider $f,h\in \FFs$ and $b\in \KT{r}$ with $b\mid g$. Assume that for some $j\in \ZZZ$ we have $$b^jf\ =\ ah + u,\qquad
a,u\ \in\ \KK[T_{k+1},\ldots,T_r]$$ where $b := \LC_{T_{k+1}}(h)$ and $\deg_{T_{k+1}}(u)<\deg_{T_{k+1}}(h)$. Then we choose the triangle mush $$\SSS'\ :=\
\{
(\mathcal F_\square\setminus\{f\}\cup \{u\}, \FFt,k,g)
\}.$$
3. [*Unused variable*]{}: If $k<r$ and $\deg_{T_{k+1}}(f)=0$ holds for each $f\in \FFs$, then we may choose $$\SSS'\ :=\
\{
(\mathcal F_\square, \FFt,k+1,g)
\}.$$
4. [*Sort polynomial*]{}: If $k<r$ holds and there is exactly one polynomial $f\in \FFs$ with $\deg_{T_{k+1}}(f)\ne 0$ and $\LC_{k(f)}(f)\mid g$, then we may choose $$\SSS'\ :=\
\left\{
(\mathcal F_\square\setminus\{f\}, \FFt\cup \{f\},k+1,g)
\right\}.$$
5. [*Last polynomial*]{}: Assume $k<r$ and there is exactly one polynomial $f\in \FFs$ with $\deg_{T_{k+1}}(f)\ne 0$. For $-1\leq j\leq d$, we write $$\begin{aligned}
f
&=&
\sum_{i=0}^d a_iT_{k+1}^i
,\qquad
f_j
\ \ :=\ \
\sum_{i=0}^j a_iT_{k+1}^i
\ \in\ R_{<k+1}[T_{k+1}],
\\
\FFt^j
&:=&
\FFt
\cup
\{f_j\},
\qquad
\FFs^j
\ \ :=\ \
\left(
\FFs
\setminus
\{f\}
\right)
\cup
\{a_{j+1},\ldots,a_d\}.\end{aligned}$$ Then we may choose $$\begin{gathered}
\SSS'\ :=\
\bigl\{
(\mathcal F_\square^1, \FFt^1,k+1,ga_1),
\ldots,
(\mathcal F_\square^d, \FFt^d,k+1,ga_d),
\\
(\mathcal F_\square^{-1}, \FFt,k+1,g)
\bigr\}.\end{gathered}$$
One directly checks that in all cases $\SSS'$ is a triangle mush. For (i), each $x\in V(\SSS)$ either satisfies $f(x)=0$ and $g(x)\ne 0$ or we have $f(x)\ne 0$ and $h\mid fg$ implies $h(x)\ne 0$, i.e., $x\in V(\SSS')$. The inclusion $V(\SSS')\subseteq V(\SSS)$ is clear from $g\mid h$. We come to (ii). Each $x\in V(\SSS)$ satisfies $$u(x)
\ \ =\ \
b(x)^jf(x) - a(x)h(x)
\ \ =\ \
0.$$ For the reverse inclusion, we use $b\mid g$ to obtain $b(x)\ne 0$. Consequently, we may infer $f(x)=0$ from $$b(x)^jf(x)
\ \ =\ \
(b^jf)(x)
\ \ =\ \
a(x)h(x) + u(x)
\ \ =\ \
0.$$ Operations (iii) and (iv) are clear. For (v), we define the following triangle mushes for $0\leq l\leq d$: $$\SSS_l\, :=\,
\{
(\FFs^l\cup \{f_l\}, \FFt,k,g)
\},
\qquad
\DDD_l\, :=\,
\{
(\FFs^j,\FFt^j,k+1,ga_j);\ l<j\leq d)
\}.$$ Observe that by an application of operation (i), we obtain an equality of solutions $$V(\SSS_l)
\ \ =\ \
V\left(
\left\{
\left(\FFs^{l-1}\cup \{f_l\},\FFt,k,g\right),\
\left(\FFs^l\cup \{f_l\},\FFt,k,ga_l\right)
\right\}
\right).$$ As the ideal $\<\FFs^{l-1}\cup \{f_l\}\>$ equals $\<\FFs^{l-1}\cup \{f_{l-1}\}\>$ and by an application of operation (iv), we obtain $$\begin{aligned}
V(\SSS_l)
&=&
V\left(
\left\{
\left(\FFs^{l-1}\cup \{f_{l-1}\},\FFt,k,g\right),\
\left(\FFs^l,\FFt^l,k+1,ga_l\right)
\right\}
\right)
\\
&=&
V\left(\SSS_{l-1}\cup (\DDD_{l-1}\setminus\DDD_l)\right).\end{aligned}$$ Adding the equations stored in $\DDD_l$ on both sides does not change the solution set, i.e., $V\left(\SSS_{l}\cup \DDD_{l}\right)$ is equal to $V\left(\SSS_{l-1}\cup \DDD_{l-1}\right)$. Iteratively, we obtain $V\left(\SSS_{d}\cup \DDD_{d}\right)
=V\left(\SSS_{0}\cup \DDD_{0}\right)$. Moreover, because of $f_0 = a_0$ and operation (iii): $$V(\SSS_0)
\ \ =\ \
V\left((\FFs^{-1},\FFt,k,g)\right)
\ \ =\ \
V\left((\FFs^{-1},\FFt,k+1,g)\right).$$ We conclude that $V(\SSS)$ equals $V(\SSS_d\cup \DDD_d)=
V(\SSS_0\cup \DDD_0)$ which in turn is the same as the solution set $V(\SSS')$.
The next algorithm transforms a triangle mush into an equivalent triangle mush consisting only of triangular systems. Given a triangular system $(\FFs,\FFt,k,g)$, the idea is to reduce $T_{k+1}$-degrees of an element $f$ of the unsorted polynomials $\FFs$ by successive polynomial divisions; afterwards, we move $f$ into the set of sorted polynomials $\FFt$.
Given a finite set of polynomials $\FF\subseteq \KT{r}$, its [*reduction*]{} is a finite subset $\red(\FF)\subseteq \KT{r}$ such that $$\begin{gathered}
\LT(f_1)\ \nmid\ \LT(f_2)
\quad\text{for all }
f_1,f_2\in \red(\FFs),
\\
\<\LT(\FF)\>\ \subseteq\ \<\red(\LT(\FF))\>,
\qquad\qquad
\<\FF\>\ =\ \<\red(\FF)\>\end{gathered}$$ where we denote by $\LT(f)$ or $\LT(M)$ the leading term of a polynomial $f$ or set of polynomials $M$ with respect to the ordering defined in Section \[sec:algotriag\]. Computing the reduction of $\FF$ means successively applying the division algorithm to the elements of $\FF$, see, e.g., [@CoLiOSh].
\[algo:triang\] [*Input:* ]{} a triangle mush $\SSS$ in $\KT{r}$.
- While there is $S:=(\mathcal F_\square, \FFt,k,g)\in\SSS$ with $k<r$, do:
- Replace $\mathcal F_\square$ by its reduction $\red(\mathcal F_\square)$.
- If there is $f\in \mathcal F_\square$ with $\deg_{T_{k+1}}(f)>0$, then:
- If there is $h\in \mathcal F_\square\setminus\{f\}$ with $\deg_{T_{k+1}}(h)>0$, then:
- Perform a polynomial division of $f$ by $h$ in the univariate polynomial ring $R:=\KK(T_{k+2},\ldots,T_r)[T_{k+1}]$ to obtain $$f \ =\ a'h + u'\ \in\ R.$$
- Set $b:=\LC_{k+1}(h)\in \KK[T_{k+2},\ldots,T_r]$ and $j:=\deg_{T_{k+1}}(h)+1\in \ZZZ$. With $a :=b^ja'$ and $u:=b^ju' \in \KK[T_{k+1},\ldots,T_r]$ we then have $$\qquad\qquad\qquad
b^jf\ =\ ah +u\ \in\ \KK[T_{k+1},\ldots,T_r].$$
- Redefine $\SSS := (\SSS\setminus \{S\}) \cup \{S',S''\}$ where $$\begin{aligned}
\qquad\qquad\qquad
S'
&:=&
(\mathcal F_\square\setminus\{f\}\cup \{u\}, \FFt,k,bg),\\
S''
&:=&
(\mathcal F_\square\cup\{b\}, \FFt,k,g).
\end{aligned}$$
- Otherwise, if there is no such $h$, then:
- Redefine $\SSS := (\SSS\setminus \{S\}) \cup \{S',S_{1},\ldots,S_{d}\}$ where with the notation of Construction \[con:mushops\] (v): $$\begin{aligned}
\qquad\qquad\qquad
S' &:=& \left(\mathcal F_\square^{-1}, \FFt,k+1,g\right),
\\
S_j &:=& \left(\mathcal F_\square^{j}, \FFt^{j},k+1,ga_j\right).
\end{aligned}$$
- Otherwise, if there is no such $f$, then:
- Redefine $\SSS := (\SSS\setminus \{S\}) \cup \{S'\}$ where $S' := (\mathcal F_\square, \FFt,k+1,g)$.
- Define $\SSS' := \SSS$.
[*Output:* ]{} $\SSS'$. Then $\SSS'$ is a triangle mush that is equivalent to $\SSS$ and consists of triangular systems.
Note that we use only operations described in Construction \[con:mushops\]; for instance, the replacement of $\SSS$ by $(\SSS\setminus \{S\}) \cup \{S',S''\}$ is an application of, first, operation (i) and then operation (ii). Therefore, $\SSS'$ is equivalent to $\SSS$. As each $S:=(\FFs,\FFt,k,g)\in \SSS'$ satisfies $k=r$, each element of $\FFs$ is constant, i.e., $S$ is triangular.
It remains to show that Algorithm \[algo:triang\] terminates. To this end, consider the infinite digraph $G'=(V',E')$ where $V'$ is the set of all semi-triangular systems over $\KT{r}$ and, given vertices $S_1,S_2\in V$, the edge $(S_1,S_2)\in E'$ exists if and only if Algorithm \[algo:triang\] replaces $S_1$ within a single iteration of the while-loop by a triangle mush $\SSS'$ with $S_2\in \SSS'$. Let $G=(V,E)$ be the subgraph induced by all semi-triangular systems that are reachable by a path starting in $\SSS$.
Consider a path $(S_1,S_2,\ldots)$ in $G$, i.e., $S_i\in V$ and $(S_i,S_{i+1})\in E$ for all $i$. We write $S_i = (\FFs^i,\FFt^i,k_i,g_i)$. By construction, $k_i\leq k_{i+1}\leq r$ holds for all $i$. This means there is $i_1\in \ZZ_{\geq 1}$ such that $k_{i+1}=k_i$ for all $i\geq i_1$ and Algorithm \[algo:triang\] will perform the polynomial division $b^jf=ah+u$, i.e., operation (ii) of Construction \[con:mushops\], for each such $S_i$. Since always $\deg_{T_{k_i+1}}(b)=0$ holds, we have $\deg_{T_{k_i+1}}(f)>\deg_{T_{k_i+1}}(u)$ and the reduction step only reduces $T_{k_i+1}$-degrees, the sequence $$(N_i)_{i\geq i_1},\qquad
N_i\ :=\ \sum_{f\in \FFs^i}\deg_{T_{k_i+1}}(f)\ \in\ \ZZZ$$ is monotonically decreasing. As $N_i \in \ZZZ$ holds, this sequence either is finite or becomes stationary. Assume the latter holds, i.e., there is $i_2\in \ZZ_{\geq i_1}$ such that $N_i = N_{i+1}$ is valid for all $i\geq i_2$. This implies, that for all $i\geq i_2$ in the polynomial division step only the “$b$-part” will be added, i.e., $$\FFs^{i+1}\ =\ \FFs^i \cup \{b\}.$$ In particular, the ideal $\<\LT(\FFs^i)\>$ is contained in $\<\LT(\FFs^{i+1})\>$ for each $i\geq i_2$. As $\KT{r}$ is noetherian, the chain $$\left\<\LT\left(\FFs^{i_2}\right)\right\>
\ \subseteq\
\left\<\LT\left(\FFs^{i_2+1}\right)\right\>
\ \subseteq\
\ldots$$ becomes stationary, i.e., there is $i_3\in \ZZ_{\geq 1}$ such that $\<\LT(\FFs^i)\>=\<\LT(\FFs^{i+1})\>$ holds for all $i\geq i_3$. Moreover, as $b = \LC_{k+1}(h)$ holds and $h\in \red(\FFs^i)$, we have $$\LT(b)
\ \notin\
\left\<\LT\left(\red\left(\FFs^i\right)\right)\right\>
\ \supseteq\
\left\<\LT\left(\FFs^i\right)\right\>.$$ Then $b$ cannot be an element of $\FFs^{i+1}$ for $i\geq i_3$, a contradiction. Thus, the sequence $(N_i)_i$ is finite. In turn, this forces the $(S_1,S_2,\ldots)$ to be finite and acyclic.
Since each vertex $S\in V$ is adjacent to only finitely many vertices, the previous argument shows that $G$ is a finite tree. In particular, the while-loop in Algorithm \[algo:triang\] will be executed at most $|G|$ times for each vertex $S\in V$, i.e., the algorithm terminates.
\[rem:simplesystems\] Algorithm \[algo:triang\] is similar to the decomposition into [*simple systems*]{} used in [@BaeGeLaRo]. Note, however, that they are interested in special properties (e.g., disjointness) of this decomposition whereas ours is weaker but needs not use operations like $\gcd$ or subresultant computations.
An example computation with Algorithm \[algo:triang\] will be performed at the end of the next section in Example \[ex:containsmon\].
Solvability {#sec:algomonomial}
===========
We now come to steps (ii) and (iii) in the list on page : as before, we assume we are given an ideal $I=\<f_1,\ldots,f_s\>\subseteq\KT{r}$ and a polynomial $g\in \KT{r}$ and want to answer the question whether there is $x\in \b\KK^r$ satisfying .
Using Algorithm \[algo:triang\] of the previous section with input $I$ and $g$, we obtain an equivalent triangle mush $\SSS$ that consists of triangular systems. Note that we can replace each system $(\FFs, \FFt, k, g)\in \SSS$ with $\FFs = \{0\}$ by the equivalent system $(\emptyset, \FFt, k, g)$; systems with $\FFs \cap \KK^*\ne \emptyset$ clearly are not solvable. Then can be rephrased as the question, whether there is $x\in \b\KK^r$ such that $$f(x)\ =\ 0
\qquad
\text{for all}\
f\in \FFt
,\qquad
g(x)\ \ne \ 0$$ holds for some $(\emptyset, \FFt, k, g)\in \SSS$. Consequently, it suffices to present methods for the case $\SSS=\{S\}$ of a single triangular system. Here is an overview of the steps to test whether $V(S)\ne \emptyset$ holds: $$\xymatrix@R=.1pt@C=8mm{
\KT{r}
&
\LL[T_{k_1},\ldots,T_{k_s}]
&
\LL[T_{k_1},\ldots,T_{k_s}]
&
\LL[T_{k_1},\ldots,T_{k_s}]/\<\FFt'\>
\\ \text{\rotatebox{90}{$\subseteq$}}
&
\text{\rotatebox{90}{$\subseteq$}}
&
\text{\rotatebox{90}{$\subseteq$}}
&
\text{\rotatebox{90}{$\in$}}
\\ S
\ar@{|->}[r]^{\ref{prop:fieldchange}}
&
\iota(S)
\ar@{|->}[r]^(.4){\ref{prop:normalize}}
&
(\emptyset, \FFt', k',g')
\ar@{|->}[r]^(.6){\ref{prop:solvable}}
&
\b{g'}
\\ &
\text{\tiny dense}
&
\text{\tiny dense, monic}
&
\text{\tiny min.~polyn.~monomial?}
}$$ Here, $\LL$ is a suitable function field. The following proposition reduces the treatment of a triangular system in $\KT{r}$ to a triangular, [*dense*]{} system in $\LL[T_{k_1},\ldots,T_{k_s}]$, i.e., a triangular system $(\emptyset, \{f_1,\ldots,f_s\}, k,g)$ such that the set $\{k_1,\ldots,k_s\}$ coincides with $\{k(f_1),\ldots,k(f_s)\}$.
\[prop:fieldchange\] Consider a triangular system $S := (\emptyset, \FFt, k, g)$ in $\KT{r}$. Write $\FFt = \{f_1,\ldots,f_s\}$ and let $k_i := k(f_i)\in \ZZ_{\geq 1}$ be as in Definition \[def:triag\]. Under the canonical embedding $$\iota\colon
\KT{r}\, \to\,
\LL[T_{k_1},\ldots,T_{k_s}],\qquad
\LL\, :=\,
\KK\left(T_{i};\ i\not\in\{k_1,\ldots,k_s\}\right)$$ we obtain a triangular system $\iota(S) := (\emptyset, \iota(\FFt),s,\iota(g))$ that is dense in the polynomial ring $\LL[T_{k_1},\ldots,T_{k_s}]$. Moreover, we have $$V(S)\ \ne \ \emptyset
\qquad{\Longleftrightarrow}\qquad
V(\iota(S))\ \ne\ \emptyset.$$
For the proof of Proposition \[prop:fieldchange\] we recall from [@ZaSa Ch. VI] the generalization of evaluation homomorphisms; we will need this to control the elements in $\b\LL$. A [*place*]{} is a $\b\KK$-homomorphism $\eps\colon R_{\varphi}\to \overline{\KK}$ with a subring $R_\eps\subseteq \overline\LL$ such that $$x\,\in\,\overline\LL\setminus R_\eps
\quad\Longrightarrow\quad
x^{-1}\in R_\eps
\ \text{ and }\
\eps(x^{-1})\,=\,0.$$
Given $x\in \overline \KK^{r-s}$, denote by ${\varepsilon}_x'\colon \b\KK[T_i;\ i\notin \{k_1,\ldots,k_s\}]\to \overline \KK$ the evaluation homomorphism. According to [@ZaSa Thm. 5 in VI.4], we have $$\xymatrix{
\b\KK[T_i;\ i\notin \{k_1,\ldots,k_s\}]
\ar@{}|(.72)\subseteq[r]
\ar[dr]_{\eps_{x}'}
&
R_{\eps_x}
\ar@{}|(.5)\subseteq[r]
\ar[d]^{\eps_x}
&
\b\LL
\\
&
\b\KK
&
}$$ with a place ${\varepsilon}_x\colon R_{{\varepsilon}_x}\to\overline \KK$ extending $\eps_x'$. Moreover, we define the [*domain*]{} of $t=(t_1,\ldots,t_s)\in \overline \LL^s$ as the intersection $$\Dom(t)\ :=\
\bigcap_{i=1}^s\Dom(t_i)
,\qquad
\Dom(t_i)\ :=\
\left\{
y\in \overline \KK^{r-s};\ t_i\in R_{{\varepsilon}_y}
\right\}.$$
\[lem:place\] In the situation of Proposition \[prop:fieldchange\], assume we have $k_1 =1,\ldots,k_s =s$. Then the following claims hold.
1. Consider $x\in \b\KK^{r-s}$ and $t_1,\ldots,t_n\in \b\LL$ satisfying $\eps_x(t_1\cdots t_n) = 0$. Then there is $1\leq j\leq n$ such that $\eps_x(t_j)=0$.
2. For each $t\in V(\iota(S))\subseteq \b\LL^s$ and each $x\in \Dom(t)\subseteq \overline \KK^{r-s}$, we have $(\eps_x(t_1),
\ldots,
\eps_x(t_s),x)\in \overline{V(S)}$ where the closure is taken in $\overline\KK^{r}$.
3. Given $x\in V(S)\subseteq \b\KK^r$, write $x=(x'',x')$ with $x'\in \b\KK^{r-s}$, $x''\in \b\KK^{s}$. Then there is $t\in V(\iota(S))\subseteq \b\LL^s$ such that $$x'\,\in\, \Dom(t)\,\subseteq\, \b\KK^{r-s}\qquad\text{and}\qquad
(\eps_{x'}(t_1),\ldots,\eps_{x'}(t_s))\,=\,x''.$$
For (i), we relabel $t_1,\ldots,t_n$ such that there is $k\in \ZZZ$ with $t_i\in R_{\eps_x}$ for all $i\leq k$ and $t_i\notin R_{\eps_x}$ for $i>k$. By definition of places, $\eps_x(t_i^{-1})=0$ for all $i>k$ and thus $$\prod_{i=1}^k \eps_x(t_i)
\,=\,
\eps_x\left(
\prod_{i=1}^n t_i
\prod_{i=k+1}^n t_i^{-1}
\right)
\,=\,
\eps_x
\left(\prod_{i=1}^n t_i\right)
\left(\prod_{i=k+1}^n \eps_x\left(t_i^{-1}\right)\right)
\,=\,
0.$$
For (ii), given $f\in \sqrt{\<\FFt\>}:g$, we have $\iota(f)\in \sqrt{\<\iota(\FFt)\>}:\iota(g)$, which means $\iota(f)(t)=0$. Write $f = \sum_{\nu} a_{\nu} T^{\nu}$. From $$f(
\eps_x(t_1),
\ldots,
\eps_x(t_s)
,x
)
\ =\
\sum_\nu
a_\nu
\prod_{i=1}^{s} \eps_x(t_i)^{\nu_i}
\prod_{j=s+1}^{r} x_{j-s}^{\nu_j}
\ =\
\eps_x
\left(
\iota(f)(t)
\right)
\ =\
0$$ we infer that $(\eps_x(t_1),
\ldots,
\eps_x(t_s),x)\in \b \KK^r$ is an element of the closure $\overline{V(S)}=V(\sqrt{\<\FFt\>}:g)$ in $\b\KK^r$.
We come to (iii). We first show by (finite) induction on $0\leq m\leq s$, that there are $t_{m+1},\ldots,t_s\in \b \LL$ such that for the evaluation homomorphism $$\theta_m\colon
\KK[T_1,\ldots,T_{r}]\,\to\,
\b\LL[T_1,\ldots,T_m],\qquad
T_j\,\mapsto\,
\begin{cases}
t_j, & m<j\leq s,\\
T_j, & \text{else}
\end{cases}$$ we have $\<f_{m+1},\ldots,f_s\>\subseteq \ker(\theta_m)$ and $\eps_{x'}(t_j) = x_j$ holds for each $m<j\leq s$. Nothing is to prove for $m=s$. Assume now that this claim holds for a fixed $1\leq m\leq s$; we show that it also holds for $m-1$. Since we have $\LC_{m}(f_i)\mid g$, $g(x)\ne 0$ and $\eps_{x'}(t_j)=x_j$ for $m<j\leq s$, setting $a := \LC_m(f_m)$, we obtain $$\eps_{x'}
\left(
\theta_m(a)
\right)
\ =\
a(x)
\bigm|
\eps_x(g)
\ =\
g(x)
\ \ne\
0.$$ In particular, $\theta_m(a)\ne 0$. Therefore, the non-zero univariate polynomial $f_m':=\theta_m(f_m)\in \b\LL[T_m]$ can be decomposed into linear factors $$f_m'
\ =\
c\prod_{j=1}^n(T_m - t_{mj})\qquad
\text{with}\
t_{mj}\,\in\,\b\LL,\quad
c\,\in\,\b\LL^*.$$ Note that $c = \theta_m(a)$ holds and thus $\eps_{x'}(c)\ne 0$. Moreover, using again $\eps_{x'}(t_j)=x_j$ for $j>m$ and $f_m'=\theta_m(f_m)$, we have $\eps_{x'}(f_m'(x_m)) = f_m(x)=0$ where the vanishing is due to $x\in V(S)$. The identity $$0
\ =\
\eps_{x'}(f_m'(x_m))
\ =\
\eps_{x'}
\left(
c\prod_{j=1}^n(x_m - t_{mj})
\right)$$ together with statement (i) provide us with $1\leq j\leq n$ such that $\eps_{x'}(t_{mj}) = x_m$. Defining $t_m := t_{mj}$, the elements $t_m,\ldots,t_s\in \b\LL$ satisfy the claims: we have $\<f_m,\ldots,f_s\>\subseteq \ker(\theta_{m-1})$ since $\theta_{m-1}(f_m) = f_m'(t_m)=0$ and $\eps_{x'}(t_m) = x_m$ holds.
Using this argument, we now have a map $\theta_0$ such that both $\<\FFt\>\subseteq \ker(\theta_0)$ and $\eps_{x'}(t_m)=x_m$ hold. Setting $t := (t_1,\ldots,t_s)$, we obtain $$t\ \in\ V(\FFt)\setminus V(g)\ =\ V(\iota(S))\ \subseteq\ \b\LL$$ because $f_m(t)=\theta_0(t_m)=0$ for each $1\leq m\leq s$ and $\eps_{x'}(\theta_0(t))=g(x)\ne 0$ implies in particular that $\theta_0(t)=g(t)\ne 0$. By construction, $\eps_{x'}(t)=x''$ holds.
Clearly, the system is dense. By Lemma \[lem:place\] (iii), $V(S)\ne \emptyset$ implies that also $V(\iota(S))$ is non-empty. If for each $t\in V(\iota(S))$, there is $x\in \Dom(t)$, then Lemma \[lem:place\] (ii) ensures $\b{V(S)}\ne \emptyset$ and therefore $V(S)\ne \emptyset$.
It thus remains to prove that $\Dom(t)\ne \emptyset$. Let $1\leq j\leq s$ be an integer. If $t_j = 0$ holds, clearly $\Dom(t_j)=\overline\KK^{r-s} \setminus V(1)$ is non-empty. If $t_j\ne 0$, we consider the product $f$ of the minimal polynomial of $t_j$ over $\LL$ with its common denominator and thereby obtain a polynomial $h$: $$\begin{aligned}
f
\ &=\
\sum_{i=0}^{m} a_iX^i,
\quad
h
\ =\
\sum_{i=0}^{m}a_{m-i}X^i
\ \ \in\ \KK\left[T_j;\ j\not\in\{k_1,\ldots,k_s\}\right][X]\end{aligned}$$ where $h(t_j^{-1}) = t_j^{-m}f(t_j)=0$. By definition, each $x\in \b\KK^{r-s}$ with $x\notin \Dom(t_j)$ must satisfy $\eps_x(t_j^{-1})=0$. For all $i>0$, from $a_{m-i}\in\KK\left[T_j;\ j\not\in\{k_1,\ldots,k_s\}\right]$ we know that $a_{m-i}\in R_{\eps_x}$ holds and therefore obtain $\eps_x(a_{m-i}t_j^{-i}) = 0$. We have $$\eps_x(a_m)
\ =\
\eps_x\left(
h(t_j^{-1})
-
\sum_{i=1}^m a_{m-i}t_j^{-i}
\right)
\ = \
0,$$ from which we infer that $a_m(x)=0$ and therefore $x\in V(a_m)\subseteq \b\KK^{r-s}$ hold; note that the inclusion $V(a_m)\subsetneq \b\KK^{r-s}$ is proper since $a_m\ne 0$. In other words, $$\Dom(t_j)\ \supseteq\
\b\KK^{r-s}\setminus V(a_m)
\ \ne\ \emptyset.$$ As finite intersection of supersets of non-empty open subsets, also the set $\Dom(t) = \Dom(t_1)\cap\ldots\cap \Dom(t_s)$ is non-empty; this completes the proof.
For the remainder of this section, we write $\LL$ for a field as in Proposition \[prop:fieldchange\]; note, however, that the following claims also hold for any field $\LL$.
The next step is to make all coefficients of a dense triangular system monic. We will call a triangular system $(\emptyset, \FFt, k, g)$ in $\LL[T_1,\ldots,T_r]$ [*monic*]{} if $\LC_{k(f)}(f) = 1$ for all $f\in \FFt$. For instance, the system in Example \[ex:triag\] is monic.
\[prop:normalize\] Consider a triangular system $S := (\emptyset, \FFt, k, g)$ in the ring $\LTT{r}$ that is dense in $\LL[T_n,\ldots,T_r]$ for a $1\leq n\leq r$. Assume there is $f\in \FFt$ with $k(f)=n$ such that $\FF := \FFt\setminus\{f\}$ is monic. Then the class $\b h\in R:=\LL[T_{n+1},\ldots,T_r]/\<\mathcal F\>$ of $h:=\LC_{T_n}(f)$ is annihilated by a polynomial $$p
\ =\
bX^j + fX^{j+1}
\, \in\, \LL[X]\setminus\{0\}
\qquad
\text{with}
\
f\,\in\, \LL[X],\ b\in \LL^*$$ where $j\in \ZZZ$ is maximal with $X^j\mid p$. Moreover, writing $f = hT_n^m+c$ with $m\in \ZZZ$ and $c\in \LL[T_n,\ldots,T_r]$ such that $\deg_{T_n}(c)<m$, we have a monic dense triangular system $S'$ that is equivalent to $S$: $$S'\ :=\
\left(\emptyset, \FF\cup \{f'\}, k, g\right)
\qquad
\text{with}
\ \
f'
\,:=\,
T_n^m -\frac{f(h)}{b}c
\ \in\
\LL[T_n,\ldots,T_r].$$
\[lem:ganz\]
1. Consider a triangular system $S:=(\emptyset,\FFt,k,g)$ in the ring $\LTT{r}$ that is dense in $\LL[T_n,\ldots,T_r]$ for a $1\leq n\leq r$. Setting $R := \LL[T_n,\ldots,T_r]/\<\FFt\>$, the ring extension $\LL\subseteq R$ is integral.
2. Let $\LL\subseteq R$ be a ring extension, $I\subseteq R$ an ideal and $h\in R$ such that $\b{h}\in R/I$ is integral over $\LL$. Define $J := \sqrt{I}:h\subseteq R$ and let $$p
\ =\
bX^j + fX^{j+1}
\, \in\, \LL[X]
\qquad
\text{with}
\
f\,\in\, \LL[X],\ b\in \LL^*$$ be the minimal polynomial of $\b h$ where $j\in \ZZZ$ is maximal with $X^j\mid p$. Then $h':=-f(h)/b\in R$ yields $hh'-1\in J$.
For (i), we write $\FFt = \{f_n,\ldots,f_r\}$ and assume $k(f_i) = i$. Define $R_j := \LL[T_n,\ldots,T_{r}]/\<f_j,\ldots,f_r\>$ for $n\leq j\leq r$ and $R_{r+1} = \LL$. The canonical projection $$\pi\colon R_{j+1}[T_j]\,\to\, R_j\ =\ R_{j+1}[T_j]/\<\b{f_j}\>,
\qquad
f\,\mapsto\,f+\<\b{f_j}\>$$ gives us $\pi(\b{f_j}(T_j))=\b{f_j}(\b{T_j})=\b 0$. Since $R_j = R_{j+1}[\b{T_j}]$ and $\b{f_j}\in R_{j+1}[X]$ is monic, the generator $\b{T_j}$ is integral over $R_{j+1}$ and non-zero. This shows that in the chain $
R = R_n\supseteq \ldots \supseteq R_{r+1}=\LL
$ each ring extension is integral, and so is $R\supseteq \LL$.
We come to (ii). Note that $p(h)\in I$ and $I\subseteq J$ ensures $p(h+J) = 0+J$. We have $$p(\b h)
\ =\
\left(\b hf(\b h) + b\right)\b h^j
\ =\ \b 0
\ \in\
R/J.$$ Observe that $\b h$ is not a zero-divisor: for each $x\in R$ with $xh\in \sqrt{I}:h$, already $x\in \sqrt{I}:h$ holds. That is $\b hf(\b h) + b=\b 0$. Setting $h' := -f(h)/b$, we obtain $h'h-1\in J$ from $$\b{h'h-1}
\ =\
\b{-\frac{f(h)}{b}h -1}
\ =\
-\frac{f(\b h)\b h + b}{b}
\ =\
\b 0\ \ \in\ \ R/J.\qedh$$
Note that the system $(\emptyset,\mathcal F,k,g)$ in $\LL[T_{1},\ldots,T_{r}]$ is dense in $\LL[T_{n+1},\ldots,T_{r}]$. By Lemma \[lem:ganz\] (i), the residue class $\b h\in R$ is integral over $\LL$, i.e., $p$ exists. Using the inclusion of the ideal $\sqrt{\<\mathcal F\>}:h\subseteq \LL[T_{n+1},\ldots,T_r]$ in the ideal $\sqrt{\<\mathcal F\>}:g\subseteq \LTT{r}$, we obtain $$hh'-1\ \in\ \sqrt{\<\mathcal F\>}:h\, \subseteq\, \sqrt{\<\mathcal F\>}:g
\qquad\text{with }
\ h'\,:=\,\frac{-f(h)}{b}\,\in\, \LL[T_{n+1},\ldots,T_r]$$ from the second statement of Lemma \[lem:ganz\]. One directly verifies the equality of ideals $$\sqrt{\<\FFt\>}:g
\ =\
\sqrt{\<\FF\>+\<f\>}:g
\ =\
\sqrt{\<\FF\>+\<f'\>}:g.$$ In particular, $V(S)=V(S')$ holds with the dense triangular system $S'$. Moreover, $\LC_1(f')=1$ by choice of $f'$ and $S'$ is monic.
In order to make Proposition \[prop:normalize\] computational, we first show how one can compute the required minimal polynomials.
\[algo:mipo\] [*Input:* ]{} an element $g\in R$ where $\LL\subseteq R$ is an integral ring extension of finite dimension $d:=\dim_\LL(R)$.
- Choosing a suitable $\LL$-vector space basis of $R$, we consider $M := [g^0, \ldots, g^d]$ as a $d\times (d+1)$ matrix over $\LL$.
- Compute the kernel $K:=\ker(M)\ne \{0\}$.
- Choose $q\in K\subseteq \LL^{d+1}$ such that $\max(1\leq j\leq d;\ q_j\ne 0)$ is minimal.
- Define $p_g := q_0X^0 + \ldots + q_dX^d\in \LL[X]$.
[*Output:* ]{} $p_g\in \LL[X]$. This is the minimal polynomial of $g\in R$.
By construction, we have $p(g) = Mq = 0$. For the minimality, let $p'=\sum_{j=0}^d q_j'X^j\in \LL[X]$ be the minimal polynomial of $g$. Then $Mq' = \sum_{j=0}^d q_j'h^j = p'(h)=0$, i.e., $q'\in K$. By choice of $q$, we have $$\deg(p')
\ =\
\max(1\leq j\leq d;\ q_j'\ne 0)
\ \geq\
\max(1\leq j\leq d;\ q_j\ne 0)
\ =\
\deg(p).\qedh$$
In Algorithm \[algo:mipo\], the element $q\in K$ can be computed using Gaussian elimination.
\[algo:normalize\] [*Input:* ]{} a triangular system $S:=(\emptyset,\FFt,k,g)$ that is dense in $\LTT{r}$. We assume $\FFt = \{f_1,\ldots,f_r\}$ with $k(f_i)=i$.
- For $n = r$ down to $1$, do:
- Set $\FFt^n := \{f_i;\ i>n\}
\subseteq\LL[T_{n+1},\ldots,T_r]$ and define the dense triangular system $(\emptyset,\FFt^n,k,g)$.
- Decompose $f_n = hT_n^d + c$ with $d\in \ZZ_{\geq 1}$ and $h\in \LL[T_{n+1},\ldots,T_r]$, $c\in \LL[T_n,\ldots,T_r]$ such that $\deg_{T_n}(c)<d$.
- Use Algorithm \[algo:mipo\] to compute the monic minimal polynomial $p_h\in \LL[X]$ of $\b h\in\LL[T_{n+1},\ldots,T_r]/\<\FFt^n\>$.
- Decompose $p_h= bX^{j+1} + aX^j$ with $b\in \LL[X]$, $a\in \LL^*$ by choosing $j\in \ZZZ$ maximal with $X^j\mid p_h$.
- Define $h' := -b(h)/a$. This yields $hh' - 1 \in \sqrt{\<\FFt^n\>}:h$.
- Redefine $f_n$ as $T_n^d+h'c\in \LL[T_n,\ldots,T_r]$. Then $S' := (\emptyset,\FFt^n\cup \{f_n\},k,g)$ is a monic triangular system that is dense in $\LL[T_n,\ldots,T_r]$.
[*Output:* ]{} $S'$. Then $S'$ is a monic triangular system that is dense in $\LTT{r}$ and is equivalent to $S$.
Note that the minimal polynomial $p_h$ exists by Lemma \[lem:ganz\] (i) since the system is dense. By Lemma \[lem:ganz\] (ii), $\b h\in \LL[T_{n+1},\ldots,T_r]/\sqrt{\<\FFt^n\>}:h$ is invertible. The remaining steps are correct by Proposition \[prop:normalize\].
We now show that the existence of solutions of a monic, dense triangular system can be tested by determining a minimal polynomial.
\[prop:solvable\] Let $S := (\emptyset, \FFt, k, g)$ be a monic triangular system that is dense in $\LTT{r}$. Set $R := \LTT{r}/\<\FFt\>$. Then $\LL\subseteq R$ is an integral extension and with the minimal polynomial $p_g\in \LL[X]$ of the residue class $\b g\in R$ we have $$V(S)\ \ne\ \emptyset
\qquad{\Longleftrightarrow}\qquad
p_g\ \in \LL[X]\ \text{ is not a monomial}.$$
\[lem:solvable\] In the situation of Proposition \[prop:solvable\], let $p\in \LL[X]$ be a polynomial with $p(g)\in \sqrt{\<\FFt\>}$. Then there is $k\in \ZZZ$ such that $p_g\mid p^k$.
By assumption, there is $k\in \ZZ_{\ge 1}$ such that $p(g)^k\in \<\FFt\>$, i.e., $p^k(\b g)=\b 0\in R$. The monic greatest common denominator $a := \gcd(p^k,p_g)\in \LL[X]$ satisfies $f(\b g)=\b 0\in R$ since $p^k(\b g) = p_g(\b g)=\b 0$. By minimality of $p_g$, we obtain $p_g = a\mid p^k$.
Given $x\in V(\FFt)\subseteq\b\LL^r$, the corresponding evaluation homomorphism $\eps_x$ fits into the commutative diagram $$\xymatrix{
\LTT{r}
\ar[rr]^{\eps_x}
\ar[dr]_{f\mapsto \b f}
&&
\b\LL
\\
&
R
\ar[ru]_{{\varphi}_x}
&
}$$
The fact, that $\LL\subseteq R$ is integral is Lemma \[lem:ganz\] (i). Assume now $p_g =X^n$ holds for some $n\in \ZZZ$, i.e., $\b g\in R$ is nilpotent. By the diagram, $g(x) = {\varphi}_x(\b g)$ then also is nilpotent for each $x\in V(\FFt)\subseteq \b\LL^r$. This means $g(x)=0$.
For the reverse direction, assume $g(x)=0$ holds for each $x\in V(\FFt)\subseteq \b\LL^r$, i.e., by the diagram, we have $p'(g(x)))=0$ with $p' := X\in \LL[X]$. By Lemma \[lem:solvable\], there is $k\in \ZZZ$ such that $p_g\mid (p')^k=X^k$.
We now put the previous propositions and algorithms together to obtain an algorithm to check the existence of solutions of a triangular system. This completes steps (ii) and (iii) of the list on page \[list:intro\].
\[algo:issolvable\] [*Input:* ]{} a triangular system $S=(\FFs, \FFt, k, g)$ in the ring $\KT{r}$.
- If $\FFs\cap \KK^*$ is non-empty, then:
- return [*false*]{}.
- Consider the triangular system $\iota(S)$ that is dense in $\LL[T_{k_1},\ldots,T_{k_s}]$ as in Proposition \[prop:fieldchange\].
- Use Algorithm \[algo:normalize\] with input $\iota(S)$ to obtain a monic, dense and equivalent system $S'= (\emptyset, \FFt', k', g')$ in $\LL[T_{k_1},\ldots,T_{k_s}]$.
- Use Algorithm \[algo:mipo\] to determine the minimal polynomial $p_{g'}\in \LL[X]$ of the residue class $\b{g'}\in \LL[T_{k_1},\ldots,T_{k_s}]/\<\FFt'\>$.
- If $p_{g'}$ is a monomial, then:
- return [*false*]{}.
- return [*true*]{}.
[*Output:* ]{} [*true*]{} if $V(S)\ne \emptyset$ and [*false*]{} otherwise.
By Proposition \[prop:fieldchange\], Algorithm \[algo:mipo\] and Algorithm \[algo:normalize\], $S'$ is equivalent, monic and dense. Proposition \[prop:solvable\] delivers the stated solvability criterion.
Monomial containment test and efficiency {#sec:implem}
========================================
Putting together steps (i)–(iii) listed on page , we are now able to test whether a given ideal $I\subseteq \KT{r}$ contains some monomial $T^\nu$, $\nu \in \ZZZ^r$. Afterwards, we explore the experimental running time of the second author’s implementation [@monomtest:implem] of the algorithm in `perl` on a series of random polynomials and compare it with Buchberger’s algorithm. Moreover, we compare its efficiency on the examples `polsys50` from [@eps] to algorithms listed in [@BaeGeLaRo Tab. 1].
\[algo:containsmonomial\] [*Input:* ]{} generators $f_1,\ldots,f_s$ for an ideal $I\subseteq \KT{r}$.
- Define the semi-triangular system $S:=(\FFs,\emptyset,0,g)$ where $g := T_1\cdots T_r$, and $\FFs := \{f_1,\ldots,f_s\}$.
- Let $\SSS$ be the output of Algorithm \[algo:triang\] applied to $\{S\}$.
- For each $S\in \SSS$, do:
- If Algorithm \[algo:issolvable\] returns [*true*]{}, then
- Return [*false*]{}.
- Return [*true*]{}.
[*Output:* ]{} [*true*]{} if $T^\mu \in I$ for some $\mu \in \ZZZ^r$. Returns [*false*]{} otherwise.
\[rem:eff\] In the second line of Algorithm \[algo:containsmonomial\] it is more efficient to modify Algorithm \[algo:triang\] such that it checks for solutions immediately after determining a new semi-triangular system.
\[ex:containsmon\] In the setting of Example \[ex:mush\], we apply Algorithm \[algo:containsmonomial\] with Remark \[rem:eff\] to test whether the ideal $I:=\<f_1,f_2\>\subseteq \KT{4}$ contains a monomial. To this end, we apply Algorithm \[algo:triang\] to the triangle mush $\SSS_0$. It will first choose the polynomial division for $(f,h):=(f_1,f_2)$ to obtain $$T_4f_1\ =\
(T_2-T_3)T_2f_2 - u,
\qquad
u\ =\
(T_2^3 - T_3T_2^2)T_4.$$ This yields a new triangle mush $\SSS_1 := \{S',S''\}$ where $S':=
(\{f_2,u\},\emptyset,0,gT_4)$ and $S'':=(\{f_1,f_2,T_4\},\emptyset,0,g)$. In the next step, we obtain triangle mushes $$\begin{aligned}
\SSS_2
&:=&
\left\{
(\{u\},\{f_2\},1,T_4g),\
(\{f_1,f_2,T_4\},\emptyset,0,g)
\right\},
\\
\SSS_3
&:=&
\left\{
(\emptyset,\{f_2,u\},4,T_4g),\
(\{f_1,f_2,T_4\},\emptyset,0,g)
\right\}.\end{aligned}$$
Algorithm \[algo:issolvable\] verifies that the zero-set $V(f_2,u)\setminus V(T_4g)$ is empty by the following steps: first, Algorithm \[algo:normalize\] with input $(\emptyset,\{f_2,u\},4,T_4g)$ will return the monic system $$\left(\emptyset, \{f_2,f_3\}, 4, T_4g\right),
\qquad
f_3
\ :=\
(T_2-T_3)T_2^2.$$ As $k(f_2)=1$ and $k(f_3)=2$, we set $\LL := \KK(T_3,T_4)$ and the ring $R:=\LL[T_1,T_2]/\<f_2,f_3\>$ is integral over $\LL$ with $\LL$-basis $(1, \b{T_2}, \b{T_2}^2)$. We have $$\b{T_4g}\ =\ \b{(T_3-T_2)T_2T_3T_4}\ \in\ R,
\qquad
\b{T_4g}^2\ =\ \b{(T_3-T_2)^2T_2^2T_3^2T_4^2}\ =\ \b 0\ \in\ R,$$ By Proposition \[prop:solvable\], the algorithm may remove this triangular set, i.e., it remains to consider $$\begin{aligned}
\SSS_4
&:=&
\left\{
(\{f_1,f_2,T_4\},\emptyset,0,g)
\right\}.\end{aligned}$$ The reduction step will remove the redundant equation $f_2$. The next steps provides us with $$\begin{aligned}
\SSS_5
&:=&
\left\{
(\emptyset,\{f_1,T_4\},4,u'g),\
(\{f_1,T_4,u'\},\emptyset,0,g)
\right\},
\qquad
u'\ :=\
(T_2-T_3)T_2.\end{aligned}$$ By Algorithm \[algo:issolvable\], the system $S:=(\emptyset,\{f_1,T_4\},4,u'g)$ has a solution: similar to before, Algorithm \[algo:normalize\] returns the monic system $$\left(\emptyset,\{f_4,T_4\},4,u'g\right),
\qquad
f_4
\ :=\
T_1-T_3$$ with $k(f_4)=1$ and $k(T_4)=4$. Setting $\LL:=\KK(T_2,T_3)$, the ring extension $\LL\subseteq R:=\LL[T_1,T_4]/\<f_4,T_4\>$ is integral with $\LL$-basis $(1)$. Since $$\b{u'g}\ =\ \b{(T_2-T_3)T_1T_2^2T_3}\ =\ \b{(T_2-T_3)T_2^2T_3^2}\ \in\ R$$ is non-zero, its minimal polynomial $p = X - (T_2-T_3)T_2^2T_3^2\in \LL[X]$ is not a monomial, i.e., $V(S)\ne \emptyset$ by Proposition \[prop:solvable\]. Thus, $\SSS_0$ has a solution as we already witnessed in Example \[ex:mush\]. In particular, $I$ contains no monomial, i.e., the algorithm returns [*false*]{}.
The remainder of this note is devoted to experimental running times. We apply the `perl` implementation [@monomtest:implem] of Algorithm \[algo:containsmonomial\] to a series of random ideals $\<f_1,\ldots,f_s\>\subseteq \KT{r}$ for fixed $2\leq s\leq 5$ and running $1\leq r\leq 10$. Moreover, setting $\FF := \{f_1,\ldots,f_s\}$, we distinguish the cases $V(\FF)=\emptyset$ and $V(\FF)\ne \emptyset$.
To make the experimental running times better comparable to Buchberger’s Gröbner basis algorithm [@CoLiOSh], we have reimplemented the latter in `perl` in two variants: the first one is the classical version whereas the second one stops as soon as a monomial could be found. Both algorithms as well as the testing sets $\FF$ are available at [@monomtest:implem]. The following graphics show the averages over the successful tests.
{width="5.5cm"} $V(\FF)\ne\emptyset$, $s=2$
{width="5.5cm"} $V(\FF)\ne\emptyset$, $s=3$
\
{width="5.5cm"} $V(\FF)\ne\emptyset$, $s=4$
{width="5.5cm"} $V(\FF)\ne\emptyset$, $s=5$
\
{width="5.5cm"} $V(\FF)=\emptyset$, $s=2$
{width="5.5cm"} $V(\FF)=\emptyset$, $s=3$
\
{width="5.5cm"} $V(\FF)=\emptyset$, $s=4$
{width="5.5cm"} $V(\FF)=\emptyset$, $s=5$
On the given set of polynomials, Algorithm \[algo:containsmonomial\] seems to be competitive when $V(\FF)\ne \emptyset$ whereas, for $V(\FF)=\emptyset$, the classical Buchberger’s algorithm usually needs less time.
Additionally, we have applied Algorithm \[algo:containsmonomial\] to the set of examples `polsys50` from [@eps]; its running time as well as the number of performed additions on a 2.66 GHz machine with time bound $300$ seconds and at most 1 GB of RAM is listed in the left-hand side part of the following table. We write “n/a” if the computation was unsuccessful either due to time reasons or because it was out of memory.
Moreover, in the right-hand part of the table, we list some of the running times listed in [@BaeGeLaRo Table 1] on the same examples. We want to stress the fact that the two sides of this table are only marginally comparable: not only is the goal different ([@BaeGeLaRo] deduces more information on the solutions whereas we test the existence of solutions), also the machines and maximal running times / memory are different.
[rrcrc|rrr]{} [ ]{}no. & time \[algo:containsmonomial\] & result & add.s & & time RC1 & time DW1 & time AT1\
[ ]{}
1 & $>$ 300 & n/a & n/a & & 3.5 & 0.4 & 3.0\
2 & $>$ 300 & n/a & n/a & & 7.4 & 7.6 & 7.1\
3 & 30.89 & 1 & 4956 & & $>$ 3h & 985.7 & 7538.0\
4 & $>$ 300 & n/a & n/a & & $>$ 4 GB & $>$ 4 GB & 0.2\
5 & 0.62 & 0 & 2449 & & &\
6 & 2.25 & 1 & 4239 & & 0.4 & 0.1 & 0.2\
7 & $>$ 1 GB & n/a & n/a & & $>$ 3h & 7352.6 & $>$ 4 GB\
8 & 0.14 & 1 & 214 & & &\
9 & 11.75 & 1 & 10149 & & &\
10 & 0.21 & 1 & 517 & & &\
11 & 0.17 & 1 & 361 & & &\
12 & 0.74 & 1 & 1909 & & 0.5 & 0.3 & 0.4\
13 & 0.15 & 1 & 214 & & &\
14 & 0.23 & 1 & 442 & & 0.5 & $>$ 3h & 1.5\
15 & 29.82 & 1 & 6655 & & &\
16 & $>$ 300 & n/a & n/a & & 0.9 & 1.4 & 1.8\
17 & $>$ 300 & n/a & n/a & & 6.5 & 4.7 & 75.5\
18 & 2.34 & 1 & 4324 & & 0.3 & 0.1 & 0.1\
19 & $>$ 300 & n/a & n/a & & 419.9 & 0.4 & 0.4\
20 & 0.25 & 1 & 668 & & &\
21 & $>$ 300 & n/a & n/a & & 1.6 & 86.6 & 4.5\
22 & $>$ 300 & n/a & n/a & & 0.6 & 1.2 & 1.5\
23 & $>$ 300 & n/a & n/a & & 0.4 & 0.1 & 29.5\
24 & $>$ 300 & n/a & n/a & & 1.2 & 1.3 & 1.0\
25 & 0.25 & 1 & 537 & & 1.2 & $>$ 3h & $>$ 4 GB\
26 & 1.07 & 0 & 4610 & & &\
27 & 1.47 & 1 & 2320 & & &\
28 & 9.89 & 1 & 6632 & & &\
29 & 0.15 & 1 & 297 & & 0,3 & 0,3 & 0,3\
30 & $>$ 300 & n/a & n/a & & $>$ 4 GB & $>$ 4 GB & 45.3\
31 & $>$ 1 GB & n/a & n/a & & $>$ 4 GB & $>$ 4 GB & $>$ 3h\
32 & 0.41 & 1 & 1200 & & &\
33 & $>$ 1 GB & n/a & n/a & & 3.4 & 1.3 & 3.5\
34 & $>$ 300 & n/a & n/a & & 911.5 & $>$ 3h & $>$ 4 GB\
35 & $>$ 300 & n/a & n/a & & 1.5 & 1.2 & 1.7\
36 & 0.13 & 1 & 160 & & &\
37 & 0.27 & 1 & 633 & & &\
38 & 11.10 & 1 & 6878 & & &\
39 & $>$ 300 & n/a & n/a & & 0.6 & 1.2 & 0.6\
40 & $>$ 300 & n/a & n/a & & &\
41 & $>$ 300 & n/a & n/a & & 1.5 & 1.5 & 7.0\
42 & 0.35 & 1 & 1028 & & &\
43 & $>$ 1 GB & n/a & n/a & & 0.7 & 3.1 & 0.2\
44 & $>$ 300 & n/a & n/a & & 24.5 & 3.4 & 1.2\
45 & $>$ 300 & n/a & n/a & & &\
46 & $>$ 300 & n/a & n/a & & &\
47 & 16.40 & 1 & 10465 & & 1.3 & 2.8 & 13.0\
48 & 0.23 & 1 & 563 & & &\
49 & $>$ 300 & n/a & n/a & & 0.3 & 610.2 & 0.5\
[ ]{}
[10]{}
P. Aubry, D. Lazard, and M. Moreno Maza. On the theories of triangular sets. , 28(1-2):105–124, 1999. Polynomial elimination—algorithms and applications.
P. Aubry and M. Moreno Maza. Triangular sets for solving polynomial systems: a comparative implementation of four methods. , 28(1-2):125–154, 1999. Polynomial elimination—algorithms and applications.
T. B[ä]{}chler, V. Gerdt, M. Lange-Hegermann, and D. Robertz. Algorithmic [T]{}homas decomposition of algebraic and differential systems. , 47(10):1233–1266, 2012.
T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas. Computing tropical varieties. , 42(1-2):54–73, 2007.
C. Chen, O. Golubitsky, F. Lemaire, M. Maza, and W. Pan. Comprehensive triangular decomposition. , pages 73–101, 2007.
D. Cox, J. Little, and D. O’Shea. . Undergraduate Texts in Mathematics. Springer, New York, third edition, 2007. An introduction to computational algebraic geometry and commutative algebra.
W. Decker and C. Lossen. , volume 16 of [*Algorithms and Computation in Mathematics*]{}. Springer-Verlag, Berlin; Hindustan Book Agency, New Delhi, 2006. A quick start using SINGULAR.
J. F. Ritt. Differential algebra. , Vol. XXXIII, 1950.
G.-M. Greuel and G. Pfister. . Springer, Berlin, extended edition, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Sch[ö]{}nemann.
S. Keicher. Computing the [GIT]{}-fan. , 22(7):1250064, 11, 2012.
T. Kremer. Ein [R]{}adikalmitgliedschaftsalgorithmus. Diploma thesis, 2015.
T. Kremer. An Algorithm for the determination of $\aa$-faces, radical membership and monomial containment. Available at <https://www.math.uni-tuebingen.de/users/kremer/afaces/>, 2015.
J. M. Thomas. Differential systems. , XXI, 1937.
J. M. Thomas. Systems and roots. , 1962.
D. Wang. ${\varepsilon}$psilon, for `Maple` including sets of examples. Available at <http://www-spiral.lip6.fr/~wang/epsilon/>, 2003.
D. Wang. Decomposing polynomial systems into simple systems. , 25(3):295–314, 1998.
W.-T. Wu. Mathematics mechanization. , 489, 2000.
O. Zariski and P. Samuel. . Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29.
[^1]: The first author was supported by the DFG Priority Program SPP 1489.
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abstract: 'The author studied the growth of the amplitude in a Mathieu-like equation with multiplicative white noise. The approximate value of the exponent at the extremum on parametric resonance regions was obtained theoretically by introducing the width of time interval, and the exponents were calculated numerically by solving the stochastic differential equations by a symplectic numerical method. The Mathieu-like equation contains a parameter $\alpha$ that is determined by the intensity of noise and the strength of the coupling between the variable and the noise. The value of $\alpha$ was restricted not to be negative without loss of generality. It was shown that the exponent decreases with $\alpha$, reaches a minimum and increases after that. It was also found that the exponent as a function of $\alpha$ has only one minimum at $\alpha \neq 0$ on parametric resonance regions of $\alpha = 0$. This minimum value is obtained theoretically and numerically. The existence of the minimum at $\alpha \neq 0$ indicates the suppression of the growth by multiplicative white noise.'
author:
- Masamichi Ishihara
bibliography:
- 'myrefs.bib'
title: Suppression of growth by multiplicative white noise in a parametric resonant system
---
Introduction
============
In past few decades, many researchers have investigated the roles of noise, and then marked phenomena were found. Such phenomena are stochastic resonance [@Gammaitoni; @Collins; @Yang; @Tessone], phase transition induced by multiplicative noise [@Broeck], etc [@Chialvo; @FUKUDA; @Zaikin; @Miyakawa; @Pikovsky]. A basic system in which multiplicative noise acts is an oscillator with varying mass. Oscillators in the presence of noise were investigated [@Stratonovich; @Landa; @Mallick2002; @Mallick2003; @Mallick2005eprint] and it was shown that the amplitude is amplified.
Another mechanism of growth is parametric resonance [@Landau]. The effects of additive white noise acting on a harmonic oscillator with a periodic coefficient has been investigated [@Zerbe]. Mean square displacement of an oscillator driven by a periodic coefficient was also studied in the presence of additive white noise [@Tashiro; @Tashiro09]. The parametric resonance induced by multiplicative colored noise was investigated in ref. [@Bobryk]. Experimentally, some physical systems which are described by the equations with a periodic coefficient and a multiplicative noise term were studied [@Berthet].
A differential equation with a periodic coefficient and a multiplicative noise term appears in some systems. Multiplicative noise may amplify or suppress the amplitude, as additive noise does. The magnitude of the amplitude is directly related to the stability of the system and the physical quantities, such as energy and the number of particle. Thus the effects of multiplicative white noise should be investigated in a parametric resonant system.
In this paper, a stochastic differential equation was analyzed by introducing the width of time interval in a parametric resonant system. The equation contains a parameter $\alpha$ that is determined by the intensity of noise and the strength of the coupling between the variable and the noise. The value of $\alpha$ was restricted not to be negative in the equation, without loss of generality. I estimated the exponent that indicates the growth of the amplitude. I showed the existence of the minimum of the exponent and estimated the minimum value as a function of $\alpha$ by deriving an approximate expression of the exponent on parametric resonance regions of $\alpha=0$. The stochastic differential equations were solved numerically by a symplectic method to avoid the growth by numerical error, and the exponent was extracted from the average of the trajectories. The behavior of the exponent as a function of $\alpha$ was displayed numerically.
I found that the exponent has only one minimum at $\alpha \neq 0$ on parametric resonance regions of $\alpha = 0$ and that the relative variation is of the order of 90%. The existence of the minimum indicates the suppression of the growth by multiplicative white noise. The results provide insight in the systems with periodically varying parameters and multiplicative noise. The multiplicative noise should suppress the growth in a parametric resonant system when the intensity of noise and the coupling strength are appropriate.
The exponent on parametric resonance regions {#sec:exponent}
============================================
An approximate equation of the exponent
---------------------------------------
An equation with a periodic coefficient and a multiplicative white noise term is interested in some branches of physics [@Berthet; @Zanchin; @Ishihara7]. A typical equation is $$\ddot{\phi} + \left[ 1 + \beta \cos \left( \gamma z \right) + \alpha r(z) \right] \phi = 0,
\label{eqn:start_equation}$$ where the dot represents the derivative with respect to $z$. The quantity $r(z)$ has the following properties: $$\langle r(z) \rangle = 0, \qquad \langle r(z) r(z') \rangle = \delta(z-z') ,
\label{eqn:properties_of_r}$$ where the notation $\langle \cdots \rangle$ represents statistical average. The value of $\alpha$ is restricted not to be negative in Eq. without loss of generality. The starting point in this study is Eq. with Eq. .
Equation is rewritten with the variable $p_{\phi}$ which is defined by $p_{\phi} = d\phi/dz$:
$$\begin{aligned}
d\phi &= p_{\phi} dz,
\label{eqn:numerical_phi} \\
dp_{\phi} &= - \left[ 1 + \beta \cos \left(\gamma z \right) \right] \phi dz
- \alpha \phi \circ dW.
\label{eqn:numerical_p_phi} \end{aligned}$$
The quantity $W(z)$ is defined by $W(z) = \int_{z_0}^{z} ds \ r(s) $ and this is a wiener process, where the quantity $z_{0}$ is an initial time. (Here, the symbol $\circ$ represents Stratonovich product.) I attempt to solve Eqs. and numerically in § \[sec:numerical\_calculation\].
Equation is just a Mathieu equation when $\alpha$ is zero, and this equation has resonance bands. With the relation $2u=\gamma z$, the Mathieu equation corresponding to Eq. is given by $$\frac{d^{2}\phi}{du^{2}} +
\left( a - 2q \cos(2u) \right) \phi =0,
\label{eqn:mathieu_equation}$$ where $a = {4}/{\gamma^{2}}$ and $-2q ={4 \beta}/{\gamma^{2}}$. Then the bands are distinguished by positive integer $n$ with the relation $n^{2} = {4}/{\gamma^2}$ . Therefore the values of $\gamma$ in resonance bands at $\alpha = 0$ are close to $2/n$.
In this paper, I attempt to estimate the growth rate of the amplitude in time. This rate is obtained from the exponent which is given by ${\displaystyle \mathop{\lim\sup}%_{z \rightarrow \infty}
\ z^{-1} \ln \left[\left|\langle \phi(z) \rangle \right| / \left|\phi_0 \right|\right]}$ , where $\phi_0$ is the initial value. I use the solution of the Mathieu equation to solve Eq. approximately in the resonance regions of $\alpha = 0$. The equation at $\alpha=0$ is $$\ddot{\Phi} + \left[ 1 + \beta \cos \left(\gamma z \right) \right] \Phi = 0,
\label{eqn:mathieu:Phi}$$ The quantity $\phi$ is represented as a product of $\Phi$ multiplied by a new variable $\psi$: $\phi = \Phi \psi$. The quantity $\psi$ satisfies the subsequent equation: $$\ddot\psi + 2 \left( {\dot{\Phi}}/{\Phi} \right) \dot{\psi} + \alpha r(z) \psi = 0.
\label{eqn:psi}$$ The exponent of $\Phi$ was investigated by many researchers in detail. Thus, the exponent of $\phi$ is estimated by obtaining the exponent of $\psi$ approximately. Here I denote the exponent of $\phi$ at $\alpha=0$ as $s \equiv s(\beta,\gamma)$ which is just the exponent of $\Phi$. The time dependence of $\Phi$ is obtained by solving Eq. . One method to solve approximately in the resonance band is performed by putting the form of $\Phi$ with the assumption $\ddot{P}_n \sim 0$ as follows:[@Landau; @Ishihara_Nonlinear; @Son; @Takimoto] $$\Phi =
\sum_{n=1} \left[ P_{n}(z) e^{i n \gamma z /2} + P_{n}^{*}(z) e^{-i n \gamma z /2} \right]
+ R(z).
\label{eqn:mode_decomposition}$$ The growth of the function $P_{m}(z)$ is largest in the $m$th resonance band. Therefore, $\Phi$ in the $m$th band is approximately given by
$$\begin{aligned}
& \Phi \sim e^{s_{m} z} F_{m}(z) ,
\label{eqn:s:m-th_band}\\
& F_{m}(z) := C e^{i m \gamma z /2} + C^{*} e^{-i m \gamma z /2} ,
\label{eqn:m-th_band}\end{aligned}$$
\[eqn:all:m-th\_band\]
where $C$ is a complex constant and $s_{m}$ is the exponent. It is conjectured that the exponent $s_{m}$ is close to the exponent $s$ in the $m$th resonance band. With Eqs. and , I obtain $${\dot{\Phi}}/{\Phi} \sim s_{m} + {\dot{F}_{m}}/{F_{m}}.
\label{eqn:dotPhi_Phi}$$ The exponent is estimated by solving Eq. with Eq. . However, it is not easy to handle Eq. . Instead, in Eq. , I replace $({\dot{\Phi}}/{\Phi})$ by the average of $(\dot{\Phi}/\Phi)$ in time. The average of $\dot{\Phi}/\Phi$ in one period of $F_{m}(z)$ is equal to $s_{m}$. Therefore, the approximate equation for $\psi$ under this approximation in the $m$th resonance band is $$\ddot{\psi} + 2 s_{m} \dot{\psi} + \alpha r(z) \psi = 0.
\label{eqn:approximated_equation_for_psi}$$
To estimate the exponent, I put the form of $\psi$ as follows: $$\psi = \psi_{0} \exp \left( \int_{z_0}^{z} dz' \sigma(z') \right).
\label{eqn:psi_exp_form}$$ Substituting Eq. into Eq. , I obtain the equation for $\sigma$: $$\dot{\sigma} + \sigma^{2} + 2s \sigma + \alpha r(z) = 0,
\label{eqn:sigma}$$ where the subscript $m$ of $s_m$ is omitted. In the next subsection, the exponent is obtained by estimating the statistical average $\langle \sigma \rangle$ approximately.
The value of the exponent at the extremum on parametric resonance regions {#subsec:Exponents_in_resonance_bands}
-------------------------------------------------------------------------
In this subsection, I estimate the minimum value of the exponent of $\phi$. It is assumed that $r(z)$ is constant in the quite small time interval to estimate $\sigma$ given by Eq. . The statistical average with respect to $r(z)$ is taken, because $r(z)$ varies randomly. The exponent of $\phi$ is estimated with the exponent $s$ and $\langle \sigma \rangle$. The existence of the extremum of the exponent is obtained by differentiating the exponent with respect to $\alpha$.
At first, I find the solution when $r(z)$ is constant. The solution of Eq. is categorized by the quantity $\cal{D}$ which is defined as $4s^{2}-4\alpha r$. I have $$\begin{aligned}
&
\int_{z_{0}}^{z} dz' \sigma(z')
=
\nonumber \\ &
\left\{
\begin{array}{ll}
\left( -s+ \frac{\sqrt{\cal{D}}}{2} \right) \left(z-z_{0}\right)
+
\ln \left| \frac{1-Ce^{-\sqrt{\cal{D}} z }}{1-Ce^{-\sqrt{\cal{D}} z_{0} }} \right|
& \quad {\cal D} > 0 \\
- s \left(z-z_{0}\right) + \ln \left| \frac{z+C'}{z_{0}+C'} \right|
& \quad {\cal D} = 0 \\
- s \left(z-z_{0}\right)
+ \ln \left|
\frac{\cos\left(\frac{\sqrt{-{\cal D}}}{2} z_{0} + C''\right)}
{\cos\left(\frac{\sqrt{-{\cal D}}}{2} z + C''\right)}\right|
& \quad {\cal D} < 0
\end{array}
\right.
,
\label{eqn:sigma_integration}\end{aligned}$$ where $C$, $C'$ and $C''$ are constants which are related to $\sigma(z_0)$. The logarithm terms of the right-hand side in Eq. do not contribute to the growth substantially.
Next, I treat the case that the quantity $r(z)$ is time dependent. For such the case, the region $[z_{0},z]$ is divided into small regions of time interval $\Delta z$. Moreover, the region of the width $\Delta z$ is divided into quite small $N$ regions numbered ’$j$’ in which the quantity $r$ is constant. I define the quantity $\Delta W_{j}$ by $r_{j} \Delta z / N$, where $r_{j}$ is the value of $r$ in the region ’$j$’. This quantity $\Delta W_{j}$ is a wiener process and the distribution function of $\Delta W_{j}$ is given by $$P(\Delta W_{j}) =
\frac{1}{\sqrt{2\pi (\Delta z)/N}}
\exp \left(- \frac{(\Delta W_{j})^{2}}{2 (\Delta z)/N} \right).$$ Then the quantity $\Delta W \equiv \displaystyle\sum_{j=1}^{N} \Delta W_{j}$ obeys the distribution function $P(\Delta W)$ which is given by $$P(\Delta W) =
\frac{1}{\sqrt{2\pi (\Delta z)}}
\exp \left(- \frac{(\Delta W)^{2}}{2 (\Delta z)} \right).$$ Therefore, the values of $\Delta W$ in the regions of time interval $\Delta z$ are distributed with the probability $P(\Delta W)$. The statistical average of a variable ${\cal O}$ is given by $\langle {\cal O} \rangle = \int_{-\infty}^{\infty} d(\Delta W) P(\Delta W) \ {\cal O}$. From Eqs. and , the exponent of $\phi$ in unit time of $z$ (I denote ${\cal G}$) should be estimated by $${\cal G} = \int_{-\infty}^{\infty} d(\Delta W) \ P(\Delta W) \Theta({\cal D})
\ \frac{\sqrt{{\cal D}}}{2},$$ where $\Theta(x)$ is the step function which is 1 for $x > 0$ and 0 for $x<0$ , and ${\cal D}= 4s^{2} - 4 \alpha (\Delta W)/\Delta z$. This integration can be performed and I obtain the following expression of ${\cal G}$: $${\cal G} = \frac{s}{2^{3/2} \kappa}
\exp \left( - \frac{\kappa^4}{4} \right)
D_{-3/2} \left( - \kappa^2 \right),
\label{eqn:res:exponent}$$ where the variable $\kappa$ is defined as $(\Delta z)^{1/4} s / \alpha^{1/2}$ and $D_{\nu}$ is the parabolic cylinder function [@Abramowitz; @Gradshteyn]. The value ${\cal G}/s$ depends only on $\kappa$, and then the parameter $\Delta z$ affects ${\cal G}/s$ through $\kappa$. A certain value $\kappa$ is realized by adjusting $\alpha$ when $\Delta z$ is given. I can read the global behavior of the exponent as a function of $\alpha$ from Eq. . The parameter $\alpha$ affects ${\cal G}/s$ through $\kappa$. The quantity ${\cal G}$ as a function of $\alpha$ has an extremum which is determined by $d{\cal G}/d\alpha$. I obtain the subsequent condition that ${\cal G}$ is extremum: $$D_{1/2} \left( - \kappa^2 \right) = 0.
\label{eqn:condition_of_minimum}$$ It is known that $D_{\nu}(x)$ for positive $\nu$ has $[\nu+1]$ zeros [@Bateman], where $[\nu+1]$ is the maximum integer which is not greater than $(\nu+1)$. Then the equation, $D_{1/2}(x)=0$, has one solution, and I write the solution as $x_{\mathrm{sol}}$. The value $x_{\mathrm{sol}}$ is negative, and then $\alpha$ is positive at the extremum of ${\cal G}$. Therefore ${\cal G}$ has one extremum surely at a positive $\alpha$. The value at the extremum of the exponent ${\cal G}$ is given by $${\cal G}_{\mathrm{min}} =
\frac{s}{2^{3/2}} \frac{1}{\left[-x_{\mathrm{sol}}\right]^{1/2}}
\exp \left( -\frac{1}{4} \left( x_{\mathrm{sol}} \right)^{2} \right)
D_{-3/2} \left( x_{\mathrm{sol}} \right) .
\label{eqn:min_val_of_G}$$ The value ${\cal G}_{\mathrm{min}}$ is smaller than $s$. That is, ${\cal G}$ has one minimum at a positive $\kappa^2$. This indicates that the exponent ${\cal G}$ is suppressed by multiplicative white noise when the value of $\alpha$ is appropriate. I must note that the expression ${\cal G}_{\mathrm{min}}$ is independent of $\Delta z$. Equation for quite small $s$ should be invalid, because the approximation of $\Phi$ given in Eq. does not work well.
Numerical calculation of the exponents by a symplectic method {#sec:numerical_calculation}
=============================================================
In this section, I attempt to solve Eqs. and numerically. Our purpose is to obtain the amplitude of $\phi$ when white noise acts multiplicatively. Therefore, the amplitude must be calculated precisely, at least, when a periodic coefficient and a white noise term are absent. The system has the symplectic structure even when noise exists if some conditions are satisfied [@Milstein_additive]. Taking this property into account, I use the symplectic method developed in ref. [@Milstein_multiplicative] to solve the stochastic differential equations with multiplicative white noise. The first-order method given in ref. [@Milstein_multiplicative] is applied to the equations in this study. The equations are solved numerically from $z=0$ to $z=500$. The time step in $z$ is set to 0.05. The initial conditions are $\phi(0) = 1$ and $\dot{\phi}(0) = 0$ in these calculations.
One trajectory of $\phi(z)$ can be calculated when one sequence of noise is given. I calculate many trajectories and take their average to obtain the mean value of the trajectories of the variable $\phi_{i}^{(j)}(z)$, where the subscript $i$ indicates the batch and the superscript $(j)$ indicates the trajectory in a certain batch $i$. In the present calculation, one batch contains 500 trajectories and 20 batches are used. I calculate the mean value ${\cal M}_{i}(z)$ of the trajectories in the batch $i$. The mean value over 20 batches, $\bar{\phi}(z)$, is given by $$\bar{\phi}(z) = \frac{1}{20} \sum_{i=1}^{20} {\cal M}_{i}(z),
\quad
{\cal M}_{i}(z) = \frac{1}{500} \sum_{j=1}^{500} \phi_{i}^{(j)}(z).$$ It is possible to perform interval estimation by using $\bar{\phi}$ and ${\cal M}_{i}$. In the case of $\alpha = 0$, there is no need to calculate many trajectories. Thus only one trajectory is calculated numerically for $\alpha=0$.
The exponent is estimated from the average $\bar{\phi}(z)$ in the range of $200<z<500$ to decrease the effects of the initial conditions. This estimation is performed as follows: 1) the sets $(z_{k},\ln\bar{\phi}(z_{k}))$ are determined, where $z_k$ is the time at which $\bar{\phi}(z_{k})$ is a local maximum and positive. 2) the sets are fit with a linear function. The coefficient of the time $z$ is adopted as the exponent.
Here, I note the reason why the values, $\ln\bar{\phi}(z_{k})$, are fit. One way to estimate the parameters is to fit the average $\bar{\phi}(z_{k})$ directly. In such the method, it is implicitly assumed that the dispersion of the distribution of the data at time $z$ and that at time $z'$ ($\neq z$) are the same (approximately). However, the dispersion is wider with time $z$ in the present case, because I treat a wiener process. The effects of non-equivalent dispersions are decreased by taking the logarithm of the data. Therefore the transformed data, $\ln\bar{\phi}(z_{k})$, are fit with the linear function. I notice that the exponents extracted by the above procedure are different generally from the Lyapunov exponents which are estimated by the mean value of the logarithm of $\phi_i^{(j)}$. The quantity, $\ln\bar{\phi}(z_{k})$, is calculated, because I focus on the enhancement of the variable $\phi$ in this study.
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Figure \[figs:exponent\_for\_various\_alpha\](a) is the map of the exponents of the Mathieu equation, Eq. , on the $\gamma$–$\beta$ plane. The step sizes in $\gamma$ and $\beta$ in the numerical calculations are taken to be 0.02 to draw this figure. I denote these step sizes as $\Delta \gamma$ and $\Delta \beta$ respectively. The color of a square is determined from the arithmetic mean of the exponents at four corners which are located at ($\gamma$,$\beta$), ($\gamma$ + $\Delta \gamma$,$\beta$), ($\gamma$,$\beta$ + $\Delta \beta$) and ($\gamma$ + $\Delta \gamma$,$\beta$ + $\Delta \beta$). The resonance band around $\gamma=2$ corresponds to the first resonance band of Eq. . The $n$th resonance band of Eq. corresponds to the band around $\gamma = 2/n$, where $n$ is positive integer.
Next, I show the map of the exponents for various values of $\alpha$ on the $\gamma$–$\beta$ plane. Figure \[figs:exponent\_for\_various\_alpha\](b) is the map at $\alpha=0.5$, \[figs:exponent\_for\_various\_alpha\](c) is at $\alpha=1.0$, \[figs:exponent\_for\_various\_alpha\](d) is at $\alpha=1.5$, \[figs:exponent\_for\_various\_alpha\](e) is at $\alpha=2.0$, and \[figs:exponent\_for\_various\_alpha\](f) is at $\alpha=2.5$. The step sizes in $\beta$ and $\gamma$ are 0.05 in the numerical calculations for Figs. \[figs:exponent\_for\_various\_alpha\](b), \[figs:exponent\_for\_various\_alpha\](c), \[figs:exponent\_for\_various\_alpha\](d), \[figs:exponent\_for\_various\_alpha\](e) and \[figs:exponent\_for\_various\_alpha\](f). The color of a square is determined in the same manner as in Fig. \[figs:exponent\_for\_various\_alpha\](a). As shown in Figs. \[figs:exponent\_for\_various\_alpha\](b), \[figs:exponent\_for\_various\_alpha\](c), \[figs:exponent\_for\_various\_alpha\](d), \[figs:exponent\_for\_various\_alpha\](e) and \[figs:exponent\_for\_various\_alpha\](f), the band structure is destroyed by noise, and the values of the exponents become large with $\alpha$ for many sets of $(\gamma,\beta)$. However it seems from these figures that the exponent on the resonance band is not a monotonically increasing function of $\alpha$. Moreover, the $\beta$ dependence of the exponent in Fig. \[figs:exponent\_for\_various\_alpha\](f) is weak as compared with those in other figures: Figs. \[figs:exponent\_for\_various\_alpha\](a), \[figs:exponent\_for\_various\_alpha\](b) and \[figs:exponent\_for\_various\_alpha\](c). This fact in Fig. \[figs:exponent\_for\_various\_alpha\](f) implies that the values of the exponents of the equation with the periodic coefficient are close to those without the periodic coefficient. (The values of the exponents at $\beta=0$ correspond to the values in the case of no periodic coefficient.) It is evident that the effects of the periodic coefficient become weak relatively. Furthermore, I investigate the $\alpha$ dependence of the exponent on the first and the second resonance bands. I draw the $\alpha$ dependence of the exponent with the fixed parameters, $\gamma$ and $\beta$. I show the exponents for the set $(\gamma=2,\beta=2)$ on the first resonance band, and the set $(\gamma=0.9,\beta=2)$ on the second resonance band. Figure \[figs::resonance:exponents:\] shows the $\alpha$ dependences of the exponents. The cross represents the data obtained by solving Eqs. and numerically. The suppression by noise is clearly seen and there is only one local minimum in each figure. The exponent decreases with $\alpha$ and reaches the minimum. It continues to increase with $\alpha$ after that. This behavior is interpreted as follows. The growth of the amplitude depends on the mechanism of parametric resonance for small $\alpha$. This mechanism is destroyed by noise with the increase of $\alpha$. Then the exponent decreases with $\alpha$. Contrarily, the amplitude is amplified by noise for large $\alpha$, as shown in many researches. In summary, the exponent decreases with $\alpha$, reaches the minimum, and increases after that. The exponents for other parameter sets, $(\gamma, \beta)$, on the resonance bands behave similarly.
Finally, the minimum value of the exponent as a function of $\alpha$ is estimated for various values of $\beta$ and $\gamma$. I denote the minimum value of the exponent estimated numerically as $s_{\mathrm{min}}$. Clearly $s_{\mathrm{min}}$ is a function of $\beta$ and $\gamma$. In these calculations, the range of $\alpha$ is set to $[0,2]$ and the step size in $\alpha$ is set to 0.01. The range of $\gamma$ is set to $[0.7,2.7]$ and the step size in $\gamma$ is set to 0.5. The exponents for various values of $\alpha$ with the fixed $\beta$ and $\gamma$ are estimated and $s_{\mathrm{min}}$ is set to the minimum value of these exponents. I calculate the quantity $s_{\mathrm{min}}/s$, because the exponent at $\alpha=0$, $s$, is also a function of $\beta$ and $\gamma$. I show the values $s_{\mathrm{min}}/s$ for $s \ge 0.3$ to compare them with the value ${\cal G}_{\mathrm{min}}/s$. The value ${\cal G}_{\mathrm{min}}/s$ is approximately 0.893 from Eq. .
Figure \[figs:ratio:SMA1\] shows the values $s_{\mathrm{min}}/s$ for $s \ge 0.3$ and the exponents $s$. The parameter $\beta$ is set to 2.0 in Fig. \[figs:ratio:SMA1\](a) and 1.5 in Fig. \[figs:ratio:SMA1\](b). Cross represents data points of $s_{\mathrm{min}}/s$ and broken line indicates ${\cal G}_{\mathrm{min}}/s$. Asterisk represents data points of $s$. As seen in Figs. \[figs:exponent\_for\_various\_alpha\] and \[figs::resonance:exponents:\], noise influences the values. Thus it is likely that the ratio $s_{\mathrm{min}}/s$ fluctuates and that the values $s_{\mathrm{min}}/s$ around the maximum of $s$ are below the value ${\cal G}_{\mathrm{min}}/s$. Thus I calculate also the simple moving average of the exponents, and attempt to find the minimum value of them. I take the average of $n$ adjoining exponents and denote this average as $s_{\mathrm{min}}^{\mathrm{SMA}n}$. For example, the minimum of the averages of three adjoining exponents is represented as $s_{\mathrm{min}}^{\mathrm{SMA3}}$. Figure \[figs:ratio:SMA3\] displays the $s_{\mathrm{min}}^{\mathrm{SMA3}}/s$ for $s \ge 0.3$ and the exponents $s$. The parameter $\beta$ is set to 2.0 in Fig. \[figs:ratio:SMA3\](a) and 1.5 in Fig. \[figs:ratio:SMA3\](b). The symbols in Figs. \[figs:ratio:SMA3\](a) and \[figs:ratio:SMA3\](b) are the same as in Figs. \[figs:ratio:SMA1\](a) and \[figs:ratio:SMA1\](b). It is found from Figs. \[figs:ratio:SMA1\] and \[figs:ratio:SMA3\] that ${\cal G}_{\mathrm{min}}/s$ is close to the values estimated by numerical calculations around the peaks of $s$ in the resonance regions.
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Discussion and Conclusion {#sec:conclusions}
=========================
I studied the growth of the amplitude in the Mathieu-like equation with multiplicative white noise. The approximate value of the exponent at the extremum was obtained by introducing the width of time interval on parametric resonance regions where parametric resonance occurs when no noise exists. The exponents were calculated by solving the stochastic differential equations numerically by the symplectic numerical method. The intensity of noise and the strength of the coupling between the noise and the variable are reflected to the value of the parameter $\alpha$. The value of $\alpha$ was restricted not to be negative in the present equation, without loss of generality. The behavior of the exponents as a function of $\alpha$ was shown roughly.
With regard to the effects of multiplicative white noise on the growth, the band structure of the Mathieu equation is destroyed when noise exists. The resonance structure survives for small values of $\alpha$, and this structure is lost for large values of $\alpha$. In the previous paper [@Ishihara8], I investigated the growth in a stochastic differential equation without a periodic coefficient, and found that the exponent is a monotone increasing function of $\alpha$. In contrast, the exponent as a function of $\alpha$ has one minimum on the parametric resonance region of $\alpha=0$. This indicates the suppression of the growth by multiplicative white noise, and this suppression occurs when the value of $\alpha$ is appropriate. Equation can roughly explain the behavior of the exponent as a function of $\alpha$: The exponent decreases with $\alpha$, reaches the minimum and increases after that.
One expects that the exponent as a function of the intensity of noise has one minimum intuitively. However the exponent may have some minima caused by noise. Theoretical expression given by Eq. indicates that only one minimum exists. This fact is numerically supported too. It is shown theoretically and numerically in the previous sections that the exponent as a function of $\alpha$ has one minimum.
The minimum value of the exponent as a function of $\alpha$ was estimated from the numerical calculations. I calculated the ratio $s_{\mathrm{min}}/s$: the minimum value divided by the exponent at $\alpha=0$, $s$. This ratio obtained numerically is in rough agreement with that obtained theoretically around the peaks of $s$ on the resonance regions. The minimum value of the exponent is approximately proportional to the exponent $s$. The relative variation is of the order of 90%, as shown in the figures and Eq. . It seems that the variation is small. Nevertheless, the amplitude is affected, because this is the variation of the exponent.
The decrease of Lyapunov exponent by noise was found in the system of an inverted Duffing oscillator with noise [@Mallick2004]. The mechanism of the growth in the present case is different from that in the case of the inverted Duffing oscillator, when noise is absent. However, the mechanism of the suppression is surely the same. In both cases, the growth is suppressed by noise when the intensity of noise is appropriate. The exponent decreases with the intensity, and reaches the minimum. After that, the exponent increases with the intensity. The decrease of the exponent by white noise implies the possibility of the large decrease by colored noise. The system in which parametric resonance occurs may be stabilized by colored noise, as found in the system of the inverted Duffing oscillator.
The expression of the exponent obtained theoretically includes the artificial parameter $\Delta z$. Then the value of $\alpha$ at the minimum of ${\cal G}$ depends on $\Delta z$, while the minimum value of ${\cal G}$ is independent of $\Delta z$. I would like to solve this problem in the future study.
|
---
abstract: 'As a network of advanced-era gravitational wave detectors is nearing its design sensitivity, efficient and accurate waveform modeling becomes more and more relevant. Understanding of the nature of the signal being sought can have an order unity effect on the event rates seen in these instruments. The paper provides a description of key elements of the Spectral Einstein Code ([SpEC]{}), with details of our spectral adaptive mesh refinement (AMR) algorithm that has been optimized for binary black hole (BBH) evolutions. We expect that the gravitational waveform catalog produced by our code will have a central importance in both the detection and parameter estimation of gravitational waves in these instruments.'
address: |
Theoretical Astrophysics 350-17,\
California Institute of Technology,\
Pasadena, CA 91125, USA\
bela@caltech.edu
author:
- 'B'' ela Szil'' agyi'
bibliography:
- 'References/References.bib'
title: Key Elements of Robustness in Binary Black Hole Evolutions using Spectral Methods
---
Introduction
============
A fundamental consequence of the field equations written down by Einstein, in his Theory of General Relativity, is the existence of singular solutions that are causally disconnected from remote observers. That is, surrounding the singularity will be an event horizon defined as the boundary of the region from within which a physical observer cannot escape. Astrophysical observations have lead to the realization that these objects are not only of theoretical importance. They are more than just the “point charge solution” of Einstein’s field equations. In fact, they are key players in how our universe functions. Given the large amount of observational evidence for the existence of black holes, it is a very natural question to ask – will these objects ever collide? If yes, what measurable quantities can be used to identify such an event. We know that galaxies collide (ours being one such “collided” galaxy). We know that galaxies host super massive black holes. Therefore black holes must collide as well.
The intent of this paper is to highlight some of the key elements of a particular numerical relativity code, the [ Sp]{}ectral [ E]{}instein [ C]{}ode ([SpEC]{}), that allowed it to reach “production-level” in simulating binary black hole (BBH) mergers.
It is important to emphasize at a very early point in the paper that, though the current manuscript has a single author, the [SpEC]{} code is the result of years of work of a large number of people within what is known as the “SXS collaboration”. While a fair portion of the paper will be describing code contributions of the author, we will also make an effort to give due credit to all others involved.
Throughout the paper we will use Latin indexes $a,b,...$ from the beginning of the alphabet for space-time quantities, while $i,j,k,...$ will stand for spatial indexes. Partial derivatives will be denoted by $\partial_a f$ while covariant derivatives will be written as $\nabla _a f$.
Evolution System
================
The Generalized Harmonic System
-------------------------------
Consider a spacetime metric tensor $\psi_{ab}$, $$ds^2 = \psi_{ab} dx^a dx^b .$$ The associated Christoffel symbol $\Gamma_{abc}$ is defined as $$\Gamma_{abc} = \frac{1}{2}\left( \partial_b \psi_{ac} + \partial_c \psi_{ab} - \partial_a \psi_{bc} \right) .
\label{eq:GammaDef}$$ We will refer to the trace of the Christoffel symbol as $$\Gamma_a \equiv \psi^{bc} \Gamma_{abc} .
\label{eq:TrGammaDef}$$
The Einstein Equations, in a concise form, can be written as $$G_{ab} = R_{ab} - \frac{1}{2} \psi_{ab} R = 0$$ where $R=\psi^{ab} R_{ab}$ is the trace of the Ricci tensor $R_{ab}$ which, in turn, is given by $$R_{ab} = - \frac{1}{2} \psi^{cd} \partial_c \partial_d \psi_{ab}
+ \nabla_{(a} \Gamma_{b)}
+ \psi^{cd} \psi^{ef} \left(\partial_e \psi_{ca} \partial_f \psi_{db} - \Gamma_{ace} \Gamma_{bdf}\right) .
\label{eq:Ricci}$$
An essential part of being able to numerically evolve a space time is to establish a well-posed initial boundary value problem. “Well-posed” here translates into saying that given some plausible initial and boundary data a unique solution will exist and that small perturbations in the freely specifiable data (on the initial slice or on the boundary) will result in small changes of the solution to the system under consideration. One approach to establish well-posedness of a system of partial differential equations is to show that the principal part (the terms containing the highest order derivatives) can be written as a first order symmetric hyperbolic system.
A careful look at the right-hand side (RHS) of Eq. (\[eq:Ricci\]) shows that the principal part consists of two terms: the wave operator acting on the metric and derivatives of $\Gamma_b$ (which itself consists of first derivatives of the metric). This, together with the identity $$\psi_{ab}\Delta^c \Delta_c x^a = - \Gamma_b$$ leads to the idea that one can think of $\Gamma_b$ as freely specifiable gauge freedom[@Friedrich1985]. That is, rather than assigning values $x^a$ to each point of the manifold in some arbitrary (but smooth) way, one can think of the coordinates being determined indirectly through a wave equation $$\psi_{ab}\Delta^c \Delta_c x^b = H_a$$ and regarding $H_a$ as freely specifiable. Specifying $H_a$, with suitable initial and boundary conditions, is equivalent to specifying $x^a$. Substituting $H_a = -\Gamma_a$ into Eq. (\[eq:RicciH\]) gives $$R^{H}_{ab} = - \frac{1}{2} \psi^{cd} \partial_c \partial_d \psi_{ab}
+ \nabla_{(a} H_{b)}
+ \psi^{cd} \psi^{ef} \left(\partial_e \psi_{ca} \partial_f \psi_{db} - \Gamma_{ace} \Gamma_{bdf}\right) .
\label{eq:RicciH}$$ If the gauge source function $H_a$ is prescribed as a function of the coordinates $(x^a)$ and the metric $\psi_{ab}$ but does not depend on derivatives of the metric, $$H_a = F_a\left(x^c,\psi_{cd}\right) ,$$ then the sole term of the principal part of $R^{H}_{ab}$ consists of the wave operator acting on the metric. This leads to a well-posed system, known as the generalized harmonic formulation of Einstein’s equations.
The identification $H_a = -\Gamma_a$ induces a constraint in the generalized harmonic system, as $H_a$ is now a free function, while the trace of the Christoffel symbol is determined by derivatives of the metric resulting form the evolution equations. These equations are equivalent to the Einstein system only in the limit in which the constraint ${\cal C}_a = H_a + \Gamma_a$ vanishes. It has been shown[@Friedrich2005] that as a consequence of the Bianchi identities, the constraints propagate according to their own set of governing equations, $$0 = \Delta^b \Delta_b {\cal C}_a + {\cal C}^b \Delta_{(a} {\cal C}_{b)} .
\label{eq:ConstraintProp}$$ Thus the constraints themselves propagate as a set of coupled scalar waves implying that their evolution system is well-posed as well. This in turn means that small perturbations of the constraints in either the initial data or boundary data will not result in uncontrollable runaway solutions of the constraint system. In addition, the form of the source term of Eq. (\[eq:ConstraintProp\]) tells us that for vanishing initial and boundary data on the constraints, they will remain zero in the entire evolved spacetime region.
In the actual numerical simulations constraint violating modes are generated at each time-step and on a variety of length-scales. Well-posedness is essential but not sufficient, as it does not exclude exponentially growing modes. Constraint propagation can further be improved by adding what is known as constraint damping terms to the evolution system. Given that the constraints are formed of first derivatives of the metric, one can add arbitrary combination of these (but not their derivatives) to the evolution system without modifying its principal part. A number of such terms have been proposed in the literature.[@Brodbeck1999; @Gundlach2005; @Pretorius2005a; @Babiuc2006] The one employed by the [SpEC]{} code can be written as
$$0 = R_ab - \nabla_{(a} {\cal C}_{b)} + \gamma_0 \left[ t_{(a} {\cal C}_{b)} - \frac{1}{2} \psi_{ab} t^c {\cal C}_c \right] ,$$
where $t_a$ is the future directed unit timelike normal to the $t=$constant surfaces of the spacetime manifold, while $\gamma_0$ is a free parameter.[^1]
Damped Harmonic Gauge {#sec:DampedHarmonicGauge}
---------------------
Next we turn our attention to various choices of the gauge source function $H_a$. One immediate choice is to set the components of $H_a$ to zero, resulting in what is known as the harmonic gauge. Substituting this into Eq. (\[eq:RicciH\]) leads to the harmonic formulation of Einstein’s equations. Harmonic coordinates determine the fields $(t,x,y,z)$ by a wave equation and suitable initial and boundary values. In flat space this is feasible, as the ‘natural’ coordinates $(t,x,y,z)$ are trivial solutions of the wave operator, $$\left( -\partial_t^2 + \partial_x^2 + \partial_y^2 + \partial_z^2 \right) \left( x^a \right) = 0$$ In a non-trivial space time, however, in the presence of an actual gravitational potential, the scalar wave operator may amplify fields propagating under its action and form singularities. The net effect would be a singular coordinate system (with metric coefficients that become very large). Better choices are needed.
In his ground-breaking work, Pretorius [@Pretorius2005a] was able to use harmonic spatial coordinates but had to prescribe the time-component $H_t$ of the gauge source function by a damped wave equation that prevented the lapse function from collapsing to zero (which would be a signature of a singular time-coordinate). For generic binary black hole mergers, we found that better gauge choices are needed.
The current favorite gauge condition for binary black hole systems evolved with [SpEC]{} is the damped harmonic gauge[@Lindblom2009c; @Szilagyi:2009qz], described by $$H_a = \mu_L \log\left(\frac{\sqrt{g}}{N} \right) t_a - \mu_S N^{-1} g_{ai} N^i
\label{eq:InertialDampedHarmonicGauge}$$ $$g_{ab} = \psi_{ab} + t_a t_b = \psi_{ab} + N^2 \delta_a^t \delta_b^t ,$$ where $g$ is the determinant of the spatial 3-metric $g_{ij}$, $N$ is the lapse function and $N^i$ is the shift, defined as $$N = (- \psi^{tt})^{-1/2}, \quad N^i = - \psi^{it} / \psi^{tt} .$$ The coefficients $\mu_L, \mu_S$ are used to control the amount of damping. We find that the choice $$\mu_S = \mu_L = \mu_0 \left[ \log \left( \frac{\sqrt{g}}{N} \right) \right]^2$$ works well, where $\mu_0$ is a time-dependent coefficient that is rolled from zero to one in a smooth time-dependent way at the start of the simulation in order to reduce numerical error induced by initial gauge-dynamics of the BBH system.
Gauge conditions are intended to impose a condition on coordinates and, as such, they cannot be coordinate-independent (or covariant). This holds for our system as well. The form given in Eq. (\[eq:InertialDampedHarmonicGauge\]) acts a damping condition on the inertial frame[^2] damps the value of $\log\left(\frac{\sqrt{g}}{N} \right)$ towards small values, effectively preventing the lapse from collapsing and/or the metric volume density $\sqrt{g}$ from becoming very large.
The drawback of this gauge choice is that near merger, when objects tend to have larger coordinate velocities, it results in distortion. An alternative would be to impose the same condition in a comoving frame[^3] (with coordinates $\tilde x^{\tilde k}$) of the binary.
When doing so, one has to be mindful of the fact that $H_a$ in general has transformation properties that depend on the particular choice of gauge condition (in our case Eq. (\[eq:InertialDampedHarmonicGauge\])), while $\Gamma_a$ transforms as prescribed by its definition (see Eqs. (\[eq:GammaDef\])-(\[eq:TrGammaDef\])). As a consequence, naively substituting the co-moving frame metric quantities $\tilde g, \tilde g_{\tilde a \tilde i}, \tilde N^{\tilde i}, \tilde t_{\tilde a}$ into Eq. (\[eq:InertialDampedHarmonicGauge\]) would not be what one might naturally think of as a corotating damped harmonic gauge condition. In order to clarify the meaning of the damped harmonic gauge in a particular frame, we start by observing that the quantity
$$\begin{aligned}
\Delta_a \equiv \psi^{bc} \psi_{ad} \left( \Gamma^d{}_{bc} - {\Gamma^{(0)}}^d{}_{bc} \right) = \Gamma_a - \psi^{bc} \psi_{ad} {\Gamma^{(0)}}^d{}_{bc}\end{aligned}$$
is a tensor, where ${\Gamma^{(0)}}^d_{bc}$ is a ‘background’ connection associated with some background metric ${\psi^{(0)}}_{ab}$. We can define the background in the inertial frame to be the Minkowski metric, $${\psi^{(0)}}_{ab} \equiv \eta_{ab} \; ,$$ while the comoving background would be the tensor-transform of the inertial frame background metric into the comoving frame, $${\tilde \psi^{(0)}}_{\tilde a\tilde b} = { \psi^{(0)}}_{ a b} \; {J^a}_{\tilde a} {J^b}_{\tilde b} ,$$ with Jacobian $${J^a}_{\tilde b} = \frac{\partial x^a}{\partial \tilde x^{\tilde b}} .$$ The transformation of $\Delta_a$ from the inertial to the corotating frame leads to $$\Gamma_a = J_a{}^{\tilde a}{} \tilde \Gamma_{\tilde a} - J_a{}^{\tilde a}{} \tilde \psi^{\tilde b\tilde c}\tilde \psi_{\tilde a\tilde d} {\tilde \Gamma^{(0)}}{}^{\tilde d}{}_{\tilde b\tilde c} .
\label{eq:GammaTransformationRule}$$
The co-moving damped harmonic gauge condition is $$-\tilde \Gamma_{\tilde a} = \mu_L \log\left(\frac{\sqrt{\tilde g}}{\tilde N} \right) \tilde t_{\tilde a} - \mu_S \tilde N^{-1} \tilde g_{\tilde a\tilde i}
\tilde N^{\tilde i} .
\label{eq:GammaCorotDampedHarmonic}$$ Substitution of Eq. (\[eq:GammaCorotDampedHarmonic\]) into Eq. (\[eq:GammaTransformationRule\]) and expressing the result in terms of inertial frame metric quantities leads to $$\begin{aligned}
H_{a} =&&
\mu_L \log\left(\frac{J \sqrt{g}}{N} \right) t_{a}
- \mu_S N^{-1} g_{ai}
\left( N^{i} - V^{i}\right)
\\ && \nonumber
+ \psi^{bc} \psi_{ad}
J_b{}^{\tilde b}
J_c{}^{\tilde c}
J^d{}_{\tilde d}
\Gamma^{(0)}{}^{\tilde d}{}_{\tilde b\tilde c} ,\end{aligned}$$ where $J=\det \left[ \frac{\partial x^{i}}{\partial x^{\tilde i}} \right]$ and $V^i = \frac{ \partial x^i }{ \partial{\tilde t} }$ is the coordinate velocity of a comoving observer in the inertial frame. When compared to the inertial-frame variant of the same gauge condition, the $J$ term effectively makes the lapse condition milder early in the inspiral; the $V^i$ term damps the comoving frame shift to small values (rather than damping the inertial frame shift to small values regardless of the object’s velocity); and the last term is a consequence of the transformation properties of $\Gamma_a$.
We have extensive experience with the inertial frame gauge condition. Whether placing this in the comoving frame helps is yet to be explored.
First Differential Order Form
-----------------------------
Historically there has been a lot more know-how available for the analytic and numeric treatment of first differential order systems. Motivated largely by this reality, the main stream Generalized Harmonic evolution system in [SpEC]{} has been written in first order differential form[@Lindblom2006]. This is done by introducing the auxiliary variables $\Phi_{iab}$ and $\Pi_{ab}$ defined by $$\begin{aligned}
N \Pi_{ab} &=& - \partial_t \psi_{ab} + \gamma_1 N^i \Phi_{iab} \\
\partial_i \psi_{ab} &=& \Phi_{iab} ,\end{aligned}$$ where $\gamma_1$ is a free parameter. The resulting evolution system takes the form $$\begin{aligned}
\partial_t\psi_{ab}&-&(1+\gamma_1)N^k\partial_k\psi_{ab}
= - N\Pi_{ab}-\gamma_1N^i\Phi_{iab},
\label{e:psiEvol}\\
\partial_t\Pi_{ab} &-& N^k\partial_k\Pi_{ab}
+ N g^{ki}\partial_k\Phi_{iab} -
\gamma_1 \gamma_2 N^k \partial_k \psi_{ab}
\nonumber\\
&=&2N\psi^{cd}\bigl(
g^{ij} \Phi_{ica} \Phi_{jdb}
- \Pi_{ca} \Pi_{db}
- \psi^{ef}\Gamma_{ace}\Gamma_{bdf}
\bigr)
\nonumber\\&&
-2N\nabla_{(a}H_{b)}
- \frac{1}{2} Nt^c t^d \Pi_{cd}\Pi_{ab}
-N t^c \Pi_{c i} g^{ij}\Phi_{jab}\nonumber\\
&&+N\gamma_0 \bigl[2\delta^c{}_{(a}t{}_{b)}-\psi_{ab}
t^c\bigr] ({H}_c+\Gamma_c)
- \gamma_1 \gamma_2 N^i \Phi_{iab},\label{e:PiEvol}\\
\partial_t\Phi_{iab}&-&N^k\partial_k\Phi_{iab}
+N\partial_i\Pi_{ab}-N\gamma_2\partial_i\psi_{ab}
\nonumber\\
&=&\frac{1}{2} N t^c t^d \Phi_{icd}\Pi_{ab}
+Ng^{jk}t^c\Phi_{ijc}\Phi_{kab}
-N\gamma_2\Phi_{iab}\label{e:PhiEvol} ,\end{aligned}$$ where $\gamma_0$ and $\gamma_2$ are additional free parameters of the system and we have included all terms, including the nonprincipal part.
The advantage of this system is that it is a manifestly first order symmetric hyperbolic system[@Lindblom2006]. One of the drawbacks is that introduction of auxiliary variables leads to additional constraints for the system. The new constraint quantities are $$\begin{aligned}
\label{eq:Ciab}
{\cal C}_{iab} &=& \partial_i \psi_{ab} - \Phi_{iab}
\\
{\cal F}_a &\approx& t^c \partial_c {\cal C}_a = N^{-1} \left(-\partial_t {\cal C}_a - N^i \partial_i {\cal C})a\right)
\\
\label{eq:Cia}
{\cal C}_{ia} &\approx& \partial_i {\cal C}_a
\\
{\cal C}_{ijab} &=& 2\partial_{[i} \Phi_{j]ab} = 2 \partial_{[j} {\cal C}_{i]ab} ,\end{aligned}$$ where full expressions for ${\cal F}_a$ and ${\cal C}_{ia}$ are given in Ref. (). It has been shown[@Lindblom2006] that the constraints associated with the extended, first order system form their own set of first order symmetric hyperbolic equations with source terms that vanish when the constraints are zero. By implication, given vanishing initial and boundary data for the constraints, the constraint propagation system will ensure that the constraints stay zero during the evolution. In addition, we find that, while the unmodified constraint propagation system may be prone to exponentially growing modes (seeded by numerical error induced at each time-step), the constraint damping terms $\gamma_0,\gamma_1,\gamma_2$ are sufficient to control these growing modes for all cases of interest. Today, in all “production-style” binary black hole runs, we use the system given by Eqs. (\[e:psiEvol\])-(\[e:PhiEvol\]).
In parallel, as a side effort, a first differential order in time, second differential order in space version of the same system has been implemented and was shown to work well for a limited number of test-cases[@Taylor:2010ki]. We expect that, if given enough effort, the second differential order form would be be at least a factor of two more efficient, given the smaller number of evolution variables the code requires. This approach has, however, received limited attention given the success of the first order system.
Boundary Conditions
-------------------
One of the benefits of expressing a system in a symmetric hyperbolic form is that this formalism provides a way of identifying the main characteristic speeds of a system across a given boundary, as well as the associated characteristic fields. This then leads to an easy-to-follow recipe for constructing boundary data. One must provide data for those quantities that are incoming through the boundary, while making sure that the outgoing quantities are updated through their evolution equations (and thus allowed to naturally leave the domain). An additional criteria is that the incoming data must not be specified in a way that injects constraint violating modes. There are a number of versions of constraint preserving boundary conditions for the generalized harmonic system. Most notably, Kreiss and Winicour has worked out a geometrically motivated, well-posed boundary system in Ref. .
The [SpEC]{} code implements a boundary algorithm based on Rinne, Lindblom and Scheel[@Lindblom2006]. The current algorithm works well for BBH evolutions of up to a few dozen orbits. However, it becomes a major limitation in evolutions lasting hundreds of light-crossing times – in these simulations our current boundary algorithm acts as a gravitational potential, causing an acceleration of the center of mass. The well-posedness and stability of this evolution-boundary system has been studied by Rinne in Ref. .
Domain Construction
===================
One key element influencing the design of the spectral domain used in the [SpEC]{} binary black hole simulations is the choice to use excision as a means of dealing with the space-time singularity inside the horizon. Finite difference codes have been successful in BBH evolutions with[@Pretorius2005a; @Szilagyi2007] and without excision[@Campanelli2006a; @Baker2006a]. While an excision-less domain is much easier to construct and an excision-less evolution algorithm is much less involved on a technical level than one with excision, spectral accuracy would be lost if the evolution field had nonsmooth features. The singularity would introduce such nonsmoothness, even if one factors out and regularizes the singularity (e.g., by evolving the conformal metric and the inverse of the conformal factor, as done in some of the puncture codes). As a result, in [SpEC]{} the computational domain needs to accommodate two inner spherical excision boundaries. In addition, the most natural outer boundary shape is spherical as well, which is easily accommodated by the wave-zone spherical grid used in our code.
![\[fig:CutSphereFullHalfGrid\] [ Left:]{} the $z\leq0$ portion of our spectral gird structure used for BBH runs. The outer region consists of a set of spherical shells centered at the coordinate origin. The angular spectral representation in these shells consists of a scalar spherical harmonic expansion, while radially we use Chebyshev Gauss Lobatto collocation points. These spherical shell subdomains are labeled as [SphereC0, SphereC1,...]{} in our code. [ Right:]{} A closer look at the interior of the grid used in our BBH runs. This plot displays the $z\leq0$ portion of the innermost wave-zone shell, [SphereC0]{}, surrounding a set of distorted cylinders and two inner spherical grid-structures, centered around the individual black-holes. The axes of the cylinders are aligned with the coordinate $x$ axis. These shapes are distorted such that the lower disk-shaped boundary of the cylinder is touching interior spherical shells centered around the individual black holes, while their outer boundary is touching the inner spherical boundary of [SphereC0]{}. The three-dimensional figure can be obtained by a $180^{\circ}$ rotation around the $x$ axis. ](Figures/CutSphere_FullHalfGrid "fig:") ![\[fig:CutSphereFullHalfGrid\] [ Left:]{} the $z\leq0$ portion of our spectral gird structure used for BBH runs. The outer region consists of a set of spherical shells centered at the coordinate origin. The angular spectral representation in these shells consists of a scalar spherical harmonic expansion, while radially we use Chebyshev Gauss Lobatto collocation points. These spherical shell subdomains are labeled as [SphereC0, SphereC1,...]{} in our code. [ Right:]{} A closer look at the interior of the grid used in our BBH runs. This plot displays the $z\leq0$ portion of the innermost wave-zone shell, [SphereC0]{}, surrounding a set of distorted cylinders and two inner spherical grid-structures, centered around the individual black-holes. The axes of the cylinders are aligned with the coordinate $x$ axis. These shapes are distorted such that the lower disk-shaped boundary of the cylinder is touching interior spherical shells centered around the individual black holes, while their outer boundary is touching the inner spherical boundary of [SphereC0]{}. The three-dimensional figure can be obtained by a $180^{\circ}$ rotation around the $x$ axis. ](Figures/CutSphere_SphereC0AndInterior "fig:")
![\[fig:CutSphereCylindersM\] (Color online) [ Left:]{} This plot reveals details of the grid in the immediate neighborhood of the two sets of spherical shells, centered around the individual black holes (shown as gray spheres). The view reveals an inner set of cylinders that touch, on one end, the outermost of these inner sets of shells, and on the other end an $x=$const. grid-plane, referred to as the [CutX]{} plane in our code. [ Right:]{} In this view we have removed the cylinders associated with the larger black hole and are providing a $3D$ grid-frame view of the cylinders around the smaller black hole. The close end of the cylinders shown here are all touching the [CutX]{} plane. The far end of the cylindrical structure touches the inner boundary of [SphereC0]{}, displayed in grey.](Figures/CutSphere_CylindersM "fig:") ![\[fig:CutSphereCylindersM\] (Color online) [ Left:]{} This plot reveals details of the grid in the immediate neighborhood of the two sets of spherical shells, centered around the individual black holes (shown as gray spheres). The view reveals an inner set of cylinders that touch, on one end, the outermost of these inner sets of shells, and on the other end an $x=$const. grid-plane, referred to as the [CutX]{} plane in our code. [ Right:]{} In this view we have removed the cylinders associated with the larger black hole and are providing a $3D$ grid-frame view of the cylinders around the smaller black hole. The close end of the cylinders shown here are all touching the [CutX]{} plane. The far end of the cylindrical structure touches the inner boundary of [SphereC0]{}, displayed in grey.](Figures/CutSphere_CylindersAroundSphereBZoomOut "fig:")
![\[fig:CutSphereCylindersAroundSphereB\] (Color online) [ Left:]{} This figure illustrates the $z<0$ portion of the spherical shells centered around the smaller black hole as well as the cylinders surrounding it (shown as a red wire frame). The large spherical surface on the close right side is the outer boundary of the shell structure around the larger black hole. The inner sets of spherical shells, around the individual black holes, are labeled [SphereA0, SphereA1...]{} and [SphereB0, SphereB1,..]{}. Both of these sets of shells are concentric, with their centers near the $x$ axis. (The exact location of the black holes is determined by the elliptic initial-data solver, as this determines the location of the black holes at $t=0$, and the shells are concentric with the associated black hole apparent horizon.) [ Right:]{} As a last illustration of our BBH grid, we show, from a close view, the excision boundary within the [SphereB0]{} subdomain (which is the innermost of the shells around the smaller black hole).](Figures/CutSphere_CylindersAroundSphereB "fig:") ![\[fig:CutSphereCylindersAroundSphereB\] (Color online) [ Left:]{} This figure illustrates the $z<0$ portion of the spherical shells centered around the smaller black hole as well as the cylinders surrounding it (shown as a red wire frame). The large spherical surface on the close right side is the outer boundary of the shell structure around the larger black hole. The inner sets of spherical shells, around the individual black holes, are labeled [SphereA0, SphereA1...]{} and [SphereB0, SphereB1,..]{}. Both of these sets of shells are concentric, with their centers near the $x$ axis. (The exact location of the black holes is determined by the elliptic initial-data solver, as this determines the location of the black holes at $t=0$, and the shells are concentric with the associated black hole apparent horizon.) [ Right:]{} As a last illustration of our BBH grid, we show, from a close view, the excision boundary within the [SphereB0]{} subdomain (which is the innermost of the shells around the smaller black hole).](Figures/CutSphere_CylindersAroundSphereBZoomIn "fig:")
After a number of attempts at the problem of domain construction, we have settled on the non-overlapping domain described in the Appendix of Ref. . This grid is composed of a small number of spherical shells around each excision boundary, labeled as [SphereA]{}$n$ and [SphereB]{}$n$, where $n=0,1,\ldots$ (see Fig. \[fig:CutSphereCylindersAroundSphereB\]). The innermost of these sets of spherical shells, also referred to as the excision subdomain, has index zero. In addition, the grid contains a third set of shells describing the gravitational wave zone (shown in Fig. \[fig:CutSphereFullHalfGrid\]), labeled as [SphereC]{}$n$, $n=0,1,\ldots$. In a typical simulation we have $20$ outer shells. The space between the innermost outer shell, [SphereC0]{} and the outer boundary of the inner shell structures is filled up by a set of cylindrical subdomains, distorted both on the lower and upper end, so that they touch their neighboring subdomain (shown in Fig. \[fig:CutSphereFullHalfGrid\], and Fig. \[fig:CutSphereCylindersM\]). The space between the two inner sets of spherical shells is filled in with another set of distorted cylinders which, on one end, are touching the spherical boundary of the outermost of [SphereA]{} and [SphereB]{}, while on the other end they touch an $x=$const grid-plane, referred to as the [CutX]{} plane, as shown on Fig. \[fig:CutSphereCylindersM\].
In conjunction with inter-block penalty boundary conditions and appropriate spectral filters (see Sec. \[sec:SpectralNum\]), a major advantage of our current compact binary grid is its robust stability with respect to high frequency noise generated by our numerical update scheme. This is a non-trivial property of a spectrally evolved BBH system where high accuracy leaves room for very little numerical dissipation. In addition, we find our domain to adapt well to binaries with very different masses.[@Mroue:2013xna; @Buchman:2012dw]
The domain construction (or control of its parameters) requires an extra amount of care near the merger of the binaries. This is to be expected, as BBH simulations tend to spend the longest amount of physical and wall-clock time in the inspiral phase. During this phase, little changes as the black holes slowly approach each other, while radiating energy via gravitational wave emission. It makes most sense to design a grid that does well during the early stage. As the binary approaches merger, the excision boundaries (kept in the near vicinity of the individual apparent horizons) approach each other. It is essential that all features of the grid are able to adjust to this deformation. In order to avoid grid singularities (or noninvertible maps, as described in Sec. \[seq:ControlSystem\]), at various stages during the plunge our algorithm defines a new grid, with modified parameters such that the position of the [CutX]{} plane remains between the spherical shells around the excision boundaries until after merger. See Secs. \[sec:rTypeAmr\] and \[sec:ShellDrop\] for details.
Control System {#seq:ControlSystem}
==============
As mentioned earlier, the [SpEC]{} code uses excision for evolving systems involving black holes. This creates its own set of challenges. In a finite difference code, in order to properly excise the interior of a black hole, one needs to locate the apparent horizon and implement an algorithm with a moving excision boundary. As the black hole moves on the grid, there will be grid-points that are evolution points on one time-step which become excision points on the next time level. In addition, there will be other points that are labeled as excision points at some time only to become evolution points as the black hole moves past them. This can be implemented on a point-wise basis, as has been done in a number of finite difference codes, e.g. in Ref. . In a spectral algorithm the removal of a single grid-point is not feasible as, in some sense, spectral differencing stencils are global within the subdomain (or block). As a result, there is no option to move an excision boundary along a particular direction by one grid-point. Rather, we employ a feed back control system[@Hemberger:2012jz] based on measurement of the position of the black holes, which rotates, translates and shrinks the grid such that the coordinate centers of the excision boundaries tightly follow the motion of the coordinate centers of the apparent horizons. In addition, we monitor the coordinate shape of the apparent horizons in the co-moving frame and apply a distortion map to the excision boundary such that it stays just inside the apparent horizon throughout the evolution, including plunge and merger. The full sequence of maps employed in a BBH evolution is given by[@Hemberger:2012jz] $x^k=\mathcal{M} (\hat x^{\hat k})$, where $\{x^k\}$ are inertial coordinates, $\{\hat x^{\hat k}\}$ are grid-frame coordinates, and $$\label{eq:MapSequence}
\begin{array}{ll}
{\mathcal M}_{{\rm Grid}\rightarrow{\rm Inertial}} =&
{\ensuremath{\mathcal{M}_{\rm Translation}}}
\circ{\ensuremath{\mathcal{M}_{\rm Rotation}}}
\circ{\ensuremath{\mathcal{M}_{\rm Scaling}}} \\
& \circ\;{\ensuremath{\mathcal{M}_{\rm Skew}}}
\circ{\ensuremath{\mathcal{M}_{\rm CutX}}}
\circ{\ensuremath{\mathcal{M}_{\rm Shape}}}.
\end{array}$$ Here
- ${\ensuremath{\mathcal{M}_{\rm Translation}}}$ is a spatial map which translates the mass-weighted average of the centers of the excision boundaries so that this point is in the neighborhood of the coordinate-center of mass of the BBH system as it evolves. This map leaves the position of the outer boundary unchanged.
- ${\ensuremath{\mathcal{M}_{\rm Rotation}}}$ rotates the grid such that the line connecting the centers of the excision boundaries is aligned with the one connecting the coordinate centers of the apparent horizons.
- ${\ensuremath{\mathcal{M}_{\rm Scaling}}}$ shrinks (or expands) the grid at the right rate such that the distance between the centers of the excision boundaries is synchronized with the distance between the coordinate centers of the apparent horizons. This map is also responsible for a slow inward motion of the outer boundary so that constraint violating modes with vanishing characteristic speed leave the domain rather than stay forever.
- [$\mathcal{M}_{\rm Skew}$]{} is responsible for distorting the [CutX]{} plane at late times, when the excision boundaries are near each other, such that this skewed grid plane stays at a finite distance from the excision boundaries even when they are non-perpendicular to the $x$ axis. See Figure 3 in Ref.
- [$\mathcal{M}_{\rm CutX}$]{} is responsible for controlling the position of the [CutX]{} plane (by moving along the $x$ axis) such that it remains at a finite distance from the point where the excision boundaries cross the $x$ axis. See Sec. \[sec:rTypeAmr\] as well as Ref. for details.
- [$\mathcal{M}_{\rm Shape}$]{} is responsible for deforming the excision boundary from its original spherical shape to that of the apparent horizon.
In Sec. \[sec:DampedHarmonicGauge\] of the paper we made reference to the [*comoving*]{} frame (with coordinates $\{\tilde x^{\tilde k}\}$) defined as the frame in which the coordinate centers of the apparent horizons are at a fixed position. This frame is connected to the inertial frame by the map $$\begin{aligned}
{\mathcal M}_{{\rm Comoving}\rightarrow{\rm Inertial}} &=& {\ensuremath{\mathcal{M}_{\rm Translation}}} \circ {\ensuremath{\mathcal{M}_{\rm Rotation}}} \circ {\ensuremath{\mathcal{M}_{\rm Scaling}}} \; ,
\label{eq:ComovingFrame}\end{aligned}$$ leading to $$\begin{aligned}
x^k_{\rm Inertial} &=&
{\ensuremath{\mathcal{M}_{\rm Translation}}} \left( {\ensuremath{\mathcal{M}_{\rm Rotation}}} \left( {\ensuremath{\mathcal{M}_{\rm Scaling}}} \left(\tilde x^{\tilde k}_{\rm Comoving}\right)\right)\right)
\; .\end{aligned}$$ Another frame of relevance is the [*distorted*]{} frame $\bar x^{\bar k}_{\rm Distorted}$, connected to the grid-frame coordinates $\hat x^{\hat k}_{\rm Grid}$ by $$\label{eq:DistortedFrame}
{\mathcal M}_{{\rm Grid}\rightarrow{\rm Distorted}} = {\ensuremath{\mathcal{M}_{\rm CutX}}} \circ {\ensuremath{\mathcal{M}_{\rm Shape}}}\; ,$$ giving $$\bar x^{\bar k}_{\rm Distorted}= {\ensuremath{\mathcal{M}_{\rm CutX}}} \left( {\ensuremath{\mathcal{M}_{\rm Shape}}} \left( \hat x^{\hat k}_{\rm Grid}\right)\right) \; .$$ This is the frame in which we search for the apparent horizons, feeding it into the control system responsible for updating the time-dependent parameters of the various maps. In terms of the maps associated with the distorted and the comoving frames the full transformation from the grid frame to the inertial frame can then be written as $$\begin{aligned}
x^k_{\rm Inertial} &=& {\mathcal M}_{{\rm Grid}\rightarrow{\rm Inertial}} \left( \hat x^{\hat k}_{\rm Grid} \right)
\\
&=&
{\mathcal M}_{{\rm Comoving}\rightarrow{\rm Inertial}} \left(
{\mathcal M}_{{\rm Skew}} \left(
{\mathcal M}_{{\rm Grid}\rightarrow{\rm Distorted}} \left(
\hat x^{\hat k}_{\rm Grid}
\right)
\right)
\right) .\end{aligned}$$
Note that for the inspiral part of our BBH simulations, where ${\ensuremath{\mathcal{M}_{\rm CutX}}}$ is inactive, the distorted frame is simply the grid frame distorted under the action of the shape-map ${\ensuremath{\mathcal{M}_{\rm Shape}}}$. Also, note that the comoving frame and the distorted frame differ by the skew-map ${\ensuremath{\mathcal{M}_{\rm Skew}}}$. The motivation of working with two different though similar frames is in their use. The comoving frame is instrumental in defining a gauge condition, as described in Sec. \[sec:DampedHarmonicGauge\]. The distorted frame is introduced for the sake of a more convenient implementation of our grid-control algorithm; thus its definition is more naturally stated through its relationship to the grid frame.
Pseudo-Spectral Numerical Algorithm {#sec:SpectralNum}
===================================
The irreducible topologies at the core of our grid construction are $$\begin{aligned}
I_1 &=& \left\{ x\in {\mathbb R}| a\leq x\leq b\right\} , \\
S_1&=&\left\{(x,y)\in {\mathbb R}^2| x^2+y^2=a^2 \right\},\\
S_2&=&\left\{(x,y,z)\in {\mathbb R}^3| x^2+y^2+z^2=a^2 \right\},\\
B_2&=&\left\{(x,y)\in {\mathbb R}^2| x^2+y^2\leq a^2 \right\} . \end{aligned}$$ The computational domain for binary black hole simulations is built of blocks that are topologically $$I_1 \times S_2,\; I_1 \times S_1 \times I_1, \quad \mbox{and} \quad I_1 \times B_2.
\label{eq:TopProduct}$$ The spectral basis is associated with the irreducible topologies $I_1, S_1, S_2, B_2$ is, in order, Chebyshev polynomials, Fourier expansion, scalar spherical harmonics and one-sided Jacobi polynomials. Each of these come with their set of collocation points so as to optimize the conversion between collocation point data and the spectral coefficient representation. Our algorithm stores collocation values of each evolved quantity, at a given time-step.
Derivatives are computed by forming the spectral coefficient representation of the same data and then recombining those with the analytic derivatives of the spectral basis functions. For example, given some smooth function $f(x)$ we approximate it as an expansion in terms of a spectral basis $b_i(x)$, $$f(x) \approx \sum_i c_i b_i(x) \, .$$ The derivative of the function will then be approximated by $$\frac{\partial}{\partial x}f(x) \approx \sum_i c_i \frac{\partial}{\partial x} b_i(x)$$ where the derivatives of $b_i(x)$ are known analytically.
Given the nonlinear nature of the evolution system of interest, we find it easier to compute the RHS of the equations, at each collocation point, using the actual function values (rather than their spectral coefficients). The nonlinearity implies coupling between neighboring coefficients. By implication, the highest coefficients of any given representation will effectively be updated by the “wrong equation.” In other words, their effective update scheme depends on whether there exists a higher coefficient in the data or not. In some cases this is not a problem since, for well-resolved, exponentially convergent representations of a smooth evolved field, the highest coefficient may already have a very small magnitude so that having (or not having) a next higher coefficient would have no appreciable impact on the evolution of the field. In order to achieve robust stability, we find that, we must filter these higher coefficients. In particular, after each time-step we set the highest four coefficients of the $Y_{lm}$ expansions to zero[@Scheel2006], $$c_{lm} \rightarrow c_{lm} \times H[L_{\max}-l-4] ,
\label{eq:HeavisideFilter}$$ where $H[n]$ is the Heaviside step-function. In addition, we use the exponential filter $$c_k \rightarrow c_k \exp\left[-\alpha \left(\frac{k}{N-1}\right)^{2p} \right], \quad k=0, \ldots, N-1
\label{eq:ExpChebFilter}$$ with $\alpha=36,p=32$ for the Chebyshev coefficients, as suggested in Refs. and . We use the same exponential filter for the Fourier coefficients as well, using $\alpha=36,p=24$, though in this case our choice is entirely heuristic. These filters are applied to the evolution variables $\psi_{ab}, \Pi_{ab}, \Phi_{iab}$ after each full time-step. In addition, each time one subdomain provides boundary data to another via interpolation, this interpolated data is filtered using the Heaviside filter Eq. (\[eq:HeavisideFilter\]) for the scalar spherical harmonic expansions on $S_2$ type boundaries and the exponential filter Eq. (\[eq:ExpChebFilter\]) for all Chebyshev and Fourier expansions. For those boundaries, where the neighboring collocation grids are aligned at the boundary, there is no need for interpolation. In these cases we are copying the data from one subdomain to the other without any filtering. Having the right filtering scheme proved crucial in being able to establish robust stability.
Accuracy Diagnostics
--------------------
Exponential convergence of the spectral representation of smooth data in a particular basis requires that, for sufficiently large mode number, the expansion coefficients are exponentially decaying functions of the mode number. As an example, for a smooth function on an $S_2$ surface, the spectral expansion in terms of scalar spherical harmonics is $$f(\theta,\phi) = \sum_{l=0}^{L} \sum_{m=-l}^{l} C_{lm} Y_{lm}(\theta,\phi) + \mbox{residual} .$$ When the function $f$ is well resolved, then the magnitude of the highest order coefficients will decay exponentially, $$C_{l,m} \sim e^{-l}, \quad \mbox{for large} \; l .$$ This suggests that one should monitor the quantity $$\label{eq:YlmPowerMonitor}
P_{l} = \sqrt{\frac{1}{2m+1} \sum_{m=-l}^l | C_{l,m} |^2}$$ in order to asses the accuracy of the spectral representation. This is done in a more generic context. Whenever dealing with tensor products of irreducible topologies, we monitor accuracy of the representation by computing the average (in an $L_2$ sense) over all other dimensions; e.g., for $I_1 \times S_1 \times I_1$, we define power-monitors according to the three indexes, $P_{k_0}, P_{k_1}, P_{k_2}$ as $$\label{eq:GenericPowerMonitor}
P_{k_0} = \sqrt{ \frac{1}{N_1 N_2} \sum_{k_1,k_2} \left|C_{k_1,k_2,k_3} \right|^2} ,$$ with similar definitions for $P_{k_1},P_{k_2}$. A time-snapshot of an example power-monitor plot is provided in Fig. \[fig:PowerMonitor\]. A few key features are immediately noticeable. The highest mode has $O($roundoff$)$ power, which results from filtering. The lowest six modes (or: the first six, counting from the left) in Fig. \[fig:PowerMonitor\] show a clean exponential decay. This is indicative of exponential convergence, and allows us to define the convergence factor as the logarithmic slope of the power in the coefficient vs. its index. The ratio of the highest to the lowest coefficient in this clean convergent set of points gives a measure of the truncation error in the expansion. And the few extra modes that are placed between the convergent modes and the filtered mode are labeled as “pile-up modes.” These indicate saturation of the grid with respect to the data that is being spectrally expanded. They are most likely related to noise in the data, possibly due to non-smooth boundary data on the subdomain boundary.
![\[fig:PowerMonitor\]Plot of a typical power-monitor. The first six points (counting from left to right) show exponential convergence of the spectral expansion. The slope (on the linear-log plot) of these points defines of the [*convergence factor*]{} of the spectral expansion. The next three points show no convergence. They are designated as [*pile-up modes*]{}. The point on the far right, with $O($roundoff$)$ value, is a coefficient that is reset by a filtering algorithm at the end of each time-step. We call this a [*filtered mode*]{}.](Figures/PowerMonitor)
For spectral Adaptive Mesh Refinement (AMR), it is important to define all of these quantities as smooth functions of the power monitor values $P_k, k=0,\ldots,N_k-1$. Let ${\cal S}[k_1,k_2]$ be the slope of the least square fit of $\log_{10}(P_k) = f(k)$ for the points $k=k_1\ldots k_2$, with $0\leq k_1<k_2\leq N_k-1$. Let ${\cal E}[k_1,k_2]$ be the error in this fit. In our current algorithm we first compute ${\cal S}[k_1,k_2]$ and ${\cal E}[k_1,k_2]$ for a variety of selections of data points. Then we define the convergence factor (up to an overall negative sign) as the average of the slopes, weighted by the inverse of their fit error, $$\label{eq:ConvFactor}
{\cal C}[P_k] \equiv -
\left.
\sum_{
\begin{subarray}{c}
k_1 = 0, 2 \\
k_2 = k_1+4 , \tilde N_k -1
\end{subarray}
}
{ \frac{ {\cal S}[k_1,k_2] } { \epsilon+ {\cal E}[k_1,k_2] } }
\right/
\sum_{
\begin{subarray}{c}
k_1 = 0, 2 \\
k_2 = k_1+4, \tilde N_k -1
\end{subarray}
}
{ \left( \epsilon+ {\cal E}[k_1,k_2] \right)^{-1} }
,$$ where $\epsilon$ is a small positive constant to avoid division by zero in case the linear fit has no error, while $\tilde N_k$ is the number of unfiltered modes, with $\tilde N_k \leq N_k$. Note that for some of the filters, in particular for the scalar $Y_{lm}$ filter used in the evolution of tensor fields, the filtering algorithm amounts to setting the highest few coefficients to zero, as shown in Eq. (\[eq:HeavisideFilter\]). In this case the number of filtered modes $N_k - \tilde N_k$ would equal the number of those coefficients that are being reset. In the case of the exponential Chebyshev filter Eq. (\[eq:ExpChebFilter\]) we approximate the number of filtered modes as those whose coefficient is below some threshold, e.g., it is $O($roundoff$)$. By calculating a large number of fits $S[k_1,k_2]$ and weighting them by the inverse of the accuracy of the fit, we give the larger weight to those sets of points that provide a clean slope. This is important on both ends of the spectrum. For the lower order modes one cannot expect, generically, a particular behavior (such as exponential decay) as we look at the spectral expansion of an arbitrary (smooth) function.[^4] The highest modes may be tainted by noise (they could be pile-up modes) which, again, would lead to inaccurate measure of the convergence factor. This algorithm also gives some level of robustness when certain coefficients show anomalies (e.g. because the data has some excess power in a given spectral mode). An example power-monitor (as a function of time), along with the associated convergence factor computed using Eq (\[eq:ConvFactor\]), is given in Fig. \[fig:ConvFactor\].
![\[fig:ConvFactor\] Convergence factor for the radial spectral expansion of the [SphereC5]{} wave-zone spherical shell for a non-spinning $22.5$ orbit BBH run, with mass ratio $q=7.1875$. The upper panel shows the power-monitor Eq. (\[eq:GenericPowerMonitor\]) associated with the radial $I_1$ expansion. The noise in the early part of the simulation is caused by the junk radiation (high frequency radiation seen in the early part of BBH evolutions, caused by unphysical content of the initial data) passing through the grid. The lower panel shows the convergence factor Eq. (\[eq:ConvFactor\]) for this same power-monitor. As the junk radiation passes through the grid, the convergence factor drops by nearly one order of magnitude. However, as this high frequency wave leaves the grid, the convergence factor rebounds and stays $O(1)$ for the rest of the evolution. ](Figures/PowerDiag_SphereC5_Bf0I1_ConvergenceFactor)
Pile-up modes are identified by a local convergence factor estimate, involving the mode under consideration and its next few higher neighbors. Once again, the AMR algorithm benefits from a pile-up mode counter that is a continuous function of the data (rather than a discrete counter). In order to measure the extent to which a given mode $j$ is a to be considered as a pile-up mode, we measure a local convergence factor around the $j$-th point, $$\tilde {\cal C}_j \equiv - {\cal S}[j,\min(\tilde N_k,j+4)] ,$$ and compare it to the overall convergence factor, ${\cal C}[P_k]$. The number of pile-up modes is then defined to be $${\cal P}[P_k] \equiv \sum_{j=2}^{\tilde N_k -1 }
\exp
\left[
-32 \left(
\frac{ \tilde {\cal C}_j}{{\cal C}[P_k] }
\right)^2
\right] .
\label{eq:PileUpMode}$$ If the local convergence factor estimate $ \tilde {\cal C}_j $ associated with the mode $j$ has a value close to that of the overall convergence factor, i.e. if $$\tilde {\cal C}_j \approx {\cal C}[P_k] ,$$ than that mode will have an $O($roundoff$)$ contribution to ${\cal P}[P_k]$. If it is near zero, i.e., $$\tilde {\cal C}_j \ll {\cal C}[P_k]$$ this indicates that we are down in the noise floor of the power-monitor plot, and the contribution of the individual mode to the number of pile-up modes will be $O(1)$. A plot showing the number of pile-up modes, and the associated power monitor, for a typical BBH production run is shown in Fig. \[fig:TruncErrAndPileUp\].
![\[fig:TruncErrAndPileUp\] The [top]{} panel shows the power-monitor Eq. (\[eq:GenericPowerMonitor\]) associated with the angular $S_2$ grid of the [SphereC5]{} wave-zone spherical shell for the first $500M$ evolution of a non-spinning $22.5$ orbit BBH run, with mass ratio $q=7.1875$. The four noisy modes with $O($roundoff$)$ values are the filtered coefficients. As the junk radiation passes through, the unfiltered modes pile-up on each other. Eventually the junk radiation leaves the grid, however, and convergence of the higher modes is recovered. [Middle]{} panel: the number of pile-up modes, Eq. (\[eq:PileUpMode\]), show very clearly when do the power spectrum coefficients clutter next to each other. [Bottom]{} panel: the truncation error estimate, Eq. (\[eq:TruncError\]) for the same data. When the junk radiation is at its peak, around $t=150M$, the truncation is as high as $-2.5$ (i.e. the actual representation error can be as large as $10^{-2.5}$). After the junk radiation passes through, the truncation error settles to a much more desirable value, around $-6$. This is consistent with the six orders of magnitude covered by the unfiltered coefficients at $t=500M$, as shown in the upper panel.](Figures/PowerDiag_SphereC5_Bf1S2_PileUp_TruncErr)
A third quantity essential in our spectral AMR algorithm is the truncation error estimate associated with the power-monitor $P_k$. This is computed using the expression $$\begin{aligned}
{\cal T}[P_k] &\equiv& \log_{10}\left(\max\left(P_1,P_2\right) \right)
- \frac{ \sum_{j=1}^{\tilde N_k} \log_{10}\left(P_j\right) w_j } { \sum_{j=1}^{\tilde N_k} w_j }, \quad \mbox{where}
\\
\quad w_j &\equiv& \exp\left(-\left(j-\tilde N_k+\frac{1}{2}\right)^2\right) .
\label{eq:TruncError}\end{aligned}$$ The aim here is to count the number of digits resolved by the given set of spectral coefficients. This is computed as the difference between the power in the larger of the two lowest order modes, $\log_{10}\left[\max\left(P_1,P_2\right) \right]$, and the power in the highest modes, which itself is computed as an exponentially weighted average, giving maximum weight to the last two points (refer to Fig. \[fig:TruncErrAndPileUp\]).
The truncation error ${\cal T}[P_k]$, the convergence factor ${\cal C}[P_k]$ and the number of pile-up modes ${\cal P}[P_k]$ are the three essential measures that our AMR algorithm relies upon.
Adaptive Mesh Refinement
========================
In finite-difference based discretization schemes one commonly used AMR algorithm is that described by Berger and Oliger in Ref. . In such a scheme the evolution would be done on a sequence of refinement levels simultaneously. The truncation error between the two highest refinement levels can then be used to monitor the accuracy of the evolution and to dynamically assign the number of refinement levels to a particular region of the grid. Each refinement level will have a factor of two smaller grid-step (and possibly time-step) than the next coarser level. For a region that has $N$ refinement levels at a particular instant of time, there will simultaneously co-exist the highest level, the coarser one, the next coarser one, to the coarsest, all used to discretize this region at varying levels of accuracy.
Our approach to mesh refinement is different. The [SpEC]{} code uses a single layer of grid for a given (BBH) simulation. On a very basic level, our mesh refinement algorithm monitors the truncation error estimates associated with each irreducible topology, within each subdomain, throughout the run and then adjusts the number of collocation points (and, accordingly, the number of spectral basis elements) in order to keep the local truncation error within some desired range. This is called $p$-type mesh refinement.
In addition, as the objects approach each other while on their trajectories, our control system and the associated maps shift the boundaries of the various subdomains with respect to each other in order to accommodate the requirement that the excision boundary must stay inside the apparent horizons at all times. As a consequence, from time to time subdomain boundaries need to be re-drawn in order to reduce the stretching or compression of the grid caused by their continuous drifting under the action of the various maps. When this change of the grid structure implies splitting or joining of subdomains, we call this as $h$-type mesh refinement. When it preserves the number of subdomains but shifts their boundaries, it is called $r$-type mesh refinement. In addition, our AMR driver splits subdomains when the number of spectral modes required for a given target accuracy exceeds some threshold. This happens regularly with the spherical shells surrounding the smaller black hole in high mass ratio mergers, and is another example of $h$-type mesh refinement in our code. In the following sections we give a description of each of these elements of our AMR algorithm.
$p$-type mesh refinement
------------------------
To zeroth order the $p$-type mesh refinement algorithm monitors the truncation error estimate ${\cal T}[P_k]$ for each power-monitor $P_k$ associated with the individual subdomains, and adjusts the number of collocation modes (by adding or removing modes) such that the accuracy of the spectral representation of the evolved quantities is within some desired range. This behavior is seen on Fig. \[fig:AmrB0Shell\]. The algorithm behind the plot contains additional elements described in the remainder of this section. Nevertheless, the plot is a good illustration of how $p$-type mesh refinement works in our code.
![\[fig:AmrB0Shell\] (Color online) The plot illustrates $p$-type mesh refinement for the innermost spherical shell around the smaller black hole (labeled [SphereB0]{}), for a $q=7.1875$ non-spinning BBH inspiral and merger. The [ top]{} panel shows the combined scalar spherical harmonic power-monitor Eq.(\[eq:YlmPowerMonitor\]) for the variables $\Phi_{iab}$ and $\Pi_{ab}$ as a function of time. The lowest four noisy modes are filtered (set to zero) before each time-step. The power monitor is evaluated after the time-step, showing that the coefficients stay small from one step to the next. The [ middle]{} panel shows the target truncation error (blue) as well as the actual numerical truncation error (red). The [ lower]{} panel shows the spectral resolution (in this case $L_{\max}+1$) as a function of time. Each time the measured numerical truncation error goes above the target value, the AMR driver adds a spectral basis element to the representation (i.e., increments $L_{\max}$ in the $Y_{lm}$ expansion), bringing the truncation error to below its target.](Figures/Amr_SphereB0_Bf1S2_GridExtentControl)
Each time a basis associated with a particular subdomain is extended, we find that the numerical evolution system needs some time to adjust and find a new quasi-equilibrium for the evolved coefficients. This can be seen both in the power-monitors and in the constraints, as shown on Fig. \[fig:AmrA3Noise\]. For this reason, our current algorithm calls the AMR driver at pre-set time-intervals (rather than every time-step). Each time a particular grid extent is changed, it will be kept at the new level for some time before it is allowed to change. During the early inspiral, grid extents are changed no more frequently than every $O(100)M$ evolution time. This time-interval is gradually decreased and during the plunge individual grid extents are allowed to change every $O(1)M$. An alternative way of dealing with this same problem would be to use a smooth filter for all spectral expansions and control the parameter of this filter as a continuous function of time so that when the AMR driver decides to remove a point the filtering coefficients gradually become stronger for the highest unfiltered coefficient, until it reaches $O($roundoff$)$ level. At this point the highest coefficient could be safely be removed (as it could only gain power from coupling to the next highest coefficient which by now is also filtered) and doing so would not present a noticeable ‘shock’ to the numerical evolution scheme. A similar approach could be applied for adding a new spectral coefficient. Implementation of such a ‘smooth’ resolution change is future work.
![\[fig:AmrA3Noise\] (Color online) Spherical and radial power-monitors as well as constraint errors during a portion of a $q=7.1875$ non-spinning BBH simulation, displaying the effects of a change in resolution. [ Top panel]{}: Angular power-monitor for the outermost spherical shell (labeled ‘[SphereA3]{}‘) around the larger black hole. At around $t=3090$ the AMR driver increases the angular resolution $L_{\max}$ from $22$ to $23$. One implication is that the $L=18$ coefficients, which had been the lowest of the modes filtered according to Eq. (\[eq:HeavisideFilter\]), will now be the highest of the un-filtered modes. The power-monitor shows this coefficient joining the evolved (or un-filtered) modes, while the newly added $L=23$ mode will be the highest of the four filtered modes. [ Middle panel]{}: Radial power-monitor of [SphereA3]{}. At the time the angular resolution is increased, the highest coefficient (which is the most sensitive as being the smallest in value) shows a temporary spike of $O(10^{-9})$. This spike is not dictated by the evolution equations, but rather is a consequence of the sudden change in the numerical algorithm (i.e., a change in the grid). [ Lower panel]{}: The $L_2$ norm of the constraint error in the spherical shells around the larger black hole. At the time of the angular resolution change of [SphereA3]{} there is a small, $O(10^{-7})$ spike seen in that same subdomain. A much smaller, $O(10^{-8})$ spike is seen in the neighboring ${\tt SphereA2}$ a short time later. This constraint error injection can be seen as a consequence of the sudden change of the evolution equation prescribing the $L=18$ coefficient resulting in a jump from near roundoff values to $O(10^{-9})$. This sudden change, dictated by the numerical scheme rather than Einstein’s equations, generates a constraint violating mode that dissipates away on a time-scale of $O(10)M$. ](Figures/Amr_SphereA3_NoiseFromGridChange)
### Target truncation error {#sec:TargetTruncError}
An important element in the robustness of our current AMR algorithm is the proper setting of the target truncation error. An essential requirement on the angular resolution of the spherical shells is that they must be able to represent the data at a level accurate enough for the needs of the horizon finder. The horizon finder, however, does not depend only on the angular representation of the data. Radial resolution is required as well. Another important physical aspect that we want to represent accurately in our simulation is the amplitude and, more importantly, the phase of the gravitational wave (as gravitational wave detectors are more sensitive to phase than amplitude). In the outer layers of the grid, these waves propagate as outgoing spherical wavefronts, which do not require a lot of resolution. It is important, however, to get the orbital phase of the binary right, since if the orbits are inaccurate then the waves (generated by the motion of the black holes) cannot be accurate either. For the sake of simplicity, we use the label ‘A’ for the black hole contained in the subdomains [SphereA]{}$n$ and ‘B’ for the black hole contained in the subdomains [SphereB]{}$n$. In the region immediately surrounding the black holes (including the subdomains discretizing their gravitational attraction between the two excision boundaries), we find that we need higher accuracy. Our target truncation error function is written as $$\label{eq:TruncErrorFunction}
{\cal T}^{\max}\left[ w_A, w_B\right] \equiv {\cal T}_0 - 4
\frac{ w_A + w_B }{w_A+w_B+1} \; ,$$ where the weighting coefficients $ w_A,w_B$ are the $L_\infty$ norm on each subdomain of the smooth weight functions $$\tilde w_X(x^i,t), \quad X=A,B$$ which are expressed in terms of inertial coordinates $x^i$, with maximum values around the time-dependent coordinate-center $x^i_X(t)$ of the individual black holes labeled ‘A’ and ‘B’: $$\begin{aligned}
&& \tilde w_X(x^i,t) \equiv \exp\left[ - \sum_i \left(\frac{x^i-x_X^i(t)}{c_X}\right)^2 \right], \quad X=A,B, \quad \mbox{and}
\\
&&
c_A \equiv \sum_i \left. \left( x_B^i(t)\right)^2\right|_{t=0}
, \quad
c_B \equiv \sum_i \left. \left( x_A^i(t)\right)^2\right|_{t=0} .\end{aligned}$$
Note that here we use the initial distance of black hole ‘A’ to the coordinate origin to set the falloff rate of the Gaussian controlling the truncation error requirement around black hole ‘B’, and vice-versa. The rationale is that if one black hole is larger, it starts off closer to the origin, i.e., the other smaller black hole will have a smaller Gaussian around it. As a further constraint on ${\cal T}^{\max}\left[ w_A, w_B\right]$, this quantity cannot differ by more than $\log_{10}2$ between two neighboring subdomains. This rule is enforced by setting stricter truncation error on those subdomains that otherwise would end up being coarser than intended. We plot both the subdomain-wise constant ${\cal T}^{\max}\left[ w_A, w_B\right]$ and the smooth function $ {\cal T}^{\max}\left[ \tilde w_A(x^i,t), \tilde w_B(x^i,t) \right]$ in Fig. \[fig:TruncErrorOnDomain\].
![\[fig:TruncErrorOnDomain\] (Color online) A $z=0$ half-slice of the target truncation error ${\cal T}^{\max}\left[ w_A, w_B\right]$ (solid gray) , shown on the subdomains near the black holes, as well as its smooth approximate ${\cal T}^{\max}\left[ \tilde w_A(x^i,t), \tilde w_B(x^i,t) \right]$, evaluated at $t=0$ (color-coded wireframe). The plot shows the start of a non-spinning, mass ratio $7.1875$ BBH simulation. The individual black holes are initially located on the $x$ axis. The elevation of the surfaces is set by the magnitude of ${\cal T}$, so that higher elevation means more accuracy. Near the individual black holes the truncation error requirements are stricter, as shown. ](Figures/TruncError_Init)
### Special rules for spherical shells {#seq:AmrRulesForShells}
In order to control the accuracy of the apparent horizons themselves (and, as such, of the numerically evolved black holes), we define three criteria that factor into the final choice of the angular resolution of the horizon finder:
1. We form a $Y_{lm}$ expansion of the shape function of the horizon surface, $r(\theta,\phi)$ which is defined as the coordinate distance between the point $(\theta,\phi)$ on the surface and its coordinate center.[^5] Given this spectral expansion, we form a power monitor Eq. (\[eq:YlmPowerMonitor\]) and then compute the truncation error estimate Eq. (\[eq:TruncError\]) and the number of pile-up modes Eq. (\[eq:PileUpMode\]). If the surface shape is coarser than required and if the number of pile-up modes is not larger than some threshold, we increase the accuracy of the horizon finder. If the number of pile-up modes is already at its limit, adding more modes would not help.
2. As a second method of determining adequacy of resolution, we form a $Y_{lm}$ expansion of the areal radius of the apparent horizon surface and require that it be represented with a truncation error that is within a desired range. Note that the surface areal radius will reduce to the coordinate radius in flat space time. Imposing an accuracy requirement on the surface areal radius requires an accurate representation of the metric quantities on the horizon surface. This in turn allows for accurate measurement of physical quantities such as areal mass and spin. Similar to the coordinate radius, the horizon finder will increase accuracy if the areal radius is under resolved (and does not have too many pile-up modes).
3. As yet another method for imposing an accuracy requirement on the horizon finder algorithm, we monitor the residual of the horizon finder. If larger than requested, the angular resolution is increased.
If the accuracy of the surface is better than required by any of these measures, the horizon finder will decrease accuracy.
The angular resolution of the horizon finder can be used to set a lower limit on the angular resolution of the underlying three-dimensional evolution grid. Given the top-4 Heaviside filters Eq. (\[eq:HeavisideFilter\]), we request that the set of spherical shells around the excision boundary associated with a given horizon must have five additional $L$ modes beyonf the surface finder’s resolution. This implies that, should any of the horizon finder accuracy criteria request higher resolution, additional data is available upon interpolation from the evolution grid. Once the horizon finder increases its resolution, the spherical shells around its excision boundary have their angular resolution increased.
The angular resolution of the horizon finder is a useful lower limit for the spherical shells around an excision boundary. There are additional, heuristic rules that we found useful not only for accuracy but also for well behaved constraints.
The first such rule is that given the angular resolution $L$ of a spherical shell subdomain, its neighbors must not have angular resolutions larger than $L+1$ or smaller than $L-1$. There is yet another heuristic rule relating the angular resolution of the excision subdomain to the spherical shells surrounding it. As the binary system evolves, the further a subdomain is from the excision boundary, the less spherical symmetry will be in the metric data. Based on truncation error estimates, this means that the AMR driver will assign larger angular resolution to the spherical shells that are further from their associated excision boundary and closer to the other black hole. In certain cases, however, the metric data on the excision boundary may itself require a large resolution (e.g., if the black hole has a very large spin or distortion). When the subdomain next to the excision boundary has a larger angular resolution that its neighbors, we find that the constraint errors grow on a short time-scale. This may be related to the fact that if a given subdomain boundary has subdomains of different resolution on its two sides, those angular modes that are represented on only one side of the boundary will be reflected. When the excision subdomain needs high resolution, it may also be responsible for a larger amount of high frequency noise generation. If this is the case and its neighbor has a lower angular resolution, all that noise can get trapped in the excision subdomain and have non-desired effects on the black hole itself. Thus, to rule this out, if the excision subdomain needs resolution then all its neighbors are given the same, or larger, resolution.
The angular resolution of the set of shells resolving the wave-zone is driven by a much simpler set of rules. In this region we base the angular resolution of the shells on the truncation error estimate Eq. (\[eq:TruncError\]) derived from the power-monitor associated with the scalar $Y_{l,m}$ expansion of the main evolution variables $\left\{\psi_{ab},\Phi_{iab},\Pi_{ab}\right\}$. Similar to the inner sets of shells,we do not allow neighboring wave-zone shells to have their angular resolution differ by more than one $L$. As a possible future improvement, it may be worth forming a power-monitor from the spherical harmonic decomposition of the Weyl scalar $\Psi_4$ and require that the truncation error associated with this power-monitor also be within limits.
### Pile-up mode treatment
An essential element of our Spectral AMR algorithm is the treatment of pile-up modes. The source of these modes is not fully understood, although we expect that inadequate filtering or unresolved high frequency modes injected through a subdomain boundary can lead to such modes. It is also expected that, when the evolved quantities have a lot of non-trivial features, a more fine-grained definition of the truncation error estimate Eq. (\[eq:TruncError\]), convergence factor Eq. (\[eq:ConvFactor\]) and pile-up modes Eq. (\[eq:PileUpMode\]) could prove helpful in better tracking and controlling the numerical representation of the evolved quantitites.
As seen in Fig. \[fig:PowerMonitor\], the presence of these pile-up modes does not contribute to the accuracy of the data. On the contrary, we find in practice that these modes are constraint violating modes. This suggests that pile-up modes are result of the numerical approximation and not a property of the underlying analytic system. In other terms, Einstein’s equations will likely not dictate the presence of such modes. Rather, these develop on the grid as a result of limitations of our particular numerical scheme. We do not have experience with other spectral evolution codes and therefore are unable to assess whether these modes are a generic property of spectrally evolved partial differential equations. In our experience, however, the less pile-up modes, the better. For this reason we make it a priority to eliminate them, or to prevent their creation.
One immediate way of handling pile-up modes is to remove them. If a power-monitor shows these modes, we simply reduce the grid-extent associated with the piled-up power-monitor. In some cases doing so would violate some of our other rules of thumb (e.g., the horizon finder may need a certain minimum resolution). In this case we keep the resolution of neighboring subdomains from increasing until the pile-up modes leave the grid either by filtering or by propagating away from the subdomain under consideration. Fig. \[fig:SphereC6\_PileUpMode\] illustrates, what we find in most cases, that removal of pile-up modes helps reduce the constraint violating modes on the grid. This suggests that these modes do not result from the underlying analytic system but are instead a numerical artifact.
![\[fig:SphereC6\_PileUpMode\] (Color online) Pile-up mode removal and its effects in a non-spinning $q=7.1875$ BBH evolution. The [ first]{} panel from the top shows the highest order modes in the radial power-monitor of the wave-zone spherical shell labeled [SphereC6]{}, for the time-interval $[850,950]$. At around $t=900$ the AMR algorithm removes a radial point in order to reduce the number of pile-up modes in this subdomain. The [ second]{} panel from the top shows that the pile-up mode diagnostic Eq. (\[eq:PileUpMode\]) is indeed decreased as the number of radial modes decreases by one. The [ third]{} panel from the top displays the truncation error Eq. (\[eq:TruncError\]) associated with both the angular representation (green curve) and the radial representation (black curve). As seen from this panel, the overall truncation error in this subdomain is dominated by the angular representation – removing a radial mode should have little effect on overall accuracy. The [ bottom]{} panel plots the $L_2$ norm of the (unnormalized) constraint error in the same subdomain. Remarkably, the constraints drop at around $t=900$ by a sizable factor, as a pile-up mode is removed. This confirms that the pile-up mode being removed was a constraint violating mode. We find, in general, that removing these modes can in fact lead to a lower amount of constraint violation on the grid. ](Figures/SphereC6_PileUpMode)
### AMR driven by projected constraint error
As an alternative way of determining whether a given grid extent needs to be changed, one can monitor the constraint quantities ${\cal C}_{ia}, {\cal C}_{iab}$, defined in Eqs. (\[eq:Cia\]),(\[eq:Ciab\]). The first of these indexes are related to derivatives, while the rest of the indexes are related to the evolution frame of the main system (i.e., the inertial frame). In order to derive an accuracy requirement based on the constraints, we make use of the fact that each subdomain is constructed as a product of topologies (see Eq. (\[eq:TopProduct\]) for a list of these topology products). We form an estimate of the contribution of the truncation error associated with the individual topologies by projecting the derivative indexes from the evolution frame $\{x^k\}$ into the frame associated with the topology product that is used to construct the grid $\{\hat x^{\hat k}\}$.
That is, for a spherical shell, we project the Cartesian $i$ index of ${\cal C}_{ia}$ onto spherical coordinates, summing over the non-derivative indexes. In general, we write $$\begin{aligned}
{\cal E}^P_{\hat k} \left[ {\cal C}_{ia} \right] &:=& \frac{1}{N_{\hat k}}
\sqrt{\sum_a \left( \sum_i \frac{\partial x^i}{\partial \hat x^{\hat k}} {\cal C}_{ia}\right)^2 }
\\
{\cal E}^P_{\hat k} \left[ {\cal C}_{iab} \right] &:=& \frac{1}{N_{\hat k}}
\sqrt{\sum_{ab} \left( \sum_i \frac{\partial x^i}{\partial \hat x^{\hat k}} {\cal C}_{iab}\right)^2 }\end{aligned}$$ where the normalization coefficient $N_k$ is set using $$N_{\hat k} =
\sqrt{\sum_i \left( \frac{\partial x^i}{\partial \hat x^{\hat k}} \right)^2 }.$$ The $L_2$ or the $L_\infty$ norm of ${\cal E}^P_{\hat k}[{\cal C}_{ia}]$ and ${\cal E}^P_{\hat k}[{\cal C}_{iab}]$ over the individual subdomains can then serve as an indicator of whether the spectral expansion associated with the topological coordinate $\hat x^{\hat k}$ has an adequate accuracy. We have tested this algorithm in conjunction with the truncation-error based AMR described in Sec. \[sec:TargetTruncError\], always applying the stricter of the two requirements. We find that this additional accuracy requirement can be helfpul in the more dynamic parts of the BBH simulation (such as during plunge) where constraint violating modes can develop faster and spoil the physical properties of the system.
$h$-type mesh refinement
------------------------
In a general spectral AMR algorithm, $h$-type mesh refinement is motivated largely by efficiency considerations. If it takes too many spectral coefficients to resolve data along a particular axis of the subdomain then differentiation may become very expensive, the time-step may be limited due to a stricter CFL limit, memory limitations may arise, and load balancing may become a problem if the application is parallelized. All of these concerns are relevant for our code as well. We monitor the convergence factor Eq. (\[eq:ConvFactor\]) of a given power-monitor and when it reaches values below some arbitrary threshold, the subdomain will be split. In our binary black hole evolutions typical values for the convergence factor are of order unity, so our threshold for splitting is set to $0.01$. In addition, we find it useful to set hard limits on the maximum number of spectral coefficients allowed for a specific irreducible topology of a give subdomain type. For instance, our rule is not to have more than 20 radial points in any spherical shell. If the $p$-type AMR driver finds that this number is not sufficient then the subdomain will be split in the radial direction.[^6]
Another role of the $h$-type AMR in our runs is around the individual excision boundaries. For a number of our binary simulations one of the apparent horizons will pick up more and more angular resolution, forcing the underlying spherical shells to add spherical harmonic coefficients with higher and higher values of $L$. When the angular resolution reaches a user-specified limit, the spherical shell is split. We find it important that when such a spherical shell is angularly split, all shells outside it (among those around the same excision boundary) be split as well. This, once again, is a rule we found useful by trial and error when optimizing for small constraint violation.
### Shell-dropping around the excision boundaries {#sec:ShellDrop}
What we call our ‘shell dropping algorithm’ is a use of $h$-type AMR which is an absolute must in our binary evolutions.
![\[fig:ShellDrop\] (Color online) This plot illustrates our shell dropping algorithm for an equal mass non-spinning head-on simulation, where the individual black holes are initially centered at $x=\pm 15,y=z=0$ in coordinate units. The [upper panel]{} shows the apparent horizon size in inertial (blue) and co-moving (black) coordinates as well as the inverse of the grid compression factor (called ‘expansion factor’ in our code). As the black holes approach each other, the distance between their centers (in inertial coordinates) decreases. In order to keep the excision boundaries concentric with the black holes, the inner portion of the grid is compressed. In this compressed coordinate system the black hole looks larger (a coordinate effect). The excision boundary is expanded, accordingly, in order to keep it near the black hole’s horizon. The [ middle panel]{} shows the co-moving frame coordinate distance between the various spherical boundaries around one of the black holes and the [CutX]{}-plane, placed at $x=0$ for this simulation. As the excision boundary radius grows in the co-moving frame, the spherical shells around it grow as well, with the distance between the outermost such shell and the [CutX]{} grid-plane decreasing. Periodically, when this distance falls below a certain threshold, the number of spherical shells is reduced by one and the radii of the remaning shells get adjusted. At around time $\approx 196M$, when only two spherical shells remain, a thin spherical shell is created just thick enough to contain the interpolation stencils for the horizon finder, but otherwise small in order to delay the outer boundary of this shell coming too close to the [CutX]{} grid-plane. The [ lower panel]{} shows the inertial-frame coordinate distance between the various spherical shells around the excision boundaries and the [CutX]{} plane. The radius of the excision sphere in this frame is related to the inertial coordinate radius of the black hole and as such it stays roughly constant during the simulation. However, in this frame there is no mapping to keep the center of the black holes at a fixed position, so that the excision sphere (and the surrounding shells) approach the grid-plane as the BBH system nears merger. This, in effect, leads to a higher resolution grid during plunge, which helps in preserving accuracy. Please refer to Fig. \[fig:ShellDropVtk\] for more details. ](Figures/HeadOn_ShellDropping)
![\[fig:ShellDropVtk\] Snapshots of the computational grid during the various stages of the shell dropping algorithm. The [ left panel]{} shows a $z=0$ cut of the grid structure around one of the black holes at $t=0$, not including any of the outer cylinders or the wave-zone spherical shells. At this initial stage the black hole radius is small compared to the distance between the excision boundary and the grid-plane on the left of the plot. In this particular simulation the excision boundary is initially surrounded by nine spherical shells. The [ middle panel]{} shows a similar cut, at the point where six out of the nine shells have been removed, as the excision boundary increases in size. Note that the remaining three shells have radii comparable to the outermost three shells of the initial configuration. The [ right panel]{} shows the grid when there is a single spherical shell left. At this point the shell control algorithm creates a thin spherical shell wide enough to fully contain the apparent horizon (as long as this is feasible) but also thin enough to keep this last outer shell at a finite distance from the grid plane. ](Figures/HeadOn_Lev0_AA_Ablock "fig:") ![\[fig:ShellDropVtk\] Snapshots of the computational grid during the various stages of the shell dropping algorithm. The [ left panel]{} shows a $z=0$ cut of the grid structure around one of the black holes at $t=0$, not including any of the outer cylinders or the wave-zone spherical shells. At this initial stage the black hole radius is small compared to the distance between the excision boundary and the grid-plane on the left of the plot. In this particular simulation the excision boundary is initially surrounded by nine spherical shells. The [ middle panel]{} shows a similar cut, at the point where six out of the nine shells have been removed, as the excision boundary increases in size. Note that the remaining three shells have radii comparable to the outermost three shells of the initial configuration. The [ right panel]{} shows the grid when there is a single spherical shell left. At this point the shell control algorithm creates a thin spherical shell wide enough to fully contain the apparent horizon (as long as this is feasible) but also thin enough to keep this last outer shell at a finite distance from the grid plane. ](Figures/HeadOn_Lev0_AH_Ablock "fig:") ![\[fig:ShellDropVtk\] Snapshots of the computational grid during the various stages of the shell dropping algorithm. The [ left panel]{} shows a $z=0$ cut of the grid structure around one of the black holes at $t=0$, not including any of the outer cylinders or the wave-zone spherical shells. At this initial stage the black hole radius is small compared to the distance between the excision boundary and the grid-plane on the left of the plot. In this particular simulation the excision boundary is initially surrounded by nine spherical shells. The [ middle panel]{} shows a similar cut, at the point where six out of the nine shells have been removed, as the excision boundary increases in size. Note that the remaining three shells have radii comparable to the outermost three shells of the initial configuration. The [ right panel]{} shows the grid when there is a single spherical shell left. At this point the shell control algorithm creates a thin spherical shell wide enough to fully contain the apparent horizon (as long as this is feasible) but also thin enough to keep this last outer shell at a finite distance from the grid plane. ](Figures/HeadOn_Lev0_AJ_Ablock "fig:")
As the black holes approach each other, the ratio of the size of the excision boundaries and the distance between their centers will grow. In the co-moving frame, where this distance is fixed by definition, the signature of the plunge is that the apparent horizons (and the excision boundaries within them) go through a rapid growth. Our implementation of the shape map ${\ensuremath{\mathcal{M}_{\rm Shape}}}$ is such that the various spherical shell boundaries around a given excision boundary stay at a constant distance from one another. As the excision boundary grows, this causes the outermost spherical shell to approach the plane cutting between the two sets of spherical shells, as seen in Fig. \[fig:ShellDrop\]. In order to prevent a grid singularity, when this condition is detected, we re-draw the computational domain, typically reducing the number of spherical shells around the excision boundary by one. When we do this, we also optimize the radii of the remaining shells so that they form a geometric sequence in the co-rotating frame. This is done several times as the black holes approach each other. Near merger typically one would have a single spherical shell left around the excision boundary of the larger hole (see the right panel of Fig. \[fig:ShellDropVtk\]), as this is quicker to approach the cutting plane. Especially for binaries with highly spinning black holes, we find that it is important to keep at least one thin spherical shell around the excision boundary. Not doing so (and thus having a set of distorted cylinders extend to the excision boundary) can introduce enough numerical noise to cause our outflow boundary condition imposed at the excision boundary to become ill-posed.[^7]
$r$-type mesh refinement {#sec:rTypeAmr}
------------------------
During the final stage of the plunge of high mass ratio BBH mergers the shell dropping algorithm described in Sec. \[sec:ShellDrop\] does not provide sufficient control of the grid. This is due to the fact that our excision boundary can be very near the individual apparent horizon surfaces and, as these approach each other, the excision boundaries must also be able to get very near each other. Given that in a typical simulation there will be at least four subdomains placed between the excision boundaries and given the number of maps connecting the grid frame with the inertial frame, each of which must stay non-singular (and invertible) during the entire simulation, one must repeatedly redraw the grid in the immediate neighborhood of the excision boundaries as the distance between these becomes a small fraction of their radii.
The one additional grid-boundary that needs to be dynamically controlled at plunge is the [CutX]{} plane. As seen in Fig. \[fig:CutXFuncOfTime\], the larger black hole tends to grow very rapidly in the distorted frame in the final few $M$ of evolution time. With its origin at a fixed location, this implies that the $x$ coordinate position of its point closest to the small black hole will rapidly approach the smaller object, with the [CutX]{} plane in the way. The dashed black line shows that if this plane were held at a fixed location, the run would have crashed at $\sim 1M$ before merger. The [CutX]{} control system described in Ref. is designed to handle this, keeping the map from the grid to the inertial frame non-singular throughout the merger. However, as stated earlier, it is essential that all individual maps in Eq. (\[eq:MapSequence\]) are also non-singular and invertible (e.g., for the sake of the horizon finder interpolator which needs to map the position of the horizon mesh points into the grid frame). This means that the excision boundary, under the action of the shape map ${\ensuremath{\mathcal{M}_{\rm Shape}}}$, must not cross the [CutX]{} plane. For this reason, at a set of discrete times, we re-position the [CutX]{} grid-plane (as shown by the black curve in Fig. \[fig:CutXFuncOfTime\]), such that it stays between the excision boundaries (and the spherical shells surrounding these) at all times, and in all frames. Fig. \[fig:CutXSnapshots\] shows the grid-structure surrounding the two excision boundaries during such a change. Given that the subdomains are not split or joined but that their shape is redefined, we regard this algorithm as $r-$type mesh refinement.
![\[fig:CutXFuncOfTime\] (Color online) [ Upper panel:]{} The plot shows the average coordinate radius of the apparent horizon, in the [*distorted*]{} frame, divided by the initial value of this average coordinate radius, for a $q=9.98875$ non-spinning BBH simulation. As the plot shows, the two black holes grow at the same rate in this frame, although they have very different masses (and sizes). This is consistent with the growth being a coordinate effect of ${\ensuremath{\mathcal{M}_{\rm Scale}}}$, which shrinks the inner region of the grid as the black holes approach in order to keep the excision boundaries concentric with them. In this shrinking frame, the black holes have larger and larger coordinate radii, as seen in the plot. [ Lower panel: ]{}This plot illustrates how the location of the [CutX]{} plane is controlled both in the [*grid*]{} frame and in the [*distorted*]{} frame for a $q=9.98875$ non-spinning BBH simulation. Let $S_A$ be the outermost spherical boundary around the larger black hole and $S_B$ be the outermost spherical boundary around the smaller black hole. The coordinate centers of both of these are near the $x$ axis in both the [*grid*]{} frame and the [*distorted*]{} frame. In the simulation, $S_A$ is to the right of the [CutX]{} plane (i.e., the $x$ coordinate value is larger than that of the [CutX]{} plane for all of its points). Similarly, $S_B$ lies left of the [CutX]{} plane. The green curve shows the $x$ coordinate of the point of $S_B$ closest to the [CutX]{} plane; similarly, the red curve shows the $x$ coordinate of the point of $S_A$ closest to the [CutX]{} plane. As the plunge proceeds, the black holes approach each other. In the [*distorted*]{} frame this translates into the green and the red curves approaching each other. The task of the [CutX]{} control system is to smoothly move the [CutX]{} plane out of the way of the black hole which approaches it faster. As it can be seen, without such a control system the code would have encountered a grid singularity shortly after $t=5290$ (at the point where the red curve crosses the dashed black line). The [CutX]{} grid-plane is moved out of the way and the grid singularity is avoided. As an additional requirement, the shape map ${\ensuremath{\mathcal{M}_{\rm Shape}}}$ must also not become singular during the simulation. This is averted by another algorithm that redefines the grid-frame location of the [CutX]{} plane, at discrete times. This is displayed by the continuous black line. Please refer to Fig. \[fig:CutXSnapshots\] for further illustration of the effects of these discrete changes in the [CutX]{} plane location. ](Figures/CutX_PlaneBehavior.pdf)
![\[fig:CutXSnapshots\] (Color online) Snapshots of the BBH grid, zoomed in around the smaller black hole, for a $q=9.98875$ non-spinning BBH simulation, at $t\approx5290.69 M$. The [ top]{} plot shows the two excision boundaries and the surrounding grid-structure, in the [*distorted*]{} frame. An important feature of the grid is the [CutX]{} plane, shown here as a vertical grid-line situated between the two excision boundaries. The [ lower first]{} plot from left to right shows the grid in the [*shape*]{} map frame. This map is responsible for keeping the shape of the excision boundary in sync with that of the individual black holes, while it leaves the [CutX]{} plane unaffected. The [ lower second]{} plot shows the grid in the same frame, after [CutX]{} plane position has been moved to the left, at a given time-instant. One result of this grid change is that the excision boundary of the larger object has more room to grow. Another effect of this grid change is that the subdomains in the immediate neighborhood of the smaller excision boundary were reduced in size (see the blue region shrink from the first plot to the second). This gives additional accuracy to the region evolving the smaller black hole. Note that the compressed grid between the excision boundary and the [CutX]{} plane does not factor into the CFL limit, as this is not the evolution frame. The next map in the sequence Eq. (\[eq:MapSequence\]), the ${\ensuremath{\mathcal{M}_{\rm CutX}}}$ map removes this compression, as seen on the fourth figure. However, the extra room between the [CutX]{} plane and the (red) excision shell of the larger black hole is essential as this object grows at a fast rate (see Fig. \[fig:CutXFuncOfTime\]). The [ lower third]{} plot is a zoomed-in version of the top plot. It shows the grid structure before the grid-change, in the [*distorted*]{} frame. This frame differs from the [*shape*]{} frame by dynamically controlling the position of the [CutX]{} plane in order to keep it at a finite distance from the larger object as it rapidly expands. The [ lower fourth]{} plot shows the [*distorted*]{} frame grid after the grid-frame position of the [CutX]{} plane has been moved. It is important to realize, in these plots, that the discrete relocation of the [CutX]{} plane in the grid-frame does not affect (by construction) the position of the [CutX]{} plane in the [*distorted*]{} frame. Similarly, the smooth, time-dependent motion of the [CutX]{} plane in the distorted frame does not imply a change of the grid-frame location of this grid-plane. The two are independent means of controlling the shape of the grid in two different frames. As one can see, the shrinking of the subdomains in the immediate neighborhood of the smaller black hole can be also seen in the distorted frame (in the fourth plot). ](Figures/CutX_CU_Distorted_Overview.pdf)
![\[fig:CutXSnapshots\] (Color online) Snapshots of the BBH grid, zoomed in around the smaller black hole, for a $q=9.98875$ non-spinning BBH simulation, at $t\approx5290.69 M$. The [ top]{} plot shows the two excision boundaries and the surrounding grid-structure, in the [*distorted*]{} frame. An important feature of the grid is the [CutX]{} plane, shown here as a vertical grid-line situated between the two excision boundaries. The [ lower first]{} plot from left to right shows the grid in the [*shape*]{} map frame. This map is responsible for keeping the shape of the excision boundary in sync with that of the individual black holes, while it leaves the [CutX]{} plane unaffected. The [ lower second]{} plot shows the grid in the same frame, after [CutX]{} plane position has been moved to the left, at a given time-instant. One result of this grid change is that the excision boundary of the larger object has more room to grow. Another effect of this grid change is that the subdomains in the immediate neighborhood of the smaller excision boundary were reduced in size (see the blue region shrink from the first plot to the second). This gives additional accuracy to the region evolving the smaller black hole. Note that the compressed grid between the excision boundary and the [CutX]{} plane does not factor into the CFL limit, as this is not the evolution frame. The next map in the sequence Eq. (\[eq:MapSequence\]), the ${\ensuremath{\mathcal{M}_{\rm CutX}}}$ map removes this compression, as seen on the fourth figure. However, the extra room between the [CutX]{} plane and the (red) excision shell of the larger black hole is essential as this object grows at a fast rate (see Fig. \[fig:CutXFuncOfTime\]). The [ lower third]{} plot is a zoomed-in version of the top plot. It shows the grid structure before the grid-change, in the [*distorted*]{} frame. This frame differs from the [*shape*]{} frame by dynamically controlling the position of the [CutX]{} plane in order to keep it at a finite distance from the larger object as it rapidly expands. The [ lower fourth]{} plot shows the [*distorted*]{} frame grid after the grid-frame position of the [CutX]{} plane has been moved. It is important to realize, in these plots, that the discrete relocation of the [CutX]{} plane in the grid-frame does not affect (by construction) the position of the [CutX]{} plane in the [*distorted*]{} frame. Similarly, the smooth, time-dependent motion of the [CutX]{} plane in the distorted frame does not imply a change of the grid-frame location of this grid-plane. The two are independent means of controlling the shape of the grid in two different frames. As one can see, the shrinking of the subdomains in the immediate neighborhood of the smaller black hole can be also seen in the distorted frame (in the fourth plot). ](Figures/CutX_CU_Shape.pdf "fig:") ![\[fig:CutXSnapshots\] (Color online) Snapshots of the BBH grid, zoomed in around the smaller black hole, for a $q=9.98875$ non-spinning BBH simulation, at $t\approx5290.69 M$. The [ top]{} plot shows the two excision boundaries and the surrounding grid-structure, in the [*distorted*]{} frame. An important feature of the grid is the [CutX]{} plane, shown here as a vertical grid-line situated between the two excision boundaries. The [ lower first]{} plot from left to right shows the grid in the [*shape*]{} map frame. This map is responsible for keeping the shape of the excision boundary in sync with that of the individual black holes, while it leaves the [CutX]{} plane unaffected. The [ lower second]{} plot shows the grid in the same frame, after [CutX]{} plane position has been moved to the left, at a given time-instant. One result of this grid change is that the excision boundary of the larger object has more room to grow. Another effect of this grid change is that the subdomains in the immediate neighborhood of the smaller excision boundary were reduced in size (see the blue region shrink from the first plot to the second). This gives additional accuracy to the region evolving the smaller black hole. Note that the compressed grid between the excision boundary and the [CutX]{} plane does not factor into the CFL limit, as this is not the evolution frame. The next map in the sequence Eq. (\[eq:MapSequence\]), the ${\ensuremath{\mathcal{M}_{\rm CutX}}}$ map removes this compression, as seen on the fourth figure. However, the extra room between the [CutX]{} plane and the (red) excision shell of the larger black hole is essential as this object grows at a fast rate (see Fig. \[fig:CutXFuncOfTime\]). The [ lower third]{} plot is a zoomed-in version of the top plot. It shows the grid structure before the grid-change, in the [*distorted*]{} frame. This frame differs from the [*shape*]{} frame by dynamically controlling the position of the [CutX]{} plane in order to keep it at a finite distance from the larger object as it rapidly expands. The [ lower fourth]{} plot shows the [*distorted*]{} frame grid after the grid-frame position of the [CutX]{} plane has been moved. It is important to realize, in these plots, that the discrete relocation of the [CutX]{} plane in the grid-frame does not affect (by construction) the position of the [CutX]{} plane in the [*distorted*]{} frame. Similarly, the smooth, time-dependent motion of the [CutX]{} plane in the distorted frame does not imply a change of the grid-frame location of this grid-plane. The two are independent means of controlling the shape of the grid in two different frames. As one can see, the shrinking of the subdomains in the immediate neighborhood of the smaller black hole can be also seen in the distorted frame (in the fourth plot). ](Figures/CutX_CV_Shape.pdf "fig:") ![\[fig:CutXSnapshots\] (Color online) Snapshots of the BBH grid, zoomed in around the smaller black hole, for a $q=9.98875$ non-spinning BBH simulation, at $t\approx5290.69 M$. The [ top]{} plot shows the two excision boundaries and the surrounding grid-structure, in the [*distorted*]{} frame. An important feature of the grid is the [CutX]{} plane, shown here as a vertical grid-line situated between the two excision boundaries. The [ lower first]{} plot from left to right shows the grid in the [*shape*]{} map frame. This map is responsible for keeping the shape of the excision boundary in sync with that of the individual black holes, while it leaves the [CutX]{} plane unaffected. The [ lower second]{} plot shows the grid in the same frame, after [CutX]{} plane position has been moved to the left, at a given time-instant. One result of this grid change is that the excision boundary of the larger object has more room to grow. Another effect of this grid change is that the subdomains in the immediate neighborhood of the smaller excision boundary were reduced in size (see the blue region shrink from the first plot to the second). This gives additional accuracy to the region evolving the smaller black hole. Note that the compressed grid between the excision boundary and the [CutX]{} plane does not factor into the CFL limit, as this is not the evolution frame. The next map in the sequence Eq. (\[eq:MapSequence\]), the ${\ensuremath{\mathcal{M}_{\rm CutX}}}$ map removes this compression, as seen on the fourth figure. However, the extra room between the [CutX]{} plane and the (red) excision shell of the larger black hole is essential as this object grows at a fast rate (see Fig. \[fig:CutXFuncOfTime\]). The [ lower third]{} plot is a zoomed-in version of the top plot. It shows the grid structure before the grid-change, in the [*distorted*]{} frame. This frame differs from the [*shape*]{} frame by dynamically controlling the position of the [CutX]{} plane in order to keep it at a finite distance from the larger object as it rapidly expands. The [ lower fourth]{} plot shows the [*distorted*]{} frame grid after the grid-frame position of the [CutX]{} plane has been moved. It is important to realize, in these plots, that the discrete relocation of the [CutX]{} plane in the grid-frame does not affect (by construction) the position of the [CutX]{} plane in the [*distorted*]{} frame. Similarly, the smooth, time-dependent motion of the [CutX]{} plane in the distorted frame does not imply a change of the grid-frame location of this grid-plane. The two are independent means of controlling the shape of the grid in two different frames. As one can see, the shrinking of the subdomains in the immediate neighborhood of the smaller black hole can be also seen in the distorted frame (in the fourth plot). ](Figures/CutX_CU_Distorted.pdf "fig:") ![\[fig:CutXSnapshots\] (Color online) Snapshots of the BBH grid, zoomed in around the smaller black hole, for a $q=9.98875$ non-spinning BBH simulation, at $t\approx5290.69 M$. The [ top]{} plot shows the two excision boundaries and the surrounding grid-structure, in the [*distorted*]{} frame. An important feature of the grid is the [CutX]{} plane, shown here as a vertical grid-line situated between the two excision boundaries. The [ lower first]{} plot from left to right shows the grid in the [*shape*]{} map frame. This map is responsible for keeping the shape of the excision boundary in sync with that of the individual black holes, while it leaves the [CutX]{} plane unaffected. The [ lower second]{} plot shows the grid in the same frame, after [CutX]{} plane position has been moved to the left, at a given time-instant. One result of this grid change is that the excision boundary of the larger object has more room to grow. Another effect of this grid change is that the subdomains in the immediate neighborhood of the smaller excision boundary were reduced in size (see the blue region shrink from the first plot to the second). This gives additional accuracy to the region evolving the smaller black hole. Note that the compressed grid between the excision boundary and the [CutX]{} plane does not factor into the CFL limit, as this is not the evolution frame. The next map in the sequence Eq. (\[eq:MapSequence\]), the ${\ensuremath{\mathcal{M}_{\rm CutX}}}$ map removes this compression, as seen on the fourth figure. However, the extra room between the [CutX]{} plane and the (red) excision shell of the larger black hole is essential as this object grows at a fast rate (see Fig. \[fig:CutXFuncOfTime\]). The [ lower third]{} plot is a zoomed-in version of the top plot. It shows the grid structure before the grid-change, in the [*distorted*]{} frame. This frame differs from the [*shape*]{} frame by dynamically controlling the position of the [CutX]{} plane in order to keep it at a finite distance from the larger object as it rapidly expands. The [ lower fourth]{} plot shows the [*distorted*]{} frame grid after the grid-frame position of the [CutX]{} plane has been moved. It is important to realize, in these plots, that the discrete relocation of the [CutX]{} plane in the grid-frame does not affect (by construction) the position of the [CutX]{} plane in the [*distorted*]{} frame. Similarly, the smooth, time-dependent motion of the [CutX]{} plane in the distorted frame does not imply a change of the grid-frame location of this grid-plane. The two are independent means of controlling the shape of the grid in two different frames. As one can see, the shrinking of the subdomains in the immediate neighborhood of the smaller black hole can be also seen in the distorted frame (in the fourth plot). ](Figures/CutX_CV_Distorted.pdf "fig:")
Overall AMR performance
-----------------------
As indicated in the various sections describing our AMR algorithm, a number of aspects would benefit from improvement. Overall, however, AMR has become an essential part of our production simulations. It avoids the need to repeat the same simulation, time after time, iterating on the grid extents of the domain. Neither do we have to contend with exponential error growth, a direct consequence of exponential convergence of our numerics and the accelerated decrease of the length-scales, as dictated by the nature of the BBH inspiral and merger problem. in Fig. \[fig:FullRunAmr\] we plot the measured truncation error level for each irreducible topology of each subdomain, the associated pile-up mode function and the overall constraint error, as measured by the global $L_2$ and $L_\infty$ norms. This figure shows that during the inspiral the AMR driver does quite well in keeping these quantities within control. The constraint error spike shortly before $t=5000M$ corresponds to merger. While one would prefer no such spike, we note that the spike vanishes as soon as ringdown part of our simulation begins, when the excision boundary is now based on the common horizon. This implies that the large errors are well within the newly formed black hole and do not affect the physics of the region outside the new excision boundary.
Aside from this spike, there are two other problem areas one can point to on this plot. The [*junk radiation*]{} has been a long-standing problem for us. Resolving it would require an unreasonable amount of computational resources for a given simulation. We are looking for alternative ways of tackling this problem. Another problem arises in the ringdown, where truncation reaches levels near $10^{-5}$ and pile-up modes are created. The constraint error in this part of the run remains small, however. One possible reason why our truncation error estimate indicates large error is that the grid used to propagate the gravitational waves generated by the quasinormal modes may not be optimal once the black hole settles down. A proper $h$-type AMR driver would possibly join the rather large number of spherical shells into a handful of subdomains once the spacetime reaches a smooth, stationary state.
Our largest concern, at the moment, is improved junk radiation treatment, as noise in the late ringdown has only a minimal effect on the primary output of our code – the gravitational waveform emitted by the BBH system.
![\[fig:FullRunAmr\] Overall AMR driver performance for the non-spinning $q=7.1875$ BBH evolution. The [ top panel]{} plots the ‘excess truncation error’ defined as the difference between the truncation error Eq. (\[eq:TruncError\]) associated with the individual power-monitors $P_k$ and the target truncation error Eq. (\[eq:TruncErrorFunction\]), ${\cal T}_{\rm excess} = {\cal T}[P_k] -{\cal T}^{\max}[w_A,w_B]$ for all spectral expansions in all subdomains. The first $\sim 500 M$ of the simulation is dominated by the ‘junk radiation’ phase, where the black holes undergo quasi-normal ringing in response to non-physical gravitational radiation content present in the initial data. As the ringing subsides and the high frequency waves leave the grid, the excess truncation error becomes negative, and relatively close to zero, showing that the AMR driver is able to control truncation error. Note that various constraints on the subdomain extents (explained in Sec. (\[seq:AmrRulesForShells\])) imply that certain subdomains have more accuracy than required. The last $\sim 1100M$ of the run corresponds to the ringdown part of the simulation. As indicated in the plot, some of the modes (the radial modes) are not well resolved, in a way similar to the initial junk radiation phase. This is, once again, due to the high frequency nature of the quasi-normal ringing of the final black hole. This problem would benefit from further improvements of the AMR algorithm. The [ bottom]{} panel shows the $L_{\infty}$ and the $L_2$ norm of the (unnormalized) constraint error. This error is largest during the junk radiation phase except for a short peak seen during the merger of the two black holes. This peak is no longer seen once the region inside the common horizon is excised, suggesting that the larger constraint violating modes are confined to the interior of the merged black hole. ](Figures/FullRunAmr)
Optimization
============
As a last element of our description we highlight one particular aspect of the optimization work that was done to [SpEC]{}. In our current parallelization scheme tasks associated with a particular subdomain are handled by a single process, with no multi-threading. Given the small number of such subdomains, the maximum number of processes one can meaningfully use in a BBH simulation is $48$ (assuming $16$ or $12$ core nodes). This on its own is a limitation we’ll possibly need to deal with by adding multi-threading to our code. But, even before that, the varying number of points per subdomain means that if each process owns a single subdomain, the work load per process would be far from even. Reducing the number of processes would help a little but, even in that case, load balancing would be very coarse grained. In order to improve this situation, we have implemented an adaptive parallel interpolation and differentiator (PID) algorithm that dynamically re-assigns as much as $30\%$ of the workload associated with a given subdomain from processes with too much work to those with too little. [MPI\_Barrier]{} calls, issued after each RHS evaluation, are used to monitor which process is working hardest. If the cost of these [MPI\_Barrier]{} calls is consistently much smaller on a particular process than on some other process, the process with the small measured [MPI\_Barrier]{} cost must be busy (see Fig. \[fig:PID\]). In this case, the process would no longer be responsible for interpolating data, as required in the inter-subdomain boundary algorithm. Neither would this process be responsible for differentiating the evolution variables. Rather, through a sequence of non-blocking [MPI]{} calls, this data would be sent to a less busy process. After all such initial messages have been sent, each process starts to compute the various pieces of the RHS (in this case: the pieces of Eq. (\[eq:RicciH\]) that do not involve derivatives). At various points during this RHS evaluation, the evolution algorithm dials back into the adaptive PID algorithm to see if any of the non-blocking messages have arrived at their destination. If so, data is processed (i.e., interpolated and/or differentiated). Then the processed data is sent on its way – the derivative data needs to get back to the process responsible for updating the corresponding subdomain, while the interpolated data is sent to those processes that need this data to complete the RHS evaluation on the subdomain boundary points. At some point in the RHS evaluation the derivative data or boundary data becomes a [*must*]{}, i.e., no more work that can be done without it. At this point blocking [MPI\_Wait]{} calls are issued, enforcing the arrival of all data needed to complete the RHS. At seen on Fig. \[fig:PID\], most communication is in fact complete during the first half of the RHS evaluation and there is virtually no waiting for data.
This, as well as other optimization techniques, have lead to significant performance improvement of the [SpEC]{} code. As stated earlier, there is more work to be done, in particular in the area of multi-threading.
![\[fig:PID\] (Color online) The plot illustrates the inner workings of our load balancing algorithm. This output is produced by the [*Intel Trace Analyzer*]{}, measuring performance of our code for two sub-time-steps of a typical BBH evolution. The horizontal direction corresponds to wall-clock time, ranging from $98.40$sec to $98.92$sec, measured from the beginning of the test-run. Each of the horizontal stripes represents an [MPI]{} process. Black lines represent [MPI]{} messages sent from one process to another. The green portions correspond to computation of the various parts of the evolution system’s RHS (in this case Eq. (\[eq:RicciH\])). Yellow portions indicate differentiation of the evolution variables, while cyan stands for interpolation of boundary data at inter-subdomain boundaries. Red stands for [MPI]{} communication. At the end of each sub-time-step an [MPI\_Barrier]{} is placed to force syncing between the various processes. The cost of this barrier on the individual processes is used as an indicator of the local process load. If the barrier cost is minimal, the processor is busy and is among the last to reach the barrier. At subsequent RHS evaluations work load is shifted away from this process. In this simulation the processor with rank zero has the largest load. By implication nearly all differentiation and interpolation task has been shifted away from this process to others with a smaller workload. As an additional feature, all [MPI]{} messages sent during the evaluation of the RHS are non-blocking (or sent in the background). The RHS evaluation algorithm then periodically calls back to the parallel adaptive interpolation/differentiator algorithm to test whether any of the non-blocking messages have reached their destination. If yes, the data is processed (e.g., differentiated) and then sent on its way in another non-blocking message. At the end of the RHS evaluation, when a given piece of the data is required for further work, an [MPI\_Wait]{} is issued for the required message. In most cases, however, the messages will have arrived by this time and there is minimal communication cost (shown in red) during RHS evaluation. ](Figures/ParallelInterpAndDeriv.pdf)
Conclusion
==========
This paper gives a tour of the main building blocks of a state-of-the art binary black hole evolution code, known as the [SpEC]{} code. We start by writing out the evolution system used in our simulations, a first order symmetric hyperbolic form of the generalized harmonic formulation. We then provide a brief description of the numerical approximation used in the code. Based on these concepts, we then detail the diagnostic quantities used to monitor (and control) the accuracy of our simulations. The definitions we provide for the truncation error, the pile-up modes and the convergence factor are then used as we give an elaborate description of our spectral adaptive mesh refinement algorithm, tuned to the problem of binary black hole evolutions. As laid out in our description, our favored method of controlling truncation error is $p$-type mesh refinement, with a target truncation error function that is tuned to use more resolution in the strong field region. In addition, we use $h$-type mesh refinement mostly as a means of preventing grid compression as the size of the apparent horizons (and the associated excision boundaries) increases relative to the distance between the coordinate centers of the binary. Late in the plunge, for binaries with sufficiently large mass ratios, we find $r$-type mesh refinement to be essential; as the black holes approach, the grid layout defined at the beginning of the simulation is no longer adequate. We update the grid boundaries as dictated by the dynamics of the binary black hole plunge and merger.
Fig. \[fig:FullRunAmr\] illustrates how well our current algorithm performs during most of the inspiral. However, our algorithm does not do well at the early stage of the run where the unphysical gravitational wave content of the initial data induces high frequency ringing of the individual black holes, which in turn generates short wavelength gravitational waves which propagate through a grid that is not designed to resolve them. As a consequence of not resolving this “junk radiation”, there is a stochastic part of the truncation error whose noise produces a floor in our convergence measurements. Our attempts to control this stochastic noise source is focused on improving the initial data algorithm, e.g., by use of the ‘joint-elimination method’ described in Ref. ().
Another phase of our run where the AMR algorithm may need improvement is the ringdown stage. The reason behind this is mostly historic – this has been the easiest part of the simulation, as evolving a single black hole is simpler than evolving a binary. We plan to improve this part of our simulation but presently it is not a the major limitation in the quality of our gravitational waveforms.
Lastly, we have provided a brief description of part of the optimization that was done on the code. This is very “raw” computer science, but it is an essential part of why are we able to perform inspirals for dozens of (or, in once case, nearly two hundred) orbits.
Future work will focus on improving the accuracy of the phase of the gravitational wave during the plunge. The dominating error source in this very dynamic part of the simulation appears to be a sub-optimal structure of the inner grid at the point where nearly all spherical shells have been removed by our shell control algorithm. This results in accumulation of orbital phase error at a faster rate than during any other part of the simulation. We expect to improve this by refining our $h$-type AMR algorithm. We also plan on exploring the use of the constraint error driven target truncation error, as this will tighten accuracy requirements when the constraints start to grow.
As yet another future project, we plan to improve our outer boundary algorithm, as we find that reflections from the outer boundary can cause unphysical effects on the binary evolution on very long time-scales (beyond $100$ orbits).
The end goal is to enable the construction of semi-analytic (or empirical) gravitational wave models and template banks by providing numerical gravitational waveforms of sufficient accuracy, length and quantity such that modeling error does not detract from the detection of gravitational waves, or the resulting physics one can discern from the observed data.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Jeffrey Winicour for his useful comments on the manuscript. We thank Dan Hemberger for providing access to data produced by the head-on BBH simulation used in some of the plots. We thank Jonathan Blackman for providing access to his unequal mass inspiral simulations, as these provided data for most of the plots. We thank Saul Teukolsky, Mark Scheel, Larry Kidder, Harald Pfeiffer and Nicholas Taylor for helpful discussions while designing the AMR code. We thank Abdul Mroue, Tony Chu, Geoffrey Lovelace and Sergei Ossokine for useful feedback on the performance of our AMR algorithm in the context of thousands of BBH simulations. A significant amount of code development was done on the UCSD cluster [ccom-boom.uscd.edu]{}. The Caltech cluster [zwicky.cacr.caltech.edu]{} is an essential element of research done with [SpEC]{}. This cluster is supported by the Sherman Fairchild Foundation and by NSF award PHY-0960291. This project was supported by the Fairchild Foundation, and the NSF grants PHY-1068881 and AST-1333520.
[^1]: In a typical BBH simulation we set $\gamma_0$ to be a smooth function of coordinates, with $\gamma_0=O(10)$ in the inner region of the simulation and $\gamma_0 = 10^{-3}$ near the outer boundary of the computational domain.
[^2]: We define [*inertial*]{} frame the coordinate system in which the evolution equations are evolved. In these coordinates the far-field metric is a perturbation around Minkowski, while the coordinate position of the individual objects is time-dependent, as the black holes orbit each other. In a more precise sense, the inertial coordinates $(t,x^i)$ are determined by the initial and boundary data of the evolved metric quantities, along with our choice of harmonic gauge source functions $H_a$ defines the coordinates.
[^3]: The [*comoving*]{} frame, defined in Eq. (\[eq:ComovingFrame\]), is the frame connected to the inertial frame by translation, rotation and scaling. In this frame the coordinate centers of the individual black holes are time-independent throughout the inspiral and plunge of the binary. Definition of a such frame is essential in our code as the underlying numerical grid is rigid, with fixed excision boundaries. These excision surfaces are kept just inside the dynamically evolved apparent horizons by the sequence of maps given in Eq. (\[eq:MapSequence\]).
[^4]: For instance, the space-time metric in the weak field region (such as the neighborhood of the outer boundary in a typical binary black hole simulation) has the form $$\psi_{ab} = \eta_{ab} \quad +\quad \mbox{small perturbation}$$ (where $\eta_{ab}$ is the Minkowski metric). In this case the lowest order coefficient corresponding to the constant element of the expansion basis will be $O(1)$ while all higher coefficients will be several orders of mangitude smaller. Resolving the perturbative part of the metric in this wave zone is essential, as this is the effect we are trying to capture and translate into the signal seen by a gravitational wave detector. The convergence factor estimate would be incorrect if it included the $O(1)$ part coming from the flat space metric $\eta_{ab}$.
[^5]: Note that the angular coordinates are defined local to the horizon center and are independent of where the horizon is located with respect to the global coordinate-systems center.
[^6]: One such scenario, found in a number of our binary black hole inspirals, is the radial splitting of some of the outermost spherical shells. This results from the action of the cubic scale map ${\ensuremath{\mathcal{M}_{\rm Scaling}}}$, which is responsible for shrinking the interior of the grid as the binary approaches merger, while some of the outer subdomains get stretched. The AMR driver monitors this through the truncation error of the radial spectral expansion and keeps adding coefficients until the user-specified limit of the maximum allowed radial collocation points is reached. Then the subdomain gets split and the evolution can proceed.
[^7]: At the excision boundary, by construction, we provide no boundary condition to any of the evolution variables. Numerically this means that we are evolving the innermost boundary points using a [*sideways stencil*]{}, i.e., using values from previously evolved points of the same subdomain, but no additional boundary data. This treament of the excision boundary is consistent with the requirements of the continuum system as long as this inner boundary is spacelike. If numerical noise becomes large enough, the inner boundary may become timelike on part of the excision boundary and our numerical treatment is no longer adequate.
|
---
abstract: 'The presence of dark energy in the Universe is inferred directly and indirectly from a large body of observational evidence. The simplest and most theoretically appealing possibility is the vacuum energy density (cosmological constant). However, although in agreement with current observations, such a possibility exacerbates the well known cosmological constant problem, requiring a natural explanation for its small, but nonzero, value. In this paper we focus our attention on another dark energy candidate, one arising from gravitational *leakage* into extra dimensions. We investigate observational constraints from current measurements of angular size of high-$z$ compact radio-sources on accelerated models based on this large scale modification of gravity. The predicted age of the Universe in the context of these models is briefly discussed. We argue that future observations will enable a more accurate test of these cosmologies and, possibly, show that such models constitute a viable possibility for the dark energy problem.'
author:
- 'J. S. Alcaniz'
title: Some Observational Consequences of Brane World Cosmologies
---
INTRODUCTION
============
Recent results based on the magnitude-redshift relation for extragalactic type Ia supernovae (SNe Ia) suggest an accelerating universe dominated by some kind of negative-pressure dark component, the so-called *quintessence* [@perlmutter; @riess]. The existence of this dark energy has also been confirmed, independently of the SNe Ia analyses, by combining the latest galaxy clustering data with cosmic microwave background (CMB) measurements [@efs]. Together, these results seem to provide the remaining piece of information connecting the inflationary flatness prediction ($\Omega_{\rm{T}} = 1$) with astronomical observations. Such a state of affairs has stimulated the interest for more general models containing an extra component describing this dark energy and, simultaneously, accounting for the present accelerated stage of the Universe. The simplest and most theoretically appealing possibility is the vacuum energy (cosmological constant). Because of their observational successes, flat models with a relic cosmological constant are considered nowadays our best description of the observed Universe. However, we face at least a serious problem when one considers a nonzero vacuum energy: in order to dominate the dynamics of the Universe only at recent times, a very small value for the cosmological constant ($\Lambda_{o} \sim 10^{-56}\rm{cm}^{-2}$) is required from observations, while naive estimates based on quantum field theories are 50-120 orders of magnitude larger, thereby originating an extreme fine tunning problem [@weinberg; @sahni] or making a complete cancellation (from an unknown physical mechanism) seem more plausible.
If the cosmological constant is null, something else must be causing the Universe to speed up. Several possible dark energy candidates have been discussed in the literature, e.g., a vacuum decaying energy density, or a time varying $\Lambda$-term [@ozer], a relic scalar field [@peebles], an extra component, the so-called “X-matter", which is simply characterized by an equation of state $p_x=\omega\rho_{x}$, where $-1 \leq \omega < 0$ [@turner] and that includes, as a particular case, models with a cosmological constant ($\Lambda$CDM) or still models based on the framework of brane-induced gravity [@dvali; @dvali1; @deff; @deff1; @deffZ]. In this last case, the basic idea is that our 4-dimensional Universe would be a surface or a brane embedded into a higher dimensional bulk space-time to which gravity can spread. Despite the fact that there is no cosmological constant on the brane, such scenarios explain the observed acceleration of the Universe because the bulk gravity sees its own curvature term on the brane as a negative-pressure dark component and accelerates the Universe [@deff1].
Brane world cosmologies have been discussed in different contexts. For example, the issue related to the cosmological constant problem has been addressed [@ccp]as well as the evolution of cosmological perturbations in the gauge-invariant formalism [@brand], cosmological phase transitions [@cpt], inflationary solutions [@cpt1], baryogenesis [@dvali99], stochastic background of gravitational waves [@hogan1], singularity, homogeneity, flatness and entropy problems [@aaa], among others (see [@hogan] for a discussion on the different perspectives of brane world models). In the observational front, some analyses [@deffZ] have shown that such models are in agreement with the most recent cosmological observations (see, however, [@avelino; @dnew]). In this case, constraints from SNe Ia + CMB data require a flat universe with $\Omega_{\rm{m}} = 0.3$ and $\Omega_{\rm{r}_c} = 0.12$, where $\Omega_{\rm{r}_c}$ is the density parameter associated with the crossover distance between the 4-dimensional and 5-dimensional gravities (see [@deff1] for details).
In the present work we focus our attention on these kinds of cosmologies. Following [@deffZ], we study models based on the framework of the brane-induced gravity of Dvali [*et al.*]{} [@dvali; @dvali1] that have been recently proposed in Refs. [@deff; @deff1]. We will also consider only the case in which the bulk space-time is 5-dimensional. Our main purpose is to investigate some observational consequences of such scenarios with emphasis on the constraints provided by observations of the angular size of high-$z$ milliarcsecond radio-sources.
We structured this paper as follows. In Section II the basic field equations and distance formulas are presented. We also briefly discuss the predicted age of the Universe. In Section III we use measurements of the angular size of high-$z$ milliarcsecond radio sources [@gurv1] to constrain the free parameters of the model. We show that a good agreement between theory and observation is possible if $\Omega_{\rm{m}} \leq 0.38$, $\Omega_{\rm{r}_c} \leq 0.29$ and $\Omega_{\rm{m}} \leq
0.09$, $\Omega_{\rm{r}_c} \leq 0.29$ ($68\%$ c.l.) for values of the characteristic length of the sources between $l \simeq 20h^{-1} - 30h^{-1}$ pc, respectively. In particular we find that a slightly closed, accelerated model with $\Omega_{\rm{m}} = 0.06$, $\Omega_{\rm{r}_c} = 0.28$, and $l = 27.06h^{-1}$ pc is the best fit for these data.
Field equations, distance formulas and the age of the universe
==============================================================
The geometry of our 4-dimensional Universe is described by the Friedmann-Robertson-Walker (FRW) line element ($c = 1$) $$ds^2 = dt^2 - R^{2}(t) \left[{dr^{2} \over 1 - kr^{2}} + r^{2} (d
\theta^2 + \rm{sin}^{2} \theta d \phi^{2})\right],$$ where $k = 0$, $\pm 1$ is the curvature parameter of the spatial section, $r$, $\theta$, and $\phi$ are dimensionless comoving coordinates, and $R(t)$ is the scale factor. The Friedmann’s equation for the kind of models we are considering is [@deff1; @deffZ] $$\left[\sqrt{\frac{\rho}{3M_{pl}^{2}} + \frac{1}{4r_{c}^{2}}} + \frac{1}{2r_{c}}\right]^{2} = H^{2}
+ \frac{k}{R(t)^{2}},$$ where $\rho$ is the energy density of the cosmic fluid, $H$ is the Hubble parameter, $M_{pl}$ is the Planck mass, and $r_c = M_{pl}^{2}/2M_{5}^{3}$ is the crossover scale defining the gravitational interaction among particles located on the brane ($M_5$ is the 5-dimensional reduced Planck mass). For distances smaller than $r_c$ the force experienced by two punctual sources is the usual 4-dimensional gravitational $1/r^{2}$ force whereas for distances larger than $r_c$ the gravitational force follows the 5-dimensional $1/r^{3}$ behavior [@comm].
Equation (2) implies that the normalization condition is given by $$\Omega_k + \left[\sqrt{\Omega_{\rm{r_c}}} + \sqrt{\Omega_{\rm{r_c}} + \Omega_{\rm{m}}}\right]^{2} =
1$$ where $\Omega_{\rm{m}}$ and $\Omega_k$ are the matter and curvature density parameters, respectively and $$\Omega_{\rm{r_c}} = 1/4r_c^{2}H^{2},$$ is the density parameter associated to the crossover radius $r_c$.
The deceleration parameter, usually defined as $q_o = -R\ddot{R}/\dot{R}^{2}|_{t_{o}}$, now takes the following form $$\begin{aligned}
q_o & = & \frac{3}{2}\Omega_{\rm{m}}\left[ 1 + \sqrt{\frac{\Omega_{\rm{r_c}}}{\Omega_{\rm{r_c}} +
\Omega_{\rm{m}}}}\right] \\ \nonumber & & \quad \quad \quad \quad \quad \quad -
\left[\sqrt{\Omega_{\rm{r_c}}} + \sqrt{\Omega_{\rm{r_c}} + \Omega_{\rm{m}}}\right]^{2} .\end{aligned}$$ For $\Omega_k = 0$ (flat case), the above expression reduces to $$q_o = \frac{3}{2}\Omega_{\rm{m}}\left[ 1 + \sqrt{\frac{\Omega_{\rm{r_c}}}{\Omega_{\rm{r_c}} +
\Omega_{\rm{m}}}}\right] - 1.$$
0.1in
Figure 1 shows the behavior of the deceleration parameter as a function of redshift for selected values of $\Omega_{\rm{m}}$ and $\Omega_{\rm{r_c}}$. As explained above, although there is no cosmological constant on the brane, brane-world models allows periods of accelerated expansion because the bulk gravity sees its own curvature term on the brane as a negative-pressure dark component [@deff1]. The best fit $\Lambda$CDM case is also showed for the sake of comparison. Note that at late times brane-cosmologies with $\Omega_{\rm{r_c}} = 0.3$ accelerates slower than $\Lambda$CDM models with $\Omega_{\Lambda} = 0.7$ and the same value of $\Omega_{\rm{m}}$. For the best fit model found in Ref. [@deffZ], i.e., $\Omega_{\rm{m}} = 0.3$ and $\Omega_{\rm{r_c}} =
0.12$, the accelerated expansion begins at $z \simeq 0.5$ whereas for $\Lambda$CDM we find $z
\simeq 0.7$. For our best fit, found in Section 3, we see that the Universe always accelerates at a faster rate than the best fit $\Lambda$CDM model. In this case, the Universe begins to accelerate at $z \simeq 2.3$ (see also [@sahini] for a more detailed discussion on this topic).
From Eqs. (1) and (2), it is straightforward to show that the comoving distance $r(z)$ is given by $$r(z) = \frac{1}{R_o H_o |\Omega_k|^{1/2}}\sum\left[|\Omega_k|^{1/2} \int_{x'}^{1} {dx \over
x^{2}f(\Omega_{j}, x)}\right],$$ where the subscript $o$ denotes present day quantities, $x' = {R(t) \over R_o} = (1 + z)^{-1}$ is a convenient integration variable and the function $\sum(r)$ is defined by one of the following forms: $\sum(r) = \mbox{sinh}(r)$, $r$, and $\mbox{sin}(r)$, respectively, for open, flat and closed geometries. The dimensionless function $f(\Omega_{j}, x)$ takes the following form: $$f(\Omega_{j}, x) = \left[\Omega_k x^{-2} + \left(\sqrt{\Omega_{\rm{r_c}}} +
\sqrt{\Omega_{\rm{r_c}}
+ \Omega_{\rm{m}}x^{-3}}\right)^{2}\right]^{1/2},$$ where $j$ stands for $m$, $r_c$ and $k$.
0.1in
Similarly, the predicted age of the Universe as a function of the redshift can be written as $$t_z = H_o^{-1}\int_{0}^{x'} {dx \over x f(\Omega_{j}, x)}.$$ As one may check from Eqs. (2), (5), (7) and (9), for $\Omega_{\rm{r_c}} = 0$, the standard relations are recovered. In Fig. 2 we show the dimensionless age parameter $H_ot_o$ as a function of $\Omega_{\rm{m}}$ for several values of $\Omega_{\rm{r_c}}$. Note that for a fixed value of $\Omega_{\rm{m}}$ the predicted age of the Universe is larger for larger values of $\Omega_{\rm{r_c}}$, thereby showing, similarly to what happens in the $\Lambda$CDM context, that the class of models studied here is efficient to solve the “already" classical age of the Universe problem. For example, if $\Omega_{\rm{m}} = 0.3$, as sugested by dynamical estimates on scales up to about $2h^{-1}$ Mpc [@calb], and $\Omega_{\rm{r_c}} = 0.15$ we find $t_o \simeq 13$ Gyr ($H_o =
70{\rm{Km.s^{-1}.Mpc^{-1}}}$) or, in terms of the age parameter, $H_ot_o \simeq 0.93$, a value that is compatible with the most recent age estimates of globular clusters [@carreta; @chab] as well as very close to some determinations based on SNe Ia data [@riess; @tonry].
\[t\]
0.1in
Constraints from high-$z$ angular size measurements
===================================================
In this section we study the constraints from angular size measurements of high-$z$ radio sources on the free parameters of the model. In the following we briefly outline our main assumptions for this analysis. Our approach is based on Ref. [@alcaniz].
The angular size-redshift relation for a rod of characteristic length $l$ can be written as [@sand] $$\theta (z) = \frac{D(1 + z)}{r(z)}.$$ In the above expression $D = 100lh$ is the angular size scale expressed in milliarcsecond (mas) for $l$ measured in parsecs (compact sources).
In order to constrain the parameters $\Omega_{\rm{m}}$ and $\Omega_{\rm{r_c}}$ we use the angular size data for milliarcsecond radio sources recently compiled by Gurvits [*et al.*]{} [@gurv1]. This data set, originally composed by 330 sources distributed over a wide range of redshifts ($0.011 \leq z \leq 4.72$), was reduced to 145 sources with spectral index $-0.38 \leq \alpha \leq
0.18$ and total luminosity $Lh^{2} \geq 10^{26}$ W/Hz in order to minimize any possible dependence of angular size on spectral index and/or linear size on luminosity. This new subsample was distributed into 12 bins with 12-13 sources per bin. Two points, however, should be stressed before discussing the resulting diagrams. First of all, the determination of cosmological parameters is strongly dependent on the characteristic length $l$ (see, e.g., [@alcaniz]). In the absence of a statistical study describing the intrinsic length distribution of the sources, we follow [@alcaniz] and, instead of assuming a specific value for the mean projected linear size, we have worked on the interval $l \simeq 20h^{-1} - 30h^{-1}$ pc, i.e., $l \sim O(40)$ pc for $h = 0.65$, or equivalently, $D = 1.4 - 2.0$ mas (see also [@gurv] for a detailed discussion on this topic). Second, following Kellermann [@kelle], we assume that compact radio sources are free of the evolutionary and selection effects that have bedevilled attempts to use extended double radio source in this context (see, for exemple, [@buc]), as they are deeply embedded in active galact nuclei, and, therefore, their morphology and kinematics do not depend considerably on the changes of the intergalactic medium. Moreover, these sources have typical ages of some tens of years, i.e., it is reasonable to suppose that a stable population is estabilished, characterized by parameters that do not change with the cosmic epoch [@jack].
Following a procedure similar to that described in [@alcaniz], we determine the cosmological parameters $\Omega_{\rm{m}}$ and $\Omega_{\rm{r_c}}$ through a $\chi^{2}$ minimization for a range of $\Omega_{\rm{m}}$ and $\Omega_{\rm{r_c}}$ spanning the interval \[0, 1\] in steps of 0.02, $$\chi^{2}(l, \Omega_{\rm{m}}, \Omega_{\rm{r_c}}) =
\sum_{i=1}^{12}{\frac{\left[\theta(z_{i}, l, \Omega_{\rm{m}}, \Omega_{\rm{r_c}}) -
\theta_{oi}\right]^{2}}{\sigma_{i}^{2}}},$$ where $\theta(z_{i}, l, \Omega_{\rm{m}}, \Omega_{\rm{r_c}})$ is given by Eqs. (7) and (10) and $\theta_{oi}$ is the observed values of the angular size with errors $\sigma_{i}$ of the $i$th bin in the sample.
\[t\]
0.1in
In Fig. 3 we show the binned data of the median angular size plotted as a function of redshift for several values of $\Omega_{\rm{m}}$ and $\Omega_{\rm{r_c}}$. For comparison we also show the standard prediction (thick line). Fig. 4 displays the $95\%$ and $68\%$ c.l. limits from angular size data on the $\Omega_{\rm{m}} - \Omega_{\rm{r_c}}$ plane for the interval $l \simeq 20h^{-1} -
30h^{-1}$ pc. Note that the limits on the plane are more restrictive for increasing values of the characteristic length $l$. It happens because for $z \sim 2$ (where most of our data points are concentrated) the parameter $\Omega_{\rm{r_c}}$ has a behavior similar to a cosmological constant or quintessence, i.e., it increases the distance between two different redshifts. In this way, according to Eq. (10), for the same $\theta_{oi}$’s the larger the value of $l$ the larger the value of $r(z)$ that is required or, equivalently, the smaller the value of $\Omega_{\rm{m}}$. For $l =
20.58h^{-1}$ pc (D = 1.4 mas) the peak of likelihood is located at $\Omega_{\rm{m}} = 0.22$ and $\Omega_{\rm{r_c}} = 0.18$. This assumption provides $\Omega_{\rm{m}} \leq 0.38$ and $\Omega_{\rm{r_c}} \leq 0.29$ at 1$\sigma$. In the subsequent panels of the same figure similar analyses are displayed for $l = 23.53h^{-1}$ pc (D = 1.6 mas), $l = 26.47h^{-1}$ pc (D = 1.8 mas) and $l = 29.41h^{-1}$ pc (D = 2.0 mas). For an analysis independent of the choice of the characteristic length $l$, i.e., minimizing Eq. (11) for $\Omega_{\rm{m}}$, $\Omega_{\rm{r_c}}$ and $l$, we obtain $\Omega_{\rm{m}} = 0.06$, $\Omega_{\rm{r_c}} = 0.28$ and $l = 27.06h^{-1}$ pc (D = 1.84 mas) as the best fit for these data with $\chi^{2} = 4.25$ and 9 degrees of freedom. We also remark that although not discussed here it is possible to determine the influence of $\Omega_{\rm{r_c}}$ on the critical redshift $z_m$ at which the angular size takes its minimal value. However as shown elsewhere [@jailS], this test cannot discriminate among world models since different scenarios provide similar values of $z_m$.
An elementary combination of our best fit with Eq. (4) enables us to estimate $r_c$ (the crossover distance between 4-dimensional and 5-dimensional gravity) in terms of the Hubble radius $H_o^{-1}$. One obtains, $$r_c \simeq 0.94 H_o^{-1}.$$ Such a value is slightly different from that one found by Deffayett [*et al.*]{} [@deff] using SNe Ia and CMB data. In their analysis it was found as a concordance model for these two tests a flat model with $\Omega_{\rm{m}} = 0.3$ and $\Omega_{\rm{r_c}} = 0.122$, leading to an estimate of the crossover radius of $r_c \simeq 1.4H_o^{-1}$. Naturally, with new results from different cosmological tests it will be possible to delimit the $\Omega_{\rm{m}} - \Omega_{\rm{r_c}}$ plane more precisely. An analysis on the observational constraints from statistics of gravitational lenses will appear in a forthcoming communication [@alc].
conclusion
==========
Based on a large body of observational evidence, a consensus is beginning to emerge that we live in a flat, accelerated universe composed of $\sim$ 1/3 of matter (barionic + dark) and $\sim$ 2/3 of a negative-pressure dark component. However, since the nature of this dark energy is still not well understood, an important task nowadays in cosmology is to investigate the existing possibilities in the light of the current observational data. In this paper we have focused our attention on some observational aspects of brane world cosmologies. These models, inspired on superstring-M theory, explain the observed acceleration of the Universe through a large scale modification of gravity arising from a gravitational leakage into an extra dimension [@deffZ]. We showed that their predicted age of the Universe is compatible with the most recent age estimates of globular clusters and, therefore, that there is no age crisis in the context of these models. By using a large sample of milliarcsecond radio sources recently updated and extended by Gurvits [ *et al.*]{} [@gurv1] we obtained, as the best fit for these data, a slightly closed, accelerated universe with $\Omega_{\rm{m}} = 0.06$ and $\Omega_{\rm{r_c}} = 0.28$. Such values lead to an estimate of the crossover radius of $r_c \simeq 0.94 H_o^{-1}$.
It is my pleasure to acknowledge helpful discussions with Professor J. A. S. Lima and Professor C. J. Hogan. I also thank Professor L. I. Gurvits for sending his compilation of the data and J. V. Cunha for producing Figure 4. This research is supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq - Brazil) and CNPq(620053/01-1-PADCT III/Milenio)
S. Perlmutter [*et al.*]{}, Nature, [**[ 391]{}**]{}, 51 (1998); S. Perlmutter [ *et al.*]{}, Astrophys. J. [**[517]{}**]{}, 565 (1999); A. Riess [*et al.*]{}, Astron. J. [**[116]{}**]{}, 1009 (1998) G. Efstathiou [*et al.*]{}, astro-ph/0109152 S. Weinberg, Rev. Mod. Phys. [**[61]{}**]{}, 1 (1989) V. Sahni and A. Starobinsky, Int. J. Mod. Phys. [**[D9]{}**]{}, 373, (2000) M. Özer and O. Taha, Nucl. Phys. [**[B287]{}**]{}, 776 (1987); K. Freese, F. C. Adams, J. A. Frieman and E. Mottola, Nucl. Phys. [**[B287]{}**]{}, 797 (1987); J. C. Carvalho, J. A. S. Lima and I. Waga, Phys. Rev. [**[D46]{}**]{} 2404 (1992); J. A. S. Lima and J. M. F. Maia, Mod. Phys. Lett. [**[A8]{}**]{}, 591 (1993); J. M. Overduin and F. I. Cooperstock, Phys. Rev. [**[D58]{}**]{}, 043506 (1998); O. Bertolami and P. J. Martins, Phys. Rev. [**[D61]{}**]{}, 064007 (2000); J. M. F. Maia and J. A. S. Lima, Phys. Rev. [**[D65]{}**]{}, 083513 (2002) B. Ratra and P. J. E. Peebles, Phys. Rev. [**[D37]{}**]{}, 3406 (1988); J. A. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett. [**[75]{}**]{}, 2077 (1995); R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. [**[80]{}**]{}, 1582 (1998) M. S. Turner and M. White, Phys. Rev. [**[D56]{}**]{}, R4439 (1997); T. Chiba, N. Sugiyama and T. Nakamura, Mon. Not. Roy. Astron. Soc. [**[289]{}**]{}, L5 (1997); G. Efstathiou, Mon. Not. Roy. Astron. Soc. [**[310]{}**]{}, 842 (1999); J. S. Alcaniz and J. A. S. Lima, Astrophys. J. [**[550]{}**]{}, L133 (2001); S. A. Bludman and M. Roos Astrophys. J. [**[547]{}**]{},77 (2001); R. Bean and A. Melchiorri, Phys. Rev. [**[D65]{}**]{}, 041302(R) (2002); D. Jain [*et al.*]{}, astro-ph/0105551; J. Kujat [*et al.*]{}, (in press). astro-ph/0112221 G. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. [**[B485]{}**]{}, 208 (2000) G. Dvali and G. Gabadadze, Phys. Rev. [**[D63]{}**]{}, 065007 (2001) C. Deffayet, Phys. Lett. B [**[502]{}**]{}, 199 (2001) C. Deffayet, G. Dvali and G. Gabadadze, Phys. Rev. [**[D65]{}**]{}, 044023 (2002) C. Deffayet [*et al.*]{}, astro-ph/0201164 S.-H. Henry Tye and I. Wasserman, Phys. Rev. Lett. [**[86]{}**]{}, 1682 (2001) K. Koyama and J. Soda, Phys. Rev. [**[D62]{}**]{}, 123502 (1997); C. van de Bruck, M. Dorca, R. H. Brandenberger and A. Lukas, Phys. Rev. [**[D62]{}**]{}, 123515 (2000); H. Kodama, A. Ishibashi and O. Seto, Phys. Rev. [**[D62]{}**]{}, 064022 (2000) S. C. Davis, W. B. Perkins, A. Davis and I. R. Vernon, Phys. Rev. [**[D63]{}**]{}, 083518 (2001) G. Dvali and S. -H. H. Tye, Phys. Lett. [**[B450]{}**]{}, 72, (1999); J. Khoury, P. J. Steinhardt and D. Waldram, Phys. Rev. [**[D63]{}**]{}, 103505 (2001) G. Dvali and G. Gabadadze, Phys. Lett. [**[B460]{}**]{}, 47 (1999) C. J. Hogan, Phys. Rev. Lett. [**[85]{}**]{}, 2044 (2000); Phys. Rev. [**[D62]{}**]{}, 121302 (2000) R. H. Brandenberger, D. A. Easson and D. Kimberly, Nucl. Phys. [**[B623]{}**]{}, 421 (2002); G. D. Starkman, D. Stojkovic and M. Trodden, Phys. Rev. Lett. [**[87]{}**]{}, 231303 (2001) C. J. Hogan, astro-ph/0104105 P. P. Avelino and C. J. A. P. Martins, Astrophys. J. [**[565]{}**]{}, 661 (2002) C. Deffayet, G. Dvali and G. Gabadadze, astro-ph/0106449 L. I. Gurvits, K. I. Kellermann and S. Frey, Astron. Astrop. [**[342]{}**]{}, 378 (1999) As is well known, according to Gauss’ law the gravitational force falls off as $r^{1
- S}$ where $S$ is the number of spatial dimensions. V. Sahni and Y. Shtanov, astro-ph/0202346 R. G. Calberg [*et al.*]{}, Astrophys. J. [**[462]{}**]{}, 32 (1996); A. Dekel, D. Burstein and S. White S., In Critical Dialogues in Cosmology, edited by N. Turok World Scientific, Singapore (1997) E. Carretta [*et al.*]{}, Astrophys. J. [**[533]{}**]{}, 215 (2000) L. M. Krauss and B. Chaboyer, astro-ph/0111597 J. L. Tonry, astro-ph/0105413 J. A. S. Lima & J. S. Alcaniz, Astrophys. J. [**[566]{}**]{}, 15 (2002) A. Sandage, Ann. Rev. Astron. Astrop. [**[26]{}**]{}, 561 (1988) L. I. Gurvits, Astrophys. J. [**[425]{}**]{}, 442 (1994) K. I. Kellermann, Nature [**[361]{}**]{}, 134 (1993) A. A. Ubachukwu, Astrop. Spac. Sci. [**[209]{}**]{}, 169 (1995); A. Buchalter, D. J. Helfland, R. H. Becker and R. L. White, Astrophys. J. [**[494]{}**]{}, 503 (1998) J. C. Jackson and M. Dodgson, Mon. Not. Roy. Astron. Soc. [**[285]{}**]{}, 806 (1997) J. A. S. Lima and J. S. Alcaniz, Astron. Astrop. [**[357]{}**]{}, 393 (2000); J. A. S. Lima and J. S. Alcaniz, Gen. Rel. Grav. [**[32]{}**]{}, 1851 (2000) D. Jain, A. Dev and J. S. Alcaniz. Submitted to Phys. Rev. D.
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abstract: 'The singularity structure and the corresponding asymptotic behavior of a 3-brane coupled to a scalar field in a five dimensional bulk is analyzed in full generality, using the method of asymptotic splittings. It is shown that the collapse singularity at a finite distance from the brane can be avoided only at the expense of making the brane world-volume positively or negatively curved.'
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CERN-PH-TH/2010-083
[Brane singularities with a scalar field bulk]{}
[Ignatios Antoniadis$^{1,*,3}$, Spiros Cotsakis$^{2,\dagger}$, Ifigeneia Klaoudatou$^{2,\ddagger}$]{}\
$^1$\
\
$^2$\
\
\
\
[(5,0)[280]{}]{}
$^3$[On leave from [*CPHT (UMR CNRS 7644) Ecole Polytechnique, F-91128 Palaiseau*]{}]{}
Introduction
============
Some time ago, an interesting idea to address the cosmological constant problem was proposed, based on the so-called self-tuning mechanism [@nima; @silver]. The simplest model consists of a 3-brane embedded in a five-dimensional bulk, in the presence of a scalar field. The later is coupled to the brane in a particular way, motivated by string theory, that allows flat brane world-volume solution independently of the brane tension value. It was however realized that a singularity appears in the bulk, at some finite distance from the brane, which can also be thought as a reservoir through which the vacuum energy decays.
An obvious question is then whether the development of such a singularity is a generic feature of these models, or under what conditions may be avoided. Here and in two subsequent papers, we investigate this question in a generalized class of models. Since in this case a general solution cannot be found analytically, we use a powerful tool developed a few years ago, called method of asymptotic splittings, that allows to compute all possible asymptotic behaviors of the equations of motion around the assumed location of a singularity [@skot]. Our main result is twofold:
- The existence of a singularity at a finite distance is unavoidable in all solutions with a flat brane. This confirms and extends the results of earlier works that made similar investigations in different models, using other methods [@Gubser; @Forste].
- The singularity can be avoided ([*e.g.*]{} moved at infinite distance) in several cases where the brane becomes curved, either positively or negatively. Thus, requiring absence of singularity brings back the cosmological constant problem, since the brane curvature depends on its tension that receives quartically divergent quantum corrections.
As mentioned already, our results are established in an extended version of the simplest model, where the bulk scalar field has a general coupling to the brane, motivated for instance from a loop corrected string effective action, allowing for curved world-volume. Besides the scalar field (or in the absence of it), we consider a generic bulk matter content parametrized by a fluid with an arbitrary equation of state. For pedagogical purposes, our analysis is separated in three parts contained in three different publications. A preliminary version of our results was published in [@iklaoud_6].
In this first paper, we give a detailed picture of the dynamical evolution of an extended version of the simplest model to include curved branes. We show that the emergence of the finite-distance singularity is the *only* possible asymptotic behavior for a flat brane, whereas for a curved brane the singularity is shifted at an infinite distance. We provide a detailed study of the asymptotics of this model using the method of asymptotic splittings expounded in [@skot].
The structure of this paper is as follows: In Section 2, we derive the form of the dynamical system on which our subsequent asymptotic analysis is applied. In Sections 3 and 4, we give the asymptotics of the models consisting of flat and curved brane respectively. In Section 5 we conclude and also comment on possible future work in various directions, considering for instance other forms of matter in the bulk. In the Appendix, we briefly outline the basic steps of the method of asymptotic splittings.
Dynamics of scalar field-brane configuration
============================================
In this Section we set up the basic equations for our braneworld. We study a braneworld model consisting of a three-brane embedded in a five-dimensional bulk space with a scalar field minimally coupled to the bulk. The total action $S_{total}$ splits in two parts, namely, the bulk action $S_{bulk}$ and the brane action $S_{brane}$, \[s\_tot\] S\_[total]{}=S\_[bulk]{}+S\_[brane]{}, with S\_[bulk]{}&=&d\^[4]{}x dY(- ()\^[2]{}),\
S\_[brane]{}&=&-d\^[4]{}xf(), Y=Y\_, where $Y$ denotes the fifth bulk dimension, $Y_{\ast}$ is the assumed initial position of the brane, $\lambda$ is a parameter defining the type of scalar field $\phi$, $\kappa^{2}_{5}=M_{5}^{-3}$, $M_{5}$ being the five-dimensional Planck mass, and $f(\phi)$ denotes the tension of the brane as a function of the scalar field.
Varying the total action (\[s\_tot\]) with respect to $g^{AB}$, we find the five-dimensional Einstein field equations in the form [@iklaoud_6], \[einst5d\] R\_[AB]{}-g\_[AB]{}R=\_[5]{}\^[2]{} (\_[A]{}\_[B]{}-g\_[AB]{}()\^[2]{}) + \_[A]{}\^\_[B]{}\^(Y), while the scalar field equation is obtained by variation of the action (\[s\_tot\]) with respect to $\phi$ [@iklaoud_6] and it is: \[scalarbr\] \_[5]{}=-(Y), where $A,B=1,2,3,4,5$ and $\a,\b=1,2,3,4$ while $\delta (Y)=1$ at $Y=Y_{\ast}$ and vanishing everywhere else, and \_[5]{}= \_[A]{}( g\^[AB]{}\_[B]{}) .
In the following we assume a bulk metric of the form \[warpmetric\] g\_[5]{}=a\^[2]{}(Y)g\_[4]{}+dY\^[2]{}, where $g_{4}$ is the four dimensional flat, de Sitter or anti de Sitter metric, i.e., \[branemetrics\] g\_[4]{}=-dt\^[2]{}+f\^[2]{}\_[k]{}g\_[3]{}, where g\_[3]{}=dr\^[2]{}+h\^[2]{}\_[k]{}g\_[2]{} and g\_[2]{}=d\^[2]{}+\^[2]{}d\^[2]{}. Here $
f_{k}=1,\cosh (H t)/H,\cos (H t)/H $ ($H^{-1}$ is the de Sitter curvature radius) and $ h_{k}=r,\sin r,\sinh r $, respectively.
The field equations (\[einst5d\])-(\[scalarbr\]) then take the form \[feq1\] &=&+,\
\[feq2\] &=&-,\
\[feq3\] ”+4’&=&0, where a prime denotes differentiation with respect to $Y$ and $k=0,\pm 1$. The variables to be determined are $a$, $a'$ and $\phi'$. These three equations are not independent since Eq. (\[feq2\]) was derived after substitution of Eq. (\[feq1\]) in the field equation $G_{\a\a}=\kappa_{5}^{2}T_{\a\a}$, $\a=1,2,3,4$, +- =-\_[5]{}\^[2]{}. In our analysis below we use the independent equations (\[feq2\]) and (\[feq3\]) to determine the unknown variables $a$, $a'$ and $\phi'$, while Eq. (\[feq1\]) will then play the role of a constraint equation for our system.
Assuming a $Y\rightarrow -Y$ symmetry and solving the Eqs. (\[einst5d\]) (the -$\a\a$- component, $\a=1,2,3,4$) and (\[scalarbr\]) on the brane we get \[bound1\] a’(Y\_)&=&-f((Y\_))a(Y\_),\
\[bound2\] ’(Y\_)&=&. The particular coupling used in [@nima] allows only for flat solutions to exist. This easily follows by using equations (\[bound1\]) and (\[bound2\]) and solving the FRW equation (\[feq1\]) on the brane for $kH^{2}$: $$kH^{2}=\frac{a^{2}(Y_{\ast})\kappa^{2}_{5}}{12}
\left(\frac{\kappa_{5}^{2}}{3}f^{2}(\phi(Y_{\ast}))-
\frac{f'^{2}(\phi(Y_{\ast}))}{4\lambda}\right).$$ Clearly, $k$ is identically zero if and only if: $$\frac{f'(\phi)}{f(\phi)}=2\sqrt{\frac{\lambda}{3}}\kappa_{5},$$ or equivalently, if and only if $f(\phi)\propto e^{2\sqrt{\lambda/3}\kappa_{5}\phi}$ (the authors of [@nima] have set $\lambda=3$ and hence the appropriate choice for the brane tension in that case is $f(\phi)\propto e^{2\kappa_{5}\phi}$). In our more general problem, the coupling function cannot be fixed this way. By working with other couplings we can allow for non-flat, maximally symmetric solutions to exist and avoid having the singularity at a finite distance away from the position of the brane.
For the rest of this paper our purpose is to find all possible asymptotic behaviours around the assumed position of a singularity, denoted by $Y_{s}$, emerging from general or particular solutions of the system (\[feq1\])-(\[feq3\]). The most useful tool for this analysis is the method of asymptotic splittings [@skot] (see the Appendix for a brief introduction) in which we start by setting x=a, y=a’, z=’, The field equations (\[feq2\]) and (\[feq3\]) become the following system of ordinary differential equations: \[syst1\_1\] x’&=&y\
\[syst1\_2\] y’&=&-Az\^[2]{}x\
\[syst1\_3\] z’&=&-4y, where $A=\kappa^{2}_{5}/4$. Hence, we have a dynamical system determined by the non-polynomial vector field =(y,-Az\^[2]{}x,-4y)\^. Equation (\[feq1\]) does not include any terms containing derivatives with respect to $\Upsilon$; it is a constraint equation which in terms of the new variables takes the form \[constraint1\] = z\^[2]{}+. Equations (\[syst1\_1\])-(\[syst1\_3\]) and (\[constraint1\]) constitute the basic dynamical system of our study. There are two major cases to be treated, the first is when we choose $k=0$ in (\[constraint1\]) and corresponds to a brane being flat, while in the second case $k\neq 0$, giving constant curvature to the brane. We shall treat these two cases independently in what follows. One important result of our analysis of this system will be that the inclusion of nonzero curvature for the brane moves the singularity an infinite distance away from the brane.
Flat brane: Finite-distance singularity
=======================================
In this Section we take $k=0$ in the basic constraint equation \[constraint\_flat\] = z\^[2]{}. We shall show that the only possible asymptotic behaviour of the solutions of this system (flat brane) is that $a\rightarrow
0$, $a'\rightarrow \infty$ and $\phi'\rightarrow \infty$, as $Y\rightarrow Y_{s}$.
We start our asymptotic analysis by inserting the forms \[dominant forms\] (x,y,z)=(\^[p]{},\^[q]{},\^[r]{}), in the system (\[syst1\_1\])-(\[syst1\_3\]), where (p,q,r)\^[3]{} (,,)\^[3]{}{}. We find that the only possible dominant balance in the neighborhood of the singularity (that is pairs of the form $\mathcal{B}=\{\mathbf{a},\mathbf{p}\}$, where $\mathbf{a}=(\alpha,\beta,\delta)$, $\mathbf{p}=(p,q,r)$, determining the dominant asymptotics as we approach the singularity) is the following balance \[sing\] \_[1]{}={(,/4,/(4)), (1/4,-3/4,-1)}. (A second balance $\mathcal{B}_{2}$ becomes only possible when we allow for non-zero curvature, $k\neq 0$, and will be analysed in the next Section. There are no other acceptable balances, hence all the possible asymptotic behaviours for a flat and curved brane can be described uniquely by the balances $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ respectively.)
Let us now focus on building a series expansion in the neighborhood of the singularity to justify the asymptotics found above. We start by calculating the Kowalevskaya exponents, eigenvalues of the matrix $\mathcal{K}=D\mathbf{f}(\mathbf{a})-\textrm{diag}(\mathbf{p}),$ where $D\mathbf{f}(\mathbf{a})$ is the Jacobian matrix of $\mathbf{f}$, which in our case reads: D(x,y,z)=(
[ccc]{} 0 & 1 & 0\
-Az\^[2]{} & 0 & -2Ax z\
& - & -\
), to be evaluated on $\mathbf{a}$. For the $\mathcal{B}_{1}$ balance we have that $\mathbf{a}=(\alpha,\alpha/4,\sqrt{3}/(4\sqrt{A\lambda}))$, and $\mathbf{p}=(1/4,-3/4,-1)$, thus =(
[ccc]{} - & 1 & 0\
- & & -\
& - & 0\
). The $\mathcal{K}$-exponents are then given by ()={-1,0,3/2}. These exponents correspond to the indices of the series coefficients where arbitrary constants first appear. The $-1$ exponent signals the arbitrary position of the singularity, $Y_{s}$. We see that the first balance $\mathcal{B}_{1}$ has two non-negative rational eigenvalues which means that it describes the asymptotics of a general solution in the form of a series expansion, i.e., a series form of the solution having the full number of arbitrary constants (which for our system equals to two). In order to construct an asymptotic expansion of this solution valid in the neighborhood of the singularity, we substitute in the system (\[syst1\_1\])-(\[syst1\_3\]) the series expansions $$\mathbf{x}=\Upsilon^{\mathbf{p}}(\mathbf{a}+
\Sigma_{j=1}^{\infty}\mathbf{c}_{j}\Upsilon^{j/s}),$$ where $\mathbf{x}=(x,y,z)$, $\mathbf{c}_{j}=(c_{j1},c_{j2},c_{j3})$, and $s$ is the least common multiple of the denominators of the positive eigenvalues; here $s=2$, and the corresponding series expansions are given by the following forms: x=\_[j=0]{}\^c\_[j1]{}\^[j/2+1/4]{},y=\_[j=0]{}\^c\_[j2]{}\^[j/2-3/4]{},z=\_[j=0]{}\^c\_[j3]{}\^[j/2-1]{}. Therefore we arrive at the following asymptotic solution around the singularity: \[Puis\_1x\] x&=&\^[1/4]{}+c\_[32]{}\^[7/4]{}+\
y&=&\^[-3/4]{}+c\_[32]{}\^[3/4]{}+\
\[Puis\_1z\] z&=&\^[-1]{}- c\_[32]{}\^[1/2]{}+.
The last step is to check whether for each $j$ satisfying $j/2=\rho$ with $\rho$ a positive eigenvalue, the corresponding eigenvector $v$ of the $\mathcal{K}$ matrix is such that the compatibility conditions hold, namely, v\^P\_[j]{}=0, where $P_{j}$ are polynomials in $\mathbf{c}_{i},\ldots, \mathbf{c}_{j-1}$ given by \_[j]{}-(j/s)\_[j]{}=P\_[j]{}. Here the relation $j/2=3/2$ is valid only for $j=3$ and the associated eigenvector is =(-, -,1). The compatibility condition, (-(3/2)\_[3]{})\_[3]{}=0, therefore indeed holds since (-(3/2)\_[3]{})\_[3]{}=c\_[32]{} (
[ccc]{} - & 1 & 0\
\
- & - & -\
\
& - & -\
) (
[c]{}\
\
1\
\
-\
)=(
[c]{} 0\
\
0\
\
0\
).This shows that a representation of the solution asymptotically by a Puiseux series as given in Eqs. (\[Puis\_1x\])-(\[Puis\_1z\]) is valid. Hence we conclude that near the singularity at finite distance $Y_{s}$ from the brane the asymptotic forms of the variables are: \[behscI\] a0, a’, ’. This is exactly the asymptotic behaviour of the solution found previously by Arkani-Hammed *et al* in [@nima]. Our analysis shows that this is *the only possible* asymptotic behaviour for a flat brane since there exist no other dominant balances in this case.
Curved brane: Infinite-distance singularity
===========================================
In this Section we show that the collapse singularity that necessarily arises in the case of a flat brane is avoided (or shifted at an infinite distance away from the brane) when we consider a curved brane instead.
The new asymptotics follow from the study of a second balance that results from the substitution of (\[dominant forms\]) in (\[syst1\_1\])-(\[syst1\_3\]). We calculate this new balance to be, \[nonsing\] \_[2]{}={(,,0),(1,0,-1)}. It corresponds to a particular solution for a *curved brane* since it satisfies Eq. (\[constraint1\]) for $k\neq 0$ and $\alpha^{2}=k H^{2}$ (here we have to sacrifice one arbitrary constant by setting it equal to $kH^{2}$), $k=\pm 1$. The $\mathcal{K}$-matrix of $\mathcal{B}_{2}$ is =D((,,0))-(1,0,-1)=(
[ccc]{} -1 & 1 & 0\
0 & 0 & 0\
0 & 0 & -3\
), with eigenvalues ()={-1,0,-3}. Thus for the balance $\mathcal{B}_{2}$ we find two distinct negative integer $\mathcal{K}$-exponents and an infinite expansion in negative powers of a *particular* solution (recall that we had to sacrifice one arbitrary constant) around the presumed singularity at $Y_{s}$, with the negative $\mathcal{K}$-exponents signaling the positions where the arbitrary constants first appear [@fordy]. We therefore expand the variables in series with descending powers of $\Upsilon$ in order to meet the two arbitrary constants occurring for $j=-1$ and $j=-3$, i.e., x=\_[j=0]{}\^[-]{}c\_[j1]{}\^[j+1]{}, y=\_[j=0]{}\^[-]{}c\_[j2]{}\^[j]{}, z=\_[j=0]{}\^[-]{}c\_[j3]{}\^[j-1]{}. Substituting these series expansions back in the system (\[syst1\_1\])-(\[syst1\_3\]) and after some manipulation, we find the following asymptotic behaviour, \[Puis\_2x\] x&=&+c\_[-11]{}+\
y&=&+\
\[Puis\_2z\] z&=&c\_[-33]{}\^[-4]{}+. Let us check the compatibility conditions for $j=-1$ and $j=-3$. We find that (+\_[3]{})\_[-1]{}=(
[ccc]{} 0 & 1 & 0\
0 & 1 & 0\
0 & 0 & -2\
) (
[c]{} c\_[-11]{}\
0\
0\
)=(
[c]{} 0\
0\
0\
), and (+3\_[3]{})\_[-3]{}=(
[ccc]{} 2 & 1 & 0\
0 & 3 & 0\
0 & 0 & 0\
) (
[c]{} 0\
0\
c\_[33]{}\
)=(
[c]{} 0\
0\
0\
), so that the compatibility conditions are indeed satisfied. The expansions given by Eqs. (\[Puis\_2x\])-(\[Puis\_2z\]) are therefore valid, and we can say that as $\Upsilon\rightarrow 0$, or equivalently as $S\equiv 1/\Upsilon\rightarrow \infty$, we have that \[behscII\] a, a’, ’. Therefore for a curved brane we find that there can be no finite-distance singularities. The only possible asymptotic behaviour is the one given in (\[behscII\]) which is only valid at an infinite distance from the brane.
Conclusions
===========
In this paper we studied a braneworld consisting of a three-brane embedded in a five-dimensional bulk space filled with a scalar field with a special emphasis in the possible formation of finite-distance singularities away from the brane into the bulk. We have shown that the dynamical behaviour of this model strongly depends on the spatial geometry of the brane, in particular whether it is flat or not. For a flat brane the model experiences a finite-distance singularity toward which all the vacuum energy decays (since $\phi'\rightarrow\infty$, as $Y\rightarrow Y_{s}$), whereas for a curved brane the model avoids the singularity which is now located at an infinite distance.
It is interesting that a third balance which initially arises from the substitution of (\[dominant forms\]) in (\[syst1\_1\])-(\[syst1\_3\]), namely, the form $$\mathcal{B}_{3}=\{(\alpha,0,0),(0,-1,-1)\},$$ is not acceptable for the model we consider in this paper since it does not give the necessary $-1$ $\mathcal{K}$-exponent. In future work [@akc2] and [@akc3], we will see that this balance although impossible in the case treated here, *does* become possible (although in a somewhat ‘mild’ form) when we replace the scalar field studied here with other matter components such as a perfect fluid or a combination of a perfect fluid and a scalar field. We therefore conclude that for the case of interest in this paper the collapse singularity found is the only type of singularity that can develop at a finite distance from a flat brane.
Acknowledgements {#acknowledgements .unnumbered}
================
S.C. and I.K. are grateful to CERN-Theory Division, where part of their work was done, for making their visits there possible and for allowing them to use its excellent facilities. The work of I.A. was supported in part by the European Commission under the ERC Advanced Grant 226371 and the contract PITN-GA-2009-237920 and in part by the CNRS grant GRC APIC PICS 3747.
Appendix: The method of asymptotic splittings
=============================================
We refer briefly here to the basic steps of the method of asymptotic splittings. A detailed analysis can be found in Ref. [@skot].
Consider a system of $n$ first order ordinary differential equations \[arb\_syst\] =(), where $\mathbf{x}=(x_{1},\ldots,x_{n})\in \mathbb{R}^{n}$, $\mathbf{f}(\mathbf{x})=(f_{1}(\mathbf{x}),\ldots,f_{n}(\mathbf{x}))$ and $'\equiv\frac{d}{dY}$, $Y$ being the independent variable. In this paper, we refrain from calling $Y$ a time variable and giving it the interpretation of time. Since we are interested in singularities located at a *distance* from the brane and into the bulk, it seems more appropriate to talk about finite-distance singularities and give to the $Y$ variable a spatial interpretation. The general solution of the above system contains $n$ arbitrary constants and describes all possible behaviours of the system starting from arbitrary initial data. Any particular solution of (\[arb\_syst\]), on the other hand, contains less than $n$ arbitrary constants and describes a possible behaviour of the system emerging from a proper subset of initial data space.
We say that a solution of the dynamical system (\[arb\_syst\]) exhibits a finite-distance singularity if there exists a $Y_{s}\in
\mathbb{R}$ and a $\mathbf{x}_{0}\in \mathbb{R}^{n}$ such that \_[YY\_[s]{}]{}(Y;\_[0]{}), where $\|\centerdot\|$ is any $L^{p}$ norm. The purpose of singularity analysis (cf. [@skot], [@goriely]) is to build series expansions of solutions around the presumed position of a singularity at $Y_{s}$ in order to study the different asymptotic behaviours of the solutions of the system (\[arb\_syst\]) as one approaches this singularity. In particular, we look for series expansions of solutions that take the form of a Puiseux series (any $\log$ terms absent), namely, a series of the form \[Puiseux\] =\^( +\_[i=1]{}\^\_[i]{}\^[i/s]{}), where $\Upsilon=Y-Y_{s}$, $\mathbf{p}\in \mathbb{Q}^{n}$, $s\in\mathbb{N}$.
The method of asymptotic splittings for any system of the form (\[arb\_syst\]) is realized by taking the following steps:
$\bullet$ First, we find all the possible *weight-homogeneous decompositions* of the vector field $\mathbf{f}$ by splitting it into components $\mathbf{f}^{(j)}$: =\^[(0)]{}+\^[(1)]{}+…+\^[(k)]{}, with each of these components being *weight homogeneous*, that is to say (\^)=\^[ +(q\^[(j)]{}-1)]{}\^[(j)]{}() j=0,…,k, where $\mathbf{a}\in\mathbb{R}^{n}$ and $q^{(j)}$ are the positive non-dominant exponents that are defined by (\[sub\_exp\]) below.
$\bullet$ We substitute the forms $\mathbf{x}=\mathbf{a}\mathbf{\Upsilon}^{\mathbf{p}}$ in the system $\mathbf{x}'=\mathbf{f}^{(0)}(\mathbf{x})$ in order to find all possible *dominant balances*, i.e., finite sets of the form $\{\mathbf{a},\mathbf{p}\}$. The *order* of each balance is defined as the number of the non-zero components of $\mathbf{a}$.
$\bullet$ For each of these balances we check the validity of the following *dominance condition*: \[dominance\] \_[0]{} =0, and define the non-dominant exponents $q^{(j)}$, $j=1,\ldots, k$ by the requirement that \[sub\_exp\] \~\^[q\^[(j)]{}]{}. The balances that cannot satisfy the condition (\[dominance\]) are then discarded.
$\bullet$ We compute the Kovalevskaya matrix $\mathcal{K}$ defined by =D\^[(0)]{}()-diag , where $D\mathbf{f}^{(0)}(\mathbf{a})$ is the Jacobian matrix of $\mathbf{f}^{(0)}$ evaluated at $\mathbf{a}$.
$\bullet$ We calculate the spectrum of the $\mathcal{K}$-matrix, $spec(\mathcal{K})$, that is the set of its $n$ eigenvalues also called the *$\mathcal{K}$-exponents*. The arbitrary constants of any particular or general solution first appear in those terms in the series (\[Puiseux\]) whose coefficients $\mathbf{c}_{k}$ have indices $k=\rho s$, where $\rho$ is a non-negative $\mathcal{K}$-exponent and $s$ is the least common multiple of the denominators of the set consisting of the non-dominant exponents $q^{(j)}$ and of the positive $\mathcal{K}$-exponents (cf. [@skot], [@goriely]). The number of non-negative $\mathcal{K}$-exponents equals therefore the number of arbitrary constants that appear in the series expansions of (\[Puiseux\]). There is always the $-1$ exponent that corresponds to the position of the singularity, $Y_{s}$. (A dominant balance corresponds thus to a general solution if it possesses $n-1$ non-negative $\mathcal{K}$-exponents (the $n$th arbitrary constant is the position of the singularity, $Y_{s}$)).
$\bullet$ We substitute the Puiseux series: x\_[i]{}=\_[j=0]{}\^c\_[ji]{}\^[p\_[i]{}+j/s]{}, i=1,…, n, in the system (\[arb\_syst\]).
$\bullet$ We find the coefficients $\mathbf{c}_{j}$ by solving the recursion relations \_[j]{}-\_[j]{}= \_[j]{}(\_[1]{},…, \_[j-1]{}) where $\mathbf{P}_{j}$ are polynomials that are read off from the original system. $\bullet$ We verify that for every $j=\rho s$, with $\rho$ a positive $\mathcal{K}$-exponent, the following compatibility conditions hold: \[comp\_cond\] \^\_[j]{}=0, where $\mathbf{\upsilon}$ is an eigenvector associated with the positive $\mathcal{K}$-exponent $\rho$.
$\bullet$ We repeat the procedure for each possible decomposition.
We note that if the compatibility condition above (Eq. (\[comp\_cond\])) is violated at some eigenvalue in the $\textrm{spec}(\mathcal{K})$, then the original Puiseux series representation of the solution cannot be admitted and instead we have to use a *$\psi$-series* for each one of the eigenvalues with this property. This is a series that includes $\log$ terms of the form =\^( +\_[i,j=1]{}\^\_[ij]{}\^[i/s]{}(\^)\^[j/s]{}), where $\rho$ is the $\mathcal{K}$-exponent for which the compatibility condition is violated. The rest of the procedure in this case is the same as before.
N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, R. Sundrum, *A small cosmological constant from a large extra dimension*, Phys. Lett. B 480 (2000) 193-199, \[arXiv:hep-th/0001197v2\].
S. Kachru, M. Schulz, E. Silverstein, *Bounds on curved domain walls in 5d gravity*, Phys. Rev. D 62 (2000) 085003, \[arXiv:hep-th/0002121\].
S. Cotsakis, J. D. Barrow, *The Dominant balance at cosmological singularities*, J. Phys. Conf. Series 68 (2007) 012004, \[arXiv:gr-qc/0608137\].
S. S. Gubser, *Curvature singularities: The good, the bad, and the naked*, Adv. Theor. Math. Phys. 4 (2000) 679, \[arXiv:hep-th/0002160\].
S. Forste, Z. Lalak, S. Lavignac and H. P. Nilles, *A comment on self-tuning and vanishing cosmological constant in the brane world*, Phys. Lett. B 481 (2000) 360 \[arXiv:hep-th/0002164\]; *The Cosmological Constant Problem from a Brane-World Perspective*, JHEP [**0009**]{} (2000) 034 \[arXiv:hep-th/0006139\].
I. Antoniadis, S. Cotsakis, I. Klaoudatou, *Braneworld cosmological singularities*, Proceedings of MG11 meeting on General Relativity, vol. 3, pp. 2054-2056, \[arXiv:gr-qc/0701033\].
A. Fordy and A. Pickering, *Analysing negative resonances in the Panlaivé test*, Phys. Lett. A 160 (1991) 347-354.
I. Antoniadis, S. Cotsakis, I. Klaoudatou, *Collapsed and ripped branes in a fluid bulk* (preprint).
I. Antoniadis, S. Cotsakis, I. Klaoudatou, *Brane singularities with non-interacting mixtures in the bulk* (preprint).
A. Goriely and C. Hyde, *Necessary and sufficient conditions for finite time singularities in ordinary differential equations*, J. Diff. Eq. 161 (2000) 422-48.
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---
abstract: 'It has been proposed that the observed nonclassical rotational inertia (NCRI) in solid helium results from the superflow of thin liquid films along interconnected grain boundaries within the sample. We have carried out new torsional oscillator measurements on large helium crystals grown under constant temperature and pressure. We observe NCRI in all samples, indicating that the phenomenon cannot be explained by a superfluid film flowing along grain boundaries.'
author:
- 'A. C. Clark'
- 'J. T. West'
- 'M. H. W. Chan'
title: Nonclassical Rotational Inertia in Helium Crystals
---
The finding of NCRI in solid helium [@science] has been replicated in torsional oscillator (TO) measurements in four other laboratories [@RR; @shkub; @kojima]. The temperature dependence of NCRI, characterized by saturation in the low temperature limit and a gradual decay to zero at higher temperature, is qualitatively reproducible in all measurements. However, the onset temperature *T$_O$*, the point where NCRI becomes resolvable from the noise, varies between 150 mK and 400 mK in the studies of commercially pure helium ($\sim$300 ppb of $^3$He). In addition, relative to the total amount of $^4$He in the cell the NCRI fraction (NCRIF) measured in the low temperature limit ranges from as little as 0.03% up to 20% [@RR2]. In a large, cubic cell with linear dimensions of $\sim$1 cm, Rittner and Reppy found the measured NCRIF $\approx$ 0.5% could be reduced to $<$ 0.05% after annealing the sample [@RR]. Although this appears consistent with two numerical simulations in which perfect crystals are insulating [@cepUmass], less dramatic results have been observed in similar annealing studies [@shkub; @prl; @ek3he].
Although the presence of crystalline defects influences NCRIF, neither the specific defects of importance, nor their relationship with NCRI are known. Three fundamentally different kinds of defects that are present in solid helium are point defects such as vacancies or interstitials, dislocation lines, and grain boundaries. Vacancies, which are likely more prevalent than interstitials, were suggested to facilitate supersolidity in early theoretical literature [@andchest]. Although recent investigations [@umass1mahan] report attractive forces between them, a small concentration of zero-point vacancies may still exist and facilitate NCRI [@reatto]. However, it is impossible that they alone can account for an NCRIF of 20% [@RR2].
There is an alternative model [@umass2balibar] in which NCRI actually results from superfluid liquid $^4$He flowing along grain boundaries. To be consistent with what is known of thin superfluid films [@KT], the $\sim$200 mK transition temperature implies an effective thickness of 0.06 nm (one-fifth of a monolayer). Thus, enormous surface areas of completely interconnected grain boundaries are necessary to support a supercurrent constituting even just one percent of the entire sample volume. This would require the crystallites to have an average grain size of $\sim$20 nm. Even for NCRIF = 0.03% [@RR2; @ek3he] the average grain size would be $<$ 1 $\mu$m, whereas samples grown by the blocked capillary (BC) method commonly result in crystals with linear dimensions $\geq$ 0.1 mm, as determined by thermal conductivities [@therm] and x-ray diffraction [@x]. In addition, annealing is found to increase this value [@therm]. The same BC method was used in all previous TO studies, and involves filling the sample cell through the hollow torsion rod with high pressure liquid helium, freezing a section of $^4$He in the filling line (the block), and then cooling the constant volume below the block along the solid-liquid coexistence boundary until solidification is complete.
Two growth techniques that are superior to the BC method [@vos; @heybey] are carried out at a fixed point anywhere on the solid-liquid coexistence curve. The first, constant pressure (CP) growth, is achieved by slowly cooling the cell while a specific freezing pressure *P$_F$* is maintained. The second, constant temperature (CT) growth, takes place at a single freezing temperature *T$_F$* for minimal overpressures above *P$_F$*. The latter technique is most often (as it is in this study) employed when growing low pressure solids from the superfluid phase [@vos; @heybey]. The first extensive investigations demonstrating that large single crystals are reliably grown at CP/CT combined either *in situ* x-ray diffraction [@greywall] or optical birefringence [@wanner; @lee] techniques with sound velocity measurements. An important experimental detail from Ref.’s [@heybey; @greywall; @wanner; @lee] is that the $^4$He filling line necessarily remained open during solidification. Also, a cold surface at one end of the cell seeded crystals, whereas the remaining surfaces were poor thermal conductors so as to avoid the nucleation of multiple crystallites. Due to such reliability, very similar methods have been incorporated in all studies of $^4$He single crystals since the early 1970s.
The motivation of the present work was to carry out a definitive experiment to test the grain boundary model. In an effort to separate the effects of isotopic impurities [@ek3he], samples were grown from both “isotopically pure” $^4$He ($\sim$1 ppb of $^3$He) and commercially pure $^4$He ($\sim$300 ppb of $^3$He). Measurements have been carried out in two torsional oscillators, one made from beryllium copper (BeCu) and one from coin silver (AgCu) (see Fig. \[fig:one\]a). Unlike other TO’s in which a hollow torsion rod serves as the $^4$He filling line [@science; @RR; @shkub; @kojima; @RR2; @prl; @ek3he], the torsion rod is solid and a CuNi capillary (o.d. = 0.3 mm, i.d. = 0.1 mm) was soldered to the opposite end of the cell. Apart from a small cold spot from which to seed crystals, the walls are coated with a thin layer of epoxy. This design allows us to seed the crystal at the cell bottom and maintain an open filling line during freezing, and thus enabled the growth of crystals at CT/CP within a TO for the first time. Having replicated the precise conditions outlined in Ref.’s [@heybey; @greywall; @wanner; @lee], we can be confident that many of our samples are single crystals or at worst comprised of just a few large crystals in the sample cell. We note that the capillary can also be intentionally blocked with solid $^4$He during the growth of a crystal, mimicking the BC method employed in previous TO studies.
 Schematic drawing of TO’s. The epoxy coating is 0.025 cm thick on the bottom of each cell, and 0.05/0.04 cm thick on the BeCu/AgCu walls. The 0.25 cm diameter BeCu cold spot is recessed (to the dotted line) from the epoxy layer. In the AgCu TO the cold surface is flush with the epoxy layer. Some relevant parameters for the BeCu/AgCu TO are *h* = 0.483/0.597 cm, *d* = 0.914/1.016 cm, $\tau$ = 0.933/1.27 ms (resonant period at 20 mK). (b) Mass loading of BeCu TO during growth of two samples. The minimum loading (solid at 25 bar) of the BeCu/AgCu TO is 740/2000 ns.](Fig1reviewed){width="1.0\columnwidth"}
The mass loading of the BeCu TO during crystal formation is displayed in Fig. \[fig:one\]b for two representative samples, one grown at CT and one by BC. For the BC sample the molar volume *V$_M$* of the solid during growth ranges from 19.3 cm$^3$$\,$mol$^{-1}$ (at *T* = 2.45 K and *P* = 55 bar) to 20.7 cm$^3$$\,$mol$^{-1}$ (*T* = 1.8 K and *P* = 30.7 bar). In contrast to this 7% change in *V$_M$*, the sample grown at CT from the superfluid was subjected to variations in *V$_M$* of $\sim$0.07% (i.e., temperature variations of $\sim$20 mK).
 Comparison of several BC, CP, and CT samples grown in the BeCu TO with two different impurity levels. *T$_F$* is the temperature where solidification is complete. For eight 1 ppb crystals grown at CT/CP, 0.26% $\leq$ NCRIF $\leq$ 0.38% ($\pm$0.01%) and *T$_O$* = 79 $\pm$ 5 mK. Among 12 BC samples, 0.46% $\leq$ NCRIF $\leq$ 2.0% and 80 mK $\leq$ *T$_O$* $\leq$ 275 mK. The two 300 ppb samples grown in the BeCu TO are also shown. The maximum speed at the rim of the cell is $<$ 5 $\mu$m$\,$s$^{-1}$ in all cases. (b) The four 300 ppb samples grown in the AgCu TO. Rim speeds are $<$ 8 $\mu$m$\,$s$^{-1}$.](Fig2reviewedALL){width="1.0\columnwidth"}
The temperature dependence of NCRIF in a number of BC and CT/CP samples is shown in Fig. \[fig:two\]. For a particular cell the NCRIF in BC samples is larger than that in samples grown at CT/CP. This is true for both $^3$He concentrations. Further, we find that *T$_O$* is reduced when employing the CT/CP growth. Surprisingly, there is an order of magnitude difference in the NCRIF measured in the two cells. It appears that for both BC and CT/CP samples, NCRIF is very sensitive to the exact internal geometry, construction materials, and thermal properties of the cells. For example, temperature gradients within the BeCu TO during growth are larger than in the AgCu TO due to the thicker epoxy layer and the much lower thermal conductivity of BeCu.
The most striking result from our study is that all eight 1 ppb samples that were grown at CT/CP (at a rate of $\sim$1 $\mu$m$\,$s$^{-1}$) collapse onto a single curve above 40 mK and thus share a common onset temperature. The vast improvement in reproducibility over that of BC samples is most likely due to the formation of single crystals within the cell. The spread in NCRIF at lower temperatures may be related to differences in crystalline defect (e.g., dislocations) densities.
 NCRIF after each sequential anneal for a BC sample quenched from the bcc phase. A CT crystal grown from the superfluid is potted for comparison. (b) Asymmetric reduction in *Q*$^{-1}$ following each anneal. All data were obtained with rim speeds $<$ 3 $\mu$m$\,$s$^{-1}$.](Fig3){width="1.0\columnwidth"}
We have carried out several annealing studies to investigate the defects in our crystals. For a BC sample (see Fig. \[fig:two\]b, *T$_F$* = 2.20 K) in the AgCu TO we found NCRIF to interestingly increase from 0.1% to 0.2% upon repeated annealing. The dissipation, which accompanies NCRI in the form of a peak [@science], also increased. In the BeCu TO we found NCRIF in BC samples to decrease with annealing. The most dramatic reduction occurred in a 1 ppb sample grown with the BC method through the bcc-hcp phase boundary (see Fig. \[fig:three\]). The sizeable tail of NCRI is such that *T$_O$* $\approx$ 275 mK. This was dramatically reduced following the first anneal at relatively low temperature. In fact, after 25 cumulative hours of annealing NCRIF asymptotically approaches that found in CT/CP samples. For a 1 ppb crystal freshly grown at CT there is no noticeable change in NCRIF even after 40 h of annealing.
Annealing the BC sample in Fig. \[fig:three\] also led to a dramatic reduction in the dissipation ($\propto$ *Q*$^{-1}$). Repeated heat treatments reduce the width of the peak, such that its position remains close to the temperature where NCRIF changes most rapidly. The fully annealed dissipation peak, just as NCRIF, approaches that found in CT/CP samples. A phenomenological model [@huse] associates the dissipation with a temperature dependent coupling between the superfluid and normal components of the solid, and predicts ($|\Delta\tau|$/$\tau$)/($\Delta$*Q*$^{-1}$) $\approx$ 1 for a homogeneous sample ($>$ 1 indicates inhomogeneity). The ratio for this sample at different stages of annealing evolves nonmonotonically from 9.5 to 12 to 10.5.
When the results of the present set of measurements are considered together with the myriad of data from earlier studies [@science; @RR; @shkub; @kojima; @RR2; @prl; @ek3he], dislocations emerge as a likely important class of defects. Dislocation lines form an entangled web throughout each crystal, and can vary in density by more than five orders of magnitude ($<$ 10$^5$ cm$^{-2}$ to 10$^{10}$ cm$^{-2}$) in solid helium samples grown above 1 K using different methods. The actual line density deduced from sound measurements [@dislocations] depends very sensitively (varying by four orders of magnitude) on the exact growth conditions of crystals [@dislocations; @hiki]. It also can vary by at least one order of magnitude from cell to cell, despite nearly identical growth procedures [@dislocations; @moredislocations]. The large range of line densities and their sensitivity to sample growth and containment can conceivably explain the very different NCRIF’s observed, even for single crystals. It is also known that only some types of dislocations can be annealed away, which may explain the unreliable effectiveness of annealing on the reduction of NCRIF.
The quantum mechanical motion of a single dislocation has recently been considered [@degennes], but a meaningful comparison with experiments requires a thorough analysis of complex dislocation networks. Recent simulations have shown that the core of some dislocations may support superflow [@umass5]. Applying Luttinger liquid theory to the dislocation network, Ref. [@umass5] predicts that if the dislocation cores alone are responsible for supersolidity then *T$_C$* $\propto$ ($\rho_S$/$\rho$)$^{0.5}$. Upon comparing several samples in Fig. \[fig:two\], the above relation is found not to be satisfied if we take *T$_C$* = *T$_O$* and $\rho_S$/$\rho$ = NCRIF. It has also been proposed that $^3$He impurities nucleate and stabilize dislocations, which in turn produce a disordered supersolid phase by providing a flow path for $^4$He interstitials [@epl].
![\[fig:four\]Normalized NCRIF in various samples. There is a wide spread in the data from the original KC experiment [@science]. BC samples presented in Fig. \[fig:two\] possess a high temperature tail of NCRI. CT/CP samples have a considerably sharper onset. A two-thirds power law is plotted for comparison.](Fig4reviewed){width="1.0\columnwidth"}
In addition to the magnitude of NCRIF, the high temperature tail and thus *T$_O$* are correlated with the way samples are prepared (see Fig. \[fig:two\]). Several points can be drawn from the data in Fig. \[fig:four\], which are scaled by the low temperature NCRIF. First, the high temperature tail of NCRI varies greatly for different BC samples, which are presumably polycrystalline. Second, the behavior of NCRIF in CT/CP samples is distinct in that the temperature dependence is much sharper, with a well-defined onset temperature. This is most apparent in crystals of 1 ppb purity. The addition of $^3$He broadens the transition and pushes the onset of NCRI to higher temperature. The sensitivity to impurities confirms the general trends observed for a large number of BC samples that were studied over a wide range of $^3$He concentrations [@ek3he].
All the data in Fig. \[fig:four\] were obtained at similar measurement frequencies. A recent TO measurement [@kojima] on the same 300 ppb sample at two different frequencies found *T$_O$* $\approx$ 250 mK at 1173 Hz and *T$_O$* $\approx$ 150 mK at 496 Hz. They also report irreversible changes in NCRIF below $\sim$40 mK upon variation of the oscillation speed. We have investigated [@TC] the thermal history of what appears to be the same phenomenon in 1 ppb crystals and find that in the low temperature limit there are in fact many metastable NCRIF’s available to the system. These results, as well as previously observed critical velocities on the order of one quantum of circulation [@prl], indicate that the likely excitations in the system are vortices.
A recent model [@anderson] capturing several aspects of the experiments equates NCRI to the rotational susceptibility of a vortex liquid phase. The high temperature tail of NCRI is said to reflect the finite response time of vortices in the sample, which are further slowed by $^3$He atoms dragged along with them. However, the low temperature behavior of NCRIF [@kojima; @TC] suggests that at least a portion of the vortices are pinned. Trace amounts of $^3$He are found to produce significant changes in the sound velocity and elastic constants measured in solid $^4$He. These results indicate that $^3$He impurities condense onto dislocation lines [@iwasaPBD]. It is reasonable to assume that these $^3$He-rich regions are the vortex pinning sites within the sample. In this scenario the interplay between vortices, impurities, and dislocations greatly impact the measured NCRIF in the solid.
A broad heat capacity peak near 75 mK was recently detected in solid $^4$He [@xi]. This finding supports the notion that the appearance of NCRI is a genuine signature of the transition between the normal and supersolid phases. It is then natural to wonder if it falls into the same universality class as that of superfluid $^4$He, i.e., the 3D XY model. The gradual onset of NCRI previously observed is not consistent with this expectation. However, the sharp onset present in CT/CP crystals of 1 ppb purity is intriguing. We noted above that the NCRIF’s in all eight samples collapse onto a single curve for *T* $>$ 40 mK (see Fig. \[fig:two\]). The NCRIF data of these crystals between 30 mK and 57 mK, as shown in Fig. \[fig:four\], can be represented by the expected two-thirds power law, with a critical temperature *T$_C$* $\approx$ 60 mK. If this highly speculative “fit” is applicable then the tail between 60 mK and 79 mK is attributable to the finite measurement frequency, residual (1 ppb) $^3$He impurities, and crystalline defects. Measurements at a much lower frequency may reveal if there is any validity to this speculation.
To summarize, we have shown that NCRI is found in large crystals of solid $^4$He grown at constant CT/CP. In contrast to the results from BC samples, the temperature dependence of NCRI in what are very likely single crystals is reproducible and exhibits a sharp onset.
We thank P. W. Anderson, J. Beamish, W. F. Brinkman, D. A. Huse, E. Kim, H. Kojima, X. Lin, N. Mulders, J. D. Reppy, and A. S. C. Rittner for informative discussions. We also thank J. A. Lipa for the 1 ppb purity helium and J. D. Reppy for the thin capillary used in our cells. Support was provided by NSF Grants DMR 0207071 and 0706339.
[31]{} E. Kim and M. H. W. Chan, Science **305**, 1941 (2004). A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. **97**, 165301 (2006). A. Penzev, Y. Yasuta, and M. Kubota, J. Low Temp. Phys. **148**, 667 (2007); M. Kondo, S. Takada, Y. Shibayama, and K. Shirahama, *ibid.* **148**, 695 (2007). Y. Aoki, J. C. Graves, and H. Kojima, Phys. Rev. Lett. **99**, 015301 (2007). A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. **98**, 175302 (2007). D. M. Ceperley and B. Bernu, Phys. Rev. Lett. **93**, 155303 (2004); N. V. Prokof’ev and B. V. Svistunov, *ibid.* **94**, 155302 (2005). E. Kim and M. H. W. Chan, Phys. Rev. Lett. **97**, 115302 (2006). E. Kim, J. S. Xia, J. T. West, X. Lin, and M. H. W. Chan, Bull. Am. Phys. Soc. **52**, 610 (2007). A. F. Andreev and I. M. Lifshitz, Sov. Phys. JETP **29**, 1107 (1969); G. V. Chester, Phys. Rev. A **2**, 256 (1970). M. Boninsegni *et al.*, Phys. Rev. Lett. **97**, 080401 (2006); G. D. Mahan and H. Shin, Phys. Rev. B **74**, 214502 (2006). D. E. Galli and L. Reatto, Phys. Rev. Lett. **96**, 165301 (2006). E. Burovski, E. Kozik, A. Kuklov, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. Lett. **94**, 165301 (2005); S. Sasaki, R. Ishiguro, F. Caupin, H. J. Maris, and S. Balibar, Science **313**, 1098 (2006); J. Low Temp. Phys. **148**, 665 (2007); L. Pollet *et al.*, Phys. Rev. Lett. **98**, 135301 (2007). J. M. Kosterlitz and D. J. Thouless, J. Phys. C: Solid State Phys. **6**, 1181 (1973); D. J. Bishop and J. D. Reppy, Phys. Rev. Lett. **40**, 1727 (1978). F. J. Webb, K. R. Wilkinson, and J. Wilks, Proc. R. Soc. A **214**, 546 (1952); G. A. Armstrong, A. A. Helmy, and A. S. Greenberg, Phys. Rev. B **20**, 1061 (1979). A. F. Schuch and R. F. Mills, Phys. Rev. Lett. **8**, 469 (1962). J. E. Vos *et al.*, Physica **37**, 51 (1967). O. W. Heybey and D. M. Lee, Phys. Rev. Lett. **19**, 106 (1967). D. S. Greywall, Phys. Rev. A **3**, 2106 (1971). R. Wanner and J. P. Franck, Phys. Rev. Lett. **24**, 365 (1970). R. H. Crepeau, O. Heybey, D. M. Lee, and S. A. Strauss, Phys. Rev. A **3**, 1162 (1971). D. A. Huse and Z. U. Khandker, Phys. Rev. B **75**, 212504 (2007). R. Wanner, I. Iwasa, and S. Wales, Solid State Commun. **18**, 853 (1976). F. Tsuruoka and Y. Hiki, Phys. Rev. B **20**, 2702 (1979). I. Iwasa, K. Araki, and H. Suzuki, J. Phys. Soc. Jpn. **46**, 1119 (1979). P.-G. de Gennes, C. R. Physique **7**, 561 (2006). M. Boninsegni *et al.*, arXiv:0705.2967v1 (2007). E. Manousakis, Europhys. Lett. **78**, 36002 (2007). A. C. Clark and M. H. W. Chan, to be published. P. W. Anderson, Nature Physics **3**, 160 (2007). H. Suzuki and I. Iwasa, J. Phys. Soc. Jpn. **49**, 1722 (1980); M. A. Paalanen, D. J. Bishop, and H. W. Dail, Phys. Rev. Lett. **46**, 664 (1981). X. Lin, A. C. Clark, and M. H. W. Chan, to be published.
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---
abstract: |
Ten years ago, Gla[ß]{}er, Pavan, Selman, and Zhang [@gla-pav-sel-zha:j:splitting] proved that if ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then all NP-complete sets can be simply split into two NP-complete sets.
That advance might naturally make one wonder about a quite different potential consequence of NP-completeness: Can the union of easy NP sets ever be hard? In particular, can the union of two non-NP-complete NP sets ever be NP-complete?
Amazingly, Ladner [@lad:j:np-incomplete] resolved this more than forty years ago: If ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then all NP-complete sets can be simply split into two non-NP-complete NP sets. Indeed, this holds even when one requires the two non-NP-complete NP sets to be disjoint. We present this result as a mini-tutorial. We give a relatively detailed proof of this result, using the same technique and idea Ladner [@lad:j:np-incomplete] invented and used in proving a rich collection of results that include many that are more general than this result: delayed diagonalization. In particular, the proof presented is based on what one can call *team diagonalization* (or if one is being playful, perhaps even *tag-team diagonalization*): Multiple sets are formed separately by delayed diagonalization, yet those diagonalizations are mutually aware and delay some of their actions until their partner(s) have also succeeded in some coordinated action.
We relatedly note that, as a consequence of Ladner’s result, if ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, there exist OptP functions $f$ and $g$ whose *composition* is NP-hard yet neither $f$ nor $g$ is NP-hard.
author:
- |
Lane A. Hemaspaandra[^1]\
Dept. of Computer Science\
University of Rochester\
Rochester, NY 14627\
USA
- |
Holger Spakowski[^2]\
Department of Mathematics\
and Applied Mathematics\
University of Cape Town\
Rondebosch 7701, South Africa
date: 'July 28, 2018'
title: 'Team Diagonalization[^3]'
---
Introduction
============
This paper is about the fact that if ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then NP-hardness can be built by in a simple way by composing/combining non-NP-hard subparts.
Our initial interest in this came from a question not about sets but about functions. At a computational biology talk one of us attended, two actions were sequentially taken on the input and the overall transformation was clearly NP-hard. During a discussion of the talk, the question came up of whether that meant that at least one of the two composed transformations must itself be NP-hard. Although for the example of that talk it probably was the case that one of the constituent transformations could be directly proved NP-hard, what we eventually realized (not nearly as quickly as we should have) is that (assuming ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$) this is not a general behavior. In particular, if ${\mbox{\rm P}}\not=
{\mbox{\rm NP}}$, then there exist NP optimization functions (i.e., ${\mbox{\rm OptP}}$ functions) $f$ and $g$ such that neither $f$ nor $g$ is even NP-Turing-hard, yet $g(f(x))$ is (functional) NP-many-one-hard. That is, non-NP-hard functions can via composition achieve NP-hardness.
The natural way to show this is simply to observe that the function case follows immediately from the language case, *and the language case already put to bed in the 1970s by Ladner [@lad:j:np-incomplete]!* So this paper presents, as a mini-tutorial on delayed diagonalization, and in honor of Professor Ladner’s career on the occasion of his retirement, a proof of the language-case result, which is in fact a special case of one of his 1975 results. (The paper also provides the observation that the (OptP) function case follows immediately from that.)
##### Credit Where Credit Is Due and Blame Where Blame Is Due
It is important to mention up front that, as this is a tutorial, the credit for the results here belongs not to the authors of this tutorial, but to the author of the underlying paper, namely Richard Ladner. All of the results of this paper are either explicitly in his seminal paper on delayed diagonalization or are implicit from or (for the case of Section \[sect:functions\]) follow easily from its results and techniques. Similarly, though the particular proof write-up of Theorem \[thm:main\] and the framing of the theorem are our attempts to frame a clear yet very accessible proof for a tutorial article, it is very important to stress that all we are doing is employing Ladner’s breakthrough technique, delayed diagonalization, that he developed to prove results of this sort and of many related sorts, and indeed the proof is his in every sense other than that any errors our version might have, which of course are ours. In fact, in some sense we are giving a rather long proof to get a result weaker than ones that his original paper gets; the reason for this is that our goal here is to be an expository paper, and to make as clear as possible, a particular flavor of delayed diagonalization. So in brief, all the ideas here and much of the detail is due to the seminal paper of Ladner—except for any errors here, as those will be due to flaws on our part in writing this tutorial.[^4]
##### Organization
The rest of this article is organized as follows. Section \[s:lit\] mentions some related or contrasting results. Section \[s:defs\] provides some preliminaries and definitions. Section \[sect:sets\] contains our write-up, but using Ladner’s delayed diagonalization, proving the result, due to Ladner, that if P and NP differ, then there exists a pair of disjoint, non-NP-complete sets in ${\mbox{\rm NP}}- {\mbox{\rm P}}$ whose union is NP-complete. The case of function composition described above will then follow easily, and is covered in Section \[sect:functions\], and Section \[s:gen\] provides a brief teaser for some of the rest of the world of results that Ladner’s work provides and/or underpins.
Related and Contrasting Work {#s:lit}
============================
The fascinating related work of Gla[ß]{}er et al. [@gla-sel-tra-wag:j:union-disjoint] has a different focus. That paper on the complexity of unions of disjoint sets primarily focuses on whether unions of disjoint NP-complete sets remain hard. That paper does have a section—Section 4.2 in that paper’s numbering—where the union of two disjoint sets is harder than its components, but the results of that section do not overlap at all with Ladner’s [@lad:j:np-incomplete] work, due to Gla[ß]{}er et al.’s focus on *equivalent* (to each other) constituent sets.
Another related paper by Gla[ß]{}er et al. [@gla-pav-sel-zha:j:splitting] shows that every nontrivial (in the sense of the set and its complement each containing at least two elements) NP-complete set is (so-called) m-mitotic. This result is interesting for us here because it implies that every nontrivial NP-complete set can be partitioned into two P-separable sets that are NP-complete. However, while Ladner’s work splits NP-complete sets into NP-*non*complete NP sets, Gla[ß]{}er et al. are interested in splitting NP-complete sets into NP-[*complete*]{} sets.
We also mention the work of Hemaspaandra et al. [@hem-jia-rot-wat:j:join-lowers] that shows that the join operator (the marked union operator) can yield a set of lower complexity (in the extended low hierarchy) than either of its constituents. This regards a focus *opposite that of Ladner’s work presented in this paper*. In particular, that work is about using combinations to lower complexity; the focus of the Ladner work that we are presenting is on using combinations to rise from non-NP-completeness to NP-completeness.
We prove Ladner’s key result using [*team diagonalization*]{}. Delayed diagonalization is the powerful technique first used by Ladner [@lad:j:np-incomplete] (see also, e.g., [@koz:j:subrecursive; @sch:j:uni; @reg:j:diag-unif-fixed-point; @for:j:diagonalization; @reg-vol:j:gap]) to show that if ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then there are incomplete sets in ${\mbox{\rm NP}}- {\mbox{\rm P}}$. Many people, rather naturally given which example one tends to see in courses and textbooks, think of delayed diagonalization as the technique of having one set handle a list (actually two lists) of requirements by looking deeply into its own history. But in fact, delayed diagonalization is far more flexible and powerful than merely being able to do that. In (what we here will call) team diagonalization, though, Ladner in effect has two sets, each with its own list of requirements to satisfy, but the two sets will take long turns as to which of them is working on its requirements, and while one is doing that, the other will politely remain simple and boring. Loosely put, the sets will each respect the goals of the other set, and will take on burdens in a completely coordinated “lock-step” fashion.
Preliminaries {#s:defs}
=============
For each string $x$, the number of characters in $x$ will be denoted $|x|$. For each set $A$ and each natural number $k$, $A^{=k}$ will denote $\{ x ~|~ x \in A \land |x| = k\}$. We take all sets and classes to be with respect to the alphabet $\Sigma = \{0,1\}$. The symmetric difference operation for sets, $(A-B) \cup (B-A)$, will be denoted $A\bigtriangleup B$.
All logarithms in this paper are base two, e.g., $\log\log i$ means $\log_2(\log_2(i))$.
We say $A \le^p_m B$ exactly if $A$ is polynomial-time many-one reducible to $B$ (i.e., there is a polynomial-time function $f$ such that, for each $x$, it holds that $x\in A \iff f(x)\in B$). A set $B$ is NP-hard (with respect to $\le^p_m$ reductions) exactly if for all $A \in {\mbox{\rm NP}}$, $A \le^p_m B$. A set $B$ is NP-complete (with respect to $\le^p_m$ reductions) exactly if $B\in{\mbox{\rm NP}}$ and $B$ is NP-hard. We say $A \le^p_T B$ if $A \in {\mbox{\rm P}}^B$. A set $B$ is NP-hard with respect to $\le^p_T$ reductions exactly if for all $A \in {\mbox{\rm NP}}$, $A \le^p_T B$. A set $B$ is NP-complete with respect to $\le^p_T$ reductions exactly if $B\in{\mbox{\rm NP}}$ and $B$ is NP-hard with respect to $\le^p_T$ reductions.
Sets $A$ and $B$ are said to be P-separable exactly if there exists a set $L \in {\mbox{\rm P}}$ such that $A \subseteq L \subseteq
\overline{B}$, i.e., there is a polynomial-time set that is a (possibly nonstrict) superset of $A$ yet has no intersection with $B$.
Given functions $f$ and $g$, we say that $f \le ^p_m g$ ($f$ functional many-one reduces to $g$) if there are polynomial-time functions $h_1$ and $h_2$ such that, for all $x$, it holds that $f(x) = h_2(g(h_1(x)))$ [@zan:j:sharp-p]. Functional many-one reductions are even more restrictive than metric reductions [@kre:j:optimization] (which have almost the same definition except they allow $h_2$ to have direct access to $x$, i.e., the definition’s key part is $f(x) = h_2(x,g(h_1(x)))$) and are most commonly studied in the context of ${\mbox{\rm \#P}}$-completeness in order to prove stronger completeness results than mere ${\mbox{\rm \#P}}$-metric-completeness or ${\mbox{\rm \#P}}$-Turing completeness. For example, Valiant’s notion of ${\mbox{\rm \#P}}$-completeness in his seminal paper [@val:j:permanent] is the Turing-reduction notion, and for the permanent of $(0,1)$ matrices the reduction he builds is not a many-one reduction (although for the permanent of $(-1,0,1,2,3)$ matrices Valiant does establish ${\mbox{\rm \#P}}$-many-one-completeness); ${\mbox{\rm \#P}}$-many-one-completeness for the permanent of $(0,1)$-matrices was obtained only more than a decade later, by Zank[ó]{} [@zan:j:sharp-p]. As another example, Deng and Papadimitriou’s [@den-pap:j:solution-concepts] proof that the Shapley–Shubik power index is ${\mbox{\rm \#P}}$-metric-complete was later strengthened to a ${\mbox{\rm \#P}}$-many-one-completeness result [@fal-hem:j:power-compare].
Let $\chi_A$ denote the characteristic function of $A$, that is, $\chi_A(x)$ equals 0 if $x\not\in A$ and equals 1 if $x\in A$. We say a function $g$ is NP-hard exactly if, for every $A\in{\mbox{\rm NP}}$, $\chi_A \le^p_m g$. (Note that this is equivalent to $\chi_{SAT} \le^p_m g$.) We say that a function $g$ is NP-Turing-hard exactly if, for all $A\in{\mbox{\rm NP}}$, $A \in {\mbox{\rm P}}^g$ (equivalently, ${\mbox{\rm SAT}}\in {\mbox{\rm P}}^g$).
We remark that in the literature when “NP-hard functions” are mentioned, the term often means what we here call NP-Turing-hard. However, for clarity, in this paper we always use, for both sets and functions, the terms NP-hard (for the many-one case) and NP-Turing-hard (for the Turing case). The key theorems statements here—in particular, Theorems \[thm:main\], \[t:optp\], and \[thm:exp\]—are stated for the most demanding choices regarding many-one versus Turing reductions (even when this requires mixing and matching within the theorem statements, as indeed happens in each of these three theorem statements), and so they imply the weaker choices.
Splitting NP-Complete Sets into NP-Incomplete Sets {#sect:sets}
==================================================
The following captures Ladner’s beautiful insight that if ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then each NP-complete set can be built from (or looked at from the other direction, partitioned into) two non-NP-complete NP sets, in a very simple way.
\[thm:main\] If ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$ and $S$ is an ${\mbox{\rm NP}}$-complete set, then there is a function $r: {\mathbb{N}}\rightarrow {\mathbb{N}}$ such that, for all $n$, $r(n)$ can be computed in time polynomial in $n$ and the following disjoint sets $A$ and $B$ belong to ${\mbox{\rm NP}}- {\mbox{\rm P}}$, satisfy $A \cup B = S$, and are not ${\mbox{\rm NP}}$-hard even under polynomial-time Turing reductions: $$\label{eq:def-of-A}
A = \left\{ x ~|~ x \in S \mbox{ and } r(|x|) \mbox{ is even} \right\}$$ and $$\label{eq:def-of-B}
B = \left\{ x ~|~ x \in S \mbox{ and } r(|x|) \mbox{ is odd} \right\}.$$
Suppose ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$. Let $M_1, M_2, \ldots $ be a standard enumeration of deterministic oracle Turing machines, each of which is explicitly polynomially clocked (upper-bounded) independently of its oracle. We assume, w.l.o.g., that the enumeration and the clocking are such that there is a universal oracle Turing machine $\mathcal{U}$ such that the following is true:
1. For each $j \geq 1$, each $x$, and each $A$, $\mathcal{U}^A(x,j)$ simulates $M^A_j(x)$ (in the sense that $x \in L(M_j^A) \iff
x \in L(\mathcal{U}^A(x,j))$), and
2. For each $j \geq 1$, each $x$, and each $A$, $\mathcal{U}^A(x,j)$ runs in time at most $|x|^j + j$.[^5]
Let $S$ be an an arbitrary NP-complete set. Let $c\ge 1$ be a constant such that $S
\in {\mbox{\rm DTIME}}(2^{n^c})$. Let $A$ and $B$ be defined by Eqns. (\[eq:def-of-A\]) and (\[eq:def-of-B\]); of course, for those definitions to be meaningful, we will need to define $r$. In particular, we will now define $r$ such that $r$ is nondecreasing, and $r(n)$ can be computed in time polynomial in $n$. It follows, keeping in mind that $S\in {\mbox{\rm NP}}$, that $A \in {\mbox{\rm NP}}$ and $B \in {\mbox{\rm NP}}$. It also follows, from the definition of $A$ and $B$, that $A$ and $B$ are disjoint and satisfy $A \cup B = S$. In addition, our definition of $r$ will be such that the other claims in the conclusion of Theorem \[thm:main\] hold, namely, that neither $A$ nor $B$ belongs to ${\mbox{\rm P}}$, and neither $A$ nor $B$ is NP-hard even under polynomial-time Turing reductions.
Let $r(0) = r(1) = r(2) = 2$. To define $r$, we describe a procedure that, for any $i \ge 2$, computes the value $r(i+1)$ in time polynomial in $i$ based on the values the values $r(0), r(1), \ldots , r(i)$.
[**Computation of $\boldsymbol{r(i+1)}$ based on $\boldsymbol{r(0), r(1), \ldots , r(i) \; (i \ge 2):}$**]{}
When we determine $r(i+1)$, we try to diagonalize against machine $M_{\lfloor r(i)/2\rfloor}$. If $r(i)$ is odd then we try to make sure that $M_{\lfloor r(i)/2\rfloor}$ does not decide SAT with the help of oracle $A$ and if $r(i)$ is even then we try to make sure that $M_{\lfloor r(i)/2\rfloor}$ does not decide SAT with the help of oracle $B$. If we succeed then we set $r(i+1) = r(i)+1$. Otherwise, we set $r(i+1) = r(i)$. (Then the same machine and oracle will be tried to diagonalize against when $r(i+2)$ is determined.)
If $$\label{eq:1}
2^{{\left( (\log\log i)^{\lfloor r(i)/2 \rfloor } + \lfloor r(i)/2 \rfloor \right)}^c} \ge i$$ then the diagonalization fails, and we set $r(i+1) = r(i)$. Otherwise there are two cases, as follows. (As one reads the cases, the fact that $A$ and $B$ are being used in text that in part is also creating them may seem circular. But why this is not fatally naughty is explained in the discussion justifying the correctness of the construction.)
Case 1: $\boldsymbol{r(i)} $ is odd.
: We try to diagonalize against $M_{\lfloor r(i)/2\rfloor}$ with oracle $A$. Determine if there exists a string $y$ of length at most $\log\log i$ satisfying $$\label{eq:diag-against-A}
y \in {\mbox{\rm SAT}}\Longleftrightarrow y \notin L(M_{\lfloor r(i)/2\rfloor}^A).$$ If such a $y$ exists then ${\mbox{\rm SAT}}$ is not polynomial-time Turing reducible to $A$ via oracle machine $M_{\lfloor r(i)/2\rfloor}$. Hence the diagonalization is successful, and so we set $r(i+1) = r(i)+1$. Otherwise, we set $r(i+1) = r(i)$.
Case 2: $\boldsymbol{r(i)}$ is even.
: We try to diagonalize against $M_{\lfloor r(i)/2 \rfloor}$ with oracle $B$. Determine if there exists a string $y$ of length at most $\log\log i$ satisfying $$\label{eq:diag-against-B}
y \in {\mbox{\rm SAT}}\Longleftrightarrow y \notin L(M_{\lfloor r(i)/2\rfloor}^B).$$ If such a $y$ exists then ${\mbox{\rm SAT}}$ is not polynomial-time Turing reducible to $B$ via oracle machine $M_{\lfloor r(i)/2\rfloor }$. Hence the diagonalization is successful, and so we set $r(i+1) = r(i)+1$. Otherwise, we set $r(i+1) = r(i)$.
For any given $n$, we compute $r(n)$ as follows: For $i = 2, 3, \ldots , n-1$, we successively compute $r(i+1)$ as described above based on the values $r(0), r(1), \ldots , r(i-1), r(i)$. Note that if for each $i \in \{ 2, 3, \ldots , n-1\}$ this is possible in time polynomial in $i$, then $r(n)$ can be computed in time polynomial in $n$.
It remains to show that this construction is correct.
1. The construction of $r(i+1)$ in cases 1 and 2 above obviously depends on the oracle sets $A$ and $B$, which according to Eqn. (\[eq:def-of-A\]) and Eqn. (\[eq:def-of-B\]) depend on $r$ being odd or even. However, the construction is not circular: To determine whether $y \notin L(M_{\lfloor r(i)/2\rfloor}^{(\cdot)})$, we only need to know the oracle up to length $(\log\log i)^{\lfloor r(i)/2 \rfloor } + \lfloor r(i)/2 \rfloor$, which is smaller than $i$ by Eqn. (\[eq:1\]). Hence the only $r(k)$ values that $r(i+1)$ may depend on are values $r(k)$ for $k$ less than than $i$.
2. \[correct-time\] We argue—and recall that, as noted above, showing this establishes that $r(n)$ can be computed (from scratch) in time polynomial in $n$—that for each $i\ge 2$, the procedure that determines $r(i+1)$ based on $r(0), r(1), \ldots , r(i-1), r(i)$ runs in time polynomial in $i$. To this end, we have to show that the conditions in Eqns. (\[eq:diag-against-A\]) and (\[eq:diag-against-B\]) can be checked for all $y$ with $|y| \le \log\log i$ in time polynomial in $i$.
First, for each $y$, checking whether $y \in {\mbox{\rm SAT}}$ can be done by brute force in time polynomial in $i$ since $y$ is of length at most $\log \log i$.
Second, the running time of $M_{\lfloor r(i)/2\rfloor}^{(\cdot)}$ on inputs of length $\log \log i$ is at most $t = (\log\log i)^{\lfloor r(i)/2 \rfloor } + \lfloor r(i)/2 \rfloor$. Hence the length of each oracle query $q$ is at most $t$. Since $S \in {\mbox{\rm DTIME}}(2^{n^c})$, there exists a constant $s > 0$ such that we can determine in time $$s\cdot 2^{t^c} \leq s \cdot 2^{{\left( (\log\log i)^{\lfloor r(i)/2 \rfloor } + \lfloor r(i)/2 \rfloor \right)}^c}$$ whether $q \in S$. And since we got past the test of Eqn. (\[eq:1\]), we thus know that the time used is at most $s \cdot i$. Finally, note that no more than $(2\log i) -1 $ different strings $y$ have to be checked in whichever one of Eqns. (\[eq:diag-against-A\]) or (\[eq:diag-against-B\]) applies. Hence the whole procedure of determining $r(i+1)$ based on $r(0), r(1), \ldots , r(i)$ takes time polynomial in $i$.
3. In the above construction, we try to diagonalize against machine $M_{\lfloor r(i)/2\rfloor}$ when we determine $r(i+1)$. Hence to check that we eventually diagonalize against all deterministic polynomial-time oracle Turing machines $M_j^{(\cdot)}$, we only have to show that $r(i)$—which, recall, is nondecreasing—grows indefinitely.
Suppose that there exist $k$ and $n_0$ such that $r(n) = k$ for all $n \ge
n_0$. Then there are two cases:
Case 1: $\boldsymbol{k}$ is odd.
: \
Then for all $i \ge n_0$ it holds that, for all $y$ with $|y| \le \log\log i$: $$\begin{aligned}
y \in {\mbox{\rm SAT}}& \Longleftrightarrow y \in L(M_{\lfloor
r(i)/2\rfloor}^A), \mbox{ and}
\end{aligned}$$ $$\begin{aligned}
y \in L(M_{\lfloor
r(i)/2\rfloor}^A) &
\Longleftrightarrow y \in L(M_{\lfloor
k/2\rfloor}^A).\end{aligned}$$ So for every string $y$, it holds that $ y \in {\mbox{\rm SAT}}\Longleftrightarrow
y \in L(M_{\lfloor
k/2\rfloor}^A)$. Furthermore, by construction, $A$ contains in this case only finitely many strings. It follows that SAT can be decided in polynomial time, which contradicts our assumption that ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$.
Case 2: $\boldsymbol{k}$ is even.
: \
Then for all $i \ge n_0$ it holds that, for all $y$ with $|y| \le \log\log i$: $$\begin{aligned}
y \in {\mbox{\rm SAT}}&\Longleftrightarrow y \in L(M_{\lfloor
r(i)/2\rfloor}^B) \mbox{ and}
\end{aligned}$$ $$\begin{aligned}
y \in L(M_{\lfloor
r(i)/2\rfloor}^B)
&\Longleftrightarrow y \in L(M_{\lfloor
k/2\rfloor}^B).
\end{aligned}$$ So for every string $y$, it holds that $ y \in {\mbox{\rm SAT}}\iff
y \in L(M_{\lfloor
k/2\rfloor}^B)$. Furthermore, by construction, $B$ contains in this case only finitely many strings. It follows that SAT can be decided in polynomial time, which contradicts our assumption that ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$.
4. Finally, we must argue that $A \not\in{\mbox{\rm P}}$ and $B\not\in{\mbox{\rm P}}$. (Recall that, at the start of the proof, we pointed out that both $A$ and $B$ belong to NP, that $A$ and $B$ are disjoint, and that $A\cup B = S$. So we do not need to re-argue those points here.) Suppose for example that $A \in {\mbox{\rm P}}$. That implies, since $S = A \cup B$, that ${\mbox{\rm P}}^S \subseteq {\mbox{\rm P}}^B$. So the fact that $S$ is NP-complete certainly yields that $B$ is NP-Turing-hard. But that contradicts the fact that, as argued above, $B$ is not NP-Turing-hard. So $A\not\in {\mbox{\rm P}}$.
By the analogous argument, it also holds that $B \not\in {\mbox{\rm P}}$.
One has the following easy corollaries. (For disjoint sets $A_1$ and $A_2$, note that $A_1\cup A_2 = A_1 \bigtriangleup A_2$, so one could equally well in the theorems below make the theorems be stated not about a union ($A_1\cup A_2$) but about a symmetric difference ($A_1 \bigtriangleup A_2$).)
\[coroll:1\] ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$ if and only if for each ${\mbox{\rm NP}}$-complete set $S$ there exist disjoint ${\mbox{\rm NP}}$ sets $A_1 \subseteq S$ and $A_2 \subseteq S$ such that $S =
A_1 \cup A_2
$ and neither $A_1$ nor $A_2$ is ${\mbox{\rm NP}}$-complete.
The direction from left to right follows directly from Theorem \[thm:main\]. Now suppose that ${\mbox{\rm P}}= {\mbox{\rm NP}}$. Let $S$ be any set such that $S \not=
\Sigma^*$ and $S \not=\emptyset$. Then $S$ is NP-complete. The only subset of $S$ that is not NP-complete is the empty set. Hence there are no NP-incomplete sets $A_1 \subseteq S$ and $A_2 \subseteq S$ such that $A_1
\cup A_2 = S$.
If ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$ then the sets $A_1$ and $A_2$ in Corollary \[coroll:1\] are both not in P.
\[coroll:2\] If ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then every ${\mbox{\rm NP}}$-complete set $S$ has the property that there is a ${\mbox{\rm P}}$ set $D$ such that both $S \cap D$ and $S \cap
\overline{D}$ are ${\mbox{\rm NP}}$-incomplete (i.e., are in ${\mbox{\rm NP}}$ yet are not ${\mbox{\rm NP}}$-complete).
That is, for every ${\mbox{\rm NP}}$-complete set $S$ there exist ${\mbox{\rm P}}$-separable sets $A_1$ and $A_2$ such that $S = A_1 \cup A_2
$ and neither $A_1$ nor $A_2$ is ${\mbox{\rm NP}}$-hard.
Let $$D = \left \{ x \in
\Sigma^* ~|~ r(|x|) \mbox{ is even} \right\}.$$ Note that $A = S \cap D$ and $B = S \cap \overline{D}$ are the sets $A$ and $B$ in Theorem \[thm:main\].
We mention a result by Gla[ß]{}er et al. [@gla-pav-sel-zha:j:splitting] that is similar in spirit to Corollary \[coroll:2\] even though it is trying to achieve the opposite. Gla[ß]{}er et al. [@gla-pav-sel-zha:j:splitting] call a set $A$ nontrivial exactly if both $A$ and $\overline{A}$ contain at least two elements. Gla[ß]{}er et al. showed (among other things) that every nontrivial set $A$ that is NP-complete is also m-mitotic. This easily implies the following theorem.
\[thm:glasser-splitting\] If ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then every ${\mbox{\rm NP}}$-complete set $S$ has the property that there is a ${\mbox{\rm P}}$ set $D$ such that both $S \cap D$ and $S \cap
\overline{D}$ are ${\mbox{\rm NP}}$-complete.
While Corollary \[coroll:2\] shows (if ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$) that every NP-complete set $S$ can be split into P-separable sets that are [*not*]{} ${\mbox{\rm NP}}$-complete, Gla[ß]{}er et al.’s theorem implies that (regardless of whether or not ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$) every nontrivial NP-complete set $S$ can be split into P-separable sets such that both sets [*are*]{} indeed NP-complete. Note that regarding ${\mbox{\rm P}}= {\mbox{\rm NP}}$ we have the following.
\[thm-was-coroll:3\]If ${\mbox{\rm P}}= {\mbox{\rm NP}}$, then no ${\mbox{\rm NP}}$-complete set $S$ has the property that there is a ${\mbox{\rm P}}$ set $D$ such that both $S \cap D$ and $S \cap
\overline{D}$ are ${\mbox{\rm NP}}$-incomplete (i.e., are in ${\mbox{\rm NP}}$ yet are not ${\mbox{\rm NP}}$-complete).
Suppose ${\mbox{\rm P}}= {\mbox{\rm NP}}$. Then the only NP-incomplete sets are $\emptyset$ and $\Sigma^*$. But the only sets that can be formed by the union of two sets chosen from $\{\emptyset,\Sigma^*\}$ are $\emptyset$ and $\Sigma^*$, which as just mentioned are NP-incomplete, yet the theorem’s claim is that $S$ is NP-complete.
Note that Corollary \[coroll:2\] and Theorem \[thm-was-coroll:3\] are not converses of each other. They are actually stronger than just giving an “if and only” statement.
The following notes that the sets being used are not merely disjoint in the sense that no string participates in both of the sets, but also differ so strongly that no length participates in both of the sets.
Two sets $A_1$ and $A_2$ are *strongly disjoint* exactly if, for every $k$, $A_1^{=k} = \emptyset$ or $A_2^{=k} = \emptyset$.
\[coroll:4\] ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$ if and only if for each ${\mbox{\rm NP}}$-complete set $S$ there exist strongly disjoint, ${\mbox{\rm P}}$-separable, ${\mbox{\rm NP}}$ sets $A_1 \subseteq S$ and $A_2 \subseteq S$ such that $A_1 \cup A_2 = S$ and neither $A_1$ nor $A_2$ is ${\mbox{\rm NP}}$-complete.
The direction from left to right follows easily from Theorem \[thm:main\] because it is easy to see that the sets $A$ and $B$ in that theorem are strongly disjoint P-separable NP-sets.
The direction from right to left follows directly from Corollary \[coroll:1\].
Composition of Functions: Hard Functions Composed from Nonhard Ones {#sect:functions}
===================================================================
Consider NPTMs where each nondeterministic path outputs one value (say, a string over the alphabet $
\Sigma^*
$). Paths that do not explicitly output a value are by convention viewed as outputting $\epsilon$, the empty string. A function $f$ is said to be in ${\mbox{\rm OptP}}$ [@kre:j:optimization] exactly if there is a thus-viewed NPTM $N$ such that, for each $x$, $f(x)$ is the lexicographically maximum value among all values output by paths of $N$ on input $x$. (By lexicographical order, we mean as is standard the order in which $\epsilon < 0 < 1 < 00 < 01 < 10 < 11 < 111 < \dots$. ${\mbox{\rm OptP}}$ is often viewed as having codomain ${\mathbb{N}}$ rather than $\Sigma^*$, but by the natural bijection, the views are the same.)
We now observe, as an easy consequence of the main result of the previous section, that the composition of non-NP-hard NP optimization functions can achieve NP-hardness. Let $g \circ f$ denote the composition of the functions, i.e., the function defined by, for each $x$, $(g \circ f)(x) = g(f(x))$.
\[t:optp\] If ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$, then there exist ${\mbox{\rm OptP}}$ functions $f$ and $g$ such that neither $f$ nor $g$ is ${\mbox{\rm NP}}$-hard (or even ${\mbox{\rm NP}}$-Turing-hard), yet $g \circ f$ is an ${\mbox{\rm NP}}$-hard ${\mbox{\rm OptP}}$ function.
Let $S = {\mbox{\rm SAT}}$ and let $A$ and $B$ be disjoint NP sets such that neither $A$ nor $B$ is NP-Turing-complete and $A \cup B=S$. Such sets exist according to Theorem \[thm:main\].
We define $f$ and $g$ as follows:
For each $x \in
\Sigma^*,$
$
f(x) =
\begin{cases}
1^{|x|+1} & \mbox{if } x \in A\\
0x & \mbox{otherwise,}
\end{cases}$
and for each $z \in
\Sigma^*$,
$ g(z) =
\begin{cases}
1 & \mbox{if } (z\mbox{'s first bit is a 1})\mbox{ or } (z = 0x \mbox{ and } x \in B) \\
0 & \mbox{otherwise.}
\end{cases}
$
It is easy to see that for each $x \in
\Sigma^*$,
$
g(f(x)) =
\begin{cases}
1 & \mbox{if } x \in A\cup B \\
0 & \mbox{otherwise.}
\end{cases}
$
Hence $\chi_{SAT} \le^p_m g\circ f$, and therefore $g\circ f$ is NP-hard under polynomial-time many-one reductions. Furthermore, $f$, $g$, and $g\circ f$ are easily seen to be ${\mbox{\rm OptP}}$ functions.
It remains to show that $f$ and $g$ are not NP-Turing-hard.
Suppose $f$ is NP-Turing-hard. Then there exists a polynomial-time oracle machine $M$ such that ${\mbox{\rm SAT}}= L(M^f)$. Since given $A$ as an oracle it is easy to compute $f$, it follows that there is a polynomial-time oracle Turing machine $M'$ such that ${\mbox{\rm SAT}}= L(M'^A)$. But this is a contradiction because we assumed that $A$ is not NP-Turing-hard.
In the same way, we can show that $g$ is not NP-Turing-hard.
Generalizations and Variants {#s:gen}
============================
Theorem \[thm:main\] can be generalized/varied in many ways. Please see Ladner’s original paper [@lad:j:np-incomplete] for results in such a general form that this can be done very broadly. But for now let us mention a few examples. First, the theorem holds for many classes other than ${\mbox{\rm NP}}$ (because Ladner proves it in a very general setting, see especially p. 160 of his paper), for instance for PSPACE, EXP, EXPSPACE, EEXP, $\Sigma_i^p$, $\Pi_i^p$, etc. Second, Theorem \[thm:main\] says that we can split $S$ into two sets that are NP-incomplete. Clearly, a straightforward adaptation of the proof shows that for any integer $k>1$, $S$ can be split into $k$ incomplete sets. As an example, we mention a variation of Theorem \[thm:main\] for the case of EEXP, with things split into three incomplete sets. Of course, here one does not need any assumption of the form ${\mbox{\rm P}}\not= {\mbox{\rm NP}}$.
\[thm:exp\] Let $S$ be any [EEXP]{}-complete set. Then there is a function $r: {\mathbb{N}}\rightarrow {\mathbb{N}}$ such that for all $n$, $r(n)$ can be computed in time polynomial in $n$ and the following pairwise-disjoint $\rm EEXP$ sets $A$, $B$, and $C$ satisfy $S = A\cup B \cup C$ and are not [EEXP]{}-hard under polynomial-time Turing reductions: $$ A = \left\{ x ~|~ x \in S \land ( {\ensuremath{{r(|x|)}\equiv {0}\left(\bmod{3}\right)}} ) \right\},$$ $$ B = \left\{ x ~|~ x \in S \land ( {\ensuremath{{r(|x|)}\equiv {1}\left(\bmod{3}\right)}} ) \right\},$$ and $$ C = \left\{ x ~|~ x \in S \land ( {\ensuremath{{r(|x|)}\equiv {2}\left(\bmod{3}\right)}} ) \right\}.$$
The proof differs from the proof of Theorem \[thm:main\] in that one now has to consider three different cases: $r(|x|) \equiv 0
\pmod{3}$, $r(|x|) \equiv 1 \pmod{3}$, and $r(|x|) \equiv 2 \pmod{3}$, and further, the length of the string $y$ that one uses in one’s diagonalization must be of length roughly triple-logarithmic in $i$, that is, computations have to look even more deeply back within the history.
##### Acknowledgments. {#acknowledgments. .unnumbered}
We thank Max Alekseyev and Daniel Štefankovič for helpful conversations. We are grateful to Richard Ladner for his work that this paper is presenting, and for his lifetime’s rich range of central contributions to computer science, which have inspired us and so many others.
[GSTW08]{}
X. Deng and C. Papadimitriou. On the complexity of comparative solution concepts. , 19(2):257–266, 1994.
P. Faliszewski and L. Hemaspaandra. The complexity of power-index comparison. , 410(1):101–107, 2009.
L. Fortnow. Diagonalization. , 71:102–112, 2000.
C. Gla[ß]{}er, A. Pavan, A. Selman, and L. Zhang. Splitting [NP]{}-complete sets. , 37:1517–1535, 2008.
C. Gla[ß]{}er, A. Selman, S. Travers, and K. Wagner. The complexity of unions of disjoint sets. , 74:1173–1187, 2008.
L. Hemaspaandra, Z. Jiang, J. Rothe, and O. Watanabe. Boolean operations, joins, and the extended low hierarchy. , 205(1–2):317–327, 1998.
L. Hemaspaandra and H. Spakowski. Team diagonalization. , 49(3), 2018. To appear.
D. Kozen. Indexings of subrecursive classes. , 11(3):277–301, 1980.
M. Krentel. The complexity of optimization problems. , 36(3):490–509, 1988.
R. Ladner. On the structure of polynomial time reducibility. , 22(1):155–171, 1975.
K. Regan. Diagonalization, uniformity, and fixed-point theorems. , 98:1–40, 1992.
K. Regan and H. Vollmer. Gap-languages and log-time complexity classes. , 188(1–2):101–116, 1997.
U. Sch[ö]{}ning. A uniform approach to obtain diagonal sets in complexity classes. , 18:95–103, 1982.
L. Valiant. The complexity of computing the permanent. , 8(2):189–201, 1979.
V. Zank[ó]{}. \#[P]{}-completeness via many-one reductions. , 2(1):76–82, 1991.
[^1]: This work was done in part at ETH Zürich’s Department of Computer Science while on a sabbatical stay, which was generously supported by that department.
[^2]: This work was done in part while visiting the University of Rochester.
[^3]: This article will appear as the Complexity Theory Column article in the September 2018 issue of *SIGACT News* [@hem-spa:jtoappear:team-diagonalization], in honor of the career of Professor Richard Ladner, who retired in 2017.
[^4]: Some Fine Print and Disclaimers: Ladner’s article is in part focused on Turing reductions, as was common back then. Yet it also weaves in explicitly, and implies, a wide range of results about many-one reductions. In particular, the reader wanting to see Ladner’s far more general treatment of the type of thing we cover here is enthusiastically pointed to Ladner’s seminal paper [@lad:j:np-incomplete], and in particular to, on his page 160, his Theorem 2 which is a quite general version of the central result that we are presenting in this article, i.e., his Theorem 2 for the case of $A$ being any P set and $B$ being an NP-complete (i.e., NP-many-one-complete) set is, in effect, the special case being presented in Section \[sect:sets\] as Theorem \[thm:main\], give or take the fact that in Theorem \[thm:main\] here we have put right into the theorem statement some details of the simple splitting’s framework, namely, the polynomial-time function $r$ and how it controls the splitting. Ladner’s Corollary 2.1 on page 160 of his paper gives the many-one analogue of his Theorem 2.
[^5]: Technical aside (that we suggest ignoring during a first reading): The w.l.o.g.s in that sentence and a few lines above are the kind of somewhat painful groundwork that is often skipped over—as we also will mostly do here. But, to touch for those interested on what is under the hood: The natural way to build a nice clocked enumeration is to pair each machine (from a truly standard enumeration of all Turing machines) together with every possible clock drawn from some nice family of polynomials that for every polynomial has at least one member that majorizes that polynomial over all natural numbers. The typical family used for that is $n^k + k$. One needs to then interlace and assign numbers in the enumeration so that even the earliest members of the enumeration meet their own claimed time bounds (such as that the $k$th machine will run in time $n^k + k$), and doing that often involves delaying (not at all in the same sense of the word as in delayed diagonalization) when machines come in (i.e., making sure their location in the enumeration has a high enough number) or even making some machines in our list dummy machines that ignore everything and in one step halt. And in doing what was just described, one has to account for the fact that clocking the time of a machine itself has overhead, since one puts a supervisor on top of the machine; but the overhead is mild—certainly at most polynomial. Beyond that, here we want to be able to uniformly simulate any given machine on the fly, and so one also has to take into account the cost of the universal machine’s own simulation of machines—itself also incurring a mild overhead—so that whatever claim one wants to make about the universal machine’s running times is correct. Despite that, all the w.l.o.g. claims above indeed can be routinely achieved. Important in that is that each (deeply) underlying machine is paired with polynomials (infinitely many) greater than particular given polynomial $p$, and so basically each machine appears infinitely often on the listing, and indeed occurs with any particular needed polynomial “headroom for simulation” even on top of the underlying time cost. So if an (deeply) underlying machine with a given oracle $A$ does happen to run in some polynomial time, one of the machines in our enumeration will capture that in such a way that even its simulation within the universal machine will duplicate, under oracle $A$, the action of the underlying machine in terms of acceptance and rejection, and will do so relative to our time claims without being cut off by time issues.
|
---
abstract: 'We extend the sphere theorem of [@CGY03] to give a conformally invariant characterization of $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $\beta(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying a ‘positivity’ condition; it follows from [@CGY03] that if $0 \leq \beta(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a ‘gap’ result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq \beta(M^4,[g]) < 4(1 + \epsilon)$ for $\epsilon > 0$ small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to pass from the bounds on $\beta$ to pointwise curvature information.'
address:
- 'Department of Mathematics, Princeton University, Princeton, NJ 08544, USA'
- 'Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA'
- 'Department of Mathematics, Princeton University, Princeton, NJ 08544, USA'
author:
- 'Sun-Yung A. Chang'
- Matthew Gursky
- Siyi Zhang
title: 'A conformally invariant gap theorem characterizing $\mathbb{CP}^2$ via the Ricci flow'
---
[^1]
[^2]
Introduction
============
In [@CGY03], the first two authors with P. Yang proved a conformally invariant sphere theorem in dimension four. In this paper we extend the results of [@CGY03] to give a characterization of complex projective space. To state our results we begin by establishing our notation and conventions.
If $(M^4,g)$ is a smooth, closed Riemannian four-manifold, we denote the Riemannian curvature tensor by $Rm$ (or $Rm_g$ if we need to specify the metric), the Ricci tensor by $Ric$, and the scalar curvature by $R$. We also denote the Weyl curvature tensor by $W$, and the Schouten tensor $$\begin{aligned}
\label{Pdef}
P = \frac{1}{2} \big( Ric - \frac{1}{6}R \cdot g \big).\end{aligned}$$ We remark that the definition of the Schouten tensor in [@CGY03] (denoted by $A$) differed from the formula in (\[Pdef\]) by a factor of two; however, in this paper we adopt the more common convention. In terms of the Weyl and Schouten tensors the Riemannian curvature tensor can be decomposed as $$\begin{aligned}
\label{rot}
Rm = W + P \bcw g\end{aligned}$$ where $\bcw$ is the Kulkarni-Nomizu product. There are two important consequences of this identity: First, since the Weyl tensor is conformally invariant, it follows that the behavior of the curvature tensor under a conformal change of metric is determined by the transformation of the Schouten tensor. The second consequence is that the splitting induces a splitting of the Euler form, so that the Chern-Gauss-Bonnet formula can be expressed as $$\begin{aligned}
\label{CGB}
8 \pi^2 \chi(M) = \int \|W \|^2 \,dv + 4 \int \sigma_2(g^{-1} P) \,dv,\end{aligned}$$ where\
$\bullet$ $\| \cdot \|$ denotes the norm of the Weyl tensor, viewed as an endormorphism of $\Omega^2(M)$, the bundle of two-forms. Note that this differs from the norm of Weyl when viewed as a four-tensor, and the two norms are related by $$\begin{aligned}
\| W \| = \frac{1}{4} | W|^2.\end{aligned}$$
$\bullet$ $g^{-1} P$ denotes the $(1,1)$-tensor (interpreted as an endomorphism of the tangent space at each point) obtained by ‘raising an index’ of the Schouten tensor, and $\sigma_2(g^{-1}P)$ is the second elementary symmetric polynomial applied to its eigenvalues. To simplify notation we will henceforth write $\sigma_2(P)$ in place of $\sigma_2(g^{-1}P)$.\
It follows from the conformal invariance of the Weyl tensor that both integrals in (\[CGB\]) are conformally invariant. While their sum is a topological invariant, their ratio can be arbitrary. As we now explain, when the scalar curvature is positive the ratio does carry geometric and topological information.
Given a Riemannian manifold $(M^n, g)$ of dimension $n \geq 3$, let $[g]$ denote the equivalence class of metrics pointwise conformal to $g$, and $Y(M^n,[g])$ denote the Yamabe invariant: $$\begin{aligned}
Y(M^n,[g]) = \inf_{ \tilde{g} \in [g] } Vol(\tilde{g})^{-\tfrac{n-2}{n}} \int R_{\tilde{g}}\,dv_{\tilde{g}}.\end{aligned}$$ We can also express the Yamabe invariant in terms of the first symmetric function of the Schouten tensor: it follows from (\[Pdef\]) that $$\begin{aligned}
\sigma_1(P) = \frac{R}{2(n-1)},\end{aligned}$$ hence $$\begin{aligned}
Y(M^n,[g]) = \inf_{ \tilde{g} \in [g] } 2 (n-1) Vol(\tilde{g})^{-\tfrac{n-2}{n}} \int \sigma_1(P_{\tilde{g}}) \,dv_{\tilde{g}}.\end{aligned}$$ With this interpretation of the Yamabe invariant, in dimension four we should view the conformal invariant $\int \sigma_2(P) \,dv$ as a kind of “second Yamabe invariant” (see [@GLW04], [@Sheng08], [@GLW10]). We therefore define $$\begin{aligned}
\label{Y1}
\mathcal{Y}_1^{+}(M^4) = \{ g\ :\ Y(M^4,[g]) > 0 \},\end{aligned}$$ and $$\begin{aligned}
\label{Y2def}
\mathcal{Y}_2^{+}(M^4) = \{ g \in \mathcal{Y}_1^{+}(M)\ :\ \int \sigma_2(P_g)\,dv_g > 0 \}.\end{aligned}$$
By a classical result of Lichnerowicz, there are topological obstructions to $\mathcal{Y}_1(M^4)$ being non-empty [@Lich]. There are also topological implications of $\mathcal{Y}_2(M^4)$ being non-empty: by [@Gur98], if $\mathcal{Y}_2^{+}(M^4) \neq \emptyset$ then the first Betti number $b_1(M^4) = 0$. In fact, it follows from [@CGY02] that $[g]$ contains a metric $\tilde{g}$ with positive Ricci curvature.
Returning to the Chern-Gauss-Bonnet formula, for metrics $g \in \mathcal{Y}_2^{+}(M^4)$ we define the conformal invariant $$\begin{aligned}
\label{betadef}
\beta(M^4,[g]) = \dfrac{ \int \| W_g \|^2 \,dV_g }{ \int \sigma_2( P_g) \, dv_g } \geq 0.\end{aligned}$$ We also define smooth invariant $$\begin{aligned}
\label{betaM}
\beta(M^4) = \inf_{[g]} \beta(M^4,[g]).\end{aligned}$$ If $\mathcal{Y}_2^{+}(M^4) = \emptyset$, we set $\beta(M^4) = -\infty$.
The main results of [@CGY03] give a (sharp) range for $\beta$ that imply the underlying manifold is the sphere:
Suppose $M^4$ is oriented. If $g \in \mathcal{Y}_2^{+}(M^4)$ with $$\begin{aligned}
\label{B1}
\beta(M^4,[g]) < 4,\end{aligned}$$ then $M^4$ is diffeomorphic to $S^4$. In particular, if $M^4$ satisfies $$\begin{aligned}
-\infty < \beta(M^4) < 4,\end{aligned}$$ then the same conclusion holds.
Furthermore, if $M^4$ admits a metric with $\beta(M^4,[g]) = 4$, then one of the following must hold:
- $M^4$ is diffeomorphic to $S^4$; or\
- $M^4$ is diffeomorphic to $\mathbb{CP}^2$ and $g \in [g_{FS}]$, where $g_{FS}$ denotes the Fubini-Study metric.
As a corollary we have the following characterization of manifolds for which $\beta(M^4) = 0$:
Assume $M^4$ is oriented. Then $\beta(M^4) = 0$ if and only if $M^4$ is diffeomorphic to $S^4$. Furthermore, $\beta(M^4,[g]) = 0$ if and only if $g \in [g_0]$, where $g_0$ denotes the round metric.
.2in
[**Remarks.**]{}
1. For the case of equality, we note that if $\beta(M^4,[g]) = 0$, then the Weyl tensor $W_{g} \equiv 0$ and it follows that $(M^4,g)$ is locally conformally flat. By our observations above, since $[g]$ admits a metric with positive Ricci curvature, by Kuiper’s theorem [@Kuiper] $(M^4,g)$ is conformally equivalent to $(S^4,g_0)$ or $(\mathbb{RP}^4, g_0)$, where $g_0$ is the standard metric.\
2. There are a number of other sphere-type theorems under integral curvature conditions; see for example [@CD10], [@CNd10], [@GLW10], [@CZ14], [@BC15], [@L16], [@BC17], and the references in [@CGY03].\
Our first goal in this paper is initiating the study of four-manifolds with $$\beta(M^4,[g]) \geq 4.$$ Suppose $M^4$ is oriented, and let $b_2(M^4)$ denote the second Betti number. Then we can write $b_2 = b_2^{+} + b_2^{-}$, where $b_2^{\pm}$ denotes the dimension of the space of self-dual/anti-self-dual harmonic two-forms. If $b_2(M^4) \neq 0$, then by changing the orientation if necessary we may assume $b_2^{+} > 0$.
Suppose $M^4$ is oriented and $b_2^{+}(M^4) > 0$. There is an $\epsilon > 0$ such that if $M^4$ admits a metric $g \in \mathcal{Y}_2^{+}(M^4)$ with $$\begin{aligned}
\label{Bep}
4 \leq \beta(M^4,[g]) < 4(1 + \epsilon),\end{aligned}$$ then $M^4$ is diffeomorphic to $\mathbb{CP}^2$.
This ‘gap’ theorem immediately gives a characterization of manifolds with $b_2(M^4) \neq 0$ and $\beta(M^4) = 4$:\
Suppose $M^4$ is oriented and $b_2^{+}(M^4) > 0$. If $$\begin{aligned}
\label{B2}
\beta(M^4) = 4,\end{aligned}$$ then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. Moreover, $\beta(M^4,[g]) = 4$ if and only if $g \in [g_{FS}]$.
The proof of Theorem A is similar in approach to the proof of Theorem 1 in [@CGY03]. The first step is to find a conformal representative satisfying a pointwise curvature condition that encodes the integral assumptions of the theorem. In [@CGY03] this involved solving a modified version of the $\sigma_2$-Yamabe problem. However, in the proof of Theorem B it is more natural to consider a modified version of the Yamabe problem introduced by the second author in [@Gur00]. In particular we show that a metric satisfying the assumptions of Theorem B can be conformally deformed to a metric that is “almost self-dual Einstein” in an $L^2$-sense, and whose scalar curvature satisfies a condition similar to the condition satisfied by the scalar curvature of a Kähler metric.
As in the proof of Theorem 1, the second step involves the Ricci flow. In [@CGY03] the weak pinching result of Margerin [@Mar98] played a crucial role. To prove Theorem B, we show that the Ricci flow with the conformal representative constructed in the first step as the initial metric, will have uniform bounds on the curvature and the Sobolev constant on a fixed time interval $[0, T_0]$, with $T_0 >0$ depending on the pinching constant. These estimates together with the convergence theory of Cheeger-Gromov-Taylor [@CGT] imply the family of Ricci flows $g_{j}(T_0)$ (up to a subsequence) will converge to the Fubini-Study metric.
In view of Theorems A and B, we make several conjectures. The first is that Theorem B remains valid if we drop the assumption on $b_2^{+}(M^4)$:
\[Con1\] If $$\begin{aligned}
\label{B2}
\beta(M^4) = 4,\end{aligned}$$ then $M^4$ is diffeomorphic to $\pm \mathbb{CP}^2$. Moreover, $\beta(M^4,[g]) = 4$ if and only if $g \in [g_{FS}]$.
It is clear that $4$ is a ‘special’ or ‘critical’ value of $\beta$, at which the topology of the underlying manifold can change. A natural question is the next critical value. As a corollary of [@Gur98], we have the following estimate for manifolds with indefinite intersection form, i.e., $b_2^{+}(M^4), b_2^{-}(M^4) > 0$.\
Suppose $M^4$ is oriented, $b_2(M^4) \neq 0$, and the intersection form of $M^4$ is indefinite. If $\mathcal{Y}_2^{+}(M^4)$ is non-empty, then $$\begin{aligned}
\label{B3}
\beta(M^4) \geq 8.\end{aligned}$$ Moreover, if $M^4$ admits a metric with $\beta(M^4,[g]) = 8$, then $g \in [g_p]$, where $g_p = g_{S^2} \oplus g_{S^2}$ is the product metric on $S^2 \times S^2$.
Our next conjecture is that we can weaken the condition $\beta(M^4) = 4$, and characterize the possible topological types of manifolds admitting metrics with $\beta$ between $4$ and $8$:
\[Con2\] If $M^4$ is oriented and admits a metric $g \in \mathcal{Y}_2^{+}(M^4)$ with $$\begin{aligned}
\label{less8}
0 \leq \beta(M^4,[g]) < 8,\end{aligned}$$ then $M^4$ is diffeomorphic to $S^4$ or $\pm \mathbb{CP}^2$.
.2in
Preliminaries {#Prelim}
=============
In this section, we state and prove some preliminary results, including the proof of Theorem C. Several of the results in this section are based on the following result in [@Gur98]:
\[ThmGur98\] Let $M^4$ be a closed oriented four-manifold with $b_2^{+}(M^4) > 0$. Then for any metric $g$ with $Y(M^4,[g])\geq0$, $$\label{Gursky theorem}
\int_M||W^+||^2dv\geq\frac{4\pi^2}{3}(2\chi(M^4)+3\tau(M^4))$$ where $\chi(M^4)$ and $\tau(M^4)$ denote the Euler characteristic and signature of $M^4$, respectively.
Furthermore:
1. Equality is achieved in (\[Gursky theorem\]) by some metric $g$ with $Y(M^4,[g])>0$ if and only if $g$ is conformal to a (positive) Kähler-Einstein metric $g_{KE}=e^{2w}g$.
2. Equality is achieved in (\[Gursky theorem\]) by some metric $g$ with $Y(M,[g])=0$ if and only if $g$ is conformal to a Ricci-flat anti-self-dual Kähler-Einstein metric $g_{KE}=e^{2w}g$.
The following lemma is the first application of this result:
\[Lemma21\] Let $M^4$ be a closed, oriented four-manifold admitting a metric $g \in \mathcal{Y}_2^{+}(M^4)$ with $$\label{Borderline comparison}
\beta(M^4,[g]) < 8.
$$ If $b_2^+(M^4)>0$, then the signature of $M^4$ satisfies $$\tau(M^4)>0.$$
The signature formula implies $$\label{W and signature}
\int ||W_g||^2 \,dv_g =\int_M||W^{+}_g||^2\,dv_g + \int ||W^{-}_g ||^2 \,dv_g = 2\int ||W_g^{+}||^2 \,dv_g -12\pi^2\tau(M^4).$$ By (\[Borderline comparison\]), $$\begin{aligned}
\int \sigma_2(P_g) \,dv_g > \frac{1}{8} \int \| W_g \|^2 \,dv_g.\end{aligned}$$ Substituting this into the Chern-Gauss-Bonnet formula, we have $$\label{W and Euler}
8\pi^2\chi(M^4)= \int ||W_g ||^2 \,dv_g + 4 \int \sigma_2(P_g) \,dv_g > \frac{3}{2} \int ||W_g||^2 \,dv_g.$$ Combining (\[W and signature\]) and (\[W and Euler\]), we get $$\label{simplification}
8\pi^2 \chi(M^4)>3\int ||W^{+}_g||^2 \,dv_g - 18\pi^2\tau(M^4),$$ and this inequality can be rewritten as $$\label{rewritten}
\int ||W_g^{+}||^2dv_g < \frac{4}{3}\pi^2(2\chi(M^4)+3\tau(M^4))+2\pi^2\tau(M^4).$$ Since $b_2^+(M^4)>0$ and $Y(M^4,[g])>0$, (\[Gursky theorem\]) in Theorem 2.1 implies $$\label{Gursky estimate}
\int ||W_g^{+}||^2 \,dv_g \geq\frac{4}{3}\pi^2(2\chi(M^4)+3\tau(M^4)).$$ Therefore, combining (\[rewritten\]) and (\[Gursky estimate\]), we conclude $$\tau(M^4)>0.$$
This lemma is sharp in the following sense: Suppose $(M,g)$ is isometric to $(S^2\times{S^2},g_{prod})$. In this case, $b_2^+(M^4)=b_2^-(M^4)=1$, $\tau(M^4)=0$ and $$\label{sigma_2 on S2S2}
\int ||W_{g} ||^2 \,dv_g= 8 \int \sigma_2(P_g) \,dv_g = \frac{64}{3}\pi^2.$$
\[Cor21\] Let $M^4$ be a closed, oriented four-manifold admitting a metric $g \in \mathcal{Y}_2^{+}(M^4)$ with $$\label{corollary 2.4}
\beta(M^4,[g]) < 8.$$ Then either $b_2(M^4)=0$, or the intersection form is definite.
Suppose $b_2(M^4) \neq 0$ and $b_2^{+}(M^4) \cdot b_2^{-}(M^4) >0$. Then Lemma \[Lemma21\] implies $\tau(M^4)=b_2^+(M^4)-b_2^-(M^4)>0$. Since $b_2^-(M^4)$ is also non-zero, we can apply Lemma \[Lemma21\] to $M^4$ endowed with the opposite orientation to show that the signature is again positive. This is a contradiction, since changing the orientation changes the sign of the signature. It follows that $b_2^{+}(M^4) \cdot b_2^{-}(M^4) = 0$, hence the intersection form is definite.
Combining the two previous results with an [*a priori*]{} upper bound for the total $\sigma_2$-curvature, we can prove the following:
\[Lemma25\] Let $M^4$ be a closed, oriented four-manifold admitting a metric $g \in \mathcal{Y}_2^{+}(M^4)$ with $$\label{8again}
\beta(M^4,[g]) < 8.$$ If $b_2(M^4) > 0$, then (after possibly changing the orientation) $b_2^{+}(M^4) = 1$ and $b_2^{-}(M^4) = 0$.
By Corollary \[Cor21\] we may choose an orientation for which the intersection form is positive definite, so $b_2^{+}(M^4) > 0$ and $b_2^{-}(M^4) = 0$. Also, by Corollary F of [@Gur98], if $g\in\mathcal{Y}_2^+(M^4)$, $b_1(M^4) = 0$. Therefore, $$\begin{aligned}
\label{chars} \begin{split}
\chi(M^4) &= 2 + b_2^{+}(M^4), \\
\tau(M^4) &= b_2^{+}(M^4) > 0.
\end{split}\end{aligned}$$ By the Chern-Gauss-Bonnet formula and (\[corollary 2.4\]), $$\begin{aligned}
\label{CG} \begin{split}
8\pi^2\chi(M^4) &= \int \| W_g \|^2 \,dv_g + 4 \int \sigma_2(P_g) \,dv_g \\
&< 12 \int \sigma_2(P_g) \,dv_g.
\end{split}\end{aligned}$$ By Theorem B of [@Gur99], we have the bound $$\label{Gursky's inequality}
\int \sigma_2(P_g) \,dv_g \leq 4 \pi^2,$$ and equality holds if and only if $(M^4,g)$ is conformally equivalent to the round sphere. More generally, since the integral is conformally invariant it is easy to show that $$\begin{aligned}
\label{S2Y}
\int \sigma_2(P_g) \,dv_g \leq \frac{1}{96} Y(M^4,[g])^2.\end{aligned}$$
Since $b_2(M^4) > 0$ strict inequality must hold, and substituting this into (\[CG\]) we get $$\begin{aligned}
\label{CG2} \begin{split}
8\pi^2\chi(M^4) &< 12 \int \sigma_2(P_g) \,dv_g \\
&< 48 \pi^2,
\end{split}\end{aligned}$$ hence $\chi(M^4)<6$. By (\[chars\]), we see that $1 \leq b_2^+(M^4) \leq 3$. It therefore suffices to rule out the possibilities $b_2^+(M^4)=2$ and $b_2^+(M^4)=3$.
If $b_2^+(M^4)=2$ then $\chi(M^4)=4$, so by the Chern-Gauss-Bonnet formula $$\int ||W_g ||^2 \,dv_g + 4 \int \sigma_2(P_g) \,dv_g = 32\pi^2.$$ Also, $b_2^+(M^4)=2$ implies $\tau(M^4) = 2$, so the signature formula gives $$\begin{aligned}
\int ||W^{+}_g ||^2 \,dv_g &= \int ||W^{-}_g ||^2 \,dv_g + 24\pi^2 \\
&\geq 24 \pi^2.\end{aligned}$$ It follows that $$\int ||W_g||^2 \,dv_g \geq 24\pi^2, \quad \int \sigma_2(P_g) \,dv_g \leq 2 \pi^2.$$ Therefore, $$\int ||W_g ||^2 \,dv_g \geq 12 \int \sigma_2(P_g) \,dv_g,$$ which contradicts (\[8again\]).
If $b_2^+(M^4)=3$, we can apply the same argument to conclude $$\int ||W_g ||^2 \,dv_g \geq 36 \int \sigma_2(P_g) \,dv_g,$$ which also contradicts (\[8again\]). Therefore, $b_2^+(M^4)=1$.
The preceding lemma implies that if we take $\epsilon\leq1$ in Theorem A, then $b_2^+(M^4)>0$ will show that $b_2^+(M^4)=1$. Note that the work [@Don] and [@Fr] of Donaldson and Freedman will imply that the manifold is *homeomorphic* to $\mathbb{CP}^2$ in this case.
We can now combine Theorem \[ThmGur98\] and Lemma \[Lemma21\] to give the proof of Theorem C. Assuming $b_2^{+}(M^4) \cdot b_2^{-}(M^4) > 0$, it follows from Lemma \[Lemma21\] that $\beta(M^4,[g]) \geq 8$ for any metric $g \in \mathcal{Y}_2^{+}(M^4)$. Moreover, if equality holds, then since $b_2^{+}(M^4) > 0$ we can argue as in the proof of Lemma \[Lemma21\] to get equality in (\[rewritten\]): $$\label{rewritten2}
\int ||W_g^{+}||^2dv_g = \frac{4}{3}\pi^2(2\chi(M^4)+3\tau(M^4))+2\pi^2\tau(M^4).$$ By Theorem \[ThmGur98\] we conclude that $\tau(M^4) \geq 0$. Reversing orientation and applying the same argument (since $b_2^{-}(M^4) > 0$) we also get $-\tau(M^4) \geq 0$, hence $\tau(M^4) = 0$. Substituting this into (\[rewritten2\]) implies that we have equality in (\[Gursky theorem\]). Therefore, $g$ is conformal to Kähler-Einstein metric $g_{KE}$. By Proposition 2 of [@Der], $\nabla W^{+}_{g_{KE}} \equiv 0$.
Applying the same argument with the opposite orientation, we see that $g$ is conformal to a Kähler-Einstein metric $g_{KE}'$. By Obata’s theorem [@Obata], Einstein metrics are unique in their conformal class (except in the case of the sphere, which is ruled out in this case). Therefore, $g_{KE} = g_{KE}'$, and since equality holds in (\[Gursky theorem\]) with the opposite orientation it follows that $\nabla W^{-}_{g_{KE}} \equiv 0$. We conclude that $g_{KE}$ is locally symmetric and Einstein; it follows from the classification of such spaces (for example, [@Jensen]) that $(M^4,g_{KE})$ is isometric to $(S^2 \times S^2, g_p)$, and Theorem C follows.
.1in
A preliminary lemma
-------------------
We end this section with a technical lemma that will be used in the proof of Theorem A.
\[L2pinchLemma\] Let $(M^4,g)$ be a closed, compact oriented Riemannian four-manifold with $b_2^+(M^4)>0$ and $$\label{CP^2 pinching}
\beta(M^4,[g])=4(1+\epsilon)$$ for some $0\leq\epsilon<1$. Then $$\label{W^-}
\int_M||W^-||^2dv=\frac{6\epsilon}{2+\epsilon}\pi^2,$$ $$\label{W^+}
\int||W^+||^2dv=12\pi^2+\int||W^-||^2 \,dv,$$ and $$\label{Yamabe constant}
Y(M^4,[g])\geq\frac{24\pi}{\sqrt{2+\epsilon}}.$$
It follows from Lemma \[Lemma25\] that $b_1(M^4)=0$, $b_2^+(M^4)=1,$ and $b_2^-(M^4)=0$. Therefore, $\chi(M^4)=3$ and $\tau(M^4)=1$. By the Chern-Gauss-Bonnet and signature formulas, we have $$\label{GBC}
24\pi^2=\int||W||^2dv+4\int\sigma_2(P)dv,$$ $$\label{Sign}
12\pi^2=\int||W^+||^2dv-\int||W^-||^2dv.$$ Since $\beta(M^4,[g])=4(1+\epsilon)$, we have $$\begin{aligned}
\label{IS2}
4\int\sigma_2(P)dv = \frac{1}{1 + \epsilon}\int \| W \|^2 dv.\end{aligned}$$ Substituting this into (\[GBC\]) we conclude $$\begin{aligned}
\label{allW}
\int \|W\|^2 dv = 24\left( \frac{1 + \epsilon}{2 + \epsilon}\right) \pi^2.\end{aligned}$$ By (\[Sign\]), $$\begin{aligned}
12\pi^2 &= \int||W^+||^2dv-\int||W^-||^2dv \\
&= \int \|W\|^2 dv - 2 \int \| W^{-}\|^2 dv \\
&= 24 \left( \frac{1 + \epsilon}{2 + \epsilon}\right) \pi^2 - 2 \int \| W^{-}\|^2 dv,\end{aligned}$$ which implies (\[W\^-\]). Also, substituting (\[W\^-\]) into the signature formula (\[Sign\]) we get (\[W\^+\]).
To prove (\[Yamabe constant\]), we fist observe that (\[IS2\]) and (\[allW\]) imply $$\begin{aligned}
\int\sigma_2(P)dv = \left( \frac{6}{2+\epsilon}\right) \pi^2.\end{aligned}$$ Therefore, by (\[S2Y\]), $$\begin{aligned}
\label{S2Y}
\left( \frac{6}{2+\epsilon}\right) \pi^2 = \int \sigma_2(P_g) \,dv_g \leq \frac{1}{96}Y(M^4,[g])^2,\end{aligned}$$ and (\[Yamabe constant\]) follows.
Modified Yamabe metrics
=======================
As mentioned in the Introduction, the proof of Theorems A and B will use the Ricci flow. We will use the fact that our assumptions are conformally invariant and choose an initial metric that satisfies certain key estimates. The metric will be a solution of a modified version of the Yamabe problem introduced in [@Gur00], which we now review.
Let $(M^4,g)$ be a Riemannian four-manifold. Define $$\label{variant of scalar curvature}
F_g^+=R_g-2\sqrt{6}||W_g^+||,$$ and $$\label{variant of conformal laplacian}
\mathcal{L}_g=-6\Delta_g+R_g-2\sqrt{6}||W^+||.$$ $\mathcal{L}_g$ is a variant of conformal Laplacian that satisfies the following conformal transformation law: $$\label{conformal tranformation law}
\mathcal{L}_{\widetilde{g}}\phi=u^{-3}\mathcal{L}_g(\phi{u}),$$ where $\widetilde{g}=u^2g\in[g]$. In analogy to the Yamabe problem, we define the functional $$\label{variant of Yamabe quotient}
\widehat{Y}_g[u]=\left\langle{u,\mathcal{L}_gu}\right\rangle_{L^2}/||u||_{L^4}^2,$$ and the associated conformal invariant $$\label{variant of Yamabe constant}
\widehat{Y}(M^4,[g])=\inf_{u\in{W^{1,2}(M,g)}}\widehat{Y}_g[u].$$ By the conformal transformation law of $\mathcal{L}_g$, the functional $u\to\widehat{Y}_g[u]$ is equivalent to the Riemannian functional $$\label{variant of definition}
\widetilde{g}=u^2g\to{vol(\widetilde{g})}^{-\frac{1}{2}}\int F_{\widetilde{g}}^+ \,dv_{\widetilde{g}}.$$ The motivation for introducing this invariant is explained in the following result (see Theorem 3.3 and Proposition 3.5 of [@Gur00]):
\(i) Suppose $M^4$ admits a metric $g$ with $F_g^+\geq0$ on $M^4$ and $F_g^+>0$ somewhere. Then $b_2^+(M^4)=0$. (ii) If $b_2^+(M^4)>0$, then $M^4$ admits a metric $g$ with $F_g^+\equiv0$ if and only if $(M^4,g)$ is a Kähler manifold with non-negative scalar curvature. (iii) If $Y(M^4,[g])>0$ and $b_2^+(M^4)>0$, then $\widehat{Y}(M^4,[g])\leq0$ and there is a metric $\widetilde{g}=u^2g$ such that $$F_{\widetilde{g}}^+=R_{\widetilde{g}}-2\sqrt{6}||W^+||_{\widetilde{g}}\equiv\hat{Y}(M,[g])\leq0$$ and $$\label{Gap W+}
\int R_{\widetilde{g}}^2 \,dv_{\widetilde{g}}\leq 24\int ||W^+_{\widetilde{g}}||^2 \,dv_{\widetilde{g}}.$$ Furthermore, equality is achieved if and only if $F_{\widetilde{g}}^+\equiv0$ and $R_{\widetilde{g}}=2\sqrt{6}||W^+_{\widetilde{g}}||\equiv{const}$.
Recall that $F_g^+\equiv0$ on a Kähler manifold $(M^4,g)$ with $R\geq0$.
In the rest of the paper, we will refer to the metric $\widetilde{g}$ in (iii) of Theorem 3.1, normalized to have unit volume, a [*modified Yamabe metric*]{}, and denote it by $g_m$. To simplify the notation, we write $\hat{Y}(M,[g])=-\mu_+$. Then $g_m$ satisfies $$R_{g_m}-2\sqrt{6}||W^+||_{g_m}=-\mu_+\leq0$$ and (\[Gap W+\]).
As a preparation for the proof of Theorem A in next section, in the rest of this section we will list some preliminary curvature estimates of the modified Yamabe metric $g_m\in[g]$ with the assumption $b_2^+(M^4)>0$ and $\beta(M,[g])=4(1+\epsilon)$.
\[initial metric\] Let $(M^4,g)$ be a closed, compact oriented Riemannian four-manifold with $b_2^+(M^4)>0$ and $$\label{CP^2 pinching}
\beta(M^4,[g])=4(1+\epsilon)$$ for some $0<\epsilon<1$, then we have for the modified Yamabe metric $g_m\in[g]$ $$\label{g_m W-}
\int_M||W^-_{g_m}||^2dv_{g_m}=\frac{6\epsilon}{2+\epsilon}\pi^2,$$ $$\label{g_m W+}
\int||W^+_{g_m}||^2dv_{g_m}=12\pi^2+\int||W^-_{g_m}||^2dv_{g_m},$$ $$\label{g_m Yamabe}
Y(M,[g_m])\geq\frac{24\pi}{\sqrt{2+\epsilon}}$$ $$\label{g_m E}
\int|E_{g_m}|^2dv_{g_m}\leq6\int||W^-_{g_m}||dv_{g_m},$$ $$\label{g_m mu}
\frac{1}{12}\mu_+{Y}\leq3\int||W^-_{g_m}||^2dv_{g_m},$$ $$\label{g_m R average}
\frac{1}{24}\int(R_{g_m}-\bar{R}_{g_m})^2dv_{g_m}\leq3\int||W^-_{g_m}||^2dv_{g_m},$$ where $\bar{R}_{g_m}=\int{R_{g_m}}dv_{g_m}$.
(\[g\_m W-\])(\[g\_m W+\])(\[g\_m Yamabe\]) follow from (\[W\^-\])(\[W\^+\])(\[Yamabe constant\]) of Lemma 2.7 and confomal invariance. The estimates (\[GBC\]) and (\[Sign\]) imply $$\label{difference}
\int||W^+||^2dv-4\int\sigma_2(P_{g_m})dv_{g_m}=3\int||W^-||_{g_m}^2dv_{g_m}.$$ Recall the modified Yamabe metric stasfies $R_{g_m}+\mu_+=2\sqrt{6}||W^+_{g_m}||$. Squaring both sides of this formula and integrating over $M^4$, we have $$\frac{1}{24}\int\left(R_{g_m}^2+2\mu_+R_{g_m}+\mu_+^2\right)dv_{g_m}=\int||W^+_{g_m}||^2dv_{g_m}.$$ With (\[difference\]), we can rewrite this equation in the following way: $$\frac{1}{2}\int|E_{g_m}|^2dv+\frac{1}{12}\mu_+\int{R_{g_m}}dv_{g_m}+\frac{1}{24}\int\mu_+^2dv_{g_m}=3\int||W^-_{g_m}||^2dv_{g_m}.$$ Since $\int{R_{g_m}}dv_{g_m}\geq Y(M^4,[g_m])>0$, (\[g\_m E\]) and (\[g\_m mu\]) follow from this equation. To see (\[g\_m R average\]), we have $$\begin{aligned}
\frac{1}{24}\int(R_{g_m}-\bar{R}_{g_m})^2dv_{g_m} & =\frac{1}{24}\left(\int{R}^2_{g_m}dv_{g_m}-\bar{R}^2_{g_m}\right)\leq\frac{1}{24}\left(\int{R}^2_{g_m}dv_{g_m}-Y^2\right) \\
& \leq{\int||W^+_{g_m}||^2dv_{g_m}-4\int\sigma_2(P_{g_m})dv_{g_m}}=3\int||W^-_{g_m}||^2dv_{g_m}.\end{aligned}$$
We end this section with a conformally invariant characterization of the Fubini-Study metric:
\[CP2\] Let $(M^4,g)$ be a closed, compact oriented Riemannian four-manifold whose metric $g$ is of positive Yamabe type. In addition, assume $b_1(M^4)=b_2^-(M^4)=0$ and $b_2^+(M^4)=1$. Then $$\int_M\sigma_2(P) \,dv \leq12\pi^2$$ and equality holds if and only if $(M^4,g)$ is conformally equivalent to $(\mathbb{CP}^2,g_{FS})$.
By our assumptions we have $\chi(M^4)=3$ and $\tau(M^4)=1$. Then the Gauss-Bonnet-Chern and signature formulas read $$24\pi^2=\int_M||W_g||^2 \,dv_g+4\int_M\sigma_2(P_g) \,dv_g,$$ and $$12\pi^2=\int_M||W^+_g||^2\,dv_g-\int_M||W^-_g||^2\,dv_g.$$ Therefore, $$\begin{aligned}
\int_M\sigma_2(P) dv &= 24 pi^2 - \int \| W \|^2 dv \\
&\leq 24\pi^2-\int_M||W^+||^2dv \\
&=12\pi^2-\int_M||W^-||^2dv\leq12\pi^2.\end{aligned}$$ If equality holds then $W^-\equiv0$ and by (\[g\_m E\]) of Lemma \[initial metric\] it immediately follows that $E_{g_m}=0$. Hence, $(M^4,g_m)$ is self-dual Einstein with positive scalar curvature and $b_2^+(M^4)=1$. It is easy to check that the equality in (\[Gursky theorem\]) is achieved, and therefore $g_m$ is conformal to a Kähler-Einstein metric. By Obata’s theorem, $(M^4,g_m)$ must be (Kähler-)Einstein with positive scalar curvature and $b_2^+(M^4)=1$.
Now $(M^4,g_m)$ is a complex surface with a positive Kähler-Einstein metric. For complex surfaces, (see Page 81 of [@B87]) $$\label{relation of topological invariants}
3c_2(M^4)-c_1(M^4)^2=\chi(M^4)-3\tau(M^4)=0.$$ Then the uniformlization of Kähler-Einstein manifolds (e.g. Theorem 2.13 in [@T]) implies that the universal cover of $(M,g_m)$ is (up to scaling) isometric to $(\mathbb{CP}^2,g_{FS})$. Hence, $(M,g_m)$ is conformally equivalent to $(\mathbb{CP}^2,g_{FS})$ since $\mathbb{CP}^2$ does not have nontrivial smooth quotient space.
The Proofs of Theorems A and B
==============================
Suppose $M^4$ is an oriented four-manifold with $b_2^{+}(M^4) > 0$ and $g$ is a metric on $M^4$ with $\beta(M^4,[g]) = 4(1 + \epsilon)$ for $\epsilon > 0$ small. We want to show that if $\epsilon > 0$ is small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. By Lemma \[initial metric\], the modified Yamabe metric $g_m \in [g]$ is close to a self-dual Einstein metric in an $L^2$-sense. Using the Ricci flow, we want to ‘smooth’ $g_m$ to obtain smallness of $W^{-}$ and $E$ in a [*pointwise*]{} sense. We do this in two stages: first, we show that for a small but uniform time, the Ricci flow applied to $g_m$ gives a metric for which $W^{-}$ and $E$ are small in an $L^p$-sense, for some $p > 2$. Next, we appeal to a parabolic Moser iteration estimate of D. Yang [@Y] to conclude $L^\infty$-smallness. The Bernstein-Bando-Shi estimates for the Ricci flow then imply bounds for $C^\infty$-norms of the curvature. To complete the proof we apply a contradiction argument using a compactness result of Cheeger-Gromov-Taylor [@CGT].
We begin with some definitions: on $(M^4,g)$, define
$$\label{Gk}
G_k(g)=|E_g|^k+|R_g-\bar{R}_g|^k+||W^-_g||^k+|(F_g^+)_-|^k,$$
where $(F_g^+)_-=\min(F_g^+,0)$. We shall suppress the subscript $g$ when there is no confusion.
\[4.1\] Under the conditions of Theorem A, the modified Yamabe metric $g_m\in[g]$ satisfies $\int{G_2(g_m)dv_{g_m}}<c(\epsilon)$, where $c(\epsilon)\to0$ as $\epsilon\to0$.
This is a direct consequence of Lemma \[initial metric\].
Now recall some basic facts about the Ricci flow: $$\label{Ricci flow}
\left\{
\begin{array}{ll}
\frac{\partial{}}{\partial{t}}g=-2Ric(g) \\
g(0)=g_0
\end{array}
\right.$$ The following short time-time existence result of Ricci flow has been established in [@Ham82].
For arbitrary smooth metric $g_0$, there exists $T=T(g_0)$ such that (\[Ricci flow\]) has a unique smooth solution for $t\in[0,T)$.
In general, the time interval $[0,T)$ depends on the initial metric $g_0$.
Along the Ricci flow, define $G_k(t)=G_k(g(t))$. The following estimate is of fundamental importance for our argument.
\[evolution of G2\] Suppose we have a solution of the Ricci flow whose initial metric satisfies $$\label{initial smallness}
\int_MG_2(0)dv_{g(0)}=\frac{1}{2}\epsilon_0$$ for some $\epsilon_0$ is sufficiently small. Let $$T=\inf\{{t}:\int_MG_2(t)dv_{g(t)}=\epsilon_0\}.$$ Assume in addition for $0\leq{t}\leq{T}$ $$\label{Yamabe}
Y(t) = Y(M^4,[g(t)]) \geq{b}>0,$$ and $$\label{average scalar}
0<\bar{R}(t)\leq{a}.$$ Then for $0\leq{t}\leq{T}$, we have $$\label{evolution G_2}
\frac{d}{dt}\int_MG_2(t)dv_{g(t)}\leq{\widetilde{a}}\int_MG_2(t)dv_{g(t)}-\widetilde{b}\left(\int_M{G_4(t)}dv_{g(t)}\right)^{1/2},$$ where $\widetilde{a}$ and $\widetilde{b}$ are uniform positive constants independent of $\epsilon_0$. Moreover, there exists $T_0$, which is independent of $\epsilon_0$ such that $T\geq{T_0}$, and we may choose $\widetilde{a}=\frac{4}{3}a$ and $\widetilde{b}=\frac{1}{12}b$.
The proof is based on the evolution of the curvature under the Ricci flow in four dimensions, along with several algebraic inequalities. We begin by summarizing the evolution formulas we will need, most of which can be found in [@CK]:
Under (\[Ricci flow\]) on Riemannnian four-manifolds, $$\label{volume element}
\frac{\partial}{\partial{t}}dv=-R{dv},$$ $$\label{E^2}
\frac{\partial}{\partial{t}}|E|^2=\Delta|E|^2-2|\nabla{E}|^2+4WEE-4trE^3+\frac{2}{3}R|E|^2,$$ $$\label{R^2}
\frac{\partial}{\partial{t}}(R^2)=\Delta{(R^2)}-2|\nabla{R}|^2+4R|E|^2+R^3,$$ $$\label{W+-^2}
\frac{\partial}{\partial{t}}||W^{\pm}||^2={\Delta||W^{\pm}||^2-2||\nabla{W^{\pm}}||^2+36\det{W^{\pm}}+W^{\pm}EE,}$$ $$\label{W+- inequality}
\frac{\partial}{\partial{t}}||W^{\pm}||\leq{\Delta||W^{\pm}||+\sqrt6||W^{\pm}||^2+\frac{\sqrt{6}}{6}|E|^2,}$$ $$\label{F inequality}
\frac{\partial}{\partial{t}}((F^+)_-)\geq\Delta{((F^+)_-)}-((F^+)_-)^2+2R(F^+)_-$$ where $$W^{\pm}EE:=W^{\pm}_{ijkl}E_{ik}E_{jl},\,\,\, F^+_g:=R_g-2\sqrt{6}||W^+||,\,\,\,(F^+)_{-}:=\min\{F_g^+,0\}.$$
For the evolution formulas of $W^{\pm}$ we rely on unpublished notes of D. Knopf [@Knopf].
As a corollary of the formulas above we have in four dimensions:
\[integral\] Under (\[Ricci flow\]), $$\label{volume}
\frac{d}{d{t}}\int{dv}=-\int{R}{dv},$$ $$\label{L2 E}
\frac{d}{dt}\int|E|^2dv=\int\left(-2|\nabla{E}|^2+4WEE-4trE^3-\frac{1}{3}R|E|^2\right)dv,$$ $$\label{L2 R-barR}
\frac{d}{dt}\int(R-\bar{R})^2dv=\int\left(-2|\nabla{R}|^2+4(R-\bar{R})|E|^2+\bar{R}(R-\bar{R})^2\right)dv,$$ $$\begin{gathered}
\label{L2 F-}
\frac{d}{dt}\int|(F^+)_{-}|^2dv\leq-2\int\left(|\nabla{(F^+)_{-}}|^2+\frac{R}{6}|(F^+)_{-}|^2\right)dv\\
-\int((F^+)_{-})^3dv+\int\left(\frac{4}{3}\bar{R}|(F^+)_{-}|^2+\frac{4}{3}(R-\bar{R})|(F^+)_{-}|^2\right)dv,\end{gathered}$$ $$\begin{gathered}
\label{L2 W-}
\frac{d}{dt}\int||W^{-}||^2dv\leq-\int\left(2|\nabla||{W^-}|||^2+R||W^-||^2\right)dv\\
+\int\left(2\sqrt{6}||W^-||^3+\frac{\sqrt6}{3}||W^-|||E|^2\right)dv,\end{gathered}$$ where $$\bar{R}=\int{R}dv\Big/\int{dv},\,\,\,F^+_g=R_g-2\sqrt{6}||W^+||,\,\,\,(F^+)_{-}=\min\{F_g^+,0\}.$$
For the proof of Proposition \[evolution of G2\] we will also need some algebraic inequalities. The first appears in ([@CGY03], Lemma 4.3), and is based on ([@Mar98], Lemma 6):
\[bew\] $$\label{WEE ||W|| and |E|}
WEE\leq\frac{\sqrt{6}}{3}\left(||W^+||+||W^-||\right)|E|^2$$
Recall the well-known decomposition of Singer-Thorpe: $$Riem=
\begin{pmatrix}
W^++\frac{R}{12}Id & B \\
B^{*} & W^-+\frac{R}{12}Id
\end{pmatrix}$$ Note the compositions satisfy $$BB^*:\Lambda^2_+\to\Lambda_+^2,\quad B^*B:\Lambda_-^2\to\Lambda_-^2.$$
Fix a point $P\in{M^4}$, and let $\lambda_1^{\pm}\leq\lambda_2^{\pm}\leq\lambda_{3}^{\pm}$ denote the eigenvalues of $W^{\pm}$, where ${W^{\pm}}$ are interpreted as endomorphisms of $\Lambda^2_{\pm}$. Also denote the eigenvalues of $BB^*:\Lambda^2_+\to\Lambda_+^2$ by $b_1^2\leq{b_2^2}\leq{b_3^2}$, where $0\leq{b_1}\leq{b_2}\leq{b_3}$. From Lemma 4.3 of [@CGY03], we have
$$\label{WEE}
WEE\leq4\left(\sum_{i=1}^3\lambda_i^+b_i^2+\sum_{i=1}^3\lambda_i^-b_i^2\right)$$
Recall from Lemma 4.2 of [@CGY03] that $|E|^2=4\sum_{i=1}^3b_i^2$. For a trace-free $3\times3$ matrix $A$, we have the sharp inequality: $$\label{sharp33}
|A(X,X)|\leq{\frac{\sqrt{6}}{3}||A|||X|^2}.$$ Apply (\[sharp33\]) to $A=diag(\lambda_1^{\pm},\lambda_2^{\pm},\lambda_3^{\pm})$ and $X=(b_1,b_2,b_3)$. We derive $$\label{sharp}
4\left(\sum_{i=1}^3\lambda_i^+b_i^2+\sum_{i=1}^3\lambda_i^-b_i^2\right)\leq\frac{\sqrt{6}}{3}\left(||W^+||+||W^-||\right)|E|^2.$$ Combining (\[WEE\]) and (\[sharp\]), we derive the desired inequality.
Now we turn to the proof of Proposition \[evolution of G2\]. From the definition of $G_k$, we have $$\frac{d}{dt}\int{G_2}dv= \frac{d}{dt}\int|E|^{2}dv+\frac{d}{dt}\int|R-\bar{R}|^{2}dv+\frac{d}{dt}\int||W^-||^{2}dv+\frac{d}{dt}\int|(F^+)_-|^{2}dv.$$
Now estimate each term of the right hand side from formulas in Corollary \[integral\].
$$\label{E2E4}
\begin{split}
\frac{d}{dt}\int|E|^2dv & \leq\int\left(-2|\nabla{E}|^2-\frac{1}{3}R|E|^2-4trE^3+\frac{4\sqrt{6}}{3}(||W^+||+||W^-||)|E|^2\right)dv\\
& \leq-\frac{Y}{3}\left(\int|E|^4dv\right)^{1/2}+4\left(\int|E|^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2} \\
& +\frac{4\sqrt{6}}{3}\left(\int||W^-||^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2}\\
& +\frac{2}{3}\left(\int|(F^+)_-|^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2}+\frac{2}{3}\int(R-\bar{R})|E|^2dv \\
& +\frac{2}{3}\int\bar{R}|E|^2dv\\
& \leq{-\frac{Y}{6}}\left(\int|E|^4dv\right)^{1/2}+\frac{2}{3}\bar{R}\int|E|^2dv.
\end{split}$$
The first inequality follows from Lemma \[bew\]. The second inequality follows from Cauchy-Schwartz inequality and the conformally invariant Sobolev inequality: $$\begin{aligned}
Y \Big( \int \phi^4 dv \Big)^{1/2} \leq \int \big( |\nabla \phi|^2 + \frac{1}{6}R \phi^2 \big) dv.\end{aligned}$$ The third inequality follows from the smallness assumption of $\epsilon_0$. Next, we esimate $$\label{R2R4}
\begin{split}
\frac{d}{dt}\int(R-\bar{R})^2dv & = \int\left(-2|\nabla(R-\bar{R})|^2-\frac{1}{3}R(R-\bar{R})^2\right)dv+\frac{1}{3}\int(R-\bar{R})^3dv\\
& +\frac{4}{3}\int\bar{R}(R-\bar{R})^2dv+4\int(R-\bar{R})|E|^2dv \\
& \leq{-\frac{Y}{3}\left(\int(R-\bar{R})^4dv\right)^{1/2}}+\frac{1}{3}\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int(R-\bar{R})^4dv\right)^{1/2}\\
& +\frac{4}{3}\bar{R}\int(R-\bar{R})^2dv+4\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2} \\
& \leq{-\frac{Y}{6}\left(\int(R-\bar{R})^4dv\right)^{1/2}}+\frac{4}{3}\bar{R}\int(R-\bar{R})^2dv\\
& +4\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2}.
\end{split}$$
The first inequality is from Sobolev inequality and Cauchy-Schwartz inequality and the second inequality is from the smallness assumption of $\epsilon_0$.
$$\label{F2F4}
\begin{split}
\frac{d}{dt}\int|(F^+)_-|^2dv & \leq-\frac{Y}{3}\left(\int|(F^+)_-|^4dv\right)^{1/2}+\left(\int|(F^+)_-|^2dv\right)^{1/2}\left(\int|(F^+)_-|^4dv\right)^{1/2} \\
& +\frac{4}{3}\bar{R}\int|(F^+)_-|^2dv +\frac{4}{3}\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int|(F^+)_-|^4dv\right)^{1/2} \\
& \leq-\frac{Y}{6}\left(\int|(F^+)_-|^4dv\right)^{1/2}+\frac{4}{3}\bar{R}\int|(F^+)_-|^2dv
\end{split}$$
The first inequality is from Sobolev inequality and Cauchy-Schwartz inequality and the second inequality is from the smallness assumption of $\epsilon_0$.
$$\label{W2W4}
\begin{split}
\frac{d}{dt}\int||W^-||^2dv & \leq -\frac{Y}{3}\left(||W^-||^4\right)^{1/2}+\frac{2}{3}\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int||W^-||^4dv\right)^{1/2}\\
& +2\sqrt{6}\left(\int||W^-||^2dv\right)^{1/2}\left(\int||W^-||^4dv\right)^{1/2}\\
& +\frac{\sqrt{6}}{3}\left(\int||W^-||^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2} \\
& \leq -\frac{Y}{6}\left(||W^-||^4\right)^{1/2}+\frac{\sqrt{6}}{3}\left(\int||W^-||^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2}
\end{split}$$
The first inequality is from Sobolev inequality and Cauchy-Schwartz inequality and the second inequality is from the smallness assumption of $\epsilon_0$.
With (\[E2E4\])(\[R2R4\])(\[F2F4\])(\[W2W4\]), it is now easy to see from the smallness assumption of $\epsilon_0$
$$\frac{d}{dt}\int_MG_2dv\leq\frac{4}{3}a\int_MG_2dv-\frac{1}{12}b\left(\int{G_4}dv\right)^{1/2}.$$
Take $\widetilde{a}=\frac{4}{3}a$ and $\widetilde{b}=\frac{1}{12}b$. Clearly, we have proved the desired inequality. Note that the differential inequality $$\frac{d}{dt}\int_MG_2dv\leq\frac{4}{3}a\int_MG_2dv$$ implies $$T\geq{T}_0=\frac{3\log2}{4}a.$$
The importance of this lemma is that $T_0$ does *not* depend on $\epsilon_0$, which implies that we may evolve the Ricci flow on a uniform time interval once we derive uniform bounds for curvatures.
With Lemma \[4.1\], it is easy to see that we may choose $\epsilon$ in Theorem A sufficiently small so that (\[initial smallness\]) is satisfied. To apply Proposition \[evolution of G2\] and Proposition \[evolution of G3\], we also need establish (\[Yamabe\]) and (\[average scalar\]). We now establish these inequalities and prove the following proposition.
\[choice of initial metric\] Suppose the initial metric of Ricci flow is chosen as the modified Yamabe metric for sufficiently small $\epsilon$ in Theorem A. Then there exists $\widetilde{T}$ which does not depend on $\epsilon$ such that for $0\leq{t}\leq{\widetilde{T}}$ all conditions of Proposition \[evolution of G2\] are satisfied.
It is clear from Lemma \[4.1\] that if we choose sufficiently small $\epsilon$ in Theorem A, we can establish (\[initial smallness\]) for arbitrary small $\epsilon_0$. On $0\leq{t}\leq{T}$, (\[initial smallness\]) implies that $$\label{smallness}
\int|E|^2dv\leq\frac{1}{2}\epsilon_0,\,\,\, \int||W^-||^2dv\leq\frac{1}{2}\epsilon_0,\,\,\,\int(R-\bar{R})^2dv\leq\frac{1}{2}\epsilon_0.$$ Recall $b_1(M)=0$, $b_2^-(M)=0$, and $b_2^+(M)=1$. From signature and Chern-Gauss-Bonnet formula, we obtain $$\int||W^+||^2dv=12\pi^2+\int||W^-||^2dv\leq12\pi^2+\frac{1}{2}\epsilon_0$$ and thereby $$\label{sigma-2}
12\pi^2\geq4\int\sigma_2(P)dv=24\pi^2-\int||W||^2dv\geq{12\pi^2-\epsilon_0}.$$ Now with the same argument in Lemma \[L2pinchLemma\], we can derive $$\frac{1}{96}Y(t)^2\geq\int\sigma_2(P_{g(t)})dv_{g(t)}\geq2\pi^2$$ if we choose sufficiently small $\epsilon_0$. Since the initial metric is of positive Yamabe type and the square of Yamabe constant has a strictly positive lower bound, we have established (\[Yamabe\]).
Note that (\[smallness\]) and (\[sigma-2\]) imply that for some $C>0$ $$\frac{1}{C^2}\leq\int\bar{R}^2_{g(t)}dv_{g(t)}=\bar{R}^2_{g(t)}vol(M,g(t))\leq{C^2},\,\,\, \frac{1}{C^2}\leq\int{R}^2_{g(t)}dv_{g(t)}\leq{C^2}..$$ To establish (\[average scalar\]), it now suffices to derive a uniform lower bound for the volume since $\int{\bar{R}^2}dv$.
\[volume\] Under conditions of Proposition \[choice of initial metric\] with $vol(M,g(0))=1$, along Ricci flow, there exists constant $T^{'}>0$ such that for $0\leq{t}\leq{T^{'}}\leq{T}$ $$\label{volume estimate}
\frac{9}{4}\geq{vol}(M,g(t))\geq\frac{1}{4}.$$ In addition, there exists $T_1>0$ which does not depend on $\epsilon_0$ such that $T^{'}>T_1$.
Recall the evolution equation for volume under the Ricci flow: $$\frac{d}{d{t}}\int{dv}=-\int{R}{dv}.$$ Hence, for $0\leq{t}\leq{T}$ $$\frac{d}{d{t}}\int{dv}\geq-\int{|R|}{dv}\geq-{\left(\int{R^2}dv\right)^{1/2}\left(\int{dv}\right)^{1/2}\geq-{C}\left(\int{dv}\right)^{1/2}}.$$ Similarly, $$\frac{d}{d{t}}\int{dv}\leq{C}\left(\int{dv}\right)^{1/2}.$$ It is then easy to derive $$|\sqrt{vol(t)}-\sqrt{vol(0)}|\leq{Ct}.$$ From this inequality, it is easy to choose $T^{'}=\min\{\frac{1}{2C},T\}$ such that (\[volume estimate\]) is satisfied. It is easy to see such a $T_1$ exists since $T\geq{T_0}$, where $T_0$ does not depend on $\epsilon_0$.
With Lemma \[volume\], we establish (\[average scalar\]) on $0\leq{t}\leq{T'}\leq{T}$. If we choose $\widetilde{T}=T'$, all conditions of Proposition \[evolution of G2\] are satisfied on $0\leq{t}\leq{\widetilde{T}}$ and clearly $\widetilde{T}$ has a positive universal lower bound.
We now derive integral estimates for $G_3$. For the sake of clearness, we first establish the estiamtes for $\int|E|^3dv$ and then derive a similar evolution inequality for $\int{G_3}dv$ as we did in Lemma \[evolution of G2\].
\[E3\] Under the conditions of Proposition \[evolution of G2\], along the Ricci flow, we have $$\int|E_{g(t)}|^3dv_{g(t)}\leq{{C\epsilon_0^{3/2}}{t^{-1}}},$$ for any $0<t\leq{\widetilde{T}}$, where $C$ is a universal constant which does not depend on $\epsilon_0$.
It is clear from (\[E2E4\]) that $$\label{|E|^2 evolution}
\frac{d}{dt}\int|E|^2dv+C\left(\int|E|^4dv\right)^{1/2}\leq{C\int|E|^2dv},$$ where $C$ is a constant which does not depend on $\epsilon_0$. From (\[E\^2\]), we can compute $$\label{E^p}
\frac{\partial}{\partial{t}}|E|^p=\frac{p}{2}|E|^{p-2}\frac{\partial}{\partial{t}}|E|^2=\frac{p}{2}|E|^{p-2}\left(\Delta|E|^2-2|\nabla{E}|^2+4WEE-4trE^3+\frac{2}{3}R|E|^2\right),$$ Note that $$\Delta|E|^p=\frac{p}{2}|E|^{p-2}\Delta|E|^2+p(p-2)|E|^{p-2}|\nabla|E||^2.$$ From this identity and (\[E\^p\]), it is easy to derive for $p\geq3$ $$\label{|E|^p integral}
\begin{split}
\frac{d}{dt}\int|E|^pdv & =\int\left(\frac{\partial}{\partial{t}}|E|^p-R|E|^p\right)dv \\
& \leq\int\left(-p(p-2)|E|^{p-2}|\nabla{|E|}|^2+2p|E|^2WEE\right)dv \\
& +\int\left(-2p|E|^{p-2}trE^3+\left(\frac{p}{3}-1\right)R|E|^p\right)dv \\
& =\int\left(-\frac{4(p-2)}{p}|\nabla{|E|^{\frac{p}{2}}}|^2-\frac{2(p-2)}{3p}R|E|^p\right)dv \\
& +\int\left(2p|E|^2WEE-2p|E|^{p-2}trE^3+C_pR|E|^p\right)dv
\end{split}$$ We now estimate the terms in last line of (\[|E|\^p integral\]) $$\int{||E|^{p-2}trE^3|}dv\leq\left(\int|E|^2dv\right)^{1/2}\left(\int|E|^{2p}dv\right)^{1/2}$$ $$\begin{split}
\int|E|^{p-2}WEEdv & \leq\frac{\sqrt{6}}{3}\int\left(||W^-||+||W^+||\right)|E|^pdv \\
& \leq\frac{\sqrt{6}}{3}\left(\int||W^-||^2dv\right)^{1/2}\left(\int|E|^{2p}dv\right)^{1/2} \\
& +\frac{1}{6}\left(\int((F^+)_-)^2dv\right)^{1/2}\left(\int|E|^{2p}dv\right)^{1/2}+\frac{1}{6}\int{R|E|^p}dv
\end{split}$$ $$\begin{split}
\int{|R||E|^p}dv & \leq\int|R-\bar{R}||E|^pdv+\bar{R}\int|E|^pdv\\
& \leq\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int|E|^{2p}dv\right)^{1/2}+\bar{R}\int|E|^pdv
\end{split}$$ From the smallness assumption of $\epsilon_0$, it is now easy to derive $$\label{|E|^p evolution}
\frac{d}{dt}\int|E|^pdv+C_p\left(\int|E|^{2p}dv\right)^{1/2}\leq{C_p\int|E|^pdv}.$$ In this proof, we shall only need this formula for $p=2,3$. Take two smooth cut-off functions $\phi_1$ and $\phi_2$ such that $0\leq\phi_i\leq1$ on $[0,\widetilde{T}]$ for $i=1,2$. Take $\tau<\tau'<\widetilde{T}$. For $\phi_1$, we choose $0\leq\phi_1\leq1$ on $[0,\tau]$ and $\phi_1\equiv1$ on $[\tau,\widetilde{T}]$. For $\phi_2$, we choose $\phi_2\equiv0$ on $[0,\tau]$, $0\leq\phi_2\leq1$ on $[\tau,\tau']$, and $\phi_1\equiv1$ on $[\tau',\widetilde{T}]$. Also assume $|\phi_i'|_{L^\infty}$ have appropriate bound. It is now easy to derive $$\label{|E|^p evolution cutoff}
\frac{d}{dt}\left(\phi_i\int|E|^pdv\right)+C\phi_i\left(\int|E|^{2p}dv\right)^{1/2}\leq{C\left(\phi_i+|\phi_i'|\right)\int|E|^pdv},$$ for $i=1,2$ and $p=2,3$.
Set $p=2$ and $i=1$. Integrate (\[|E|\^p evolution cutoff\]) over $[0,t]$ for some $t>\tau'$:
$$\label{E^2 integration}
\int|E|^2dv+C\int_\tau^t\left(\int|E|^4dv\right)^{1/2}ds\leq{C\left(1+\frac{1}{\tau}\right)\int_0^t\left(\int|E|^2dv\right)ds}$$
Set $p=3$ and $i=2$. Integrate (\[|E|\^p evolution cutoff\]) over $[0,t]$ for some $t>\tau'$:
$$\label{E^3 integration}
\int|E|^3dv+C\int_{\tau'}^t\left(\int|E|^6dv\right)^{1/2}ds\leq{C\left(1+\frac{1}{\tau'-\tau}\right)\int_\tau^t\left(\int|E|^3dv\right)ds}$$
Now we have $$\begin{aligned}
\int_\tau^t\left(\int|E|^3dv\right)ds &\leq \int_\tau^t\left(\int|E|^2dv\right)^{1/2}\left(\int|E|^4dv\right)^{1/2}ds \\
&\leq C\epsilon_0^{1/2}\int_\tau^t\left(\int|E|^4\right)^{1/2}ds \\
&\leq C\epsilon_0^{1/2}\left(1+\frac{1}{\tau}\right)\int_0^t\left(\int|E|^2dv\right)ds\end{aligned}$$ where second line follow from taking $\epsilon=\frac{1}{2}\epsilon_0$ in Lemma 5.5 and third line follows from (\[E\^2 integration\]). It then follows $$\begin{aligned}
\int|E|^3dv &\leq C\left(1+\frac{1}{\tau'-\tau}\right)\int_\tau^t\left(\int|E|^3dv\right)ds \\
&\leq C\epsilon_0^{1/2}\left(1+\frac{1}{\tau}\right)\left(1+\frac{1}{\tau'-\tau}\right)\int_0^t\left(\int|E|^2dv\right)ds \\
&\leq C\epsilon_0^{3/2}\left(1+\frac{1}{\tau}\right)\left(1+\frac{1}{\tau'-\tau}\right)t\end{aligned}$$ Take $\tau=\frac{1}{4}t$ and $\tau'=\frac{1}{2}t$ and we get desired estimate. In particular, if we choose $t\in[\frac{1}{4}\widetilde{T},\widetilde{T}]$, we have $$\label{E3 small}
\sup_{\widetilde{T}/4\leq{t}\leq{\widetilde{T}}}\int|E|^3dv\leq{C\epsilon_0^{3/2}}.$$
Now we prove an evolution inequality for $\int{G_3}dv$ similar as (\[evolution G\_2\]).
\[evolution of G3\] Under the same conditions of Proposition \[evolution of G2\], for $\frac{1}{4}\widetilde{T}\leq{t}\leq{\widetilde{T}}$, we have $$\label{evolution G_3}
\frac{d}{dt}\int_MG_3(t)dv_{g(t)}\leq{\widetilde{a}'}\int_MG_3(t)dv_{g(t)}-\widetilde{b}'\left(\int_M{G_6(t)}dv_{g(t)}\right)^{1/2},$$ where $\widetilde{a}'$ and $\widetilde{b}'$ are uniform positive constants independent of $\epsilon_0$ and we may choose $\widetilde{a}'=c_1a$ and $\widetilde{b}'=c_2b$, where $c_1$ and $c_2$ are universal positive constants.
The proof is similar to that of (\[evolution G\_2\]). From the definition of $G_k$, we have $$\label{G_3}
\frac{d}{dt}\int{G_3}dv= \frac{d}{dt}\int|E|^{3}dv+\frac{d}{dt}\int|R-\bar{R}|^{3}dv+\frac{d}{dt}\int||W^-||^{3}dv+\frac{d}{dt}\int|(F^+)_-|^{3}dv.$$ Note that we shrink the time interval to $[\frac{1}{4}\widetilde{T},\widetilde{T}]$, so from the previous lemma, we have known that $\int|E|^3$ is bounded by $c(\epsilon_0)$, where $c(\epsilon_0)\to0$ as $\epsilon_0\to0$.
We now estimate the right hand side of (\[G\_3\]) term by term
$$\label{E3E6}
\begin{split}
\frac{d}{dt}\int|E|^3dv & =\int\left(\frac{\partial}{\partial{t}}|E|^3-R|E|^3\right)dv \\
& = \int\left(\frac{3}{2}|E|\left(\Delta|E|^2-2|\nabla{E}|^2+4WEE-4trE^3\right)\right)dv \\
& \leq{-c_2Y}\left(\int|E|^6dv\right)^{1/2}+c_1{\bar{R}}\int|E|^3dv,
\end{split}$$
where the inequality is established similarly to (\[E2E4\]).
$$\label{W3W6}
\begin{split}
\frac{d}{dt}\int||W^-||^3dv & =\int\left(\frac{\partial}{\partial{t}}||W^-||^3-R||W^-||^3\right)dv \\
& \leq\int\left(3||W^-||^2\Delta||W^-||+3\sqrt{6}||W^-||^3+\frac{\sqrt{6}}{2}||W^-||^2|E|^2-R||W^-||^3\right)dv
\end{split}$$
Recall convexity inequality: $$\label{convex}
ab\leq{\frac{a^p}{p}+\frac{b^q}{q}}$$ for $a,b\geq0$ and $1/p+1/q=1$. Take $p=3/2$, $q=3$, $a=\left(\int||W^-||^6dv\right)^{1/3}$ and $b=\left(\int|E|^3dv\right)^{2/3}$. We have $$\label{convex W}
\begin{split}
\int||W^-||^2|E|^2dv& \leq\left(\int||W^-||^6dv\right)^{1/3}\left(\int|E|^3dv\right)^{2/3} \\
& \leq{\frac{2K}{3}\left(\int||W^-||^6dv\right)^{1/2}+\frac{1}{3K}\left(\int|E|^3dv\right)^2}
\end{split}$$ We may take $K$ to be a small multiple of the Yamabe constant and absorb the first term of (\[convex W\]) by the Soblev inequality. The second term of (\[convex W\]) is bounded by a constant multiple of $\int|E|^3dv$ from (\[E3 small\]) and the smallness assumption of $\epsilon_0$. It is then easy to derive from Sobolev inequality and Cauchy-Schwartz inequality $$\frac{d}{dt}\int||W^-||^3dv \leq {-c_2Y}\left(\int||W^-||^6dv\right)^{1/2}+c_1{\bar{R}}\int||W^-||^3dv+C\int|E|^3dv$$
$$\label{F3F6}
\begin{split}
\frac{d}{dt}\int|(F^+)_-|^3dv & =-\frac{d}{dt}\int((F^+)_-)^3dv\\
& =\int\left(-3((F^+)_-)^2\frac{\partial}{\partial{t}}((F^+)_-)-R|(F^+)_-|^3\right)dv\\
& \leq {-c_2Y}\left(\int|(F^+)_-|^6dv\right)^{1/2}+c_1{\bar{R}}\int|(F^+)_-|^3dv
\end{split}$$
The inequality follows from evolution inequality (\[F inequality\]), Sobolev inequality and the following trick: $$\begin{split}
\int{R}|(F^+)_-|^3dv & \leq{\int{(R-\bar{R})}|(F^+)_-|^3dv}+\bar{R}\int|(F^+)_-|^3dv \\
& \leq{\left(\int(R-\bar{R})^2\right)^{1/2}\left(\int|(F^+)_-|^6dv\right)^{1/2}}+C\int|(F^+)_-|^3dv
\end{split}$$ Note the first term of the last line can be absorbed by Sobolev inequality with the smallness asssumption of $\epsilon_0$.
$$\label{R3R6}
\begin{split}
\frac{d}{dt}\int|R-\bar{R}|^3dv & = \int\left(\frac{\partial}{\partial{t}}|R-\bar{R}|^3-R|R-\bar{R}|^3\right)dv \\
& = \int\left(\frac{3}{2}|R-\bar{R}|\frac{\partial}{\partial{t}}(R-\bar{R})^2-R|R-\bar{R}|^3\right)dv
\end{split}$$
$$\label{(R-Rbar)^2}
\begin{split}
\frac{\partial}{\partial{t}}(R-\bar{R})^2 & = \frac{\partial}{\partial{t}}R^2 -2\frac{\partial}{\partial{t}}(R\bar{R})+\frac{\partial}{\partial{t}}\bar{R}^2\\
& = \Delta{(R^2)}-2|\nabla{R}|^2+4R|E|^2+R^3-2R\frac{\partial}{\partial{t}}\bar{R}-2\bar{R}\frac{\partial}{\partial{t}}R+2\bar{R}\frac{\partial}{\partial{t}}\bar{R} \\
& = \Delta{(R-\bar{R})^2}-2|\nabla{(R-\bar{R})}|^2+4(R-\bar{R})|E|^2+R^2(R-\bar{R})-2(R-\bar{R})\frac{\partial}{\partial{t}}\bar{R}
\end{split}$$
Recall the evolution equation of $\bar{R}$ under Ricci flow: $$\label{Rbar}
\frac{\partial}{\partial{t}}\bar{R}=\frac{1}{vol}\int\left(2|E|^2-\frac{1}{2}R^2\right)dv+\bar{R}^2.$$ Plugging (\[(R-Rbar)\^2\]) and (\[Rbar\]) into (\[R3R6\]), we can derive $$\label{R3R6 plugin}
\begin{split}
\frac{d}{dt}\int|R-\bar{R}|^3dv & \leq\int\left(\frac{3}{2}|R-\bar{R}|\Delta(R-\bar{R})^2-R|R-\bar{R}|^3\right)dv \\
& +6\int(R-\bar{R})^2|E|^2dv+\frac{3}{2}\int|R^2-\bar{R}^2|(R-\bar{R})^2dv \\
& +\frac{3}{2}\left|\bar{R}^2-\frac{\int{R^2}dv}{vol}\right|\int(R-\bar{R})^2dv+\frac{6}{vol}\int|E|^2dv\int(R-\bar{R})^2dv.
\end{split}$$ Now we estimate the terms in second and third line of (\[R3R6 plugin\]): $$\begin{split}\label{R2E2}
\int(R-\bar{R})^2|E|^2dv & \leq\left(\int|R-\bar{R}|^6dv\right)^{1/3}\left(\int|E|^3dv\right)^{2/3} \\
& \leq{\frac{2K}{3}\left(\int|R-\bar{R}|^6dv\right)^{1/2}+\frac{1}{3K}\left(\int|E|^3dv\right)^2}
\end{split}$$ We may take $K$ to be a small multiple of the Yamabe constant and absorb the first term of (\[R2E2\]) by the Soblev inequality. The second term of (\[R2E2\]) is bounded by a constant multiple of $\int|E|^3dv$ from (\[E3 small\]) and the smallness assumption of $\epsilon_0$. $$\label{R2E2 integral}
\begin{split}
\int|E|^2dv\int(R-\bar{R})^2dv & \leq{C}{\left(\int|E|^3dv\right)^{2/3}\left(\int(R-\bar{R})^6dv\right)^{1/3}}\\
& \leq{{C}{K}\left(\int(R-\bar{R})^6dv\right)^{1/2}+\frac{C}{K}\left(\int|E|^3dv\right)^2}
\end{split}$$ The $C$ in first line just depends on volume. For the second line, we may take $K$ to be a small multiple of the Yamabe constant and absorb the first term of (\[R2E2 integral\]) by the Soblev inequality. The second term of (\[R2E2 integral\]) is bounded by a constant multiple of $\int|E|^3dv$ from (\[E3 small\]) and the smallness assumption of $\epsilon_0$. $$\begin{split}
\int|R^2-\bar{R}^2|(R-\bar{R})^2dv & =\int|R+\bar{R}||R-\bar{R}|^3dv \\
& \leq{\bar{R}\int(R-\bar{R})^3dv+\int|R||R-\bar{R}|^3dv} \\
& \leq{2\bar{R}\int(R-\bar{R})^3dv+\int|R-\bar{R}|^4dv} \\
& \leq{2\bar{R}\int(R-\bar{R})^3dv+\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int|R-\bar{R}|^6dv\right)^{1/2}}
\end{split}$$ The last term can be absorbed by Sobolev inequality from the smallness assumption of $\epsilon_0$. $$\begin{split}
\left|\bar{R}^2-\frac{\int{R^2}dv}{vol}\right|\int(R-\bar{R})^2dv &=\frac{1}{vol}\left(\int(R-\bar{R})^2dv\right)^2\\
& \leq{C}\left(\int(R-\bar{R})^2dv\right)^{1/2}\left(\int(R-\bar{R})^6dv\right)^{1/2}
\end{split}$$ This term can be absorbed by Sobolev inequality from the smallness assumption of $\epsilon_0$.
Combining all these estimates for $|R-\bar{R}|$, we derive $$\label{R3R6 final}
\frac{d}{dt}\int|R-\bar{R}|^3dv \leq {-c_2Y}\left(\int|R-\bar{R}|^6dv\right)^{1/2}+c_1{\bar{R}}\int|{R-\bar{R}}|^3dv+C\int|E|^3dv$$
Now we combine (\[E3E6\])(\[W3W6\])(\[F3F6\])(\[R3R6 final\]) to derive \[evolution G\_3\].
\[L3\] With the modified Yamabe metric chosen as initial metric, under the Ricci flow, we have $$\sup_{\widetilde{T}/2\leq{t}\leq{\widetilde{T}}}\int{G}_3(t)dv_{g(t)}\leq{C\epsilon_0^{3/2}}.$$
The proof is fundamentally the same as that of Lemma \[E3\]. The only difference is to replace $|E|^k$ by $G_k(t)$ since we have evolution equations of same type as is shown in Proposition \[evolution of G2\] and Proposition (\[evolution of G3\]).
To derive the $L^\infty$-boundedness, we shall apply the following result established by Deane Yang in [@Y].
\[Yang\] Assume that with respect to the metric $g=g(t)$, $0\leq{t}\leq{T}$, the following Sobolev inequality holds: $$\label{Sobolev Inequality}
\left(\int|\varphi|^{\frac{2n}{n-2}}dv\right)^{\frac{n-2}{n}}\leq{C_S\left[\int|\nabla\varphi|^2dv+\int\varphi^2dv\right]},\varphi\in{W^{1,2}(M^n)}.$$ Also, let $b\geq0$ on $M^n\times[0,T]$ satisfy $$\frac{\partial}{\partial{t}}dv\leq{bdv}.$$ Let $q>n$, and suppose $u\geq0$ is a function on $M^n\times[0,T]$ satisfying $$\frac{\partial{u}}{\partial{t}}\leq\Delta{u}+bu,$$ and that $$\label{b L^q/2}
\sup_{0\leq{t}\leq{T}}|b|_{L^{q/2}}\leq\beta.$$ Given $p_0>1$, there exists a constant $C=C(n,q,p_0,C_S,\beta)$ such that for $0\leq{t}\leq{T}$, $$\label{Yang's boundedness}
|u(t,\cdot)|_{\infty}\leq{C}e^{Ct}t^{-\frac{n}{2p_0}}|u(0,\cdot)|_{p_0}.$$ Moreover, given $p\geq{p_0}>1$, the following inequality holds for $0\leq{t}\leq{T}$: $$\label{Yang's evolution}
\frac{d}{dt}\int{u^p}dv+\int\left|\nabla\left(u^{p/2}\right)\right|^2dv\leq{Cp^{\frac{2n}{q-n}}}\int{u^p}dv$$ where $C=C(n,q,p_0,C_S)$.
\[Linfty\] With the modified Yamabe metric chosen as the initial metric, we have $$\sup_{3\widetilde{T}/4\leq{t}\leq{\widetilde{T}}}\{|E|+|R-\bar{R}|+||W^-||+|F^+_-|\}\leq{C}\epsilon_0.$$
Apply Lemma \[Yang\] to $u=G_2$, $q=6>n=4$ and $p_0=3/2$ on $t\in[\widetilde{T}/2,\widetilde{T}]$. Condition (\[b L\^q/2\]) is satisfied by Lemma \[L3\]. Hence, we can prove the desired estimate.
Now recall the Bernstein-Bando-Shi estimate (see for example Chapter 7 of [@CK]).
\[Cinfty\] Let $(M^4,g(t))$ be a solution to the Ricci flow. For every $m\in\mathbb{N}$, there exists a constant $C_m$depending only on $m$ such that if $$\sup_{x\in{M}}|Rm(x,t)|_{g(t)}\leq{K},\quad t\in\left[0,\frac{1}{K}\right],$$ then $$\sup_{x\in{M}}|\nabla^mRm(x,t)|_{g(t)}\leq{\frac{C_mK}{t^{m/2}}},\quad t\in\left[0,\frac{1}{K}\right],$$
Now we are at the position to prove Theorem A.
We argue by contradiction. Suppose there is a sequence of manifolds $(M_j,g_j)$ satisfying $\beta(M_j,[g_j])<4(1+\epsilon_j)$ with $\epsilon_j\to0$ and each of them is *not* diffeomorphic to standard $\mathbb{CP}^2$. For each conformal class $[g_j]$, we choose the modified Yamabe metric $(g_j)_{G}$ as initial metric and evolve the metric along Ricci flow. Then Lemma \[Linfty\] and Lemma \[Cinfty\] will imply that there is a time $\widetilde{T}$ such that the curvatures of $g_j(\widetilde{T})$ are uniformly bounded in $C^{\infty}$-norm and the Sobolev constants are also uniformly bounded. The convergence theory [@CGT] established by Cheeger, Gromov and Taylor then shows that there is a subsequence of $\{(M_j,g_j(\widetilde{T}))\}$ which converges smoothly to a manifold $(M_\infty,g_\infty)$. As $\epsilon_j\to0$, we obtain that $(M_\infty,g_\infty)$ satisfies $$\int_{M_\infty}||W||^2dv_\infty=\int_{M_\infty}\sigma_2dv_\infty$$ Note that we also have $b_1(M_\infty)=0$, $b_2^+(M_\infty)=1$ and $b_2^-(M_\infty)=0$. Hence, by Chern-Gauss-Bonnet and signature formula, we can easily derive that $(M_\infty,g_\infty)$ is self-dual Einstein. The same argument in Lemma \[CP2\] will show that $(M_\infty,g_\infty)$ is conformal equivalent to $(\mathbb{CP}^2,g_{FS})$. Since the convergence is smooth, we thereby obtain that $(M_j,g_j)$ must be diffeomorphic to $\mathbb{CP}^2$ with standard differentiable structure when $j$ is sufficiently large. This is clearly a contradiction to our assumption. Hence, we have proved the theorem.
.1in
To prove Theorem B, suppose $M^4$ is oriented with $b_2^{+}(M^4) > 0$. If $\beta(M^4) = 4$, then by definition we can find a metric $g$ with $$\begin{aligned}
\beta(M^4,[g]) < 4(1 + \epsilon/2),\end{aligned}$$ where $\epsilon > 0$ is from Theorem A. From Theorem A we conclude that $M^4$ is diffeomorphic to $\mathbb{CP}^2$. In addition, if $g$ is a metric on $\mathbb{CP}^2$ for which $\beta(M^4,[g]) = 4$, then taking $\epsilon = 0$ in Lemma \[L2pinchLemma\] we see that $g$ is self-dual. It follows, for example, from [@Poon86] that $(M^4,[g])$ is conformally equivalent to $(\mathbb{CP}^2,g_{FS})$.
[10]{}
A. Besse, Einstein Manifolds. Berlin: Springer-Verlag (1987).
V. Bour and G. Carron, *Optimal integral pinching results*, Ann. Sci. Éc. Norm. Supér. (4) **48** (2015), no. 1, 41–70.
V. Bour and G. Carron, *A sphere theorem for three dimensional manifolds with integral pinched curvature*, Comm. Anal. Geom. **25** (2017), no. 1, 97–124.
G. Catino and Z. Djadli, *Conformal deformations of integral pinched 3-manifolds*, Adv. Math. **223** (2010), no. 2, 393–404.
G. Catino and C.B. Ndiaye, *Integral pinching results for manifolds with boundary*, *Ann. Sc. Norm. Super. Pisa Cl. Sci.* (5) **9** (2010), no. 4, 785–813.
S. Y. A. Chang, M. J. Gursky, and P. Yang, *An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature*, Ann. Math. **155** (2002), 709-787.
S. Y. A. Chang, M. J. Gursky, and P. Yang, *A conformally invariant sphere theroem in four dimensions*, Publ. Math. Inst. Hautes Études Sci. **98** (2003), 105-143.
S. Y. A. Chang, J. Qing, and P. Yang, *On a conformal gap and finiteness theorem for a class of four-manifolds*, Geom. Func. Anal. **17** (2007), 404-434.
J. Cheeger, M. Gromov, and M. Taylor, *Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds*, J. Diff. Geom. **17**, (1982), 15-53.
B.-L. Chen and X.-P. Zhu, *A conformally invariant classification theorem in four dimensions*, Comm. Anal. Geom. **22** (2014), no. 5, 811–831.
B. Chow and D. Knopf, The Ricci Flow: An Introduction. American Mathematical Society (2004). A. Derdzinski, *Self-dual Kähler manifolds and Einstein manifolds of dimension four*, Compositio Math. [**49**]{} (1983), no. 3, 405–-433.
S. Donaldson, *An application of gauge theory to four-dimensional topology*, J. Diff. Geom. (1983), 279-315.
M. Freedman, *The topology of four-dimensional manifolds*, J. Diff. Geom. **17** (1982), 357-453.
Y. Ge, Yuxin, C.-S. Lin, and G. Wang, *On the $\sigma_2$-scalar curvature*, J. Diff. Geom. **84** (2010), 45–86.
P. Guan, C.-S. Lin, G. Wang, *Application of the method of moving planes to conformally invariant equations*, Math. Z. **247** (2004), no. 1, 1–19.
M. J. Gursky, *The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics*, Ann. Math. **148** (1998), 315-337.
M. J. Gursky, *The principal eigenvalue of a conformally invariant differential operator, with an Application to semilinear elliptic PDE*, Comm. Math. Phys. **207** (1999), 131-143.
M. J. Gursky, *Four-manifolds with $\delta{W}^+=0$ and Einstein constants of the sphere*, Math. Ann. **318** (2000), 417-431.
M. J. Gursky and C. LeBrun *On Einstein manifolds of positive sectional curvature*, Ann. Glob. Anal. Geom. **17** (1999), 315-328.
R. Hamilton, *Three-manifolds with positive Ricci curvature*, J. Diff. Geom. **17** (1982), 255-306.
N. Hitchin, *Compact four-dimensional Eintein manifolds*, J. Diff. Geom. **9** (1974), 435-441.
M. Itoh, *The Weitzenböck formula for the Bach operator*, Nagoya Math. J. **137** (1995), 149–181.
N. H. Kuiper, *On conformally flat spaces in the large*, Ann. Math. [**50**]{} (1949), 916–-924.
G. R. Jensen, *Homogeneous Einstein spaces of dimension four*, J. Differential Geometry [**3**]{} (1969), 309-–349.
D. Knopf, Unpublished notes.
M. Lai, *A note on a three-dimensional sphere theorem with integral curvature condition*, Commun. Contemp. Math. **18** (2016), no. 4, 1550070, 8 pp.
C. Lebrun, *On the topology of self-dual 4-manifolds*, Proc. Amer. Math. Soc. **98** (1986), 637-640.
C. LeBrun, S. Nayatani, and T. Nitta, *Self-dual manifolds with positive Ricci curvature*, Math. Zeit. **224** (1997), 49-63.
G. Li, J. Qing, and Y. Shi, *Gap phenomena and curvature estimates for conformally compact Einstein manifolds*, Trans. Amer. Math. Soc. **369** (2017), 4385-4413.
A. Lichnerowicz, *Spineurs harmoniques*, [*C. R. Acad. Sci. Paris*]{} [**257**]{} (1963), 7-9.
C. Margerin, *A sharp characterization of the smooth 4-sphere in curvature terms*, Comm. Anal. Geom. **6** (1998), no. 1, 21–65.
M. Obata, *The conjectures on conformal transformations of Riemannian manifolds*, J. Differential Geometry [**6**]{} (1971/72), 247-–258.
Y. S. Poon, *Compact self-dual manifolds with positive scalar curvature*, J. Differential Geom. **24** (1986), no. 1, 97–132.
W. Sheng, *Admissible metrics in the $\sigma_k$-Yamabe equation*, Proc. Amer. Math. Soc. **136** (2008), no. 5, 1795–1802.
G. Tian, Canonical Metrics in Kähler Geometry. Berlin: Birkhäuser (2000). D. Yang, *$L^p$ pinching and compactness theorems for compact Riemannian manifolds*, Forum Math. **4** (1992), 323-333.
[^1]: The first author was supported in part by NSF Grant MPS-1509505.
[^2]: The second author was supported in part by NSF Grant DMS-1811034.
|
---
abstract: 'In this paper we categorify the $q$-Schur algebra $\mathbf{S}_q(n,d)$ as a quotient of Khovanov and Lauda’s diagrammatic $2$-category $\Ucat$ [@K-L3]. We also show that our $2$-category contains Soergel’s [@S] monoidal category of bimodules of type $A$, which categorifies the Hecke algebra $H_q(d)$, as a full sub-$2$-category if $d\leq n$. For the latter result we use Elias and Khovanov’s diagrammatic presentation of Soergel’s monoidal category of type $A$ [@E-Kh].'
address:
- |
Departamento de Matemática\
Universidade do Algarve\
Campus de Gambelas\
8005-139 Faro\
Portugal and CAMGSD\
Instituto Superior Técnico\
Avenida Rovisco Pais\
1049-001 Lisboa\
Portugal
- |
Instituto de Sistemas e Robótica and CAMGSD\
Instituto Superior Técnico\
Avenida Rovisco Pais\
1049-001 Lisboa\
Portugal and\
Mathematical Institute SANU\
Knez Mihailova 36\
11000 Beograd\
Serbia
- |
CAMGSD\
Instituto Superior Técnico\
Avenida Rovisco Pais\
1049-001 Lisboa\
Portugal
author:
- Marco Mackaay
- Marko Stošić
- Pedro Vaz
title: 'A diagrammatic categorification of the q-Schur algebra'
---
Introduction {#sec:intro}
============
There is a well-known relation, called [*Schur-Weyl duality*]{} or [ *reciprocity*]{}, between the polynomial representations of homogeneous degree $d$ of the general linear group $\mbox{GL}(n,\mathbb{Q})$ and the finite-dimensional representations of the symmetric group on $d$ letters $S_d$. Recall that all irreducible polynomial representations of $\mbox{GL}(n,\mathbb{Q})$ of homogeneous degree $d$ occur in the decomposition of $V^{\otimes d}$, where $V=\mathbb{Q}^n$ is the natural representation of $\mbox{GL}(n,\mathbb{Q})$. Instead of the $\mbox{GL}(n,\mathbb{Q})$-action, we can consider the ${\mathbf U}(\mathfrak{gl}_n)$-action, without loss of generality. A key observation for Schur-Weyl duality is that the permutation action of $S_d$ on $V^{\otimes d}$ commutes with the action of ${\mathbf U}(\mathfrak{gl}_n)$. Furthermore, we have $$\mathbb{Q}[S_d]\cong \mbox{End}_{{\mathbf U}(\mathfrak{gl}_n)}(V^{\otimes d})$$ if $n\geq d$.
By definition, the [*Schur algebra*]{} is the other centralizer algebra $$S(n,d):=\mbox{End}_{S_d}(V^{\otimes d}).$$ It is well known that both ${\mathbf U}(\mathfrak{sl}_n)$ and ${\mathbf U}(\mathfrak{gl}_n)$ map surjectively onto $S(n,d)$, for any $d>0$. Therefore we can also define $S(n,d)$ as the image of the map $${\mathbf U}(\mathfrak{gl}_n)\to \End_{\mathbb{Q}}(V^{\otimes d}),$$ which is the definition used in this paper. Both $S(n,d)$ and $\mathbb{Q}[S_d]$ are split semi-simple finite-dimensional algebras, and the [*double centralizer property*]{} above implies that the categories of finite-dimensional modules $S(n,d)-\mbox{mod}$ and $S_d-\mbox{mod}$ are equivalent, for $n\geq d$.
There are two more facts of interest to us. The first is that there actually exists a concrete functor which gives rise to the above mentioned equivalence. For $n\geq d$, there exists an embedding of $\mathbb{Q}[S_d]$ in $S(n,d)$, which induces the so called [*Schur functor*]{} $$S(n,d)-\mbox{mod}\longrightarrow S_d-\mbox{mod}.$$ As it turns out, this functor is an equivalence.
The second fact of interest to us is that the Schur algebras $S(n,d)$ for various values of $n$ and $d$ are related. If $n\leq m$, then $S(n,d)$ can be embedded into $S(m,d)$. A more complicated relation is the following: for any $k\in\mathbb{N}$, there is a surjection $$S(n,d+nk)\to S(n,d).$$ This surjection is compatible with the projections of ${\mathbf U}(\mathfrak{gl}_n)$ and ${\mathbf U}(\mathfrak{sl}_n)$ onto the Schur algebras. With these surjections, the Schur algebras form an inverse system. As it turns out, the projections of ${\mathbf U}(\mathfrak{sl}_n)$ onto the Schur algebras give rise to an embedding $${\mathbf U}(\mathfrak{sl}_n)\subset \oplus_{d=0}^{n-1}\lim_{\leftarrow k}S(n,d+nk).$$ To get a similar embedding for ${\mathbf U}(\mathfrak{gl}_n)$, one needs to consider generalized Schur algebras. We do not give the details of this generalization, because we will not need it. We refer the interested reader to [@D-G]. 0.5cm
All the facts recollected above have $q$-analogues, which involve the quantum groups ${\mathbf U}_q(\mathfrak{gl}_n)$ and ${\mathbf U}_q(\mathfrak{sl}_n)$, the Hecke algebra $H_q(d)$, the $q$-Schur algebra $S_q(n,d)$, and their respective finite-dimensional representations over $\mathbb{Q}(q)$.
If one is only interested in the finite-dimensional representations of ${\mathbf U}_q(\mathfrak{gl}_n)$ and ${\mathbf U}_q(\mathfrak{sl}_n)$, which can all be decomposed into weight spaces, it is easier to work with Lusztig’s idempotented version of these quantum groups, denoted $\Ugl$ and $\U$. In these idempotented versions, the Cartan subalgebras are “replaced” by algebras generated by orthogonal idempotents corresponding to the weights. The kernel of the surjection $\Ugl\to S_q(n,d)$ is simply the ideal generated by all idempotents corresponding to the $\mathfrak{gl}_n$-weights which do not appear in the decomposition of $V^{\otimes d}$. The same is true for the kernel of $\U\to S_q(n,d)$, using $\mathfrak{sl}_n$-weights. We will say more about $\Ugl$ and $\U$ in the next section. 0.5cm
We are interested in the [*categorification*]{} of the $q$-algebras above, the relations between them and the applications to low-dimensional topology. By a categorification of a $q$-algebra we mean a monoidal category or a 2-category whose Grothendieck group, tensored by $\mathbb{Q}(q)$, is isomorphic to that $q$-algebra.
As a matter of fact, all of them have been categorified already, and some of them in more than one way. Soergel defined a category of bimodules over polynomial rings in $d$ variables, which he proved to categorify $H_q(d)$. Elias and Khovanov gave a diagrammatic version of the Soergel category. Grojnowski and Lusztig [@G-L] were the first to categorify $S_q(n,d)$, using categories of perverse sheaves on products of partial flag varieties. Subsequently Mazorchuk and Stroppel constructed a categorification using representation theoretic techniques [@M-S1] and so did Williamson [@Will] for $n=d$ using singular Soergel bimodules. Khovanov and Lauda have provided a diagrammatic 2-category $\Ucat$ which categorifies $\U$. Rouquier [@R2] followed a more representation theoretic approach to the categorification of the quantum groups. The precise relation of his work with Khovanov and Lauda’s remains unclear. We note that the categorifications mentioned above have been obtained for arbitrary root data. However, this paper is only about type $A$ and we will not consider other types.
Our interest is in the diagrammatic approach, by which $H_q(d)$ and ${\mathbf U}_q(\mathfrak{sl}_n)$ have already been categorified. The goal of this paper is to define a diagrammatic categorification of $S_q(n,d)$. Recall that the objects of $\Ucat$ are the weights of $\mathfrak{sl}_n$, which label the regions in the diagrams which constitute the 2-morphisms. Our idea is quite simple: define a new 2-category $\glcat$ just as $\Ucat$ but switch to $\mathfrak{gl}_n$-weights, which we conjecture to give a categorification of $\Ugl$. Next we mod out $\glcat$ by all diagrams which have regions labeled by weights not appearing in the decomposition of $V^{\otimes d}$. This way we obtain a 2-category $\Scat(n,d)$ and the main result of this paper is the proof that it indeed categorifies $S_q(n,d)$.
There are two good reasons for switching to $\mathfrak{gl}_n$-weights, besides giving a conjectural categorification of $\Ugl$. It is easier to say explicitly which $\mathfrak{gl}_n$-weights do not appear in $V^{\otimes d}$, as we will show in the next section. Also, while working on our paper we found a sign mistake in what Khovanov and Lauda call their signed categorification of $\U$ [@K-L:err]. Fortunately it does not affect their unsigned version, but the corrected signed version loses a nice property, the cyclicity. We discovered that with $\mathfrak{gl}_n$-weights there is a different sign convention which solves the problem, at least for $\Scat(n,d)$.
On our way of proving the main result of this paper we obtain some other interesting results:
- For $n\geq d$, we define a fully faithful 2-functor from Soergel’s category of bimodules to $\Scat(n,d)$, which categorifies the well-known inclusion $H_q(d)\subset S_q(n,d)$ explained in Section \[sec:hecke-schur\].
- We define functors $\Scat(n,d)
\to \Scat(m,d)$ when $n\leq m$. We are not (yet) able to prove that these are faithful, although we strongly suspect that they are. We know that they are not full, but suspect that they are “almost full” in a sense that we will explain in Section \[sec:grothendieck\].
- We define essentially surjective full 2-functors $$\Scat(n,d+kn)\to \Scat(n,d)$$ which categorify the surjections above.
- We show that Khovanov and Lauda’s 2-representation of $\Ucat$ on the equivariant cohomology of flag varieties descends to $\Scat(n,d)$.
- We conjecture how to categorify the irreducible representations of $S_q(n,d)$ using $\Scat(n,d)$. Khovanov and Lauda’s categorification of these representations, using the so-called cyclotomic quotients, should be equivalent to a quotient of ours.
Understanding the precise relation with the other categorifications of $S_q(n,d)$ would be very important, but is left for the future. As a matter of fact, Brundan and Stroppel have already established a link between the category $\mathcal{O}$ approach to categorification and Khovanov and Lauda’s (see for example [@B-S]), which perhaps can be used to obtain an equivalence between Mazorchuk and Stroppel’s categorification of the $q$-Schur algebra and ours. For $n=d$, Williamson’s 2-category of Soergel’s singular bimodules is equivalent to Khovanov and Lauda’s 2-category build out of the equivariant cohomology of partial flag varieties (of flags in $\mathbb{Q}^d$) and we expect both to be equivalent to $\Scat(d,d)$. 0.5cm
Besides the intrinsic interest of $\Scat(n,d)$, with its combinatorics and its link to representation theory, there is also a potential application to knot theory. First recall that there is a natural surjection of the braid group onto $H_q(d)$. The Jones-Ocneanu trace of the image of a braid in $H_q(d)$ is equal to the so called HOMFLYPT knot polynomial of the braid closure. This construction has been categorified: Rouquier defined a complex of Soergel bimodules for each braid and Khovanov discovered that its Hochschild homology categorifies the Jones-Ocneanu trace, showing that in this way one obtains a homology which is isomorphic to the Khovanov-Rozansky HOMFLYPT-homology. Using Elias and Khovanov’s work, Elias and Krasner [@E-Kr] worked out the diagrammatic version of Rouquier’s complex. Their work still remains to be extended to include the Hochschild homology. Besides this approach, which is the one most directly related to the results in this paper, we should also mention a geometric approach due to Webster and Williamson in [@W-W1] and a representation theoretic approach due to Mazorchuk and Stroppel [@M-S2].
More generally, there is a natural homomorphism from the colored braid group, with $n$ strands colored by natural numbers whose sum is equal to $d$, to $S_q(n,d)$. It is not as widely advertised as the non-colored version, but one can easily obtain it from Lusztig’s formulas in Section 5.2.1 in [@Lu] or from the second part of the paper by Murakami-Ohtsuki-Yamada [@M-O-Y]. One can also define a colored version of the Jones-Ocneanu trace on $S_q(n,d)$ to obtain the colored HOMFLYPT knot invariant. Naturally the question arises how to categorify the colored HOMFLYPT knot polynomial. In [@C-R] Chuang and Rouquier defined a colored version of Rouquier’s complex for a braid, using a representation theoretic approach. They proved invariance under the second braid-like Reidemeister move and conjectured invariance under the third move. In [@M-S-V] we defined a complex of singular Soergel bimodules, which is equivalent to the Chuang-Rouquier complex. We conjectured that the Hochschild homology of such a complex categorifies the colored HOMFLYPT knot polynomial of the braid closure. We were only able to prove our conjecture for the colors 1 and 2, due to the complexity of the calculations for general colors. Webster and Williamson subsequently showed our conjecture to be true, using a generalization of their geometric approach [@W-W2]. Cautis, Kamnitzer and Licata [@C-K-L] also studied the Chuang-Rouquier complex from a geometric point of view. By the above mentioned 2-representation of $\Scat(n,d)$ into singular Soergel bimodules, it is natural to expect that one should be able to define the Chuang-Rouquier complex in $\Scat(n,d)$, such that its 2-representation gives exactly the complex of singular Soergel bimodules which we conjectured. In a forthcoming paper we will come back to this. In the meanwhile, papers have appeared in which the colored HOMFLYPT homology has been constructed using matrix factorizations (see [@Wu1; @Wu2; @Wu3; @Yo1; @Yo2]). 0.5cm The outline of this paper is as follows:
- In Section \[sec:hecke-schur\] we recall some results on the above mentioned $q$-algebras. Our choice has been highly selective in an attempt to prevent this paper from becoming too long. We have only included those results which we categorify or which we need in order to categorify. We hope that this introduction makes up for what we left out.
- In Section \[sec:scat\] we define the 2-categories $\mathcal{U}(\mathfrak{gl}_n)$ and $\Scat(n,d)$. As said before, the first one is just a copy of Khovanov and Lauda’s definition of $\Ucat$, but with a different set of weights and a different sign convention. The second one is a quotient of the first one.
- To understand some of the properties of $\Scat(n,d)$, we first define its $2$-representation in the $2$-category of bimodules over polynomial rings in Section \[sec:2rep\]. Except for the different sign convention, it is the factorization of the 2-representation of [@K-L3] through $\Scat(n,d)$. The only new feature is our interpretation of this 2-representation in terms of the categorified MOY-calculus, which we developed in [@M-S-V].
- Section \[sec:struct\] is devoted to comparing the structure of the 2-HOM spaces of $\Ucat$ to those of $\Scat(n,d)$. The latter ones remain a bit of a mystery to us and we can only prove just enough about them for what we need in the rest of this paper.
- In Section \[sec:soergel\] we define a fully faithful embedding of Soergel’s categorification of $H_q(d)$ into $\Scat(n,d)$. We have not yet attributed any notation to Soergel’s category in this introduction, because there are actually two slightly different versions of it and we will need both, one for $d=n$ and the other for $d<n$.
- In Section \[sec:grothendieck\] we prove that $\Scat(n,d)$ indeed categorifies $S_q(n,d)$. We also conjecture how to categorify the Weyl modules of $S_q(n,d)$.
Hecke and $q$-Schur algebras {#sec:hecke-schur}
============================
In this section we recollect some facts about the $q$-algebras mentioned in the introduction. For details and proofs see [@D] and [@Mar] unless other references are mentioned. We work over the field $\bQ(q)$, where $q$ is a formal parameter.
The quantum general and special linear algebras
-----------------------------------------------
Let us first recall the quantum general and special linear algebras. The $\mathfrak{gl}_n$-weight lattice is isomorphic to $\bZ^n$. Let $\epsilon_i=(0,\ldots,1,\ldots,0)\in \bZ^n$, with $1$ being on the $i$th coordinate, and $\alpha_i=\epsilon_i-\epsilon_{i+1}\in\bZ^{n}$, for $i=1,\ldots,n-1$. We also define the Euclidean inner product on $\bZ^n$ by $(\epsilon_i,\epsilon_j)=\delta_{i,j}$.
The [*quantum general linear algebra*]{} ${\mathbf U}_q(\mathfrak{gl}_n)$ is the associative unital $\bQ(q)$-algebra generated by $K_i,K_i^{-1}$, for $1,\ldots, n$, and $E_{\pm i}$, for $i=1,\ldots, n-1$, subject to the relations $$\begin{gathered}
K_iK_j=K_jK_i\quad K_iK_i^{-1}=K_i^{-1}K_i=1
\\
E_iE_{-j} - E_{-j}E_i = \delta_{i,j}\dfrac{K_iK_{i+1}^{-1}-K_i^{-1}K_{i+1}}{q-q^{-1}}
\\
K_iE_{\pm j}=q^{\pm (\epsilon_i,\alpha_j)}E_{\pm j}K_i
\\
E_{\pm i}^2E_{\pm j}-(q+q^{-1})E_{\pm i}E_{\pm j}E_{\pm i}+E_{\pm j}E_{\pm i}^2=0
\qquad\text{if}\quad |i-j|=1
\\
E_{\pm i}E_{\pm j}-E_{\pm j}E_{\pm i}=0\qquad\text{else}.\end{gathered}$$
\[defn:qsln\] The [*quantum special linear algebra*]{} ${\mathbf U}_q(\mathfrak{sl}_n)\subseteq {\mathbf U}_q(\mathfrak{gl}_n)$ is the unital $\bQ(q)$-subalgebra generated by $K_iK^{-1}_{i+1}$ and $E_{\pm i}$, for $i=1,\ldots, n-1$.
Recall that the ${\mathbf U}_q(\mathfrak{sl}_n)$-weight lattice is isomorphic to $\bZ^{n-1}$. Suppose that $V$ is a ${\mathbf U}_q(\mathfrak{gl}_n)$-weight representation with weights $\lambda=(\lambda_1,\ldots,\lambda_n)\in\bZ^n$, i.e. $$V\cong \bigoplus_{\lambda}V_{\lambda}$$ and $K_i$ acts as multiplication by $q^{\lambda_i}$ on $V_{\lambda}$. Then $V$ is also a ${\mathbf U}_q(\mathfrak{sl}_n)$-weight representation with weights $\overline{\lambda}=(\overline{\lambda}_1,\ldots,\overline{\lambda}_{n-1})\in
\bZ^{n-1}$ such that $\overline{\lambda}_j=\lambda_j-\lambda_{j+1}$ for $j=1,\ldots,n-1$. Conversely, given a ${\mathbf U}_q(\mathfrak{sl}_n)$-weight representation with weights $\mu=(\mu_1,\ldots,\mu_{n-1})$, there is not a unique choice of ${\mathbf U}_q(\mathfrak{gl}_n)$-action on $V$. We can fix this by choosing the action of $K_1\cdots K_n$. In terms of weights, this corresponds to the observation that, for any $d\in\bZ$ the equations $$\begin{aligned}
\label{eq:sl-gl-wts1}
\lambda_i-\lambda_{i+1}&=\mu_i\\
\label{eq:sl-gl-wts2}
\qquad \sum_{i=1}^{n}\lambda_i&=d\end{aligned}$$ determine $\lambda=(\lambda_1,\ldots,\lambda_n)$ uniquely, if there exists a solution to and at all. To fix notation, we define the map $\phi_{n,d}\colon \bZ^{n-1}\to \bZ^{n}\cup \{*\}$ by $$\phi_{n,d}(\mu)=\lambda$$ if and have a solution, and put $\phi_{n,d}(\mu)=*$ otherwise.
Recall that ${\mathbf U}_q(\mathfrak{gl}_n)$ and ${\mathbf U}_q(\mathfrak{sl}_n)$ are both Hopf algebras, which implies that the tensor product of two of their representations is a representation again.
Both ${\mathbf U}_q(\mathfrak{gl}_n)$ and ${\mathbf U}_q(\mathfrak{sl}_n)$ have plenty of non-weight representations, but we are not interested in them. Therefore we can restrict our attention to the Beilinson-Lusztig-MacPherson [@B-L-M] idempotented version of these quantum groups, denoted $\Ugl$ and $\U$ respectively. To understand their definition, recall that $K_i$ acts as $q^{\lambda_i}$ on the $\lambda$-weight space of any weight representation. For each $\lambda\in\bZ^n$ adjoin an idempotent $1_{\lambda}$ to ${\mathbf U}_q(\mathfrak{gl}_n)$ and add the relations $$\begin{aligned}
1_{\lambda}1_{\mu} &= \delta_{\lambda,\nu}1_{\lambda}
\\
E_{\pm i}1_{\lambda} &= 1_{\lambda\pm\alpha_i}E_{\pm i}
\\
K_i1_{\lambda} &= q^{\lambda_i}1_{\lambda}.\end{aligned}$$
\[defn:Uglndot\] The idempotented quantum general linear algebra is defined by $$\Ugl=\bigoplus_{\lambda,\mu\in\bZ^n}1_{\lambda}{\mathbf U}_q(\mathfrak{gl}_n)1_{\mu}.$$
For $\ii=(\alpha_1 i_1,\ldots,\alpha_{n-1}i_{n-1})$, with $\alpha_j=\pm$, define $$E_{\ii}:=E_{\alpha_1 i_1}\cdots E_{\alpha_{n-1} i_{n-1}}$$ and define $\ii_{\Lambda}\in\bZ^n$ to be the $n$-tuple such that $$E_{\ii}1_{\mu}=1_{\mu + \ii_{\Lambda}}E_{\ii}.$$
Similarly for ${\mathbf U}_q(\mathfrak{sl}_n)$, adjoin an idempotent $1_{\mu}$ for each $\mu\in\bZ^{n-1}$ and add the relations $$\begin{aligned}
1_{\mu}1_{\nu} &= \delta_{\mu,\nu}1_{\lambda}
\\
E_{\pm i}1_{\mu} &= 1_{\mu\pm\overline{\alpha}_i}E_{\pm i}
\\
K_iK^{-1}_{i+1}1_{\mu} &= q^{\mu_i}1_{\mu}.\end{aligned}$$
The idempotented quantum special linear algebra is defined by $$\U=\bigoplus_{\mu,\nu\in\bZ^{n-1}}1_{\mu}{\mathbf U}_q(\mathfrak{sl}_n)1_{\nu}.$$
Note that $\Ugl$ and $\U$ are both non-unital algebras, because their units would have to be equal to the infinite sum of all their idempotents. Furthermore, the only ${\mathbf U}_q(\mathfrak{gl}_n)$ and ${\mathbf U}_q(\mathfrak{sl}_n)$-representations which factor through $\Ugl$ and $\U$, respectively, are the weight representations. Finally, note that there is no embedding of $\U$ into $\Ugl$, because there is no embedding of the $\mathfrak{sl}_n$-weights into the $\mathfrak{gl}_n$-weights.
The $q$-Schur algebra
---------------------
Let $d\in\bN$ and let $V$ be the natural $n$-dimensional representation of ${\mathbf U}_q(\mathfrak{gl}_n)$. Define $$\Lambda(n,d)=\{\lambda\in \bN^n\colon\,\,
\sum_{i=1}^{n}\lambda_i=d\}$$ $$\Lambda^+(n,d)=\{\lambda\in\Lambda(n,d)\colon d\geq
\lambda_1\geq\lambda_2\geq\cdots
\geq\lambda_n\geq 0\}.$$ Recall that the weights in $V^{\otimes d}$ are precisely the elements of $\Lambda(n,d)$, and that the highest weights are the elements of $\Lambda^+(n,d)$. The highest weights correspond exactly to the irreducibles $V_{\lambda}$ that show up in the decomposition of $V^{\otimes d}$.
As explained in the introduction, we can define the $q$-Schur algebra as follows:
The $q$-Schur algebra $S_q(n,d)$ is the image of the representation $\psi_{n,d}\colon {\mathbf U}_q(\mathfrak{gl}_n)\to \End_{\mathbb{Q}}(V^{\otimes d})$.
For each $\lambda\in\Lambda^+(n,d)$, the ${\mathbf U}_q(\mathfrak{gl}_n)$-action on $V_{\lambda}$ factors through the projection $\psi_{n,d}\colon {\mathbf U}_q(\mathfrak{gl}_n)\to S_q(n,d)$. This way we obtain all irreducible representations of $S_q(n,d)$. Note that this also implies that all representations of $S_q(n,d)$ have a weight decomposition. As a matter of fact, it is well known that $$S_q(n,d)\cong \prod_{\lambda\in\Lambda^+(n,d)}\End_{\mathbb{Q}}(V_{\lambda}).$$ Therefore $S_q(n,d)$ is a finite-dimensional split semi-simple unital algebra and its dimension is equal to $$\sum_{\lambda\in\Lambda^+(n,d)}\dim(V_{\lambda})^2=\binom{n^2+d-1}{d}.$$
Since $V^{\otimes d}$ is a weight representation, $\psi_{n,d}$ gives rise to a homomorphism $\Ugl\to S_q(n,d)$, for which we use the same notation. This map is still surjective and Doty and Giaquinto, in Theorem 2.4 of [@D-G], showed that the kernel of $\psi_{n,d}$ is equal to the ideal generated by all idempotents $1_{\lambda}$ such that $\lambda\not\in\Lambda(n,d)$. Let $\SD(n,d)$ be the quotient of $\Ugl$ by the kernel of $\psi_{n,d}$. Clearly we have $\SD(n,d)\cong S_q(n,d)$. By the above observations, we see that $\SD(n,d)$ has a Serre presentation. As a matter of fact, by Corollary 4.3.2 in [@C-G], this presentation is simpler than that of $\Ugl$: one does not need to impose the last two Serre relations, involving cubical terms, because they are implied by the other relations and the finite dimensionality.[^1]
$\SD(n,d)$ is isomorphic to the associative unital $\bQ(q)$-algebra generated by $1_{\lambda}$, for $\lambda\in\Lambda(n,d)$, and $E_{\pm i}$, for $i=1,\ldots,n-1$, subject to the relations $$\begin{aligned}
1_{\lambda}1_{\mu} &= \delta_{\lambda,\mu}1_{\lambda}
\\[0.5ex]
\sum_{\lambda\in\Lambda(n,d)}1_{\lambda} &= 1
\\[0.5ex]
E_{\pm i}1_{\lambda} &= 1_{\lambda\pm\alpha_i}E_{\pm i}
\\[0.5ex]
E_iE_{-j}-E_{-j}E_i &= \delta_{ij}\sum\limits_{\lambda\in\Lambda(n,d)}
[\overline{\lambda}_i]1_{\lambda}.\end{aligned}$$ We use the convention that $1_{\mu}X1_{\lambda}=0$, if $\mu$ or $\lambda$ is not contained in $\Lambda(n,d)$. Recall that $[a]$ is the $q$-integer $(q^a-q^{-a})/(q-q^{-1})$.
Although there is no embedding of $\U$ into $\Ugl$, the projection $$\psi_{n,d}\colon{\mathbf U}_q(\mathfrak{gl}_n)\to S_q(n,d)$$ can be restricted to ${\mathbf U}_q(\mathfrak{sl}_n)$ and is still surjective. This gives rise to the surjection $$\psi_{n,d}\colon \U\to \SD(n,d),$$ defined by $$\label{eq:psi}
\psi_{n,d}(E_{\pm i}1_{\lambda})=E_{\pm i}1_{\phi_{n,d}(\lambda)},$$ where $\phi_{n,d}$ was defined below equations and . By convention we put $1_{*}=0$.
As mentioned in the introduction, the $q$-Schur algebras for various values of $n$ and $d$ are related. Let $m\geq n$ and $d$ be arbitrary. There is an obvious embedding of the set of ${\mathbf U}_q(\mathfrak{gl}_n)$-weights into the set of ${\mathbf U}_q(\mathfrak{gl}_m)$-weights, given by $$(\lambda_1,\ldots,\lambda_n)\mapsto(\lambda_1,\ldots,\lambda_n,0,\ldots,0).$$ For fixed $d$, this gives an inclusion $\Lambda(n,d)\subseteq\Lambda(m,d)$, which we can use to define $$\xi_{n,m}=\sum_{\lambda\in\Lambda(n,d)}1_{\lambda}\in \SD(m,d).$$ Note that $\xi_{n,m}\ne 1$ unless $n=m$.
There is a well-defined homomorphism $${\iota}_{n,m}\colon \SD(n,d)\to\xi_{n,m}\SD(m,d)\xi_{n,m}$$ given by $$E_{\pm i}\mapsto\xi_{n,m}E_{\pm i}\xi_{n,m}\qquad\mbox{and}\qquad
1_{\lambda}\mapsto \xi_{n,m}1_{\lambda}\xi_{n,m}=1_{\lambda}.$$
It is easy to see that this is an isomorphism.
Suppose $d'=d+nk$, for a certain $k\in\bN$. Then we define a homomorphism $$\pi_{d',d}\colon \SD(n,d')\to \SD(n,d)$$ by $$1_{\lambda}\mapsto 1_{\lambda-(k^n)}\quad\mbox{and}\quad
E_{\pm i}\mapsto E_{\pm i}.$$
It is easy to check that $\pi_{d',d}$ is well-defined and surjective. It is also easy to see that $$\pi_{d',d}\psi_{n,d'}=\psi_{n,d}$$ and that $\pi_{d',d}$ induces a linear isomorphism $$V_{\lambda}\to V_{\lambda-(k^n)},$$ which intertwines the $\SD(n,d')$ and $\SD(n,d)$ actions, if $\lambda-(k^n)\in\Lambda^+(n,d)$. Of course $V_{\lambda}$ and $V_{\lambda-(k^n)}$ are isomorphic as ${\mathbf U}_q(\mathfrak{sl}_n)$ representations. Furthermore, note that for any $d=0,\ldots,n-1$ the set $$\label{eq:inversesystem}
\left(S_q(n,d+nk),\pi_{d+nk,d}\right)_{k\in\bN}$$ forms an inverse system, so we can form the inverse limit algebra $$\lim_{\longleftarrow k}S_q(n,d+nk).$$ The following lemma is perhaps a bit surprising.
\[lem:inverselimit\] The map $\sum_d\prod_k\psi_{n,d+nk}$, with $d=0,\ldots, n-1$ and $k\in\bN$, gives an embedding $${\mathbf U}_q(\mathfrak{sl}_n)\subset \bigoplus_{d=0}^{n-1}
\lim_{\longleftarrow k}S_q(n,d+nk).$$
We also have $$\label{eq:inverselimit}
\U\subset\bigoplus_{d=0}^{n-1}
\lim_{\longleftarrow k}S_q(n,d+nk).$$ The reader should remember this embedding when reading Corollary \[cor:inverselimit\]. The results in this paragraph were taken from [@B-L-M].
We need to recall two more facts about $q$-Schur algebras and their representations. The first is that the irreducibles $V_{\lambda}$, for $\lambda\in\Lambda^+(n,d)$, can be constructed as subquotients of $\SD(n,d)$, called Weyl modules. Let $<$ denote the lexicographic order on $\Lambda(n,d)$.
\[lem:weyl\] For any $\lambda\in\Lambda^+(n,d)$, we have $$V_{\lambda}\cong \SD(n,d)1_{\lambda}/[\mu>\lambda].$$ Here $[\mu>\lambda]$ is the ideal generated by all elements of the form $1_{\mu}x1_{\lambda}$, for some $x\in \SD(n,d)$ and $\mu>\lambda$.
Finally, we recall a well known anti-involution on $\SD(n,d)$, which we will need in this paper.
\[defn:tau\] We define an algebra anti-involution $$\tau\colon \SD(n,d)\to \SD(n,d)^{\mbox{\scriptsize op}}$$ by $$\tau(1_{\lambda})=1_{\lambda},
\quad \tau(1_{\lambda+\alpha_i}E_i1_{\lambda})=
q^{-1-\overline{\lambda}_i}1_{\lambda}E_{-i}1_{\lambda+\alpha_i},
\quad \tau(1_{\lambda}E_{-i}1_{\lambda+\alpha_i})= q^{1+\overline{\lambda}_i}
1_{\lambda+\alpha_i}E_i1_{\lambda}.$$
Note that up to a shift $t'$, we have $$\begin{aligned}
1_{\mu}E_{s_1}E_{s_2}\cdots E_{s_{m-1}}E_{s_m}1_{\lambda}q^t &\mapsto
1_{\lambda}E_{-s_m}E_{-s_{m-1}}\cdots E_{-s_2}E_{-s_1}1_{\mu}q^{-t+t'}.\end{aligned}$$ Our $\tau$ is the analogue of the one in [@K-L3].
The Hecke algebra
-----------------
Recall that $H_q(n)$ is a $q$-deformation of the group algebra of the symmetric group on $n$ letters.
The Hecke algebra $H_q(n)$ is the unital associative $\bQ(q)$-algebra generated by the elements $T_i$, $i=1,\ldots, n-1$, subject to the relations $$\begin{aligned}
T_i^2 &= (q^2-1)T_i+q^2
\\
T_iT_j &= T_jT_i\qquad\text{if}\quad|i-j|>1
\\
T_iT_{i+1}T_i &=T_{i+1}T_iT_{i+1}.\end{aligned}$$
Note that some people write $q$ where we write $q^2$ and use $v=q^{-1}$ in their presentation of the Hecke algebra. It is also not uncommon to find $t$ instead of our $q$.
For $q=1$ we recover the presentation of $\bQ[S_n]$ in terms of the simple transpositions $\sigma_i$. For any element $\sigma\in S_n$ we can define $T_{\sigma}=T_{i_1}\cdots T_{i_k}$, choosing a reduced expression $\sigma=\sigma_{i_1}\cdots \sigma_{i_k}$. The relations above guarantee that all reduced expressions of $\sigma$ give the same element $T_{\sigma}$. The $T_{\sigma}$, for $\sigma\in S_n$, form a linear basis of $H_q(n)$.
There is a simple change of generators, which is convenient for categorification purposes. Write $b_i=q^{-1}(T_i+1)$. Then the relations above become $$\begin{aligned}
b_i^2 &= (q+q^{-1})b_i
\\
b_ib_j &= b_jb_i \qquad\text{if}\quad|i-j|>1
\\
b_ib_{i+1}b_i+b_{i+1} &= b_{i+1}b_ib_{i+1}+b_i.\end{aligned}$$ These generators are the simplest elements of the so called [*Kazhdan-Lusztig basis*]{}. Although the change of generators is simple, the whole change of linear bases is very complicated.
As mentioned in the introduction, there is a $q$-version of Schur-Weyl duality. There is a $q$-permutation action of $H_q(d)$ on $V^{\otimes d}$, which is induced by the $R$-matrix of ${\mathbf U}_q(\mathfrak{gl}_n)$ or ${\mathbf U}_q(\mathfrak{sl}_n)$ and commutes with the actions of these quantum enveloping algebras. With respect to these actions, $H_q(d)$ and $\SD(n,d)$ have the double centralizer property. Furthermore, their respective categories of finite-dimensional representations are equivalent.
Suppose $n\geq d$. We explicitly recall the embedding of $H_q(d)$ into $\SD(n,d)$. Let $1_d=1_{(1^d)}$. Note that the ${\mathbf U}_q(\mathfrak{gl}_n)$-weight $(1^d)$ gives the zero ${\mathbf U}_q(\mathfrak{sl}_n)$-weight, for $n=d$, and a fundamental ${\mathbf U}_q(\mathfrak{sl}_n)$-weight for $n>d$. We define the following map $$\sigma_{n,d}\colon H_q(d)\to 1_{d}\SD(n,d)1_{d}$$ by $$\sigma_{n,d}(b_i)=1_dE_{-i}E_i1_d=1_dE_iE_{-i}1_d,$$ for $i=1,\ldots,d-1$. It is easy to check that $\sigma_{n,d}$ is well-defined. It turns out that $\sigma_{n,d}$ is actually an isomorphism, which induces the [*$q$-Schur functor*]{} $\SD(n,d)-\mbox{mod}\to H_q(d)-\mbox{mod}$, where mod denotes the category of finite-dimensional modules. This functor is an equivalence. Let us state explicitly an easy implication of this equivalence, which we need in the sequel.
\[lem:emb\] Let $0<d\leq n$ and let $A$ be a unital associative $\bQ(q)$-algebra. Suppose $\pi\colon \SD(n,d)\to A$ is a surjection of $\bQ(q)$-algebras, such that $\pi\circ \sigma_{n,d}\colon H_q(d)\to A$ is an embedding. Then $A\cong \SD(n,d)$.
Recall that $$\SD(n,d)\cong \prod_{\lambda\in\Lambda^+(n,d)}\mbox{End}_{\Q(q)}(V_{\lambda}).$$ The fact that the $q$-Schur functor is an equivalence means that the projection of $\sigma_{n,d}(H_q(d))$ onto $\mbox{End}_{\Q(q)}(V_{\lambda})$ is non-zero, for any $\lambda\in\Lambda^+(n,d)$. Since all $\mbox{End}_{\Q(q)}(V_{\lambda})$ are simple algebras, $A$ has to be isomorphic to the product $$\prod_{\lambda\in\Lambda'}\mbox{End}_{\Q(q)}(V_{\lambda}),$$ for a certain subset $\Lambda'\subseteq \Lambda^+(n,d)$. But $\pi\circ \sigma_{n,d}$ is an embedding, so $\Lambda'=\Lambda^+(n,d)$ has to hold.
The 2-categories $\glcat$ and $\Scat(n,d)$ {#sec:scat}
==========================================
In this section we define two $2$-categories, $\mathcal{U}(\mathfrak{gl}_n)$ and $\Scat(n,d)$, using a graphical calculus analogous to Khovanov and Lauda’s in [@K-L3]. We thank Khovanov and Lauda for letting us copy their definition of $\mathcal{U}_{\to}(\mathfrak{sl}_n)$. Taking their definition, we first introduce a change of weights to obtain $\mathcal{U}(\mathfrak{gl}_n)$. Then we divide by an ideal to obtain $\Scat(n,d)$.
As remarked in the introduction, our signs are slightly different from those in [@K-L3]. Khovanov and Lauda [@K-L:err] corrected their sign convention in $\mathcal{U}_{\to}(\mathfrak{sl}_n)$. As it turns out, the corrected $\mathcal{U}_{\to}(\mathfrak{sl}_n)$ is no longer cyclic, which makes working with that sign convention awkward. Fortunately Khovanov and Lauda’s non-signed version, $\Ucat$, is still correct and cyclic and is isomorphic to the corrected $\mathcal{U}_{\to}(\mathfrak{sl}_n)$ [@K-L3; @K-L:err]. However, the sign convention in $\Ucat$ is not so practical for the 2-representation into bimodules, so we have decided to stick to our own sign convention in this paper. To get from our signs back to Khovanov and Lauda’s (corrected) signs in $\mathcal{U}_{\to}(\mathfrak{sl}_n)$, apply the $2$-isomorphism which is the identity on all objects, $1$- and $2$-morphisms except the left cups and caps, on which it is given by $$\label{eq:signs}
\Ucapli_{i,\lambda}\mapsto (-1)^{\lambda_{i+1}+1}\,\,\Ucapli_{i,\lambda}
\quad\mbox{and}\quad \Ucupli_{i,\lambda}\mapsto (-1)^{\lambda_{i+1}}\,\,
\Ucupli_{i,\lambda}.$$
The various parts of our definition of $\mathcal{U}(\mathfrak{gl}_n)$ and $\Scat(n,d)$ below have exactly the same order as the corresponding parts of Khovanov and Lauda’s definition of $\mathcal{U}_{\to}(\mathfrak{sl}_n)$, so the reader can compare them in detail. From now on we will always write $\Ucat$, instead of $\mathcal{U}_{\to}(\mathfrak{sl}_n)$, for the corrected signed categorification of $\U$. Since we will never work with the unsigned version, there should be no confusion.
The 2-category $\glcat$
-----------------------
As already remarked in the introduction, the idea underlying the definition of $\mathcal{U}(\mathfrak{gl}_n)$ is very simple: it is obtained from $\Ucat$ by passing from $\mathfrak{sl}_n$-weights to $\mathfrak{gl}_n$-weights.
From now on let $n\in\N_{>1}$ be arbitrary but fixed and let $I=\{1,2,\ldots,n-1\}$. In the sequel we use [*signed sequences*]{} $\ii=(\alpha_1i_1,\ldots,\alpha_mi_m)$, for any $m\in\N$, $\alpha_j\in\{\pm 1\}$ and $i_j\in I$. The set of signed sequences we denote $\sseq$. For $\ii=(\alpha_1i_1,\ldots,\alpha_mi_m)\in\sseq$ we define $\ii_{\Lambda}:=\alpha_1 (i_1)_{\Lambda}+\cdots+\alpha_m (i_m)_{\Lambda}$, where $$(i_j)_{\Lambda}=(0,0,\ldots,1,-1,0\ldots,0),$$ such that the vector starts with $i_j-1$ and ends with $k-1-i_j$ zeros. To understand these definitions, the reader should recall our definition of $E_{\ii}$ and $\ii_{\Lambda}$ below Definition \[defn:Uglndot\]. We also define the symmetric $\Z$-valued bilinear form on $\Q[I]$ by $i\cdot i=2$, $i\cdot (i+1)=-1$ and $i\cdot j=0$, for $\vert i-j\vert>1$. Recall that $\overline{\lambda}_i=\lambda_i-\lambda_{i+1}$.
\[def\_glcat\] $\glcat$ is an additive $\Q$-linear 2-category. The 2-category $\glcat$ consists of
- objects: $\lambda\in\bZ^n$.
The hom-category $\glcat(\lambda,\lambda')$ between two objects $\lambda$, $\lambda'$ is an additive $\Q$-linear category consisting of:
- objects[^2] of $\glcat(\lambda,\lambda')$: a 1-morphism in $\glcat$ from $\lambda$ to $\lambda'$ is a formal finite direct sum of 1-morphisms $$\cal{E}_{\ii} \onel\{t\} = \onelp \cal{E}_{\ii} \onel\{t\}
:= \cal{E}_{\alpha_1 i_1}\dotsm\cal{E}_{\alpha_m i_m} \onel\{t\}$$ for any $t\in \Z$ and signed sequence $\ii\in\sseq$ such that $\lambda'=\lambda+\ii_{\Lambda}$ and $\lambda$, $\lambda'\in\bZ^n$.
- morphisms of $\glcat(\lambda,\lambda')$: for 1-morphisms $\cal{E}_{\ii} \onel\{t\}$ and $\cal{E}_{\jj} \onel\{t'\}$ in $\glcat$, the hom sets $\glcat(\cal{E}_{\ii} \onel\{t\},\cal{E}_{\jj} \onel\{t'\})$ of $\glcat(\lambda,\lambda')$ are graded $\Q$-vector spaces given by linear combinations of degree $t-t'$ diagrams, modulo certain relations, built from composites of:
i) Degree zero identity 2-morphisms $1_x$ for each 1-morphism $x$ in $\glcat$; the identity 2-morphisms $1_{\cal{E}_{+i} \onel}\{t\}$ and $1_{\cal{E}_{-i} \onel}\{t\}$, for $i \in I$, are represented graphically by $$\begin{array}{ccc}
1_{\cal{E}_{+i} \onel\{t\}} &\quad & 1_{\cal{E}_{-i} \onel\{t\}} \\ \\
\xy
(0,0)*{\dblue\xybox{(0,8);(0,-8); **\dir{-} ?(.5)*\dir{>}+(2.3,0)*{\scriptstyle{}};}};
(-1,-11)*{ i};(-1,11)*{ i};
(6,2)*{ \lambda};
(-8,2)*{ \lambda +i_{\Lambda}};
(-10,0)*{};(10,0)*{};
\endxy
& &
\;\;
\xy
(0,0)*{\dblue\xybox{(0,8);(0,-8); **\dir{-} ?(.5)*\dir{<}+(2.3,0)*{\scriptstyle{}};}};
(-1,-11)*{ i};(-1,11)*{ i};
(6,2)*{ \lambda};
(-8.5,2)*{ \lambda -i_{\Lambda}};
(-12,0)*{};(12,0)*{};
\endxy
\\ \\
\;\;\text{ {\rm deg} 0}\;\;
& &\;\;\text{ {\rm deg} 0}\;\;
\end{array}$$ for any $\lambda + i_{\Lambda} \in\bZ^n$ and any $\lambda - i_{\Lambda} \in \bZ^n$, respectively.
More generally, for a signed sequence $\ii=(\alpha_1i_1, \alpha_2i_2, \ldots
\alpha_mi_m)$, the identity $1_{\cal{E}_{\ii} \onel\{t\}}$ 2-morphism is represented as $$\begin{array}{ccc}
\xy
(-12,0)*{\dblue\xybox{(-12,8);(-12,-8); **\dir{-};}};
(-4,0)*{\dred\xybox{(-4,8);(-4,-8); **\dir{-};}};
(4,0)*{\cdots};
(12,0)*{\dgreen\xybox{(12,8);(12,-8); **\dir{-};}};
(-12,11)*{i_1}; (-4,11)*{ i_2};(12,11)*{ i_m };
(-12,-11)*{ i_1}; (-4,-11)*{ i_2};(12,-11)*{ i_m};
(18,2)*{ \lambda}; (-20,2)*{ \lambda+\ii_{\Lambda}};
\endxy
\end{array}$$ where the strand labeled $i_{k}$ is oriented up if $\alpha_{k}=+$ and oriented down if $\alpha_{k}=-$. We will often place labels with no sign on the side of a strand and omit the labels at the top and bottom. The signs can be recovered from the orientations on the strands.
ii) For each $\lambda \in \bZ^n$ the 2-morphisms $$\begin{tabular}{|l|c|c|c|c|}
\hline
{\bf Notation:} \xy (0,-5)*{};(0,7)*{}; \endxy&
$\dblue\Uup_{\black i,\lambda}$ & $\dblue\Udown_{\black i,\lambda}$
&$\Ucrossij_{i,j,\lambda}$
&$\Ucrossdij_{i,j,\lambda}$ \\
\hline
{\bf 2-morphism:} & \xy
(0,0)*{\dblue\xybox{(0,7);(0,-7); **\dir{-} ?(.75)*\dir{>}+(2.3,0)*{\scriptstyle{}}
?(.1)*\dir{ }+(2,0)*{\black \scs i};
(0,-2)*{\txt\large{$\bullet$}};}};
(4,4)*{ \lambda};
(-8,4)*{ \lambda +i_{\Lambda}};
(-10,0)*{};(10,0)*{};
\endxy
&
\xy
(0,0)*{\dblue\xybox{(0,7);(0,-7); **\dir{-} ?(.75)*\dir{<}+(2.3,0)*{\scriptstyle{}}
?(.1)*\dir{ }+(2,0)*{\black\scs i};
(0,-2)*{\txt\large{$\bullet$}};}};
(-6,4)*{ \lambda};
(8,3.9)*{ \lambda +i_{\Lambda}};
(-10,0)*{};(10,9)*{};
\endxy
&
\xy
(0,0)*{\xybox{
(0,0)*{\dblue\xybox{(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} ;}};
(0,0)*{\dgreen\xybox{(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};}};
(-5,-3)*{\scs i};
(5.1,-3)*{\scs j};
(8,1)*{ \lambda};
(-12,0)*{};(12,0)*{};
}};
\endxy
&
\xy
(0,0)*{\xybox{
(0,0)*{\dgreen\xybox{(-4,4)*{};(4,-4)*{} **\crv{(-4,1) & (4,-1)}?(1)*\dir{>} ;}};
(0,0)*{\dblue\xybox{(4,4)*{};(-4,-4)*{} **\crv{(4,1) & (-4,-1)}?(1)*\dir{>};}};
(-6,-3)*{\scs i};
(6,-3)*{\scs j};
(8,1)*{ \lambda};
(-12,0)*{};(12,0)*{};
}};
\endxy
\\ & & & &\\
\hline
{\bf Degree:} & \;\;\text{ $i\cdot i$ }\;\;
&\;\;\text{ $i\cdot i$}\;\;& \;\;\text{ $-i \cdot j$}\;\;
& \;\;\text{ $-i \cdot j$}\;\; \\
\hline
\end{tabular}$$
$$\begin{tabular}{|l|c|c|c|c|}
\hline
{\bf Notation:} \xy (0,-5)*{};(0,7)*{}; \endxy
& \text{$\Ucupri_{i,\lambda}$}
& \text{$\Ucupli_{i,\lambda}$}
& \text{$\Ucapli_{i,\lambda}$}
& \text{$\Ucapri_{i,\lambda}$} \\
\hline
{\bf 2-morphism:} & \xy
(0,-3)*{\dblue\bbpef{\black i}};
(8,-3)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy
& \xy
(0,-3)*{\dblue\bbpfe{\black i}};
(8,-3)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy
& \xy
(0,0)*{\dblue\bbcef{\black i}};
(8,3)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy
& \xy
(0,0)*{\dblue\bbcfe{\black i}};
(8,3)*{ \lambda};
(-12,0)*{};(12,0)*{};(8,8)*{};
\endxy\\& & & &\\ \hline
{\bf Degree:} & \;\;\text{ $1+\llambda_i$}\;\;
& \;\;\text{ $1-\llambda_i$}\;\;
& \;\;\text{ $1+\llambda_i$}\;\;
& \;\;\text{ $1-\llambda_i$}\;\;
\\
\hline
\end{tabular}$$
- Biadjointness and cyclicity:
i) \[it:sl2i\] $\mathbf{1}_{\lambda+i_{\Lambda}}\cal{E}_{+i}\onel$ and $\onel\cal{E}_{-i}\mathbf{1}_{\lambda+i_{\Lambda}}$ are biadjoint, up to grading shifts: $$\label{eq_biadjoint1}
\text{$
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{}="1";
(0,0)*{}="2";
(8,0)*{}="3";
(-8,-10);"1" **\dir{-};
"1";"2" **\crv{(-8,8) & (0,8)} ?(0)*\dir{>} ?(1)*\dir{>};
"2";"3" **\crv{(0,-8) & (8,-8)}?(1)*\dir{>};
"3"; (8,10) **\dir{-};}};
(12,-9)*{\lambda};
(-6,9)*{\lambda+i_{\Lambda}};
\endxy
\; =
\;
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{}="1";
(0,0)*{}="2";
(8,0)*{}="3";
(0,-10);(0,10)**\dir{-} ?(.5)*\dir{>};}};
(5,8)*{\lambda};
(-9,8)*{\lambda+i_{\Lambda}};
\endxy
\qquad \quad \xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{}="1";
(0,0)*{}="2";
(8,0)*{}="3";
(-8,-10);"1" **\dir{-};
"1";"2" **\crv{(-8,8) & (0,8)} ?(0)*\dir{<} ?(1)*\dir{<};
"2";"3" **\crv{(0,-8) & (8,-8)}?(1)*\dir{<};
"3"; (8,10) **\dir{-};}};
(12,-9)*{\lambda+i_{\Lambda}};
(-6,9)*{ \lambda};
\endxy
\; =
\;
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{}="1";
(0,0)*{}="2";
(8,0)*{}="3";
(0,-10);(0,10)**\dir{-} ?(.5)*\dir{<};}};
(9,8)*{\lambda+i_{\Lambda}};
(-6,8)*{ \lambda};
\endxy
$}$$
$$\label{eq_biadjoint2}
\text{$
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(8,0)*{}="1";
(0,0)*{}="2";
(-8,0)*{}="3";
(8,-10);"1" **\dir{-};
"1";"2" **\crv{(8,8) & (0,8)} ?(0)*\dir{>} ?(1)*\dir{>};
"2";"3" **\crv{(0,-8) & (-8,-8)}?(1)*\dir{>};
"3"; (-8,10) **\dir{-};}};
(12,9)*{\lambda};
(-5,-9)*{\lambda+i_{\Lambda}};
\endxy
\; =
\;
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(8,0)*{}="1";
(0,0)*{}="2";
(-8,0)*{}="3";
(0,-10);(0,10)**\dir{-} ?(.5)*\dir{>};}};
(5,-8)*{\lambda};
(-9,-8)*{\lambda+i_{\Lambda}};
\endxy
\qquad \quad \xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(8,0)*{}="1";
(0,0)*{}="2";
(-8,0)*{}="3";
(8,-10);"1" **\dir{-};
"1";"2" **\crv{(8,8) & (0,8)} ?(0)*\dir{<} ?(1)*\dir{<};
"2";"3" **\crv{(0,-8) & (-8,-8)}?(1)*\dir{<};
"3"; (-8,10) **\dir{-};}};
(12,9)*{\lambda+i_{\Lambda}};
(-6,-9)*{ \lambda};
\endxy
\; =
\;
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(8,0)*{}="1";
(0,0)*{}="2";
(-8,0)*{}="3";
(0,-10);(0,10)**\dir{-} ?(.5)*\dir{<};}};
(9,-8)*{\lambda+i_{\Lambda}};
(-6,-8)*{ \lambda};
\endxy
$}$$
ii) $$\label{eq_cyclic_dot}
\text{$
\xy
(0,0)*{\dblue\xybox{
(-8,5)*{}="1";
(0,5)*{}="2";
(0,-5)*{}="2'";
(8,-5)*{}="3";
(-8,-10);"1" **\dir{-};
"2";"2'" **\dir{-} ?(.5)*\dir{<};
"1";"2" **\crv{(-8,12) & (0,12)} ?(0)*\dir{<};
"2'";"3" **\crv{(0,-12) & (8,-12)}?(1)*\dir{<};
"3"; (8,10) **\dir{-};
(0,4)*{\txt\large{$\bullet$}};}};
(15,-9)*{ \lambda+i_{\Lambda}};
(-12,9)*{\lambda};
(10,8)*{\scs };
(-10,-8)*{\scs i};
\endxy
\quad = \quad
\xy
(0,0)*{\dblue\xybox{
(0,10);(0,-10); **\dir{-} ?(.75)*\dir{<}+(2.3,0)*{\scriptstyle{}}
?(.1)*\dir{ }+(2,0)*{\scs };
(0,0)*{\txt\large{$\bullet$}};}};
(-6,5)*{ \lambda};
(8,5)*{ \lambda +i_{\Lambda}};
(-10,0)*{};(10,0)*{};(-2,-8)*{\scs i};
\endxy
\quad = \quad
\xy
(0,0)*{\dblue\xybox{
(8,5)*{}="1";
(0,5)*{}="2";
(0,-5)*{}="2'";
(-8,-5)*{}="3";
(8,-10);"1" **\dir{-};
"2";"2'" **\dir{-} ?(.5)*\dir{<};
"1";"2" **\crv{(8,12) & (0,12)} ?(0)*\dir{<};
"2'";"3" **\crv{(0,-12) & (-8,-12)}?(1)*\dir{<};
"3"; (-8,10) **\dir{-};
(0,4)*{\txt\large{$\bullet$}};}};
(15,9)*{\lambda+i_{\Lambda}};
(-12,-9)*{\lambda};
(-10,8)*{\scs };
(10,-8)*{\scs i};
\endxy
$}$$
iii) All 2-morphisms are cyclic with respect to the above biadjoint structure.[^3] This is ensured by the relations , and the relations $$\label{eq_cyclic_cross-gen}
\text{$
\xy 0;/r.19pc/:
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(-4,-4)*{};(18,-4)*{} **\crv{(-4,-16) & (18,-16)} ?(1)*\dir{<}?(0)*\dir{<};
(18,-4);(18,12) **\dir{-};
(4,4)*{};(-18,4)*{} **\crv{(4,16) & (-18,16)} ?(1)*\dir{>};
(-18,4);(-18,-12) **\dir{-};
}};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)};
(12,-4);(12,12) **\dir{-};
(-12,4);(-12,-12) **\dir{-};
(4,-4)*{};(12,-4)*{} **\crv{(4,-10) & (12,-10)}?(1)*\dir{<}?(0)*\dir{<};
(-4,4)*{};(-12,4)*{} **\crv{(-4,10) & (-12,10)}?(1)*\dir{>}?(0)*\dir{>};
}};
(8,1)*{ \lambda};
(-10,0)*{};(10,0)*{};
(20,11)*{\scs j};(10,11)*{\scs i};
(-20,-11)*{\scs j};(-10,-11)*{\scs i};
% }};
\endxy
\quad = \quad \xy
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<};}};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};}};
(-5,3)*{\scs i};
(5.1,3)*{\scs j};
(-8,0)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy \quad := \quad
\xy 0;/r.19pc/:
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(18,4)*{} **\crv{(-4,16) & (18,16)} ?(1)*\dir{>};
(4,-4)*{};(-18,-4)*{} **\crv{(4,-16) & (-18,-16)} ?(1)*\dir{<}?(0)*\dir{<};
(18,4);(18,-12) **\dir{-};
(-18,-4);(-18,12) **\dir{-};
}};
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)};
(12,4);(12,-12) **\dir{-};
(-10,0)*{};(10,0)*{};
(-4,-4)*{};(-12,-4)*{} **\crv{(-4,-10) & (-12,-10)}?(1)*\dir{<}?(0)*\dir{<};
(4,4)*{};(12,4)*{} **\crv{(4,10) & (12,10)}?(1)*\dir{>}?(0)*\dir{>};
(-12,-4);(-12,12) **\dir{-};
}};
(8,1)*{ \lambda};
(-20,11)*{\scs i};(-10,11)*{\scs j};
(20,-11)*{\scs i};(10,-11)*{\scs j};
\endxy
$}$$ Note that we can take either the first or the last diagram above as the definition of the up-side-down crossing. We have chosen the last one above, because it is the one which matches Khovanov and Lauda’s signs. The cyclic condition on 2-morphisms expressed by and ensures that diagrams related by isotopy represent the same 2-morphism in $\glcat$.
It will be convenient to introduce degree zero 2-morphisms: $$\label{eq_crossl-gen}
\text{$
\xy
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};}};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};}};
(-5,-3)*{\scs j};
(-5,3)*{\scs i};
(8,2)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy
:=
\xy 0;/r.19pc/:
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4);(-4,12) **\dir{-};
(4,-4);(4,-12) **\dir{-};
}};
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)};
(-12,-4);(-12,12) **\dir{-};
(12,4);(12,-12) **\dir{-};
(-10,0)*{};(10,0)*{};
(-4,-4)*{};(-12,-4)*{} **\crv{(-4,-10) & (-12,-10)}?(1)*\dir{<}?(0)*\dir{<};
(4,4)*{};(12,4)*{} **\crv{(4,10) & (12,10)}?(1)*\dir{>}?(0)*\dir{>};
}};
(16,1)*{\lambda};
(-14,11)*{\scs i};(-2,11)*{\scs j};
(14,-11)*{\scs i};(2,-11)*{\scs j};
\endxy
\quad = \quad
\xy 0;/r.19pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{<};
(4,4);(4,12) **\dir{-};(-4,-4);(-4,-12) **\dir{-};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)};
(12,-4);(12,12) **\dir{-};
(-12,4);(-12,-12) **\dir{-};
(10,0)*{};(-10,0)*{};
(4,-4)*{};(12,-4)*{} **\crv{(4,-10) & (12,-10)}?(1)*\dir{>}?(0)*\dir{>};
(-4,4)*{};(-12,4)*{} **\crv{(-4,10) & (-12,10)}?(1)*\dir{<}?(0)*\dir{<};
}};
(16,1)*{\lambda};
(14,11)*{\scs j};(2,11)*{\scs i};
(-14,-11)*{\scs j};(-2,-11)*{\scs i};
\endxy
$}$$ $$\label{eq_crossr-gen}
\text{$
\xy
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<};}};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};}};
(5.1,-3)*{\scs i};
(5.1,3)*{\scs j};
(-8,2)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy
:=
\xy 0;/r.19pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,4);(4,12) **\dir{-};
(-4,-4);(-4,-12) **\dir{-};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)};
(12,-4);(12,12) **\dir{-};
(-12,4);(-12,-12) **\dir{-};
(10,0)*{};(-10,0)*{};
(4,-4)*{};(12,-4)*{} **\crv{(4,-10) & (12,-10)}?(1)*\dir{<}?(0)*\dir{<};
(-4,4)*{};(-12,4)*{} **\crv{(-4,10) & (-12,10)}?(1)*\dir{>}?(0)*\dir{>};
}};
(-16,1)*{\lambda};
(14,11)*{\scs j};(2,11)*{\scs i};
(-14,-11)*{\scs j};(-2,-11)*{\scs i};
\endxy
\quad = \quad
\xy 0;/r.19pc/:
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{<};
(-4,4);(-4,12) **\dir{-}; (4,-4);(4,-12) **\dir{-};
}};
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)};
(-12,-4);(-12,12) **\dir{-};
(12,4);(12,-12) **\dir{-};
(-10,0)*{};(10,0)*{};
(-4,-4)*{};(-12,-4)*{} **\crv{(-4,-10) & (-12,-10)}?(1)*\dir{>}?(0)*\dir{>};
(4,4)*{};(12,4)*{} **\crv{(4,10) & (12,10)}?(1)*\dir{<}?(0)*\dir{<};
}};
(-16,1)*{\lambda};
(-14,11)*{\scs i};(-2,11)*{\scs j};
(14,-11)*{\scs i};(2,-11)*{\scs j};
\endxy
$}$$ where the second equality in and follow from . Again we have indicated which choice of twists we use to define the sideways crossings, which is exactly the choice which matches Khovanov and Lauda’s sign conventions.
iv) All dotted bubbles of negative degree are zero. That is, $$\label{eq_positivity_bubbles}
\xy
(-12,0)*{\dblue\cbub{\black m}{\black i}};
(-8,8)*{\lambda};
\endxy
= 0
\qquad
\text{if $m<\llambda_i-1$} \qquad
\xy
(-12,0)*{\dblue\ccbub{\black m}{\black i}};
(-8,8)*{\lambda};
\endxy = 0\quad
\text{if $m< -\llambda_i-1$}$$ for all $m \in \Z_+$, where a dot carrying a label $m$ denotes the $m$-fold iterated vertical composite of $\Uup_{i,\lambda}$ or $\Udown_{i,\lambda}$ depending on the orientation. A dotted bubble of degree zero equals $\pm 1$: $$\label{eq:bubb_deg0}
\xy 0;/r.18pc/:
(0,-1)*{\dblue\cbub{\black\llambda_i-1}{\black i}};
(4,8)*{\lambda};
\endxy
= (-1)^{\laii} \quad \text{for $\llambda_i \geq 1$,}
\qquad \quad
\xy 0;/r.18pc/:
(0,-1)*{\dblue\ccbub{\black -\llambda_i-1}{\black i}};
(4,8)*{\lambda};
\endxy
= (-1)^{\laii-1} \quad \text{for $\llambda_i \leq -1$.}$$
v) For the following relations we employ the convention that all summations are increasing, so that a summation of the form $\sum_{f=0}^{m}$ is zero if $m < 0$. $$\begin{aligned}
\label{eq:redtobubbles}
\text{$\xy 0;/r.18pc/:
(10,8)*{\lambda};
%(0,0)*{\twoIu{i}};
(0,-3)*{\dblue\xybox{
(-3,-8)*{};(3,8)*{} **\crv{(-3,-1) & (3,1)}?(1)*\dir{>};?(0)*\dir{>};
(3,-8)*{};(-3,8)*{} **\crv{(3,-1) & (-3,1)}?(1)*\dir{>};
(-3,-12)*{\bbsid};
(-3,8)*{\bbsid};
(3,8)*{}="t1";
(9,8)*{}="t2";
(3,-8)*{}="t1'";
(9,-8)*{}="t2'";
"t1";"t2" **\crv{(3,14) & (9, 14)};
"t1'";"t2'" **\crv{(3,-14) & (9, -14)};
"t2'";"t2" **\dir{-} ?(.5)*\dir{<};}};
(9,0)*{}; (-7.5,-12)*{\scs i};
\endxy$} \; = \; -\sum_{f=0}^{-\llambda_i}
\xy
(19,4)*{\lambda};
(0,0)*{\dblue\bbe{}};(-2,-8)*{\scs i};
(12,-2)*{\dblue\cbub{\black\llambda_i-1+f}{\black i}};
(0,6)*{\dblue\bullet}+(6,1)*{\scs -\llambda_i-f};
\endxy
\qquad \quad
\text{$ \xy 0;/r.18pc/:
(-12,8)*{\lambda};
(0,-2)*{\dblue\xybox{
(-3,-8)*{};(3,8)*{} **\crv{(-3,-1) & (3,1)}?(1)*\dir{>};?(0)*\dir{>};
(3,-8)*{};(-3,8)*{} **\crv{(3,-1) & (-3,1)}?(1)*\dir{>};
(3,-12)*{\bbsid};
(3,8)*{\bbsid}; %(6,-8)*{\scs i};
(-9,8)*{}="t1";
(-3,8)*{}="t2";
(-9,-8)*{}="t1'";
(-3,-8)*{}="t2'";
"t1";"t2" **\crv{(-9,14) & (-3, 14)};
"t1'";"t2'" **\crv{(-9,-14) & (-3, -14)};
"t1'";"t1" **\dir{-} ?(.5)*\dir{<};}};(7.5,-11)*{\scs i};
\endxy$} \; = \;
\sum_{g=0}^{\llambda_i}
\xy
(-12,8)*{\lambda};
(0,0)*{\dblue\bbe{}};(2,-8)*{\scs i};
(-12,-2)*{\dblue\ccbub{\black -\llambda_i-1+g}{\black i}};
(0,6)*{\dblue\bullet}+(8,-1)*{\scs \llambda_i-g};
\endxy\end{aligned}$$ $$\begin{aligned}
\label{eq:EF}
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{};
(8,0)*{};
(-4,10)*{}="t1";
(4,10)*{}="t2";
(-4,-10)*{}="b1";
(4,-10)*{}="b2";
"t1";"b1" **\dir{-} ?(.5)*\dir{<};
"t2";"b2" **\dir{-} ?(.5)*\dir{>};}};
(-6,-8)*{\scs i};
(6,-8)*{\scs i};
(10,2)*{\lambda};
(-10,2)*{\lambda};
\endxy}
&\quad = \quad&
\vcenter{ \xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{<};?(0)*\dir{<};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)}?(1)*\dir{>};}};
(8,8)*{\lambda};(-6,-7)*{\scs i};(6.8,-7)*{\scs i};
\endxy}
\quad - \quad
\sum_{f=0}^{\llambda_i-1} \sum_{g=0}^{f}
\vcenter{\xy 0;/r.18pc/:
(-10,10)*{\lambda};
(0,0)*{\dblue\xybox{
(-8,0)*{}; (8,0)*{};
(-4,-15)*{}="b1";
(4,-15)*{}="b2";
"b2";"b1" **\crv{(5,-8) & (-5,-8)}; ?(.05)*\dir{<} ?(.93)*\dir{<}
?(.8)*\dir{}+(0,-.1)*{\bullet}+(-5,2)*{\black\scs f-g};
(-4,15)*{}="t1";
(4,15)*{}="t2";
"t2";"t1" **\crv{(5,8) & (-5,8)}; ?(.15)*\dir{>} ?(.95)*\dir{>}
?(.4)*\dir{}+(0,-.2)*{\bullet}+(3,-2)*{\black\scs \mspace{38mu}\;\;\; \llambda_i-1-f};
(0,0)*{\ccbub{\black\scs \quad\;\;\;-\llambda_i-1+g}{i}};}};
\endxy}
\label{eq_ident_decomp0}
%\nn
% \\ \; \nn \\
\\[2ex]
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{};(8,0)*{};
(-4,10)*{}="t1";
(4,10)*{}="t2";
(-4,-10)*{}="b1";
(4,-10)*{}="b2";
"t1";"b1" **\dir{-} ?(.5)*\dir{>};
"t2";"b2" **\dir{-} ?(.5)*\dir{<};}};
(-6,-8)*{\scs i};(6,-8)*{\scs i};
(10,2)*{\lambda};
(-10,2)*{\lambda};
\endxy}
&\quad = \quad&
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{<};?(0)*\dir{<};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)}?(1)*\dir{>};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)};}};
(8,8)*{\lambda};(-6.8,-7)*{\scs i};(6,-7)*{\scs i};
\endxy}
\quad - \quad
\sum_{f=0}^{-\llambda_i-1} \sum_{g=0}^{f}
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{}; (8,0)*{};
(-4,-15)*{}="b1";
(4,-15)*{}="b2";
"b2";"b1" **\crv{(5,-8) & (-5,-8)}; ?(.1)*\dir{>} ?(.95)*\dir{>}
?(.8)*\dir{}+(0,-.1)*{\bullet}+(-5,2)*{\black\scs f-g};
(-4,15)*{}="t1";
(4,15)*{}="t2";
"t2";"t1" **\crv{(5,8) & (-5,8)}; ?(.15)*\dir{<} ?(.97)*\dir{<}
?(.4)*\dir{}+(0,-.2)*{\bullet}+(3,-2)*{\black\scs \mspace{32mu}\;\;-\llambda_i-1-f};
(0,0)*{\cbub{\black\scs \quad\; \llambda_i-1+g}{i}};}};
(-10,10)*{\lambda};
\endxy} \label{eq_ident_decomp}\end{aligned}$$ for all $\lambda\in \bZ^n$ (see and for the definition of sideways crossings). Notice that for some values of $\lambda$ the dotted bubbles appearing above have negative labels. A composite of $\dblue\Uup_{\black\!\! i,\lambda}$ or $\dblue\Udown_{\black i,\lambda}$ with itself a negative number of times does not make sense. These dotted bubbles with negative labels, called [*fake bubbles*]{}, are formal symbols inductively defined by the equation $$\makebox[0pt]{ $
\left(\ \xy 0;/r.15pc/:
(0,0)*{\dblue\xybox{
(0,0)*{\ccbub{\black\mspace{-32mu}-\llambda_i-1}{\black i}};}};
(4,8)*{\lambda};
\endxy
+
\xy 0;/r.15pc/:
(0,0)*{\dblue\xybox{
(0,0)*{\ccbub{\black\mspace{-12mu}-\llambda_i-1+1}{\black i}};}};
(4,8)*{\lambda};
\endxy t
+ \cdots +
\xy 0;/r.15pc/:
(0,0)*{\dblue\xybox{
(0,0)*{\ccbub{\black\mspace{-12mu}-\llambda_i-1+r}{\black i}};}};
(4,8)*{\lambda};
\endxy t^{r}
+ \cdots
\right)
%%%%
%
%%%%
\left( \xy 0;/r.15pc/:
(0,0)*{\dblue\xybox{
(0,0)*{\cbub{\black\mspace{-22mu}\llambda_i-1}{\black i}};}};
(4,8)*{\lambda};
\endxy
+ \cdots +
\xy 0;/r.15pc/:
(0,0)*{\dblue\xybox{
(0,0)*{\cbub{\black\mspace{-8mu}\llambda_i-1+r}{\black i}};}};
(4,8)*{\lambda};
\endxy t^{r}
+ \cdots
\right) =-1 $ } %\nn \\
\label{eq_infinite_Grass}$$ and the additional condition $$\xy 0;/r.18pc/:
(0,0)*{\dblue\cbub{\black -1}{\black i}};
(4,8)*{\lambda};
\endxy
\quad = (-1)^{\laii},
\qquad
\xy 0;/r.18pc/:
(0,0)*{\dblue\ccbub{\black -1}{\black i}};
(4,8)*{\lambda};
\endxy
\quad = (-1)^{\laii-1} \qquad \text{if $\llambda_i =0$.}$$ Although the labels are negative for fake bubbles, one can check that the overall degree of each fake bubble is still positive, so that these fake bubbles do not violate the positivity of dotted bubble axiom. The above equation, called the infinite Grassmannian relation, remains valid even in high degree when most of the bubbles involved are not fake bubbles. See [@L1] for more details.
vi) NilHecke relations: $$\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)}?(1)*\dir{>};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)}?(1)*\dir{>};}};
(8,8)*{\lambda};(-5,-7)*{\scs i};(5.1,-7)*{\scs i};
\endxy}
=0, \qquad \quad
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(4,4)*{};(12,12)*{} **\crv{(4,7) & (12,9)}?(1)*\dir{>};
(12,4)*{};(4,12)*{} **\crv{(12,7) & (4,9)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(-4,4)*{}; (-4,12) **\dir{-};
(12,-4)*{}; (12,4) **\dir{-};
(12,12)*{}; (12,20) **\dir{-};}};
(-9.5,-11)*{\scs i};(1.5,-11)*{\scs i};(9.5,-11)*{\scs i};
(12,0)*{\lambda};
\endxy}
\;\; =\;\;
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(-4,4)*{};(-12,12)*{} **\crv{(-4,7) & (-12,9)}?(1)*\dir{>};
(-12,4)*{};(-4,12)*{} **\crv{(-12,7) & (-4,9)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(4,4)*{}; (4,12) **\dir{-};
(-12,-4)*{}; (-12,4) **\dir{-};
(-12,12)*{}; (-12,20) **\dir{-};}};
(-1.5,-11)*{\scs i};(9.5,-11)*{\scs i};(-9.5,-11)*{\scs i};
(12,0)*{\lambda};
\endxy} \label{eq_nil_rels}$$ $$\begin{aligned}
\xy
(0,1)*{\dblue\xybox{
(4,4);(4,-4) **\dir{-}?(0)*\dir{<}+(2.3,0)*{};
(-4,4);(-4,-4) **\dir{-}?(0)*\dir{<}+(2.3,0)*{};}};
(6,2)*{\lambda};(-6.2,-2)*{\scs i}; (4,-2)*{\scs i};
\endxy
\quad =
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>}?(.25)*{\bullet};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-5,-3)*{\black\scs i};
(5.1,-3)*{\black\scs i};
(8,1)*{\black\lambda};
(-10,0)*{};(10,0)*{};
}};
\endxy
\;\; -
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>}?(.75)*{\bullet};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-5,-3)*{\black\scs i};
(5.1,-3)*{\black\scs i};
(8,1)*{\black\lambda};
(-10,0)*{};(10,0)*{};
}};
\endxy
\;\; =
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>}?(.75)*{\bullet};
(-5,-3)*{\black\scs i};
(5.1,-3)*{\black\scs i};
(8,1)*{\black\lambda};
(-10,0)*{};(10,0)*{};
}};
\endxy
\;\; -
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} ;
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>}?(.25)*{\bullet};
(-5,-3)*{\black\scs i};
(5.1,-3)*{\black\scs i};
(8,1)*{\black\lambda};
(-10,0)*{};(10,0)*{};
}};
\endxy %\nn %\\
\label{eq_nil_dotslide}\end{aligned}$$ We will also include for $i =j$ as an $\mf{sl}_2$-relation.
- For $i \neq j$ $$\label{eq_downup_ij-gen}
\text{$
\vcenter{ \xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)}?(1)*\dir{>};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{<};?(0)*\dir{<};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)};
}};
(7,4)*{\lambda};(-6,-7)*{\scs i};(6.5,-7)*{\scs j};
\endxy}
\;\;= \;\;
\xy 0;/r.18pc/:
(3.5,0)*{\dred\xybox{
(3,9);(3,-9) **\dir{-}?(.55)*\dir{>}+(2.3,0)*{};}};
(-3.5,0)*{\dblue\xybox{
(-3,9);(-3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{};}};
(7,2)*{\lambda};(-6,-6)*{\scs i};(3.8,-6)*{\scs j};
\endxy
\qquad
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{<};?(0)*\dir{<};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)}?(1)*\dir{>};
}};
(7,4)*{\lambda};(-6.5,-7)*{\scs i};(6,-7)*{\scs j};
\endxy}
\;\;=\;\;
\xy 0;/r.18pc/:
(3.5,0)*{\dred\xybox{
(3,9);(3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{};
}};
(-3.5,0)*{\dblue\xybox{
(-3,9);(-3,-9) **\dir{-}?(.55)*\dir{>}+(2.3,0)*{};
}};
(7,2)*{\lambda};(-6.5,-6)*{\scs i};(3.6,-6)*{\scs j};
\endxy
$}$$
- The analogue of the $R(\nu)$-relations:
i) For $i \neq j$ $$\begin{aligned}
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)}?(1)*\dir{>};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)}?(1)*\dir{>};
}};
(8,8)*{\lambda};(-5,-6)*{\scs i};
(5.3,-6)*{\scs j};
\endxy}
\qquad = \qquad
\left\{
\begin{array}{ccc}
\xy 0;/r.18pc/:
(4,0)*{\dred\xybox{
(3,9);(3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{};
}};
(-2.5,0)*{\dblue\xybox{
(-3,9);(-3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{};
}};
(8,2)*{\lambda};(-5,-6)*{\scs i};(5.1,-6)*{\scs j};
\endxy & & \text{if $i \cdot j=0$,}\\ \\
(i-j)\left(
\vcenter{\xy 0;/r.18pc/:
(4.5,0)*{\dred\xybox{
(3,9);(3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{};
}};
(-2.5,0)*{\dblue\xybox{
(-3,9);(-3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{};
(-3,4)*{\bullet};
}};
(8,2)*{\black\lambda};
(-5,-6)*{\bscs i}; (5.1,-6)*{\bscs j};
\endxy} \quad
- \quad
\vcenter{\xy 0;/r.18pc/:
(4,0)*{\dred\xybox{
(3,9);(3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{}; (3,4)*{\bullet};
}};
(-2,0)*{\dblue\xybox{
(-3,9);(-3,-9) **\dir{-}?(.5)*\dir{<}+(2.3,0)*{};
}};
(9,2)*{\black\lambda};
(-5,-6)*{\bscs i}; (5.1,-6)*{\bscs j};
\endxy}\right)
& & \text{if $i \cdot j =-1$.}
\end{array}
\right. %\nn \\
\label{eq_r2_ij-gen}\end{aligned}$$ Notice that $(i-j)$ is just a sign, which takes into account the standard orientation of the Dynkin diagram.
$$\begin{aligned}
\label{eq_dot_slide_ij-gen}
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>}?(.75)*{\bullet};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-5,-3)*{\scs i};
(5.1,-3)*{\scs j};
(8,1)*{ \lambda};
(-10,0)*{};(10,0)*{};
\endxy
\;\; =
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>}?(.25)*{\bullet};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-5,-3)*{\scs i};
(5.1,-3)*{\scs j};
(8,1)*{ \lambda};
(-10,0)*{};(10,0)*{};
\endxy
\qquad \xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>}?(.75)*{\bullet};
}};
(-5,-3)*{\scs i};
(5.1,-3)*{\scs j};
(8,1)*{ \lambda};
(-10,0)*{};(10,0)*{};
\endxy
\;\; =
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} ;
}};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>}?(.25)*{\bullet};
}};
(-5,-3)*{\scs i};
(5.1,-3)*{\scs j};
(8,1)*{ \lambda};
(-10,0)*{};(12,0)*{};
\endxy\end{aligned}$$
ii) Unless $i = k$ and $i \cdot j=-1$ $$\text{$
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,4)*{};(12,12)*{} **\crv{(4,7) & (12,9)}?(1)*\dir{>};
(12,12)*{}; (12,20) **\dir{-};
}};
(-4,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(-4,4)*{}; (-4,12) **\dir{-};
}};
(0,0)*{\dgreen\xybox{
(12,4)*{};(4,12)*{} **\crv{(12,7) & (4,9)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(12,-4)*{}; (12,4) **\dir{-};
}};
(12,0)*{\lambda};
(-10,-11)*{\scs i};
( 2,-11)*{\scs j};
(10.5,-11)*{\scs k};
\endxy}
\;\; =\;\;
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dgreen\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(-12,12)*{} **\crv{(-4,7) & (-12,9)}?(1)*\dir{>};
(-12,12)*{}; (-12,20) **\dir{-};
}};
(4,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(4,4)*{}; (4,12) **\dir{-};
}};
(0,0)*{\dblue\xybox{
(-12,4)*{};(-4,12)*{} **\crv{(-12,7) & (-4,9)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(-12,-4)*{}; (-12,4) **\dir{-};
}};
(12,0)*{\lambda};
(10,-11)*{\scs k};
(-1.5,-11)*{\scs j};
(-9.5,-11)*{\scs i};
\endxy} \label{eq_r3_easy-gen}
$}$$
For $i \cdot j =-1$ $$\text{$
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,4)*{};(12,12)*{} **\crv{(4,7) & (12,9)}?(1)*\dir{>};
(12,4)*{};(4,12)*{} **\crv{(12,7) & (4,9)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(12,-4)*{}; (12,4) **\dir{-};
(12,12)*{}; (12,20) **\dir{-};
}};
(-4,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(-4,4)*{}; (-4,12) **\dir{-};
}};
(12,0)*{\lambda};
(-10,-11)*{\scs i};
(1.5,-11)*{\scs j};
(9.5,-11)*{\scs i};
\endxy}
\quad - \quad
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(-12,12)*{} **\crv{(-4,7) & (-12,9)}?(1)*\dir{>};
(-12,4)*{};(-4,12)*{} **\crv{(-12,7) & (-4,9)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(-12,-4)*{}; (-12,4) **\dir{-};
(-12,12)*{}; (-12,20) **\dir{-};
}};
(4,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(4,4)*{}; (4,12) **\dir{-};
}};
(12,0)*{\lambda};
(10,-11)*{\scs i};
(-1.5,-11)*{\scs j};
(-9.5,-11)*{\scs i};
\endxy}
\;\; =\;\;\;\; (i-j)
\xy 0;/r.18pc/:
(10,0)*{\dblue\xybox{
(4,12);(4,-12) **\dir{-}?(.5)*\dir{<};
(22,12);(22,-12) **\dir{-}?(.5)*\dir{<}?(.25)*\dir{}+(0,0)*{}+(10,0)*{\scs};
}};
(3.5,0)*{\dred\xybox{
(-4,12);(-4,-12) **\dir{-}?(.5)*\dir{<}?(.25)*\dir{}+(0,0)*{}+(-3,0)*{\scs };
}};
(20,0)*{\lambda};
(-5.6,-11)*{\scs i};
(3.1,-11)*{\scs j};
(15.2,-11)*{\scs i};
\endxy
\label{eq_r3_hard-gen}
$}.$$
- The additive $\Z$-linear composition functor $\glcat(\lambda,\lambda')
\times \glcat(\lambda',\lambda'') \to \glcat(\lambda,\lambda'')$ is given on 1-morphisms of $\glcat$ by $$\cal{E}_{\jj}\mathbf{1}_{\lambda'}\{t'\} \times \cal{E}_{\ii}\onel\{t\} \mapsto
\cal{E}_{\jj\ii}\onel\{t+t'\}$$ for $\ii_{\Lambda}=\lambda-\lambda'$, and on 2-morphisms of $\glcat$ by juxtaposition of diagrams $$\text{$
\left(\figleft{\lambda'}{\lambda''}\right)
\;\; \times \;\;
\left(\figright{\lambda'}{\lambda}\right)
\;\;\mapsto \;\
\figleft{}{\lambda''}
\figright{}{\lambda}
$} .$$
This concludes the definition of $\glcat$. In the next subsection we will show some further relations, which are easy consequences of the ones above.
### Further relations in $\glcat$
The following $\glcat$-relations follow from the relations in Definition \[def\_glcat\] and are going to be used in the sequel.
*Bubble slides*: $$\label{eq:bub_slides}
\text{$
\xy
(14,8)*{\lambda};
(0,0)*{\dgreen\bbe{}};
(0,-12)*{\scs j};
(12,-2)*{\dblue\ccbub{\black -\llambda_i-1+m}{\black i}};
(0,6)*{ }+(7,-1)*{\scs };
\endxy
$}
\quad = \quad
\begin{cases}
\ \xsum{f=0}{m}(f-m-1)
\xy
(0,8)*{\lambda+j_{\Lambda}};
(12,0)*{\dgreen\bbe{}};
(12,-12)*{\scs j};
(0,-2)*{\dblue\ccbub{\black - \overline{(\lambda + j_\Lambda)}_i -1 + f}{\black i}};
(12,6)*{\dgreen\bullet}+(5,-1)*{\scs m-f};
\endxy
& \text{if $i=j$}
\\ \\
\ \qquad \qquad \xy
(0,8)*{\lambda+j_{\Lambda}};
(12,0)*{\dgreen\bbe{}};
(12,-12)*{\scs j};
(0,-2)*{\dblue\ccbub{\black -\overline{(\lambda + j_\Lambda)_i} -1+m }{\black i}};
\endxy & \text{if $i \cdot j=0$}
\end{cases}$$ $$\begin{aligned}
\label{eq:2ndbubbslide}
\text{$
\xy
(14,8)*{\lambda};
(0,0)*{\dred\bbe{}};
(0,-12)*{\scs i+1};
(12,-2)*{\dblue\ccbub{\black -\llambda_i-1+m}{\black i}};
(0,6)*{ }+(7,-1)*{\scs };
\endxy
$}
&= \quad
\xy
(-4,8)*{\lambda+(i+1)_{\Lambda}};
(12,0)*{\dred\bbe{}};
(12,-12)*{\scs i+1};
(-6,-2)*{\dblue\ccbub{\black -(\overline{\lambda +(i+1)_{\Lambda}})_i-2+m}{\black i}};
(12,6)*{\dred\bullet}+(5,-1)*{\scs };
\endxy
\quad - \quad
\xy
(-4,8)*{\lambda+(i+1)_{\Lambda}};
(12,0)*{\red\bbe{}};
(11,-12)*{\scs i+1};
(-6,-2)*{\dblue\ccbub{\black -(\overline{\lambda+(i+1)_{\Lambda}})_i -1+m}{\black i}};
\endxy
%\\
%\intertext{\textcolor{red}{NEED NOTATION HERE:
%$(\overline{\lambda+(i+1)_{\Lambda}})_i $
%IN TERMS OF $\overline{\lambda}_i$! MAYBE $(\mu+\overline{(i+1)}_{\Lambda})_i$}}\nn
\\ \displaybreak[0]
\text{$
\xy
(6,8)*{\lambda};
(12,0)*{\dred\bbe{}};
(12,-12)*{\scs i+1};
(0,-2)*{\dblue\ccbub{\black -\llambda_i-1+m}{\black i}};
(-12,6)*{ }+(7,-1)*{\scs };
\endxy
$}
&=
-\sum\limits_{f+g=m}
\xy
(18,8)*{\lambda-(i+1)_{\Lambda}};
(0,0)*{\dred\bbe{}};
(0,-12)*{\scs i+1};
(16,-2)*{\dblue\ccbub{\black -(\overline{\lambda -(i+1)_{\Lambda}})_i-2+g}{\black i}};
(0,6)*{\dred\bullet}+(-3,-1)*{\scs f};
\endxy
\\\nn\\
\label{eq:extrabubble4}
\displaybreak[0]
\text{$
\xy
(14,8)*{\lambda};
(0,0)*{\dred\bbe{}};
(0,-12)*{\scs i+1};
(12,-2)*{\dblue\cbub{\black \llambda_i-1+m}{\black i}};
(0,6)*{ }+(7,-1)*{\scs };
\endxy
$}
&= -\sum\limits_{f+g=m}\ \
\xy
(-4,8)*{\lambda+(i+1)_{\Lambda}};
(12,0)*{\dred\bbe{}};
(12,-12)*{\scs i+1};
(-6,-2)*{\dblue\ccbub{\black (\overline{\lambda +(i+1)_{\Lambda}})_i-1+g}{\black i}};
(12,6)*{\dred\bullet}+(2,-1)*{\scs f};
\endxy
\\\nn \\ \displaybreak[0]
\text{$
\xy
(6,8)*{\lambda};
(12,0)*{\dred\bbe{}};
(12,-12)*{\scs i+1};
(0,-2)*{\dblue\cbub{\black \llambda_i-1+m}{\black i}};
(-12,6)*{ }+(7,-1)*{\scs };
\endxy
$}
&=
\quad
\xy
(18,8)*{\lambda-(i+1)_{\Lambda}};
(0,0)*{\dred\bbe{}};
(0,-12)*{\scs i+1};
(16,-2)*{\dblue\cbub{\black (\overline{\lambda -(i+1)_{\Lambda}})_i-2+m}{\black i}};
(0,6)*{\dred\bullet}+(-3,-1)*{\scs };
\endxy
\quad-\quad
\xy
(18,8)*{\lambda-(i+1)_{\Lambda}};
(0,0)*{\dred\bbe{}};
(0,-12)*{\scs i+1};
(16,-2)*{\dblue\cbub{\black (\overline{\lambda -(i+1)_{\Lambda}})_i-1+m}{\black i}};
\endxy\end{aligned}$$ If we switch labels $i$ and $i+1$, then the r.h.s. of the above equations gets a minus sign. Bubble slides with the vertical strand oriented downwards can easily be obtained from the ones above by rotating the diagrams 180 degrees.
*More Reidemeister 3 like relations*. Unless $i=k=j$ we have $$\label{eq_other_r3_1}
\text{$
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,4)*{};(12,12)*{} **\crv{(4,7) & (12,9)}?(1)*\dir{>};
(12,12)*{}; (12,20) **\dir{-};
}};
(-4,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(0)*\dir{<};
(-4,4)*{}; (-4,12) **\dir{-};
}};
(0,0)*{\dgreen\xybox{
(12,4)*{};(4,12)*{} **\crv{(12,7) & (4,9)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(12,-4)*{}; (12,4) **\dir{-};
}};
(12,0)*{\lambda};
( -10,-11)*{\scs i};
( 2.5,-11)*{\scs j};
(10.5,-11)*{\scs k};
\endxy}
\;\; =\;\;
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dgreen\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,4)*{};(-12,12)*{} **\crv{(-4,7) & (-12,9)}?(1)*\dir{>};
(-12,12)*{}; (-12,20) **\dir{-};
}};
(4,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(0)*\dir{<};
(4,4)*{}; (4,12) **\dir{-};
}};
(0,0)*{\dblue\xybox{
(-12,4)*{};(-4,12)*{} **\crv{(-12,7) & (-4,9)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(-12,-4)*{}; (-12,4) **\dir{-};
}};
(12,0)*{\lambda};
(10,-11)*{\scs k};
(-2.5,-11)*{\scs j};
(-9.5,-11)*{\scs i};
\endxy}
$}$$ and when $i=j=k$ we have $$\label{eq_r3_extra}
\text{$
\vcenter{
\xy 0;/r.17pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{<};
?(0)*\dir{<};
(4,4)*{};(12,12)*{} **\crv{(4,7) & (12,9)}?(1)*\dir{>};
(12,4)*{};(4,12)*{} **\crv{(12,7) & (4,9)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(-4,4)*{}; (-4,12) **\dir{-};
(12,-4)*{}; (12,4) **\dir{-};
(12,12)*{}; (12,20) **\dir{-};
}};
(12,0)*{\lambda};
(-10,-11)*{\scs i};
( 2.5,-11)*{\scs i};
( 9.5,-11)*{\scs i};
\endxy}
-\;
\vcenter{
\xy 0;/r.17pc/:
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<};
(-4,4)*{};(-12,12)*{} **\crv{(-4,7) & (-12,9)}?(1)*\dir{>};
(-12,4)*{};(-4,12)*{} **\crv{(-12,7) & (-4,9)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(4,4)*{}; (4,12) **\dir{-} ?(.5)*\dir{<};
(-12,-4)*{}; (-12,4) **\dir{-};
(-12,12)*{}; (-12,20) **\dir{-};
}};
(12,0)*{\lambda};
(10 ,-11)*{\scs i};
(-2.5,-11)*{\scs i};
(-9.5,-11)*{\scs i};
\endxy}
%%
\; = \;
\sum_{} \; \xy 0;/r.17pc/:
(0,0)*{\dblue\xybox{
(-4,12)*{}="t1";
(4,12)*{}="t2";
"t2";"t1" **\crv{(5,5) & (-5,5)}; ?(.15)*\dir{} ?(.9)*\dir{>}
?(.2)*\dir{}+(0,-.2)*{\bullet}+(3,-2)*{\bscs f_1};
(-4,-12)*{}="t1";
(4,-12)*{}="t2";
"t2";"t1" **\crv{(5,-5) & (-5,-5)}; ?(.05)*\dir{} ?(.9)*\dir{<}
?(.15)*\dir{}+(0,-.2)*{\bullet}+(3,2)*{\bscs f_3};
(-8.5,0.5)*{\ccbub{\bscs -\llambda_i-3+f_4}{\black i}};
(13,12)*{};(13,-12)*{} **\dir{-} ?(.5)*\dir{<};
(13,8)*{\bullet}+(3,2)*{\bscs f_2};
}};
(18,-6)*{\lambda};
\endxy
+\;
\sum_{}
\; \xy 0;/r.17pc/:
(0,0)*{\dblue\xybox{
(-10,12)*{};(-10,-12)*{} **\dir{-} ?(.5)*\dir{<};
(-10,8)*{\bullet}+(-3,2)*{\bscs g_2};
(-4,12)*{}="t1";
(4,12)*{}="t2";
"t1";"t2" **\crv{(-4,5) & (4,5)}; ?(.15)*\dir{>} ?(.9)*\dir{>}
?(.4)*\dir{}+(0,-.2)*{\bullet}+(3,-2)*{\bscs \;\; g_1};
(-4,-12)*{}="t1";
(4,-12)*{}="t2";
"t2";"t1" **\crv{(4,-5) & (-4,-5)}; ?(.12)*\dir{>} ?(.97)*\dir{>}
?(.8)*\dir{}+(0,-.2)*{\bullet}+(1,4)*{\bscs g_3};
(16.6,-4.5)*{\cbub{\bscs \llambda_i-1+g_4}{\black i}};
}};
(18,6)*{\lambda};
\endxy
$}$$ where the first sum is over all $f_1, f_2, f_3, f_4 \geq 0$ with $f_1+f_2+f_3+f_4=\llambda_i$ and the second sum is over all $g_1, g_2,
g_3, g_4 \geq 0$ with $g_1+g_2+g_3+g_4=\llambda_i -2$. Note that the first summation is zero if $\llambda_i<0$ and the second is zero when $\llambda_i<2$.
Reidemeister 3 like relations for all other orientations are determined from , , and the above relations using duality.
### Enriched $\Hom$ spaces
For any shift $t$, there are 2-morphisms $$\begin{aligned}
\xy
(0,0)*{\dblue\xybox{
(0,7);(0,-7); **\dir{-} ?(.75)*\dir{>}+(2.3,0)*{\scriptstyle{}};
(0.1,-2)*{\txt\large{$\bullet$}};
(6,4)*{ \black\lambda};
(-10,0)*{};(10,0)*{};(0,-9)*{\bscs i };}};
\endxy
\maps \cal{E}_{+i}\onel\{t\} \To \cal{E}_{+i}\onel\{t-2\}\quad
\xy
(0,0)*{\dblue\xybox{
(-4,-6)*{};(4,6)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
}};
(0,0)*{\dred\xybox{
(4,-6)*{};(-4,6)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-4,-8)*{\bscs i};
(4,-8)*{\bscs j};
(8,1)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy
\maps \cal{E}_{+i+j}\onel\{t\} \To \cal{E}_{+j+i}\onel\{t-i\cdot j\} \nn \\
\text{$
\xy
(0,-3)*{\dblue\bbpef{\black i}};
(8,-5)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy \maps
\onel\{t\} \To \cal{E}_{-i+i}\onel\{t-(1+\llambda_i)\} \quad \xy
(0,0)*{\dblue\bbcfe{\black i}};
(7,4)*{ \lambda};
(-12,0)*{};(12,0)*{};
\endxy \maps
\cal{E}_{-i+i}\onel\{t\} \To \onel\{t-(1-\llambda_i)\}
$}\nn\end{aligned}$$ in $\glcat$, and the diagrammatic relation $$\text{$
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(4,4)*{};(12,12)*{} **\crv{(4,7) & (12,9)}?(1)*\dir{>};
(12,4)*{};(4,12)*{} **\crv{(12,7) & (4,9)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(-4,4)*{}; (-4,12) **\dir{-};
(12,-4)*{}; (12,4) **\dir{-};
(12,12)*{}; (12,20) **\dir{-}; (-5.5,-3)*{\bscs i};
(5.5,-3)*{\bscs i};(14,-3)*{\bscs i};
(18,8)*{\black\lambda};}};
\endxy}
\;\; =\;\;
\vcenter{
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(-4,4)*{};(-12,12)*{} **\crv{(-4,7) & (-12,9)}?(1)*\dir{>};
(-12,4)*{};(-4,12)*{} **\crv{(-12,7) & (-4,9)}?(1)*\dir{>};
(4,12)*{};(-4,20)*{} **\crv{(4,15) & (-4,17)}?(1)*\dir{>};
(-4,12)*{};(4,20)*{} **\crv{(-4,15) & (4,17)}?(1)*\dir{>};
(4,4)*{}; (4,12) **\dir{-};
(-12,-4)*{}; (-12,4) **\dir{-};
(-12,12)*{}; (-12,20) **\dir{-};(-5.5,-3)*{\bscs i};
(5.5,-3)*{\bscs i};(-14,-3)*{\bscs i};
(10,8)*{\black\lambda};}};
\endxy}$}$$ gives rise to relations in $\glcat\big(\cal{E}_{iii}\onel\{t\},\cal{E}_{iii}\onel\{t+3i\cdot i\}\big)$ for all $t\in \Z$.
Note that for two $1$-morphisms $x$ and $y$ in $\glcat$ the 2hom-space $\HomGL(x,y)$ only contains 2-morphisms of degree zero and is therefore finite-dimensional. Following Khovanov and Lauda we introduce the graded 2hom-space $$\HOMGL(x,y)=\oplus_{t\in\Z}\HomGL(x\{t\},y),$$ which is infinite-dimensional. We also define the $2$-category $\glcat^*$ which has the same objects and $1$-morphisms as $\glcat$, but for two $1$-morphisms $x$ and $y$ the vector space of 2-morphisms is defined by $$\label{eq:ast}
\glcat^*(x,y)=\HOMGL(x,y).$$
The 2-category $\Scat(n,d)$
---------------------------
Fix $d\in\N_{>0}$. As explained in Section \[sec:hecke-schur\], the $q$-Schur algebra $\SD(n,d)$ can be seen as a quotient of $\glcat$ by the ideal generated by all idempotents corresponding to the weights that do not belong to $\Lambda(n,d)$. It is then natural to define the 2-category $\Scat(n,d)$ as a quotient of $\glcat$ as follows.
The 2-category $\Scat(n,d)$ is the quotient of $\glcat$ by the ideal generated by all 2-morphisms containing a region with a label not in $\Lambda(n,d)$.
We remark that we only put real bubbles, whose interior has a label outside $\Lambda(n,d)$, equal to zero. To see what happens to a fake bubble, one first has to write it in terms of real bubbles with the opposite orientation using the infinite Grassmannian relation .
A 2-representation of $\Scat(n,d)$ {#sec:2rep}
==================================
In this section we define a $2$-functor $$\fbim: \Scat{(n,d)}^{*}\to {\bim}^{*},$$ where $\bim$ is the graded $2$-category of bimodules over polynomial rings with rational coefficients. Recall that in the previous section (formula ), we have defined the $^*$ version of a graded $2$-category, as the $2$-category with the same objects and $1$-morphisms, while the $2$-morphisms between two $1$-morphisms can have arbitrary degree.
In [@K-L3] Khovanov and Lauda defined a 2-functor $\Gamma^G_d$ from $\Ucat$ to a $2$-category equivalent to a sub-2-category of $\bim^*$. As one can easily verify, $\Gamma^G_d$ kills any diagram with labels outside $\Lambda(n,d)$, so it descends to $\Scat(n,d)$. In this section we have rewritten this 2-functor, which we denote $\fbim$, in terms of categorified MOY-diagrams, because we think it might help some people to understand its definition more easily. For further comments see Section \[sec:2-functor\].
Categorified MOY diagrams
-------------------------
Before proceeding with the definition of $\fbim$, we first specify our notation for MOY diagrams and their categorification.
A colored MOY diagram [@M-O-Y], is an oriented trivalent graph whose edges are labeled by natural numbers (this label is also called the *color* or the *thickness* of the corresponding edge). At each trivalent vertex we have at least one incoming and one outgoing edge, and we require that at each vertex the sum of the labels of the incoming edges is equal to the sum of the labels of the outgoing edges. Moreover, in this paper we assume that all edges in MOY diagrams are oriented upwards.\
To obtain a bimodule corresponding to a given colored MOY diagram, we proceed in the following way: To each edge labeled $a$, we associate $a$ variables, say $\underline{x}=(x_1,\ldots,x_a)$, and to different edges we associate different variables. At every vertex (like the ones in Figure \[fig:triv\]), we impose the relations $$\begin{aligned}
e_i(z_1,\ldots,z_{a+b})
&= e_i(x_1,\ldots,x_a,y_1,\ldots,y_{b})\\
e_i(z'_1,\ldots,z'_{a+b})
&= e_i(x'_1,\ldots,x'_a,y'_1,\ldots,y'_{b})\end{aligned}$$ for all $i\in\{1,\ldots,a+b\}$, where $e_i$ is the $i$th elementary symmetric polynomial. In other words, at every vertex we require that an arbitrary symmetric polynomial in the variables corresponding to the incoming edges, is equal to the same symmetric polynomial in the variables corresponding to the outgoing edges.\
$$\labellist
\tiny\hair 2pt
\pinlabel $a$ at 16 93 \pinlabel $b$ at 100 95 \pinlabel $a+b$ at 90 42
%
\pinlabel $x_1,\dotsc,x_a$ at -8 155 \pinlabel $y_1,\dotsc,y_b$ at 128 155
\pinlabel $z_1,\dotsc,z_{a+b}$ at 68 -10
\endlabellist
\figins{0}{0.7}{vertexup}
\mspace{140mu}
\labellist
\tiny\hair 2pt
\pinlabel $a$ at 12 42 \pinlabel $b$ at 114 44 \pinlabel $a+b$ at 94 95
%
\pinlabel $x'_1,\dotsc,x'_a$ at -8 -10 \pinlabel $y'_1,\dotsc,y'_b$ at 128 -10
\pinlabel $z'_1,\dotsc,z'_{a+b}$ at 68 155
\endlabellist
\figins{0}{0.7}{vertexdwn}$$
Now, to an arbitrary diagram $\Gamma$, we associate the ring $R_{\Gamma}$ of polynomials over $\bQ$ which are symmetric in the variables on each strand separately, modded out by the relations corresponding to all trivalent vertices.
In particular, to a graph without trivalent vertices (just strands): $$\labellist
\tiny\hair 2pt
\pinlabel $c$ at -10 189 \pinlabel $b$ at 220 191 \pinlabel $a$ at 330 189
\pinlabel $\dotsc$ at 125 100
\pinlabel $\underline{z}$ at 7 -20
\pinlabel $\underline{y}$ at 235 -20
\pinlabel $\underline{x}$ at 349 -20
\endlabellist
\figins{-21}{0.7}{IDweb}\vspace*{2ex}$$
we associate the ring of partially symmetric polynomials $\bQ[\underline{x},\underline{y},\ldots,\underline{z}]^{S_a\times S_b\times\cdots\times S_c}$.
In this way, the ring $R_{\Gamma}$ associated to a MOY diagram $\Gamma$, is a bimodule over the rings of partially symmetric polynomials associated to the top (right action) and bottom end (left action) strands, respectively (remember that we are assuming that all MOY diagrams are oriented upwards, so they have a top and a bottom end). Bimodules are graded by setting the degree of any variable equal to 2.
In the rest of the paper, we will often identify the MOY diagram and the corresponding bimodule. Also, by abuse of notation, we shall call the elements of the bimodule $R_{\Gamma}$ polynomials.\
There is another way to describe these bimodules associated to MOY diagrams (see e.g. [@Kh; @M-S-V; @Will]). Fix the polynomial ring $R:=\mathbb{Q}[x_1,\ldots,x_d]$. For any $(a_1,\ldots,a_n)\in\Lambda(n,d)$, let $R^{a_1,\ldots,a_n}$ be the sub-ring of polynomials which are invariant under $S_{a_1}\times\cdots\times S_{a_n}$. To the first diagram in Figure \[fig:triv\] one associates the $R^{a+b}-R^{a,b}$-bimodule $$\mbox{Res}_{R^{a,b}}^{R^{a+b}}R^{a,b},$$ where one simply restricts the left action on $R^{a,b}$ to $R^{a+b}\subseteq R^{a,b}$. To the second diagram in Figure \[fig:triv\] one associates the $R^{a,b}-R^{a+b}$-bimodule $$\mbox{Ind}_{R^{a+b}}^{R^{a,b}}R^{a+b}:=R^{a,b}\otimes_{R^{a+b}}R^{a+b}.$$ In this way, to every MOY-diagram $\Gamma$ one associates a tensor product of bimodules, which is isomorphic to the bimodule $R_{\Gamma}$ that we described in the paragraph above.
In this paper we always use $R_{\Gamma}$, since it is computationally easier to use polynomials than to use tensor products of polynomials.
Definition of $\fbim$
---------------------
Now we can proceed with the definition of $\fbim \colon \Scat(n,d)^*\to \bim^*$.
Let $z_1,\ldots,z_d$ be variables. For convenience we shall use Khovanov and Lauda’s notation $k_i=\lambda_1+\cdots+\lambda_i$, for $i=1,\ldots,n$.
On objects $\lambda\in\Lambda(n,d)$, the 2-functor $\fbim$ is given by: $$\lambda=(\lambda_1,\cdots,\lambda_n)
\mapsto \bQ[z_1,\ldots ,z_d]^{S_{\lambda_1}\times\cdots\times S_{\lambda_n}}.$$
On $1$-morphisms we define $\fbim$ as follows: $$1_{\lambda}\{t\}\mapsto \bQ[z_1,\ldots,z_d]^{S_{\lambda_1}\times\cdots\times S_{\lambda_n}}\{t\}.$$
In terms of MOY diagrams this is presented by: $$1_{\lambda}\{t\}\quad \longmapsto\qquad
\labellist
\tiny\hair 2pt
\pinlabel $\lambda_n$ at -17 189
\pinlabel $\lambda_2$ at 210 189
\pinlabel $\lambda_1$ at 322 189
\pinlabel $\dotsc$ at 125 100
\endlabellist
\figins{-21}{0.7}{IDweb}$$
Note that we are drawing the entries of $\lambda$ from right to left, which is compatible with Khovanov and Lauda’s convention.
The remaining generating $1$-morphisms are mapped as follows: $$\begin{aligned}
\cal{E}_{+i}\onel\{t\}\ \
&\mapsto
\qquad
\labellist
\tiny\hair 2pt
\pinlabel $\lambda_n$ at -22 193
\pinlabel $\laii$ at 150 193
\pinlabel $\lai$ at 275 193
\pinlabel $\lambda_1$ at 460 193
\pinlabel $1$ at 243 74
\pinlabel $\laii-1$ at 150 -15
\pinlabel $\lai+1$ at 310 -15
\pinlabel $\dotsc$ at 120 100
\pinlabel $\dotsc$ at 400 100
\endlabellist
\figins{-23}{0.7}{HweblID}\quad \{t+1+k_{i-1}+k_i-k_{i+1}\}
\displaybreak[0] \\[4ex]
\cal{E}_{-i}\onel\{t\}\ \
&\mapsto
\qquad
\labellist
\tiny\hair 2pt
\pinlabel $\lambda_n$ at -22 193
\pinlabel $\laii$ at 150 193
\pinlabel $\lai$ at 275 193
\pinlabel $\lambda_1$ at 460 193
\pinlabel $1$ at 243 74
\pinlabel $\laii+1$ at 150 -15
\pinlabel $\lai-1$ at 310 -15
\pinlabel $\dotsc$ at 120 100
\pinlabel $\dotsc$ at 400 100
\endlabellist
\figins{-23}{0.7}{HwebrID}\quad \{t+1-k_i\}\end{aligned}$$
In both cases, the partition corresponding to the bottom strands is $\lambda+j_{\Lambda}$ (with $j$ being $+i$ or $-i$). Thus, the condition we imposed on $\Scat(n,d)$ that all regions have labels from $\Lambda(n,d)$ (i.e. no region can have labels with negative entries), ensures that on the RHS above we really have MOY diagrams.
The composite $\fbim(\mathcal{E}_i1_{\lambda+j_{\Lambda}}\mathcal{E}_j1_{\lambda})$ is given by stacking the MOY diagram corresponding to $\mathcal{E}_j1_{\lambda}$ on top of the one corresponding to $\mathcal{E}_i1_{\lambda+j_{\Lambda}}$. The shifts add under composition.\
To define $\fbim$ on $2$-morphisms, we give the image of the generating $2$-morphisms. In the definitions the divided difference operator $\partial_{xy}$ is used. For $p\in Q[x,y,\ldots]$ it is given by $$\partial_{xy}p=\frac{p-p_{\mid x\leftrightarrow y}}{x-y},
\label{novo}$$ where $p_{\mid x\leftrightarrow y}$ is the polynomial obtained from $p$ by swapping the variables $x$ and $y$. Moreover, for $\underline{x}=(x_1,\ldots,x_a)$, we use the shorthand notation $$\begin{aligned}
\partial_{\underline{x}y}
&= \partial_{x_1y}\partial_{x_2y}\cdots\partial_{x_ay}\\
\partial_{y\underline{x}}
&= \partial_{yx_1}\partial_{yx_2}\cdots\partial_{yx_a}.\end{aligned}$$ 0.3cm
Before listing the definition of $\fbim$, we explain the notation we are using. We denote a bimodule map as a pair, the first term showing the corresponding MOY diagrams (of the source and target 1-morphism), and the second being an explicit formula of the map in terms of the (classes of) polynomials that are the elements of the corresponding rings. In a few cases we have added an intermediate MOY-diagram, in order to clarify the definition. Finally, in order to simplify the pictures, in each formula we only draw the strands that are affected, while on the others we just set the identity. Also in every line we require that the polynomial rings corresponding to the top (respectively bottom) end strands are the same throughout the movie. Furthermore, we only write explicitly the variables of the strands that are relevant in the definition of the corresponding bimodule map.
$$\begin{aligned}
\xy
(0,-1.5)*{\dblue\xybox{
(0,7);(0,-7); **\dir{-} ?(1)*\dir{>};
(4,4)*{ \bscs \lambda};
(0,-9)*{\bscs i };}};
\endxy
\quad
& \mapsto
\quad
\id
\left(\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -30 183 \pinlabel $\lai$ at 150 185 \pinlabel $1$ at 62 74
\endlabellist
\quad
\figins{-19}{0.6}{Hwebl}\ \quad
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\xy
(0,-1.5)*{\dblue\xybox{
(0,7);(0,-7); **\dir{-} ?(1)*\dir{>};
(0.1,-2)*{\txt\large{$\bullet$}}+(2.3,1)*{\bscs r};
(4,4)*{ \bscs \lambda};
(0,-9)*{\bscs i };}};
\endxy
\quad
& \mapsto
\quad
\left(\quad\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -30 183 \pinlabel $\lai$ at 150 185 \pinlabel $1$ at 62 74
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 56 134 \pinlabel $x$ at 56 162
\endlabellist
\figins{-19}{0.6}{Hwebl}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -30 183
\pinlabel $\lai$ at 150 185 \pinlabel $1$ at 62 74
\endlabellist
\figins{-19}{0.6}{Hwebl}\ \
,\quad
p \mapsto x^rp
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\xy
(0,-1.5)*{\dblue\xybox{
(0,7);(0,-7); **\dir{-} ?(0)*\dir{<};
(4,4)*{ \bscs \lambda};
(0,-9)*{\bscs i };}};
\endxy
\quad
& \mapsto
\quad
\id
\left(\quad\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -33 183 \pinlabel $\lai$ at 146 185 \pinlabel $1$ at 66 74
\endlabellist
\figins{-19}{0.6}{Hwebr}\ \quad
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\xy
(0,-1.5)*{\dblue\xybox{
(0,7);(0,-7); **\dir{-} ?(0)*\dir{<};
(0.1,0)*{\txt\large{$\bullet$}}+(2.3,-1)*{\bscs r};
(4,4)*{ \bscs \lambda};
(0,-9)*{\bscs i };}};
\endxy
\quad
& \mapsto
\quad
\left(\quad\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -33 183 \pinlabel $\lai$ at 146 185 \pinlabel $1$ at 66 74
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 70 134 \pinlabel $x$ at 70 162
\endlabellist
\figins{-19}{0.6}{Hwebr}\ \
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -33 183 \pinlabel $\lai$ at 146 185 \pinlabel $1$ at 66 74
\endlabellist
\figins{-19}{0.6}{Hwebr}\ \
,\quad
p\mapsto x^rp
\right)\end{aligned}$$
$$\begin{aligned}
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-5,-3)*{\scs i};
(5.1,-3)*{\scs i};
(7,1)*{\scs \lambda};
\endxy
\quad
& \mapsto
\quad
\left(\quad
\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 193 \pinlabel $\lai$ at 138 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 169 \pinlabel $x_2$ at 50 197
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 50 52 \pinlabel $x_1$ at 50 20
\endlabellist
\figins{-19}{0.6}{HHwebll}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 193 \pinlabel $\lai$ at 146 195 \pinlabel $2$ at 62 74
\endlabellist
\figins{-19}{0.6}{Hwebl}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 193 \pinlabel $\lai$ at 140 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 169 \pinlabel $x_1$ at 50 197
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 50 52 \pinlabel $x_2$ at 50 20
\endlabellist
\figins{-19}{0.6}{HHwebll}
\ \
, \quad
p\mapsto \partial_{x_1x_2}p
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
}};
(-6,-3)*{\scs i};
(6.1,-3)*{\scs i};
(7,1)*{\scs\lambda};
\endxy
\quad
&\mapsto
\quad
\left(\quad
\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -36 193 \pinlabel $\lai$ at 130 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 65 171 \pinlabel $x_1$ at 65 199
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 65 52 \pinlabel $x_2$ at 65 20
\endlabellist
\figins{-19}{0.6}{HHwebrr}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 193 \pinlabel $\lai$ at 142 195 \pinlabel $2$ at 66 74
\endlabellist
\figins{-19}{0.6}{Hwebr}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 193 \pinlabel $\lai$ at 130 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 65 171 \pinlabel $x_2$ at 65 199
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 65 52 \pinlabel $x_1$ at 65 20
\endlabellist
\figins{-19}{0.6}{HHwebrr}
\ \
, \quad
p\mapsto \partial_{x_1x_2}p
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%% sideways ii %%%%%%%%%
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
}};
(-6,-3)*{\scs i};
(6.1,-3)*{\scs i};
(7,1)*{\scs\lambda};
\endxy
\quad
& \mapsto
\quad
\left(\quad
\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -36 193 \pinlabel $\lai$ at 130 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 65 176 \pinlabel $x_1$ at 65 204
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 65 42 \pinlabel $y$ at 65 10
\endlabellist
\figins{-19}{0.6}{sqweblr}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 193 \pinlabel $\lai$ at 130 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 70 171 \pinlabel $y$ at 70 201
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 70 52 \pinlabel $x_2$ at 70 20
\endlabellist
\figins{-19}{0.6}{sqwebrl}
\ \
, \quad
p\mapsto p\vert_{x_1\mapsto x_2}
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-6,-3)*{\scs i};
(6.1,-3)*{\scs i};
(7,1)*{\scs\lambda};
\endxy
\quad
& \mapsto
\quad
\left(\quad
\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -36 193 \pinlabel $\lai$ at 130 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 65 176 \pinlabel $x_1$ at 65 204
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 65 42 \pinlabel $y$ at 65 10
\endlabellist
\figins{-19}{0.6}{sqwebrl}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 193 \pinlabel $\lai$ at 130 195
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 65 176 \pinlabel $y$ at 65 206
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 65 42 \pinlabel $x_2$ at 65 10
\endlabellist
\figins{-19}{0.6}{sqweblr}
\ \
, \quad
p\mapsto p\vert_{x_1\mapsto x_2}
\right)\end{aligned}$$
$$\begin{aligned}
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} }};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-5,-3)*{\scs i};
(6.9,-3)*{\scs i+1};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -25 189 \pinlabel $\laii$ at 170 191 \pinlabel $\lai$ at 255 189
\pinlabel $1$ at 66 135 \pinlabel $1$ at 165 45
\endlabellist
\figins{-19}{0.6}{Hhwebl}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 70 191 \pinlabel $\lai$ at 255 189
\pinlabel $1$ at 166 125 \pinlabel $1$ at 66 50
\endlabellist
\figins{-19}{0.6}{hHwebl}\ \
,\ \
p\mapsto p
\right)
\nn\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\xy
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} }};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-6.9,-3)*{\scs i+1};
(5.1,-3)*{\scs i};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 70 191 \pinlabel $\lai$ at 250 189
\pinlabel $1$ at 166 125 \pinlabel $1$ at 66 50
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 114 \pinlabel $x$ at 50 142
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 150 184 \pinlabel $y$ at 150 214
\endlabellist
\figins{-19}{0.6}{hHwebl}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 165 191 \pinlabel $\lai$ at 255 189
\pinlabel $1$ at 66 135 \pinlabel $1$ at 165 45
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 186 \pinlabel $x$ at 50 214
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 146 108 \pinlabel $y$ at 146 138
\endlabellist
\figins{-19}{0.6}{Hhwebl}\ \
,\ \
p\mapsto (x-y)p
\right)
\nn\displaybreak[0]
\\[1.5ex]
%%%%% pointing down %%%%%
\xy
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<} }};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
}};
(-7.9,-3)*{\scs i+1};
(6.1,-3)*{\scs i};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 70 191 \pinlabel $\lai$ at 250 189
\pinlabel $1$ at 166 125 \pinlabel $1$ at 66 40
%\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 104 \pinlabel $x$ at 50 132
%\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 150 184 \pinlabel $z$ at 150 212
\endlabellist
\figins{-19}{0.6}{hHwebr}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -34 189 \pinlabel $\laii$ at 166 191 \pinlabel $\lai$ at 255 189
\pinlabel $1$ at 66 125 \pinlabel $1$ at 165 45
%\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 186 \pinlabel $x$ at 50 214
%\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 146 108 \pinlabel $z$ at 146 136
\endlabellist
\figins{-19}{0.6}{Hhwebr}\ \
,\ \
p\mapsto p
\right)
\nn\displaybreak[0]
\\[1.5ex]
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<} }};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
}};
(-6.4,-3)*{\scs i};
(7.9,-3)*{\scs i+1};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -25 189 \pinlabel $\laii$ at 170 191 \pinlabel $\lai$ at 255 189
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 55 181 \pinlabel $x$ at 55 214
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 151 103 \pinlabel $y$ at 151 136
\pinlabel $1$ at 66 125 \pinlabel $1$ at 165 45
\endlabellist
\figins{-19}{0.6}{Hhwebr}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 70 191 \pinlabel $\lai$ at 255 189
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 151 181 \pinlabel $y$ at 151 214
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 55 103 \pinlabel $x$ at 55 136
\pinlabel $1$ at 163 125 \pinlabel $1$ at 71 45
\endlabellist
\figins{-19}{0.6}{hHwebr}\ \
,\ \
p\mapsto (x-y)p
\right)
\nn\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{eq:assoc1}
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} }};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
}};
(-5,-3)*{\scs i};
(7.8,-3.1)*{\scs i+1};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 195 \pinlabel $\laii$ at 168 197 \pinlabel $\lai$ at 258 195
\pinlabel $1$ at 70 132 \pinlabel $1$ at 168 46
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 80 189 \pinlabel $x_1$ at 80 217
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 160 104 \pinlabel $y$ at 160 132
\endlabellist
\figins{-19}{0.6}{Hhwebrl}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -40 195 \pinlabel $\laii$ at 73 198 \pinlabel $\lai$ at 255 195
\pinlabel $1$ at 70 50 \pinlabel $1$ at 168 126
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 80 109 \pinlabel $x_2$ at 80 137
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 160 189 \pinlabel $y$ at 160 217
\endlabellist
\figins{-19}{0.6}{hHwebrl}
,\ \
p\mapsto p\vert_{x_1\mapsto x_2}
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{eq:assoc2}
\xy
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} }};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
}};
(-6.9,-3)*{\scs i+1};
(6.0,-3)*{\scs i};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 75 191 \pinlabel $\lai$ at 250 189
\pinlabel $1$ at 166 115 \pinlabel $1$ at 66 50
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 114 \pinlabel $y$ at 50 143
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 176 179 \pinlabel $x_1$ at 176 207
\endlabellist
\figins{-19}{0.6}{hHweblr}\
\ra\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 165 191 \pinlabel $\lai$ at 255 189
\pinlabel $1$ at 66 115 \pinlabel $1$ at 165 45
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 176 \pinlabel $y$ at 50 205
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 176 108 \pinlabel $x_2$ at 176 136
\endlabellist
\figins{-19}{0.6}{Hhweblr}\ \
,\ \
p\mapsto p_{\vert {x_1\mapsto x_2}}
\right)
\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<} }};
(0,0)*{\dred\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-6.5,-3)*{\scs i};
(7.5,-3.1)*{\scs i+1};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 195 \pinlabel $\laii$ at 168 197 \pinlabel $\lai$ at 258 195
\pinlabel $1$ at 70 112 \pinlabel $1$ at 168 46
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 75 169 \pinlabel $x_1$ at 75 197
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 160 104 \pinlabel $y$ at 160 132
\endlabellist
\figins{-19}{0.6}{Hhweblr}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -40 195 \pinlabel $\laii$ at 73 198 \pinlabel $\lai$ at 255 195
\pinlabel $1$ at 70 50 \pinlabel $1$ at 168 111
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 80 109 \pinlabel $x_2$ at 80 137
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 165 174 \pinlabel $y$ at 165 202
\endlabellist
\figins{-19}{0.6}{hHweblr}
,\ \
p\mapsto p_{\vert {x_1\mapsto x_2}}
\right)
\nn\displaybreak[0]
\\[1.5ex]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\xy
(0,0)*{\dred\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<} }};
(0,0)*{\dblue\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-7.9,-3)*{\scs i+1};
(6.0,-3)*{\scs i};
(9,1)*{ \scs\lambda};
\endxy
\quad
&
\mapsto
\quad
\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 75 191 \pinlabel $\lai$ at 250 189
\pinlabel $1$ at 166 125 \pinlabel $1$ at 66 50
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 60 104 \pinlabel $y$ at 60 140
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 176 179 \pinlabel $x_1$ at 176 207
\endlabellist
\figins{-19}{0.6}{hHwebrl}\
\ra\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laiii$ at -35 189 \pinlabel $\laii$ at 165 191 \pinlabel $\lai$ at 255 189
\pinlabel $1$ at 66 125 \pinlabel $1$ at 165 45
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 50 176 \pinlabel $y$ at 50 204
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 176 103 \pinlabel $x_2$ at 176 131
\endlabellist
\figins{-19}{0.6}{Hhwebrl}\ \
,\ \
p\mapsto p_{\vert {x_1\mapsto x_2}}
\right)
\nn\end{aligned}$$
For $\vert i-j\vert\geq 2$: $$\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>} }};
(0,0)*{\dgreen\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{>};
}};
(-5,-3)*{\scs i};
(5.1,-3)*{\scs j};
(9,1)*{ \scs\lambda};
\endxy
\quad
\mapsto
\quad
\id\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 189 \pinlabel $\lai$ at 142 191 \pinlabel $1$ at 55 65
\pinlabel $\dotsm$ at 200 110
\pinlabel $\lambda_{j+1}$ at 210 191 \pinlabel $\laj$ at 394 191 \pinlabel $1$
at 305 115
\endlabellist
\figins{-19}{0.64}{HHweblL}
\quad \right)$$ $$\xy
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(0)*\dir{<} }};
(0,0)*{\dgreen\xybox{
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(0)*\dir{<};
}};
(-6.1,-3)*{\scs i};
(6,-3)*{\scs j};
(9,1)*{ \scs\lambda};
\endxy
\quad\mapsto\quad
\id\left(\ \
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 189 \pinlabel $\lai$ at 146 191 \pinlabel $1$ at 65 65
\pinlabel $\dotsm$ at 210 110
\pinlabel $\lambda_{j+1}$ at 240 191 \pinlabel $\laj$ at 440 191 \pinlabel $1$ at 355 100
\endlabellist
\figins{-19}{0.6}{HHwebrR}
\quad
\right)$$ Sideways crossings for $\vert i-j\vert\geq 2$ are defined in the same way as in the case of $\vert i-j\vert=1$.
$$\begin{aligned}
\xy
(0,0)*{\dblue\bbpef{\bscs i}};
(6,0)*{\scs\lambda};
\endxy
\quad
\mapsto &
\quad
\left(\ \
\begin{aligned}
&
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -38 189 \pinlabel $\lai$ at 150 191
\endlabellist
\quad
\figins{-19}{0.6}{id2web} \ \
%\qra
%\labellist
%\tiny\hair 2pt
%\pinlabel $\laii$ at -38 189 \pinlabel $\lai$ at 150 191
%\pinlabel $1$ at 60 65
%\pinlabel $\color[rgb]{.5,.5,.5}{\swarrow}$ at 50 160 \pinlabel $x_{\!\laii}$ at 73 189
%\endlabellist
%\figins{-19}{0.6}{digwebl}
\qra\ \
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -30 189 \pinlabel $\lai$ at 125 191
\pinlabel $1$ at 60 50 \pinlabel $1$ at 50 130
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 55 191 \pinlabel $x$ at 55 230
\pinlabel $\color[rgb]{.5,.5,.5}{\swarrow}$ at 130 41
\pinlabel $\und{t}$ at 163 60
\endlabellist
\figins{-19}{0.6}{sqwebrl}\qquad\
\\[1.0ex]
&
p\mapsto \sum\limits_{\ell=0}^{\lai}(-1)^{\ell}x^{\lai-\ell}
e_{\ell}(\und{t})p
\end{aligned}
\right)\end{aligned}$$
$$\begin{aligned}
\xy
(0,0)*{\dblue\bbpfe{\bscs i}};
(6,0)*{\scs\lambda};
\endxy
\quad
\mapsto &
\quad
\left(\ \
\begin{aligned}
&
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -37 189 \pinlabel $\lai$ at 150 191
\endlabellist
\figins{-19}{0.6}{id2web}
%\qra
%\labellist
%\tiny\hair 2pt
%\pinlabel $\laii$ at -35 189 \pinlabel $\lai$ at 148 191
%\pinlabel $1$ at 75 150
%\pinlabel $\color[rgb]{.5,.5,.5}{\nearrow}$ at 75 45 \pinlabel $y_1$ at 50 12
%\endlabellist
%\figins{-19}{0.6}{digwebr}
\qra\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 189 \pinlabel $\lai$ at 130 191
\pinlabel $1$ at 60 90 \pinlabel $1$ at 50 170
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 55 41 \pinlabel $y$ at 53 7
\pinlabel $\color[rgb]{.5,.5,.5}{\nearrow}$ at -15 146
\pinlabel $\und{z}$ at -50 105
\endlabellist
\figins{-19}{0.6}{sqweblr}
\quad
\\[1.0ex]
&
p\mapsto \sum\limits_{\ell=0}^{\lambda_{i+1}}(-1)^{\ell}e_{\laii-\ell}(\und{z})y^{\ell}p\ \
\end{aligned}
\right)\end{aligned}$$
$$\begin{aligned}
\xy
(0,0)*{\dblue\bbcef{\bscs i}};
(6,0)*{\scs\lambda};
\endxy
\quad
\mapsto &
\quad
\left(\ \
\begin{aligned}
&
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -30 189 \pinlabel $\lai$ at 130 191
\pinlabel $1$ at 50 90 \pinlabel $1$ at 50 130
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 55 181 \pinlabel $y$ at 55 212
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 55 31 \pinlabel $x$ at 54 -10
\pinlabel $\color[rgb]{.5,.5,.5}{\ra}$ at -10 106
\pinlabel $\und{u}$ at -50 105
\endlabellist
\figins{-19}{0.6}{sqwebrl}
\qra\quad
%\labellist
%\tiny\hair 2pt
%\pinlabel $\laii$ at -40 189 \pinlabel $\lai$ at 153 191
%\pinlabel $1$ at 60 65
%\pinlabel $\color[rgb]{.5,.5,.5}{\nearrow}$ at -20 71
%\pinlabel $\und{x}(\laii-1)$ at -140 41
%\endlabellist
%\figins{-19}{0.6}{digwebl}
%\qra
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -38 189 \pinlabel $\lai$ at 150 191
\endlabellist
\figins{-19}{0.6}{id2web}\quad
\\[1.0ex]
&
\qquad p\mapsto \partial_{\und{u}x}(p_{\vert {y=x}})
\end{aligned}
\right)\end{aligned}$$
$$\begin{aligned}
\xy
(0,0)*{\dblue\bbcfe{\bscs i}};
(6,0)*{\scs\lambda};
\endxy
\quad
\mapsto &
\quad
\left(\ \
\begin{aligned}
&
\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -35 189 \pinlabel $\lai$ at 130 191
\pinlabel $1$ at 65 90 \pinlabel $1$ at 65 130
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 55 181 \pinlabel $y$ at 55 212
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 55 41 \pinlabel $x$ at 55 10
\pinlabel $\color[rgb]{.5,.5,.5}{\leftarrow}$ at 123 86
\pinlabel $\und{u}$ at 158 85
\endlabellist
\figins{-19}{0.6}{sqweblr}
\quad\qra\quad
%\labellist
%\tiny\hair 2pt
%\pinlabel $\laii$ at -35 189 \pinlabel $\lai$ at 148 191
%\pinlabel $1$ at 70 65
%\pinlabel $\color[rgb]{.5,.5,.5}{\nwarrow}$ at 150 71
%\pinlabel $\und{x}(1,\lai)$ at 245 38
%\pinlabel $\color[rgb]{.5,.5,.5}{\searrow}$ at 70 150
%\pinlabel $x_1$ at 45 180
%\endlabellist
%\figins{-19}{0.6}{digwebr}
%\quad\ra\quad
\labellist
\tiny\hair 2pt
\pinlabel $\laii$ at -38 189 \pinlabel $\lai$ at 150 191
\endlabellist
\figins{-19}{0.6}{id2web}\quad
\\[1.0ex]
& \qquad\ \
p\mapsto \partial_{x\und{u}}(p_{\vert_{y=x}})
\end{aligned}
\right)\end{aligned}$$
This ends the definition of $\fbim$. Without giving any details, we remark that the bimodule maps above can be obtained as composites of elementary ones, called [*zip, unzip, associativity, digon creation*]{} and [*annihilation*]{}, which can be found in [@M-S-V].
$\fbim$ is a $2$-functor {#sec:2-functor}
------------------------
We are now able to explain the relation between our $\fbim$ and Khovanov and Lauda’s (see Subsection 6.3 in [@K-L3]) $$\Gamma^G_d\colon \Ucat^*\to
\mbox{\bf EqFLAG}^*_d\subset \bim^*.$$
In the first place, we categorify the homomorphism $\psi_{n,d}$ from Section \[sec:hecke-schur\]. Note that all the relations in $\Scat(n,d)$ only depend on $\mathfrak{sl}_n$-weights, except the value of the degree zero bubbles, which truly depend on $\mathfrak{gl}_n$-weights.
We define a 2-functor \[defn:Psi\] $$\Psi_{n,d}\colon \Ucat\to \Scat(n,d).$$ On objects and $1$-morphisms $\Psi_{n,d}$ is defined just as $\psi_{n,d}\colon
\U \to \SD(n,d)$ in . On $2$-morphisms we define $\Psi_{n,d}$ as follows. Let $D$ be a string diagram representing a $2$-morphism in $\Ucat$ (from now on we will simply say that $D$ is a diagram in $\Ucat$). Then $\Psi_{n,d}$ maps $D$ to the same diagram, multiplied by a power of $-1$ depending on the left cups and caps in $D$ according to the rule in . The labels in $\bZ^{n-1}$ of the regions of $D$ are mapped by $\phi_{n,d}$ to labels in $\bZ^n$ of the corresponding regions of $\Psi_{n,d}(D)$, or to $*$. This means that, if $D$ has a region labeled by $\lambda$ such that $\phi_{n,d}(\lambda)\not\in\Lambda(n,d)$, then $\Psi_{n,d}(D)=0$ by definition. Finally, extend this definition to all $2$-morphisms by linearity.
It is easy to see that $\Psi_{n,d}$ is well-defined, full and essentially surjective.
In the second place, recall that there is a well-known isomorphism $$\bQ[x_1,\ldots,x_d]^{S_{\lambda_1}\times\cdots\times S_{\lambda_n}}\cong
H_{GL(d)}(Fl(\underline{k})),$$ with $\underline{k}=(k_0,k_1,k_2,k_3,\ldots,k_n)= (0,\lambda_1,\lambda_1+\lambda_2,
\lambda_1+\lambda_2+\lambda_3,\ldots,d)$, for any $\lambda\in\Lambda(n,d)$ (see (6.25) in [@K-L3], for example). Using this isomorphism, it is straightforward to check that the following lemma holds by comparing the images of the generators. Recall that $\Gamma_d^G$ kills all diagrams with labels outside $\Lambda(n,d)$.
\[lem:rep\] The following triangle is commutative $$\xymatrix{
\Ucat^*\ar[rr]^{\Gamma^G_d}\ar[dr]_{\Psi_{n,d}} && \bim^*
\\
& \Scat(n,d)^*\ar[ur]_{\fbim} &
}$$
The following result is now an immediate consequence of Khovanov and Lauda’s Theorem 6.13.
\[prop:fbim\] $\fbim$ defines a 2-functor from $\Scat(n,d)^*$ to $\bim^*$.
One could of course prove Proposition \[prop:fbim\] by hand. We will just give two sample calculations. The result of the second one, the image of the dotted bubbles, will be needed in a later section.
### Examples of the direct proof of Proposition \[prop:fbim\]
We shall give the proof for the zig-zag relation of biadjointness and compute the images of the bubbles by $\fbim$.
Before proceeding, we give some useful relations that are used in the computations. First of all, both the kernel and the image of the divided difference operator $\partial_{xy}$ consist of the polynomials that are symmetric in the variables $x$ and $y$. If $p$ is symmetric in the variables $x$ and $y$ then $$\partial_{xy}(p\,q)=p\,\partial_{xy}q$$ for any polynomial $q$. Also, note that $\partial_{yx}=-\partial_{xy}$.
We shall frequently use the following useful identities (see for example [@F] for the proofs). For $\underline{x}=(x_1,\ldots,x_k)$, let $h_j(\underline{x})$ denote the $j$-th complete symmetric polynomial in the variables $x_1,\ldots,x_k$. Then we have $$\label{use1}
\partial_{y\underline{x}}(y^{N})=h_{N-k}(y,\underline{x}),$$ and $$\label{use2}
\sum_{j=0}^k(-1)^je_j(\underline{x})h_{k-j}(\underline{x})=\delta_{k,0}.$$
Moreover, if $x$, $\underline{u}=(u_1,\ldots,u_{a})$ and $\underline{t}=(t_1,\ldots,t_{a+1})$ are variables such that $$e_l(x,\underline{u})=e_l(\underline{t}),\quad l=1,\ldots,a+1,$$ then for every $l=1,\ldots,a+1$, we have $$\label{use3}
e_l(\underline{u})=\sum_{j=0}^l(-1)^jx^je_{l-j}(\underline{t}),$$ and $$\label{use4}
e_l(\underline{t})=e_l(\underline{u})+xe_{l-1}(\underline{u}).$$
$\bullet$ The zig-zag relations.\
In order to reduce the number of subindices (to keep the notation as concise as possible), we denote $\lambda_i=a$ and $\lambda_{i+1}=b$.
Then the left hand side of the first of the relations is mapped by $\fbim$ as follows: $$\begin{aligned}
\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{}="1";
(0,0)*{}="2";
(8,0)*{}="3";
(-8,-10);"1" **\dir{-};
"1";"2" **\crv{(-8,8) & (0,8)} ?(0)*\dir{>} ?(1)*\dir{>};
"2";"3" **\crv{(0,-8) & (8,-8)}?(1)*\dir{>};
"3"; (8,10) **\dir{-};}};
(12,5)*{\lambda};(-10,-8)*{\scs i};
\endxy
\quad
\mapsto &
\quad
\left(
\begin{aligned}
&
\quad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -12 193 \pinlabel $a$ at 128 191
\pinlabel $1$ at 55 85
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 55 31 \pinlabel $x_1$ at 55 0
\pinlabel $\color[rgb]{.5,.5,.5}{\searrow}$ at -16 51
\pinlabel $\und{u}$ at -46 70
\pinlabel $\color[rgb]{.5,.5,.5}{\swarrow}$ at 126 41
\pinlabel $\und{t}$ at 153 62
\endlabellist
\figins{-19}{0.7}{Hweblbot}
\quad \qra
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -10 193 \pinlabel $a$ at 124 191
\pinlabel $1$ at 38 45 \pinlabel $1$ at 72 94 \pinlabel $1$ at 38 190
\pinlabel $\color[rgb]{.5,.5,.5}{\leftarrow}$ at 120 71
\pinlabel $\und{v}$ at 154 70
\pinlabel $\color[rgb]{.5,.5,.5}{\downarrow}$ at 70 182
\pinlabel $x_2$ at 70 210
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 70 30
\pinlabel $x_1$ at 70 -5
\pinlabel $y$ at 40 120
\pinlabel $\color[rgb]{.5,.5,.5}{\searrow}$ at -16 41
\pinlabel $\und{u}$ at -46 60
\pinlabel $\color[rgb]{.5,.5,.5}{\leftarrow}$ at 132 11
\pinlabel $\und{t}$ at 161 12
\endlabellist
\figins{-19}{0.7}{sqweblrl}
\qquad\ra\quad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -10 193 \pinlabel $a$ at 124 191
\pinlabel $1$ at 38 150
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 70 135
\pinlabel $x_2$ at 70 105
\endlabellist
\figins{-19}{0.7}{Hwebltop}\ \ \
\\[1.0ex]
& \qquad
p\mapsto
\partial_{x_1\und{v}}
\left(\sum\limits_{\ell=0}^{a}(-1)^{\ell}x_2^{a-\ell}
e_{\ell}(\und{v}) p\right)
\end{aligned}
\right)\end{aligned}$$
Note that $p=p(x_1,\underline{u},\underline{t})$ is symmetric in the variables $\underline{u}$ and $\underline{t}$ separately. Also, the lowest trivalent vertex on the right strand in the middle picture of the movie, implies that $e_l(x_1,\underline{v})=e_l(\underline{t})$, for every $l=1,\ldots,a+1$. So, $x_1^j$ for $j>a$ is a symmetric polynomial in the variables $\underline{t}$ (e.g. this follows from (\[use3\]) for $l=a+1$). Thus we can write $p$ as: $$p=\sum_{j=0}^a {x_1^jq_j(\underline{u},\underline{t})},$$ where $q_j=q_j(\underline{u},\underline{t})$, $j=0,\ldots,a$, are polynomials symmetric in $\underline{u}$ and $\underline{t}$ separately.
Then we have: $$\partial_{x_1\underline{v}}(\sum_{l=0}^a(-1)^lx_2^{a-l}
e_l(\underline{v})p)=\sum_{l=0}^a(-1)^lx_2^{a-l}\partial_{x_1\underline{v}}
(e_l(\underline{v})\sum_{j=0}^ax_1^jq_j){=}_{(l\mapsto a-l)}$$ $$\label{pom11}
=\sum_{l=0}^a\sum_{j=0}^a(-1)^{a-l}x_2^lq_j\partial_{x_1\underline{v}}(x_1^je_{a-l}
(\underline{v})).$$
Since $e_l(x_1,\underline{v})=e_l(\underline{t})$, for every $l=1,\ldots,a+1$, by (\[use3\]) we have $e_{a-l}(\underline{v})=\sum_{k=0}^{a-l}(-1)^k x_1^k e_{a-l-k}(\underline{t})$. After replacing this in (\[pom11\]), we get
$$\begin{aligned}
&= \sum_{l=0}^a\sum_{j=0}^ax_2^lq_j\sum_{k=0}^{a-l}(-1)^{a-l-k}e_{a-l-k}
(\underline{t})\partial_{x_1\underline{v}}(x_1^{j+k})=_{(\ref{use1})}\\
&= \sum_{l=0}^a\sum_{j=0}^ax_2^lq_j\sum_{k=0}^{a-l}(-1)^{a-l-k}e_{a-l-k}
(\underline{t})h_{j+k-a}(x_1,\underline{v})=\\
&= \sum_{l=0}^a\sum_{j=0}^ax_2^lq_j\sum_{k=0}^{a-l}(-1)^{a-l-k}e_{a-l-k}
(\underline{t})h_{j+k-a}(\underline{t})=_{(k\mapsto a-l-k)}\\
&= \sum_{l=0}^a\sum_{j=0}^ax_2^lq_j\sum_{k=0}^{a-l}(-1)^{k}e_{k}
(\underline{t})h_{j-l-k}(\underline{t}).\end{aligned}$$
Since $h_p(\underline{t})=0$ for $p<0$, we must have $k\le j-l(\le a-l)$ in the innermost summation, and so by (\[use2\]) the last expression above is equal to $$\begin{aligned}
&= \sum_{l=0}^a\sum_{j=0}^ax_2^lq_j\sum_{k=0}^{j-l}(-1)^{k}
e_{k}(\underline{t})h_{j-l-k}(\underline{t})=\sum_{l=0}^a\sum_{j=0}^ax_2^lq_j
\delta_{j-l,0}=\\
&= \sum_{j=0}^ax_2^jq_j=p_{\mid x_1\mapsto x_2},\end{aligned}$$ which is just the identity map, as wanted.\
$\bullet$ Images of bubbles by $\fbim$.\
Again we denote $\lambda_i=a$ and $\lambda_{i+1}=b$.
The clockwise oriented bubble with $r\ge 0$ dots on it is mapped by $\fbim$ as follows $$\begin{aligned}
\xy 0;/r.18pc/:
(0,0)*{\dblue\cbub{\black r}{\black i}};
(4,8)*{\scs\lambda};
\endxy
\quad
\mapsto &
\quad
\left(
\begin{aligned}
&
\quad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -20 193 \pinlabel $a$ at 150 191
\pinlabel $\color[rgb]{.5,.5,.5}{\nearrow}$ at -18 58
\pinlabel $\und{t}$ at -52 20
\pinlabel $\color[rgb]{.5,.5,.5}{\nwarrow}$ at 148 58
\pinlabel $\und{u}$ at 185 20
\endlabellist
\figins{-19}{0.6}{id2web}
\qquad\ra\qquad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -15 197 \pinlabel $a$ at 128 196
\pinlabel $\color[rgb]{.5,.5,.5}{\swarrow}$ at 130 30
\pinlabel $\und{u}$ at 162 48
\pinlabel $\color[rgb]{.5,.5,.5}{\swarrow}$ at 120 115
\pinlabel $\und{v}$ at 160 138
\pinlabel $x$ at 55 45
\pinlabel $y$ at 65 170
\pinlabel $\color[rgb]{.5,.5,.5}{\nearrow}$ at -18 148
\pinlabel $\und{t}$ at -50 110
\endlabellist
\figins{-19}{0.6}{sqweblr}
\qquad\ra\quad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -15 191 \pinlabel $a$ at 144 193
\endlabellist
\figins{-19}{0.6}{id2web}\ \ \
\\[1.0ex]
& \qquad\
p\mapsto
\partial_{x\und{v}}
\biggl(\sum\limits_{\ell=0}^{b}(-1)^{\ell}e_{b-\ell}(\und{t})x^{\ell+r} p\biggr)
\end{aligned}
\right)\end{aligned}$$
The polynomial $p=p(\underline{t},\underline{u})$ is symmetric in the variables $\underline{t}$ and $\underline{u}$ separately. In particular, we have $\partial_{x\underline{v}} (p\,q)=p\,\partial_{x\underline{v}}(q)$, for any polynomial $q$. We have: $$\begin{aligned}
\partial_{x\underline{v}}(\sum_{l=0}^b {(-1)^l e_{b-l}(\underline{t})x^{l+r}p})
&= \sum_{l=0}^b {(-1)^l
e_{b-l}(\underline{t})p\partial_{x\underline{v}}(x^{l+r})}=_{(\ref{use1})}\\
&= p\sum_{l=0}^b {(-1)^l e_{b-l}(\underline{t})h_{l+r-a+1}(\underline{u})}=_{(l\mapsto b-l)}\\
&= p (-1)^b\sum_{l=0}^b {(-1)^l e_{l}(\underline{t})h_{b-a+r+1-l}(\underline{u})}.\end{aligned}$$ Since $e_l(\underline{t})=0$ for $l>b$, and $h_{b-a+r+1-l}(\underline{u})=0$, for $l>b-a+r+1$, we have that the clockwise oriented bubble is mapped by $\fbim$ to the following bimodule map: $$\label{pom12}
p\mapsto p (-1)^b\sum_{l=0}^{b-a+r+1} {(-1)^l e_{l}
(\underline{t})h_{b-a+r+1-l}(\underline{u})}.$$ In particular, if $b-a+r+1<0$, i.e. if $r<a-b-1$, the bubble is mapped to zero, and if $r=a-b-1$, the bubble is mapped to $(-1)^b$ times the identity (note that $a-b=\lambda_i-\lambda_{i+1}$ is $\mathfrak{sl}_n$ weight). Also, $r$ can be naturally extended to $r\ge a-b-1$ (in (\[pom12\])), i.e. to include fake bubbles in the case $a\le b$.\
The counter-clockwise oriented bubble with $r\ge 0$ dots on it is mapped by $\fbim$ as follows\
$$\begin{aligned}
\xy 0;/r.18pc/:
(0,0)*{\dblue\ccbub{\black r}{\black i}};
(4,8)*{\scs\lambda};
\endxy
\quad
\mapsto &
\quad
\left(
\begin{aligned}
&
\quad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -15 194 \pinlabel $a$ at 144 192
\pinlabel $\color[rgb]{.5,.5,.5}{\nearrow}$ at -18 58
\pinlabel $\und{t}$ at -52 20
\pinlabel $\color[rgb]{.5,.5,.5}{\nwarrow}$ at 148 58
\pinlabel $\und{u}$ at 183 20
\endlabellist
\figins{-19}{0.6}{id2web}
\qquad\ra\qquad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -15 198 \pinlabel $a$ at 130 197
\pinlabel $\color[rgb]{.5,.5,.5}{\searrow}$ at -15 35
\pinlabel $\und{t}$ at -54 58
\pinlabel $\color[rgb]{.5,.5,.5}{\nwarrow}$ at 130 155
\pinlabel $\und{u}$ at 166 122
\pinlabel $\color[rgb]{.5,.5,.5}{\uparrow}$ at 58 38
\pinlabel $x$ at 58 0
\pinlabel $y$ at 45 172
\pinlabel $\color[rgb]{.5,.5,.5}{\searrow}$ at -10 121
\pinlabel $\und{v}$ at -50 140
\endlabellist
\figins{-19}{0.6}{sqwebrl}
\qquad\ra\quad
\labellist
\tiny\hair 2pt
\pinlabel $b$ at -15 191 \pinlabel $a$ at 145 193
\endlabellist
\figins{-19}{0.6}{id2web}\quad
\\[1.0ex]
&
\qquad
p\mapsto
\partial_{\und{v}x}
\left(\sum\limits_{\ell=0}^{a}(-1)^{\ell}x^{a-\ell+r}
e_{\ell}(\und{u})p\right)
\end{aligned}
\right)\end{aligned}$$
Completely analogously as above, we have that the counter-clockwise oriented bubble is mapped by $\fbim$ to the following bimodule map: $$\label{pom13}
p\mapsto p (-1)^{b+1}\sum_{l=0}^{a-b+r+1} {(-1)^l e_{l}(\underline{u})h_{a-b+r+1-l}
(\underline{t})}.$$ Again, from $r<b-a-1$, the bubble is mapped to zero, and if $r=b-a-1$, it is mapped to $(-1)^{b+1}$ times the identity. Moreover, $r$ can be naturally extended to $r\ge b-a-1$, i.e. to include fake bubbles in the case $b\le a$.\
Our reason for changing the signs from [@K-L3], was to make the signs in the image of the degree zero bubbles, i.e. $(-1)^b$ for the clockwise bubble and $(-1)^{b+1}$ for the counter-clockwise bubble, coincide with those of .
Finally, by the Giambelli and the dual Giambelli formulas (see e.g. [@F]), from (\[pom12\]) and (\[pom13\]) the infinite Grassmannian relation follows directly.
Comparisons with ${\mathcal U}(\mathfrak{sl}_n)$ {#sec:struct}
================================================
In this section we show the analogues for $\Scat(n,d)$ of some of Khovanov and Lauda’s results on the structure of $\Ucat$. Our results are far from complete. More work will need to be done to understand the structure of $\Scat(n,d)$ better.
To simplify terminology, by a $2$-functor we will always mean an additive $\Q$-linear degree preserving $2$-functor.
Categorical inclusions and projections
--------------------------------------
In the first place, we categorify the homomorphisms $\pi_{d',d}$ from Section \[sec:hecke-schur\].
Let $d'=d+kn$, with $k\in\bN$. We define a $2$-functor $$\Pi_{d',d}\colon \Scat(n,d')\to\Scat(n,d).$$ On objects and $1$-morphisms $\Pi_{d',d}$ is defined as $\pi_{d',d}$. On $2$-morphisms $\Pi_{d',d}$ is defined as follows. For any diagram $D$ in $\Scat(n,d')$ with regions labeled $\lambda\in\Lambda(n,d')$ such that $\lambda-(k^n)\in\Lambda(n,d)$, let $\Pi_{d',d}(D)$ be given by the same diagram with labels of the form $\lambda-(k^n)$, multiplied by $(-1)^k$ for every left cap and left cup in $D$. For any other diagram $D$, let $\Pi_{d',d}(D)=0$. Extend this definition to all $2$-morphisms by linearity.
Note that $\Pi_{d',d}$ is well-defined, because $\overline{\lambda}=\overline{\lambda-(k^n)}$. The extra $(-1)^k$ for left cups and caps is necessary to match our normalization of the degree zero bubbles. It also ensures that we have $$\Pi_{d',d}\Psi_{n,d'}=\Psi_{n,d},$$ where $$\Psi_{n,d}\colon \Ucat\to \Scat(n,d)$$ is the 2-functor defined in Definition \[defn:Psi\].
Note also that the $\Pi_{d',d}$ form something like an inverse system of $2$-functors between $2$-categories, because $$\Pi_{d',d}\Pi_{d'',d'}=\Pi_{d'',d}$$ (compare to ). We say “something like” an inverse system, because we have not been able to find a precise definition of such a structure in the literature on $n$-categories. Also one would have to think carefully if the “inverse limit” of the $\Scat(n,d)$ would still be Krull-Schmidt. Finally, there appears to be no general theorem that says that the Grothendieck group of an inverse limit is the inverse limit of the Grothendieck groups (even for algebras there is no such theorem). So we cannot (yet) reasonably conjecture the categorification of the embedding . All we can say at the moment is the following:
We have: \[cor:inverselimit\]
1. Let $f1_{\alpha}$ be a $2$-morphism in $\Ucat$. Let $d_0>0$ be the minimum value such that $\alpha=\overline{\beta}$ with $\beta\in\Lambda(n,d_0)$. Then $f=0$ if and only if $\Psi_{n,d_0+nk}(f)=0$ for any $k\geq 0$.
2. Let $\{f_i1_{\alpha}\}_{i=1}^{s}$ be a finite set of $2$-morphisms in $\HomU(x,y)$. Then the $f_i1_{\alpha}$ are linearly independent if and only if there exists a $d>0$ such that the $\Psi_{n,d}(f_i1_{\lambda})$ are linearly independent in $\HomS(x,y)$.
The proof of Corollary \[cor:inverselimit\] follows from Khovanov and Lauda’s Lemma 6.16 in [@K-L3], which implies Theorem 1.3 in [@K-L3], our Lemma \[lem:rep\] and the remarks above Corollary \[cor:inverselimit\].
The main reason for trying to categorify is the following: if the inverse limit of the $\Scat(n,d)$ turns out to exist, perhaps it contains a sub-2-category which categorifies ${\mathbf U}_q(\mathfrak{sl}_n)$.
The structure of the 2HOM-spaces
--------------------------------
We now turn our attention to the structure of the 2HOM-spaces in $\Scat(n,d)$. The reader should compare our results to Khovanov and Lauda’s in [@K-L3]. We first show the analogue of Lemma 6.15.
### Bubbles for $n=2$
For starters suppose that $n=2$. Let $\lambda=(a,b)\in\Lambda(2,d)$. Recall that a partially symmetric polynomial $p(\underline{x},\underline{y})=p(\underline{x},\underline{y})\in
\Q[x_1,\ldots,x_a,y_1,\ldots,y_b]^{S_a\times S_b}$ is called [*supersymmetric*]{} if the substitution $x_1=t=y_1$ gives a polynomial independent of $t$ (see [@F-P] and [@McD] for example). We let $R_{a,b}^{ss}$ denote the ring of supersymmetric polynomials. The [*elementary*]{} supersymmetric polynomials are $$e_{j}(\underline{x},\underline{y})=\sum_{s=0}^j (-1)^s h_{j-s}(\underline{x})\varepsilon_s(\underline{y}),$$ where $h_{j-s}(\underline{x})$ is the $j-s$th complete symmetric polynomial in $a$ variables and $\varepsilon_s(\underline{y})$ the $s$th elementary symmetric polynomial in $b$ variables, which we put equal to zero if $s>b$ by convention. It is easy to see that $e_j(\underline{x},\underline{y})$ is supersymmetric, because we have $$\prod_{r=1}^a\prod_{s=1}^b\dfrac{1-y_rZ}{1-x_sZ}=\sum_j e_j(\underline{x},\underline{y})Z^j.$$ Using the supersymmetric analogue of the Giambelli formula we can define the [*supersymmetric Schur polynomials*]{} $$\pi_{\alpha}(\underline{x},\underline{y})=\det(e_{\alpha_i+j-i}(\underline{x},\underline{y}))$$ for $1\leq i,j\leq m$ and $\alpha$ a partition of length $m$. In the following lemma we give the basic facts about supersymmetric Schur polynomials, which are of interest to us in this paper. For the proofs see [@F-P; @McD] and the references therein. Let $\Gamma(a,b)$ be the set of partitions $\alpha$ such that $\alpha_j\leq b$ for all $j>a$.
\[lem:ss\] We have
1. If $\alpha\not\in\Gamma(a,b)$, then $\pi_{\alpha}(\underline{x},\underline{y})=0$.
2. The set $\{\pi_{\alpha}(\underline{x},\underline{y})\,|\,\alpha\in\Gamma(a,b)\}$ is a linear basis of $R_{a,b}^{ss}$.
3. We have $$\pi_{\alpha}(\underline{x},\underline{y})\pi_{\beta}(\underline{x},\underline{y})=\sum_{\gamma}C_{\alpha\beta}^{\gamma}\pi_{\gamma}(\underline{x},\underline{y}),$$ where $C_{\alpha\beta}^{\gamma}$ are the Littlewood-Richardson coefficients.
4. We have $$\pi_{\alpha}(\underline{x},\underline{y})=(-1)^{|\alpha|}\pi_{\alpha'}(\underline{y},\underline{x}),$$ where $|\alpha|=\sum_i\alpha_i$ and $\alpha'$ is the conjugate partition.
5. We also get the ordinary Schur polynomials as special cases $$\begin{aligned}
\pi_{\alpha}(\underline{x},0) &= \pi_{\alpha}(\underline{x})
\\
\pi_{\beta}(0,\underline{y}) &= (-1)^{|\beta|}\pi_{\beta'}(\underline{y}).\end{aligned}$$
In [@K-L-M-S] the [*extended calculus*]{} in ${\mathcal U}(\mathfrak{sl}_2)$ was developed. Here we only use a little part of it. Below, for partitions $\alpha,\beta$ with length $m$, we write $\alpha^{\spadesuit}=\alpha-(a-b)-m$ for counter-clockwise oriented bubbles of thickness $m$ in a region labeled $(a,b)$, and $\beta^{\spadesuit}=\beta+(a-b)-m$ for clockwise oriented bubbles of thickness $m$. Recall that thick bubbles labeled by a spaded Schur polynomial can be written as Giambelli type determinants (see Equations (3.33) and (3.34) in [@K-L-M-S], but bear our sign conventions in mind): $$\label{eq_ccbub_det}
\xy
(0,0)*{\stccbub{m}{\alpha}};
(0,9)*{\scs (a,b)};
\endxy \quad := \quad
\left|
\begin{array}{ccccc}
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_1+1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_1+2}{}};(5,7)*{\scs (a,b)};
\endxy & \cdots &
\xy 0;/r.18pc/:
(0,0)*{\laudaccbub{\spadesuit+\alpha_1+(m-1)}{}};(5,7)*{\scs (a,b)}; \endxy \\ \\
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_2-1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_2}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_2+1}{}};(5,7)*{\scs (a,b)};
\endxy & \cdots &
\xy 0;/r.18pc/:
(0,0)*{\laudaccbub{\spadesuit+\alpha_2+(m-2)}{}};(5,7)*{\scs (a,b)}; \endxy \\ \\
\cdots & \cdots & \cdots & \cdots & \cdots \\ \\
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_m-m+1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_m-m+2}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_m-m+3}{}};(5,7)*{\scs (a,b)};
\endxy & \cdots &
\xy 0;/r.18pc/: (0,0)*{\laudaccbub{\spadesuit+\alpha_m}{}};(5,7)*{\scs (a,b)}; \endxy
\end{array}
\right|$$
$$\label{eq_cbub_det}
\xy
(0,0)*{\stcbub{m}{\beta}};
(0,9)*{\scs (a,b)};
\endxy \quad := \quad
\left|
\begin{array}{ccccc}
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_1+1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_1+2}{}};(5,7)*{\scs (a,b)};
\endxy & \cdots &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_1+(m-1)}{}};(5,7)*{\scs (a,b)};
\endxy \\ \\
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_2-1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_2}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_2+1}{}};(5,7)*{\scs (a,b)};
\endxy & \cdots &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_2+(m-2)}{}};(5,7)*{\scs (a,b)};
\endxy \\ \\
\cdots & \cdots & \cdots & \cdots & \cdots \\ \\
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_m-m+1}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_m-m+2}{}};(5,7)*{\scs (a,b)}; \endxy &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_m-m+3}{}};(5,7)*{\scs (a,b)};
\endxy & \cdots &
\xy 0;/r.18pc/: (0,0)*{\laudacbub{\spadesuit+\beta_m}{}};(5,7)*{\scs (a,b)}; \endxy
\end{array}
\right|.$$
The reader unfamiliar with [@K-L-M-S] can interpret the above simply as definitions. In Proposition 4.10 in [@K-L-M-S] it is proved that the clockwise thick bubbles form a linear basis of $\ENDU(1_{a-b})$ and that they obey the Littlewood-Richardson rule under multiplication. Of course the counter-clockwise thick bubbles form another basis and also obey the L-R rule. Proposition 4.10 in [@K-L-M-S] also shows the relation between the two bases (recall that we have slightly different sign conventions in this paper and that $\alpha'$ is the partition conjugate to $\alpha$): $$\label{eq:c-cc}
\xy
(0,0)*{\stcbub{m}{\alpha}};
(0,9)*{\scs (a,b)};
\endxy \quad =(-1)^{|\alpha|+m}\quad
\xy
(0,0)*{\stccbub{m}{\alpha'}};
(0,9)*{\scs (a,b)};
\endxy.$$ Therefore, in our case the non-zero clockwise thick bubbles also form a nice basis of $\ENDS(1_{(a,b)})$.
\[lem:superschur\] $\fbim\colon \ENDS(1_{(a,b)})\to R_{a,b}^{ss}$ is a ring isomorphism, mapping the clockwise thick bubbles to the corresponding supersymmetric Schur polynomials.
It is clear that the thick bubbles generate $\ENDS(1_{(a,b)})$, because they are the image of the thick bubbles in $\ENDU(1_{a-b})$, which form a linear basis. Since $\Psi_{n,d}$ is a $2$-functor, we see that the multiplication of bubbles in $\ENDS(1_{(a,b)})$ satisfies the Littlewood-Richardson rule. In Section \[sec:2rep\] we showed that using $\fbim$ we get $$\begin{aligned}
\xy 0;/r.18pc/:
(0,0)*{\dblue\cbub{\black r}{\black i}};
(4,8)*{\scs(a,b)};
\endxy
\quad
\mapsto &\quad
(-1)^b e_{-(a-b)+1+r}(\underline{x},\underline{y}).\end{aligned}$$ This implies that $$\xy
(0,0)*{\stcbub{m}{\beta}};
(0,9)*{\scs (a,b)};
\endxy \qquad\mapsto\quad (-1)^{mb}\pi_{\beta}(\underline{x},\underline{y}).$$ Therefore, by Lemma \[lem:ss\], all we have to show is that $$\xy
(0,0)*{\stcbub{m}{\beta}};
(0,9)*{\scs (a,b)};
\endxy\qquad = \quad 0$$ if $\beta\not\in\Gamma(a,b)$. We proceed by induction on $m$. Note that if $m<a+1$, then $\beta\in\Gamma(a,b)$, so the induction starts at $m=a+1$. If $m=a+1$, then $\beta_{a+1}=\beta_m>b$ implies that $\beta_i>b$ holds for all $i=1,\ldots,m$, because $\beta$ is a partition. Therefore, for any $i=1,\ldots,m$, we have $$\beta_i+a-b-m=\beta_i+a-b-(a+1)=\beta_i-b-1\geq 0.$$ Thus the bubble is real and equals zero because its inner region is labeled $(-1,a+b+1)\not\in\Lambda(2,d)$.
Suppose that $m>a+1$ and that the result has been proved for bubbles of thickness $<m$. Using induction, we will prove that it holds for bubbles of thickness $m$. The trouble is that in this case the bubble can be fake, so we cannot repeat the argument above. Instead we use a second induction, this time on $\beta_m$. Write $\beta'=(\beta_1,\ldots,\beta_{m-1})$. First suppose $\beta_m=0$. Then
$$\xy
(0,0)*{\stcbub{m}{\beta}};
(0,9)*{\scs (a,b)};
\endxy\qquad = \quad (-1)^b
\xy
(0,0)*{\stcbub{m-1\mspace{40mu}}{\beta'}};
(0,9)*{\scs (a,b)};
\endxy\qquad = \quad 0$$
by induction on $m$. Now suppose $\beta_m>0$. Then we have $$\xy
(0,0)*{\stcbub{m-1\mspace{40mu}}{\beta'}};
(8,9)*{\scs (a,b)};
(20,0)*{\dblue\ccbub{\black a-b-1+\beta_m}{}};
\endxy\ = \quad 0$$ by induction on $m$. By Pieri’s rule, the left-hand side equals $$\qquad \sum_{\beta<\gamma\leq\beta+(\beta_m)}
\xy
(0,0)*{\stcbub{m}{\gamma}};
(8,9)*{\scs (a,b)};
\endxy
\ +\quad
\xy
(0,0)*{\stcbub{m}{\beta}};
(8,9)*{\scs (a,b)},
\endxy$$ where $\beta+(\beta_m)=(\beta_1+\beta_m,\beta_2,\ldots,\beta_{m-1},0)$. Note that for any $\beta<\gamma\leq \beta+(\beta_m)$, we have $\gamma\not\in\Gamma(a,b)$ and $\gamma_m<\beta_m$. Thus, by induction on $\beta_m$, all the thick bubbles labeled with $\pi^{\spadesuit}_\gamma$ are zero. This implies that $$\xy
(0,0)*{\stcbub{m}{\beta}};
(8,9)*{\scs (a,b)}.
\endxy\ =\quad 0.$$
Note that for bubbles with the opposite orientation we have
$$\begin{aligned}
\xy 0;/r.18pc/:
(0,0)*{\dblue\ccbub{\black r}{\black i}};
(4,8)*{\scs(a,b)};
\endxy
\
\mapsto &\quad
(-1)^{b+1} e_{(a-b)+1+r}(y,x).\end{aligned}$$
This implies that
$$\xy
(0,0)*{\stccbub{m}{\alpha}};
(0,9)*{\scs (a,b)};
\endxy \quad \mapsto \quad (-1)^{m(b+1)}\pi_{\alpha}(y,x).$$
This way we get another isomorphism between $\ENDS(1_{(a,b)})$ and $R^{ss}_{a,b}$.
### Bubbles for $n>2$
For $n>2$, we get polynomials in thick bubbles of $n-1$ colors. Unfortunately we have not been able to find anything in the literature on a generalization of supersymmetric polynomials to more than two alphabets. Nor has the extended calculus for $\Ucat$ been worked out and written up for $n>2$ so far. Therefore all we can say is the following. Let $$S\Pi_{\lambda}=\bigotimes_{i=1}^{n-1}R_{\lambda_i,\lambda_{i+1}}^{ss}.$$ There is a surjective homomorphism $$S\Pi_{\lambda}\to
\ENDS(1_{\lambda})$$ sending supersymmetric polynomials to the corresponding clockwise oriented thick bubbles. Note that $\Psi_{n,d}\colon \Pi_{\overline{\lambda}}\cong \ENDU(1_{\overline{\lambda}})\to
\ENDS(1_{\lambda})$ factors through $S\Pi_{\lambda}$. Recall that $$\Pi_{\overline{\lambda}}\cong \bigotimes_{i=1}^{n-1}\Lambda(\underline{x}),$$ where $\Lambda(\underline{x})$ is the ring of symmetric functions in infinitely many variables $\underline{x}=(x_1,x_2,\ldots)$ (see (3.24) and Lemma 6.15 in [@K-L3]). The map $\Pi_{\overline{\lambda}}\to S\Pi_{\lambda}$ referred to above is defined by $$\pi^i_{\alpha}(\underline{x})\mapsto \pi_{\alpha}(\underline{x},\underline{y}),$$ where $(\underline{x},\underline{y})
=(x_1,\ldots,x_{\lambda_i},y_1,\ldots,y_{\lambda_{i+1}})$ and $\pi^i_{\alpha}(\underline{x})=1\otimes\cdots\otimes
\pi_{\alpha}(\underline{x})\otimes\cdots\otimes 1 $ belongs to the $i$-th tensor factor and .
Note also that the projection $$S\Pi_{\lambda}\to
\ENDS(1_{\lambda})$$ is not an isomorphism in general. For example, with blue bubbles colored 1 and red bubbles colored 2, we have $$\begin{aligned}
\xy 0;/r.18pc/:
(0,0)*{\dred\cbub{\black 1}{2}};
(4,9)*{\scs(0,1,0)};
\endxy
- \ \
\xy 0;/r.18pc/:
(0,-1)*{\dblue\cbub{\black -1}{1}};
(4,8)*{\scs(0,1,0)};
\endxy
\
=&\quad 0.
\label{eq:twistbub3}\end{aligned}$$ To see why this holds, first use $$\label{eq:non-inj}
\xy 0;/r.18pc/:
(0,0)*{\dred\cbub{\black 1}{2}};
(4,9)*{\scs(0,1,0)};
\endxy
=\quad
\xy 0;/r.18pc/:
(0,0)*{\dblue\ccbub{\black 0}{1}};
(14,0)*{\dred\cbub{\black 1}{2}};
(7,9)*{\scs(0,1,0)};
\endxy$$ This equation holds because $$\xy 0;/r.18pc/:
(0,0)*{\dblue\ccbub{\black 0}{1}};
(4,9)*{\scs(0,1,0)};
\endxy
=1.$$ Then slide the red bubble inside the blue one on the r.h.s. of with bubble-slide . Note that we have to switch the colors $i$ and $i+1$ in , but that only changes the sign on the r.h.s. of that bubble-slide, as remarked below the list of bubble-slides. After doing that, only one blue bubble with one dot survives, because in the interior of that bubble, which is labeled $(1,0,0)$, only a degree zero red bubble is non-zero. This holds because the red bubbles of positive degree are real bubbles and their interior is labeled $(1,-1,1)\not\in\Lambda(3,1)$. The degree zero red bubble is equal to 1, by . Thus we have obtained $$\xy 0;/r.18pc/:
(0,0)*{\dred\cbub{\black 1}{2}};
(4,9)*{\scs(0,1,0)};
\endxy
=
\xy 0;/r.18pc/:
(0,0)*{\dblue\ccbub{\black 1}{1}};
(4,9)*{\scs(0,1,0)};
\endxy,$$ which is equal to $$\xy 0;/r.18pc/:
(0,-1)*{\dblue\cbub{\black -1}{1}};
(4,8)*{\scs(0,1,0)};
\endxy$$ by the infinite Grassmannian relation and relation .
The relation above between bubbles of different colors generalizes. Using the extended calculus for $\Scat(n,d)$ [@K-L-M-S], we can see that whenever $\lambda$ is of the form $(\ldots,0,\lambda_i,0,\ldots)$, bubbles of the same degree of colors $i-1$ and $i$ are equal up to a sign. This also has to do with the fact that compositions of $d$ of the form $(\ldots,a,0,\dots)$ and $(\ldots,0,a,\ldots)$ are equivalent as objects in the Karoubi envelope $\ScatD(n,d)$. We will explain this in Remark \[rem:zeros\]. Here we just leave a conjecture about $\EndS(1_{\lambda})$.
\[conj:bubbles\] Let $\lambda\in\Lambda(n,d)$ be arbitrary and let $\mu\in\Lambda(n,d)$ be obtained from $\lambda$ by placing all zero entries of $\lambda$ at the end, but without changing the relative order of the non-zero entries, e.g. for $\lambda=(2,0,1)$ we get $\mu=(2,1,0)$. Then we conjecture that $$\EndS(1_{\lambda})\cong S\Pi_{\mu}.$$
Note that if $\mu_k\ne 0$ and $\mu_{k+1}=0$ for a certain $1\leq k\leq n-1$ in Conjecture \[conj:bubbles\], then $S\Pi_{\mu}$ is isomorphic to the algebra of all partially symmetric polynomials $\Q[x_1,\ldots,x_d]^{S_{\mu_1}\times\cdots\times S_{\mu_k}}$. This follows from the fact that $R^{ss}_{\mu_k,0}$ is the algebra of symmetric polynomials in $\mu_k$ variables. For example, suppose $\mu=(1,1,0)$. Then $R^{ss}_{1,1}\cong \Q[x-y]$ and $R^{ss}_{1,0}\cong\Q[y]$, so $S\Pi_{(1,1,0)}\cong \Q[x-y]\otimes\Q[y]$. The latter algebra is isomorphic to $\Q[x,y]$ by $$(x-y)\otimes 1+1\otimes y \leftrightarrow x,\qquad
1\otimes y\leftrightarrow y.$$
### More general 2-morphisms
There is not all that much that we know about more general $2$-hom spaces in $\Scat(n,d)$. Let us give a conjecture about an “analogue” of Lemma 3.9 from [@K-L3] for $\Scat(n,d)$. Let $\nu\in\N[I]$ and ${\mathbf i},{\mathbf j}\in\nu$. Recall (see Section 2 in [@K-L1] and Subsection 3.2.2 in [@K-L3]) that ${}_{\mathbf i}R(\nu)_{\mathbf j}$ is the vector space of upwards oriented braid-like diagrams as in $\Ucat$ whose lower boundary is labeled by $\mathbf i$ and upper boundary by $\mathbf j$, modulo the braid-like relations in $\Ucat$. Note that all strands of such a diagram have labels uniquely determined by $\mathbf i$ and $\mathbf j$. Note also that the braid-like relations in $\Ucat$ are independent of the weights, so the definition of ${}_{\mathbf i}R(\nu)_{\mathbf j}$ does not involve weights. Unfortunately, we cannot define the analogue of ${}_{\mathbf i}R(\nu)_{\mathbf j}$ for $\Scat(n,d)$, because there the braid-like diagrams with a region labeled by a weight outside $\Lambda(n,d)$ are equal to zero, creating a weight dependence. However, we will be able to use ${}_{\mathbf i}R(\nu)_{\mathbf j}$ and the fact that $\Psi_{n,d}$ is full. Khovanov and Lauda (Lemma 3.9, Definition 3.15 and the remarks thereafter, and Theorem 1.3 in [@K-L3]) showed that the obvious map $$\Psi_{{\mathbf i},{\mathbf j},\overline{\lambda}}\colon
{}_{\mathbf i}R(\nu)_{\mathbf j}\otimes \Pi_{\overline{\lambda}}\to
\HOMU({\mathcal E}_{\mathbf i}1_{\overline{\lambda}},
{\mathcal E}_{\mathbf j}1_{\overline{\lambda}})$$ is an isomorphism. Unfortunately it is also impossible to factor $\HOMS({\mathcal E}_{\mathbf i}1_{\lambda}, {\mathcal E}_{\mathbf j}1_{\lambda})$ so nicely into braid-like diagrams and bubbles. For example, let $\lambda=(0,1)$ and look at the following reduction to bubble relation
$$0 \; =\; \text{$\xy 0;/r.18pc/:
(12,8)*{\scs (0,1)};
%(0,0)*{\twoIu{i}};
(0,-3)*{\dblue\xybox{
(-3,-8)*{};(3,8)*{} **\crv{(-3,-1) & (3,1)}?(1)*\dir{>};?(0)*\dir{>};
(3,-8)*{};(-3,8)*{} **\crv{(3,-1) & (-3,1)}?(1)*\dir{>};
(-3,-12)*{\bbsid};
(-3,8)*{\bbsid};
(3,8)*{}="t1";
(9,8)*{}="t2";
(3,-8)*{}="t1'";
(9,-8)*{}="t2'";
"t1";"t2" **\crv{(3,14) & (9, 14)};
"t1'";"t2'" **\crv{(3,-14) & (9, -14)};
"t2'";"t2" **\dir{-} ?(.5)*\dir{<};}};
(9,0)*{}; (-7.5,-12)*{\scs i};
\endxy$} \; = \; -\sum_{f=0}^{1}
\xy
(19,4)*{\scs (0,1)};
(0,0)*{\dblue\bbe{}};(-2,-8)*{\scs i};
(12,-2)*{\dblue\cbub{\black f}{\black i}};
(0,6)*{\dblue\bullet}+(6,1)*{\scs 1-f};
\endxy.$$ This result generalizes to any $\lambda$, using the extended calculus in [@K-L-M-S]. Thus, given any $\lambda$, there is an upper bound $t_r$ for the number of dots on the arcs of the $r$-strands. Any braid-like diagram in $\HOMS({\mathcal E}_{\mathbf i}1_{\lambda},
{\mathcal E}_{\mathbf j}1_{\lambda})$ with more than $t_r$ dots on an $r$-colored strand can be written as a linear combination of braid-like diagrams whose $r$-strands have $\leq t_r$ dots with coefficients in $\ENDS(1_{\lambda})$. By the fullness of $\Psi_{n,d}$ and the fact that ${}_{\mathbf i}R(\nu)_{\mathbf j}$ has a basis ${}_{\mathbf i}B_{\mathbf j}$ which only contains a finite number of braid-like diagrams if one forgets the dots (see Theorem 2.5 in [@K-L1]), it follows that $\HOMS({\mathcal E}_{\mathbf i}1_{\lambda},
{\mathcal E}_{\mathbf j}1_{\lambda})$ is finitely generated over $\ENDS(1_{\lambda})$. In Section \[sec:soergel\] we will say a little more about the image of $$B_{{\mathbf i},{\mathbf j},\overline{\lambda}}=
\Psi_{{\mathbf i},{\mathbf j},\overline{\lambda}}
({}_{\mathbf i}B_{\mathbf j})\subseteq \HOMU({\mathcal E}_{\mathbf i}
1_{\overline{\lambda}},{\mathcal E}_{\mathbf j}1_{\overline{\lambda}})$$ in $\HOMS({\mathcal E}_{\mathbf i}1_{\lambda},
{\mathcal E}_{\mathbf j}1_{\lambda})$ under $\Psi_{n,d}$. Recall again that ${}_{\mathbf i}B_{\mathbf j}$ is Khovanov and Lauda’s basis of ${}_{\mathbf i}R(\nu)_{\mathbf j}$ in Theorem 2.5 in [@K-L1]. Unfortunately, all we can give for now is a conjecture.
\[conj:struct\] We conjecture that $\HOMS({\mathcal E}_{\mathbf i}1_{\lambda},
{\mathcal E}_{\mathbf j}1_{\lambda})$ is a free right module of finite rank over $\ENDS(1_{\lambda})$.
Note that if ${\mathcal E}_{\mathbf i}1_{\lambda}=1_{\mu}{\mathcal E}_{\mathbf i}$ and ${\mathcal E}_{\mathbf j}1_{\lambda}=1_{\mu}{\mathcal E}_{\mathbf j}$, then we also conjecture that $\HOMS(1_{\mu}{\mathcal E}_{\mathbf i},
1_{\mu}{\mathcal E}_{\mathbf j})$ is a free left module of finite rank over $\ENDS(1_{\mu})$. However, it is not hard to give examples which show that, if the conjectures are true at all, the ranks of $\HOMS(1_{\mu}{\mathcal E}_{\mathbf i}1_{\lambda},1_{\mu}{\mathcal E}_{\mathbf j}1_{\lambda})$ as a right $\ENDS(1_{\lambda})$-module and as a left $\ENDS(1_{\mu})$-module are not equal in general. This is not surprising, because the graded dimensions of $\ENDS(1_{\lambda})$ and $\ENDS(1_{\mu})$ are not equal in general either.
The categorical anti-involution
-------------------------------
The last part of this section is dedicated to the categorification of the anti-involution $\tau\colon \SD(n,d)\to\SD(n,d)^{\mbox{\scriptsize op}}$ in Section \[sec:hecke-schur\]. We simply follow Khovanov and Lauda’s Subsection 3.3.2. Let $\Scat(n,d)^{\mbox{\scriptsize coop}}$ denote the 2-category which the same objects as $\Scat(n,d)$, but with the directions of the 1- and 2-morphisms reversed. We define a strict degree preserving 2-functor $\tilde{\tau}\colon \Scat(n,d)\to \Scat(n,d)^{\mbox{\scriptsize coop}}$ by $$\begin{aligned}
\lambda&\mapsto\lambda\\
1_{\mu}\mathcal{E}_{s_1}\mathcal{E}_{s_2}\cdots\mathcal{E}_{s_{m-1}}
\mathcal{E}_{s_m}1_{\lambda}\{t\}&\mapsto
1_{\lambda}\mathcal{E}_{-s_m}\mathcal{E}_{-s_{m-1}}\cdots\mathcal{E}_{-s_2}
\mathcal{E}_{-s_1}1_{\mu}\{-t+t'\}\\
\zeta&\mapsto\zeta^*.\end{aligned}$$ Let $D$ be a diagram, then $D^*$ is obtained from $D$ by rotating the latter $180^{\circ}$. Since $\Scat(n,d)$ is cyclic, it does not matter in which way you rotate. By linear extension this defines $\zeta^*$ for any $2$-morphism. The shift $t'$ is defined by requiring that $\tilde{\tau}$ be degree preserving. One can easily check that $\tilde{\tau}$ is well-defined. For more details on the analogous $\tilde{\tau}$ defined on $\Ucat$ see Subsection 3.3.2 in [@K-L3]. As a matter of fact $\tilde{\tau}$ is a functorial anti-involution. The most important result about $\tilde{\tau}$ is the analogue of Remark 3.20 in [@K-L3].
\[lem:tildetau\] There are degree zero isomorphisms of graded $\Q$-vector spaces $$\begin{aligned}
\HOMS(fx,y)&\cong \HOMS(x,\tilde{\tau}(f)y)\\
\HOMS(xg,y)&\cong \HOMS(x,y\tilde{\tau}(g)),\end{aligned}$$ for any $1$-morphisms $x,y,f,g$.
The diagrammatic Soergel categories and $\Scat(n,d)$ {#sec:soergel}
====================================================
The diagrammatic Soergel category revisited {#ssec:soergel}
-------------------------------------------
In this subsection we recall the diagrammatics for Soergel categories introduced by Elias and Khovanov in [@E-Kh]. Actually we first recall the version sketched by Elias and Khovanov in Section 4.5 and used by Elias and Krasner in [@E-Kr]. After that we will comment on how to alter it in order to get the original version by Elias and Khovanov. Note that both versions categorify the Hecke algebra, although they are not equivalent as categories. In this paper we will need both versions.
Fix a positive integer $n$. The category $\mathcal{SC}_1(n)$ is the category whose objects are finite length sequences of points on the real line, where each point is colored by an integer between $1$ and $n-1$. We read sequences of points from left to right. Two colors $i$ and $j$ are called [*adjacent*]{} if $\vert i-j\vert=1$ and [*distant*]{} if $\vert i-j\vert >1$. The morphisms of $\mathcal{SC}_1(n)$ are given by generators modulo relations. A morphism of $\mathcal{SC}_1(n)$ is a $\Q$-linear combination of planar diagrams constructed by horizontal and vertical gluings of the following generators (by convention no label means a generic color $j$):
- Generators involving only one color: $$\xymatrix@R=1.0mm{
\figins{-15}{0.5}{enddot}
&
\figins{-9}{0.5}{startdot}
&
\figins{-15}{0.55}{merge}
&
\figins{-15}{0.55}{split}
\\
\text{EndDot} & \text{StartDot} & \text{Merge} & \text{Split}
}$$
It is useful to define the cap and cup as $$\figins{-17}{0.55}{dvert-u}\
\equiv\
\figins{-17}{0.55}{cap-v}\
\mspace{50mu}
\figins{-17}{0.55}{dvert-d}\
\equiv\
\figins{-17}{0.55}{cup-v}$$
- Generators involving two colors:
- The 4-valent vertex, with distant colors, $$\labellist
\tiny\hair 2pt
\pinlabel $i$ at -4 -10
\pinlabel $j$ at 134 -10
\pinlabel $i$ at 134 140
\pinlabel $j$ at -4 140
\endlabellist
\figins{-15}{0.55}{4vert}\vspace{1.5ex}$$
- and the 6-valent vertex, with adjacent colors $i$ and $j$ $$\labellist
\tiny\hair 2pt
\pinlabel $i$ at -4 -10
\pinlabel $j$ at 66 -12
\pinlabel $i$ at 136 -10
\pinlabel $j$ at -4 140
\pinlabel $i$ at 66 140
\pinlabel $j$ at 136 140
\pinlabel $j$ at 250 -10
\pinlabel $i$ at 320 -10
\pinlabel $j$ at 390 -10
\pinlabel $i$ at 250 140
\pinlabel $j$ at 320 140
\pinlabel $i$ at 390 140
\endlabellist
\figins{-17}{0.55}{6vertd}
\mspace{55mu}
\figins{-17}{0.55}{6vertu}\ .
\vspace{1.5ex}$$
In this setting a diagram represents a morphism from the bottom boundary to the top. We can add a new colored point to a sequence and this endows $\mathcal{SC}_1(n)$ with a monoidal structure on objects, which is extended to morphisms in the obvious way. Composition of morphisms consists of stacking one diagram on top of the other.
We consider our diagrams modulo the following relations.
*”Isotopy” relations:* $$\label{eq:adj}
\figins{-17}{0.55}{biadj-l}\
=\
\figins{-17}{0.55}{line}\
=\
\figins{-17}{0.55}{biadj-r}$$
$$\label{eq:curldot}
\figins{-17}{0.55}{dcurl-ul}\
=\
\figins{-17}{0.55}{enddot}\
=\
\figins{-17}{0.55}{dcurl-ur}$$
$$\label{eq:v3rot}
\figins{-17}{0.55}{yl}\
=\
\figins{-17}{0.55}{merge}\
=\
\figins{-17}{0.55}{ly}$$
$$\label{eq:v4rot}
\figins{-17}{0.55}{4vertr-l}\
=\
\figins{-17}{0.55}{4vert}\
=\
\figins{-17}{0.55}{4vertr-r}$$
$$\label{eq:v6rot}
\figins{-17}{0.55}{6vertu-l}\
=\
\figins{-17}{0.55}{6vertu}\
=\
\figins{-17}{0.55}{6vertu-r}$$
The relations are presented in terms of diagrams with generic colorings. Because of isotopy invariance, one may draw a diagram with a boundary on the side, and view it as a morphism in $\mathcal{SC}_1(n)$ by either bending strands up or down. By the same reasoning, a horizontal line corresponds to a sequence of cups and caps.
*One color relations:*
$$\label{eq:dumbrot}
\figins{-16}{0.5}{dumbells}\
=\
\figins{-14}{0.45}{dumbellh}$$
$$\label{eq:lollipop}
\figins{-17}{0.55}{lollipop-u}\
=\
0$$
$$\label{eq:deltam}
\figins{-17}{0.55}{startenddot-edge}\
+\
\figins{-17}{0.55}{edge-startenddot}\
=\ 2\
\figins{-17}{0.55}{matches-ud}$$
*Two distant colors:* $$\label{eq:reid2dist}
\figins{-32}{0.9}{reid2}\
=\
\figins{-32}{0.9}{line-br}$$
$$\label{eq:slidedotdist}
\figins{-16}{0.5}{4vertdot}\
=\
\figins{-16}{0.5}{4vertnodot}$$
$$\label{eq:slide3v}
\figins{-17}{0.55}{splitslide-u}\
=\
\figins{-17}{0.55}{splitslide-d}$$
*Two adjacent colors:* $$\label{eq:dot6v}
\figins{-16}{0.5}{6vertdotd}\
=\
\figins{-16}{0.5}{mergedots}\
+\
\figins{-16}{0.5}{capcupdot}$$
$$\label{eq:reid3}
\figins{-30}{0.85}{id-r3}\
=\
\figins{-30}{0.85}{reid3}\
-\
\figins{-30}{0.85}{dumbell-dd}$$
$$\label{eq:dumbsq}
\figins{-30}{0.85}{dumbellsquare}\
=\
\figins{-30}{0.85}{dumbellsquareh}$$
$$\label{eq:slidenext}
\labellist
\tiny\hair 2pt
\pinlabel $j$ at -15 35
\pinlabel $i$ at 46 -10
\endlabellist
\figins{-17}{0.55}{sedot-edge-d}\
-\
\labellist
\tiny\hair 2pt
\pinlabel $j$ at 63 35
\pinlabel $i$ at 5 -10
\endlabellist
\figins{-17}{0.55}{edge-sedot-d}\quad
=\
\frac{1}{2}
\Biggl(\
\labellist
\tiny\hair 2pt
\pinlabel $i$ at 58 35
\pinlabel $i$ at 5 -10
\endlabellist
\figins{-17}{0.55}{edge-startenddot}\
-\
\labellist
\tiny\hair 2pt
\pinlabel $i$ at -5 35
\pinlabel $i$ at 48 -10
\endlabellist
\figins{-17}{0.55}{startenddot-edge}\
\Biggr)$$
*Relations involving three colors:* (adjacency is determined by the vertices which appear) $$\label{eq:slide6v}
\figins{-18}{0.6}{6vert-slidel}\
=\
\figins{-18}{0.6}{6vert-slider}$$
$$\label{eq:slide4v}
\figins{-18}{0.6}{4vert-slidel}\
=\
\figins{-18}{0.6}{4vert-slider}$$
$$\label{eq:dumbdumbsquare}
\figins{-30}{0.85}{dumbdumbsquare}\
=\
\figins{-30}{0.85}{dumbdumbsquareh}.$$
Introduce a grading on $\mathcal{SC}_1(n)$ by declaring dots to have degree $1$, trivalent vertices degree $-1$ and $4$- and $6$-valent vertices degree $0$.
The category $\mathcal{SC}_2(n)$ is the category containing all direct sums and grading shifts of objects in $\mathcal{SC}_1(n)$ and whose morphisms are the grading preserving morphisms from $\mathcal{SC}_1(n)$.
The category $\mathcal{SC}(n)$ is the Karoubi envelope of the category $\mathcal{SC}_2(n)$.
The extension $\mathcal{SC}'(n)$ of $\mathcal{SC}(n)$ {#ssec:usoergel}
-----------------------------------------------------
In [@E-Kh] Elias and Khovanov give a slightly different diagrammatic Soergel category, denoted $\mathcal{SC}'(n)$, which is a faithful extension of $\mathcal{SC}(n)$. The objects of $\mathcal{SC}'_1(n)$ are the same as those of $\mathcal{SC}_1(n)$. The vector spaces of morphisms are an extension of the ones of $\mathcal{SC}_1(n)$ in the following sense. Regions can be decorated with boxes colored by $i$ for $1\leq i\leq n$, which we depict as $${\bbox{i}}$$ For $f$ a polynomial in the set of boxes colored from $1$ to $n$ we use the shorthand notation $${\bbox{f}}$$ The set of boxes is therefore in bijection with the polynomial ring in $n$ variables. Let $s_i$ be the transposition that switches $i$ and $i+1$. Define the formal symbol $$\bbox{\partial_{i}f}
=\bbox{\partial_{x_ix_{i+1}}f}
%=\frac{\bbox{f}-\bbox{s_if} }{ \bbox{i}-\bbox{i+1} },$$ where $\partial_{x_ix_{i+1}}$ was defined in Equation . This way any box $\bbox{f}$ can be written as $$\bbox{f}=
\bbox{P_i(f)} + \bbox{i}\,\bbox{\partial_if}$$ where $P_i(f)$ is a polynomial which is symmetric in $\bbox{i}$ and $\bbox{i+1}$ (we will take this formula as a definition of $P_i(f)$).
The boxes are related to the previous calculus by the *box relations* $$\begin{aligned}
\label{eq:box1}
\figins{-6}{0.25}{startenddot}_{\,i}\
&= \
\bbox{i} -\bbox{i+1}
\\[1ex] \displaybreak[0]
%\bbox{f}\
%\underset{i}{\figins{-20}{0.65}{vedge}}
%&=\
%\underset{i}{\figins{-20}{0.65}{vedge}}\ \bbox{P_i(f)}\
%+\
%\bbox{i}\
%\underset{i}{\figins{-20}{0.65}{vedge}}\ \bbox{\partial_if}
%\\[1ex] \displaybreak[0]
\biggl(
\bbox{i}\
+\
\bbox{i+1}
\biggr)\
\underset{i}{\figins{-20}{0.65}{vedge}}
&=
\underset{i}{\figins{-20}{0.65}{vedge}}
\biggl(
\bbox{i}\
+\
\bbox{i+1}
\biggr)
\\[1ex] \displaybreak[0]
\bbox{i}\ \bbox{i+1}\
\underset{i}{\figins{-20}{0.65}{vedge}}
&=
\underset{i}{\figins{-20}{0.65}{vedge}}\ \bbox{i}\ \bbox{i+1}
\\[1ex] \displaybreak[0]
\bbox{j}\
\underset{i}{\figins{-20}{0.65}{vedge}}
&=
\underset{i}{\figins{-20}{0.65}{vedge}}\ \bbox{j}
\rlap{\hspace*{10ex} for $j\neq i, i+1$.}\end{aligned}$$
It is clear that $\mathcal{SC}(n)$ is a faithful monoidal subcategory of $\mathcal{SC}'(n)$. As explained in Section 4.5 of [@E-Kh], the category $\mathcal{SC}(n)$ is also isomorphic to the quotient of $\mathcal{SC}'(n)$ by the central morphism $$\raisebox{1pt}{\bbox{e_1}}
\overset{def}{=}
\sum\limits_{i=1}^n{\bbox{i}}.$$ This result depends subtly on the base field, which in our case is $\bQ$.
The category $\mathcal{SC}'_1(n)$ has a grading induced by the one of $\mathcal{SC}_1(n)$, if we declare that a box colored $i$ has degree 2 for all $1\geq i\geq n$.
The category $\mathcal{SC}'_2(n)$ is the category containing all direct sums and grading shifts of objects in $\mathcal{SC}'_1(n)$ and whose morphisms are the grading preserving morphisms from $\mathcal{SC}'_1(n)$. The category $\mathcal{SC}'(n)$ is the Karoubi envelope of the category $\mathcal{SC}'_2(n)$.
Elias and Khovanov’s main result in [@E-Kh] is that $\mathcal{SC}(n)$ and $\mathcal{SC}'(n)$ are equivalent to the corresponding Soergel categories. A corollary to that is the following theorem, where $K_0$ is the split Grothendieck group and $K_0^{\bQ(q)}(-)=K_0(-)\otimes_{\bZ[q,q^{-1}]}\bQ(q)$.
\[thm:e-k-s\] We have $$K_0^{\bQ(q)}(\mathcal{SC}(n))\cong K_0^{\bQ(q)}(\mathcal{SC}'(n))\cong H_q(n).$$
As explained in [@E-Kh], this result also depends on the fact that we are working over $\bQ$. Recall that $\mathcal{SC}(n)$ and $\mathcal{SC}'(n)$ are monoidal categories, with the monoidal structure defined by concatenation. Therefore their Grothendieck groups are algebras indeed.
Let $\bim(n)^*=\mbox{End}_{\bim^*}(\Q[x_1,\ldots,x_n])$. Elias and Khovanov defined functors from $\mathcal{SC}(n)$ and $\mathcal{SC}'(n)$ to $\bim(n)^*$ (see [@E-Kh; @E-Kr]) which we denote by $\fek$ and $\fek'$ respectively.
A functor from $\mathcal{SC}(n)$ to $\Scat(n,n)^*((1^n),(1^n))$ {#ssec:scqs}
---------------------------------------------------------------
Let $n\geq 1$ be arbitrary but fixed. In this subsection we define an additive $\Q$-linear monoidal functor $$\Sigma_{n,n}\colon \mathcal{SC}_1(n)\to \Scat(n,n)^*((1^n),(1^n)),$$ where the target is the monoidal category whose objects are the $1$-endomorphisms of $(1^n)$ in $\Scat(n,n)^*$ and whose morphisms are the $2$-morphisms between such $1$-morphisms in $\Scat(n,n)^*$. This monoidal functor categorifies the homomorphism $\sigma_{n,n}$ from Section \[sec:hecke-schur\].
*On objects:* $\Sigma_{n,n}$ sends the empty sequence in $\mathcal{SC}_1(n)$ to $1_n=1_{(1^n)}$ in $\Scat(n,n)^*$ and the one-term sequence $(i)$ to $\mathcal{E}_{-i}\mathcal{E}_{+i}1_n$, with $\Sigma_{n,n}(jk)$ given by the horizontal composite $\mathcal{E}_{-j}\mathcal{E}_{+j}\mathcal{E}_{-k}\mathcal{E}_{+k}1_n$.
*On morphisms:*
- The empty diagram is sent to the empty diagram in the region labeled $(1^n)$.
- The vertical line coloured $i$ is sent to the identity $2$-morphism on $\mathcal{E}_{-i}\mathcal{E}_{+i}1_n$. $$\labellist
\tiny\hair 2pt
\pinlabel $i$ at -10 60
\endlabellist
\figins{-16}{0.5}{line}\ \
\longmapsto\ \ \
\text{$
\xy
(0,0)*{\dblue\xybox{
(-5,7);(-5,-7); **\dir{-} ?(.5)*\dir{>}+(2.3,0)*{\scriptstyle{}};}};
(-6.3,-9)*{\scs i};
(0,0)*{\dblue\xybox{
(10,7);(10,-7); **\dir{-} ?(.5)*\dir{<}+(12.3,0)*{\scriptstyle{}};}};
( -1.2,-9)*{\scs i};
(6,0)*{ (1^n)};
\endxy
$}
\vspace*{2ex}$$
- The *StartDot* and *EndDot* morphisms are sent to the cup and the cap respectively: $$\labellist
\tiny\hair 2pt
\pinlabel $i$ at -10 60
\endlabellist
\figins{-16}{0.5}{startdot}
\longmapsto\
{\dblue \xy
(0,2)*{\bbpef{\black i}};
(10,2)*{\black (1^n) };
\endxy}
\mspace{140mu}
\labellist
\tiny\hair 2pt
\pinlabel $i$ at -10 60
\endlabellist
\figins{-16}{0.5}{enddot}
\longmapsto\
{\dblue \xy
(0,-2.5)*{\bbcef{\black i}};
(8,0.5)*{ \black (1^n) };
\endxy}
\vspace*{2ex}$$
- *Merge* and *Split* are sent to diagrams involving cups and caps: $$\labellist
\hair 2pt
\pinlabel $\scs i$ at 45 95
\endlabellist
\figins{-16}{0.6}{merge}
\longmapsto\
\xy 0;/r.16pc/;
(0,-1.5)*{\dblue\bbcfe{\black i}};
(14,4)*{(1^n) };
(-12,3)*{};(12,3)*{};
( 7.5,2)*{\dblue\xybox{
(-3,-5)*{}; (-10,8.5) **\crv{(-3,1) & (-10,3)}?(1)*\dir{>};}};
( -7.5,2)*{\dblue\xybox{
( 3,-5)*{}; ( 10,8.5) **\crv{( 3,1) & ( 10,3)}?(0)*\dir{<};}};
(-11,-7)*{\scs i};
( 11,-7)*{\scs i};
\endxy
\mspace{80mu}
\labellist
\hair 2pt
\pinlabel $\scs i$ at 45 45
\endlabellist
\figins{-16}{0.6}{split}
\longmapsto \
\xy 0;/r.16pc/;
(0,5)*{\dblue\bbpfe{\black i}};
(14,0)*{(1^n) };
(-12,3)*{};(12,3)*{};
(-7.5,2)*{\dblue\xybox{
(-3,-5)*{}; (-10,8.5) **\crv{(-3,1) & (-10,3)}?(0)*\dir{<};}};
( 7.5,2)*{\dblue\xybox{
( 3,-5)*{}; ( 10,8.5) **\crv{( 3,1) & ( 10,3)}?(1)*\dir{>};}};
(-4,-7)*{\scs i};
( 4,-7)*{\scs i};
\endxy
\vspace*{2ex}$$
- The *4-valent vertex* with distant colors. For $i$ and $j$ distant we have: $$\labellist
\hair 2pt
\pinlabel $\scs j$ at -5 -12
\pinlabel $\scs i$ at 128 -10
\endlabellist
\figins{-16}{0.6}{4vert}
\longmapsto\ \ \
\text{$
\xy 0;/r.16pc/;
(14,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
( 0,-11)*{}; (-15,8.5) **\crv{( 0,-11) & (-15,8.5)}?(0)*\dir{<};
( 5,-11)*{}; (-10,8.5) **\crv{( 5,-11) & (-10,8.5)}?(1)*\dir{>};}};
( 0,0)*{\dred\xybox{
( 0,-11)*{}; ( 15,8.5) **\crv{( 0,-11) & ( 15,8.5)}?(0)*\dir{<};
( 5,-11)*{}; ( 20,8.5) **\crv{( 5,-11) & ( 20,8.5)}?(1)*\dir{>};}};
(-10,-12)*{\scs j}; (-5,-12)*{\scs j};
( 10,-12)*{\scs i}; ( 5,-12)*{\scs i};
\endxy
$}
\vspace*{2ex}$$
- For the *6-valent vertices* we have: $$\label{eq:sixval}
\labellist
\tiny\hair 2pt
\pinlabel $i+1$ at -5 -10
\pinlabel $i$ at 65 -10
\endlabellist
\figins{-18}{0.6}{6vertu}\
\longmapsto\ \ \
\text{$
\xy 0;/r.16pc/;
(16,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,10)*{}; ( 5,-10) **\crv{(-4.5, 7) & ( 7.5,0) & ( 5,-9)}?(0)*\dir{<};
(12.5,10)*{}; ( 0,-10) **\crv{( 9.5, 7) & (-2.5,0) & ( 0,-9)}?(1)*\dir{>};
(17.5,10)*{}; (-12.5, 10) **\crv{( 8, 0) & ( 2.5,-6) & (-3,0)}?(0)*\dir{<};
}};
( 0,0)*{\dred\xybox{
(-10,-20)*{};( 10,-20) **\crv{(-9,-19) & (0,-12) & (8,-19)}?(.2)*\dir{>} ?(.8)*\dir{>};
( 2.5, 0)*{};( 15,-20) **\crv{( 2.5,0) & ( 2,-10) & ( 15,-20)}?(0)*\dir{<};
(-2.5, 0)*{};(-15,-20) **\crv{(-2.5,0) & (-2,-10) & (-15,-20)}?(1)*\dir{>};
}};
( -17,-12)*{\scs i+1}; (-10,-12)*{\scs i+1};
(-2.5,-12)*{\scs i }; (2.5,-12)*{\scs i };
( 16,-12)*{\scs i+1};
( 16, 12)*{\scs i};
\endxy
$}
\vspace*{2ex}$$ and $$\labellist
\tiny\hair 2pt
\pinlabel $i$ at -5 -10
\pinlabel $i+1$ at 65 -10
\endlabellist
\figins{-18}{0.6}{6vertd}\
\longmapsto\ \ \
\text{$
\xy 0;/r.16pc/;
(16,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,-10)*{}; ( 5, 10) **\crv{(-4.5,-7) & ( 7.5,0) & ( 5,9)}?(1)*\dir{>};
(12.5,-10)*{}; ( 0, 10) **\crv{( 9.5,-7) & (-2.5,0) & ( 0,9)}?(0)*\dir{<};
(17.5,-10)*{}; (-12.5,-10) **\crv{( 8, 0) & (2.5,6) & (-3, 0)}?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
(-10,0)*{};(10,0) **\crv{(-9,-1) & (0,-8) & (8,-1)}?(.2)*\dir{<} ?(.8)*\dir{<};
( 2.5,-20)*{}; ( 15, 0) **\crv{( 2.5,-20) & ( 2,-10) & ( 15,0)}?(1)*\dir{>};
(-2.5,-20)*{}; ( -15, 0) **\crv{(-2.5,-20) & (-2,-10) & (-15,0)}?(0)*\dir{<};
}};
( -16,-12)*{\scs i}; (-11,-12)*{\scs i};
(-3.5,-12)*{\scs i+1}; (3.5,-12)*{\scs i+1};
( 11,-12)*{\scs i }; ( 16,-12)*{\scs i};
( 10, 12)*{\scs i+1};
\endxy
$}
\vspace*{2ex}$$
It is clear that $\Sigma_{n,n}$ respects the gradings of the morphisms. Moreover, let us remark that, in the decategorified picture, the image of $H_q(n)$ lies in the projection of the zero weight space of $\U$ onto $\SD(n,n)$, so we have $E_iE_{-i}=E_{-i}E_i$. Using the $2$-isomorphism $\mathcal{E}_i\mathcal{E}_{-i}\cong \mathcal{E}_{-i}
\mathcal{E}_i$ given by the crossing, we obtain a $2$-functor naturally isomorphic to $\Sigma_{n,n}$. However, this $2$-functor cannot be obtained by simply inverting the orientation of the diagrams defining $\Sigma_{n,n}$, as can be easily checked. As a matter of fact, inverting the orientations does not even give a $2$-functor, e.g. relation is not preserved.
$\Sigma_{n,n}$ is a monoidal functor.
The assignment given by $\Sigma_{n,n}$ clearly respects the monoidal structures of the categories $\mathcal{SC}_1(n)$ and $\mbox{End}_{\Scat(n,n)^*}(1^n)$. So we only need to show that $\Sigma_{n,n}$ is a functor, i.e. it respects the relations to .
*”Isotopy relations”:* Relations to are straightforward to check and correspond to isotopies of their images under $\Sigma_{n,n}$.
*One color relations:* To check the one color relations we only need to use the $\mathfrak{sl}_2$ relations. Relation corresponds to an easy isotopy of diagrams in $\Scat(n,n)$. For relation we have $$\labellist
\hair 2pt
\pinlabel $\scs i$ at 26 45
\endlabellist
\Sigma_{n,n}\Biggl(\
\figins{-16}{0.5}{lollipop-u}\
\Biggr)
=\ \
\text{$
\xy
(0,3)*{\dblue\xybox{%
(-6,0)*{};
(6,0)*{};
(4,0);(-4,0) **\crv{(4,6) & (-4,6)};?(.7)*\dir{}+(-2,0)*{\bscs i};
?(0)*\dir{<} ?(.95)*\dir{<};
(4,0);(-4,0) **\crv{(4,-6) & (-4,-6)};}};
(0,3)*{\dblue\xybox{
(8,0);(-8,0) **\crv{(8,12) & (-8,12)};
(8,0);(4,-12) **\crv{(8,-4) & (4,-6)};?(0.8)*\dir{<}+(2,-1)*{\bscs i};
(-8,0);(-4,-12) **\crv{(-8,-4) & (-4,-6)};
}};
(12,8)*{\scs (1^n)};(0,-5)*{\scs (1_{+i}^n)};
\endxy
=\
0
$}$$ because the bubble in the diagram on the r.h.s. has negative degree. We have used the notation $1_{+i}^n=(1,\ldots,2,0,1,\ldots,1)$, with the $2$ on the $i$th coordinate.
Relation requires some more work. First notice that from relations and it follows that $$\text{$
0\ =\
\xy
(1,0)*{\dblue\xybox{%
(4,4);(-4,4) **\crv{(4,2) & (8,-4) & (0,-9) & (-8,-4) & (-4,2)};
?(.7)*\dir{}+(-1,8)*{\bscs i};?(0)*\dir{<} ?(.95)*\dir{<};
(4,-12);(-4,-12) **\crv{(4,-10) & (8,-4) & (0,1) & (-8,-4) & (-4,-10)};
?(.7)*\dir{}+(-1,-8)*{\bscs i};?(1)*\dir{>} ?(.05)*\dir{>};
}};
(12,4)*{\scs (1^n)};(3,0)*{}
\endxy\
=\
\xy
(1,0)*{\dblue\xybox{%
(4,4);(-4,4) **\crv{(4,-3) & (-4,-3)};?(.7)*\dir{}+(-4,3.5)*{\bscs i};
?(0)*\dir{<} ?(.95)*\dir{<};
(4,-12);(-4,-12) **\crv{(4,-5) & (-4,-5)};?(.7)*\dir{}+(-4,-3.5)*{\bscs i};
?(1)*\dir{>} ?(.05)*\dir{>};
}};
(6,0)*{\scs (1^n)};
\endxy\
-\
\xy
(4,0)*{\dblue\xybox{%
( 3,9);( 3,-9) **\crv{( 4,1) & ( 4,-1)};?(0)*\dir{<}; (-3.75,0)*{\bullet};
(-3,9);(-3,-9) **\crv{(-4,1) & (-4,-1)};?(1)*\dir{>};
}};
(12,4)*{\scs (1^n)};(-1.5,-7)*{\scs i}; (9.2,-7)*{\scs i}
\endxy
-\
\xy
(4,0)*{\dblue\xybox{%
( 3,9);( 3,-9) **\crv{( 4,1) & ( 4,-1)};?(0)*\dir{<}; (3.75,0)*{\bullet};
(-3,9);(-3,-9) **\crv{(-4,1) & (-4,-1)};?(1)*\dir{>};
}};
(12,4)*{\scs (1^n)};(-1.5,-7)*{\scs i}; (8.2,-7)*{\scs i}
\endxy
+\ \ \ \ \
\xy
(0,0)*{\dblue\xybox{%
(3,0);(-3,0) **\crv{(3,4.2) & (-3,4.2)};?(.7)*\dir{}+(-2,0)*{\bscs i};
?(.05)*\dir{>} ?(1)*\dir{>}; (0,-3.1)*{\bullet}+(-0.4,-2)*{\bscs -2};
(3,0);(-3,0) **\crv{(3,-4.2) & (-3,-4.2)};%}};
(3,9);(3,-9) **\crv{(4,6) & (6,4) & (6,-3) & (4,-6)};?(0)*\dir{<};
(-3,9);(-3,-9) **\crv{(-4,6) & (-6,4) & (-6,-3) & (-4,-6)};?(1)*\dir{>};
}};
(10,4)*{\scs (1^n)};(-5.5,-7)*{\scs i}; (5.5,-7)*{\scs i}
\endxy\ $}.$$ The first diagram is zero, because the middle region has label $(1,\ldots,3,-1,\ldots,1)\not\in\Lambda(n,n)$, with $3$ on the $i$th coordinate. Therefore $$\labellist
\hair 2pt
\pinlabel $\scs i$ at -2 22
\pinlabel $\scs i$ at -2 130
\endlabellist
\Sigma_{n,n}\Biggl(\
\figins{-17}{0.55}{matches-ud}\
\Biggr)
=\
\xy
(1,0)*{\dblue\xybox{%
(4,4);(-4,4) **\crv{(4,-3) & (-4,-3)};?(.7)*\dir{}+(-4,3.5)*{\bscs i};
?(0)*\dir{<} ?(.95)*\dir{<};
(4,-12);(-4,-12) **\crv{(4,-5) & (-4,-5)};?(.7)*\dir{}+(-4,-3.5)*{\bscs i};
?(1)*\dir{>} ?(.05)*\dir{>};
}};
(6,0)*{\scs (1^n)};
\endxy\
=\
\xy
(4,0)*{\dblue\xybox{%
( 3,9);( 3,-9) **\crv{( 4,1) & ( 4,-1)};?(0)*\dir{<}; (-3.75,0)*{\bullet};
(-3,9);(-3,-9) **\crv{(-4,1) & (-4,-1)};?(1)*\dir{>};
}};
(12,4)*{\scs (1^n)};(-1.5,-7)*{\scs i}; (9.2,-7)*{\scs i}
\endxy
%%
+\
\xy
(4,0)*{\dblue\xybox{%
( 3,9);( 3,-9) **\crv{( 4,1) & ( 4,-1)};?(0)*\dir{<}; (3.75,0)*{\bullet};
(-3,9);(-3,-9) **\crv{(-4,1) & (-4,-1)};?(1)*\dir{>};
}};
(12,4)*{\scs (1^n)};(-1.5,-7)*{\scs i}; (8.2,-7)*{\scs i}
\endxy
-\ \ \ \ \
%%
\xy
(0,0)*{\dblue\xybox{%
(3,0);(-3,0) **\crv{(3,4.2) & (-3,4.2)};?(.7)*\dir{}+(-2,0)*{\bscs i};
?(.05)*\dir{>} ?(1)*\dir{>}; (0,-3.1)*{\bullet}+(-0.4,-2)*{\bscs -2};
(3,0);(-3,0) **\crv{(3,-4.2) & (-3,-4.2)};%}};
(3,9);(3,-9) **\crv{(4,6) & (6,4) & (6,-3) & (4,-6)};?(0)*\dir{<};
(-3,9);(-3,-9) **\crv{(-4,6) & (-6,4) & (-6,-3) & (-4,-6)};?(1)*\dir{>};
}};
(10,4)*{\scs (1^n)};(-5.5,-7)*{\scs i}; (5.5,-7)*{\scs i}
\endxy$$ Using and the bubble evaluation we obtain $$\begin{aligned}
\label{eq:edge-dots}
\labellist
\hair 2pt
\pinlabel $\scs i$ at -10 60
\endlabellist
\Sigma_{n,n}\Biggl(\
\figins{-17}{0.55}{edge-startenddot}\
\Biggr)
&=\ 2\
\xy
(4,0)*{\dblue\xybox{%
( 3,9);( 3,-9) **\crv{( 4,1) & ( 4,-1)};?(0)*\dir{<}; (3.75,0)*{\bullet};
(-3,9);(-3,-9) **\crv{(-4,1) & (-4,-1)};?(1)*\dir{>};
}};
(12,4)*{\scs (1^n)};(-1.5,-7)*{\scs i}; (8.2,-7)*{\scs i}
\endxy
-\ \ \ \ \
\xy
(0,0)*{\dblue\xybox{%
(3,0);(-3,0) **\crv{(3,4.2) & (-3,4.2)};?(.7)*\dir{}+(-2,0)*{\bscs i};
?(.05)*\dir{>} ?(1)*\dir{>}; (0,-3.1)*{\bullet}+(-0.4,-2)*{\bscs -2};
(3,0);(-3,0) **\crv{(3,-4.2) & (-3,-4.2)};%}};
(3,9);(3,-9) **\crv{(4,6) & (6,4) & (6,-3) & (4,-6)};?(0)*\dir{<};
(-3,9);(-3,-9) **\crv{(-4,6) & (-6,4) & (-6,-3) & (-4,-6)};?(1)*\dir{>};
}};
(10,4)*{\scs (1^n)};(-5.5,-7)*{\scs i}; (5.5,-7)*{\scs i}
\endxy
\displaybreak[0]
\intertext{and}
\label{eq:dots-edge}
\labellist
\hair 2pt
\pinlabel $\scs i$ at 65 60
\endlabellist
\Sigma_{n,n}\Biggl(\
\figins{-17}{0.55}{startenddot-edge}\
\Biggr)
&=\ 2\
\xy
(4,0)*{\dblue\xybox{%
( 3,9);( 3,-9) **\crv{( 4,1) & ( 4,-1)};?(0)*\dir{<}; (-3.75,0)*{\bullet};
(-3,9);(-3,-9) **\crv{(-4,1) & (-4,-1)};?(1)*\dir{>};
}};
(12,4)*{\scs (1^n)};(-1.5,-7)*{\scs i}; (9.2,-7)*{\scs i}
\endxy
-\ \ \ \ \
\xy
(0,0)*{\dblue\xybox{%
(3,0);(-3,0) **\crv{(3,4.2) & (-3,4.2)};?(.7)*\dir{}+(-2,0)*{\bscs i};
?(.05)*\dir{>} ?(1)*\dir{>}; (0,-3.1)*{\bullet}+(-0.4,-2)*{\bscs -2};
(3,0);(-3,0) **\crv{(3,-4.2) & (-3,-4.2)};%}};
(3,9);(3,-9) **\crv{(4,6) & (6,4) & (6,-3) & (4,-6)};?(0)*\dir{<};
(-3,9);(-3,-9) **\crv{(-4,6) & (-6,4) & (-6,-3) & (-4,-6)};?(1)*\dir{>};
}};
(10,4)*{\scs (1^n)};(-5.5,-7)*{\scs i}; (5.5,-7)*{\scs i}
\endxy.\end{aligned}$$ This establishes that $$\Sigma_{n,n}
\Biggl(
\figins{-17}{0.55}{startenddot-edge}\
\Biggr)
+\
\Sigma_{n,n}
\Biggl(\
\figins{-17}{0.55}{edge-startenddot}\
\Biggr)
=\ 2\
\Sigma_{n,n}
\Biggl(\
\figins{-17}{0.55}{matches-ud}\
\Biggr).$$
*Two distant colors:* Checking relations to is straightforward and only uses relations and with distant colors $i$ and $j$.
*Adjacent colors:* To prove relation we first notice that using we get $$\labellist
\hair 2pt
\pinlabel $\scs i$ at 0 -10
\pinlabel $\scs i+1$ at 69 -11
\endlabellist
\Sigma_{n,n}\Biggl(\
\figins{-12}{0.4}{6vertdotd}\
\Biggr)\
=
\xy 0;/r.16pc/;
(16,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,-10)*{}; ( 5, 10) **\crv{(-4.5,-7) & ( 7.5,0) & ( 5,9)}?(1)*\dir{>};
(12.5,-10)*{}; ( 0, 10) **\crv{( 9.5,-7) & (-2.5,0) & ( 0,9)}?(0)*\dir{<};
(17.5,-10)*{}; (-12.5,-10) **\crv{( 8, 0) & (2.5,6) & (-3, 0)}?(1)*\dir{>};
}};
( 0,1)*{\dred\xybox{
(-10,0)*{};(10,0) **\crv{(-9,-1) & (0,-8) & (8,-1)}?(.2)*\dir{<} ?(.8)*\dir{<};
(-15,0)*{}; (15, 0) **\crv{(-4,-10) & (-2.5,-18) & (2.5,-18) & (4,-10)};
?(1)*\dir{>};?(0.01)*\dir{>};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( 11,-12)*{\scs i }; ( 16,-12)*{\scs i};
( -9, 12)*{\scs i+1}; (-16, 12)*{\scs i+1};
\endxy\
=\
\xy 0;/r.16pc/;
(16,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,-10)*{}; (12.5,-10) **\crv{(-4,-5) & ( 9,-5)}?(1)*\dir{>};
(5, 10)*{}; ( 0, 10) **\crv{( 6,0) & (2.5,-4) & ( -1,0)}?(0)*\dir{<};
(17.5,-10)*{}; (-12.5,-10) **\crv{( 8, 0) & (2.5,6) & (-3, 0)}?(1)*\dir{>};
}};
( 0,2.5)*{\dred\xybox{
(-10,0)*{};(10,0) **\crv{(-9,-1) & (0,-8) & (8,-1)}?(.2)*\dir{<} ?(.8)*\dir{<};
(-15,0)*{}; (15, 0) **\crv{(-6,-8) & (-2.5,-15) & (2.5,-15) & (6,-8)};
?(1)*\dir{>};?(0.01)*\dir{>};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( 11,-12)*{\scs i }; ( 16,-12)*{\scs i};
( -9, 12)*{\scs i+1}; (-16, 12)*{\scs i+1};
\endxy\ .
\vspace*{2ex}$$ Note that the other term on the r.h.s. of is equal to zero, because it contains a region whose label has a negative entry, i.e. does not belong to $\Lambda(n,n)$.
Using followed by and gives $$\xy 0;/r.16pc/;
(16,2)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,-10)*{}; (12.5,-10) **\crv{(-2,-6) & (2.5,-5) & ( 7,-6)}?(1)*\dir{>};
(5, 10)*{}; ( 0, 10) **\crv{( 4.5,9) & (2.5,8) & ( .5,9)}?(0)*\dir{<};
(17.5,-10)*{}; (-12.5,-10) **\crv{(8, -3) & (2.5,-2) & (-3, -3)}?(1)*\dir{>};
}};
( 0,5.5)*{\dred\xybox{
(-10,0)*{};(10, 0) **\crv{( -8,-2) & (0, -9) & ( 8,-2)}?(0)*\dir{<};
(-15,0)*{};(15, 0) **\crv{(-10,-5) & (0,-13) & (10,-5)}?(1)*\dir{>};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( -9, 12)*{\scs i+1}; (-16, 12)*{\scs i+1};
\endxy\
\ +\
\xy 0;/r.16pc/;
(16,2)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,-10)*{}; (12.5,-10) **\crv{(-2,-6) & (2.5,-5) & ( 7,-6)}?(1)*\dir{>};
(5,10)*{};( 17.5,-10) **\crv{(5,10) & ( 4.5,0) & ( 17.5,-10)}?(0)*\dir{<};
(0,10)*{};(-12.5,-10) **\crv{(0,10) & ( 0.5,0) & (-12.5,-10)}?(1)*\dir{>};
}};
( 0,5.5)*{\dred\xybox{
(-10,0)*{};(10, 0) **\crv{( -8,-2) & (0, -9) & ( 8,-2)}?(0)*\dir{<};
(-15,0)*{};(15, 0) **\crv{(-10,-5) & (0,-13) & (10,-5)}?(1)*\dir{>};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( -9, 12)*{\scs i+1}; (-16, 12)*{\scs i+1};
\endxy\ .$$ Applying to the two red strands in the middle region of the second term (only one term survives) followed by and gives $$\labellist
\hair 2pt
\pinlabel $\scs i$ at 0 -10
\pinlabel $\scs i+1$ at 69 -11
\endlabellist
\Sigma_{n,n}\Biggl(\
\figins{-12}{0.4}{6vertdotd}\
\Biggr)\
=\
\xy 0;/r.16pc/;
(16,2)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,-10)*{}; (12.5,-10) **\crv{(-2,-6) & (2.5,-5) & ( 7,-6)}?(1)*\dir{>};
(5, 10)*{}; ( 0, 10) **\crv{( 4.5,9) & (2.5,8) & ( .5,9)}?(0)*\dir{<};
(17.5,-10)*{}; (-12.5,-10) **\crv{(8, -3) & (2.5,-2) & (-3, -3)}?(1)*\dir{>};
}};
( 0,5.5)*{\dred\xybox{
(-10,0)*{};(10, 0) **\crv{( -8,-2) & (0, -9) & ( 8,-2)}?(0)*\dir{<};
(-15,0)*{};(15, 0) **\crv{(-10,-5) & (0,-13) & (10,-5)}?(1)*\dir{>};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( -9, 12)*{\scs i+1}; (-16, 12)*{\scs i+1};
\endxy\
\ +\
\xy 0;/r.16pc/;
(18,2)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,-10)*{}; (12.5,-10) **\crv{(-2,-6) & (2.5,-5) & ( 7,-6)}?(1)*\dir{>};
(5,10)*{};( 17.5,-10) **\crv{(5,10) & ( 4.5,0) & ( 17.5,-10)}?(0)*\dir{<};
(0,10)*{};(-12.5,-10) **\crv{(0,10) & ( 0.5,0) & (-12.5,-10)}?(1)*\dir{>};
}};
( 0,7.5)*{\dred\xybox{
(-15,0)*{};(-10, 0) **\crv{(-15,-4) & (-12.5,-6) & (-10,-4)}?(1)*\dir{>};
( 15,0)*{};( 10, 0) **\crv{( 15,-4) & ( 12.5,-6) & ( 10,-4)}?(0)*\dir{<};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( 10, 12)*{\scs i+1}; (-15, 12)*{\scs i+1};
\endxy\ ,$$ which is equal to $\Sigma_{n,n}\bigl(\figins{-5}{0.2}{capcupdot}\bigr)+\Sigma_{n,n}\bigl(\figins{-5}{0.2}{mergedots}\bigr)$ .
The corresponding relation with colors switched is not difficult to prove. We have $$\labellist
\hair 2pt
\pinlabel $\scs i+1$ at 5 -12
\pinlabel $\scs i$ at 69 -10
\endlabellist
\Sigma_{n,n}\Biggl(\ \
\figins{-12}{0.4}{6vertdotdd}\
\Biggr)\
=
\xy 0;/r.16pc/;
(20,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
% (-7.5,10)*{}; ( 5,-10) **\crv{(-4.5, 7) & ( 7.5,0) & ( 5,-9)}?(0)*\dir{<};
% (12.5,10)*{}; ( 0,-10) **\crv{( 9.5, 7) & (-2.5,0) & ( 0,-9)}?(1)*\dir{>};
(17.5,10)*{}; (-12.5, 10) **\crv{( 8, 0) & ( 2.5,-6) & (-3,0)}?(0)*\dir{<};
(12.5,10)*{}; ( -7.5,10) **\crv{(9.5, 7) & (-2.5,0) & (0,-11) & (5,-11) & (7.5,0) & (-4.5,7) }?(1)*\dir{>}?(0.49)*\dir{>};
}};
( 0,0)*{\dred\xybox{
(-10,-20)*{};( 10,-20) **\crv{(-9,-19) & (0,-12) & (8,-19)}?(.2)*\dir{>} ?(.8)*\dir{>};
( 2.5, 0)*{};( 15,-20) **\crv{( 2.5,0) & ( 2,-10) & ( 15,-20)}?(0)*\dir{<};
(-2.5, 0)*{};(-15,-20) **\crv{(-2.5,0) & (-2,-10) & (-15,-20)}?(1)*\dir{>};
}};
( -17,-12)*{\scs i+1}; (-10,-12)*{\scs i+1};
% (-2.5,-12)*{\scs i }; (2.5,-12)*{\scs i };
( 10, 12)*{\scs i };
( 16,-12)*{\scs i+1};
( 16, 12)*{\scs i};
\endxy
.
%\vspace*{2ex}$$ Use on the bottom part of the diagram. Only one of the resulting terms survives (use the first relation in ), which in turn equals $$\xy 0;/r.16pc/;
(20,0)*{(1^n) };
( 0,2)*{\dblue\xybox{
% (17.5,10)*{}; (-12.5, 10) **\crv{( 8, 0) & ( 2.5,-6) & (-3,0)}?(0)*\dir{<};
% (12.5,10)*{}; ( -7.5,10) **\crv{(9.5, 7) & (-2.5,0) & (0,-11) & (5,-11) & (7.5,0) & (-4.5,7) }?(1)*\dir{>}?(0.49)*\dir{>};
(7.5,15)*{}; (12.5,15) **\crv{(4.5,12) & (-6.5,5) & (-3,1) & (0,1) & (2.5,5) }?(1)*\dir{>};
(-12.5,15)*{}; (-17.5,15) **\crv{(-9.5,12) & (1.5,5) & (-2,1) & (-5,1) & (-7.5,5) }?(0)*\dir{<};
}};
( 0,0)*{\dred\xybox{
(-10,-20)*{};( 10,-20) **\crv{(-9,-19) & (0,-12) & (8,-19)}?(.2)*\dir{>} ?(.8)*\dir{>};
( 2.5, 0)*{};( 15,-20) **\crv{( 2.5,0) & ( 2,-10) & ( 15,-20)}?(0)*\dir{<};
(-2.5, 0)*{};(-15,-20) **\crv{(-2.5,0) & (-2,-10) & (-15,-20)}?(1)*\dir{>};
}};
( -17,-12)*{\scs i+1}; (-10,-12)*{\scs i+1};
% (-2.5,-12)*{\scs i }; (2.5,-12)*{\scs i };
( 10, 12)*{\scs i };
( 16,-12)*{\scs i+1};
( 16, 12)*{\scs i};
\endxy$$ (use the first relation in combined with ). Applying we get two terms, one of which is $$\xy 0;/r.16pc/;
(18,2)*{(1^n) };
( 0,0)*{\dred\xybox{
(-7.5,-10)*{}; (12.5,-10) **\crv{(-2,-6) & (2.5,-5) & ( 7,-6)}?(1)*\dir{>};
(5,10)*{};( 17.5,-10) **\crv{(5,10) & ( 4.5,0) & ( 17.5,-10)}?(0)*\dir{<};
(0,10)*{};(-12.5,-10) **\crv{(0,10) & ( 0.5,0) & (-12.5,-10)}?(1)*\dir{>};
}};
( 0,7.5)*{\dblue\xybox{
(-15,0)*{};(-10, 0) **\crv{(-15,-4) & (-12.5,-6) & (-10,-4)}?(1)*\dir{>};
( 15,0)*{};( 10, 0) **\crv{( 15,-4) & ( 12.5,-6) & ( 10,-4)}?(0)*\dir{<};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( 10, 12)*{\scs i+1}; (-15, 12)*{\scs i+1};
\endxy\$$ (this follows easily from ) and the other equals $$\xy 0;/r.16pc/;
(16,2)*{(1^n) };
( 0,0)*{\dred\xybox{
(-7.5,-10)*{}; (12.5,-10) **\crv{(-2,-6) & (2.5,-5) & ( 7,-6)}?(1)*\dir{>};
(5,10)*{};( 17.5,-10) **\crv{(5,10) & ( 4.5,0) & ( 17.5,-10)}?(0)*\dir{<};
(0,10)*{};(-12.5,-10) **\crv{(0,10) & ( 0.5,0) & (-12.5,-10)}?(1)*\dir{>};
}};
( 0,5.5)*{\dblue\xybox{
(-10,0)*{};(10, 0) **\crv{( -8,-2) & (0, -9) & ( 8,-2)}?(0)*\dir{<};
(-15,0)*{};(15, 0) **\crv{(-10,-5) & (0,-13) & (10,-5)}?(1)*\dir{>};
}};
(-16,-12)*{\scs i}; (-11,-12)*{\scs i};
( -9, 12)*{\scs i+1}; (-16, 12)*{\scs i+1};
\endxy\
.$$ Here we used . The rest of the computation is the same as in the previous case.
We now prove relation . We only prove the case where ”blue“ corresponds to $i$ and ”red“ corresponds to $i+1$. The relation with the colors reversed is proved in the same way. Start with $$\Sigma_{n,n}
\left(\
\labellist
\tiny\hair 2pt
\pinlabel $i+1$ at 70 -14
\pinlabel $i$ at 0 -12
\endlabellist
\figins{-26}{0.8}{reid3-inv}\
\right)
=\
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-7,-26) **\crv{(-4.5, 8) & ( 4.5,3) & ( 5.5,-8.5) & (4.5,-20) & (-4.5,-24)}
?(0)*\dir{<}?(0.47)*\dir{<};
( 12,10)*{};(12,-26) **\crv{(9.5, 8) & (0.5,3) & (-0.5,-8.5) & (0.5,-20) & (9.5,-24)}
?(1)*\dir{>}?(0.5)*\dir{>};
(17.5,10)*{};(-12.5, 10) **\crv{(7,1.5) & (6,1) & ( 2.5,-2) & (-1,1) & (-2,1.5)}
?(0)*\dir{<};
(17.5,-26)*{};(-12.5,-26) **\crv{(7,-17.5) & (6,-17) & (2.5,-14) & (-1,-17) & (-2,-17.5)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5, 16)
**\crv{( 2.5,-18) & ( 6,-14) & (12,-6) & ( 12,-2) & (12,2) & (6,10) & (2.5,14) }
?(1)*\dir{>}?(0.5)*\dir{>};
(-2.5,-20)*{}; (-2.5, 16)
**\crv{(-2.5,-18) & (-6,-14) & (-12,-6) & (-12,-2) & (-12,2) & (-6,10) & (-2.5,14) }
?(0)*\dir{<}?(0.5)*\dir{<};
(-6,-2)*{}; (6,-2) **\crv{(-6,0) & (0,5) & (6,0)}?(0.01)*\dir{>};
(-6,-2)*{}; (6,-2) **\crv{(-6,-4) & (0,-9) & (6,-4)};
}};
(16,10)*{(1^n) };
( -16,-20)*{\scs i}; (-10,-20)*{\scs i};
(-3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( 10,-20)*{\scs i}; ( 16,-20)*{\scs i};
( -16, 20)*{\scs i}; (7,-4)*{\scs i+1};
\endxy\
= \
%%%%%%%%%%%%%%
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-7,-26) **\crv{(-4.5, 7) & ( 7.5,-1) & ( 11.5,-8.5) & (7.5,-16) & (-4.5,-23)}
?(0)*\dir{<}?(0.5)*\dir{<};
( 12,10)*{};(12,-26) **\crv{(9.5, 7) & (-2.5,-1) & (-6.5,-8.5) & (-2.5,-16) & (9.5,-23)}
?(1)*\dir{>}?(0.5)*\dir{>};
(17.5,10)*{};(-12.5, 10) **\crv{(7, 0) & (6,-1) & ( 2.5,-4) & (-1,-1) & (-2,0)}
?(0)*\dir{<};
(17.5,-26)*{};(-12.5,-26) **\crv{(7,-16) & (6,-15) & (2.5,-13) & (-1,-15) & (-2,-16)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5, 16)
**\crv{( 2.5,-18) & ( 6,-14) & (12,-6) & ( 12,-2) & (12,2) & (6,10) & (2.5,14) }
?(1)*\dir{>}?(0.5)*\dir{>};
(-2.5,-20)*{}; (-2.5, 16)
**\crv{(-2.5,-18) & (-6,-14) & (-12,-6) & (-12,-2) & (-12,2) & (-6,10) & (-2.5,14) }
?(0)*\dir{<}?(0.5)*\dir{<};
}};
(16,10)*{(1^n) };
( -16,-20)*{\scs i}; (-10,-20)*{\scs i};
(-3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( 10,-20)*{\scs i}; ( 16,-20)*{\scs i};
( -16, 20)*{\scs i};
\endxy,$$ where the second equality follows from and . Now notice that $$\label{eq:reid3proof}
0 =
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-7,-26) **\crv{(-4.5, 7) & ( 7.5,-1) & ( 11.5,-8.5) & (7.5,-16) & (-4.5,-23)}
?(0)*\dir{<}?(0.5)*\dir{<};
( 12,10)*{};(12,-26) **\crv{(9.5, 7) & (-2.5,-1) & (-6.5,-8.5) & (-2.5,-16) & (9.5,-23)}
?(1)*\dir{>}?(0.5)*\dir{>};
(17.5,10)*{};(-12.5, 10) **\crv{(7, 0) & (7.5,-4) & ( 2.5,-14) & (-2.5,-4) & (-2,0)}
?(0)*\dir{<};
(17.5,-26)*{};(-12.5,-26) **\crv{(7,-16) & (7.5,-12) & (2.5,-2) & (-2.5,-12) & (-2,-16)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5, 16)
**\crv{( 2.5,-18) & ( 6,-14) & (12,-6) & ( 12,-2) & (12,2) & (6,10) & (2.5,14) }
?(1)*\dir{>}?(0.5)*\dir{>};
(-2.5,-20)*{}; (-2.5, 16)
**\crv{(-2.5,-18) & (-6,-14) & (-12,-6) & (-12,-2) & (-12,2) & (-6,10) & (-2.5,14) }
?(0)*\dir{<}?(0.5)*\dir{<};
}};
(16,10)*{(1^n) };
( -16,-20)*{\scs i}; (-10,-20)*{\scs i};
(-3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( 10,-20)*{\scs i}; ( 16,-20)*{\scs i};
( -16, 20)*{\scs i};
\endxy
=
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-7,-26) **\crv{(-4.5, 7) & ( 7.5,-1) & ( 11.5,-8.5) & (7.5,-16) & (-4.5,-23)}
?(0)*\dir{<}?(0.5)*\dir{<};
( 12,10)*{};(12,-26) **\crv{(9.5, 7) & (-2.5,-1) & (-6.5,-8.5) & (-2.5,-16) & (9.5,-23)}
?(1)*\dir{>}?(0.5)*\dir{>};
(17.5,10)*{};(-12.5, 10) **\crv{(7, 0) & (6,-4) & ( 2.5,-8) & (-1,-4) & (-2,0)}
?(0)*\dir{<};
(17.5,-26)*{};(-12.5,-26) **\crv{(7,-16) & (6,-12) & (2.5,-8) & (-1,-12) & (-2,-16)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5, 16)
**\crv{( 2.5,-18) & ( 6,-14) & (12,-6) & ( 12,-2) & (12,2) & (6,10) & (2.5,14) }
?(1)*\dir{>}?(0.5)*\dir{>};
(-2.5,-20)*{}; (-2.5, 16)
**\crv{(-2.5,-18) & (-6,-14) & (-12,-6) & (-12,-2) & (-12,2) & (-6,10) & (-2.5,14) }
?(0)*\dir{<}?(0.5)*\dir{<};
}};
(16,10)*{(1^n) };
( -16,-20)*{\scs i}; (-10,-20)*{\scs i};
(-3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( 10,-20)*{\scs i}; ( 16,-20)*{\scs i};
( -16, 20)*{\scs i};
\endxy
-
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-7,-26) **\crv{(-4.5, 7) & ( 7.5,-1) & ( 11.5,-8.5) & (7.5,-16) & (-4.5,-23)}
?(0)*\dir{<};%?(0.47)*\dir{<};
(-12,10)*{};(-12,-26)
**\crv{(-11.5,7) & (-2.5,-1) & (0.5,-8.5) & (-2.5,-16) & (-11.5,-23)}
?(1)*\dir{>}?(0.5)*{\bullet};
( 12,10)*{};(12,-26) **\crv{(9.5, 7) & (-2.5,-1) & (-6.5,-8.5) & (-2.5,-16) & (9.5,-23)}
?(1)*\dir{>};%?(0.5)*\dir{>};
( 17,10)*{};(17,-26) **\crv{(16.5, 7) & (6.5,-1) & (4.5,-8.5) & (6.5,-16) & (16.5,-23)}
?(0)*\dir{<}?(0.5)*{\bullet};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5, 16)
**\crv{( 2.5,-18) & ( 6,-14) & (12,-6) & ( 12,-2) & (12,2) & (6,10) & (2.5,14) }
?(1)*\dir{>}?(0.5)*\dir{>};
(-2.5,-20)*{}; (-2.5, 16)
**\crv{(-2.5,-18) & (-6,-14) & (-12,-6) & (-12,-2) & (-12,2) & (-6,10) & (-2.5,14) }
?(0)*\dir{<}?(0.5)*\dir{<};
}};
(16,10)*{(1^n) };
( -16,-20)*{\scs i}; (-10,-20)*{\scs i};
(-3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( 10,-20)*{\scs i}; ( 16,-20)*{\scs i};
\endxy.$$ The first equality in comes from the fact that the inner most region of the diagram has a label outside $\Lambda(n,n)$. The second equality follows from . The last term is the only non-zero term coming from the sum in (this is a consequence of ).
Applying and to the last term, we obtain a diagram that can be simplified further by successive application of , , and again . $$\begin{aligned}
&
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-12,8)*{};(12,-24) **\crv{(-6,4) & (-8,-6) & (2,-16)}?(1)*\dir{>};
(-12,-24)*{};(12, 8) **\crv{(-6,-20) & (-8,-10) & (2,0)}?(0)*\dir{<};
( 17,8)*{};(-7,-24) **\crv{( 11,4) & ( 13,-6) & (3,-16)}?(0)*\dir{<};
( 17,-24)*{};(-7,8) **\crv{( 11,-20) & ( 13,-10) & (3,0)}?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-18)*{}; ( 2.5, 14)
**\crv{( 2.5,-15) & ( 6,-14) & (12,-4) & ( 12,-2) & (12,2) & (6,8) & (2.5,11) }
?(1)*\dir{>}?(0.5)*\dir{>};
(-2.5,-18)*{}; (-2.5, 14)
**\crv{(-2.5,-15) & (-6,-14) & (-12,-4) & (-12,-2) & (-12,2) & (-6,8) & (-2.5,11) }
?(0)*\dir{<}?(0.45)*\dir{<};
}};
(18,10)*{(1^n) };
( -15,-18)*{\scs i}; (-10,-18)*{\scs i};
(-3.5,-18.1)*{\scs i+1}; (3.5,-18.1)*{\scs i+1};
( 10,-18)*{\scs i}; ( 16,-18)*{\scs i};
\endxy\
%
\overset{\eqref{eq_r3_hard-gen}}{=}\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-12,8)*{};(-12,-24) **\crv{(-11,4) & (-13,-8) & (-11,-20)}?(1)*\dir{>};
( -7,8)*{};(-7,-24) **\crv{( 3,0) & ( 14,-8) & (3,-16)}?(0)*\dir{<};
( 12,8)*{};(12,-24) **\crv{( 2,0) & (-9,-8) & (2,-16)}?(1)*\dir{>};
( 17,8)*{};( 17,-24) **\crv{(16,4) & (19,-8) & (16,-20)}?(0)*\dir{<};
}};
( 0,0)*{\dred\xybox{
( 2.5,-18)*{}; ( 2.5, 14)
**\crv{( 2.5,-15) & ( 6,-14) & (12,-4) & ( 12,-2) & (12,2) & (6,8) & (2.5,11) }
?(1)*\dir{>}?(0.5)*\dir{>};
(-2.5,-18)*{}; (-2.5, 14)
**\crv{(-2.5,-15) & (-6,-14) & (-12,-4) & (-12,-2) & (-12,2) & (-6,8) & (-2.5,11) }
?(0)*\dir{<}?(0.45)*\dir{<};
}};
(20,10)*{(1^n) };
( -15,-18)*{\scs i}; (-10,-18)*{\scs i};
(-3.5,-18.1)*{\scs i+1}; (3.5,-18.1)*{\scs i+1};
( 10,-18)*{\scs i}; ( 14.5,-18)*{\scs i};
\endxy\
\\[1.5ex]\displaybreak[0]
&
\mspace{45mu}
\overset{\eqref{eq_ident_decomp0}+\eqref{eq:bubb_deg0}}{=}\
\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-12,8)*{};(-12,-24) **\crv{(-11,4) & (-13,-8) & (-11,-20)}?(1)*\dir{>};
( -7,8)*{};(-7,-24) **\crv{(-5,0) & ( 2,-8) & (-5,-16)}?(0)*\dir{<};
( 12,8)*{};(12,-24) **\crv{(10,0) & (3,-8) & (10,-16)}?(1)*\dir{>};
( 17,8)*{};(17,-24) **\crv{(16,4) & (19,-8) & (16,-20)}?(0)*\dir{<};
}};
(0,0)*{\dred\xybox{
( 7,8)*{};( 7,-24) **\crv{(9,0) & (16,-8) & (9,-16)}?(0)*\dir{<};
( -2,8)*{};(-2,-24) **\crv{(-4,0) & (-11,-8) & (-4,-16)}?(1)*\dir{>};
}};
(20,10)*{(1^n) };
( -15,-18)*{\scs i}; (-10,-18)*{\scs i};
(-3.5,-18.1)*{\scs i+1}; (3.5,-18.1)*{\scs i+1};
( 10,-18)*{\scs i}; ( 14.5,-18)*{\scs i};
\endxy\
+\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-12,8)*{};(-12,-24) **\crv{(-11,4) & (-13,-8) & (-11,-20)}?(1)*\dir{>};
( 17,8)*{};( 17,-24) **\crv{(16,4) & (19,-8) & (16,-20)}?(0)*\dir{<};
(-7, 8)*{};(12, 8) **\crv{(-5, 4) & ( 2.5, -5) & (12, 4)}?(0)*\dir{<};
(-7,-24)*{};(12,-24) **\crv{(-5,-20) & ( 2.5,-11) & (12,-20)}?(1)*\dir{>};
}};
(0,0)*{\dred\xybox{
( 7,8)*{};( 7,-24) **\crv{(9,0) & (11,-8) & (9,-16)}?(0)*\dir{<};
( -2,8)*{};(-2,-24) **\crv{(-4,0) & (-6,-8) & (-4,-16)}?(1)*\dir{>};
}};
(20,10)*{(1^n) };
( -15,-18)*{\scs i}; (-10,-18)*{\scs i};
(-3.5,-18.1)*{\scs i+1}; (3.5,-18.1)*{\scs i+1};
( 10,-18)*{\scs i}; ( 14.5,-18)*{\scs i};
\endxy\
\\[1.5ex]\displaybreak[0]
&
\mspace{60mu}
\overset{\eqref{eq_downup_ij-gen}}{=}
\mspace{20mu}
\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-12,8)*{};(-12,-24) **\crv{(-11,4) & (-13,-8) & (-11,-20)}?(1)*\dir{>};
( -7,8)*{};( -7,-24) **\crv{( -8,4) & ( -6,-8) & ( -8,-20)}?(0)*\dir{<};
( 12,8)*{};( 12,-24) **\crv{(13,4) & (11,-8) & (13,-20)}?(1)*\dir{>};
( 17,8)*{};( 17,-24) **\crv{(16,4) & (19,-8) & (16,-20)}?(0)*\dir{<};
}};
(0,0)*{\dred\xybox{
( 7,8)*{};( 7,-24) **\crv{(7,0) & ( 6,-8) & (7,-16)}?(0)*\dir{<};
( -2,8)*{};(-2,-24) **\crv{(-2,0) & (0,-8) & (-2,-16)}?(1)*\dir{>};
}};
(20,10)*{(1^n) };
( -15,-18)*{\scs i}; (-10,-18)*{\scs i};
(-3.5,-18.1)*{\scs i+1}; (3.5,-18.1)*{\scs i+1};
( 10,-18)*{\scs i}; ( 14.5,-18)*{\scs i};
\endxy\
+\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-12,8)*{};(-12,-24) **\crv{(-11,4) & (-13,-8) & (-11,-20)}?(1)*\dir{>};
( 17,8)*{};( 17,-24) **\crv{(16,4) & (19,-8) & (16,-20)}?(0)*\dir{<};
(-7, 8)*{};(12, 8) **\crv{(-5, 4) & ( 2.5, -5) & (12, 4)}?(0)*\dir{<};
(-7,-24)*{};(12,-24) **\crv{(-5,-20) & ( 2.5,-11) & (12,-20)}?(1)*\dir{>};
}};
(0,0)*{\dred\xybox{
( 7,8)*{};( 7,-24) **\crv{(9,0) & (11,-8) & (9,-16)}?(0)*\dir{<};
( -2,8)*{};(-2,-24) **\crv{(-4,0) & (-6,-8) & (-4,-16)}?(1)*\dir{>};
}};
(20,10)*{(1^n) };
( -15,-18)*{\scs i}; (-10,-18)*{\scs i};
(-3.5,-18.1)*{\scs i+1}; (3.5,-18.1)*{\scs i+1};
( 10,-18)*{\scs i}; ( 14.5,-18)*{\scs i};
\endxy\ .\end{aligned}$$ Applying to the vertical red strands in the second term, followed by and , we get that it is equal to $$\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-12,10)*{};(-12,-16) **\crv{(-11,4) & (-13,-3) & (-11,-10)}?(1)*\dir{>};
( 17,10)*{};( 17,-16) **\crv{(16,4) & (19,-3) & (16,-10)}?(0)*\dir{<};
(-7, 10)*{};(12, 10) **\crv{(-7, 4) & ( 2.5, -5) & (12, 4)}?(0)*\dir{<};
(-7,-16)*{};(12,-16) **\crv{(-7,-10) & ( 2.5,-1) & (12,-10)}?(1)*\dir{>};
}};
(0,0)*{\dred\xybox{
( -2, 10)*{};(7, 10) **\crv{(-2,8) & (2.5,1) & (7,8)}?(1)*\dir{>};
( -2,-16)*{};(7,-16) **\crv{(-2,-14) & (2.5,-7) & (7,-14)}?(0)*\dir{<};
}};
(20,8)*{(1^n) };
( -15,-15)*{\scs i}; (-10,-15)*{\scs i};
(-3.5,-15.1)*{\scs i+1}; (3.5,-15.1)*{\scs i+1};
( 10,-15)*{\scs i}; ( 14.5,-15)*{\scs i};
\endxy\ ,$$ which equals $\Sigma_{n,n}\bigl(\figins{-5}{0.22}{dumbell-dd-short-inv}\bigr)$. Therefore $\Sigma_{n,n}\bigl(\figins{-5}{0.22}{reid3-short-inv}\bigr)
= \Sigma_{n,n}\bigl(\figins{-5}{0.22}{id3-short-inv}\bigr) +
\Sigma_{n,n}\bigl(\figins{-5}{0.22}{dumbell-dd-short-inv}\bigr)$.
We now prove relation . We denote the left and right hand sides of $L$ and $R$, respectively. We have $$\Sigma_{n,n}(L)\
=\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-7,-26) **\crv{(-4.5, 8) & ( 4.5,3) & ( 5.5,-8.5) & (4.5,-20) & (-4.5,-24)}
?(0)*\dir{<}?(0.47)*\dir{<};
( 12,10)*{};(12,-26) **\crv{(9.5, 8) & (0.5,3) & (-0.5,-8.5) & (0.5,-20) & (9.5,-24)}
?(1)*\dir{>}?(0.5)*\dir{>};
(17.5,10)*{};(-12.5, 10) **\crv{(7,1.5) & (6,1) & ( 2.5,-2) & (-1,1) & (-2,1.5)}
?(0)*\dir{<};
(17.5,-26)*{};(-12.5,-26) **\crv{(7,-17.5) & (6,-17) & (2.5,-14) & (-1,-17) & (-2,-17.5)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; (15.5,-4.5) **\crv{( 2.5,-18) & ( 6,-13) & (10,-6) & ( 12,-4.5)}
?(1)*\dir{>};
( 2.5, 16)*{}; (15.5, 0.5) **\crv{( 2.5,14) & ( 6,9) & (10,2) & ( 12, 0.5)}
?(0)*\dir{<};
(-2.5,-20)*{}; (-15.5,-4.5) **\crv{(-2.5,-18) & (-6,-13) & (-10,-6) & (-12,-4.5)}
?(0)*\dir{<};
(-2.5, 16)*{}; (-15.5, 0.5) **\crv{(-2.5,14) & (-6,9) & (-10,2) & (-12, 0.5)}
?(1)*\dir{>};
(-6,-2)*{}; (6,-2) **\crv{(-6,0) & (0,5) & (6,0)}?(0.01)*\dir{>};
(-6,-2)*{}; (6,-2) **\crv{(-6,-4) & (0,-9) & (6,-4)};
}};
(18,10)*{(1^n) };
( -16,-20)*{\scs i}; (-10,-20)*{\scs i};
(-3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( 10,-20)*{\scs i}; ( 16,-20)*{\scs i};
( -16, 20)*{\scs i}; (7,-4)*{\scs i+1};
\endxy\
=\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7.5,-26)*{};( 17.5,10) **\crv{(-7.5,-26) & (7,-10) & ( 17.5,10)}?(1)*\dir{>};
(-12.5,-26)*{};( 12.5,10) **\crv{(-12.5,-26) & (-2,-6) & ( 12.5,10)}?(0)*\dir{<};
(12,-26)*{};(-12.5,10) **\crv{(12,-26) & (-2,-10) & (-12.5,10)}?(0)*\dir{<};
(17.5,-26)*{};(-7.5,10) **\crv{(17.5,-26) & (7,-6) & (-7.5,10)}?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; (15.5,-4.5) **\crv{( 2.5,-18) & ( 6,-13) & (10,-6) & ( 12,-4.5)}
?(1)*\dir{>};
( 2.5, 16)*{}; (15.5, 0.5) **\crv{( 2.5,14) & ( 6,9) & (10,2) & ( 12, 0.5)}
?(0)*\dir{<};
(-2.5,-20)*{}; (-15.5,-4.5) **\crv{(-2.5,-18) & (-6,-13) & (-10,-6) & (-12,-4.5)}
?(0)*\dir{<};
(-2.5, 16)*{}; (-15.5, 0.5) **\crv{(-2.5,14) & (-6,9) & (-10,2) & (-12, 0.5)}
?(1)*\dir{>};
}};
(20,10)*{(1^n) };
( -16,-20)*{\scs i}; (-10,-20)*{\scs i};
(-3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( 10,-20)*{\scs i}; ( 16,-20)*{\scs i};
( -16, 20)*{\scs i};
\endxy$$ The second equality is obtained as in Equation . The same argument shows that this equals $\Sigma_{n,n}(R)$.
Relation is straightforward to check (it only uses bubble slides).
*Relations involving three colors:* Relations and are easy because the green strands have to be distant from red and blue and so we have all Reidemeister 2 and Reidemeister 3 like moves between green and one of the other colors.
It remains to prove that $\Sigma_{n,n}$ respects relation . First notice that the diagrams on the left- and right-hand side of are invariant under $180^{\circ}$ rotations and that they can be obtained from one another using a $90^{\circ}$ rotation. Therefore it suffices to show that the image of one of them is invariant under $90^{\circ}$ rotations. Denote by $L$ the diagram on the left-hand-side of . Then $$\Sigma_{n,n}(L)\
=\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-15,-26) **\crv{(-4.5, 8) & ( 4.5,3) & ( 5.5,-8.5) & (4.5,-20) & (-6.5,-22)}
?(0)*\dir{<};
( 20,10)*{};(12,-26) **\crv{(11.5, 6) & (0.5,3) & (-0.5,-8.5) & (0.5,-20) & (9.5,-24)}
?(1)*\dir{>};
(25.5,10)*{};(-12.5, 10) **\crv{(11,1) & (6,-1) & ( 2,-2) & (-1,1) & (-2,1.5)}
?(0)*\dir{<};
(17.5,-26)*{};(-20.5,-26) **\crv{(7,-17.5) & (6,-17) & (2.5,-14) & (-1,-16) & (-6,-17.5)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5,16) **\crv{( 2.5,-14) & ( 22,-3) & (2.5,10)}
?(1)*\dir{>};
(-2.5,-20)*{}; (-2.5,16) **\crv{(-2.5,-14) & (-22,-3) & (-2.5,10)}
?(0)*\dir{<};
(-23,-5)*{}; (23,-5) **\crv{(-9,-4.5) & (-2.8,3) & (3.0,3) & (9,-4.5)}
?(1)*\dir{>};
(-23,0)*{}; (23,0) **\crv{(-9,1.5) & (-2.8,-7) & (3.0,-7) & (9,1.5)}
?(0)*\dir{<};
}};
(0,0)*{\dgreen\xybox{
(-20.5, 10)*{}; (-10.5,-26) **\crv{(-14.5,4) & (-18.5,-16)}?(1)*\dir{>};
( 15.5, 10)*{}; ( 25.5,-26) **\crv{( 25.5,-2) & (19.5,-20)}?(0)*\dir{<};
(-16.5, 10)*{}; ( 11.5, 10) **\crv{(-13,-2) & (-8.5,-4) & (1,-8) & (5.5,-7) & (22,-2)}?(0)*\dir{<};
(- 6.5,-26)*{}; ( 21.5,-26) **\crv{(-16.5,-14) & (-1,-9) & (4,-9) & (13,-11) & (17.5,-13)}?(1)*\dir{>};
}};
( 24, 10)*{(1^n) };
(-24.5,-20)*{\scs i}; (-18,-20)*{\scs i};
( 24 , 20)*{\scs i}; ( 18,20)*{\scs i};
(- 3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( -27, -3)*{\scs i+1};(-27,2)*{\scs i+1};
(-26.5, 17)*{\scs i+2};(-20,20)*{\scs i+2};
( 26.5,-17)*{\scs i+2};(20,-20)*{\scs i+2};
\endxy\ .$$ Taking into account that the green strands are distant from the blue ones, we apply and a sequence of Reidemeister 3 like moves to obtain $$\Sigma_{n,n}(L)\
=\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-15,-26) **\crv{(-4.5, 8) & ( 4.5,3) & ( 5.5,-8.5) & (4.5,-20) & (-6.5,-22)}
?(0)*\dir{<};
( 20,10)*{};(12,-26) **\crv{(11.5, 6) & (0.5,3) & (-0.5,-8.5) & (0.5,-20) & (9.5,-24)}
?(1)*\dir{>};
(25.5,10)*{};(-12.5, 10) **\crv{(11,1) & (6,-1) & ( 2,-2) & (-1,1) & (-2,1.5)}
?(0)*\dir{<};
(17.5,-26)*{};(-20.5,-26) **\crv{(7,-17.5) & (6,-17) & (2.5,-14) & (-1,-16) & (-6,-17.5)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5,16) **\crv{( 2.5,-14) & ( 18,-3) & (2.5,10)}
?(1)*\dir{>};
(-2.5,-20)*{}; (-2.5,16) **\crv{(-2.5,-14) & (-18,-3) & (-2.5,10)}
?(0)*\dir{<};
(-23,-5)*{}; (23,-5) **\crv{(-9,-5) & (0,4.5) & (9,-5)}
?(1)*\dir{>};
(-23,0)*{}; (23,0) **\crv{(-9,0) & (0,-9.5) & (9,0)}
?(0)*\dir{<};
}};
(0,0)*{\dgreen\xybox{
(-20.5, 10)*{}; (-10.5,-26) **\crv{(-14.5,4) & (-10.5,-20)}?(1)*\dir{>};
( 15.5, 10)*{}; ( 25.5,-26) **\crv{( 15.5,4) & (19.5,-20)}?(0)*\dir{<};
(-16.5, 10)*{}; ( 11.5, 10) **\crv{(-11.5,7) & (4.5,7)}?(0)*\dir{<};
(- 6.5,-26)*{}; ( 21.5,-26) **\crv{( 1.5,-23) & (14.5,-23)}?(1)*\dir{>};
}};
( 24, 10)*{(1^n) };
(-24.5,-20)*{\scs i}; (-18,-20)*{\scs i};
( 24 , 20)*{\scs i}; ( 18,20)*{\scs i};
(- 3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( -27, -3)*{\scs i+1};(-27,2)*{\scs i+1};
(-26.5, 17)*{\scs i+2};(-20,20)*{\scs i+2};
( 26.5,-17)*{\scs i+2};(20,-20)*{\scs i+2};
\endxy\ .$$ Using twice between the two horizontal red lines and a vertical blue line, followed by gives $$\Sigma_{n,n}(L)\
=\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-15,-26) **\crv{(-4.5, 8) & ( 4.5,3) & ( 5.5,-8.5) & (4.5,-20) & (-6.5,-22)}
?(0)*\dir{<};
( 20,10)*{};(12,-26) **\crv{(11.5, 6) & (0.5,3) & (-0.5,-8.5) & (0.5,-20) & (9.5,-24)}
?(1)*\dir{>};
(25.5,10)*{};(-12.5, 10) **\crv{(11,1) & (6,-1) & ( 2,-2) & (-1,1) & (-2,1.5)}
?(0)*\dir{<};
(17.5,-26)*{};(-20.5,-26) **\crv{(7,-17.5) & (6,-17) & (2.5,-14) & (-1,-16) & (-6,-17.5)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5,16) **\crv{( 2.5,-14) & ( 18,-3) & (2.5,10)}
?(1)*\dir{>};
(-2.5,-20)*{}; (-2.5,16) **\crv{(-2.5,-14) & (-18,-3) & (-2.5,10)}
?(0)*\dir{<};
(-23,-5)*{}; (23,-5) **\crv{(-9,-5) & (0,-3) & (9,-5)}?(1)*\dir{>};
(-23,0)*{}; (23,0) **\crv{(-9,0) & (0,-2) & (9,0)}?(0)*\dir{<};
}};
(0,0)*{\dgreen\xybox{
(-20.5, 10)*{}; (-10.5,-26) **\crv{(-14.5,4) & (-10.5,-20)}?(1)*\dir{>};
( 15.5, 10)*{}; ( 25.5,-26) **\crv{( 15.5,4) & (19.5,-20)}?(0)*\dir{<};
(-16.5, 10)*{}; ( 11.5, 10) **\crv{(-11.5,7) & (4.5,7)}?(0)*\dir{<};
(- 6.5,-26)*{}; ( 21.5,-26) **\crv{( 1.5,-23) & (14.5,-23)}?(1)*\dir{>};
}};
( 24, 10)*{(1^n) };
(-24.5,-20)*{\scs i}; (-18,-20)*{\scs i};
( 24 , 20)*{\scs i}; ( 18,20)*{\scs i};
(- 3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( -27, -3)*{\scs i+1};(-27,2)*{\scs i+1};
(-26.5, 17)*{\scs i+2};(-20,20)*{\scs i+2};
( 26.5,-17)*{\scs i+2};(20,-20)*{\scs i+2};
\endxy\ .$$ Notice that the sums in are not increasing and therefore there are no terms with dots here. Applying and to the top and bottom we can pass the top and bottom $(i,i)$ crossings to the middle of the diagram (the terms coming from the sums in are zero). We get $$\Sigma_{n,n}(L)\
=\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-15,-26) **\crv{(-3.5, 8) & ( 5.5,-6) & ( 5,-8.5) & (5.5,-11) & (-4.5,-22)}
?(0)*\dir{<};
( 20,10)*{};(12,-26) **\crv{(11.5, 6) & (-0.5,-6) & ( 0,-8.5) & (-0.5,-11) & (9.5,-24)}
?(1)*\dir{>};
(25.5,10)*{};(-12.5, 10) **\crv{(11,3) & (6,1) & ( 2,2) & (-1,3) & (-2,3.5)}
?(0)*\dir{<};
(17.5,-26)*{};(-20.5,-26) **\crv{(7,-19.5) & (6,-19) & (2.5,-17.5) & (-1,-18) & (-6,-20)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5,16) **\crv{( 6,-17) & ( 12,-14) & (12, 8) & (6,13)}
?(1)*\dir{>};
(-2.5,-20)*{}; (-2.5,16) **\crv{(-6,-17) & (-12,-14) & (-12, 8) & (-6,13)}
?(0)*\dir{<};
(-23,-5)*{}; (23,-5) **\crv{(-19,-5) & (0,-16) & (19,-5)}?(1)*\dir{>};
(-23,0)*{}; (23,0) **\crv{(-19,0) & (0,11) & (19,0)}?(0)*\dir{<};
}};
(0,0)*{\dgreen\xybox{
(-20.5, 10)*{}; (-10.5,-26) **\crv{(-14.5,4) & (-10.5,-20)}?(1)*\dir{>};
( 15.5, 10)*{}; ( 25.5,-26) **\crv{( 15.5,4) & (19.5,-20)}?(0)*\dir{<};
(-16.5, 10)*{}; ( 11.5, 10) **\crv{(-11.5,8) & (4.5,6)}?(0)*\dir{<};
(- 6.5,-26)*{}; ( 21.5,-26) **\crv{( 1.5,-22) & (14.5,-24)}?(1)*\dir{>};
}};
( 24, 10)*{(1^n) };
(-24.5,-20)*{\scs i}; (-18,-20)*{\scs i};
( 24 , 20)*{\scs i}; ( 18,20)*{\scs i};
(- 3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( -27, -3)*{\scs i+1};(-27,2)*{\scs i+1};
(-26.5, 17)*{\scs i+2};(-20,20)*{\scs i+2};
( 26.5,-17)*{\scs i+2};(20,-20)*{\scs i+2};
\endxy\ .$$ Using in the middle of the diagram followed by and a sequence of Reidemeister 3 like moves to pass the vertical red strands to the middle gives $$\Sigma_{n,n}(L)\
=\ \ \
\xy 0;/r.16pc/;
( 0,0)*{\dblue\xybox{
(-7,10)*{};(-15,-26) **\crv{(-6.5, 8) & (-4.5,-6) & (-4,-11) & (-4.5,-22)}
?(0)*\dir{<};
( 20,10)*{};(12,-26) **\crv{(11.5, 7) & (11,-6) & ( 11,-8.5) & (11,-11)}
?(1)*\dir{>};
(24.5,8)*{};(-12.5, 10) **\crv{(11,3) & (6,1) & ( 2,2) & (-1,3) & (-2,3.5)}
?(0)*\dir{<};
(17.5,-26)*{};(-19.5,-24) **\crv{(7,-19.5) & (6,-19) & (2.5,-17.5) & (-1,-18) & (-6,-20)}
?(1)*\dir{>};
}};
( 0,0)*{\dred\xybox{
( 2.5,-20)*{}; ( 2.5,16) **\crv{( 3.5,-6) & ( 3.5,6)}
?(1)*\dir{>};
(-2.5,-20)*{}; (-2.5,16) **\crv{(-3.5,-6) & (-3.5,6)}
?(0)*\dir{<};
(-23,-5)*{}; (23,-5) **\crv{(-9,-6) & (9,-6)}?(1)*\dir{>};
(-23,0)*{}; (23,0) **\crv{(-9,1) & (9,1)}?(0)*\dir{<};
}};
(0,0)*{\dgreen\xybox{
(-19.5, 8)*{}; (-10.5,-26) **\crv{(-14.5,4) & (-10.5,-20)}?(1)*\dir{>};
( 15.5, 10)*{}; ( 24.5,-24) **\crv{( 15.5,4) & (19.5,-20)}?(0)*\dir{<};
(-16.5, 10)*{}; ( 11.5, 10) **\crv{(-11.5,8) & (4.5,6)}?(0)*\dir{<};
(- 6.5,-26)*{}; ( 21.5,-26) **\crv{( 1.5,-22) & (14.5,-24)}?(1)*\dir{>};
}};
( 24, 10)*{(1^n) };
(-24.5,-20)*{\scs i}; (-18,-20)*{\scs i};
( 24 , 20)*{\scs i}; ( 18,20)*{\scs i};
(- 3.5,-20.1)*{\scs i+1}; (3.5,-20.1)*{\scs i+1};
( -27, -3)*{\scs i+1};(-27,2)*{\scs i+1};
(-26.5, 17)*{\scs i+2};(-20,20)*{\scs i+2};
( 26.5,-17)*{\scs i+2};(20,-20)*{\scs i+2};
\endxy\ ,$$ which is symmetric under $90^{\circ}$ rotations.
$\mathcal{SC}_1(n)$ is a full sub-2-category of $\Scat(n,n)$
------------------------------------------------------------
\[lem:comm\] The following diagram commutes $$\xymatrix{
\mathcal{SC}_1(n)\ar[rr]^{\fek}\ar[dr]_{\Sigma_{n,n}} && \bim(n)^*
\\
& \Scat(n,n)^*((1^n),(1^n))\ar[ur]_{\fbim} &
}$$
The commutativity of the diagram can be checked by direct computation. Most of the computation is straightforward except for the 6-valent vertex. To compute its image under $\mathcal{F}_{Bim}\Sigma_{n,n}$ we divide it in layers and compute the bimodule maps for each layer. We do the case with the colors as in Equation \[eq:sixval\], the other case being similar. Remember that $$\labellist
\tiny\hair 2pt
\pinlabel $i+1$ at -5 -10
\pinlabel $i$ at 65 -10
\endlabellist
\figins{-18}{0.6}{6vertu}\ \ \
\xmapsto{\Sigma_{n,n}} \ \
\text{$
\xy 0;/r.16pc/;
(16,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
(-7.5,10)*{}; ( 5,-10) **\crv{(-4.5, 7) & ( 7.5,0) & ( 5,-9)}?(0)*\dir{<};
(12.5,10)*{}; ( 0,-10) **\crv{( 9.5, 7) & (-2.5,0) & ( 0,-9)}?(1)*\dir{>};
(17.5,10)*{}; (-12.5, 10) **\crv{( 8, 0) & ( 2.5,-6) & (-3,0)}?(0)*\dir{<};
}};
( 0,0)*{\dred\xybox{
(-10,-20)*{};( 10,-20) **\crv{(-9,-19) & (0,-12) & (8,-19)}?(.2)*\dir{>} ?(.8)*\dir{>};
( 2.5, 0)*{};( 15,-20) **\crv{( 2.5,0) & ( 2,-10) & ( 15,-20)}?(0)*\dir{<};
(-2.5, 0)*{};(-15,-20) **\crv{(-2.5,0) & (-2,-10) & (-15,-20)}?(1)*\dir{>};
}};
( -17,-12)*{\scs i+1}; (-10,-12)*{\scs i+1};
(-2.5,-12)*{\scs i }; (2.5,-12)*{\scs i };
( 16,-12)*{\scs i+1};
( 16, 12)*{\scs i};
\endxy
$}\ \ .
\vspace*{2ex}$$ It is easy to see that the map corresponding to the layer $$\text{$
\xy 0;/r.16pc/;
(20,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
( 8.5,8)*{}; (5,-8) **\crv{( 8.5, 7) & ( 7.5,0) & ( 5,-8)}?(0)*\dir{<};
(-3.5,8)*{}; (0,-8) **\crv{(-3.5, 7) & (-2.5,0) & ( 0,-8)}?(1)*\dir{>};
}};
( 0,0.5)*{\dred\xybox{
(-10,-18)*{};( 10,-18) **\crv{(-8,-14) & (0,0) & (8,-14)}?(.2)*\dir{>} ?(.83)*\dir{>};
( 12, -2)*{};( 15,-18) **\crv{(12,-10) & (15,-18)}?(0)*\dir{<};
(-12, -2)*{};(-15,-18) **\crv{(-12,-10) & (-15,-18)}?(1)*\dir{>};
}};
( -17,-10)*{\scs i+1}; (-10,-10)*{\scs i+1};
(-2.5,-10)*{\scs i }; (2.5,-10)*{\scs i };
( 16,-10)*{\scs i+1};
\endxy
$}$$ consists only of a relabeling of variables. The next one is $$\begin{aligned}
\xy 0;/r.12pc/;
(0,5)*{\dblue\bbpef{\black i}};
(16,-2)*{(1^n) };
(-12,3)*{};(12,3)*{};
(-6.5,2)*{\dblue\xybox{
(-3,-5)*{}; (-10,8.5) **\crv{(-3,1) & (-10,3)}?(0)*\dir{<};}};
( 6.5,2)*{\dblue\xybox{
( 3,-5)*{}; ( 10,8.5) **\crv{( 3,1) & ( 10,3)}?(1)*\dir{>};}};
(-12.5,2)*{\dred\xybox{
(-3,-5)*{}; (-10,8.5) **\crv{(-3,1) & (-10,3)}?(0)*\dir{<};}};
(12.5,2)*{\dred\xybox{
( 3,-5)*{}; ( 10,8.5) **\crv{( 3,1) & ( 10,3)}?(1)*\dir{>};}};
(-3,-7)*{\scs i};(3,-7)*{\scs i};
(-9,-7.1)*{\scs i+1};(9,-7.1)*{\scs i+1};
\endxy
\ \
\mapsto &
\ \
\left(\quad
\begin{aligned}
\labellist
\tiny\hair 2pt
\pinlabel $1$ at -20 483 \pinlabel $1$ at 140 483 \pinlabel $1$ at 295 483
\pinlabel $y$ at 235 90 \pinlabel $x$ at 140 245 \pinlabel $z$ at 235 385
\endlabellist
\figins{-42}{1.20}{web-v6cup0}
\qra
\labellist
\tiny\hair 2pt
\pinlabel $1$ at -20 483 \pinlabel $1$ at 140 483 \pinlabel $1$ at 295 483
\pinlabel $y$ at 235 90 \pinlabel $x'$ at 140 320 \pinlabel $x$ at 140 180
\pinlabel $z$ at 235 385 \pinlabel $t_1,t_2$ at 370 170
\endlabellist
\figins{-42}{1.20}{web-v6cup}
&\quad\
\\[1.0ex]\
p\mapsto \sum\limits_{\ell=0}^{2}(-1)^{\ell}x'^{2-\ell}\varepsilon_{\ell}(t_1,t_2)p
\quad &
\end{aligned}
\right).\end{aligned}$$ The next step consists of the two crossings between strands labeled $i$, $$\text{$
\xy 0;/r.16pc/;
(20,0)*{(1^n) };
( 0,0)*{\dblue\xybox{
( 12.5,8)*{}; (5,-8) **\crv{(8.5,0)}?(0)*\dir{<};
( 12.5,-8)*{}; (5,8) **\crv{(8.5,0)}?(1)*\dir{>};
(-7.5,8)*{}; (0,-8) **\crv{(-3.5,0)}?(1)*\dir{>};
(-7.5,-8)*{};(0,8) **\crv{(-3.5,0)}?(0)*\dir{<};
}};
( 0,0.5)*{\dred\xybox{
( 15, -2)*{};( 15,-18) **\crv{(15,-10) & (15,-18)}?(0)*\dir{<};
(-15, -2)*{};(-15,-18) **\crv{(-15,-10) & (-15,-18)}?(1)*\dir{>};
% (-10,-18)*{};(-2.5,-2) **\crv{(-5.5,-9)}?(1)*\dir{>};
% (10,-18)*{};(2.5,-2) **\crv{(5.5,-9)}?(1)*\dir{>};
}};
( -17,-10)*{\scs i+1}; (-10,-10)*{\scs i};
(-2.5,-10)*{\scs i }; (2.5,-10)*{\scs i };
(10,-10)*{\scs i};
( 16,-10)*{\scs i+1};
\endxy
$}$$ corresponding to the map $p\mapsto \partial_{zx'}\partial_{xy}p$. The left pointing $(i,i)$-crossing and the remaining $(i,i+1)$ crossings consist only of relabeling of variables and shifts. Putting everything together, the reader can check that this map coincides with the one obtained from $\cF_{EK}$ by a straightforward computation.
We now get to the main result of this subsection.
\[prop:fulfaith\] The functor $\Sigma_{n,n}$ is an equivalence of categories.
We have to show that $\Sigma_{n,n}$ is essentially surjective and fully faithful. By the commutation 2-isomorphisms, i.e. the relations involving Reidemeister II and III type moves between diagrams in $\Scat(n,n)$, we can commute the factors of any object $x$ in $\Scat(n,n)^*((1^n),(1^n))$ so that it becomes a direct sum of objects whose factors are all of the form $\mathcal{E}_{-i}\mathcal{E}_{+i}1_n$. This is always possible because $x$ has to have as many factors $\mathcal{E}_{-j}$ as $\mathcal{E}_{+j}$, for any $j=1,\ldots,n-1$, or else $x$ contains a factor $1_{\lambda}$ with $\lambda\not\in\Lambda(n,n)$ and is therefore equal to zero. This shows that $\Sigma_{n,n}$ is essentially surjective.
Since the functor $\fek$ is faithful [@E-Kh], it follows from Lemma \[lem:comm\] that $\Sigma_{n,n}$ is faithful too. Therefore it only remains to show that $\Sigma_{n,n}$ is full. To this end we first note that $$\tilde{\tau}(\mathcal{E}_{-i}\mathcal{E}_{+i}1_n)=
\mathcal{E}_{-i}\mathcal{E}_{+i}1_n.$$ By simply checking the definitions one sees that the natural isomorphisms in Corollary 4.12 in [@E-Kh] and the ones in Lemma \[lem:tildetau\] in this paper intertwine $\Sigma_{n,n}$. For example, we have a commutative square $$\begin{CD}
\HOM_{\mathcal{SC}_1(n)}(i\kk,\jj)@>{\cong}>>\HOM_{\mathcal{SC}_1(n)}(\kk,i\jj)\\
@V{\Sigma_{n,n}}VV @V{\Sigma_{n,n}}VV\\
\HOM_{\Scat(n,n)}(\mathcal{E}_{-i}\mathcal{E}_{+i}1_n\Sigma_{n,n}(\kk),\Sigma_{n,n}(\jj))
@>{\cong}>>
\HOM_{\Scat(n,n)}(\Sigma_{n,n}(\kk),
\mathcal{E}_{-i}\mathcal{E}_{+i}1_n\Sigma_{n,n}(\jj)).
\end{CD}$$ This observation together with the results after Corollary 4.12 in Section 4.3 in [@E-Kh] and the fact that $\Sigma_{n,n}$ is additive and $\Q$-linear implies that it is enough to prove that $$\Sigma_{n,n}\colon \HOM_{\mathcal{SC}_1(n)}(\emptyset,\ii)\to
\HOM_{\Scat(n,n)}(1_n,{\mathcal E}_{-i_1}{\mathcal E}_{+i_1}\cdots
{\mathcal E}_{-i_t}{\mathcal E}_{+i_t}1_n)$$ is surjective, where $\ii=(i_1,\ldots,i_t)$ is a sequence of $t$ points of strictly increasing color $1\leq i_1 < i_2 <\cdots<i_t\leq n-1$. If $t=0$, then this is true, because $\HOM_{\mathcal{SC}_1(n)}(\emptyset,\emptyset)\cong
\Q[x_1-x_2,x_2-x_3,\ldots,x_{n-1}-x_n]$ by Elias and Khovanov’s Theorem 1. Note that $$\Scat(n,d)^*((1^n),(1^n))\cong \Q[x_1-x_2,x_2-x_3,\ldots,x_{n-1}-x_n]$$ is exactly the ring generated by the colored bubbles, as we proved in Lemma \[lem:superschur\]. The functor $\Sigma_{n,n}$ sends double dots to colored bubbles.
Note also that $S\Pi_{(1^n)}\cong \Q[x_1-x_2,x_2-x_3,\ldots,x_{n-1}-x_n]$ and the surjective map $S\Pi_{(1^n)}\to \END_{\Scat(n,n)}(1_n)$, which we explained in Section \[sec:struct\], is equal to $\Sigma_{n,n}$. This actually shows that $\END_{\Scat(n,n)}(1_n)\cong S\Pi_{(1^n)}$, which is compatible with our Conjecture \[conj:bubbles\].
For $t>0$, note that by Corollary 4.11 in [@E-Kh] $\HOM_{\mathcal{SC}_1(n)}(\emptyset,\ii)$ is a free left $\HOM_{\mathcal{SC}_1(n)}(\emptyset,\emptyset)$-module of rank one, generated by the diagram consisting of $t$ StartDots colored $i_1,\ldots,i_t$ respectively. Note also that, by the fullness of $\Psi_{n,n}$ and by Theorem 1.3, Proposition 1.4 and Theorem 2.7 in [@K-L3], we know that $$\HOM_{\Scat(n,n)}(1_n,{\mathcal E}_{-i_1}{\mathcal E}_{+i_1}\cdots
{\mathcal E}_{-i_t}{\mathcal E}_{+i_t}1_n)$$ is a free $\END_{\Scat(n,n)}(1_n)$-module of rank one generated by the diagram consisting of $t$ cups colored $i_1,\ldots,i_t$ respectively. Our functor $\Sigma_{n,n}$ maps the StartDots to the cups, so we get that $$\Sigma_{n,n}\colon \HOM_{\mathcal{SC}_1(n)}(\emptyset,\ii)\to
\HOM_{\Scat(n,n)}(1_n,{\mathcal E}_{-i_1}{\mathcal E}_{+i_1}\cdots
{\mathcal E}_{-i_t}{\mathcal E}_{+i_t}1_n)$$ is an isomorphism.
A functor from $\mathcal{SC}_1'(d)$ to $\Scat(n,d)^*((1^d),(1^d))$ for $d<n$ {#ssec:uscqs}
----------------------------------------------------------------------------
Let $d<n$ be arbitrary but fixed. For $(1^d)\in\Lambda(n,d)$, we write $1_d=1_{(1^d)}$. We define a monoidal additive $\Q$-linear functor $$\Sigma_{n,d}\colon\mathcal{SC}_1'(d)\to \Scat(n,d)^*((1^d),(1^d)),$$ which is very similar to $\Sigma_{n,n}$ from the previous subsection and categorifies $\sigma_{n,d}$ of Section \[sec:hecke-schur\]. Recall that $\mathcal{SC}_1(d)\subseteq\mathcal{SC}_1'(d)$ is a faithful subcategory. So we define $\Sigma_{n,d}$ in exactly the same way as $\Sigma_{n,n}$, but restricting to the colors $1\leq i\leq d-1$ and sending $\emptyset$ to the empty diagram in the region labeled $(1^d)$ instead of $(1^n)$. The only new ingredient for the definition of $\Sigma_{n,d}$ is the image of the boxes, which we define by $$\Sigma_{n,d}\bigl(\;\bbox{i}\;\bigr) =
\sum\limits_{j=i}^{d-1}\
\xy
(0,0)*{\dblue\xybox{%
(3,0);(-3,0) **\crv{(3,4.2) & (-3,4.2)};
?(.05)*\dir{>} ?(1)*\dir{>};
(3,0);(-3,0) **\crv{(3,-4.2) & (-3,-4.2)} ?(.3)*\dir{}+(2,0)*{\bscs j};
}};
(6,3)*{\scs (1^d)},
\endxy-\xy 0;/r.18pc/:
(0,-1)*{\dred\ccbub{\black -1}{\black d}};
(8,4)*{\scs(1^d)};
\endxy$$ for any $i=1,\ldots,d$. Note that we have $$\Sigma_{n,d}\bigl(\;\bbox{i}-\bbox{i+1}\;\bigr) = \ \
\xy
(0,0)*{\dblue\xybox{%
(3,0);(-3,0) **\crv{(3,4.2) & (-3,4.2)};
?(.05)*\dir{>} ?(1)*\dir{>};
(3,0);(-3,0) **\crv{(3,-4.2) & (-3,-4.2)} ?(.3)*\dir{}+(2,0)*{\bscs i};
}};
(6,3)*{\scs (1^d)},
\endxy$$ which agrees with the first box relation . One easily checks that $\Sigma_{n,d}$ preserves the other box relations as well. The rest of the proof that $\Sigma_{n,d}$ is well-defined uses the same arguments as in the previous subsection.
As in Subsection \[ssec:scqs\] we have
\[lem:ucomm\] There is a commutative diagram $$\xymatrix{
\mathcal{SC}_1'(d)\ar[rr]^{\fek'}\ar[dr]_{\Sigma_{n,d}} && \bim(d)^*
\\
& \Scat(n,d)^*((1^d),(1^d))\ar[ur]_{\fbim} &
}$$
\[prop:ufull\] The functor $\Sigma_{n,d}$ is an equivalence of categories.
Note that $\mathcal{E}_{+k}1_d=0$, for any $k\geq d$, so by the commutation isomorphisms in $\Scat(n,d)$ we see that any object $x$ in $\Scat(n,d)^*((1^d),(1^d))$ is isomorphic to a direct sum of objects whose factors are all of the form $\mathcal{E}_{-i}\mathcal{E}_{+i}1_d$ with $1\leq i\leq d-1$. This is a consequence of the commutation relations on the decategorified level [@D-G] which become commutation isomorphisms on the category level. Therefore $\Sigma_{n,d}$ is essentially surjective. Faithfulness follows from Elias and Khovanov’s results and the commuting triangle in Lemma \[lem:ucomm\], just as in the previous subsection.
The arguments which show that $\Sigma_{n,d}$ is full are almost identical to the ones in the previous subsection. The only difference is that we now have $$\HOM_{\mathcal{SC}_1'(d)}(\emptyset,\emptyset)\cong\Q[x_1,\ldots,x_d]\cong
\ENDS(1_d).$$ The first isomorphism follows from Elias and Khovanov’s results in [@E-Kh]. The second isomorphism follows from the fact that the $i$-colored bubbles of positive degree are all zero for $i>d$, since their inner regions are labeled by elements that do not belong to $\Lambda(n,d)$, and the $d$-colored bubble with a dot is mapped to $x_d$. Therefore we have $$\ENDS(1_d)\cong
\Q[x_1-x_2,\ldots,x_{d-1}-x_d,x_d]\cong \Q[x_1,\ldots,x_d].$$
Grothendieck algebras {#sec:grothendieck}
=====================
The Grothendieck algebra of $\Scat(n,d)$
----------------------------------------
To begin with, let us introduce some notions and notations analogous to Khovanov and Lauda’s in Section 3.5 in [@K-L3]. Let $\UcatD$ and $\ScatD(n,d)$ denote the Karoubi envelopes of $\Ucat$ and $\Scat(n,d)$ respectively. We define the objects of $\ScatD(n,d)$ to be the elements in $\Lambda(n,d)$ and we define the hom-category $\ScatD(n,d)(\lambda,\mu)$ to be the usual Karoubi envelope of $\Scat(n,d)(\lambda,\mu)$, for any $\lambda,\mu\in\Lambda(n,d)$. There exist idempotents $e\in\EndS({\mathcal E}_{\mathbf i}1_{\lambda})$, so that $({\mathcal E}_{\mathbf i},e)$ is a direct summand of ${\mathcal E}_{\mathbf i}$ in $\ScatD(n,d)$. For example, we can define the idempotents $$e_{+i,m,\lambda} = \xy 0;/r.15pc/:
(0,0)*{\dblue\xybox{
(-12,-20)*{}; (12,20) **\crv{(-12,-8) & (12,8)}?(1)*\dir{>};
(-4,-20)*{}; (4,20) **\crv{(-4,-13) & (12,2) & (12,8)&(4,13)}?(1)*\dir{>};?(.88)*\dir{}+(0.1,0)*{\bullet};
(4,-20)*{}; (-4,20) **\crv{(4,-13) & (12,-8) & (12,-2)&(-4,13)}?(1)*\dir{>}?(.86)*\dir{}+(0.1,0)*{\bullet};
?(.92)*\dir{}+(0.1,0)*{\bullet};
(12,-20)*{}; (-12,20) **\crv{(12,-8) & (-12,8)}?(1)*\dir{>}?(.70)*\dir{}+(0.1,0)*{\bullet};
?(.90)*\dir{}+(0.1,0)*{\bullet};?(.80)*\dir{}+(0.1,0)*{\bullet};
(16,0)*{\lambda};
}};
\endxy
,
\qquad e_{-i,m,\lambda} = (-1)^{\frac{m(m-1)}{2}}\;\xy 0;/r.15pc/:
(0,0)*{\dblue\xybox{
(-12,-20)*{}; (12,20) **\crv{(-12,-8) & (12,8)}?(0)*\dir{<};
(-4,-20)*{}; (4,20) **\crv{(-4,-13) & (12,2) & (12,8)&(4,13)}?(0)*\dir{<};?(.88)*\dir{}+(0.1,0)*{\bullet};
(4,-20)*{}; (-4,20) **\crv{(4,-13) & (12,-8) & (12,-2)&(-4,13)}?(0)*\dir{<}?(.86)*\dir{}+(0.1,0)*{\bullet};
?(.92)*\dir{}+(0.1,0)*{\bullet};
(12,-20)*{}; (-12,20) **\crv{(12,-8) & (-12,8)}?(0)*\dir{<}?(.70)*\dir{}+(0.1,0)*{\bullet};
?(.90)*\dir{}+(0.1,0)*{\bullet};?(.80)*\dir{}+(0.1,0)*{\bullet};
(16,0)*{\lambda} }};
\endxy$$ in $\EndS(\mathcal{E}_{+i^m}1_{\lambda})$ and $\EndS(\mathcal{E}_{-i^m}1_{\lambda})$ respectively. We can define the $1$-morphisms in $\ScatD(n,d)$ $${\mathcal E}_{\pm i^{(m)}}1_{\lambda}:=({\mathcal E}_{\pm i^m}1_{\lambda},
e_{\pm i,m,\lambda})\left\{ \dfrac{m(1-m)}{2}\right\}$$ and have $${\mathcal E}_{\pm i^m}1_{\lambda}\cong \left({\mathcal E}_{\pm i^{(m)}}1_{\lambda}
\right)^{\oplus [m]!}.$$ Recall that $[m]!\in\bN[q,q^{-1}]$ is the $q$-factorial $[m][m-1]\cdots 1$, with $[s]=(q^s-q^{-s})/(q-q^{-1})$. For any $q$-integer $\oplus_{n=-j}^{k} a_{n}q^{n}\in\bN[q,q^{-1}]$, we define $$A^{\oplus_{n=-j}^{k} a_{n}q^{n}}= \bigoplus_{n=-j}^{k}
\left(\oplus_{i=1}^{a_n}A\{n\}\right).$$ Note that $e_{+i,m,\lambda}=0$ for $m>\lambda_{i+1}$ and $e_{-i,m,\lambda}=0$ for $m>\lambda_i$, because for those values of $m$ the left-most region of their defining diagrams has a label with a negative entry. This shows that these idempotents depend on $\lambda$, which was not the case in [@K-L3]. Note that these lower bounds for $m$ are sharp, i.e. $$\begin{aligned}
{\mathcal E}_{+i^{(m)}}1_{\lambda}=0&\Leftrightarrow
m>\lambda_{i+1}\\
{\mathcal E}_{-i^{(m)}}1_{\lambda}=0&\Leftrightarrow m>\lambda_i. \end{aligned}$$ This follows from observing the image of ${\mathcal E}_{\pm i^{(m)}}1_{\lambda}$ under the $2$-functor $\fbim\colon \Scat(n,d)^*\to \bim^*$.
Before we go on, let us make the remark alluded to above Conjecture \[conj:bubbles\], when we showed that $$\begin{aligned}
\label{eq:non-inj_two}
\xy 0;/r.18pc/:
(0,0)*{\dred\cbub{\black 1}{2}};
(4,9)*{\scs(0,1,0)};
\endxy
- \ \
\xy 0;/r.18pc/:
(0,-1)*{\dblue\cbub{\black -1}{1}};
(4,8)*{\scs(0,1,0)};
\endxy
\
=&\quad 0.\end{aligned}$$
\[rem:zeros\] Suppose $\lambda=(\ldots,a,0,\ldots)\in\Lambda(n,d)$, with $a$ in the $i$th position. Let $\mu=(\ldots,0,a,\ldots)$ be obtained from $\lambda$ by switching $a$ and $0$. From Theorem 5.6 and Corollary 5.8 in [@K-L-M-S] it follows that $$\mathcal{E}_{i^{(a)}}\mathcal{E}_{{-i}^{(a)}}1_{\lambda}\cong
1_{\lambda}\quad\mbox{and}\quad
\mathcal{E}_{{-i}^{(a)}}\mathcal{E}_{i^{(a)}}1_{\mu}\cong 1_{\mu},$$ because we have $\mathcal{E}_{i^{(j)}}1_{\lambda}=0$ and $\mathcal{E}_{{-i}^{(j)}}1_{\mu}=0$ in $\ScatD(n,d)$ for any $j>0$. Therefore $\lambda$ and $\mu$ are isomorphic objects in the 2-category $\ScatD(n,d)$. Our proof of used the $2$-isomorphism between $1_{(0,1,0)}$ and $\mathcal{E}_{-1}\mathcal{E}_11_{(0,1,0)}$ explicitly in the first step.
Note that $\ScatD(n,d)$ is Krull-Schmidt, just as $\UcatD$. Therefore, we can take the split Grothendieck algebras/categories $K_0^{\Q(q)}(\UcatD)$ and $K_0^{\bQ(q)}(\ScatD(n,d))$. Considering the latter as a category, we follow Khovanov and Lauda [@K-L3] and define $\Lambda(n,d)$ to be the set of objects. The hom-space $\hom(\lambda,\mu)$ we define to be the split Grothendieck algebra of the additive category $\ScatD(\lambda,\mu)$. Alternatively, we can see this as an (idempotented) algebra rather than a category. In the sequel we will use both points of view interchangeably. Note that the remark above shows that there are objects in $K_0^{\bQ(q)}(\ScatD(n,d))$ which are isomorphic, e.g. $(1,0,0), (0,1,0)$ and $(0,0,1)$ are all isomorphic in $K_0^{\bQ(q)}(\ScatD(3,1))$.
Analogous to Khovanov and Lauda’s homomorphism $\gamma=\gamma_U\colon
\dot{\mathbf U}(\mathfrak{sl}_n)\to K_0^{\bQ(q)}(\UcatD)$, we define a homomorphism $\gamma_S\colon \SD(n,d)\to K_0^{\bQ(q)}(\ScatD(n,d))$ by $$E_{s_1}\cdots E_{s_m}1_{\lambda}\mapsto
\left[\mathcal{E}_{s_1}\cdots\mathcal{E}_{s_m}
1_{\lambda}\right].$$ Our main goal in this section is to prove that $\gamma_S$ is an isomorphism. Recall that in order to show that $\gamma_U$ is an isomorphism, Khovanov and Lauda had to determine the indecomposable direct summands of certain $1$-morphisms $x$ in $\UcatD$. They did this by looking at $K_0^{\bQ(q)}(\ENDU(x))$, which is the Grothendieck group of the finitely generated graded projective $\ENDU(x)$-modules. This allowed them to use known results about the Grothendieck groups of graded algebras, which we recall below. The connection between the two sorts of Grothendieck groups relies on the fact that a finitely-generated graded projective $\ENDU(x)$-module is determined by an idempotent $e$ in $\ENDU(x)$ and $[(x,e)]$ is an element of $K_0^{\bQ(q)}(\UcatD)$. The isomorphism classes of indecomposable projective modules form a basis of $K_0^{\bQ(q)}(\ENDU(x))$ and correspond to the minimal idempotents in $\ENDU(x)$. We refer to [@K-L3] for more details. We will follow Khovanov and Lauda’s approach closely to show that $\gamma_S$ is surjective, but will use a completely different method to show that $\gamma_S$ is injective. Although we have tried to explain our results clearly, we suspect that the part of this section which deals with the surjectivity of $\gamma_S$ will be quite hard to understand for someone unfamiliar with [@K-L1; @K-L2; @K-L3; @L1]. The part on the injectivity of $\gamma_S$ can probably be read independently.
Before we move on to our results in this section, we should recall the basic facts about Grothendieck groups of (graded) algebras which Khovanov and Lauda explained in Subsections 3.8.1 and 3.8.2 in [@K-L3]. If $A$ is a finite-dimensional algebra over a field, let $K_0(A)$ be the Grothendieck group of the category of the finitely generated projective $A$-modules.
\[prop:surj\] Let $f\colon A\to B$ be a surjective homomorphism between two finite-dimensional algebras, then $K_0(f)\colon K_0(A)\to K_0(B)$ is surjective.
Unfortunately in the applications in [@K-L3] and in our paper, the algebras involved are not finite-dimensional. But fortunately they are $\bZ$-graded and we can resort to finite-dimensional quotients which do not alter the Grothendieck groups. Let $A$ be a $\bZ$-graded algebra over a field, such that in each degree it has finite dimension and the grading is bounded from below.
\[defn:virtnilpot\] Let $I\subset A$ be a two-sided homogeneous ideal. We say that $I$ is [*virtually nilpotent*]{} if for each degree $a\in\bZ$ there exists an $N>0$ such that the degree $a$ summand of $I^N$ is equal to zero.
\[lem:virtnilpot\] Let $I\subset A$ be a virtually nilpotent ideal. Then $K_0(A)\cong K_0(A/I)$.
\[cor:virtnilpot\] Let $f\colon A\to B$ be a degree preserving homomorphism of $\bZ$-graded algebras of the type described above, and $I\subset A$ a virtually nilpotent ideal of finite codimension. If $f$ is surjective, then $K_0(f)\colon K_0(A)\cong K_0(A/I)\to K_0(B/f(I))\cong
K_0(B)$ is surjective.
We also need a fact about the split Grothendieck group of Krull-Schmidt categories. This result is not recalled in [@K-L3], but is well known in homological algebra. We thank Mikhail Khovanov for explaining it to us. To help the reader, we briefly sketch the proof below.
\[prop:inj\] Let ${\mathcal F}\colon C\to D$ be an additive $\Q$-linear degree preserving functor between two graded Krull-Schmidt categories, whose hom-spaces are finite-dimensional in each degree and whose gradings are bounded from below. If ${\mathcal F}$ is fully faithful, then $K_0({\mathcal F})\colon K_0(C)\to K_0(D)$ is injective.
Since $C$ and $D$ are Krull-Schmidt, each object in $C$ or $D$ can be uniquely decomposed into indecomposables, which generate $K_0(C)$ and $K_0(D)$ respectively. Being fully faithful, $\mathcal F$ maps the set of indecomposables in $C$ injectively into the set of indecomposables in $D$.
We now get to the main part of this section. By simply checking the definitions, we see that the following square commutes: $$\begin{CD}
\label{cd:groth}
\U @>{\gamma_U}>>K_0^{\bQ(q)}(\UcatD)\\
@V{\phi_{n,d}}VV @VV{K_0^{\bQ(q)}(\Psi_{n,d})}V\\
\SD(n,d)@>{\gamma_S}>> K_0^{\bQ(q)}(\ScatD(n,d)).
\end{CD}$$ We know that $\phi_{n,d}$ is surjective and $\gamma_U$ is an isomorphism. We also know that $\Psi_{n,d}$ is full, but we cannot automatically conclude that $K_0^{\bQ(q)}(\Psi_{n,d})$ is surjective, because $\ENDS(x)$ is infinite-dimensional for any $1$-morphism $x$. We want to prove that $K_0^{\bQ(q)}(\Psi_{n,d})$ and $\gamma_S$ are surjective. Of course it suffices to prove that $\gamma_S$ is surjective.
Let us first sketch the chain of arguments that leads to the proof of the surjectivity of $\gamma_U$ in Theorem 1.1 in [@K-L3]. The proof is by induction with respect to the width of an indecomposable $1$-morphism $P$ in $\Ucat$, which by definition is the smallest non-negative integer $m$ such that $P$ is isomorphic to a direct summand of ${\mathcal E}_{\mathbf i}1_{\lambda}\{t\}$ with $||{\mathbf i}||=m$. In Lemma 3.38 Khovanov and Lauda prove that any indecomposable object of width $m$ is isomorphic to a direct summand of $\mathcal{E}_{\nu,-\nu'}1_{\lambda}\{t\}$, for certain $\lambda\in\bZ^{n-1}$, $t\in\bZ$ and $\nu,\nu'\in\bN[I]$, such that $\vert\vert\nu\vert\vert+\vert\vert\nu'\vert\vert=m$. This narrows down the number of cases that need to be considered in the proof of Theorem 1.1.
Next, suppose $P$ has width zero, then $P\cong 1_{\lambda}$ up to a shift, and $K_0^{\bQ(q)}(\ENDU(1_{\lambda}))$ lies in the image of $\gamma_U$, because it is isomorphic to $\Q$ with generator $[1_{\lambda}]$. The induction step relies on the exact sequence of rings (3.38) $$\label{eq:KLExSeq}
0\to I_{\nu,-\nu',\lambda}\to \ENDU({\mathcal E}_{\nu,-\nu'}
1_{\lambda})\to R_{\nu,-\nu',\lambda}\to 0.$$ Recall that for $\mathfrak{g}=\mathfrak{sl}_n$, the ring $R_{\nu,-\nu',\lambda}$ is isomorphic to that of 2-morphisms whose diagrams are split into upward strands with source and target belonging to $\nu$, downward strands with source and target belonging to $-\nu'$, and bubbles on the right-hand side. The ideal $I_{\nu,-\nu',\lambda}$ is generated by diagrams which contain at least one cup or cap between $\nu$ and $-\nu'$. Note that the latter are precisely the 2-morphisms which factor through a direct sum of objects with width smaller than $\vert\vert\nu\vert\vert+\vert\vert\nu'\vert\vert$. As they remark in Remark 3.18, this exact sequence is split for $\mathfrak{g}=\mathfrak{sl}_n$. Therefore there is a direct sum decomposition $$\label{eq:KLDirSum}
K_0^{\bQ(q)}(\ENDU({\mathcal E}_{\nu,-\nu'}
1_{\lambda}))\cong K_0^{\bQ(q)}(I_{\nu,-\nu',\lambda})
\oplus K_0^{\bQ(q)}(R_{\nu,-\nu',\lambda}).$$ The fact that $K_0^{\bQ(q)}(R_{\nu,-\nu',\lambda})$ lies in the image of $\gamma_U$ is essentially a consequence of the results in [@K-L1; @K-L2] and a technical result involving a virtually nilpotent ideal, the details of which we do not need here. On the other hand, The 2-morphisms in $I_{\nu,-\nu',\lambda}$ factor through direct sums of objects of smaller width, so any minimal idempotent in this ideal corresponds to an object of smaller width. Therefore $K_0^{\bQ(q)}(I_{\nu,-\nu',\lambda})$ lies in the image of $\gamma_U$ by induction. This shows that $K_0^{\bQ(q)}(\ENDU({\mathcal E}_{\nu,-\nu'}
1_{\lambda}))$ lies in the image of $\gamma_U$, as had to be proved. We should warn the reader that, contrary to what might seem at a first reading, the direct sum decomposition in does not preserve indecomposability. For example, consider the direct sum $EF1_1\cong FE1_1\oplus 1_1$ for $n=2$. This corresponds to the diagrammatic equation $$\begin{aligned}
\label{eq:EFagain}
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{};
(8,0)*{};
(-4,10)*{}="t1";
(4,10)*{}="t2";
(-4,-10)*{}="b1";
(4,-10)*{}="b2";
"t1";"b1" **\dir{-} ?(.5)*\dir{<};
"t2";"b2" **\dir{-} ?(.5)*\dir{>};}};
(-6,-8)*{};
(6,-8)*{};
(10,2)*{(1)};
(-10,2)*{(1)};
\endxy}
&\quad = \quad&
\vcenter{ \xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{<};?(0)*\dir{<};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)}?(1)*\dir{>};}};
(10,2)*{(1)};(-6,-7)*{};(6.8,-7)*{};
\endxy}
\quad - \quad
\vcenter{\xy 0;/r.18pc/:
(10,2)*{(1)};
(0,0)*{\dblue\xybox{
(-4,-8)*{}="b1";
(4,-8)*{}="b2";
"b2";"b1" **\crv{(5,-2) & (-5,-2)}; ?(.05)*\dir{<} ?(.93)*\dir{<}
?(.8)*\dir{}+(0,-.1)*{}+(-5,2)*{};
(-4,8)*{}="t1";
(4,8)*{}="t2";
"t2";"t1" **\crv{(5,2) & (-5,2)}; ?(.15)*\dir{>} ?(.95)*\dir{>}
?(.4)*\dir{};}}
\endxy}\end{aligned}$$ The identity on $EF1_1$ is an indecomposable idempotent in $R_{+,-,(1)}$, but can be decomposed in $\ENDU(\mathcal{E}_{+,-}1_1)$ into the two indecomposable idempotents on the right-hand side of , which have width 2 and 0 respectively. Note that the second term on the right-hand side belongs to $I_{+,-,(1)}$. So Khovanov and Lauda’s homomorphism $$\beta\colon \ENDU(\mathcal{E}_{+,-}1_1)\to R_{+,-,(1)}$$ maps the first term on the right-hand side to the identity on $EF1_1$. The map backwards, which they call $\alpha$, is simply the inclusion, so it maps the identity to the identity. In the induction step above, one therefore writes $$\begin{aligned}
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-4,-4)*{};(4,4)*{} **\crv{(-4,-1) & (4,1)}?(1)*\dir{>};
(4,-4)*{};(-4,4)*{} **\crv{(4,-1) & (-4,1)}?(1)*\dir{<};?(0)*\dir{<};
(-4,4)*{};(4,12)*{} **\crv{(-4,7) & (4,9)};
(4,4)*{};(-4,12)*{} **\crv{(4,7) & (-4,9)}?(1)*\dir{>};}};
(10,2)*{(1)};(-6,-7)*{};(6.8,-7)*{};
\endxy}
&\quad = \quad
\vcenter{\xy 0;/r.18pc/:
(0,0)*{\dblue\xybox{
(-8,0)*{};
(8,0)*{};
(-4,10)*{}="t1";
(4,10)*{}="t2";
(-4,-10)*{}="b1";
(4,-10)*{}="b2";
"t1";"b1" **\dir{-} ?(.5)*\dir{<};
"t2";"b2" **\dir{-} ?(.5)*\dir{>};}};
(-6,-8)*{};
(6,-8)*{};
(10,2)*{(1)};
(-10,2)*{(1)};
\endxy}
\quad - \left( - \quad
\vcenter{\xy 0;/r.18pc/:
(10,2)*{(1)};
(0,0)*{\dblue\xybox{
(-4,-8)*{}="b1";
(4,-8)*{}="b2";
"b2";"b1" **\crv{(5,-2) & (-5,-2)}; ?(.05)*\dir{<} ?(.93)*\dir{<}
?(.8)*\dir{}+(0,-.1)*{}+(-5,2)*{};
(-4,8)*{}="t1";
(4,8)*{}="t2";
"t2";"t1" **\crv{(5,2) & (-5,2)}; ?(.15)*\dir{>} ?(.95)*\dir{>}
?(.4)*\dir{};}}
\endxy}\right)\end{aligned}$$ to prove that the class of the indecomposable summand of $EF1_1$ of width 2 corresponding to the idempotent on the left-hand side, belongs to the image of $\gamma_U$.
Next let us see how Khovanov and Lauda’s proofs can be adapted to our setting. In the first place, note that all results in Section 3.5 of [@K-L3] continue to be true. More precisely, the statements in their Propositions 3.24, 3.25 and 3.26 are still true, although some direct summands might now be zero depending on the labels of the regions in the diagrams. The crucial Lemma 3.38 in Section 3.8 in [@K-L3] holds literally true in our case just as well.
Let us now prove the analogue of their Theorem 1.1. Our proof is essentially the same, except that we use the fact that $\gamma_U$ is an isomorphism and $\Psi_{n,d}$ is full to avoid having to formulate and use analogues of the results in [@K-L1] and [@K-L2], which might be hard. This is the reason why we did not go into the details of those results above.
\[lem:surj\] The homomorphism $$\gamma_S\colon \SD(n,d)\to K_0^{\bQ(q)}(\ScatD(n,d))$$ is surjective.
For the basis of the induction, recall our surjection $S\Pi_{\lambda}\to \ENDS(1_{\lambda})$ explained in Section \[sec:struct\]. The ideal of elements of positive degree $S\Pi_{\lambda}^+$ is virtually nilpotent of codimension one, so by Corollary \[cor:virtnilpot\] it follows that $$\Q\cong K_0^{\bQ(q)}(S\Pi_{\lambda})\to K_0^{\bQ(q)}(\ENDS(1_{\lambda}))$$ is surjective. Therefore $K_0^{\bQ(q)}(\ENDS(1_{\lambda}))$ is generated by $[1_{\lambda}]$, i.e. $1_{\lambda}$ is also indecomposable in our case. Since $\gamma_S(1_{\lambda})=[1_{\lambda}]$, we see that $K_0^{\bQ(q)}(\ENDS(1_{\lambda}))$ lies in the image of $\gamma_S$. Note that we have not yet proved that $[1_{\lambda}]\ne 0$. After we have proved that $K_0^{\bQ(q)}(\ScatD(n,d))\cong \SD(n,d)$ in Theorem \[thm:groth\], it follows that $K_0^{\bQ(q)}(\ENDS(1_{\lambda}))\cong \Q$ with $[1_{\lambda}]\ne 0$ being the generator.
For the induction step, note that $\Psi_{n,d}$ maps the exact sequence (\[eq:KLExSeq\]) surjectively onto the exact sequence $$\label{eq:MSVExSeq}
0\to \Psi_{n,d}(I_{\nu,-\nu',\overline{\lambda}})\to
\ENDS({\mathcal E}_{\nu,-\nu'}
1_{\lambda})\to \ENDS({\mathcal E}_{\nu,-\nu'}1_{\lambda})/
\Psi_{n,d}(I_{\nu,-\nu',\overline{\lambda}})\to 0.$$ We do not know if this exact sequence is split, but fortunately it does not matter for our purpose.
Note also that $\Psi_{n,d}$ induces a surjective map $$R_{\nu,-\nu',\lambda}\to \ENDS({\mathcal E}_{\nu,-\nu'}1_{\lambda})/
\Psi_{n,d}(I_{\nu,-\nu',\overline{\lambda}}).$$ Recall that Khovanov and Lauda defined a virtually nilpotent ideal $\beta\alpha(J)\subset R_{\nu,-\nu',\overline{\lambda}}$ of codimension one in Section 3.8.3 in [@K-L3], alluded to above. By Corollary \[cor:virtnilpot\] this implies that $$\label{eq:Rsurj}
K_0^{\bQ(q)}(R_{\nu,-\nu',\overline{\lambda}})\to
K_0^{\bQ(q)}(\ENDS
({\mathcal E}_{\nu,-\nu'}1_{\lambda})/
\Psi_{n,d}(I_{\nu,-\nu',\overline{\lambda}}))$$ is surjective. Now, just as in the proof of Theorem 1.1, let $e\in \ENDS({\mathcal E}_{\nu,-\nu'}1_{\lambda})$ be a minimal idempotent of width $m$, with $||\nu||+||\nu'||=m$. We have to show that $[({\mathcal E}_{\nu,-\nu'}1_{\lambda},e)]$ lies in the image of $\gamma_S$. Let $\overline{e}$ be the image of $e$ in $\ENDS
({\mathcal E}_{\nu,-\nu'}1_{\lambda})/
\Psi_{n,d}(I_{\nu,-\nu',\overline{\lambda}})$. Note that we do not know a priori that $\overline{e}$ is indecomposable, but that does not matter. By the surjectivity of , we can lift $\overline{e}$ to an idempotent $e'\in
R_{\nu,-\nu',\overline{\lambda}}$. By Khovanov and Lauda’s results, we know that $$[({\mathcal E}_{\nu,-\nu'}1_{\overline{\lambda}},e')]\in
K_0^{\bQ(q)}(\ENDU
({\mathcal E}_{\nu,-\nu'}1_{\lambda}))\subseteq
K_0^{\bQ(q)}(\UcatD)$$ is in the image of $\gamma_U$. By the commutativity of the square in , this implies that $$[({\mathcal E}_{\nu,-\nu'}1_{\lambda},\Psi_{n,d}(e'))]\in
K_0^{\bQ(q)}(\ENDS
({\mathcal E}_{\nu,-\nu'}1_{\lambda}))\subseteq
K_0^{\bQ(q)}(\ScatD(n,d))$$ is in the image of $\gamma_S$. Note that $e-\Psi_{n,d}(e')$ maps to zero in $\ENDS
({\mathcal E}_{\nu,-\nu'}1_{\lambda})/
\Psi_{n,d}(I_{\nu,-\nu',\overline{\lambda}})$. By the minimality of $e$, we therefore have $\Psi_{n,d}(e')=e+e''$, with $e''$ an orthogonal idempotent in $\Psi_{n,d}(I_{\nu,-\nu',\overline{\lambda}})$ which can be decomposed into minimal idempotents of width $<m$. By induction $[({\mathcal E}_{\nu,-\nu'}1_{\lambda},e'')]$ is contained in the image of $\gamma_S$. This shows that $[({\mathcal E}_{\nu,-\nu'}1_{\lambda},e)]$ is contained in the image of $\gamma_S$ too, as we had to show.
The following two corollaries are immediate.
\[cor:surj1\] The homomorphism $$K_0^{\bQ(q)}(\Psi_{n,d})\colon K_0^{\bQ(q)}(\dot{\mathcal U}(\mathfrak{sl}(n)))
\to K_0^{\bQ(q)}(\ScatD(n,d))$$ is surjective.
\[cor:surj2\] $K_0^{\bQ(q)}(\ScatD(n,d))$ is a quotient of $\SD(n,d)$. In particular $K_0^{\bQ(q)}(\ScatD(n,d))$ is finite-dimensional and semi-simple.
Before we prove the main result of this paper, we first categorify the homomorphism ${\iota}_{n,m}$ from Section \[sec:hecke-schur\]. Let $m\geq n$ and $d$ arbitrary. Let $\Xi_{n,m}=\oplus_{\lambda\in
\Lambda(n,d)}1_{\lambda}\in \Scat(m,d)$. Let $\Scat(n,m,d)$ be the full sub-2-category of $\Scat(m,d)$ whose objects belong to $\Lambda(n,d)\subseteq \Lambda(m,d)$.
\[defn:INCL\] Let $m\geq n$ and $d$ arbitrary. We define a functor $${\mathcal I}_{n,m}\colon \Scat(n,d)\to \Scat(n,m,d)$$ by mapping any diagram in $\Scat(n,d)$ to itself, using the inclusion $\Lambda(n,d)\subseteq\Lambda(m,d)$ for the labels.
It is easy to see that ${\mathcal I}_{n,m}$ is well-defined and essentially surjective. We conjecture it to be faithful, but have no proof. It is certainly not full, because $\Scat(n,m,d)$ contains $n$-colored bubbles for example. Perhaps there is a virtually nilpotent ideal $I\subset \Scat(n,m,d)$ such that $\Scat(n,d)\cong \Scat(n,m,d)/I$, e.g. the ideal generated by all diagrams with $n$-colored bubbles of positive degree on the right-hand side.
\[thm:groth\] The homomorphism $$\gamma_S\colon \SD(n,d)\to K_0^{\bQ(q)}(\ScatD(n,d))$$ is an isomorphism.
After the result of Lemma \[lem:surj\] it only remains to show that $K_0^{\bQ(q)}(\Scat(n,d))$ and $\SD(n,d)$ have the same dimension.
We first show the case $n=d$. Let $1_n=1_{(1^n)}$. In Proposition \[prop:fulfaith\] we proved that $\mathcal{SC}_1(n)\cong \Scat(n,n)^*((1^n),(1^n))$ is a full sub-2-category of $\Scat(n,n)^*$. By Proposition \[prop:inj\] this implies $$K_0^{\bQ(q)}(\mathcal{SC}(n))\cong K_0^{\bQ(q)}(\ScatD(n,n)((1^n),(1^n)))\subseteq
K_0^{\bQ(q)}(\ScatD(n,n)).$$ By Theorem \[thm:e-k-s\] we know that $K_0^{\bQ(q)}(\mathcal{SC}(n))$ is isomorphic to $H_q(n)$. Thus Lemma \[lem:emb\] implies that $K_0^{\bQ(q)}(\ScatD(n,n))\cong \SD(n,n)$.
Now let $d<n$. In Proposition \[prop:ufull\] we proved that $\mathcal{SC}'_1(d)\cong \Scat(n,d)^*((1^d),(1^d))$ is a full sub-2-category of $\Scat(n,d)^*$. By Proposition \[prop:inj\] this implies $$K_0^{\bQ(q)}(\mathcal{SC}'(d))\cong K_0^{\bQ(q)}(\ScatD(n,d)((1^d),(1^d)))
\subseteq K_0^{\bQ(q)}(\ScatD(n,d)).$$ By Theorem \[thm:e-k-s\] we know that $K_0^{\bQ(q)}(\mathcal{SC}'(d))$ is isomorphic to $H_q(d)$. Thus Lemma \[lem:emb\] shows that $K_0^{\bQ(q)}(\ScatD(n,d))\cong \SD(n,d)$.
Next, assume that $n<d$. Consider the functor $${\cal I}_{n,d}\colon \Scat(n,d)\to \Scat(n,d,d).$$ We have the following commuting square $$\begin{CD}
\SD(n,d)@>{\iota_{n,d}}>>\xi_{n,d}\SD(d,d)\xi_{n,d}\\
@V{\gamma_S(n)}VV @VV{\gamma_S(d)}V\\
K_0^{\bQ(q)}(\ScatD(n,d))@>{K_0^{\bQ(q)}({\mathcal I}_{n,d})}>>K_0^{\bQ(q)}(\ScatD(n,d,d)).
\end{CD}$$ We already know that $\gamma_S(d)\colon \SD(d,d)\to K_0^{\bQ(q)}(\Scat)(d,d))$ is an isomorphism from the first case we proved. Therefore $\gamma_S(d)\colon\xi_{n,d}\SD(n,d)\xi_{n,d}\to
[\Xi_{n,d}]K_0^{\bQ(q)}(\ScatD(d,d))[\Xi_{n,d}]\cong K_0^{\bQ(q)}(\ScatD(n,d,d))$ is an isomorphism as well. Recall that $\iota_{n,d}$ is an isomorphism. It follows that $\gamma_S(n)$ is injective. Recall that $\gamma_S(n)$ is surjective, by Lemma \[lem:surj\]. It follows that $K_0^{\bQ(q)}(\ScatD(n,d))\cong \SD(n,d)$.
Note that we did not follow Khovanov and Lauda’s approach to prove injectivity of $\gamma_S$. Recall that they defined a non-degenerate $\Q$-semilinear form on $\dot{U}(\mathfrak{sl}_n)$, which is closely related to Lusztig’s bilinear form in [@Lu], and defined an inner product on $K_0^{\bQ(q)}(\UcatD)$ by $$\langle[x],[y]\rangle=\dim_q(\HOMUD(x,y)).$$ They showed that $\gamma_U$ is injective by proving that it is an isometry. We could not prove that $\gamma_S$ is injective in this way, because we could not find such a $\Q$-semilinear form on $\SD(n,d)$ in the literature.[^4] By our Theorem \[thm:groth\], we can define one now. We first define a non-degenerate $\Q$-semilinear form on $K_0^{\bQ(q)}(\ScatD(n,d))$ as above $$\langle [x],[y]\rangle=\dim_q(\HOMSD(x,y)).$$
We define a non-degenerate $\Q$-semilinear form on $\SD(n,d)$ by $$\langle x,y\rangle=\langle\gamma_S(x),\gamma_S(y)\rangle.$$
By definition $\gamma_S$ is an isometry. It is easy to see that the semilinear form on $\SD(n,d)$ has the following properties (compare to Proposition 2.4 in [@K-L3]):
We have
1. $\langle 1_{\lambda_1}x1_{\lambda_2},1_{\lambda'_1}x1_{\lambda'_2}\rangle=0$ for all $x,y$ unless $\lambda_1=\lambda'_1$ and $\lambda_2=\lambda'_2$.
2. $\langle ux,y\rangle=\langle x,\tau(u)y\rangle.$
However, Khovanov and Lauda’s interpretation of the semilinear form on $\dot{U}(\mathfrak{sl}_n)$ in Theorem 2.7 in [@K-L3], which shows that $\langle[x],[y]\rangle=\dim_q(\HOMUD(x,y))$ can be obtained by counting the number of minimal diagrams in each degree in $\HOMUD(x,y)$, does not hold in our case. This is because minimal diagrams in $\Scat(n,d)$ are not linearly independent in general. For example, consider relation for $n=2$ and $\lambda=(1,0)$. Note that the sum on the right-hand side only contains one term. The first term on the right-hand side, i.e. the one with the two crossings, has a middle region with label $(2,-1)\not\in\Lambda(2,1)$, so it is equal to zero. This shows that the minimal diagram on the left-hand side is equivalent to the minimal diagram on the right-hand side.
Categorical Weyl modules
------------------------
We conjecture that it is easy to categorify the irreducible representations $V_{\lambda}$, for $\lambda\in\Lambda^+(n,d)$, using the category $\Scat(n,d)$. Recall from Lemma \[lem:weyl\] that $$V_{\lambda}\cong \SD(n,d)1_{\lambda}/[\mu>\lambda].$$
For any $\lambda\in\Lambda^+(n,d)$, let $\Scat(n,d)1_{\lambda}$ be the category whose objects are the $1$-morphisms in $\Scat(n,d)$ of the form $x1_{\lambda}$ and whose morphisms are the $2$-morphisms in $\Scat(n,d)$ between such 1-morphisms. Note that $\Scat(n,d)1_{\lambda}$ does not have a monoidal structure, because two $1$-morphisms $x1_{\lambda}$ and $y1_{\lambda}$ cannot be composed in general. Alternatively one can see $\Scat(n,d)1_{\lambda}$ as a graded ring, whose elements are the morphisms.
Let ${\mathcal V}_{\lambda}$ be the quotient of $\Scat(n,d)1_{\lambda}$ by the ideal generated by all diagrams which contain a region labeled by $\mu>\lambda$.
Note that there is a natural categorical action of $\Scat(n,d)$, and therefore of $\Ucat$, on ${\mathcal V}_{\lambda}$, defined by putting a diagram in $\Scat(n,d)$ on the left-hand side of a diagram in ${\mathcal V}_{\lambda}$. This action descends to an action of $\SD(n,d)\cong K_0^{\bQ(q)}(\ScatD(n,d))$ on $K_0^{\bQ(q)}(\dot{{\mathcal V}}_{\lambda})$, where $\dot{{\mathcal V}}_{\lambda}$ is the Karoubi envelope of ${\mathcal V}_{\lambda}$. Note that $\gamma_S$ induces a well-defined linear map $\gamma_{\lambda}\colon V_{\lambda}\to K_0^{\bQ(q)}(\dot{{\mathcal V}}_{\lambda})$, which intertwines the $\SD(n,d)$-actions.
\[lem:repsurj\] The linear map $\gamma_{\lambda}$ is surjective.
We first show that $K_0^{\bQ(q)}(\ScatD(n,d)1_{\lambda})\to
K_0^{\bQ(q)}(\dot{{\mathcal V}}_{\lambda})$ is surjective. Again, we want to use Proposition \[prop:surj\], but have to be careful because the graded rings involved are not finite-dimensional. Choose an object $x\in \Scat(n,d)1_{\lambda}$. Recall that $\ENDS(x)$ is finitely generated as a right module over $\ENDS(1_{\lambda})$. Let $\ENDS(1_{\lambda})^+\subseteq
\ENDS(1_{\lambda})$ be the two-sided ideal of $2$-morphisms of strictly positive degree. Note that $\ENDS(1_{\lambda})^+$ is a codimension one virtually nilpotent ideal. Let $\mbox{END}^+_{\Scat(n,d)}(x)\subseteq \ENDS(x)$ be the image of $\ENDS(x)\otimes \ENDS(1_{\lambda})^+$ under the right action. Then $\mbox{END}^+_{\Scat(n,d)}(x)$ is a two-sided ideal of finite codimension and is virtually nilpotent, because the grading of $\ENDS(1_{\lambda})$ is bounded from below.
Now let $\mbox{END}^{>\lambda}_{\Scat(n,d)}(x)
\subseteq \ENDS(x)$ be the two-sided ideal generated by all diagrams with at least one region labeled by a $\mu>\lambda$. By Corollary \[cor:virtnilpot\], the projection $$\ENDS(x)\to \ENDS(x)/\mbox{END}^{>\lambda}_{\Scat(n,d)}(x)$$ induces a surjective homomorphism $$K_0^{\bQ(q)}(\ENDS(x))\to K_0^{\bQ(q)}(\ENDS(x)/\mbox{END}^{>\lambda}_{\Scat(n,d)}(x)).$$ Since $x$ was arbitrary, it follows that $$K_0^{\bQ(q)}(\ScatD(n,d)1_{\lambda})\to K_0^{\bQ(q)}(\dot{{\mathcal V}}_{\lambda})$$ is surjective. Thus, the composite linear map $$\SD(n,d)1_{\lambda}\cong K_0^{\bQ(q)}(\ScatD(n,d)1_{\lambda})\to
K_0^{\bQ(q)}(\dot{{\mathcal V}}_{\lambda})$$ is surjective. Note that $[\mu>\lambda]$ is contained in the kernel of this map, which proves this lemma.
For any $\lambda\in\Lambda^+(n,d)$, we have $$K_0^{\bQ(q)}(\dot{{\mathcal V}}_{\lambda})\cong V_{\lambda}.$$
We do not know how to prove the conjecture in general. Note that by Lemma \[lem:repsurj\], we have a surjective linear map $\gamma_{\lambda}\colon V_{\lambda}\to K_0^{\bQ(q)}(\dot{\mathcal V}_{\lambda})$, which intertwines the $\SD(n,d)$-actions. Since $V_{\lambda}$ is irreducible, we have $K_0^{\bQ(q)}(\dot{\mathcal V}_{\lambda})\cong V_{\lambda}$ or $K_0^{\bQ(q)}(\dot{\mathcal V}_{\lambda})=0$. So it suffices to show that $K_0^{\bQ(q)}(\dot{\mathcal V}_{\lambda})\ne 0$. Particular cases can be proved easily. For example, if $\lambda=(d)$, then ${\mathcal V}_{\lambda}=
\Scat(n,d)1_{\lambda}$, because there are no weights higher than $(d)$. By Theorem \[thm:groth\] we have $K_0^{\bQ(q)}(\ScatD(n,d)1_{\lambda})\cong
\SD(n,d)1_{\lambda}$, which proves the conjecture in this case.
We can also prove the case $n=2$. If $\lambda=(d,0)$, then the result follows from the previous case. Suppose $\lambda=(d-c,c)$, for $0<2c\leq d$. Note that $(d-2c,0)=(d-c,c)-(c,c)\in\Lambda^+(2,d-2c)$. Recall that we have a functor $$\Pi_{d,d-2c}\colon \Scat(2,d)\to\Scat(2,d-2c),$$ which induces a functor $$\Pi_{d,d-2c}\colon {\mathcal V}_{(d-c,c)}\to {\mathcal V}_{(d-2c,0)}.$$ Thus we have the following commuting square: $$\begin{CD}
V_{(d-c,c)}@>{\pi_{d,d-2c}}>> V_{(d-2c,0)}\\
@V{\gamma_{(d-c,c)}}VV @VV{\gamma_{(d-2c,0)}}V\\
K_0^{\bQ(q)}(\dot{\mathcal V}_{(d-c,c)})@>{K_0^{\bQ(q)}(\Pi_{d,d-2c})}>>K_0^{\bQ(q)}
(\dot{\mathcal V}_{(d-2c,0)})
\end{CD}$$ We know that $\pi_{d,d-2c}$ and $\gamma_{(d-2c,0)}$ are isomorphisms and $\gamma_{(d-c,c)}$ is surjective. Therefore $\gamma_{(d-c,c)}$ is an isomorphism too, so $K_0^{\bQ(q)}(\dot{\mathcal V}_{(d-c,c)})\cong V_{(d-c,c)}$.
There is an obvious functor from the Khovanov-Lauda [@K-L1] cyclotomic quotient category $R(*,\lambda)$ to a quotient of our ${\mathcal V}_{\lambda}$. The quotient is obtained by putting all bubbles of positive degree in the right-most region of the diagrams, labeled $\lambda$, equal to zero. By our observations above about $\END^+_{\Scat(n,d)}(x)$, this quotient has the same Grothendieck group as ${\mathcal V}_{\lambda}$. The functor is the “identity” on objects and morphisms. The reduction to bubbles argument before Conjecture \[conj:struct\] shows that our quotient satisfies the cyclotomic condition. The functor is clearly essentially surjective and full and we conjecture it to be faithful, so that it would be an equivalence of categories.
[**Acknowledgements**]{}
Foremost, we want to thank Mikhail Khovanov for his remark that we should be able to define the Chuang-Rouquier complex for a colored braid using his and Lauda’s diagrammatic calculus [@K-L3]. That observation formed the starting point of this paper. We also thank Mikhail Khovanov and Aaron Lauda for further inspiring conversations and for letting us copy their LaTeX version of the definition of $\Ucat$ almost literally.
Secondly, we thank the anonymous referee for his or her detailed suggestions and comments which have helped us to improve the readability of the paper significantly (we hope).
Furthermore, the authors were supported by the Fundação para a Ciência e Tecnologia (ISR/IST plurianual funding) through the programme “Programa Operacional Ciência, Tecnologia, Inovação” (POCTI) and the POS Conhecimento programme, cofinanced by the European Community fund FEDER.
PV was also financially supported by the Fundação para a Ciência e Tecnologia through the post-doctoral fellowship SFRH/BPD/46299/ 2008.
MS was also partially supported by the Ministry of Science of Serbia, project 174012.
[15]{}
A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of $\mathfrak{gl}_n$. Duke Math. J. 61(2) (1990), 655-677.
J. Brundan and C. Stroppel, Khovanov’s diagram algebra III: category $\mathcal{O}$. Represent. Theory 15 (2011), 170-243.
S. Cautis, J. Kamnitzer and A. Licata, Derived equivalences for cotangent bundles of Grassmannians via categorical sl(2) actions. arXiv:0902.1797.
N. Chris and V. Ginzburg, [*Representation theory and complex geometry*]{}. Birkhäuser, 1997.
J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and sl(2)-categorification. Annals of Math. 167 (2008), 245-298.
S. Donkin, [*The $q$-Schur algebra*]{}. Lond. Math. Soc. Lect. Not. Ser. 253, Cambridge University Press, 1998.
S. Doty and A. Giaquinto, Presenting Schur algebras. *International Mathematics Research Notices* 36 (2002), 1907-1944.
B. Elias and M. Khovanov, Diagrammatics for Soergel categories. arXiv: 0902.4700v1 \[math.RT\].
B. Elias and D. Krasner, Rouquier complexes are functorial over braid cobordisms. Homology, Homotopy Appl. 12 (2010), no. 2, 109-146.
W. Fulton, [*Young tableaux*]{}. Lond. Math. Soc. Stud. Texts 35, Cambridge University Press, 1997.
W. Fulton and P. Pragacz, [*Schubert varieties and degeneracy loci*]{}. Lect. Notes in Math. 1689, Springer, 1998.
I. Grojnowski and G. Lusztig, On bases of irreducible representations of quantum $GL_n$. Contemp. Math. 139 (1992), 167-174.
M. Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules. Int. J. Math. 18(8) (2007), 869-885.
M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I. *Representation Theory* 13 (2009), 309-347.
M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups II. Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685-2700.
M. Khovanov and A. Lauda, A categorification of quantum ${\rm sl}(n)$. Quantum Topol. 1 (2010), no. 1, 1-92.
M. Khovanov and A. Lauda, Erratum to: "A categorification of quantum ${\rm sl}(n)$”. Quantum Topol. 2 (2011), no. 1, 97-99.
M. Khovanov, A. Lauda, M. Mackaay and M. Stošić, Extended graphical calculus for categorified quantum sl(2). arXiv:1006.2866 \[math.QA\].
D. Krasner, Integral HOMFLT-PT and $sl(n)$-link homology. Int. J. Math. Math. Sci. 2010, Art. ID 896879, 25 pp.
A. Lauda, A categorification of quantum $\mathfrak{sl}_2$. Adv. Math. 225 (2010), no. 6, 3327-3424.
A. Lauda, Categorified quantum $sl(2)$ and equivariant cohomology of iterated flag varieties. Algebr. Represent. Theory 14 (2011), no. 2, 253-282,
G. Lusztig, [*Introduction to quantum groups*]{}. Prog. in Math. 110, Birkhäuser, 1993.
I.G. Macdonald, [*Symmetric functions and Hall polynomials*]{}. Oxford Mathematical Monographs, Oxford University Press, 1995.
M. Mackaay, $sl(3)$-Foams and the Khovanov-Lauda categorification of quantum $sl(k)$. arXiv:0905.2059 \[math.QA\].
M. Mackaay,M. Stošić and P. Vaz, The 1,2-coloured HOMFLY-PT link homology. Trans. Amer. Math. Soc. 363 (2011), no. 4, 2091-2124.
M. Mackaay and P. Vaz, The diagrammatic Soergel category and $sl(N)$ foams for $N\geq 4$. Int. J. Math. Math. Sci. 2010, Art. ID 468968, 20 pp.
S. Martin, [*Schur algebras and representation theory*]{}. Cambridge Tracts in Math. 112, Cambridge University Press, 1993.
V. Mazorchuk and C. Stroppel, Categorification of (induced) cell modules and the rough structure of generalised Verma modules. [*Adv. Math.*]{} 219: 1363-1426, 2008.
V. Mazorchuk and C. Stroppel, A combinatorial approach to functorial quantum $sl(k)$ knot invariants. Amer. J. Math. 131(6) (2009), 1679-1713.
H. Murakami, T. Ohtsuki and S. Yamada, HOMFLY polynomial via an invariant of colored plane graphs. [*Enseign. Math.*]{} (2) 44: 325-360, 1998.
R. Rouquier, Categorification of ${\mathfrak{sl}}_2$ and braid groups. Trends in representation theory of algebras and related topics, 137-167, Contemp. Math., 406, Amer. Math. Soc., Providence, RI, 2006.
R. Rouquier, 2-Kac-Moody algebras. arXiv:0812.5023.
W. Soergel, The combinatorics of Harish-Chandra bimodules. *Journal Reine Angew. Math.* 429 (1992) 49-74.
P. Vaz, The diagrammatic Soergel category and $sl(2)$ and $sl(3)$ foams. Int. J. Math. Math. Sci. 2010, Art. ID 612360, 23 pp.
B. Webster, Knots and higher representation theory I: diagrammatic and geometric categorification of tensor products. arXiv:1001.2020 \[math.GT\].
B. Webster, Knots and higher representation theory II: the categorification of quantum knot invariants. arXiv:1005:4559 \[math.GT\].
B. Webster and G. Williamson, A geometric model for Hochschild homology of Soergel bimodules. Geometry and Topology 12 (2008), 1243-1263.
B. Webster and G. Williamson, A geometric construction of the colored HOMFLYPT homology. arXiv: 0905.0486 \[math.GT\].
G. Williamson, [*Singular Soergel bimodules*]{}. Doctoral thesis, Albert-Ludwigs-Universität, Freiburg, Germany, 2008.
H. Wu, Matrix factorizations and colored MOY graphs. arXiv:0803.2071 \[math.GT\].
H. Wu, A colored $\mathfrak{sl}_n$-homology for links in $S^3$. arXiv:0907.0695 \[math.GT\].
H. Wu, Equivariant colored $\mathfrak{sl}_n$-homology for links. arXiv:1002.2662 \[math.GT\].
Y. Yonezawa, Matrix factorizations and intertwiners of the fundamental representations of quantum group $U_q(\mathfrak{sl}_n)$. arXiv:0806.4939 \[math.QA\].
Y. Yonezawa, Quantum $(\mathfrak{sl}_n,\bigwedge V_n)$ link invariants and matrix factorizations. Doctoral thesis, Nagoya University, Nagoya, Japan, 2010.
[^1]: We thank Raphaël Rouquier for pointing this out to us and giving us the reference.
[^2]: We refer to objects of the category $\glcat(\lambda,\lambda')$ as 1-morphisms of $\glcat$. Likewise, the morphisms of $\glcat(\lambda,\lambda')$ are called 2-morphisms in $\glcat$.
[^3]: See [@L1] and the references therein for the definition of a cyclic 2-morphism with respect to a biadjoint structure.
[^4]: Williamson defines such a form in [@Will] for $n=d$, but we do not know of any diagrammatic interpretation of his form even in that restricted case. We conjecture that his form is equivalent to ours for $n=d$. This is the only related form in the literature that we could find, even after asking numerous experts.
|
---
author:
- |
Yifan Wu[^1]\
Carnegie Mellon University\
`yw4@cs.cmu.edu`\
George Tucker\
Google Brain\
`gjt@google.com`\
Ofir Nachum\
Google Brain\
`ofirnachum@google.com`
bibliography:
- 'iclr2019\_conference.bib'
title: 'The Laplacian in RL: Learning Representations with Efficient Approximations'
---
[^1]: Work performed while an intern at Google Brain.
|
---
abstract: 'We evaluate the radiative decay into a vector a pseudoscalar and a photon of several resonances dynamically generated from the vector vector interaction. The process proceeds via the decay of one of the vector components into a pseudoscalar and a photon, which have an invariant mass distribution very different from phase space as a consequence of the two vector structure of the resonances. Experimental work along these lines should provide useful information on the nature of these resonances.'
author:
- |
J. Yamagata-Sekihara$^1$ and E. Oset$^1$\
\
\
title: '$VP\gamma$ radiative decay of resonances dynamically generated from the vector meson-vector meson interaction'
---
Introduction
============
The success of the chiral unitary approach generating resonances from the interaction of pseudoscalar mesons or pseudoscalar mesons with baryons, providing the properties of these resonances and their production cross sections in different reactions [@review; @puri] has stimulated work replacing pseudoscalar mesons by vector mesons. A natural extension of the chiral Lagrangians to incorporate vector mesons and their interaction is provided by the hidden local gauge formalism for vector interactions with pseudoscalar mesons, vectors and photons [@hidden1; @hidden2; @hidden3; @hidden4]. Once again the skilful combination of the interaction provided by these Lagrangians with unitary techniques in coupled channels allows one to obtain a realistic approach to study the vector-vector interaction, and work in this direction has been already done in [@raquel; @geng] studying the vector-vector interaction up to about 2000 MeV. The nonperturbative unitary techniques are essential there and many resonances are generated within the scheme. In practice one solves a set of coupled channels Bethe Salpeter equations using as Kernel the interaction provided by the hidden gauge Lagrangians, regularizing the loops with a natural scale [@ollerulf]. Several mesonic resonances are found as poles in the scattering matrices, indicating that one has some kind of molecular states. Actually, there are strong experimental arguments to suggest that the $f_0(1370)$ is a $\rho \rho$ molecule [@klempt; @crede]. The results of [@raquel] show that the $f_0(1370)$ and $f_2(1270)$ mesons are dynamically generated from the $\rho \rho$ interaction.
The work of [@geng] extends that of [@raquel] to the interaction of all members of the vector nonet resulting in the dynamical generation of eleven resonances, some of which can be associated to known resonances ($f_0(1370)$, $f_2(1270)$, $f'_2(1525)$, $f_0(1710)$ and $K^*_2(1430)$), while others are predictions.
The nature of these resonances as molecular states of a pair of vector mesons, allows one to evaluate many observables like the radiative decay of the $f_0(1370)$ and $f_2(1270)$ mesons into $\gamma \gamma$ [@yamagata], were agreement with the experimental data is found. Similarly, the $J/\psi$ decay into $\phi (\omega)$ and one of those resonances [@chinacola], and the $J/\psi$ radiative decay into $\gamma$ and one of the those resonances [@chinavalgerman], were also found consistent with experiment.
The idea of vector-vector molecules has also found support in alternative studies. In [@gutsche] the Y(3940) and Y(4140) are assumed to be bound states of $D^*$ and $\bar{D}^*$ and $D^*_s$ and $\bar{D}^*_s$ respectively and the Weinberg compositness condition [@weinberg1; @weinberg2; @Hanhart:2007yq; @Baru:2003qq] is invoked to get the coupling of the Y resonances to these components. Another approach, based on chiral symmetry and heavy quark symmetry has been used in [@shilinzhu], where also bound states of the $D^* \bar{D^*}$ systems are found in some cases.
The particular structure of these states induced the idea [@liuke] that the study of their decay into $D^*\bar{D}\gamma$, or $D^*_s\bar{D_s} \gamma$ should provide o good test for the claimed structure of these resonances. The idea was caught up in [@weihong] and applied to several X,Y,Z resonances of hidden charm nature which are dynamically generated within the hidden gauge approach from the interaction of vector mesons [@xyz].
In the present paper we shall further pursue this idea for the states generated in [@raquel; @geng] and study the decay of the $f_0(1370)$, $f_2(1270)$, $f'_2(1525)$, $f_0(1710)$ and the $K^*_2(1430)$ resonances into several channels involving one vector, one pseudoscalar and a photon. While we find similar results for the shapes of the invariant mass distributions of the pseudoscalar-photon pair as in [@weihong], we find that the decay rates obtained in this case are far larger than those obtained in the charm sector. The experimental investigation of these decay rates would provide further information concerning the claimed nature of those resonances as vector-vector molecules, and provide further support for the similar nature claimed for some of the X,Y,Z resonances [@gutsche; @shilinzhu; @xyz].
Formalism
=========
In [@geng] the $f_0(1370),~f_0(1710),~f_2(1270),~f_2'(1520)$, and $K^*_2(1430)$ were dynamically generated by the vector-vector interaction. The states were identified by observing poles in the vector-vector scattering matrix with certain quantum numbers. The real part of the pole position provides the mass of the resonance and the imaginary part one half its width. In addition the residues at the poles provide the product of the coupling of the resonance to the initial and final channels, from where, by looking at the scattering amplitudes in different channels, we can obtain the coupling of the resonance to all channels up to an irrelevant global sign for just one coupling. In Table \[tab:1\] the couplings to the most relevant channel are shown.
Resonance spin
--------------- ------ ----------------- ---------------- ------------------
$f_0(1370)$ J=0 $ \rho \rho $
$$ (7920,-$i$1071)
$f_0(1710)$ J=0 $K^*{\bar K^*}$ $\phi \omega $ $\omega \omega $
$$ (7124,$i$96) (3010,-$i$210) (-1763,$i$108)
$f_2(1270)$ J=2 $\rho \rho$
$$ (10889,-$i$99)
$f_2'(1520)$ J=2 $K^*{\bar K^*}$ $\phi \omega$ $\omega \omega$
$$ (10121,$i$101) (5016,-$i$17) (-2709,$i$8)
$K^*_2(1430)$ J=2 $\rho K^*$ $K^*\omega$
$$ (10901,-$i$71) (2267,-$i$13)
: \[tab:1\]Coupling constans of main decay channel of the resonances of [@geng]. All quantities are in units of MeV.
In [@geng] these couplings are given in isospin basis. However, we need them now in charge basis, which are readily obtained for the isospin combinations $$\begin{aligned}
|\rho\rho,I=0\rangle &=&-\frac{1}{\sqrt{3}}(|\rho^{+}\rho^{-}\rangle +|\rho^-\rho^+\rangle+
|\rho^0\rho^0\rangle),\nonumber \\
|K^*K^*,I=0\rangle &=&-\frac{1}{\sqrt{2}}(|K^{*+}K^{*-}\rangle -
|K^{*-}K^{*+}\rangle)~~.\end{aligned}$$ In addition, the couplings of [@geng] are calculated with the unitary normalization (extra $1/\sqrt{2}$ factor to account for identical particles in the sum over intermediate states). Thus, the couplings of $\rho\rho$, $K^*{\bar K^*}$ and $\omega\omega$ components must be multiplied by $(\sqrt{2/3})$, $1$ and $\sqrt{2}$ to get the appropriate coupling for the charged or neutral states (a sign is irrelevant for the width).
In the present work we address the problem of the decay mode of the resonance when one vector meson further decays into a pseudoscalar meson and a photon. The corresponding Feynman diagram is shown in Fig. \[fig:diagram\].
![\[fig:diagram\]](diagram.eps){width="5.5cm" height="4.5cm"}
The spin projection operators on $J=0,2$, evaluated assuming the three momenta of $V$ to be small with respect to the mass of the vector mesons, which is indeed the case here, are given in terms of the polarization vectors by $$\begin{aligned}
P'^{(0)}&=&\frac{1}{\sqrt{3}}\epsilon_i^{(1)}\epsilon_i^{(2)},\nonumber \\
P'^{(2)}&=&\left\{\frac{1}{2}\left(\epsilon_i^{(1)}\epsilon_j^{(2)}+\epsilon_j^{(1)}\epsilon_i^{(2)}\right)-\frac{1}{3}\epsilon_l^{(1)}\epsilon_l^{(2)}\delta_{ij}\right\},
\label{eq:projectors}\end{aligned}$$ where $i,j$ are spatial indices. On the other hand the anomalous vertex for the $V$ decay into $P \gamma$ is given by $$-it_{V \to P \gamma}=-ig_{V\gamma P
}\epsilon_{\mu\nu\alpha\beta}p^{\mu}\epsilon^{\nu}(V)k^{\alpha}\epsilon^{\beta}(\gamma),$$ which gives rise to a width $$\Gamma_{V \to P \gamma}=\frac{1}{48\pi}g^2_{V\gamma P
}\frac{k}{M^2_{V}}(M^2_{V}-m^2_{P})^2.
\label{eq:radwidth}$$ Using Eq. (\[eq:radwidth\]) and values of the PDG for the vector radiative widths, we obtain the coupling of $g_{VP\gamma}$ for the $V\to P \gamma$ decay, which are given by $$\begin{aligned}
g_{\rho^\pm \to \pi^\mp \gamma}&=& 2.19\times10^{-4}~{\rm MeV}^{-1}\nonumber\\
g_{\rho^0 \to \pi0 \gamma}&=& 2.52\times10^{-4}~{\rm MeV}^{-1}\nonumber\\
g_{K^{*\pm} \to K^\pm \gamma}&=& 2.53\times10^{-4}~{\rm MeV}^{-1}\nonumber\\
g_{K^{*0} \to K^0 \gamma}&=& 2.19\times10^{-4}~{\rm MeV}^{-1}\nonumber\\
g_{\omega \to \pi^0 \gamma}&=& 6.96\times10^{-4}~{\rm MeV}^{-1}\nonumber~~.\end{aligned}$$
Let us begin with the $f_0(1370)$ case. With the previous information we can already write the amplitude for the decay of the $f_0(1370)$ into $\rho^+\pi^- \gamma$, which is given by $$\begin{aligned}
-it&=&-i\frac{\sqrt{2}}{\sqrt{3}}\tilde{g}\frac{1}{\sqrt{3}}\epsilon_i^{(1)}\epsilon_i^{(2)}\frac{i}{p^2-M^2_{\rho}+iM_{\rho}\Gamma_{\rho}}\nonumber \\
&&\times (-i)g_{\rho\pi\gamma}\epsilon_{\mu\nu\alpha\beta}p^{\mu}\epsilon^{\nu
(2)}k^{\alpha}\epsilon^{\beta}(\gamma),\end{aligned}$$ where the indices (1), (2) indicate the $\rho^+$ and the $\rho^-$ respectively. The sum over the intermediate $\rho^-$ polarizations can be readily done as $$\sum\limits_{\lambda}\epsilon_i^{(2)}\epsilon^{\nu
(2)}=-g_i^\nu=-\delta_{i\nu},$$ where we have neglected the three momenta of the intermediate $\rho^-$ which is in average very small compared with the $\rho^-$ mass, particularly at large invariant masses of the $\pi^- \gamma$ system which concentrates most of the strength, as we shall see. The sum of $|t|^2$ over the final polarizations of the vector and the photon is readily done and, neglecting again terms of order $\vec{p}\,^2/M_{\rho}^2$, we get the result $$\begin{aligned}
\sum|t|^2&=&\frac{2}{3}\frac{1}{3}\tilde{g}^2g^2_{\rho\pi\gamma}\left|\frac{1}{p^2-M^2_{\rho}+iM_{\rho}\Gamma_{\rho}}\right|^2
2(p\cdot k)^2\nonumber \\
&=&\frac{2}{9}\frac{1}{2}\tilde{g}^2g^2_{\rho\pi\gamma}\left|\frac{p^2-m^2_\pi}{p^2-M^2_{\rho}+iM_{\rho}\Gamma_{\rho}}\right|^2.
\label{eq:tdos}
\end{aligned}$$ The differential mass distribution with respect to the invariant mass of the $\rho^{-} \gamma$ system, which is equal to $p^2$, is finally given by $$\frac{d
\Gamma_R}{dM_{\rm inv}}=\frac{1}{4M_R^2}\frac{1}{(2\pi)^3}p_{\rho}
\tilde{p}_\pi\sum|t|^2,
\label{eq:dgamma}$$ where $p_{\rho}$ is the momentum of the $\rho^{+}$ in the rest frame of the resonance $R$ and $\tilde{p}_\pi$ is the momentum of the $\pi^-$ in the rest frame of the final $\pi^{-} \gamma$ system given by $$\begin{aligned}
p_{\rho}&=&\frac{\lambda^{1/2}(M_R^2,M^2_{\rho},M^2_{\rm inv})}{2M_R},\nonumber \\
\tilde{p}_\pi&=&\frac{M^2_{\rm inv}-m^2_\pi}{2M_{\rm inv}}.\end{aligned}$$
One further step must be taken since the resonance and vector meson have decay widths. To take this into account, we consider the mass distribution of these two states and convolute the expression of the width with the mass distribution of the two particles. The differential mass distribution of the radiative decay of the resonance is then given as, $$(\frac{d\Gamma_R}{dM_{\rm inv}})'=\frac{F}{G}~~,$$ where $F$ and $G$ are given by $$\begin{aligned}
F&=&\int^{m_\rho+\Gamma_\rho/2}_{m_\rho-\Gamma_\rho/2}d\tilde{m_\rho}(-\frac{1}{\pi}){\rm Im}\frac{1}{{\tilde m_\rho}^2-m_\rho^2+i\Gamma_{\rho}m_\rho}\nonumber\\
&\times&\int^{M_R+\Gamma_R/2}_{M_R-\Gamma_R/2}(-\frac{1}{\pi}){\rm Im}\frac{1}{{\tilde M_R}^2-M_R^2+i\Gamma_RM_R}\times\frac{d\Gamma_R}{dM_{\rm inv}}({\tilde m_\rho},{\tilde M_R},M_{\rm inv})\\
G&=&\int^{m_\rho+\Gamma_\rho/2}_{m_\rho-\Gamma_\rho/2}d\tilde{m_\rho}(-\frac{1}{\pi}){\rm Im}\frac{1}{{\tilde m_\rho}^2-m_\rho^2+i\Gamma_{\rho}m_\rho}\nonumber\\
&\times&\int^{M_R+\Gamma_R/2}_{M_R-\Gamma_R/2}(-\frac{1}{\pi}){\rm Im}\frac{1}{{\tilde M_R}^2-M_R^2+i\Gamma_RM_R}\end{aligned}$$
In the case of the tensor state we must do extra work since the projector operators are different. In this case we must keep the indices $i, j$ in $t$ and multiply with $t^*$ with the same indices $i, j$. This sums over all possible final polarizations but also the initial $R$ polarizations, so in order to take the sum and average over final and initial polarizations, respectively, one must divide the results of the $i,j$ sum of $tt^*$ by $(2J+1)$, where $J$ is the spin of the resonance R. The explicit evaluation for the case of the tensor states, $J=2$, of $\rho\rho$ proceeds as follows: The $t$ matrix is now written as $$\begin{aligned}
t&=&\frac{1}{\sqrt{2}}\tilde{g}g_{\rho\pi\gamma} \left\{
\frac{1}{2}\left(\epsilon_i^{(1)}\epsilon_j^{(2)}+\epsilon_j^{(1)}\epsilon_i^{(2)}
\right)
-\frac{1}{3}\epsilon_l^{(1)}\epsilon_l^{(2)}\delta_{ij}\right\}\nonumber \\
&&\times\frac{1}{p^2-M^2_{\rho}+iM_{\rho}\Gamma_{\rho}}\epsilon_{\mu\nu\alpha\beta}p^{\mu}\epsilon^{\nu
(2)} k^\alpha\epsilon^\beta(\gamma).\end{aligned}$$ As mentioned above, we must multiply $t_{i,j}$ by $t^*_{i,j}$, recalling that the indices $i,j$ are spatial indices and divide by $(2J+1)$ (5 in this case) in order to obtain the modulus squared of the transition matrix, summed and averaged over the final and initial polarizations. Neglecting again terms that go like $\vec{p}\,^2/m_\rho$ we obtain the same expression as in Eq. (\[eq:tdos\]).
Results
=======
We show here the results for different cases:
$f_0(1370)$ and $f_0(1710)$
---------------------------
The decay modes considered are $$\begin{aligned}
f_0(1370)&\to&\rho\pi\gamma\nonumber\\
f_0(1710)&\to&K^*{\bar K}\gamma\nonumber\\
f_0(1710)&\to&\phi\pi\gamma\\
f_0(1710)&\to&\omega\pi\gamma\nonumber~~.
\label{eq:f0}\end{aligned}$$
In Fig. \[fig:noconv\] we show the calculated differential mass distribution of $f_0(1370)\to\rho^+\pi^-\gamma$. This calculation has done without the decay widths of $f_0(1370)$ and $\rho^+$ meson, namely Eq. (\[eq:dgamma\]). Since the $f_0(1370)$ resonance is below the $\rho^+\rho^-$ threshold, the peak position appears below the $\rho^-$ mass energy.
![\[fig:noconv\]Differential mass distribution for $f_0(1370)\to\rho^+\pi^-\gamma$ without convolution.](fig2_rhorho.eps){width="8.cm" height="6.0cm"}
In Fig. \[fig:I0\], we consider the decay widths of the resonance and vector meson. For the $f_0(1370)$ resonance the width is poorly determined experimentally. We take a slice of the resonance around its peak and use $\Gamma_{f_0(1370)}=100~{\rm MeV/c}$. With this window we prevent that the resonance goes into two physical $\rho$ mesons, one of which would decay into $\pi \gamma$. For the $f_0(1710)$ the decay into two vector mesons is forbidden except in the case of $\omega \omega$ final state. This is the reason why the decay rate into $\omega \pi^0 \gamma$ is exceptionally large, since it corresponds to the decay of the resonance into $\omega \omega$ followed by the decay of either of the two $\omega$ into $\pi^0 \gamma$. The phase space is not restricted, unlike in the other cases where the intermediate vector meson that decays into a pseudoscalar meson and a photon is necessarily off shell. In the figure we compare the mass distribution with the phase space distribution of each decay mode (dashed dotted line), obtained omitting the $p^2$ dependent terms in eq. (\[eq:tdos\]). As we can see, the shapes obtained with the dynamical picture of the resonances is very different from what one gets using simple phase space.\
![\[fig:I0\]Differential mass distribution for the case of $f_0(1370)$ and $f_0(1710)$. The phase space calculation corresponds to the decay channel of the solid line with the same normalization.](I0.eps){width="13.cm" height="9.0cm"}
$f_2(1270)$ and $f_2'(1520)$ case
---------------------------------
The decay modes considered are: $$\begin{aligned}
f_2(1270)&\to&\rho\pi\gamma\nonumber\\
f_2'(1520)&\to&K^*{\bar K}\gamma\nonumber\\
f_2'(1520)&\to&\phi\pi\gamma\\
f_2'(1520)&\to&\omega\pi\gamma\nonumber~~.
\label{eq:f2}\end{aligned}$$ The $f_2(1270)$ and $f_2'(1520)$ decay invariant mass distributions are calculated as before and the results are shown in Fig. \[fig:I2\]. Once again we see the striking difference between the results obtained from the dynamical picture of the resonances and phase space (dashed dotted lines).
![\[fig:I2\]Differential mass distribution for the case of $f_2(1270)$ and $f_2(1520)$. The phase space calculation corresponds to the decay channel of the solid line with the same normalization.](I2.eps){width="13.cm" height="9.0cm"}
$K_{2}^*(1430)$ case
--------------------
The decay modes considered are: $$\begin{aligned}
K^*_2(1430)&\to&K^{*}\rho\to K^{*}\pi\gamma\nonumber\\
K^*_2(1430)&\to&\rho K\gamma\nonumber\\
K^*_2(1430)&\to&K^{*}\omega\to K^{*}\pi\gamma\\
K^*_2(1430)&\to&\omega K\gamma\nonumber~~.
\label{eq:f2_K}\end{aligned}$$ We show the results of the $K^*_2(1430)$ decay in Fig. \[fig:I2K\], where once again we see the striking differences with the results obtained with those with just phase space.
![\[fig:I2K\]Differential mass distribution for the case of $K^*_2(1430)$. The phase space calculation corresponds to the decay channel of the solid line with the same normalization.](I2K.eps){width="13.cm" height="9.0cm"}
In Table \[tab:gamma\], we show the results of the width of vector-vector meson $\Gamma_{R\to VV}$ and the radiative decay width $\Gamma_{R\to VP\gamma}$ The radiative decay width $\Gamma_{R\to VP\gamma}$ is obtained integrating the differential mass distribution.
The large radiative decay width of the $f_0(1710) \to \omega \pi^0 \gamma$ comes, as indicated before, because the decay into $\omega \omega $ is now allowed. This case serves us to make a test of the calculation. The radiative decay width should be twice the product of the $f_0(1710) \to
\omega \omega$ width times the branching ratio of the $\omega \to \pi^0 \gamma$, which is experimentally 8.28 %. $$\Gamma_{f_0(1710)\to \omega\pi^0\gamma}=\Gamma_{f_0\to \omega\omega}\times 2\times B_{\omega\to\pi^0\gamma}
\label{eq:a}$$ with $$\Gamma_{f_0\to\omega\omega}=\frac{1}{8\pi M^2_{f_0}}g^2k
\label{eq:b}$$ where $k$ is the $\omega$ momentum in the $f_0((1710)\to \omega\omega$ decay, and $g$ is the $f_0(1710)$ coupling to $\omega\omega$ from [@geng], shown in Table \[tab:1\]. This coupling incorporates the unitary normalization of the $\omega\omega$ (extra factor $1/\sqrt{2}$) which makes unnecessary to divide by a factor of two the $f_0(1710)\to \omega\omega$ width in Eq. (\[eq:b\]). The factor 2 in Eq. (\[eq:a\]) accounts for the $\omega\to \pi^0\gamma$ decay of each of the two omegas. The result of Table \[tab:gamma\] for $f_0(1710)\to\omega\pi^0\gamma$ fulfills the relationship of Eq. (\[eq:a\]).
Decay mode $\Gamma_{R\to VP\gamma}$ \[keV\]
------------------------------------------------------------------- ----------------------------------
$f_0(1370)\to \rho^\pm\rho^\mp\to\rho^\pm\pi^\mp\gamma$ 1.06
$f_0(1370)\to \rho^0\rho^0\to \rho^0\pi^0\gamma$ 1.42
$f_0(1710)\to K^{*\pm}K^{*\mp}\to K^{*\pm}K^{\mp}\gamma$ 7.30
$f_0(1710)\to K^{*0}{\bar K^{*0}}\to K^{*0}{\bar K^0}\gamma$ 12.73
$f_0(1710)\to \phi\omega\to \phi\pi^0\gamma$ 14.45
$f_0(1710)\to \omega\omega\to \omega\pi^0\gamma$ 2.40$\times 10^3$
$f_2(1270)\to \rho^\pm\rho^\mp\to\rho^\pm\pi^\mp\gamma$ 2.63$\times 10^{-1}$
$f_2(1270)\to \rho^0\rho^0\to\rho^0\pi^0\gamma$ 3.55$\times 10^{-1}$
$f_2'(1520)\to K^{*\pm}K^{*\mp}\to K^{*\pm}K^{\mp}\gamma$ $2.18\times 10^{-2}$
$f_2'(1520)\to K^{*0}{\bar K^{*0}}\to K^{*0}{\bar K^0}\gamma$ $3.77\times 10^{-2}$
$f_2'(1520)\to \phi\omega\to \phi\pi^0\gamma$ $4.60\times 10^{-1}$
$f_2'(1520)\to \omega\omega\to \omega\pi^0\gamma$ $38.16$
$K^{*}_2(1430)\to K^*\rho^\pm \to K^*\pi^\pm\gamma$ $4.07\times10^{-1}$
$K^{*}_2(1430)\to K^*\rho^0 \to K^*\pi^0\gamma$ $5.81\times10^{-1}$
$K^{*}_2(1430)\to \rho^+K^{*-} \to \rho^+K^-\gamma$ $8.99\times10^{-2}$
$K^{*}_2(1430)\to \rho^0{\bar K^{*0}} \to \rho^0{\bar K^0}\gamma$ $2.36\times10^{-1}$
$K^{*}_2(1430)\to K^*\omega \to K^*\pi^0\gamma$ $1.99\times10^{-1}$
$K^{*}_2(1430)\to \omega K^{*-} \to \omega K^-\gamma$ $1.78\times10^{-3}$
$K^{*}_2(1430)\to \omega{\bar K^{*0}} \to \omega{\bar K^0}\gamma$ $4.79\times10^{-3}$
: \[tab:gamma\] Radiative decay width of vector-vector meson $\Gamma_{R\to VP\gamma}$.
Summary
=======
We have studied the radiative decay width of $f_0(1370)$, $f_0(1710)$, $f_2(1250)$, $f_2'(1520)$ and $K^*(1430)$ resonances, which are dynamically generated by the vector-vector interaction, into a vector, a pseudoscalar and a photon. Except in one case, the $f_0(1710) \to \omega \pi^0 \gamma$, where all the strength of the $\pi^0 \gamma$ invariant mass accumulates at the $\omega$ mass value, in all the other cases we find wider distributions, quite different from what one expects in terms of phase space. The memory of the resonance as been a bound state of two vector mesons is responsible for this shape, and the strength of the pseudoscalar plus photon invariant mass acumulates as close to the mass of the vector meson as possible, within the phase space availability. The case of the $f_0(1710) \to \omega \pi^0 \gamma$ is special because what one sees is the decay into $\omega \omega$. Since the branching ratio of the $\omega$ to $\pi^0 \gamma$ is relatively big (it is actually used to detect $\omega$ in the TAPS detector [@mariana]), this decay mode would be a direct measurement of the $f_0(1710) \to \omega \omega$ decay, allowing to test predictions of the theoretical framework of the vector-vector coupled channels approach.
The rates obtained for the radiative decays are relatively large in some cases, of the order of the keV or tens of keV. These magnitudes are easily measurable and the results obtained here should stimulate experimental work in this direction, which should teach us much regarding the nature of the resonances studied.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is partly supported by DGICYT contract number FIS2006-03438 and the Generalitat Valenciana in the Program Prometeo. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity Study of Strongly Interacting Matter (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU. J. Y. is the Yukawa Fellow and this work is partially supported by Yukawa Memorial Foundation.
[99]{}
J. A. Oller, E. Oset and A. Ramos, Prog. Part. Nucl. Phys. [**45**]{}, 157 (2000) E. Oset, D. Cabrera, V. K. Magas, L. Roca, S. Sarkar, M. J. Vicente Vacas and A. Ramos, Pramana [**66**]{}, 731 (2006) \[arXiv:nucl-th/0504033\].
M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. [**54**]{}, 1215 (1985).
M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. [**164**]{}, 217 (1988). M. Harada and K. Yamawaki, Phys. Rept. [**381**]{}, 1 (2003) U. G. Meissner, Phys. Rept. [**161**]{}, 213 (1988).
R. Molina, D. Nicmorus and E. Oset, Phys. Rev. D [**78**]{}, 114018 (2008) L. S. Geng and E. Oset, Phys. Rev. D [**79**]{}, 074009 (2009)
J. A. Oller and U. G. Meissner, Phys. Lett. B [**500**]{}, 263 (2001)
E. Klempt and A. Zaitsev, Phys. Rept. [**454**]{}, 1 (2007)
V. Crede and C. A. Meyer, Prog. Part. Nucl. Phys. [**63**]{}, 74 (2009)
H. Nagahiro, J. Yamagata-Sekihara, E. Oset, S. Hirenzaki and R. Molina, Phys. Rev. D [**79**]{}, 114023 (2009)
A. Martinez Torres, L. S. Geng, L. R. Dai, B. X. Sun, E. Oset and B. S. Zou, Phys. Lett. B [**680**]{}, 310 (2009) L. S. Geng, F. K. Guo, C. Hanhart, R. Molina, E. Oset and B. S. Zou, arXiv:0910.5192 \[hep-ph\]. T. Branz, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D [**80**]{}, 054019 (2009) S. Weinberg, Phys. Rev. [**130**]{}, 776 (1963). S. Weinberg, Phys. Rev. [**137**]{}, B672 (1965).
C. Hanhart, Yu. S. Kalashnikova, A. E. Kudryavtsev and A. V. Nefediev, Phys. Rev. D [**76**]{}, 034007 (2007) \[arXiv:0704.0605 \[hep-ph\]\]. V. Baru, J. Haidenbauer, C. Hanhart, Yu. Kalashnikova and A. E. Kudryavtsev, Phys. Lett. B [**586**]{}, 53 (2004) \[arXiv:hep-ph/0308129\]. X. Liu, Z. G. Luo, Y. R. Liu and S. L. Zhu, Eur. Phys. J. C [**61**]{}, 411 (2009) X. Liu and H. W. Ke, Phys. Rev. D [**80**]{}, 034009 (2009) \[arXiv:0907.1349 \[hep-ph\]\]. W. H. Liang, R. Molina and E. Oset, arXiv:0912.4359 \[hep-ph\].
R. Molina and E. Oset, Phys. Rev. D [**80**]{}, 114013 (2009)
T. Branz, L. S. Geng and E. Oset, arXiv:0911.0206 \[hep-ph\]. R. Molina, H. Nagahiro, A. Hosaka and E. Oset, Phys. Rev. D [**80**]{}, 014025 (2009)
C. Amsler [*et al.*]{} \[Particle Data Group\], Phys. Lett. B [**667**]{}, 1 (2008).
M. Kotulla [*et al.*]{} \[CBELSA/TAPS Collaboration\], Phys. Rev. Lett. [**100**]{}, 192302 (2008) \[arXiv:0802.0989 \[nucl-ex\]\].
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abstract: 'A list of open problems on global behavior in time of some evolution systems, mainly governed by P.D.E, is given together with some background information explaining the context in which these problems appeared. The common characteristic of these problems is that they appeared a long time ago in the personnal research of the author and received almost no answer till then at the exception of very partial results which are listed to help the readers’ understanding of the difficulties involved.'
author:
- |
Alain HARAUX\
Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598,\
Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France.\
haraux@ann.jussieu.fr\
date:
title: 'Some simple problems for the next generations.'
---
35B15, 35B40, 35L10, 37L05, 37L15
Evolution equations, bounded solutions, compactness, oscillation theory, almost periodicity, weak convergence, rate of decay, .
[**Introduction**]{}
Will the next generations go on studying mathematical problems? This in itself is an open question, but the growing importance of computer’s applications in everyday’s life together with the fundamental intrication of computer science, abstract mathematical logic and the developments of new mathematical methods makes the positive answer rather probable.
This text does not comply with the usual standards of mathematical papers for two reasons: it is a survey paper in which no new result will be presented and the results which we recall to motivate the open questions will be given without proof.
It is not so easy to introduce an open question in a few lines. Giving the statement of the question is not enough, we must also justify why we consider the question important and explain why it could not be solved until now. Both points are delicate because the importance of a problem is always questionable and the difficulty somehow disappears when the problem is solved.
The questions presented here concern the theory of differential equations and mostly the case of PDE. They were encountered by the author during his research and some of them are already 40 years old. They might be considered purely academical by some of our colleagues more concerned by real world applications, but they are selected, among a much wider range of open questions, since their solution probably requires completely new approaches and will likely open the door towards a new mathematical landscape.
Compactness and almost periodicity
==================================
Throughout this section, the terms “maximal monotone operator" and “almost periodic function" will be used without having been defined. Although both terms are by now rather well known, the definitions and main properties of these objects will be found respectively in the reference texts [@HB] and [@AP2].\
One of my first fields of investigation was, in connection with the abstract oscillation theory, the relationship between (pre-)compactness and asymptotic almost periodicity for the trajectories of an almost periodic contractive process. The case of autonomous processes (contraction semi-groups on a metric space) had been studied earlier in the Hilbert space framework by Dafermos and Slemrod [@D-S], the underlying idea being that on the omega-limit set of a precompact trajectory, the semi-group becomes an isometry group. Then the situation resembles the simpler case of the isometry group generated on a Hilbert space $H$ by the equation $$u'+Au(t) = 0$$ with $$A^* = -A$$ for which almost periodicity of precompact trajectories was known already from L. Amerio quite a while ago (the case of vibrating membranes and vibrating plates with fixed bounded edge are special cases of this general result).
The case of a non-autonomous process, associated with a time-dependent evolution equation of the form $$u'+A(t) u(t)\ni 0$$ is not so good in general. In [@H1] I established an almost periodicity result for precompact trajectories of a [*periodic*]{} contraction process on a complete metric space, and in the same paper I exhibited a simple almost periodic (linear) isometry process on $\R^2$, generated by an equation of the form $$u'+c(t)Ju(t) = 0$$ with $J$ a $\pi\over2$- rotation around $0$, for which no trajectory except $0$ is almost periodic.
Actually, while writing my thesis dissertation, I was specifically interested in the so-called “quasi-autonomous" problem, and I met the following general question
[**Problem 1.1.**]{} (1977) Let A be a maximal monotone operator on a real Hilbert space H, let $ f:{\R}\longrightarrow H$ be almost periodic and let $u$ be a solution of de $$u'+A u(t)\ni f(t)$$ on $[0,
+\infty)$ with a [**precompact range**]{}. Can we conclude that $u$ is asymptotically almost periodic?
After studying a lot of particular cases in which the answer is positive ( $A = L$ linear, $A$ a subdifferential $\partial \Phi $ and some operators of the form $ L + \partial \Phi $ ) , I proved in [@HPP] that the answer is positive if $ H={\R}^N$ with $ N \le 2$. But the answer is unknown for general maximal monotone operators even if $ H={\R}^3.$
[In [@wrong] it is stated that the answer is positive for all $N$, but there is a mistake in the proof, relying on a geometrical property which is not valid in higher dimensions, more specifically in 3D the intersection of the (relative) interiors of two arbitrarily close isometric proper triangles can be empty. Therefore the argument from [@HPP] cannot be used in the same way for $N\ge3$. ]{}
[The problem is also open even when $ A\in C^1(H, H)$, in which case the monotonicity just means $$\forall u\in H,\quad \forall v\in H, \quad (A'(u), v) \ge 0.$$ ]{}
[The answer is positive if $f$ is periodic, as a particular case of the main result of [@H1]. ]{}
Since an almost periodic function has precompact range, studying the existence of almost periodic solutions requires some criteria for precompactness of bounded orbits. In the case of evolution PDE, precompactness is classically derived from higher regularity theory. For parabolic equations the smoothing effect provides some higher order regularity for $t>0$ for bounded semi-orbits defined on $\R^+$. In the hyperbolic case, although there is no smoothing effect in finite time, precompactness of orbits was derived by Amerio and Prouse [@AP] from higher regularity of the source and strong coercivity of the damping operator $g$ in the case of the semilinear hyperbolic problem $$u_{tt} -
\Delta u + g(u_t) = f(t, x)
\hbox { in }
{\R^+}\times
\Omega,\ \ \ \ u = 0 \hbox { on } {\bb
R^+}\times
{\partial}\Omega$$ where $ \Omega $ be a bounded domain of ${\R}^N$. But this method does not apply even in the simple case $g(v) = cv^3$ for $c>0, N\le 3$, a case where boundedness of all trajectories is known. The following question makes sense even when the source term is periodic in t and $g$ is globally Lipschitz continuous.
[**Problem 1.2.**]{} (1978) Let $ \Omega $ be a bounded domain of ${\R}^N$ and $g$ a nonincreasing Lipschitz function. We consider the semilinear hyperbolic problem $$u_{tt} -
\Delta u + g(u_t) = f(t, x)
\hbox { in }
{\R^+}\times
\Omega,\ \ \ \ u = 0 \hbox { on } {\bb
R^+}\times
{\partial}\Omega$$ We assume that $ f:{\R}\longrightarrow L^2(\Omega)$ is continuous and periodic in t. Assuming $$u\in C_b({\R}^+,
H^1_0(\Omega))\cap C^1_b({\R}^+, L^2(\Omega))$$ can we conclude that $$\bigcup_{t\geq 0} \{(u(t,.),u_t(t,.))\}\hbox{ is
precompact in }
H^1_0(\Omega)\times L^2({\Omega}) ?$$
The answer is positive in the following extreme cases
1\) If g = 0 (by Browder-Petryshyn’s theorem, there is a periodic solution, hence compact, and all the others are precompact by addition.)
2\) If $g^{-1}$ is uniformly continuous, cf. [@Edin] , the result does not require Lipschitz continuity of $g$ and applies for instance to $g(v) = cv^3$ for $c>0, N\le 3$
It would be tempting to “interpolate", but even the case $g(v) = v^+$ and $N=1$ already seems to be non-trivial.
[The same question is of course also relevant when $f$ is almost periodic, and the result of [@Edin] is true in this more general context. Moreover precompactness of bounded trajectories when $g = 0$ is also true when $f$ is almost periodic. This is related to a fundamental result of Amerio stating that if the primitive of an almost periodic function: $\R\longrightarrow H $ is bounded, it is also almost periodic. More precisely, if $ H $ is a Hilbert space and $L$ is a (possibly unbounded ) skew-adjoint linear operator with compact resolvent, let us consider a bounded solution (on $\R$ with values in $H$) of the equation $$U'+ AU = F$$ where $F: \R\longrightarrow H$ is almost periodic . Then $ \exp (tA)U := V$ is a bounded solution of $$V' = \exp (tA)F$$ and, since $\exp (tA)\psi $ is almost periodic as well as $\exp (- tA)\psi $ for any $\psi\in H$, by a density argument on generalized trigonometric polynomials, it is immediate to check that a function $W: \R\longrightarrow H$ is almost periodic if and only if $\exp (tA)W: \R\longrightarrow H$ is almost periodic. Then Amerio’s Theorem applied to $V$ gives the result, and this property applies in particular to the wave equation written as a system in the usual energy space. Then starting from a solution bounded on $\R^+$, a classical translation-(weak)compactness argument of Amerio gives a solution bounded on $\R$ of the same equation. We skip the details since this remark is mainly intended for experts in the field.]{}
Oscillation theory
==================
Apart from the almost periodicity of solutions which provides a starting point to describe precisely the global time behavior of vibrating strings and membranes with fixed edge, it is natural to try a description of sign changes of the solutions on some subset of the domain. Let us first consider the basic equation $$\label{basic-eq}
u''+ Au(t)= 0,$$ where $V$ is a real Hilbert space, $A\in L(V, V')$ is a symmetric, positive, coercive operator and there is a second real Hilbert space $H$ for which $V\hookrightarrow H
=H' \hookrightarrow V'$ where the imbedding on the left is compact. In this case it is well known that all solutions $u\in C(\R , V) \cap C^1(\R , H)$ of are almost periodic : $\R\rightarrow V $ with mean-value $0$. Then for any form $\zeta\in V'$, the function $g(t): = \langle \zeta, u(t)\rangle$ is a real-valued continuous almost periodic function with mean-value $0$. It is then easy to show that either $g\equiv 0$, or there exists $M>0$ such that on each interval $J$ with $ |J | \ge M $, g takes both positive and negative values. We shall say that a number $M>0$ is a strong oscillation length for a numerical function $g\in L^1_{loc} (\R)$ if the following alternative holds: either $g(t) = 0$ almost everywhere, or for any interval $J$ with $ |J | \ge M $, we have $$meas \{ t\in J, f(t) >0\} >0 \quad \hbox {and}\quad meas \{ t\in J, f(t) <0\} >0.$$ As a consequence of the previous argument , under the above conditions on $H, V$ and $A$, for any solution $u\in C(\R , V) \cap C^1(\R , H)$ of and for any $\zeta\in V'$, the function $g(t): = \langle \zeta, u(t)\rangle $ has some finite strong oscillation length $M = M(u, \zeta)$.
In the papers [@CH1; @CH2; @HK] the main objective was to obtain a strong oscillation length independent of the solution and the observation in various cases, including non-linear perturbations of equation . A basic example is the vibrating string equation $$\label{NLwave}
u_{tt} - u_{xx}+ g(t, u) = 0 \quad \hbox {in} \,\, \R\times (0, l), \quad u = 0 \,\, \hbox {on}\,\, \R\times \{0, l\}$$ where $l>0$ and $g(t, .)$ is an odd non-decreasing function of $u$ for all $t$. Here the function spaces are $H= L^2(0, l)$ and $V= H^1_0(0, l)$. Since any function of V is continous, a natural form $\zeta\in V'$ is the Dirac mass $\delta_{x_0}$ for some ${x_0}\in (0, l).$ It turns out that $2l$ is a strong oscillation length independent of the solution and the observation point ${x_0}$, exactly as in the special case $g= 0$, the ordinary vibrating string. Since in this case all solutions are $2l$-periodic with mean-value $0$ functions with values in $V$, it is clear that $2l$ is a strong oscillation length independent of the solution and the observation point ${x_0}$. The slightly more complicated $g(t, u) = au$ with $a>0$ is immediately more difficult since the general solution is no longer time-periodic, it is only almost periodic in $t$. The time-periodicity is too unstable and for an almost periodic function, the determination of strong oscillation lengths is not easy in general, as was exemplified in [@HK]. The oscillation result of [@CH1; @CH2] is consequently not so immediate even in the linear case. In the nonlinear case, it becomes even more interesting because the solutions are no longer known to be almost periodic.
In dimensions $N\ge 2$, even the linear case becomes difficult. It has been established in [@HK] that even for analytic solutions of the usual wave equation in a rectangle, there is no uniform pointwise oscillation length common to all solutions at some points of the domain. One would imagine that it becomes true if the point is replaced by an open subset of the domain, but apparently nobody knows the answer to the exceedingly simpler following question:
[**Problem 2.1.**]{} (1985) Let $ \Omega = (0, 2l)\times (0, 2l)\subset
{\R}^2$. We consider the linear wave equation $$u_{tt} -
\Delta u = 0
\hbox { in }
{\R}\times
\Omega,\ \ \ \ u = 0 \hbox { on } {\bb
R}\times
{\partial}\Omega$$ Given $T>0$ , can we find a solution $u$ for which $$\forall (t, x) \in [0, T]\times (0, l)\times (0, l),\quad u(t, x) >0 ?$$ Or does this become impossible for $T$ large enough?
Another simple looking intriguing question concerns the pointwise oscillation of solutions to semi linear beam equations, since the solutions of the corresponding linear problem oscillate at least as fast as those of the string equation:
\(1985) We consider the semilinear beam equation $$u_{tt} + u_{xxxx} + g(u) = 0
\hbox { dans } {\R}\times (0, 1) , \quad u =
u_{xx} = 0 \hbox { on } {\R}\times \{0,1\}$$ with g odd and nonincreasing with respect to u. Is it possible for a solution $u(t, .)$ to remain positive at some point $x_0$ on an arbitrarily long (possibly unbounded) time interval?
Finally, let us mention a question on spatial oscillation of solutions to parabolic problems. Since the heat equation has a very strong smoothing effect on the data, and all solutions are analytic inside the domain for $t>0$, it seems natural to think that they do no accumulate oscillations and for instance in 1D, the zeroes of $u(t,.)$ will be isolated for $t>0. $ A very general result of this type, valid for semi linear problems as well has been proved by Angenent [@Ang]. But as soon as $N\ge 2$, even the linear case is not quite understood. The answer to the following question seems to be unknown:
[**Problem 2.3.**]{} (1997) Let $ \Omega \subset {\R}^N$ be a bounded open domain. We consider the heat equation $$u_{t} -
\Delta u = 0
\hbox { in }
{\R}\times
\Omega,\ \ \ \ u = 0 \hbox { on } {\bb
R}\times
{\partial}\Omega$$ For $t>0$ , we consider $${\cal E} = \{x\in \Omega,\quad u(t, x)
\not= 0 \}$$ Is it true that $ {\cal E}$ has a finite number of connected components?
[The solutions $u$ of the [*elliptic* ]{}problem $$- \Delta u + f(u)=0 \hbox { in } \Omega,\ \ \ \ u = 0 \hbox { on }
{\partial}\Omega$$ are such that $ \{x\in \Omega,\quad u( x)
\not= 0 \}$ has a finite number of connected components for a large class of functions $f$, cf. e.g. [@CHM]. Hence stationary solutions cannot provide a counterexample. ]{}
A semi-linear string equation
=============================
There are in the Literature a lot of results on global behavior of solutions to Hamiltonian equations in finite and infinite dimensions. Apart from Poincar[é]{}’s recurrence theorem and the classical results of Liouville on quasi-periodicilty for most solutions of completely integrale finite dimensional hamiltonians, none of the recent results is easy and there is essentially nothing on PDE except in 1D. Even the case of semi linear string equations is not at all well-understood. While looking for almost-periodic solutions (trying to generalize the Rabinowitz theorem on non-trivial periodic solution) I realized that even precompactness of general solutions is unknown for the simplest semi linear string equation in the usual energy space :
\(1976) For the simple equation $$u_{tt} - u_{xx} + u^3 = 0 \hbox { in } {\R} \times (0,1) , \,\,
u = 0 \hbox { on } {\R} \times \{0,1 \}$$ the following simple looking questions seem to be still open
Question 1. Are there solutions which converge weakly to 0 as time goes to infinity?
Question 2. If $(u(0,.), u_t(0, .))\in H^2((0, 1)) \cap
H_0^1((0, 1))\times H_0^1((0, 1)): = {\cal V} $, does $(u(t,.), u_t(t, .))$ remain bounded in ${\cal V} $ for all times?
[To understand the difficulty of the problem, let us just mention that the equation $$iu_{t} + |u|^2 u = 0 \hbox { in } {\R} \times (0,1) ,\ \
u = 0 \hbox { on } {\R} \times \{0,1 \}$$ has many solutions tending weakly to $0$ and, although the calculations are less obvious, the same thing probably happens to $$u_{tt} + u^3 = 0 \hbox { in } {\R} \times (0,1) , \,\,
u = 0 \hbox { on } {\R} \times \{0,1 \}$$ Hence the problem appears as a competition between the “good" behavior of the linear string equation and the bad behavior of the distributed ODE associated to the cubic term. ]{}
[If the answer to question 2 is negative, it means that, following the terminology of Bourgain [@Bourg], the cubic wave equation on an interval is a weakly turbulent system. Besides, weak convergence to $0$ might correspond to an accumulation of steep spatial oscillations of weak amplitude, not contradictory with the energy conservation of solutions.]{}
[In [@CHW1]-[@CHW3], the authors investigated the problem $$\label{intterm}u_{tt} - u_{xx}+ u\int_0^lu^2(t, x) dx = 0 \quad \hbox {in} \,\, \R\times (0, l), \quad u = 0 \,\, \hbox {on}\,\, \R\times \{0, l\}$$ which can be viewed as a simplified model to understand the above equation. In this case, there is no solution tending weakly to $0$, and the answer to question 2 is positive. Interestingly enough, in this case the distributed ODE takes the form $ u_{tt} + c^2(t) u= 0 $ , so that the solution has the form $a(x) u_1(t) + b(x) u_2(t)$ and remains in a two-dimensional vector space! This precludes both weak convergence to $0$ and weak turbulence. ]{}
Rate of decay for damped wave equations
=======================================
Let us consider the semilinear hyperbolic problem $$u_{tt} -
\Delta u + g(u_t) = 0\hbox { in }
{\R^+}\times
\Omega,\ \ \ \ u = 0 \hbox { on } {\bb
R^+}\times
{\partial}\Omega$$ where $ \Omega $ be a bounded domain of ${\R}^N$ and $g$ is a nondecreasing function with $g(0) =0$. Under some natural growth conditions on $g$, the initial value problem is well-posed and can be put in the framework of evolution equations generated by a maximal monotone operator in the energy space $$H^1_0(\Omega) \times L^2(\Omega)$$
An immediate observation is the formal identity $$\frac {d}{dt} [\int_\Omega (u_t^2 + | \nabla u|^2 ) dx] = - 2\int_\Omega g(u_t)u_t dx \le 0$$ showing that the energy of the solution is non-increasing. When $g(s) = cs$ with $c>0$, one can prove the exponential decay of the energy by a simple calculation involving a modified energy function $$E_\varepsilon (t) = \int_\Omega (u_t^2 + |\nabla u|^2 ) dx + \varepsilon \int_\Omega uu_t dx$$ The exponential decay is of course optimal since $$\frac {d}{dt} [\int_\Omega (u_t^2 + | \nabla u|^2 ) dx] = - 2\int_\Omega cu_t ^2dx \ge -2c \int_\Omega (u_t^2 + | \nabla u|^2 ) dx$$
A similar calculation can be performed if $ 0< c\le g'(s) \le C $ , and the result is even still valid for $g(s) = cs + a |s|^\alpha s $ under a restriction on $\alpha>0 $ depending on the dimension.
More difficult, and somehow more interesting, is the case
$$g(s) = a |s|^\alpha s, \quad a>0, \alpha>0$$ in which under a restriction relating $\alpha$ and $N$, various authors (cf. e.g. [@Nakao], [@HZ] and the references therein) obtained the energy estimate
$$\int_\Omega (u_t^2 + |\nabla u|^2 ) dx \le C (1+t) ^{- \frac{2}{\alpha}}$$ But now the energy identity only gives $$\frac {d}{dt} [\int_\Omega (u_t^2 + | \nabla u|^2 ) dx] = - 2\int_\Omega a|u_t |^{\alpha + 2}dx$$ while to prove the optimality of the decay we would need something like $$\frac {d}{dt} [\int_\Omega (u_t^2 + | \nabla u|^2 ) dx] \ge - C( \int_\Omega u_t ^{ 2}dx) ^{1 +\frac{\alpha}{2} }$$ Unfortunately the norm of $ u_t$ in $L^{\alpha + 2}$ cannot be controlled in terms of the $L^2$ norm, even if strong restrictions on $u_t$ are known. If $u_t$ is known to be bounded in a strong norm, let us say an $L^p$ norm with $p$ large, we can derive a lower estimate of the type $$[\int_\Omega (u_t^2 + | \nabla u|^2 ) dx]\ge \delta (1+t) ^{- \beta}$$ for some $\beta > \frac{2}{\alpha}$. But even $p= \infty$ does not allow to reach the right exponent.
In 1994, using special Liapunov functions only valid for $N = 1$, the author ( cf. [@Below]) showed that for all sufficiently regular non-trivial initial data, we have the estimate $$\int_\Omega (u_t^2 + |\nabla u|^2 ) dx \ge C (1+t) ^{- \frac{3}{\alpha}}$$ In general, for $N>2$ , some estimate of the form
$$\int_\Omega (u_t^2 + |\nabla u|^2 ) dx \ge C (1+t) ^{- K}$$ will be obtained if the initial data belong to $D(-\Delta)\times H^1_0(\Omega)$ and $\alpha < \frac{4}{N-2}$ . But we shall have in all cases $K > \frac{4}{\alpha}$ and $K$ tends to infinity when $\alpha $ approaches the value $\frac{4}{N-2}$ .
It is perfectly clear that none of the above partial results is satisfactory, since for analogous systems in finite dimensions, of the type
$$u '' + Au + g(u')$$ with $A$ symmetric, coercive , $ (g(v), v) \ge c |v|^{\alpha + 2}$ and $ |g(v)|\le C |v|^{\alpha + 1}$, the exact asymptotics of any non-trivial solution is $$|u'|^2 + |u|^2\sim (1+t) ^{- \frac{2}{\alpha}}$$ Moreover, an optimality result of the decay estimate has been obtained in 1D by P. Martinez and J. Vancostenoble [@Van-Mart] in the case of a boundary damping for which the same upper estimate holds. The difference is that inside the domain, an explicit formula gives a lot of information on the solution.
For the equation $$u_{tt} -
\Delta u + g(u_t) = 0\hbox { in }
{\R^+}\times
\Omega,\ \ \ \ u = 0 \hbox { on } {\bb
R^+}\times
{\partial}\Omega$$ with $$g(s) = a |s|^\alpha s, \quad a>0, \alpha>0$$ Question 1. Can we find a solution $u$ for which $$|\int_\Omega (u_t^2 + |\nabla u|^2 ) dx \sim (1+t) ^{- \frac{2}{\alpha}}?$$ Question 2. Can we find a solution $u$ for which the above property is [*not*]{} satisfied?
[Both questions seem to be still open for any domain and any $\alpha>0$. ]{}
The resonance problem for damped wave equations with source term
================================================================
To close this short list, we consider the semilinear hyperbolic problem with source term $$u_{tt} -
\Delta u + g(u_t) = f(t, x)
\hbox { in }
{\R^+}\times
\Omega,\ \ \ \ u = 0 \hbox { on } {\bb
R^+}\times
{\partial}\Omega$$ where $ \Omega $ be a bounded domain of ${\R}^N$. We assume that the exterior force $f(t, x)$ is bounded with values in $L^2(\Omega)$, In this case, all solutions $U = (u, u_t)$ are locally bounded on $(0, T)$ with values in the energy space $ H^1_0(\Omega) \times L^2(\Omega)$. The question is what happens as $t $ tends to infinity.
When $g(s) $ behaves like a super linear power $ |s|^\alpha s$ for large values of the velocity, it follows from a method introduced by G. Prouse [@Prouse] and extended successively by many authors, among which M. Biroli [@Biroli], [@Biroli-H] and the author of this survey, that the energy of any weak solution remains bounded for $t$ large, under the restriction $ \alpha (N-2) \le 4$ . Then many attempts were tried to avoid this growth assumption. Many partial results were obtained under additional conditions ($ f $ bounded in stronger norms, $f $ anti-periodic, higher growths for $N\le 2$, cf e.g. [@Manus], [@Manus2], [@TG] ). But the following basic question remains open:
Assume $N\ge 3$, $$g(s) = a |s|^\alpha s, \quad a>0, \,\,\alpha>\frac {4}{N-2}$$ Is it still true that the energy of all solutions remains bounded for any exterior force $f(t, x)$ bounded with values in $L^2(\Omega)$?
The positive boundedness results require a weaker boundedness condition on $f$, it is sufficient that it belongs to a Stepanov space $S^p(\R, L^2(\Omega)$ with $p>1$. The first results in the direction were actually published by G. Prodi in 1956, so that the problem is about 60 years old...
[10]{}
L. Amerio and G. Prouse, *Uniqueness and almost-periodicity theorems for a non linear wave equation*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. **8**, 46 (1969) 1–8.
L. Amerio and G. Prouse, *Almost-periodic functions and functional equations*, Van Nostrand Reinhold Co., New York-Toronto, Ont.-Melbourne 1971 viii+184 pp.
S. Angenent, *The zero set of a solution of a parabolic equation*, J. Reine Angew. Math. **390** (1988), 79–96.
M. Biroli , *Bounded or almost periodic solution of the non linear vibrating membrane equation*, Ricerche Mat. **22** (1973), 190–202.
M. Biroli and A. Haraux, *Asymptotic behavior for an almost periodic, strongly dissipative wave equation*, J. Differential Equations **38** (1980), no. 3, 422–440.
S. Bochner and J. Von Neumann, *On compact solutions of operational-differential equations*, Ann. of Math. **2**, 36 (1935), 255–291.
Jean Bourgain, *On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian* , Internat. Math. Res. Notices **6** (1996), 277–304.
H. Brezis, *Op[' e]{}rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert* (French) North-Holland Mathematics Studies, No. 5., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. vi+183 pp.
T. Cazenave and A. Haraux, *Propri[' e]{}t[' e]{}s oscillatoires des solutions de certaines [' e]{}quations des ondes semi-lin[' e]{}aires*, C. R. Acad. Sci. Paris Ser. I Math. **298** (1984), no. 18, 449–452.
T. Cazenave and A. Haraux, *Oscillatory phenomena associated to semilinear wave equations in one spatial dimension*, Trans. Amer. Math. Soc. **300** (1987), no. 1, 207–233.
T. Cazenave, A. Haraux and F.B. Weissler, *Une [' e]{}quation des ondes compl[\` e]{}tement int[' e]{}grable avec non-lin[' e]{}arit[' e]{} homog[\` e]{}ne de degr[' e]{} trois*, C. R. Acad. Sci. Paris Ser. I Math. **313** (1991), no. 5, 237–241.
T. Cazenave, A. Haraux and F.B. Weissler, *A class of nonlinear, completely integrable abstract wave equations*, J. Dynam. Differential Equations **5** (1993), no. 1, 129–154.
T. Cazenave, A. Haraux and F.B. Weissler, *Detailed asymptotics for a convex Hamiltonian system with two degrees of freedom.*, J. Dynam. Differential Equations **5** (1993), no. 1, 155–187.
M. Comte, A. Haraux and P. Mironescu, *Multiplicity and stability topics in semilinear parabolic equations*, Differential Integral Equations **13** (2000), no. 7-9, 801–811.
C.M. Dafermos and M. Slemrod, *Asymptotic behavior of nonlinear contraction semigroups*, J. Functional Analysis **13** (1973), 97–106.
T. Gallouet, *Sur les injections entre espaces de Sobolev et espaces d’Orlicz et application au comportement [à]{} l’infini pour des [' e]{}quations des ondes semi-linéaires* (French), Portugal. Math. **42** (1983/84), no. 1, 97– 112 (1985).
A. Haraux, *Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes*, [*J. Diff. Eq.* ]{} [**49**]{}(1983), no. 3, 473–483.
A. Haraux, *A simple almost-periodicity criterion and applications*, [*J. Diff. Eq.* ]{} [**66**]{} (1987), no. 1, 51–61.
A. Haraux, *Asymptotic behavior for two-dimensional, quasi-autonomous, almost periodic evolution equations*, [*J. Diff. Eq.* ]{} [**66**]{} (1987), no. 1, 62–70.
A. Haraux, *$ L^p$ estimates of solutions to some non-linear wave equations in one space dimension*, Int.J. Math. Modelling and Numerical Optimization [**1**]{} (2009), Nos 1-2, p. 146– 154.
A. Haraux, *On the strong oscillatory behavior of all solutions to some second order evolution equations*, Port. Math. [**72**]{} (2015), no. 2, 193–206.
A. Haraux, *Almost-periodic forcing for a wave equation with a nonlinear, local damping term*, Proc. Roy. Soc. Edinburgh Sect. A [**94**]{} (1983), no. 3–4, 195–212.
A. Haraux, *Nonresonance for a strongly dissipative wave equation in higher dimensions*, Manuscripta Math. [**53** ]{}(1985), no. 1-2, 145–166.
A. Haraux, *Anti-periodic solutions of some nonlinear evolution equations*, Manuscripta Math. 63 (1989), no. 4, 479–505.
A. Haraux, *Semi-linear hyperbolic problems in bounded domains*, Math. Rep. 3 (1987), no. 1, i–xxiv and 1–281.
A. Haraux and V. Komornik, *Oscillations of anharmonic Fourier series and the wave equation*, Rev. Mat. Iberoamericana **1** (1985), no. 4, 57–77.
A. Haraux and E. Zuazua, *Decay estimates for some semilinear damped hyperbolic problems*, Arch. Rational Mech. Anal. [**100**]{} (1988), no. 2, 191–206.
Z. Hu and A.B. Mingarelli, *Almost periodicity of solutions for almost periodic evolution equations*, [*Differential Integral Equations*]{} [**18**]{} (2005), no. 4, 469–480.
C. F. Muckenhoupt, *Almost periodic functions and vibrating systems*, Journal of Mathematical Physics **8** (1928–1929), 163–199.
M.Nakao, *Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term*, J. Math. Anal. Appl. [**58**]{} (1977), no. 2, 336–343.
G. Prodi, *Soluzioni periodiche di equazioni a derivate parziali di tipo iperbolico non lineari* (Italian), Ann. Mat. Pura Appl. (4) [**42**]{} (1956), 25–49.
G. Prouse, *Soluzioni limitate dell’equazione delle onde non omogenea con termine dissipativo quadratico* (Italian), Ricerche Mat. [**14**]{} (1965), 41–48.
J. Vancostenoble and P. Martinez, *Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks*, SIAM J. Control Optim. 39 (2000), no. 3, 776–797 (electronic).
S. Zaidman, *Solutions presque-p[é]{}riodiques des [é]{}quations diff[é]{}rentielles abstraites*(French), Enseign. Math. (2) [**24**]{} (1978), no. 1-2, 87–110.
|
---
abstract: 'We propose an alternative formalism to simulate CMB temperature maps in $\Lambda$CDM universes with nontrivial spatial topologies. This formalism avoids the need to explicitly compute the eigenmodes of the Laplacian operator in the spatial sections. Instead, the covariance matrix of the coefficients of the spherical harmonic decomposition of the temperature anisotropies is expressed in terms of the elements of the covering group of the space. We obtain a decomposition of the correlation matrix that isolates the topological contribution to the CMB temperature anisotropies out of the simply connected contribution. A further decomposition of the topological signature of the correlation matrix for an arbitrary topology allows us to compute it in terms of correlation matrices corresponding to simpler topologies, for which closed quadrature formulae might be derived. We also use this decomposition to show that CMB temperature maps of (not too large) multiply connected universes must show “patterns of alignment”, and propose a method to look for these patterns, thus opening the door to the development of new methods for detecting the topology of our Universe even when the injectivity radius of space is slightly larger than the radius of the last scattering surface. We illustrate all these features with the simplest examples, those of flat homogeneous manifolds, i.e., tori, with special attention given to the cylinder, i.e., $T^1$ topology.'
author:
- 'W.S. Hipólito–Ricaldi'
- 'G.I. Gomero'
title: Topological signatures in CMB temperature anisotropy maps
---
Introduction {#S:Intro}
============
It is becoming widely recognized that universes with a nontrivial spatial topology may be more natural models for our Universe than the traditional simply connected ones. This naturalness can be invoked from the mathematical point of view by arguing that there is an infinity of locally homogeneous and isotropic multiply connected 3–spaces, while there are only three simply connected ones; or with physical arguments coming from incursions into the nobody’s territories of Quantum Gravity and Quantum Cosmology. On the other hand, from a more pragmatic point of view, we can argue that cosmological models with a nontrivial spatial topology offer a very rich field of research, and are particularly well suited to explain certain reported “anomalous” features in CMB temperature maps, such as the alignments of their low $\ell$–modes [@Aligne1; @Aligne2].
Conversely, the full sky CMB temperature maps produced by the space missions COBE and WMAP provide us with an amazingly rich and high quality amount of data with which we can look for the topology of space. This is very compelling for those who wish to unmask our Universe and see its shape, since Cosmic Topology is at present an almost exclusively observational and phenomenological issue, due to the lack of an accepted fundamental physical theory which can predict the global topology of space.
Theory demands topology of space to leave several different kinds of marks in CMB temperature maps. Two of them have been largely studied and exploited to try to unveil the shape of our Universe, the distorsion of the angular power spectrum with respect to that of a simply connected universe [@PowSpec]–[@Weeks04], and the existence of “circles in the sky” [@CinSky; @CSSK04]. Two other closely related signatures, a non–null bipolar power spectrum [@StatAnis; @SH05], and alignments of the low $\ell$–modes [@Aligne1; @Aligne2; @Weeks04], have been only marginally used. Our main motivation for deciding to adventure into cosmic topology with the CMB was the desire to get a deeper understanding of the nature and properties of these alignments as a topological signature.
One indispensable tool for a project like this is a software facility to produce simulated CMB temperature maps in multiply connected universes, so that we could systematically study the effects of different sizes and topologies on these alignments. These simulation procedures exist and have been used in several studies in cosmic topology [@Aurich], [@LevinB]–[@BPS], so we could expect that this issue of the project would not present any problem.
However, almost all known methods for computing CMB temperature anisotropies in multiply connected universes need to solve the Helmholtz equation in the manifold modelling our 3–dimensional space, the only exception to our knowledge being the work of Bond et al. [@BPS]. To solve the Helmholtz equation is a relatively easy problem in Euclidean manifolds [@LevinA; @RWULL04], but a very difficult task in spherical [@SphEigen; @Lachieze] and hyperbolic 3–spaces [@HypEigen]. Indeed, the spherical case has been completely solved analitically only recently by Lachièze–Rey [@Lachieze], while for compact hyperbolic manifolds the only possible approach is numerical [@HypEigen].
Among other results, in this paper we present a new approach to the computation of the correlation matrix $\langle a_{\ell m} a_{\ell' m'}^*
\rangle$ of the coefficients of the spherical harmonic decomposition of CMB temperature anisotropies in a universe with nontrivial spatial topology. The main feature of this approach is that it avoids the explicit computation of the solutions of the Helmholtz equation in the spatial sections of spacetime. Instead, we express the correlation matrix in terms of the covering group alone. Incidentally, the idea of generating a CMB map exploiting the symmetries of the quotient space was already suggested time ago by Janna Levin and collaborators [@LevinB; @LSS98]. In particular we wish to quote a citation of a nontrivial claim in [@LSS98] (p.2695) which we have, in our opinion, succeeded in achieving: “By understanding the symmetries of the fundamental polyhedron and the identification rules, a CBR pattern can be deduced without the need to explicitly obtain the spectrum mode by mode.”
Our main formal result is a generic decomposition of the form $$\label{GenDecomp}
X^\Gamma = X^{s.c.} + X^{t.s.} \; ,$$ where $X$ may be any covariance function which can be related to the two–point correlation function of the Newtonian potential (see Sec. \[S:CycDec\]), $\Gamma$ is the covering group of (the multiply connected) space, $s.c.$ stands for “simply connected”, and $t.s.$ means “topological signature”.
Thus, in Eq. (\[GenDecomp\]), $X^\Gamma$ is the covariance function computed in the manifold $M=\widetilde{M}/\Gamma$, and $X^{s.c.}$ the same covariance function but computed in the universal covering space $\widetilde{M}$. It means that all the topological information is encoded in the “perturbative” term $X^{t.s.}$, and that is why we refer to it, generically, as the topological signature of $X^\Gamma$. This decomposition is always possible whenever one can express $X^\Gamma$ in terms of $\Gamma$, as for example in the PSH method for detecting multiple copies of standard candles [@CosCris]. In the present case we succeeded in writing the correlation matrix of the $a_{\ell m}$’s in this way by formally manipulating the two–point correlation function of the Newtonian potential derived by Bond, Pogosyan and Souradeep [@BPS].
Our approach to compute the correlation matrix $\langle a_{\ell m} a_{\ell'
m'}^* \rangle$ has some methodological advantages in the simulation of CMB temperature maps. Indeed, by means of a suitable decomposition of the covering group $\Gamma$ in cyclic subgroups, we are able to write down a formula for the correlation matrix of a complicated topology in terms of the correlation matrices of the cyclic topologies (topologies with a cyclic covering group) that cover it maximally. Since correlation matrices for cyclic manifolds are relatively easy to compute (we obtain a closed quadrature formula for the cylinder), we expect to obtain in the near future more efficient ways to simulate maps for complicated manifolds.
The decomposition of $\Gamma$ in cyclic subgroups describes in a transparent way the symmetries of the manifold, and this fact gives rise to another advantage of our approach, in this case from the observational side. Universes with a cyclic topology present an alignment along the “direction of the generator isometry”. It follows that a CMB map in a universe with a nontrivial topology might present “patterns of alignment” (one for each $\ell$–mode) characteristic of its shape and size. Indeed, and this shall become “obvious”, symmetries of the quotient manifolds translate into symmetries of their patterns of alignment. We propose a method to search for these patterns by constructing maps of the dispersion of the squares $|a_{\ell m}|^2$ around the power spectrum. This opens the door to the development of methods to look for topology by searching these patterns, instead of limiting ourselves to considerations concerning only the special directions defined by the alignments.
For the sake of brevity, other advantages of our approach to compute the matrix $\langle a_{\ell m} a_{\ell' m'}^* \rangle$ are discussed in Sec. \[S:Discuss\] only. We prefer now to make a few remarks on some limitations of our work. We have considered here a few simplifications to develop the formalism, and worked out the details for the very simplest nontrivial topologies. In fact, we (i) have considered the Sachs–Wolfe effect as the only source of temperature anisotropies in CMB maps, (ii) have written the correlation matrix $\langle a_{\ell m} a_{\ell' m'}^* \rangle$ formally only for Euclidean 3-spaces, (iii) have worked out the details for homogeneous flat manifolds, (iv) have made a detailed analysis and some simulations only for cylinders ($T^1$ topology), and (v) these simulations were done using the Einstein–de Sitter model. We wish to close this introduction by justifying each one of these simplifications.
The shape of space is a global property, thus we expect the topological signatures in CMB to show themselves on very large scales only. Although a proof is missing, we believe that the main features of these signals would observationally appear if we restrict the searches to the low $\ell$–modes in the temperature maps, i.e., we do not need high resolution CMB maps in Cosmic Topology! Since the Sachs–Wolfe effect is the main source of temperature anisotropies at these scales [@RULW04], we expect that theoretical explorations considering only this effect will put in evidence the main features of the topological signals that we would observe in a real map. The addition of the missing part of the anisotropies will only modify quantitatively the predictions made with our approximation, and thus, will only be important when adjusting theoretical models with data.
We consider this paper technically hard, so much care has been taken to write it in a clear and pedagogical way. The main features of our formalism and of the topological signatures we predict in CMB can be understood by restricting the presentation to Euclidean topologies. The inclusion of nonzero spatial curvature will only introduce additional technical considerations (and nothing qualitatively new), thus we decided to leave the spherical and hyperbolic cosmological models for a future paper.
The same applies to the lack in the paper of detailed explicit correlation matrices for nonhomogeneous flat manifolds. Indeed, the price we pay for simplicity and transparency in the presentation of the results for each specific cyclic topology is the need for very hard calculations in the middle steps, as can be seen in the appendices. Explicit calculations for nonhomogeneous flat manifolds will only add one page to the main body of the paper, and one or two more appendices to the already large list of them (see the end of the introduction). An exhaustive presentation for all the Euclidean manifolds is left for a future work.
The last but not the least, we performed the simulations with the Einstein–de Sitter model for simplicity. Nothing is lost from the theoretical point of view with this simplification since, as discussed in the paper, the structure of the topological signatures in CMB is captured in this oversimplified and old fashioned model of our Universe. However, more realistic $\Lambda$CDM models will be required to confront theory with observations quantitatively.
We close this introduction by giving a detailed description of the structure of the paper. In Sec. \[S:CMBSimul\] we briefly review the two most common methods to simulate CMB temperature maps in multiply connected universes, as well as present the method we have developed, and implemented for the Euclidean case. In Sec. \[S:CycDec\] we define the topological signature in a correlation function, perform the decomposition of the covering group of a quotient space in its cyclic subgroups, and write the topological signature in terms of this decomposition. We also show here that the symmetries of a quotient space appear transparently in the decomposition of its covering group in cyclic subgroups.
In Sec. \[S:tori\] we apply our formalism to the homogeneous Euclidean manifolds, which are the simplest. We first compute the correlation matrix and the angular power spectrum for the cylinder, and apply the general results of the previous section in order to write down the correlation matrix of the spherical harmonic coefficients and the angular power spectrum for a generic torus. Finally, as an illustration, we write those expressions explicitly for the chimney ($T^2$ topology).
In Sec. \[S:TopSign\] we first show, by means of simulations, that in a universe with the topology of a cylinder the low $\ell$–modes are aligned in a similar fashion as they are in the WMAP data. We then use the results in the previous sections to argue that CMB temperature maps in a universe with a nontrivial topology must present characteristic patterns of alignment, and propose the method of mapping on the sphere the dispersion of the squares $|a_{\ell m}|^2$ to look for them. Finally, in Sec. \[S:Discuss\] we discuss in detail the results of this paper and suggest further lines of research.
In brief, the main goal of this paper is to show that our approach to the computation of the correlation matrix $\langle a_{\ell m} a_{\ell' m'}^*
\rangle$ (Sec. \[S:CMBSimul\]), together with the decomposition of the covering group of a manifold in cyclic subgroups (Sec. \[S:CycDec\]), led to the discovery of a new topological signature in CMB temperature maps, i.e., the “patterns of alignment” (Sec. \[S:TopSign\]). To illustrate this we have used the simplest example, i.e., that of flat homogeneous manifolds (Sec. \[S:tori\]).
The paper has four appendices. In Appendix \[Ap:SphHarm\] we collect standard definitions and results related to spherical harmonics in order to set the conventions used in this paper. The technical calculations needed for the computation of the correlation matrix $\langle a_{\ell m} a_{\ell' m'}^*
\rangle$ for the cylinder are presented in Appendices \[Ap:Clausen\] and \[Ap:EvalFllm\]. In the former we develop from scratch the theory of Clausen $\varphi$–functions, for which we have not found any suitable reference in the literature. In the latter we compute a function that is the key part for computing efficiently the topological signature of the correlation matrix for the cylinder. Finally, in Appendix \[S:OldRes\] we briefly reproduce known results for the correlation matrix of five out of the six compact orientable Euclidean 3–spaces. These formulae have been obtained previously in the literature by considering explicitly the solutions of the Helmholtz equation in these manifolds, and are written in terms of the $k$–modes [@LevinA; @RWULL04]. Our derivation avoids the need for considering these solutions.
Simulating CMB temperature maps {#S:CMBSimul}
===============================
In this section we briefly describe two methods currently available for simulating CMB temperature maps in universes with non–trivial spatial topology, and proceed to develop our own formulation. We consider $\Lambda$CDM universes, where the background metric of spacetime is of the Robertson–Walker type, and include scalar and adiabatic perturbations as the seeds for the temperature anisotropies of the CMB. In the Newtonian gauge we have $$ds^2 = a^2(\eta) \left[(1 + 2\Phi)d\eta^2 - (1 - 2\Phi)\gamma_{ij} dx^i dx^j
\right]$$ for the metric, where $\eta$ is the conformal time, $a(\eta)$ is the scale factor, $\Phi$ is the Newtonian potential, and $$\gamma_{ij} = \left( 1 + \frac{K}{4} (x^2 + y^2 + z^2) \right)^{-2}
\delta_{ij}$$ is the metric of the spatial section of the background with sectional curvature $K=0$, $\pm 1$.
The matter content consists of radiation ($\Omega_r$), baryonic and cold dark matter ($\Omega_m = \Omega_b + \Omega_{cdm}$), and dark energy in the form of a cosmological constant ($\Omega_{\Lambda}$). Since we are interested on fluctuations on large angular scales, we make the assumption of instantaneous recombination and do not consider finite thickness effects. The main contribution to the temperature anisotropy observed at the direction ${\mathbf{n}}$ comes from the complete (ordinary plus integrated) Sachs–Wolfe effect $$\begin{aligned}
\label{SW}
\frac{\delta T}{T} ({\mathbf{n}}) & = & \frac{1}{3} \, \Phi(\eta_{LSS},R_{LSS}
{\mathbf{n}}) + \\
& & \hspace{1.5cm} 2 \int_{\eta_{LSS}}^{\eta_0} d\eta \left. \frac{\partial
\Phi}{\partial \eta} \right|_{(\eta,R(\eta){\mathbf{n}})} \; , \nonumber\end{aligned}$$ where the index $LSS$ stands for ‘last scattering surface’, the index $0$ for present time, and $R(\eta)$ is the comoving distance at instant $\eta$ between a photon, scattered at $\eta_{LSS}$, and the observer.
The Newtonian potential is written as $$\label{SepVar}
\Phi(\eta,{\mathbf{x}}) = \int dq F(\eta,q) \xi(q,{\mathbf{x}}) \; .$$ The temporal part satisfies the equation $$\begin{aligned}
\label{Temp}
F''(\eta) + 3 \mathcal{H} \left( 1+c_s^2 \right) F'(\eta) + \left[ 2
\mathcal{H}' + \right. \hspace{1cm} & & \\
\left. \left(1+3c_s^2 \right) (\mathcal{H}^2 - K) + c_s^2 q^2 \right] F(\eta)
& = & 0 \; , \nonumber\end{aligned}$$ where $c_s$ is the speed of sound in the fluid and $\mathcal{H} = a'/a$ is the Hubble parameter in conformal time. On the other hand, the spatial part consists of solutions of the Helmholtz equation $$\label{Helmholtz}
(\Delta + q^2) \, \xi(q,{\mathbf{x}}) = 0 \; ,$$ where the index $q$ has been put as a variable in $\xi$ for simplicity of notation.
The integral in Eq. (\[SepVar\]) has to be understood in a measure theoretic sense. Indeed, for multiply connected spaces the measure $dq$ is not the usual one but a combination of a discrete and an absolutely continuous measures, reducing the integral in (\[SepVar\]) to a sum and an integral in the usual sense. In particular, if the space is compact, the measure reduces to a discrete one. This comes from the well–known fact that not every eigenmode of the Laplacian operator in the universal covering space $\widetilde{M}$ is also an eigenmode in a quotient space $M = \widetilde{M}/\Gamma$. In fact, only eigenmodes in $\widetilde{M}$ satisfying the invariance conditions $$\label{automorph}
\xi(q,g {\mathbf{x}}) = \xi(q,{\mathbf{x}})$$ for any $g \in \Gamma$ project to eigenmodes in $M$.
The most straightforward way of simulating CMB temperature maps is by solving (\[Temp\]) and (\[Helmholtz\]), performing the sum in (\[SepVar\]), and then evaluating the SW effect (\[SW\]). However, one has to consider that a temperature anisotropy map is a realization of a random field on the 2–sphere, and this randomness is inherited from that of the Newtonian potential (\[SepVar\]). There are currently two ways to implement this random character in the simulations, and one goal of this paper is to propose a third one.
The first and most direct method is to consider the randomness in the temporal part of the decomposition (\[SepVar\]) of the Newtonian potential. The two–point correlation function of the Newtonian potential at fixed time $\eta$ can then be written as $$\begin{aligned}
\label{twopointAur}
\langle \Phi(\eta,{\mathbf{x}}) \, \Phi(\eta,{\mathbf{x}}') \rangle & = & \int dq \,
dq' f(\eta,q,q') \times \\
& & \hspace{1.2cm} \xi(q,{\mathbf{x}}) \, \xi^*(q',{\mathbf{x}}') \; , \nonumber\end{aligned}$$ where $f(\eta,q,q') = \langle F(\eta,q) \, F(\eta,q') \rangle$ is the two–point correlation function for the amplitudes of the scalar perturbation modes and $\xi(q,{\mathbf{x}})$ are normalized solutions of the Helmholtz equation.
Assuming that the Newtonian potential is a homogeneous and isotropic random field, the two point correlation function (\[twopointAur\]) reduces to a function of time $\eta$ and the distance $d({\mathbf{x}}, {\mathbf{x}}')$, and thus we get $f(\eta,q,q') = P_\Phi(\eta,q) \, \delta(q-q')$, where $P_\Phi(\eta,q)$ is the gravitational power spectrum. If, in addition, the Newtonian potential is assumed to be gaussian, its random character is completely encoded in the variance of the temporal part $$\label{VarianceTemp}
\langle F^2(\eta,q) \rangle = P_\Phi(\eta,q) \; .$$
Specifying this function, one then takes as an initial condition, $F(\eta_{init},q)$, a realization of a normal distribution with zero mean and variance given by Eq. (\[VarianceTemp\]), and some suitable condition for the initial first derivative. With these initial conditions one solves for (\[Temp\]), so one can now compute the potential (\[SepVar\]). Topology is considered by restricting in (\[SepVar\]) to normalized solutions of the Helmholtz equation satisfying the invariance conditions (\[automorph\]). This method has been extensively used in [@Aurich; @LevinB; @LSS98], although in the former the authors do not consider the randomness of the function $F(\eta,q)$. Instead the random character of the CMB maps is attributed exclusively to the random character of the eigenmodes of the Laplacian in compact hyperbolic spaces.
The second method to produce simulated maps of CMB temperature anisotropies in universes with nontrivial spatial topology was first described in [@RULW04], and used in [@URLW04; @RWULL04]. It is based in considering the randomness of the Newtonian potential in the eigenmodes of the Laplacian operator. We begin by decomposing the general solution of the Helmholtz equation, in the universal covering space, as a sum of fundamental solutions $$\label{HelmGen}
\xi(q,{\mathbf{x}}) = \sum_{\ell,m} \widehat{\xi}_{\ell m}(q)
\mathcal{Y}_{\ell m}(q,{\mathbf{x}}) \; ,$$ where $$\label{HelmRadAng}
\mathcal{Y}_{\ell m}(q,{\mathbf{x}}) = \rho_\ell(q,x) \, Y_{\ell m}({\mathbf{n}})$$ is the normalized solution of the Helmholtz equation after separating it in radial and angular variables. Here we have put $x = |{\mathbf{x}}|$, ${\mathbf{n}}$ is the unit vector in the direction of ${\mathbf{x}}$, and the $Y_{\ell
m}({\mathbf{n}})$ are the spherical harmonic functions (see appendix \[Ap:SphHarm\]). Since the solutions (\[HelmRadAng\]) are normalized, the randomness of the eigenmodes’ amplitudes relies on the coefficients $\widehat{\xi}_{\ell m}(q)$.
Introducing (\[SepVar\]) in (\[SW\]), and using (\[HelmGen\]) and (\[HelmRadAng\]), we arrive at the decomposition of the temperature anisotropy map in spherical harmonics $$\label{TempSphHarm}
\frac{\delta T}{T} ({\mathbf{n}}) = \sum_{\ell,m} a_{\ell m} Y_{\ell m}({\mathbf{n}}) \; ,$$ with multipole coefficients $$\label{HarmCoeff}
a_{\ell m} = \int dq \, \widehat{\xi}_{\ell m}(q) \, G_\ell(q) \; ,$$ and the effects of physical cosmology given by $$\begin{aligned}
\label{Gl(q)}
G_\ell(q) & = & \frac{1}{3} F(\eta_{LSS},q) \, \rho_\ell(q,R_{LSS}) + \\
& & \hspace{1cm} 2 \int_{\eta_{LSS}}^{\eta_0} \hspace{-0.2cm} d\eta \left.
\frac{\partial F}{\partial \eta} \right|_{(\eta,q)} \! \rho_\ell(q,R(\eta))
\; . \nonumber\end{aligned}$$
At this point it is convenient to recall how does topology enter in the story. Note that, due to the invariance conditions (\[automorph\]), not every solution of the form (\[HelmGen\]) is a solution in a quotient space. However, one would expect that any solution in a quotient space could be written in this form, the only problem being to find the correct coefficients $\widehat{\xi}_{\ell m}(q)$. These coefficients are not independent one from the other, since the invariance conditions (\[automorph\]) establish certain relations among them. In what follows we will assume that these relations can always be found, so that we will always represent an eigenmode in a quotient space by Eq. (\[HelmGen\]), with suitable coefficients.
A crucial point in [@RULW04] is the decomposition of these coefficients as $$\label{Xi_Decomp}
\widehat{\xi}_{\ell m}(q) = \sqrt{P_\Phi(q)} \, {\widehat{{\mathbf{e}}}_{\ell m}}(q) \; ,$$ where $P_\Phi(q)$ is the gravitational initial power spectrum, and the ${\widehat{{\mathbf{e}}}_{\ell m}}(q)$ form a multivariate gaussian random variable, with a non–diagonal covariance matrix due to the relations among the coefficients $\widehat{\xi}_{\ell m}(q)$ coming from the invariance conditions.
The simulation procedure can now be described. First we solve Eq. (\[Temp\]) using the initial condition $F(\eta_{init},q) = 1$, and a suitable condition for the first derivative, and use this in (\[Gl(q)\]) to compute $G_\ell(q)$. Then generate a realization of the random variable ${\widehat{{\mathbf{e}}}_{\ell m}}(q)$ and use (\[Xi\_Decomp\]) to compute $\widehat{\xi}_{\ell m}(q)$. The map is now simulated by computing the coefficients $a_{\ell m}$ using (\[HarmCoeff\]), and performing the sum in (\[TempSphHarm\]).
An alternative method of simulation, also proposed in [@RULW04], is to construct the covariance matrix of the $a_{\ell m}$’s as $$\begin{aligned}
\label{CovalmGen}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle & = & \int dq \, dq' G_\ell(q) \,
G_{\ell'}(q') \times \\
& & \hspace{2cm} \langle \widehat{\xi}_{\ell m}(q) \, \widehat{\xi}^*_{\ell'
m'}(q') \rangle \; . \nonumber\end{aligned}$$ The substitution of (\[Xi\_Decomp\]) into (\[CovalmGen\]), and the evaluation of the resulting integral give rise to expressions for the covariance matrix in terms of the eigenvalues and eigenmodes of the Laplacian operator. The multipolar coefficients are then obtained directly as a realization of a gaussian distribution with zero mean and covariance given by (\[CovalmGen\]).
The method we propose in this paper lies along these lines, but we are able to manipulate the correlation function for the $\widehat{\xi}_{\ell m}(q)$ in a way that avoids the need for an explicit determination of the eigenmodes of the Laplacian. Instead, the final expression after the integration of (\[CovalmGen\]) is given in terms of the isometries of the corresponding covering group.
Our starting point is an expression, obtained by Bond et al. in [@BPS], that relates the two–point correlation function of the Newtonian potential in a simply connected universe, and that in a multiply connected universe, when both potentials have the same initial power spectrum. For a homogeneous and isotropic random field the expression is $$\label{two-point}
\langle \Phi(\eta,{\mathbf{x}}) \Phi(\eta,{\mathbf{x}}') \rangle^{\Gamma} =
\sum_{g \in \Gamma} |g| \, \langle \Phi(\eta,{\mathbf{x}}) \Phi(\eta,g
{\mathbf{x}}') \rangle^{s.c.} \; ,$$ where $|g| =1$ if $g$ is orientation preserving, and $-1$ otherwise.
We now show how to use Eq. (\[two-point\]) in order to express (\[CovalmGen\]) in terms of the covering group, and for simplicity we will restrict the presentation to flat topologies. In Euclidean space, the most general solution of Eq. (\[Helmholtz\]) is written in the form $$\label{SolHelmEuc}
\xi(q,{\mathbf{x}}) = \int d^3 k \, \delta(q-k) \, \widehat{\xi}({\mathbf{k}}) \,
e^{i {\mathbf{k}} \cdot {\mathbf{x}}} \; .$$ If we now expand the plane waves in spherical harmonics as $$\label{PlaneW}
e^{i {\mathbf{k}} \cdot {\mathbf{x}}} = 4 \pi \sum_{\ell,m} i^\ell j_\ell(kx)
Y_{\ell m}^*({\mathbf{n}}_{{\mathbf{k}}}) Y_{\ell m}({\mathbf{n}}) \; ,$$ where $j_\ell(x)$ is the spherical Bessel function of order $\ell$, and introduce it in (\[SolHelmEuc\]) we obtain $\xi(q,{\mathbf{x}})$ expressed as in Eq. (\[HelmGen\]) with $$\label{CoefHelm}
\widehat{\xi}_{\ell m}(q) = 4 \pi i^\ell \int \! d^3k \, \delta(q-k) \,
\widehat{\xi}({\mathbf{k}}) Y_{\ell m}^*({\mathbf{n}}_{{\mathbf{k}}}) \; ,$$ where ${\mathbf{n}}_{{\mathbf{k}}}$ is the unit vector in the direction of ${\mathbf{k}}$, and $\rho_\ell(q,x) = j_\ell(qx)$.
Note that we have not decomposed $\widehat{\xi}_{\ell m}(q)$ as in Eq. (\[Xi\_Decomp\]). Instead, the decomposition (\[CoefHelm\]) allows us to implement the randomness of the Newtonian potential in the modes $\widehat{\xi}({\mathbf{k}})$. In fact, introducing (\[CoefHelm\]) in (\[CovalmGen\]), the covariance matrix for the $a_{\ell m}$’s now reads $$\begin{aligned}
\label{CovalmEucGen}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle & = & (4 \pi)^2 i^{\ell - \ell'}
\hspace{-0.2cm} \int \! d^3k \, d^3k' G_\ell(k) G_{\ell'}(k') \times
\nonumber \\
& & \hspace{0.2cm} \langle \widehat{\xi}({\mathbf{k}}) \,
\widehat{\xi}^*({\mathbf{k'}}) \rangle Y_{\ell m}^*({\mathbf{n}}_{{\mathbf{k}}})
Y_{\ell' m'}({\mathbf{n}}_{{\mathbf{k'}}}) \; . \nonumber \\
& &\end{aligned}$$ It is the correlation function of the modes $\widehat{\xi}({\mathbf{k}})$ that carries all the topological information, as we will see in the following.
Introducing (\[SolHelmEuc\]) in (\[SepVar\]) we obtain $$\Phi(\eta, {\mathbf{x}}) = \int d^3k \, F(\eta,k) \, \widehat{\xi}({\mathbf{k}}) \,
e^{i {\mathbf{k}} \cdot {\mathbf{x}}} \; ,$$ thus the two–point correlation function of the Newtonian potential now reads $$\begin{aligned}
\label{TwoPointEuc}
\langle \Phi(\eta,{\mathbf{x}}) \, \Phi(\eta,{\mathbf{x}}') \rangle & = & \int d^3k
\, d^3k' F(\eta,k) \, F(\eta,k') \times \nonumber \\
& & \langle \widehat{\xi}({\mathbf{k}}) \,
\widehat{\xi}^*({\mathbf{k'}}) \rangle \, e^{i ({\mathbf{k}} \cdot {\mathbf{x}} -
{\mathbf{k'}} \cdot {\mathbf{x'}})} \; .\end{aligned}$$
At this point we have to recall that an Euclidean isometry can always be written as $g = (R,{\mathbf{r}})$, where $R$ is an orthogonal transformation and ${\mathbf{r}}$ is an Euclidean vector, and that this isometry acts on a vector ${\mathbf{x}}$ as $g {\mathbf{x}} = R {\mathbf{x}} + {\mathbf{r}}$. It is now an easy task to deduce from Eqs. (\[two-point\]) and (\[TwoPointEuc\]) that $$\label{twopointEucGen}
\langle \widehat{\xi}({\mathbf{k}}) \, \widehat{\xi}^*({\mathbf{k'}})
\rangle^{\Gamma} = \sum_{g \in \Gamma} \langle \widehat{\xi}({\mathbf{k}}) \,
\widehat{\xi}^*(R{\mathbf{k'}}) \rangle^{s.c} \, e^{-i R{\mathbf{k'}} \cdot
{\mathbf{r}}} \; .$$
In most inflationary models the initial perturbations of the gravitational field are homogeneous and isotropic Gaussian random fields, thus the correlation matrix of the ${\mathbf{k}}$–modes in a simply connected universe takes the form $$\langle \widehat{\xi}({\mathbf{k}}) \widehat{\xi}^*({\mathbf{k'}}) \rangle^{s.c} =
\frac{P_{\Phi}(k)}{k^3} \, \delta({\mathbf{k}} - {\mathbf{k}}') \; .$$ The use of (\[twopointEucGen\]) now yields $$\langle \widehat{\xi}({\mathbf{k}}) \widehat{\xi}^*({\mathbf{k'}}) \rangle^\Gamma =
\frac{P_{\Phi}(k)}{k^3} \, \sum_{g \in \Gamma} \delta({\mathbf{k}} - R{\mathbf{k}}')
\, e^{-i R{\mathbf{k}}' \cdot {\mathbf{r}}} \; ,$$ for the correlation matrix of the ${\mathbf{k}}$–modes in the quotient space $M
= \widetilde{M}/\Gamma$, which when substituted in (\[CovalmEucGen\]) finally gives $$\begin{aligned}
\label{MCCorr}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma} & = & (4 \pi)^2 \,
i^{\ell - \ell'} \int \! \frac{d^3k}{k^3} \, \Psi_{\ell \ell'}(k) \times \\
& & \hspace{2cm} \Upsilon^{\Gamma}_{\ell' m'}({\mathbf{k}}) \,
Y_{\ell m}^*({\mathbf{n}}_{{\mathbf{k}}}) \; , \nonumber\end{aligned}$$ where the physical effects are encoded in $$\label{PhysSign}
\Psi_{\ell \ell'}(k) = P_{\Phi}(k) \, G_\ell(k) \, G_{\ell'}(k) \; ,$$ and the topological information in $$\label{DefTopSign}
\Upsilon^{\Gamma}_{\ell m}({\mathbf{k}}) = \sum_{g \in \Gamma} e^{-i
{\mathbf{k}} \cdot {\mathbf{r}}} \, Y_{\ell m}({\mathbf{n}}_{R^T{\mathbf{k}}}) \; .$$
The integration in (\[MCCorr\]) is over the whole ${\mathbf{k}}$–space. The topological information is carried in Eq. (\[DefTopSign\]), which automatically selects the eigenvalues of the Laplacian operator in $M$. This can be seen in Appendix \[S:OldRes\], where $\Upsilon^{\Gamma}_{\ell
m}({\mathbf{k}})$ is expressed in terms of Dirac’s delta functions centered in the eigenvalues of the Laplacian operator in the corresponding quotient spaces.
Decomposition of $\Gamma$ in cyclic subgroups {#S:CycDec}
=============================================
In this section we develop some formal results in order to proceed further. Especifically, we define the topological signature of any covariance function that can be decomposed as $$X^\Gamma = \sum_{g \in \Gamma} X^g \; ,$$ as for example, the two–point correlation function of the Newtonian potential and the correlation matrix $\langle a_{\ell m} \, a^*_{\ell' m'}
\rangle$. Then we work out a suitable decomposition of a covering group in their cyclic subgroups, and write down the topological signature in terms of this decomposition. We also show here that the symmetries of a quotient space appear transparently in the decomposition of its covering group in cyclic subgroups. It follows that the main result of this section is the ellucidation of how these symmetries manifest themselves in the topological signature of CMB temperature anisotropy maps.
Let us begin by writing the obvious decomposition $$\label{TopSignGen}
X^\Gamma = X^{s.c.} + X^{\widehat{\Gamma}} \; ,$$ where $\widehat{\Gamma} = \Gamma \setminus \{id\}$. The second term in the right hand side is the topological signature in the covariance function. The expressions we present in the following are analogous to (\[TopSignGen\]) and are also rather obvious.
It is convenient to introduce a notation, so natural, that has been used in (\[TopSignGen\]) without any previous definition. Let $S$ be any subset of isometries of the covering space $\widetilde{M}$, then a superscript $S$ in the covariance function means $$X^S = \sum_{g \in S} X^g \; .$$
Then if $M = \widetilde{M}/\Gamma$ is a quotient space and $\Gamma_1 \subset
\Gamma$ is any subset of the covering group, we can immediately write $X^\Gamma = X^{\Gamma_1} + X^{\Gamma \setminus \Gamma_1}$. This expression is nothing but the simplest generalization of Eq. (\[TopSignGen\]), which corresponds to the trivial case $\Gamma_1 = \{id\}$. We get a further generalization as follows, let $\Gamma_1$ and $\Gamma_2$ be any two subsets of the covering group $\Gamma$, such that $\Gamma_1 \cap
\Gamma_2 = \Gamma_3$, then $$\label{SecDecomp}
X^\Gamma = X^{\Gamma_1} + X^{\Gamma_2} - X^{\Gamma_3} + X^{\Gamma \setminus
(\Gamma_1 \cup \Gamma_2)} \; .$$ We can now write the formal result we are interested in. Consider the subsets $G_1, \dots, G_n \subset \Gamma$ such that for any $i \neq j$, $G_i \cap G_j =
H$, then by induction on (\[SecDecomp\]) we get $$\label{FinDecomp}
X^\Gamma = \sum_{i=1}^n X^{G_i} - (n-1) X^H + X^{\Gamma \setminus G} \; ,$$ where $G = \cup G_i$.
To move forward, let $G_1 = \langle g_1 \rangle$ and $G_2 = \langle g_2
\rangle$ be two cyclic subgroups of $\Gamma$, and let ${\mathbf{0}} \in
\widetilde{M}$ be a lift to $\widetilde{M}$ of the position of the observer in $M$. We will say that $g_1$ and $g_2$ are conjugate by an isometry that “does not move the observer” if there exists an isometry $\eta$ fixing ${\mathbf{0}}$ and such that $g_1 = \eta^{-1} g_2 \eta$. Note that, as a consequence, we have that $d({\mathbf{0}}, g_1 {\mathbf{0}}) = d({\mathbf{0}}, g_2
{\mathbf{0}})$, where $d({\mathbf{x}},{\mathbf{y}})$ is the distance between two points ${\mathbf{x}}$ and ${\mathbf{y}}$ in $\widetilde{M}$. By extension, we will also say that the groups $G_1$ and $G_2$ are conjugate by an isometry that does not move the observer. In addition, we will say that $g_1$ is a minimal distance generator of $G_1$ (with respect to the observer) if $d({\mathbf{0}},g_1
{\mathbf{0}}) \leq d({\mathbf{0}},\gamma{\mathbf{0}})$ for any other generator $\gamma
\in G_1$.
Now consider the isometries $g_1, \dots, g_n \in \Gamma$ that generate the cyclic groups $G_i = \langle g_i \rangle$, and such that if $i \neq j$ then $G_i \cap G_j = \{id\}$. By using Eq. (\[FinDecomp\]) we immediately obtain that the topological signature of the covariance function can be decomposed as $$X^{\widehat{\Gamma}} = \sum_{i=1}^n X^{\widehat{G}_i} + X^{\Gamma \setminus G}
\; .$$ In the following we will be particularly interested in the case where the $g_i$’s are minimal distance generators of the $G_i$’s, and the latter form a complete set of groups mutually conjugate by isometries that do not move the observer.
Decomposing the topological signature further along these lines, let $g_1,
\dots, g_n, h_1, \dots, h_m \in \Gamma$ be minimal distance generators of the groups $G_i = \langle g_i \rangle$ and $H_j = \langle h_j \rangle$, and let $G
= \cup G_i$ and $H = \cup H_j$. Moreover, suppose that the $G_i$’s and the $H_j$’s form two complete sets of groups mutually conjugate by isometries that do not move the observer, and such that $G \cap H = \{id\}$ and $d({\mathbf{0}},
g_1 {\mathbf{0}}) \leq d({\mathbf{0}}, h_1 {\mathbf{0}})$. Then the topological signature can be decomposed as $$X^{\widehat{\Gamma}} = \sum_{i=1}^n X^{\widehat{G}_i} + \sum_{i=1}^m
X^{\widehat{H}_i} + X^{\Gamma \setminus (G \cup H)} \; .$$
We can proceed along these lines again and again, and obtain the following decomposition, in cyclic subgroups, of the covering group $\Gamma$, $$\label{DecCyclic}
\Gamma = \bigcup_{i=1}^\infty \bigcup_{j=1}^{k_i} \Gamma_{ij} \; ,$$ where $g_{ij} \in \Gamma$ is a minimal distance generator of the cyclic group $\Gamma_{ij}$, and such that
1. For each $i \in \mathbb{N}$, the set $\{\Gamma_{i1}, \dots,
\Gamma_{ik_i}\}$ is a complete set of groups mutually conjugate by isometries that do not move the observer.
2. If $i \neq i'$, the sets $\cup_{j=1}^{k_i} \Gamma_{ij}$ and $\cup_{j=1}^{k_{i'}} \Gamma_{i'j}$ have the identity as the only common element.
3. If $i < i'$, then $d({\mathbf{0}}, g_{i1} {\mathbf{0}}) \leq d({\mathbf{0}},
g_{i'1} {\mathbf{0}})$.
Then the topological signature of the covariance function can be written as $$\label{DecTSCyclic}
X^{\widehat{\Gamma}} = \sum_{i=1}^\infty \sum_{j=1}^{k_i}
X^{\widehat{\Gamma}_{ij}} \; .$$ Thus, to compute the topological signature of the covariance function for any multiply connected manifold, it is enough to know how to compute it for manifolds whose covering groups are cyclic groups. It is now obvious that this decomposition will be particularly useful for calculating the correlation matrix of the $a_{\ell m}$’s for any compact manifold once we know how to calculate it for cyclic flat (twisted cylinders), spherical (lens spaces), and hyperbolic manifolds.
Let us now show that the decomposition (\[DecCyclic\]) describes transparently the symmetries of the quotient manifold $M =
\widetilde{M}/\Gamma$. Actually, this decomposition contains the symmetries of the Dirichlet fundamental polyhedron of $M$ centered at the observer’s position ${\mathbf{0}} \in \widetilde{M}$, which is what one expects to reconstruct with cosmological observations. Recall that the Dirichlet fundamental polyhedron centered at ${\mathbf{0}} \in \widetilde{M}$ is the set $\mathcal{D}_{{\mathbf{0}}} \subset \widetilde{M}$ defined by (see [@Beardon]) $$\mathcal{D}_{{\mathbf{0}}} = \{ {\mathbf{x}} \in \widetilde{M} : d({\mathbf{0}},
{\mathbf{x}}) \leq d(g{\mathbf{0}},{\mathbf{x}}) \; \mbox{ for any } \; g \in \Gamma \}
\; .$$ The first thing to be noted is that, although the whole covering group enters in this definition, it turns out that, in order to effectively construct the Dirichlet polyhedron, we only need the minimal distance generators (and maybe the first few positive powers) of the first few cyclic groups $\Gamma_{ij}$ and their inverses.
In fact, for each $g \in \Gamma$ consider the semi–space $$H_g = \{ {\mathbf{x}} \in \widetilde{M} \; : \; d({\mathbf{0}},{\mathbf{x}}) \leq
d(g{\mathbf{0}},{\mathbf{x}}) \} \; .$$ Then it is obvious that the Dirichlet polyhedron is the intersection of all of these semi–spaces. However, there is a high redundancy here, since for a sufficiently large positive power $n$, we may have $$\bigcap_{k=1}^{n-1} H_{g_{ij}^k} \subset H_{g_{ij}^n} \; ,$$ and so this and further powers of $g_{ij}$ do not effectively contribute to the polyhedron $\mathcal{D}_{{\mathbf{0}}}$. Additionally, if some $H_g$ effectively contributes to the polyhedron, so does $H_{g^{-1}}$, thus the same argument holds for the inverses of the minimal distance generators. Moreover, note that due to condition 3. above, for a sufficiently large $i$, it may be the case that the semi–spaces $H_{g_{ij}}$ do not contribute effectively to the polyhedron.
The faces of the Dirichlet polyhedron are subsets of the boundary planes of the semi–spaces effectively contributing to it. In fact, for each $H_g$ effectively contributing, the corresponding face is orthogonal to the geodesic joining ${\mathbf{0}}$ and $g{\mathbf{0}}$, and cuts it at its middle point. It follows that the decomposition (\[DecCyclic\]) describes the symmetries of the Dirichlet fundamental polyhedron of $M$ centered at the observer.
Flat homogeneous manifolds {#S:tori}
==========================
We have seen in the previous section that, to compute the topological signature of CMB temperature anisotropies in a given manifold, we just need to know how to compute it for the cyclic manifolds that cover it maximally. In the flat orientable case the cyclic manifolds are twisted cylinders, i.e., manifolds with covering group generated by a screw motion. We will now focus on the simplest case, the flat homogeneous manifolds, which are generated by translations only.
The flat homogeneous manifolds are 3–tori or $T^3$ manifolds (generated by three linearly independent translations), chimneys or $T^2$ manifolds (generated by two linearly independent translations), and cylinders or $T^1$ manifolds (generated by one translation). Thus, we first compute the correlation matrix $\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma}$ for cylinders, and then show how the decomposition (\[DecTSCyclic\]) is used to compute the topological signature in this matrix for two– and three–dimensional tori. We also show that the computation of the angular power spectrum in tori is greatly simplified by this decomposition.
The cylinder {#Ss:Cylinder}
------------
Let us consider a cylinder orthogonal to the $z$–direction, that is with covering group generated by the translation $g = (I, {\mathbf{a}})$, with ${\mathbf{a}} = L {\widehat{{\mathbf{e}}}_{z}}$, where distances are measured in units of the radius of the last scattering surface $R_{LSS}$. This choice of the coordinate system is very convenient since here the cylinder appears invariant under (i) arbitrary rotations around the $z$–axis, (ii) the parity transformation, and (iii) the reflection on the $y=0$ plane. Thus, according to Sec. \[Ss:Symmetry\], we will end with a real correlation matrix with no $m$–dependent correlations and the multipoles $\ell$ and $\ell'$ correlated only when both are even or odd.
It is convenient to recall here that our cylinder has injectivity radius equal to $L/2$, thus cylinders with $L<2$ are “small” and might present topological copies of discrete sources and/or circles in the sky. On the other hand, “large” cylinders, i.e., those with $L>2$, have undetectable topology with the methods currently available [@Detect].
The covering group of the cylinder is labeled by the integers as $g^n = (I,n
{\mathbf{a}})$, with $n \in \mathbb{Z}$. We then have from (\[DefTopSign\]) that all the topological information is encoded in $$\Upsilon^{\Gamma}_{\ell m} ({\mathbf{k}}) = \sum_{n \in \mathbb{Z}} e^{-i nk_zL}
\, Y_{\ell m}({\mathbf{n}}_{{\mathbf{k}}}) \; ,$$ and thus the correlation matrix of the $a_{\ell m}$’s for the cylinder is simply
$$\label{CorrMatCylGen}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma} = (4 \pi)^2 \, i^{\ell - \ell'} \int \!
\frac{d^3k}{k^3} \, \Psi_{\ell \ell'}(k) \left( \sum_{n \in \mathbb{Z}} e^{-i nk_zL}
\right) Y_{\ell' m'}({\mathbf{n}}_{{\mathbf{k}}}) \, Y_{\ell m}^*({\mathbf{n}}_{{\mathbf{k}}})\; .$$
To reduce this integral we may use any of the following two identities, either $$\label{FirstId}
\sum_{n \in \mathbb{Z}} e^{-i nk_zL} = 2\pi \sum_{p \in \mathbb{Z}} \delta(k_zL
- 2\pi p) \; ,$$ or $$\label{SecondId}
\sum_{n \in \mathbb{Z}} e^{-i nk_zL} = 1 + 2\sum_{n=1}^\infty \cos(nk_zL) \; .$$ The first identity is obvious since the left hand side is the Fourier expansion of the right hand side. This option yields a formula of the kind obtained in Appendix \[S:OldRes\], which expresses the correlation matrix in terms of the eigenvalues of the Laplacian operator on the cylinder. The second one still uses a parametrization in terms of the covering group, and thus can be used to isolate the topological signature.
(10,4.8)
In fact, using (\[SecondId\]) to evaluate (\[CorrMatCylGen\]), and integrating in spherical coordinates, we get $$\label{CorrMatCyl}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma} = \langle a_{\ell m} \,
a^*_{\ell' m'} \rangle^{s.c.} + \langle a_{\ell m} \, a^*_{\ell' m'}
\rangle^{\widehat{\Gamma}} \; ,$$ where the simply connected part is as usual $$\label{CorrMatSC}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{s.c.} = C_\ell^{s.c.}
\delta_{\ell \ell'} \delta_{mm'} \; ,$$ with the (simply connected) angular power spectrum given by $$\label{PowSpecSC}
C_\ell^{s.c.} = (4\pi)^2 \int_0^\infty \frac{dx}{x} \, \Psi_{\ell \ell}(x)
\; ,$$ and the topological signature for the correlation matrix is given by $$\begin{aligned}
\label{TopSignCyl}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\widehat{\Gamma}} & = & (4\pi)^2
\, i^{\ell - \ell'} \delta_{\ell \ell'}^{\mbox{\tiny mod(2)}} \, \delta_{mm'}
\times \\
& & \hspace{1cm} \int_0^\infty \frac{dx}{x} \, \Psi_{\ell \ell'} \left(
\frac{x}{L} \right) F_{\ell \ell'}^m(x) \; , \nonumber\end{aligned}$$ with $$\label{CylFllmInit}
F_{\ell \ell'}^m(x) = 2 \sum_{n=1}^\infty \int_{-1}^1 \! dy \, \cos(nxy)
\mathcal{P}_\ell^m(y) \mathcal{P}_{\ell'}^m(y) \; ,$$ where $\mathcal{P}_\ell^m(x)$ is the normalized associated Legendre function (see Appendix \[Ap:SphHarm\]). As expected, we have ended up with a real correlation matrix with factors $\delta_{\ell \ell'}^{\mbox{\tiny mod(2)}}$ and $\delta_{m m'}$.
After evaluating the series in (\[CylFllmInit\]), it turns out that $F_{\ell
\ell'}^m(x)$ is a piecewise continuous function. In fact, in each interval $[2\pi q, 2\pi (q+1)]$, it is a polynomial of degree $(\ell + \ell' + 1)$ in $\pi/x$. Indeed, the final result is $$\begin{aligned}
\label{CylFllmFin}
F_{\ell \ell'}^m(x) & = & \sum_{q \in \mathbb{Z}} \mathcal{F}_{\ell
\ell'}^m(x,q) \, \Theta(x - 2\pi q) \times \\
& & \hspace{2.5cm} \Theta(2\pi(q+1) - x) \; , \nonumber\end{aligned}$$ where $\Theta(x)$ is the Heaviside step function, and the form of the polynomial $\mathcal{F}_{\ell \ell'}^m(x,q)$ in the $q$–th interval of length $2\pi$ is $$\begin{aligned}
\label{PolyqCylFllmFin}
\mathcal{F}_{\ell \ell'}^m(x,q) & = & 4 \sum_{k=0}^{\frac{\ell + \ell'}{2}}
(-1)^k \mathcal{P}_{\ell \ell' m}^{(2k)}(0) \times \\
& & \hspace{1cm} g_{2k+1}(q) \, \left( \frac{\pi}{x} \right)^{2k+1} -
\delta_{\ell \ell'} \; . \nonumber\end{aligned}$$ Here $g_k(q)$ are polynomials of degree $k$ in $q$, and $\mathcal{P}_{\ell
\ell' m}^{(k)}(0)$ is the $k$–th derivative of the polynomial $$\label{PllmDef}
\mathcal{P}_{\ell \ell' m}(x) = \mathcal{P}_\ell^m(x) \mathcal{P}_{\ell'}^m(x)$$ evaluated at the origin. In Appendix \[Ap:Clausen\] we present recurrence relations for the polynomials $g_k(q)$, and all the technical steps that take us from (\[CylFllmInit\]) to (\[CylFllmFin\]) can be found in Appendix \[Ap:EvalFllm\].
(8,7.5)
The integrals appearing in the topological signature (\[TopSignCyl\]) can be easily evaluated since Eqs. (\[CylFllmFin\]) and (\[PolyqCylFllmFin\]) allow an exact and very fast computation of the function $F_{\ell
\ell'}^m(x)$, and the integrands decay very fast, as illustrated in figs.\[fig:fm22\] and \[fig:fm55\]. In this figures we have adopted, for simplicity, a scale invariant Einstein–de Sitter model, thus $$\label{PsiEdS}
\Psi_{\ell \ell'}(x) \propto j_\ell (x)j_{\ell'}(x) \; .$$ The nice behavior of the integrands in (\[TopSignCyl\]) is not a consequence of this particular choice of $\Psi_{\ell \ell'}(x)$. Actually, the integrand in (\[TopSignCyl\]) always decays very fast because $\Psi_{\ell \ell'}(x)$ and $F_{\ell \ell'}^m(x)$ are both decaying functions, thus the evaluation of the topological signature for the cylinder is always very efficient.
The computation of the topological signature of the power spectrum reduces to a simple integral. In fact we obtain $$C_\ell^{\widehat{\Gamma}} = (4\pi)^2 \int_0^\infty \frac{dx}{x} \, \Psi_{\ell
\ell} \left( \frac{x}{L} \right) f_\ell(x) \; ,$$ with $$f_\ell(x) = \frac{1}{2\ell + 1} \sum_{m=-l}^\ell F_{\ell \ell}^m(x) \; .$$ Using (\[CylFllmInit\]) to perform this sum, the Addition Theorem for Spherical Harmonics (see Appendix \[Ap:SphHarm\]) yields immediately $$\label{TopSignClFin}
C_\ell^{\widehat{\Gamma}} = 2(4\pi)^2 \int_0^\infty \frac{dx}{x^2} \,
\Psi_{\ell \ell} \left( \frac{x}{L} \right) \varphi_1(x) \; ,$$ where $\varphi_1(x)$ is the first Clausen $\varphi$–function given in Appendix \[Ap:Clausen\].
In fig.\[fig:senhalCl\_L\]a we show the low $\ell$–modes of the topological signature of the angular power spectrum of a cylinder, normalized w.r.t. $C_{\ell}^{s.c.}$, as a function of its size $L$. We can see that the topological signature is typically much smaller than the cosmic variance, even for small cylinders which have already been discarded observationally as candidates for the shape of our Universe because of the lack of antipodal matched circles in WMAP data [@Aligne1; @CSSK04]. Thus it is apparent that the angular power spectrum is not a good indicator to look for topology in this case.
The correlation matrix given by (\[CorrMatCyl\])–(\[PolyqCylFllmFin\]) corresponds to a cylinder for which the direction of compactification is parallel to the $z$–axis. The correlation matrix corresponding to a cylinder with a different orientation can be easily obtained from the previous one by simply rotating the celestial sphere. Thus, parametrizing the rotations with Euler angles, if $R(\alpha, \beta, \gamma) \in SO(3)$ takes the $z$–axis to the direction of compactification of the cylinder, the topological signature of the corresponding correlation matrix can be computed using the expressions (\[a\_lmRot\])–(\[Wigner-Euler\]) of Appendix \[Ap:SphHarm\] yielding $$\begin{aligned}
\label{CorrMatCylRot}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\widehat{\Gamma}}_R & = &
e^{i(m'-m) \alpha} \times \\
& & \hspace{0.5cm} \sum_{m_1} d_{mm_1}^\ell(\beta) \,
d_{m'm_1}^{\ell'}(\beta) \times \nonumber \\
& & \hspace{2cm} \langle a_{\ell m_1} \, a^*_{\ell' m_1}
\rangle^{\widehat{\Gamma}} \; , \nonumber\end{aligned}$$ since $\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{s.c.}$ is rotationally invariant. Moreover, the $\gamma$ angle does not appear in this expression since $R_z(\gamma)$ in (\[Euler\]) does not move the $z$–axis, and $\langle
a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma}$ is invariant under such rotations.
It should be noted here that, no matter its orientation, the cylinder is always invariant under parity, thus its correlation matrix will always conserve the factor $\delta_{\ell \ell'}^{\mbox{\tiny mod(2)}}$. On the other hand, the correlation matrix will remain real as far as we perform rotations with $\alpha = 0$, since in this case we do not rotate the cylinder around the $z$–axis, and thus it remains invariant under reflections on the plane $y=0$. However, any rotation of the cylinder (other than one with $\beta = \pi$) makes it non–invariant under azimuthal rotations, thus the correlation matrix of an arbitrarily oriented cylinder has $m$–dependent correlations. All these features can be seen explicitly in (\[CorrMatCylRot\]).
tori {#Ss:tori}
----
In order to calculate the correlation matrix of the $a_{\ell m}$’s for a two– or a three–torus we use the decomposition (\[DecCyclic\]) of its covering group in cyclic subgroups. Let $\Gamma_{ij} = \langle g_{ij} \rangle$ be the covering group of the cylinder generated by the element $g_{ij} \in \Gamma$, and let us write $L_i = d({\mathbf{0}}, g_{ij} {\mathbf{0}})$, $g_i = (I, L_i
{\widehat{{\mathbf{e}}}_{z}})$, and $\Gamma_i = \langle g_i \rangle$. In the Euclidean case, the orientation preserving isometries that do not move the observer are rotations, thus let $R_{ij} \in SO(3)$ be the rotation taking ${\widehat{{\mathbf{e}}}_{z}}$ to the unit vector along $g_{ij} {\mathbf{0}}$.
(6,5.3)
Using the decomposition (\[DecTSCyclic\]), we easily write the topological signature for the torus as a superposition of topological signatures of rotated cylinders. In fact, $$\label{TopSignTorusDef}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\widehat{\Gamma}} =
\sum_{i=1}^\infty \sum_{j=1}^{k_i} \langle a_{\ell m} \, a^*_{\ell' m'}
\rangle^{\widehat{\Gamma}_i}_{R_{ij}} \; ,$$ where the correlation matrices of the rotated cylinders are written in terms of the Wigner $D$–functions and Euler angles, according to (\[CorrMatCylRot\]), as $$\begin{aligned}
\label{TopSignTorusAux}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\widehat{\Gamma}_i}_{R_{ij}}
& = & e^{i(m'-m) \alpha_{ij}} \times \\
& & \sum_{m_1} d_{mm_1}^\ell(\beta_{ij}) \, d_{m'm_1}^{\ell'}(\beta_{ij})
\times \nonumber \\
& & \hspace{2cm} \langle a_{\ell m_1} \, a^*_{\ell' m_1}
\rangle^{\widehat{\Gamma}_i} \; , \nonumber\end{aligned}$$ and $(\beta_{ij}, \alpha_{ij})$ are the angular spherical coordinates of the vector $g_{ij} {\mathbf{0}}$, and $k_i$ is the number of cylinders of size $L_i$. Since any group of translations is invariant under parity, from Sec. \[Ss:Symmetry\] we know that the correlation matrix for a homogeneous flat manifold has always the factor $\delta_{\ell \ell'}^{\mbox{\tiny mod(2)}}$, and this is evident from (\[TopSignTorusDef\]), since it is just a sum of correlation matrices of cylinders.
The power spectrum is rotationally invariant, thus from (\[TopSignTorusDef\]) one can easily write down the expression for the topological signature of the power spectrum of the torus as a superposition of topological signatures of power spectra of cylinders, $$\label{ClTorus}
C_\ell^{\widehat{\Gamma}} = \sum_{i=1}^\infty k_i \,
C_\ell^{\widehat{\Gamma}_i} \; .$$
Let us consider a chimney with square base for the sake of illustration. It is convenient to orient the chimney so that its covering group consists of translations in the horizontal plane. We take as generators of the covering group the translations $g_1 = (I, {\mathbf{a}})$ and $g_2 = (I, {\mathbf{b}})$, with ${\mathbf{a}} = L {\widehat{{\mathbf{e}}}_{x}}$ and ${\mathbf{b}} = L {\widehat{{\mathbf{e}}}_{y}}$.
It is more convenient to reparametrize the cyclic decomposition as follows. Parametrize each cyclic subgroup by a pair of integer numbers $(p,q)$ as $G_{pq} = \langle g_2^q g_1^p \rangle$. Clearly, if the greatest common divisor of $(p,q)$ is $r$, then $$G_{pq} < G_{\frac{p}{r} \frac{q}{r}} \; ,$$ where ‘$<$’ means ‘subgroup of’. Thus we must restrict the labels to pairs $(p,q)$ of coprime numbers.
The only exceptions are when (i) $p=\pm 1$ and $q=0$ and viceversa, and (ii) when $p=\pm 1$ and $q=\pm 1$. Thus the first two complete sets of cyclic subgroups conjugate by a rotation are $\{ G_{1,0} , G_{0,1} \}$ and $\{
G_{1,1} , G_{-1,1} \}$. In both cases the conjugation is performed by a rotation of $\pi/2$ around the $z$–axis. The compactification lengths of the corresponding cylinders are $L_{1,0} = L_{0,1} = L$ and $L_{1,1} = L_{-1,1} =
\sqrt{2} L$ respectively. The Euler angles $(\beta, \alpha)$ to rotate the corresponding cylinders from the $z$–axis to their orientation in the chimney, according to (\[TopSignTorusAux\]), are $\beta = \pi/2$ in all cases, and $\alpha_{1,0} = 0$, $\alpha_{0,1} = \pi/2$, $\alpha_{1,1} = \pi/4$ and $\alpha_{-1,1} = 3\pi/4$, respectively.
To write the remaining complete sets of cyclic subgroups conjugate by a rotation let us define, for a pair of coprime natural numbers $(p,q)$, with $p>q\geq1$, the groups $$\begin{aligned}
G_{pq}^{(1)} = G_{pq} = \langle g_2^q g_1^p \rangle & , & G_{pq}^{(3)} =
G_{-q,p} = \langle g_2^p g_1^{-q} \rangle \; , \\
G_{pq}^{(2)} = G_{qp} = \langle g_2^p g_1^q \rangle & , & G_{pq}^{(4)} =
G_{-p,q} = \langle g_2^q g_1^{-p} \rangle \; .\end{aligned}$$ The compactification lengths are all equal to $L_{pq} = \sqrt{p^2+q^2}L$, and the Euler angles $(\beta, \alpha)$ to rotate the corresponding cylinders from the $z$–axis to their orientation in the chimney, according to (\[TopSignTorusAux\]), are $\beta = \pi/2$ in all cases, and $$\begin{aligned}
\alpha_{pq}^{(1)} = \arctan \frac{q}{p} & , & \alpha_{pq}^{(3)} =
\frac{\pi}{2} + \alpha_{pq}^{(1)} \; , \\
\alpha_{pq}^{(2)} = \frac{\pi}{2} - \alpha_{pq}^{(1)} & , & \alpha_{pq}^{(4)}
= \pi - \alpha_{pq}^{(1)} \; ,\end{aligned}$$ respectively.
Let us denote by $\Gamma_{pq}$ the covering group of the cylinder with compactification scale $L_{pq}$ and oriented along the $z$–axis. Then, putting all this together, using (\[TopSignTorusDef\]) and (\[TopSignTorusAux\]), and taking into account the invariance properties derived in Sec. \[Ss:Symmetry\], the topological signature of the chimney with square base is $$\begin{aligned}
\label{TopSignChim}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\widehat{\Gamma}} & = &
\delta_{mm'}^{\mbox{\tiny mod(4)}} \sum_{m_1} d_{mm_1}^\ell(\pi/2) \times \\
& & \hspace{1.5cm} d_{m'm_1}^{\ell'}(\pi/2) \, \mathcal{W}_{\ell \ell'
m_1}^{m' - m} \; , \nonumber\end{aligned}$$ with
$$\label{WTopSignChim}
\mathcal{W}_{\ell \ell' m_1}^m = 2 \left( \langle a_{\ell m_1} \, a^*_{\ell'
m_1} \rangle^{\widehat{\Gamma}_{1,0}} + (-1)^{m/4} \langle a_{\ell m_1} \,
a^*_{\ell' m_1} \rangle^{\widehat{\Gamma}_{1,1}} \right) + 4 \sum_{(p,q)}
\cos m \alpha_{pq}^{(1)} \, \langle a_{\ell m_1} \, a^*_{\ell' m_1}
\rangle^{\widehat{\Gamma}_{pq}} \; ,$$
where the sum in $(p,q)$ is evaluated only for pairs of coprime natural numbers $(p,q)$ such that $p>q\geq1$.
(8,10.1)
The topological signature of the power spectrum of the chimney with square base is simply $$\label{PowSpecChim}
C_\ell^{\widehat{\Gamma}} = 2 \left( C_\ell^{\widehat{\Gamma}_{1,0}} +
C_\ell^{\widehat{\Gamma}_{1,1}} \right) + 4 \sum_{(p,q)}
C_\ell^{\widehat{\Gamma}_{pq}} \; .$$ Since the topological signature of the power spectrum of a cylinder converges quickly to zero as a function of the compactification scale (see fig.\[fig:senhalCl\_L\]a), it follows that the sum in (\[PowSpecChim\]) also converges quickly.
Moreover, the topological signature of the power spectrum of the chimney is larger than that of the cylinder. This is so because the $\ell$-th mode of the topological signature of the angular power spectrum of the cylinder oscillates very slowly. Thus from (\[PowSpecChim\]) this signature is slightly higher in the chimney, as can be seen in fig.\[fig:senhalCl\_L\]b. Actually, this is a general result that holds for manifolds whose covering groups are not cyclic.
Patterns of alignment {#S:TopSign}
=====================
The nondiagonal character of the topological signature of the correlation matrix of the $a_{\ell m}$’s in multiply connected universes and their $m$–dependence are manifestations of their globally anisotropic nature. They manifest themselves in statistically anisotropic temperature maps, i.e., realizations of random temperature fluctuations for which mean values of functions of the temperature over ensembles of universes depend on the orientation [@StatAnis].
In this section we analyze an expected consequence of the topology of space on the temperature anisotropies of the CMB that has not received the deserved attention up to the present, namely the existence of preferred directions in space. We show that the decomposition of the topological signature of the correlation matrix of the $a_{\ell m}$’s in a universe with a complex topology, in signatures corresponding to cyclic topologies, demands the existence of “patterns of alignments” along these directions. For the sake of simplicity we consider the Einstein–de Sitter model, thus from now on we will take (\[PsiEdS\]) to perform all our calculations.
We want to call attention to the existence of alignments of the low $\ell$–modes of the CMB temperature maps in multiply connected universes. Indeed, in fig.\[fig:mapas\] we show a low resolution temperature map simulation for a cylinder with $L=2$ (in units of $R_{LSS}$), together with the maps corresponding to the first four $\ell$–modes. One can see that these $\ell$–maps present alignments along the $z$–direction, which in this case is the unique direction of compactification of space.
(8,8.4)
Similar alignments as those present in our simulations have been reported as being observed in WMAP data, and have been attributed to a possible nontrivial topology of space with the shape of a cylinder [@Aligne1]. These models have been quickly abandoned due to the lack of circles in the sky which should be present if the Universe were small [@Aligne1; @CSSK04]. However our simulations show that even in universes slightly larger, and so not presenting such circles, these alignments should still be observable. Thus whether these observed alignments are a consequence of a nontrivial shape of our Universe is still an open question [@LocShape].
We will show here that if our Universe had a nontrivial topology, its CMB temperature map will present characteristic patterns of alignment, even if its size is somewhat larger than the observable universe. Moreover, from the observed patterns of alignment, we might be able to reconstruct the shape of space.
In fig.\[fig:alm\] we show the topological signature of the low $\ell$–modes of the diagonal part of the correlation matrix of the $a_{\ell m}$’s, normalized w.r.t. $C_{\ell}^\Gamma$, for a cylinder oriented along the polar axis, as a function of the size $L$ of compactification. It is apparent that, for a given $\ell$–mode, there are multipole coefficients for which their expected values are above the mean (the angular power spectrum), and others for which these expected values are below it. This is the reason why the low $\ell$–modes in a cylinder are aligned. Actually, the expectation values $\langle |a_{\ell m}|^2 \rangle$ are all equal to $C_\ell$ only in the simply connected case, thus the dispersion around the mean $$\label{Dispalm}
\sigma_\ell = \sqrt{\frac{1}{2 \ell + 1} \sum_m \left( \langle |a_{\ell m}|^2
\rangle - C_\ell \right)^2}$$ is null. However, in a multiply connected universe, this dispersion is non zero, and in a particular map, it adds to the cosmic variance. Thus it seems natural to propose the dispersion of the squares $|a_{\ell m}|^2$ around their mean value as a measure of these alignments in a map.
(5,5.1)
In fig.\[fig:dispcyl\] we show a plot of the dispersion (\[Dispalm\]), normalized w.r.t. $C_\ell^\Gamma$, for a cylinder oriented along the polar axis, as a function of $L$, for low multipoles. Note that even for a large cylinder ($L \approx 2$) the dispersion is larger than $15 \%$ of the power for multipoles up to $\ell = 5$. Indeed, on these scales the dispersion is of the order of the cosmic variance, and thus might be detectable.
(8,8.5)
In order to show that this is a good measure of the alignment of multipoles, and that it provides an efficient method to determine the directions of possible alignments in real or simulated maps, let us compute the dispersion of the squares $\langle |a_{\ell m}|^2 \rangle$, Eq. (\[Dispalm\]), for a cylinder which is oriented along a direction making an angle $\beta$ with the $z$–axis. Each one of these squares can be computed with $$\langle |a_{\ell m}|^2 \rangle^{\Gamma}_R = \sum_{m_1} \left[ d_{mm_1}^\ell(\beta)
\right]^2 \langle |a_{\ell m_1}|^2\rangle^{\Gamma} \; ,$$ which is nothing but (\[CorrMatCylRot\]) restricted to the diagonal part.
In fig.\[fig:dispcylRot\] it is shown this dispersion as a function of $\beta$ for different multipole coefficients and for different values of $L$. One can see that the dispersion has a maximum when the cylinder is oriented along the $z$–axis. Thus in order to look for the alignments in a hypothetical universe with the shape of a cylinder, one should just rotate the celestial sphere around different directions until find those two opposite ones along which the dispersion of the squares $|a_{\ell m}|^2$ is maximum.
However, in order to collect definitive evidence that the universe is indeed a cylinder, one should map the dispersion of the squares $|a_{\ell m}|^2$ on the sphere for each $\ell$–mode, i.e. one should determine the dispersion (\[Dispalm\]) as a function of the orientation of the celestial sphere. If the universe had the topology of a cylinder, these dispersion maps should be axially symmetric around a special direction, where the dispersion is maximum. Moreover, this direction should be identified with the direction of compactification of the cylinder.
If the universe has the topology of a flat homogeneous manifold note, from (\[TopSignTorusDef\]) and (\[TopSignTorusAux\]), that the topological signature is a superposition of rotated cylinders of different sizes. Thus a CMB map for a universe with this kind of topology might present alignments along the directions corresponding to these cylinders. In fact, rotating the celestial sphere and computing the dispersion of the squares $|a_{\ell m}|^2$, an easy computation shows that whenever we perform the rotation $R(0,-\theta,-\varphi)$, with $\theta = \beta_{ij}$ and $\varphi =
\alpha_{ij}$, one has the cylinder labeled by $(i,j)$ oriented along the polar axis, and thus *dispersion maps might present local maxima along these directions*.
Whether these local maxima are observable in a given dispersion map will depend on (i) the scale of compactification of the corresponding cylinder $L_i$, (ii) the background due to the simply connected part, and (iii) the other cylinders’ topological signatures. For large values of $L_i$ the corresponding local maxima will not be observable, however one can expect those maxima corresponding to the smaller cylinders to be detectable. The existence and distribution of these maxima in each dispersion map, together with their relative intensities is what we call a *pattern of alignment*.
It might seem that the problem of constructing dispersion maps for manifolds that are not flat homogeneous is more involved, since general cyclic manifolds do not have axial symmetry as the cylinder has. Eq. (\[CorrMatCylRot\]) depends on two angles only because the cylinder is axially symmetric, but in the general case the expression for the correlation matrix in a rotated frame depends on the three Euler angles. Thus it seems at first sight that in these cases, a dispersion map should be a function on the 3–sphere. Fortunately, the diagonal elements of the rotated correlation matrix depend only on the last two Euler angles as $$\begin{aligned}
\langle |a_{\ell m}|^2 \rangle^{\Gamma}_R & = & \sum_{m_1,m_2} e^{i(m_2 -
m_1)\gamma} \times \\
& & \hspace{1.5cm} d_{mm_1}^\ell(\beta) d_{mm_2}^\ell(\beta) \times \\
& & \hspace{3.5cm} \langle a_{\ell m_1} a_{\ell m_2}^* \rangle^{\Gamma} \; ,\end{aligned}$$ thus the same conclusion holds in the general case. *Dispersion maps on the 2–sphere for low $\ell$–modes should display patterns of alignment showing the symmetries of our Universe if it has a (not too large) nontrivial topology*.
Discussion {#S:Discuss}
==========
In order to study systematically the effects of a nontrivial spatial topology in the temperature fluctuations of the CMB, we need to have the ability to simulate efficiently temperature maps in multiply connected $\Lambda$CDM cosmologies. Almost all the usual methods to perform these simulations use explicitly the solutions of the Helmholtz equation in 3–manifolds with nontrivial topology. The computation of the eigenfunctions and eigenvalues of the Laplacian operator is simple only in Euclidean manifolds, while in spherical and hyperbolic spaces it is a nontrivial problem. In fact, it is only recently that an analytical computation has been achieved for all the spherical manifolds. The hyperbolic cases still have to be done numerically.
In this paper we have developed a simulation procedure that avoids the explicit use of the solutions of the Helmholtz equation. Instead, our results are expressed in terms of the covering group $\Gamma$ of the corresponding manifold. In this section we summarize the details of the method, its efficiency, the simple applications performed here, and discuss future related work.
The formalism {#Ss:Formal}
-------------
The cornerstone of our method is formula (\[two-point\]), which is the two–point correlation function of the scalar perturbations in a multiply connected universe expressed in terms of the covering group of the manifold [@BPS]. By means of simple formal manipulations we obtain an expression for the correlation matrix of the spherical harmonic expansion coefficients of the temperature maps, Eqs. (\[MCCorr\])–(\[DefTopSign\]), which contain all the topological information expressed as a sum over the elements of the covering group.
Former applications of (\[two-point\]) required a regularization procedure in order to account for divergences of the series, as well as some resummation techniques for accelerating the convergence. We do not have these problems here because the divergent series, which are actually distributions, appear only inside integrals. Indeed, on the one hand, we show in Appendix \[S:OldRes\] that our formalism easily reproduces results previously reported in the literature, as well as some simple generalizations, without the need of any regularization procedure. On the other hand, elementary decompositions of the two–point correlation function (\[two-point\]), shown in Sec. \[S:CycDec\], guarantee that our final expressions are highly convergent, as discussed below.
Two decompositions of a generic covariance function which can be written as a sum over the covering group are crucial for the efficiency of our formalism. The first one, a trivial decomposition given by (\[TopSignGen\]), defines the topological signature of the covariance function. When written for the correlation matrix of the harmonic expansion coefficients, it yields the topological signature in the temperature anisotropy maps, as illustrated for the cylinder by (\[CorrMatCyl\]). This expression shows that the topological signature is nothing but a “perturbation” of the correlation matrix corresponding to the simply connected case. Since, as discussed in Sec. \[Ss:Cylinder\], these “perturbations” are small, the efficiency of the calculation follows.
The second decomposition given by (\[DecTSCyclic\]) allows us to write the topological signature of any manifold in terms of the topological signatures of its maximal covering manifolds with cyclic covering groups. The example of the tori illustrates the power of this approach, since we can write down explicit formulae for a general torus whether its generating translations are orthogonal and/or equal. Trying to do this with the explicit use of eigenfunctions of the Laplacian (or with the method used in Appendix \[S:OldRes\]) turns out to be tedious if not difficult.
Moreover, by construction, this decomposition is invariant under the symmetries of the manifold, thus it carries information on how shall these symmetries manifest in individual CMB temperature anisotropy maps, as will be discussed in Sec. \[Ss:TopSign\].
Another advantage of this second decomposition is the simplicity for writing down the power spectrum for complicated manifolds. Expressions like (\[ClTorus\]) and (\[PowSpecChim\]) are computationally very efficient once we have saved the power spectrum for cyclic manifolds as a function of its scale of compactification, since we have just to perform a weighted sum of power spectra for cyclic manifolds at different scales considering the multiplicity of the decomposition.
Topological signatures {#Ss:TopSign}
----------------------
A further advantage of splitting the correlation matrix of the multipole coefficients into its simply connected part and its topological signature is that we can identify very easily the geometric features of the signature. Although we have made explicit calculations for flat homogeneous manifolds only, qualitatively these results are general.
Universes with cylindrical topology of size $L \approx 2$ present clear alignments of their low $\ell$–modes along the direction of compactification. A dispersion map of the squares $\langle |a_{\ell m}|^2 \rangle$, for a given low $\ell$, exhibits an axial symmetry around this direction, thus it reduces to a function of the polar angle. These dispersion maps are shown in fig.\[fig:dispcylRot\].
By decomposing the covering group $\Gamma$ in cyclic subgroups one can see that, whatever the shape of our Universe, and if it is not too large, dispersion maps (one for each individual low multipole) might show patterns of alignment. In the general case such maps are functions on the two–dimensional projective space or, by a lifting, on the 2–sphere. Although we have shown the existence of patterns of alignment explicitly only for homogeneous flat manifolds, it follows from the exposition of the general formalism that the same conclusions hold for any manifold of constant curvature. Thus, we propose the construction of these dispersion maps in the WMAP data, and so the search for patterns of alignment, as a new method for detecting a possible nontrivial topology of our Universe.
It is interesting to comment on some features relating Levin and collaborators’ proposal of pattern formation in CMB temperature maps and the results we present in this paper. The patterns proposed by Levin et al. [@LevinB; @LSS98] are due to individual eigenmodes ($k$–modes), the patterns we have identified here are due to multipole modes ($\ell$–modes). In either case the modes compete to form their patterns in a CMB temperature map, however the observable modes in a map on the sphere are the latter, since spherical harmonics form a base on the space of functions on the sphere. On the other side, the association between real space perturbations and angular temperature fluctuations requires some averaging over the $k$–modes [@Inoue03], thus these patterns appear mixed in a map and their observation might demand more elaborated techniques.
Further remarks and future research {#Ss:Future}
-----------------------------------
The formalism we have developed in this paper reveals new insights on the problem of characterizing the marks that topology leaves in CMB maps, and opens up new possibilities for developing further methods for unvealing the shape of our Universe. It makes explicit that the multipole alignments observed in COBE and WMAP full sky CMB temperature maps may be a manifestation of its global shape, provides details of the nature and features of these alignments, and gives at least one methodology to test this hypothesis. As a consequence, further work is much needed.
One line of further research is the implementation of our formalism in the spherical and hyperbolic cases. One way to do this requires first to identify the radial part of the fundamental solution (\[HelmRadAng\]) of the Helmholtz equation in the universal covering, as well as the analog of the “plane wave expansion” solution (\[SolHelmEuc\]) in these geometries, and to write the expansion of the corresponding “plane waves” in spherical harmonics as in (\[PlaneW\]). The difficult part seems to be expressing the “plane wave expansion” in a suitable form to reproduce the formal steps used in the Euclidean case.
Moreover we have to compute the topological signature of all other cyclic manifolds, in order to extend the computations to any quotient space that could be a candidate for the shape of space. We also need to include acoustic oscillations, and Doppler and finite width effects in $\Lambda$CDM models so that we could determine the relevant angular scales in Cosmic Topology, i.e. the angular scales at which the topological signatures appear. This is a crucial step in order to confront quantitatively the theory with real CMB maps in an efficient and rigurous way. An ultimate goal may be to implement all this methodology in a software package for public use.
The identification of the “topological signature” of the correlation matrix $\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma}$ also opens up a path for solving a problem raised by Riazuelo et al. in [@RULW04]. The correlation matrix for a multiply connected universe is non diagonal and, tipically, $m$–dependent. In fact, this is the source of the statistical anisotropy of the CMB in these universes. However, for very large manifolds this correlation matrix becomes effectively diagonal, and equal to that corresponding to the universal covering counterpart. A natural question raises up: at what typical scales does the correlation matrix “becomes diagonal”? In terms of our formalism this problem can be stated as finding the scales at where the topological signature becomes observationally negligible compared to the simply connected part. A closed analysis of the topological signature might give some answers to this and related questions. For example, establishing bounds on the integral in (\[TopSignCyl\]) might solve the problem for the cylinder.
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank Cristiane Camilo Hernandez for her unvaluable help with fig.\[fig:mapas\], Wanderson Wanzeller for his continuous help in computational issues, and Carlos Alexandre Wensche for showing us the papers [@Aligne1] which triggered our interest in this topic. We would also like to thank the Brazilian federal institutions CBPF and INPE for warm hospitality in several ocassions, and to the participants of the Seminar of Cosmic Topology held monthly at IFT and the Workshop New Physics from Space held every year in Campos do Jordão, São Paulo, where we had lots of opportunities to discuss this work at the several stages of its development during the last two years. W.S. Hipólito–Ricaldi acknowledges CAPES and G.I. Gomero aknowledges FAPESP (contract 02/12328-6) for financial support.
Spherical harmonics {#Ap:SphHarm}
===================
In order to be self contained and to set the notation used in the paper, in this appendix we present basic definitions, some useful formulae of spherical harmonic functions and Wigner rotation matrices, as well as some invariance properties of the correlation matrix $\langle a_{\ell m} \, a^*_{\ell' m'}
\rangle$ under coordinate transformations. For a complete treatment of spherical harmonic functions the reader can consult [@VMK].
Basic definitions {#Ss:BasicDef}
-----------------
Let us denote by ${\mathbf{n}} = (\theta,\varphi)$ a point in a 2–sphere parametrized in the usual spherical coordinates, then the spherical harmonic functions are defined as $$Y_{\ell m}({\mathbf{n}}) = \sqrt{\frac{2\ell + 1}{4\pi} \, \frac{(\ell - m)!}{(\ell + m)!}} \,
P_\ell^m(\cos \theta) \, e^{im \varphi} \; ,$$ where $$P_\ell^m(x) = (-1)^m \, \left( 1-x^2 \right)^{m/2} \frac{d^m}{dx^m} \, P_\ell(x)$$ are the associated Legendre functions with non–negative index $0 \leq m \leq
\ell$, and with $$P_\ell(x) = \frac{1}{2^\ell \, \ell !} \, \frac{d^\ell}{dx^\ell} \, \left( x^2
- 1 \right)^\ell$$ being the Legendre polynomials. The associated Legendre functions with negative index $m$ are defined by $$P_\ell^{-m}(x) = (-1)^m \, \frac{(\ell - m)!}{(\ell + m)!} \, P_\ell^m(x) \; .$$ Moreover, it is often convenient to introduce the normalized associated Legendre functions $$\mathcal{P}_\ell^m(x) = \sqrt{\frac{2\ell + 1}{2} \, \frac{(\ell - m)!}{(\ell + m)!}} \,
P_\ell^m(x) \; .$$
It can easily be seen that the Legendre polynomial $P_\ell(x)$ is an $\ell$–th degree polynomial of parity $\ell$, and thus the associated Legendre function $P_\ell^m(x)$ is a function of parity $\ell - m$. It follows that the function $\mathcal{P}_{\ell \ell' m}(x)$ defined in (\[PllmDef\]) is an $(\ell +
\ell')$–th degree polynomial of parity $\ell + \ell'$, and thus the expression for $F_{\ell \ell'}^m(x)$ in (\[CylFllmInit\]), which is evaluated in Appendix \[Ap:EvalFllm\], contains only even polynomials.
Spherical harmonics form a complete orthonormal set of functions on the sphere, thus their most common application is in the expansion of functions, like a CMB temperature anisotropy map, in multipoles as in (\[TempSphHarm\]), where the coefficients $a_{\ell m}$, called the multipole coefficients, are given by $$a_{\ell m} = \int_{\mathbb{S}^2} d \Omega \, \frac{\delta T}{T}({\mathbf{n}}) \,
Y_{\ell m}^*({\mathbf{n}}) \; .$$ Since the temperature map is a real function on the sphere, the multipole coefficients obey the constraint $a_{\ell m}^* = (-1)^m \, a_{\ell , -m}$.
A very useful formula is given by the Addition Theorem for Spherical Harmonics $$\label{AddFormula}
P_\ell ({\mathbf{n}} \cdot {\mathbf{n}}') = \frac{4\pi}{2 \ell + 1} \sum_{m = -
\ell}^\ell Y_{\ell m}({\mathbf{n}}) Y_{\ell m}^*({\mathbf{n}}') \; ,$$ which for the particular case ${\mathbf{n}} = {\mathbf{n}}'$ yields the identity $$\sum_{m = - \ell}^\ell \left[ \mathcal{P}_\ell^m(x) \right]^2 = \frac{2 \ell +
1}{2} \; .$$
Wigner $D$–functions {#Ss:Wigner}
--------------------
In several ocassions it is convenient to rotate the sphere and compute the multipole coefficients in this new coordinate system. This can be achieved by means of the Wigner $D$–functions which can be defined operationally as the functions $D_{mm_1}^\ell(R)$ such that, for any rotation $R \in SO(3)$, then $$Y_{\ell m}(R{\mathbf{n}}) = \sum_{m_1} D_{mm_1}^\ell(R) \, Y_{\ell m_1}({\mathbf{n}}) \; .$$ In this case, it can be shown that the multipole coefficients of the temperature anisotropy map in the rotated reference frame are $$\widetilde{a}_{\ell m} = \sum_{m_1} D_{mm_1}^{*\ell}(R) \, a_{\ell m_1} \; .$$ This expression can be used to compute the correlation matrix of the $a_{\ell
m}$’s in a rotated frame simply as $$\label{a_lmRot}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle_R = \hspace{-0.2cm} \sum_{m_1,
m_1'} \hspace{-0.2cm} D_{mm_1}^{*\ell}(R) D_{m'm_1'}^{\ell'}(R) \langle
a_{\ell m_1} a^*_{\ell' m_1'} \rangle \; .$$
The Wigner $D$–functions take a very simple form when we express the rotation matrix $R$ in terms of its Euler angles as $$\label{Euler}
R(\alpha, \beta, \gamma) = R_z(\alpha) \cdot R_y(\beta) \cdot R_z(\gamma) \; .$$ Indeed, for this decomposition we have $$\label{Wigner-Euler}
D_{mm'}^\ell (R(\alpha, \beta, \gamma)) = e^{i(m \alpha + m' \gamma)}
d_{mm'}^\ell(\beta) \; ,$$ where $d_{mm'}^\ell(\beta) = D_{mm'}^\ell(R_y(\beta))$ is a real matrix with the following symmetries $$\begin{aligned}
d_{mm'}^\ell(\beta) & = & (-1)^{m-m'} d_{m'm}^\ell(\beta) \; , \\
d_{mm'}^\ell(\beta) & = & d_{-m',-m}^\ell(\beta) \; , \\
d_{mm'}^\ell(\pi-\beta) & = & (-1)^{\ell - m'} d_{-m,m'}^\ell(\beta) \; , \\
d_{mm'}^\ell(-\beta) & = & (-1)^{m'-m} d_{m,m'}^\ell(\beta) \; .\end{aligned}$$ There exist several explicit and recursive formulae to compute these matrices (see [@VMK]). A very efficient recursive procedure can be found in [@BFB97]. The following formula will be enough to reproduce the results presented in this paper.
$$d_{mm'}^\ell(\beta) = \sqrt{(\ell + m)!(\ell - m)!(\ell + m')!(\ell - m')!} \,
\sum_k{(-1)^k \, \frac{\left( \cos \frac{\beta}{2} \right)^{2\ell-2k+m-m'} \,
\left( \sin \frac{\beta}{2} \right)^{2k-m+m'}}{k! (\ell + m-k)! (\ell - m'-k)!
(m'-m+k)!}} \; ,$$
where the sum in $k$ is evaluated whenever the arguments inside the factorials are non–negative.
Symmetry considerations {#Ss:Symmetry}
-----------------------
Some consequences of the symmetries of the quotient manifold on the invariance structure of the correlation matrix of the $a_{\ell m}$’s can be deduced directly from the transformation rules of the spherical harmonic functions under coordinate transformations. The results obtained in this way are formal, generic, and are very useful in practical computations. We end this Appendix by deducing the invariance properties the correlation matrix must have, given some symmetries of the corresponding quotient manifold. These invariance properties have been used in [@RULW04] to simplify the correlation matrix for the 3–torus, however we want to remark here that they are general and do not depend on the geometry of the universal covering space.
Let us begin with the invariance properties of $\langle a_{\ell m} \,
a^*_{\ell' m'} \rangle^{\Gamma}$ under rotations around the $z$–axis. Under a rotation $R_z(\alpha): \varphi \to \varphi + \alpha$, the function $Y_{\ell
m}({\mathbf{n}})$ transforms as $$Y_{\ell m}(R_z(\alpha) {\mathbf{n}}) = e^{im \alpha} Y_{\ell m}({\mathbf{n}}) \; .$$ As a consequence the transformation rules for the multipole coefficients of a CMB temperature map are of the form $\widetilde{a}_{\ell m} = e^{-im \alpha}
a_{\ell m}$, and so the correlation matrix transforms under this rotation as $$\label{PhiTransfCorrMat}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma}_{R_z(\alpha)} = e^{i(m'
- m) \alpha} \langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma} \; .$$
We extract two consequences out of (\[PhiTransfCorrMat\]). First, if the quotient space is invariant under a rotation of $\alpha = 2 \pi/s$ around the $z$–axis, then the correlation matrix must be zero unless $m = m'$ mod $s$. Second, if the quotient space is invariant under “any” rotation around the $z$–axis, the correlation matrix must be zero unless $m = m'$. In practice, if we take our coordinate system such that the fundamental polyhedron of the quotient manifold is oriented so that it is invariant under a $2\pi/s$ rotation around the polar axis, the correlation matrix will present a factor $\delta_{m m'}^{\mbox{\tiny mod(s)}}$, and correspondingly, if the orientation is such that the polyhedron is invariant under arbitrary rotations around the $z$–axis, the correlation matrix will present a factor $\delta_{m m'}$.
Let us now take a look at invariance under the inversion transformation $P :
{\mathbf{n}} \to - {\mathbf{n}}$. Under this transformation the spherical harmonic functions change as $$Y_{\ell m}(P {\mathbf{n}}) = (-1)^{\ell} Y_{\ell m}({\mathbf{n}}) \; ,$$ thus the multipole coefficients $a_{\ell m}$ change as $\widetilde{a}_{\ell m}
= (-1)^{\ell} a_{\ell m}$, and as a consequence the transformation rule for the correlation matrix is $$\label{InvTransfCorrMat}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma}_P = (-1)^{\ell + \ell'}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma} \; .$$ Thus, if the fundamental polyhedron is oriented such that it appears invariant under the parity transformation, the correlation matrix must be zero unless $\ell = \ell'$ mod 2, i.e., the correlation matrix will present a factor $\delta_{\ell \ell'}^{\mbox{\tiny mod(2)}}$.
To end this section, let us consider the reflection on the $y=0$ plane. This operation changes only the azimuthal angle as $P_y : \varphi \to - \varphi$, thus the transformation rule for the spherical harmonics are $$Y_{\ell m}(P_y {\mathbf{n}}) = Y_{\ell m}^*({\mathbf{n}}) \; ,$$ the multipole coefficients $a_{\ell m}$ change as $\widetilde{a}_{\ell m} =
a_{\ell m}^*$, and thus, the transformation rule for the correlation matrix is $$\label{y0TransfCorrMat}
\langle a_{\ell m} \, a^*_{\ell' m'} \rangle^{\Gamma}_{P_y} = \langle a_{\ell
m} \, a^*_{\ell' m'} \rangle^{* \Gamma} \; .$$ It immediately follows that if the fundamental polyhedron is oriented such that it appears invariant under the reflection on the $y=0$ plane, the correlation matrix must be real.
Clausen functions {#Ap:Clausen}
=================
In this appendix we briefly present some computational aspects of the theory of Clausen functions, as far as we need them for our purposes. Clausen functions are periodic functions of period $2\pi$. There are two kinds of Clausen functions, the $\varphi$–class and the $\psi$–class. Clausen $\varphi$–functions can be expressed in terms of polynomials, while Clausen $\psi$–functions involve higher transcendental functions, the so–called Clausen integrals. Fortunately, we are interested exclusively in the Clausen $\varphi$–functions, thus we will develop the details of the theory only for them. The Clausen $\varphi$–functions are defined as $$\begin{aligned}
\label{ClausenDefPhi}
\varphi_{2s-1}(x) & = & \sum_{n=1}^\infty \frac{\sin nx}{n^{2s-1}} \nonumber
\\ \\
\varphi_{2s}(x) & = & \sum_{n=1}^\infty \frac{\cos nx}{n^{2s}} \nonumber\end{aligned}$$ for $s=1,2,\dots$, and can be calculated recursively with the formulae, $$\begin{aligned}
\label{ClausenRecurse}
\varphi_{2s}(x) & = & \zeta(2s) - \int_0^x \varphi_{2s-1}(y) \, dy \nonumber
\\ \\
\varphi_{2s+1}(x) & = & \int_0^x \varphi_{2s}(y) \, dy \; , \nonumber\end{aligned}$$ where $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ is the Riemann Zeta function.
These recurrence relations are complemented by the initial condition $$\begin{aligned}
\label{FirstClausen}
\varphi_1(x) & = & \sum_{n=1}^\infty \frac{\sin nx}{n} \nonumber \\
& = & \frac{1}{2} \sum_{q \in \mathbb{Z}} \, [(2q+1)\pi -x]
\times \\
& & \hspace{1cm} \Theta(x - 2\pi q) \, \Theta(2\pi(q+1) - x) \; , \nonumber\end{aligned}$$ where $\Theta(x)$ is the Heaviside step function. Formula (\[FirstClausen\]) can be verified by computing the Fourier series of the second right hand side.
Since the Clausen functions are periodic of period $2 \pi$, we can write $$\begin{aligned}
\varphi_s(x) & = & \sum_{q \in \mathbb{Z}} \, f_s(x - 2 \pi q) \times \\
& & \hspace{1cm} \Theta(x - 2\pi q) \, \Theta(2\pi(q+1) - x) \; ,\end{aligned}$$ with $f_1(x) = \frac{1}{2} (\pi -x)$. The recurrence formulae (\[ClausenRecurse\]) yield the following expressions for the Clausen functions in the period $[0,2\pi]$, $$\begin{aligned}
\label{OddPerClaus}
f_{2s+1}(x) & = & \sum_{r=0}^{s-1} \frac{(-1)^r}{(2r+1)!} \, \zeta(2(s-r)) \,
x^{2r+1} + \\
& & \hspace{1.5cm} \frac{(-1)^s}{2} \left( \frac{\pi x^{2s}}{(2s)!} -
\frac{x^{2s+1}}{(2s+1)!} \right) \nonumber\end{aligned}$$ for $s=0,1,2,\dots$, and $$\begin{aligned}
\label{EvenPerClaus}
f_{2s}(x) & = & \sum_{r=0}^{s-1} \frac{(-1)^r}{(2r)!} \, \zeta(2(s-r)) \,
x^{2r} + \\
& & \hspace{1.5cm} \frac{(-1)^s}{2} \left( \frac{\pi x^{2s-1}}{(2s-1)!} -
\frac{x^{2s}}{(2s)!} \right) \nonumber\end{aligned}$$ for $s=1,2,3,\dots$.
From the definitions (\[ClausenDefPhi\]) we get $f_{2s+1}(\pi) = 0$, which can be used to obtain a recurrence formula for the Riemann Zeta function of even argument, $$\begin{aligned}
\label{RiemannEven}
\zeta(2s) & = & \sum_{r=1}^{s-1} \frac{(-1)^{r+1}}{(2r+1)!} \, \zeta(2(s-r))
\, \pi^{2r} - \\
& & \hspace{3.5cm} \frac{(-1)^s \, s}{(2s+1)!} \, \pi^{2s} \; . \nonumber\end{aligned}$$ Writing $\zeta(2s) = g_{2s}(0) \pi^{2s}$, and substituting this into (\[RiemannEven\]) we have $$g_{2s}(0) = \sum_{r=1}^{s-1} \frac{(-1)^{r+1}}{(2r+1)!} \, g_{2(s-r)}(0) -
\frac{(-1)^s \, s}{(2s+1)!} \; .$$ The convenience for introducing this notation will be apparent in what follows.
We will now seek for generalizations of the formulae (\[OddPerClaus\]) and (\[EvenPerClaus\]), i.e., we look for explicit expressions for the Clausen functions in the $q$–th interval $[2\pi q, 2\pi (q+1)]$. Since the Clausen functions satisfy the periodicity conditions $\varphi_{2s-1}(2\pi q) = 0$ and $\varphi_{2s}(2\pi q) = \zeta(2s)$, the recurrence relations (\[ClausenRecurse\]) can be rewritten in the form $$\begin{aligned}
\label{q-ClausenRecurse}
\varphi_{2s}(x) & = & \zeta(2s) - \int_{2\pi q}^x \varphi_{2s-1}(y) \, dy
\nonumber \\ \\
\varphi_{2s+1}(x) & = & \int_{2\pi q}^x \varphi_{2s}(y) \, dy \; , \nonumber\end{aligned}$$
Defining the polynomials $f_s^q(x) = f_s(x-2\pi q)$, we notice that $\varphi_s(x)$ coincides with $f_s^q(x)$ in the interval $[2\pi q, 2\pi (q+1)]$. This fact, and the expressions (\[q-ClausenRecurse\]), allow us to write recurrence formulae analog to (\[ClausenRecurse\]) for the polynomials $f_s^q(x)$ as follows $$\begin{aligned}
f_{2s}^q(x) & = & g_{2s}(q) \pi^{2s} - \int_0^x f_{2s-1}^q(y) \, dy \; ,
\nonumber \\
\label{q-PerClausRecurse}
& & \\
f_{2s+1}^q(x) & = & g_{2s+1}(q) \pi^{2s+1} + \int_0^x f_{2s}^q(y) \, dy \; ,
\nonumber\end{aligned}$$ where $$\begin{aligned}
g_{2s}(q) & = & g_{2s}(0) + \frac{1}{\pi^{2s}} \int_0^{2\pi q} f_{2s-1}^q(y)
\, dy \; , \nonumber \\
\label{q-gRecurse}
& & \\
g_{2s+1}(q) & = & - \frac{1}{\pi^{2s+1}} \int_0^{2\pi q} f_{2s}^q(y) \, dy
\; , \nonumber\end{aligned}$$ with initial conditions, given by the first Clausen function, $f_1^q(x) =
g_1(q) \pi - \frac{x}{2}$ and $g_1(q) = q + \frac{1}{2}$.
The expressions (\[q-PerClausRecurse\]) can be written in a unified way as $$f_s^q(x) = g_s(q) \pi^s - (-1)^s \int_0^x f_{s-1}^q(y) \, dy \; .$$ Using this expression we readily obtain the explicit formula, which is the generalization of (\[OddPerClaus\]) and (\[EvenPerClaus\]) we were looking for, $$\begin{aligned}
\label{q-PerClausRecurse2}
f_s^q(x) & = & \sum_{r=0}^{s-1} \frac{(-1)^{\mu(r,s)}}{r!} \, g_{s-r}(q)
\pi^{s-r} x^r - \\
& & \hspace{3.5cm} \frac{(-1)^{\mu(s,1)}}{2} \, \frac{x^s}{s!} \; , \nonumber\end{aligned}$$ where $$\mu(r,s) = \left \lfloor \frac{r}{2} + \frac{1+ (-1)^s}{4} \right \rfloor \; ,$$ and $\lfloor x \rfloor$ is the floor function of $x$, i.e., the largest integer smaller than $x$.
The expressions (\[q-gRecurse\]) can also be written in a unified way as $$g_s(q) = g_s(0) + \frac{(-1)^s}{\pi^s} \int_0^{2\pi q} f_{s-1}^q(y) \, dy \; ,$$ where $$g_s(0) = \left\{ \begin{array}{l@{\qquad}l}
\frac{\zeta(s)}{\pi^s} & \mbox{if $s$ is even} \\
0 & \mbox{if $s>1$ is odd} \; .
\end{array} \right.$$ From this we get the expression analogous to (\[q-PerClausRecurse2\])
$$\label{q-gRecurse2}
g_s(q) = g_s(0) + (-1)^s \left[ \sum_{r=1}^{s-1} (-1)^{\mu(r-1,s-1)} \,
\frac{2^r}{r!} \, g_{s-r}(q) \, q^r - (-1)^{\mu(s-1,1)} \, \frac{2^{s-1}}{s!}
\, q^s \right] \; .$$
The polynomials $g_s(q)$ can also be written in the canonical form $$\label{q-gExplicit}
g_s(q) = \sum_{k=0}^s A_k^s q^k \; ,$$ where the coefficients are given by $A_0^s = g_s(0)$, $$A_n^s = (-1)^s \sum_{r=1}^n (-1)^{\mu(r-1,s-1)} \frac{2^r}{r!} \,
A_{n-r}^{s-r}$$ for $0 < n < s$, and $$\begin{aligned}
A_s^s & = & (-1)^s \left[ \sum_{r=1}^{s-1} (-1)^{\mu(r-1,s-1)} \frac{2^r}{r!}
\, A_{s-r}^{s-r} \right. - \\
& & \hspace{3.5cm} \left. (-1)^{\mu(s-1,1)} \frac{2^{s-1}}{s!} \right] \; ,\end{aligned}$$ with initial conditions $A_0^1 = \frac{1}{2}$ and $A_1^1 = 1$. These coefficients are obtained by just introducing (\[q-gExplicit\]) into (\[q-gRecurse2\]) and collecting terms.
The function $F_{\ell \ell'}^m(x)$ {#Ap:EvalFllm}
==================================
In this Appendix we evaluate the function $F_{\ell \ell'}^m(x)$ given by (\[CylFllmInit\]). We first observe (see Appendix \[Ap:SphHarm\]) that the function $\mathcal{P}_{\ell \ell' m}(x)$, given by (\[PllmDef\]), is an even polynomial of $(\ell + \ell')$–degree. Thus, we begin by considering the integral $$I(\alpha) = \int_{-1}^1 P(y) \cos \alpha y \, dy \; ,$$ where $P(y)$ is an even analytical function. Integrating succesively by parts we get $$\begin{aligned}
\label{AlphaInt}
I(\alpha) & = & 2 \left[ \frac{\sin \alpha}{\alpha} \sum_{s=0}^\infty
\frac{(-1)^s}{\alpha^{2s}} P^{(2s)}(1) + \right. \\
& & \hspace{1.5cm} \left. \frac{\cos \alpha}{\alpha^2}
\sum_{s=0}^\infty \frac{(-1)^s}{\alpha^{2s}} P^{(2s+1)}(1) \right] \; ,
\nonumber\end{aligned}$$ where $P^{(k)}(x)$ is the $k$–th derivative of $P(x)$.
Making $\alpha = nx$ and $P(x) = \mathcal{P}_{\ell \ell' m}(x)$ in (\[AlphaInt\]), substituting (\[AlphaInt\]) in (\[CylFllmInit\]), and performing the sum in $n$ we get $$\begin{aligned}
\label{CylFllmClaus}
F_{\ell \ell'}^m(x) & = & 4 \sum_{s=0}^{\frac{\ell+\ell'}{2}} (-1)^s \left[
\frac{\mathcal{P}_{\ell \ell' m}^{(2s)}(1)}{x^{2s+1}} \, \varphi_{2s+1}(x)
\right. + \\
& & \hspace{2.5cm} \left. \frac{\mathcal{P}_{\ell \ell'
m}^{(2s+1)}(1)}{x^{2s+2}} \, \varphi_{2s+2}(x) \right] \; , \nonumber\end{aligned}$$ where $\varphi_k(x)$ is the $k$–th Clausen $\varphi$–function defined in Appendix \[Ap:Clausen\].
Since the Clausen functions are periodic functions of period $2\pi$, analytic in each period, it follows that $F_{\ell \ell'}^m(x)$ is a piecewise continuous function, analytic in each period as well. Thus we will now show how the explicit expression for $F_{\ell \ell'}^m(x)$, in the $q$–th interval $[2\pi
q, 2\pi (q+1)]$, given in (\[PolyqCylFllmFin\]), comes out.
Introducing the explicit form for the Clausen $\varphi$–functions (\[q-PerClausRecurse2\]), in the sum of (\[CylFllmClaus\]) yields a huge expresion, but a close inspection reveals that it is a polynomial in $\pi/x$. The independent term is simply $$\begin{aligned}
- \frac{1}{2} \sum_{s=0}^{\frac{\ell+\ell'}{2}} \frac{(-1)^s}{(s+1)!} \,
\mathcal{P}_{\ell \ell' m}^{(s)}(1) & = & - \frac{1}{4} \int_{-1}^1
\mathcal{P}_{\ell \ell' m}(x) \, dx \\
& = & - \frac{1}{4} \, \delta_{\ell \ell'} \; ,\end{aligned}$$ where the first equality can be deduced by writing the Taylor expansion of the integrand of the right hand side, and integrating. On the other hand, summing up all the coefficients of the $(r+1)$–th odd term, and proceeding as before, we have this term equal to $$(-1)^r \, \mathcal{P}_{\ell \ell' m}^{(2r)}(0) \, g_{2r+1}(q) \left(
\frac{\pi}{x} \right)^{2r+1} \; ,$$ while the $(r+1)$–th even term is equal to $$(-1)^r \, \mathcal{P}_{\ell \ell' m}^{(2r+1)}(0) \, g_{2r+2}(q) \left(
\frac{\pi}{x} \right)^{2r+2} \; ,$$ which by the parity of $\mathcal{P}_{\ell \ell' m}(x)$ is zero. Summing up all the terms we finally get (\[CylFllmFin\]) and (\[PolyqCylFllmFin\]).
Known results for closed flat 3–manifolds {#S:OldRes}
=========================================
In this section we briefly show how we can obtain the formulae for the correlation matrix of the $a_{\ell m}$’s and the angular power spectrum, currently available in the literature, for some closed flat manifolds, as well as a simple generalization, i.e., considering the observer out of the axis of rotations of the screw motions of the covering group. We present explicit derivations and formulae for the correlation matrix $\langle a_{\ell m}
a^*_{\ell' m'} \rangle$, as well as for the power spectrum, in order to allow the interested reader to perform their own simulations confidently.
We first give a brief description of flat orientable closed 3–manifolds and their covering groups. The versions of the diffeomorphic and isometric classifications of flat 3–manifolds we present here were given by Wolf in [@Wolf], and previous descriptions in the context of cosmic topology were given in [@Gomero] (see [@LevinA; @RWULL04] for alternative descriptions).
There are six diffeomorphic classes of compact orientable Euclidean 3–manifolds. The generators for the covering groups of the first five classes, ${\mathcal{G}_1}-{\mathcal{G}_5}$, are $\gamma_1 = (I, {\mathbf{a}})$, $\gamma_2 = (I,
{\mathbf{b}})$ and $\gamma_3 = (A_i, {\mathbf{c}})$, where $A_1 = I$ is the identity and $$\begin{aligned}
A_2 = \left( \begin{array}{ccc}
-1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{array} \right) & , &
A_4 = \left( \begin{array}{ccc}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array} \right) \; , \\
A_3 = \left( \begin{array}{ccc}
0 & -1 & 0 \\
1 & -1 & 0 \\
0 & 0 & 1
\end{array} \right) & , &
A_5 = \left( \begin{array}{ccc}
0 & -1 & 0 \\
1 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) \; ,\end{aligned}$$ for the classes ${\mathcal{G}_1}-{\mathcal{G}_5}$ respectively. It is important to remark that these matrices for the rotations are written in the basis formed by the set $\{{\mathbf{a}},{\mathbf{b}},{\mathbf{c}}\}$ of linearly independent vectors. Thus, the torus ${\mathcal{G}_1}$ is generated by three independent translations, while for the other manifolds the generators are two independent translations and a screw motion along a linearly independent direction. The manifold ${\mathcal{G}_6}$ is the most involved since their generators are all screw motions. In the following we present some general considerations concerning the classes ${\mathcal{G}_1}-{\mathcal{G}_5}$ only.
For space forms of the classes ${\mathcal{G}_2}-{\mathcal{G}_5}$, the following facts are easily derivable (see [@Gomero] for details):
1. The vector ${\mathbf{c}}$ is orthogonal to both ${\mathbf{a}}$ and ${\mathbf{b}}$.
2. \[angle\] The angle between ${\mathbf{a}}$ and ${\mathbf{b}}$ is a free parameter for the class ${\mathcal{G}_2}$, while its value is fixed to be $2\pi/3$, $\pi/2$ and $\pi/3$ for the classes ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ and ${\mathcal{G}_5}$ respectively.
3. Denoting by $|{\mathbf{c}}|$ the length of the vector ${\mathbf{c}}$, and similarly for any other vector, one has that $|{\mathbf{a}}| = |{\mathbf{b}}|$ for the classes ${\mathcal{G}_3}-{\mathcal{G}_5}$, while both lengths are independent free parameters in the class ${\mathcal{G}_2}$. Moreover, in all classes ${\mathcal{G}_2}-{\mathcal{G}_5}$, $|{\mathbf{c}}|$ is an independent free parameter.
4. Denoting the canonical unitary basis vectors in Euclidean space by $\{{\widehat{{\mathbf{e}}}_{x}},{\widehat{{\mathbf{e}}}_{y}},{\widehat{{\mathbf{e}}}_{z}}\}$, one can always write ${\mathbf{a}} =
|{\mathbf{a}}| \, {\widehat{{\mathbf{e}}}_{x}}$, ${\mathbf{b}} = |{\mathbf{b}}| \cos \varphi \, {\widehat{{\mathbf{e}}}_{x}}
+ |{\mathbf{b}}| \sin \varphi \, {\widehat{{\mathbf{e}}}_{y}}$, and ${\mathbf{c}} = |{\mathbf{c}}| \,
{\widehat{{\mathbf{e}}}_{z}}$, for the basis $\{{\mathbf{a}},{\mathbf{b}},{\mathbf{c}}\}$, where $\varphi$ is the angle between ${\mathbf{a}}$ and ${\mathbf{b}}$, as established in the item \[angle\].
Thus in dealing with manifolds of classes ${\mathcal{G}_2}-{\mathcal{G}_5}$, the axis of rotation of the generator screw motion can be taken to be the $z$–axis, and the orthogonal part of this generator, in the basis $\{{\widehat{{\mathbf{e}}}_{x}}, {\widehat{{\mathbf{e}}}_{y}},
{\widehat{{\mathbf{e}}}_{z}}\}$, is $$\label{Rot}
A = \left( \begin{array}{ccc}
\cos \alpha & - \sin \alpha & 0 \\
\sin \alpha & \cos \alpha & 0 \\
0 & 0 & 1
\end{array} \right) \; ,$$ with $\alpha=\pi$, $2\pi/3$, $\pi/2$ and $\pi/3$ respectively. Since the axis of rotation passes through the origin, the translational part of the generator $\gamma_3$ is ${\mathbf{c}} = (0,0,L_z)$, where we have put $|{\mathbf{c}}| = L_z$ as is usual in cosmic topology.
However, in cosmological applications we need to consider the arbitrariness of the position of the observer inside space. Thus if the axis of rotation is at a distance $\rho$ from the origin (the observer), and its intersection with the horizontal plane makes an angle $\phi$ with the positive $x$–axis, the translational part of the screw motion $\gamma_3 = (A,{\mathbf{c}})$ is $$\begin{aligned}
\label{transGen}
{\mathbf{c}} & = & \rho [\cos \phi -\cos(\phi + \alpha)] \, {\widehat{{\mathbf{e}}}_{x}} + \nonumber
\\
& & \hspace{0.3cm} \rho [\sin \phi - \sin(\phi + \alpha)] \, {\widehat{{\mathbf{e}}}_{y}} + L_z
\, {\widehat{{\mathbf{e}}}_{z}} \; .\end{aligned}$$
In order to perform calculations of the topological signature of CMB temperature maps, we need to write the covering group for the manifold under study in a compact form. For a torus ${\mathcal{G}_1}$ the problem is trivial, since the covering group is generated by three independent translations, and thus any two isometries commute (see Sec.\[Ss:G1\] below), while the covering groups for the other closed flat manifolds are noncommutative since they contain screw motions.
The generators of the covering groups for the classes ${\mathcal{G}_2}-{\mathcal{G}_5}$ satisfy certain relations of the form $$\gamma_3 \gamma_1^{n_1} \gamma_2^{n_2} = \gamma_1^{m_1} \gamma_2^{m_2}
\gamma_3 \; ,$$ where $n_1,n_2,m_1,m_2 \in \mathbb{Z}$, and they hold whether the axis of rotation passes through the origin or not. It follows that a generic isometry can always be put in the form $$\label{GenIso}
\gamma = \gamma_1^{n_1} \gamma_2^{n_2} \gamma_3^{n_3} \; ,$$ with $\gamma_1^{n_1} = (I,n_1 {\mathbf{a}})$, $\gamma_2^{n_2} = (I,n_2
{\mathbf{b}})$, and $$\gamma_3^{n_3} = (A^h, n_3 {\mathbf{c}}_\parallel + \mathcal{O}_h
{\mathbf{c}}_\perp) \; ,$$ where $A$ is given by (\[Rot\]), $\alpha = 2\pi/s$, $n_3 = sq+h$, with $q$ and $h$ integers such that $0 < h \leq s$, the parameter $s$ being $2,3,4$ and 6 corresponding to ${\mathcal{G}_2}$, ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ and ${\mathcal{G}_5}$ respectively, $${\mathcal{O}}_h = \sum_{j=0}^{h-1} A^j \; ,$$ ${\mathbf{c}}_\parallel = L_z {\widehat{{\mathbf{e}}}_{z}}$, and ${\mathbf{c}}_\perp = \rho [\cos \phi -
\cos(\phi + \alpha)] \, {\widehat{{\mathbf{e}}}_{x}} + \rho [\sin \phi - \sin(\phi + \alpha)] \,
{\widehat{{\mathbf{e}}}_{y}}$.
It is now straightforward to compute both the correlation matrix $\langle
a_{\ell m} a^*_{\ell' m'} \rangle$ and the angular power spectrum $C_\ell$. They all have a simple structure. We first describe the general procedure for obtaining these expressions and present the results in a unified form. We finally specify each case separately. Note that, due to (\[transGen\]), in all of our calculations we are considering that the observer may be off an axis of rotation of the screw motions of $\Gamma$.
Upon introducing (\[GenIso\]) into (\[DefTopSign\]), we transform the series of exponentials in a series of Dirac’s delta functions by using (\[FirstId\]). The integration of (\[MCCorr\]) is then immediate in Cartesian coordinates, the general result being $$\begin{aligned}
\langle a_{\ell m} a^*_{\ell' m'} \rangle & = & \frac{(4 \pi)^2}{V} \, i^{\ell
- \ell'} \sum_{{\mathbf{p}} \in \widehat{\mathbb{Z}}^3} \frac{1}{\beta^3} \,
\Psi_{\ell \ell'}(2\pi \beta) \times \\
& & \hspace{1cm} Y_{\ell' m'}({\mathbf{n}}_{\vec{\beta}})
Y_{\ell m}^*({\mathbf{n}}_{\vec{\beta}}) \, f_{m'}^\Gamma(2\pi \vec{\beta}) \; ,\end{aligned}$$ where $V$ is the volume of the manifold, and $\widehat{\mathbb{Z}}^3 =
\mathbb{Z}^3 \setminus (0,0,0)$, since the term corresponding to ${\mathbf{p}} =
0$ represents a constant perturbation, and thus is neglected.
The function $$f_{m}^\Gamma({\mathbf{k}}) = \frac{1}{s} \! \left[ 1 + \sum_{h=1}^{s-1}
\omega_s^{-hm} e^{-i {\mathbf{k}} \cdot {\mathcal{O}}_h {\mathbf{c}}} \right] \; ,$$ where $\omega_s$ is the first complex $s$th root of unity, is a complex modulating term characteristic of the geometry and topology of the spatial section of the universe model, and depends only on the screw motion generators. The vector $\vec{\beta}({\mathbf{p}})$ comes from the discretization of the wavevector ${\mathbf{k}}$ due to the Dirac’s deltas (each $2\pi \beta$ is an eigenvalue of the Laplacian operator), and ${\mathbf{n}}_{\vec{\beta}}$ is the unit vector in the direction of $\vec{\beta}$.
Using the property $\langle a_{\ell m} a^*_{\ell' m'} \rangle = \langle
a_{\ell' m'} a^*_{\ell m} \rangle^*$ one can easily show, by resumming the series, that the variances of the multipole moments can be put in the general form $$\begin{aligned}
\langle |a_{\ell m}|^2 \rangle & = & \frac{(4\pi)^2}{V} \sum_{{\mathbf{p}} \in
\widehat{\mathbb{Z}}^3} \frac{1}{\beta^3} \, \Psi_{\ell \ell}(2\pi \beta)
\times \\
& & \hspace{1.7cm} \left| Y_{\ell m}({\mathbf{n}}_{\vec{\beta}}) \right|^2 \,
\Re\left( f_{m}^\Gamma(\vec{2\pi \beta}) \right) \; ,\end{aligned}$$ where $\Re$ stands for the real part of a complex number. The angular power spectrum is then $$C_\ell = \frac{4\pi}{V} \sum_{{\mathbf{p}} \in \widehat{\mathbb{Z}}^3}
\frac{1}{\beta^3} \, \Psi_{\ell \ell}(2\pi \beta) \, \Xi_\ell(2\pi
\vec{\beta}) \; ,$$ where $$\Xi_\ell({\mathbf{k}}) = \frac{4\pi}{2\ell + 1} \sum_{m=-l}^\ell \left|
Y_{\ell m}({\mathbf{n}}_{{\mathbf{k}}}) \right|^2 \Re\left(
f_{m}^\Gamma({\mathbf{k}}) \right)$$ can be evaluated using the Addition Theorem for Spherical Harmonics yielding $$\Xi_\ell({\mathbf{k}}) = \frac{1}{s} \left[ 1 + \sum_{h=1}^{s-1} P_\ell(\cos
\theta_{{\mathbf{k}},h}) \cos({\mathbf{k}} \cdot {\mathcal{O}}_h {\mathbf{c}}) \right] \; ,$$ where $$\cos \theta_{{\mathbf{k}},h} = \cos^2 \theta_{{\mathbf{k}}} + \sin^2
\theta_{{\mathbf{k}}} \, \cos \frac{2\pi h}{s} \; .$$
Rectangular torus ${\mathcal{G}_1}$ {#Ss:G1}
-----------------------------------
The generators for the rectangular torus are the translations ${\mathbf{a}} = L_x
{\widehat{{\mathbf{e}}}_{x}}$, ${\mathbf{b}} = L_y {\widehat{{\mathbf{e}}}_{y}}$ and ${\mathbf{c}} = L_z {\widehat{{\mathbf{e}}}_{z}}$, thus a generic isometry of its covering group can be written as $\gamma =
(I,{\mathbf{r}})$, with ${\mathbf{r}} = n_xL_x {\widehat{{\mathbf{e}}}_{x}} + n_yL_y {\widehat{{\mathbf{e}}}_{y}} + n_zL_z
{\widehat{{\mathbf{e}}}_{z}}$, and $n_x,n_y,n_z \in \mathbb{Z}$, i.e., the covering group of ${\mathcal{G}_1}$ is parametrized by $\mathbb{Z}^3$.
It follows immediately that, for a rectangular torus, the expression (\[DefTopSign\]) takes the form $$\begin{aligned}
\Upsilon^{\Gamma}_{\ell m} ({\mathbf{k}}) & = & \sum_{{\mathbf{n}} \in
\mathbb{Z}^3} \! e^{-i(n_xk_xL_x + n_yk_yL_y + n_zk_zL_z)}
Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \\
& = & (2 \pi)^3 \sum_{{\mathbf{p}} \in \mathbb{Z}^3} \,
\delta(k_xL_x \! - \! 2 \pi p_x) \times \\
& & \delta(k_yL_y \! - \! 2 \pi p_y) \,
\delta(k_zL_z \! - \! 2 \pi p_z) Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \; .\end{aligned}$$ Following the procedure described above one gets $f_{m}^\Gamma({\mathbf{k}}) =
1$, and $\beta_x = \frac{p_x}{L_x}$, $\beta_y = \frac{p_y}{L_y}$, and $\beta_z
= \frac{p_z}{L_z}$. In particular, we have the well known result $$C_\ell = \frac{4\pi}{V} \sum_{{\mathbf{p}} \in \widehat{\mathbb{Z}}^3}
\frac{1}{\beta^3} \, \Psi_{\ell \ell}(2\pi \beta) \; .$$
Rectangular ${\mathcal{G}_2}$ {#Ss:G2}
-----------------------------
The generators for the rectangular ${\mathcal{G}_2}$ are $\gamma_1 = (I, {\mathbf{a}})$, $\gamma_2 = (I, {\mathbf{b}})$ and $\gamma_3 = (A, {\mathbf{c}})$, with ${\mathbf{a}}
= L_x {\widehat{{\mathbf{e}}}_{x}}$, ${\mathbf{b}} = L_y {\widehat{{\mathbf{e}}}_{y}}$, ${\mathbf{c}} = 2 \rho \cos \phi \,
{\widehat{{\mathbf{e}}}_{x}} + 2 \rho \sin \phi \, {\widehat{{\mathbf{e}}}_{y}} + L_z {\widehat{{\mathbf{e}}}_{z}}$, and $A$ given in (\[Rot\]) with $\alpha = \pi$. They satisfy the relations $\gamma_1 \gamma_3
\gamma_1 = \gamma_3$ and $\gamma_2 \gamma_3 \gamma_2 = \gamma_3$, which allow to write any isometry of the covering group by (\[GenIso\]) with $$\label{G2gamma3}
\gamma_3^{n_3} = \left\{ \begin{array}{lcc}
(I, n_3 {\mathbf{c}}_\parallel) & & \mbox{if $n_3$ is even} \\
(A, n_3 {\mathbf{c}}_\parallel + {\mathbf{c}}_\perp) & & \mbox{if $n_3$ is odd}
\end{array} \right. \; ,$$ where ${\mathbf{c}}_\perp = 2 \rho \cos \phi \, {\widehat{{\mathbf{e}}}_{x}} + 2 \rho \sin \phi \,
{\widehat{{\mathbf{e}}}_{y}}$.
It follows from (\[GenIso\]) and (\[G2gamma3\]) that the expression (\[DefTopSign\]) takes the form
$$\begin{aligned}
\Upsilon^{\Gamma}_{\ell m} ({\mathbf{k}}) & = & \sum_{{\mathbf{n}} \in
\mathbb{Z}^3} e^{-i(n_xk_xL_x + n_yk_yL_y + 2n_zk_zL_z)} \left[ 1 + (-1)^m
e^{-i {\mathbf{k}} \cdot {\mathbf{c}}} \right] \, Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \\
& = & (2 \pi)^3 \sum_{{\mathbf{p}} \in \mathbb{Z}^3} \, \delta(k_xL_x - 2 \pi
p_x) \, \delta(k_yL_y - 2 \pi p_y) \, \delta(k_zL_z - \pi p_z) \,
Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \, f_{m}^\Gamma({\mathbf{k}}) \; ,\end{aligned}$$
where we have put $n_1 = n_x$, $n_2 = n_y$, and $n_3 = 2n_z$ or $2n_z + 1$, depending on whether $n_3$ is even or odd. The components of $\vec{\beta}$ are $\beta_x = \frac{p_x}{L_x}$, $\beta_y = \frac{p_y}{L_y}$, and $\beta_z =
\frac{p_z}{2L_z}$.
${\mathcal{G}_3}$ {#Ss:G3}
-----------------
The generators for a manifold of class ${\mathcal{G}_3}$ are $\gamma_1 = (I,
{\mathbf{a}})$, $\gamma_2 = (I, {\mathbf{b}})$, and $\gamma_3 = (A, {\mathbf{c}})$, with ${\mathbf{a}} = L \, {\widehat{{\mathbf{e}}}_{x}}$, ${\mathbf{b}} = - \frac{L}{2} \, ({\widehat{{\mathbf{e}}}_{x}} -
\sqrt{3} \, {\widehat{{\mathbf{e}}}_{y}})$, ${\mathbf{c}} = \frac{\rho}{2} \, (3 \cos \phi + \sqrt{3}
\sin \phi) \, {\widehat{{\mathbf{e}}}_{x}} + \frac{\rho}{2} \, (3 \sin \phi - \sqrt{3} \cos \phi)
\, {\widehat{{\mathbf{e}}}_{y}} + L_z {\widehat{{\mathbf{e}}}_{z}}$, and $A$ given in (\[Rot\]) with $\alpha =
2\pi/3$. They satisfy the relations $\gamma_2^{-1} \gamma_3 \gamma_1 =
\gamma_3$ and $\gamma_1 \gamma_2 \gamma_3 \gamma_2 = \gamma_3$, which allow us to write any isometry of the covering group by (\[GenIso\]) with $$\label{G3gamma3}
\gamma_3^{n_3} = \left\{ \hspace{-0.1cm} \begin{array}{lcc}
(I, n_3 {\mathbf{c}}_\parallel) & & \hspace{-0.2cm} \mbox{if } n_3 = 0 \mbox{ mod
3} \\
(A, n_3 {\mathbf{c}}_\parallel + {\mathbf{c}}_\perp) & & \hspace{-0.2cm} \mbox{if }
n_3 = 1 \mbox{ mod 3} \\
(A^2, n_3 {\mathbf{c}}_\parallel + {\mathcal{O}}_2 {\mathbf{c}}_\perp) & & \hspace{-0.2cm}
\mbox{if } n_3 = 2 \mbox{ mod 3}
\end{array} \right. \; ,$$ where ${\mathbf{c}}_\perp = \frac{\rho}{2} \, (3 \cos \phi + \sqrt{3} \sin \phi)
\, {\widehat{{\mathbf{e}}}_{x}} + \frac{\rho}{2} \, (3 \sin \phi - \sqrt{3} \cos \phi) \,
{\widehat{{\mathbf{e}}}_{y}}$.
It follows from (\[GenIso\]) and (\[G3gamma3\]) that the expression (\[DefTopSign\]) takes the form
$$\begin{aligned}
\Upsilon^{\Gamma}_{\ell m} ({\mathbf{k}}) & = & \sum_{{\mathbf{n}} \in
\mathbb{Z}^3} e^{-i\left[n_xk_xL + n_y \left(\frac{\sqrt{3}}{2}k_y -
\frac{1}{2}k_x\right)L + 3n_zk_zL_z\right]} \left[ 1 + \omega_3^{-m} e^{-i
{\mathbf{k}} \cdot {\mathbf{c}}} + \omega_3^{-2m} e^{-i {\mathbf{k}} \cdot {\mathcal{O}}_2
{\mathbf{c}}} \right] Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \\
& = & (2 \pi)^3 \sum_{{\mathbf{p}} \in \mathbb{Z}^3} \, \delta(k_xL - 2 \pi p_x)
\, \delta \left( \left[ {\textstyle \frac{\sqrt{3}}{2}} k_y - {\textstyle
\frac{1}{2}} k_x \right]L - 2 \pi p_y \right) \, \delta \left( k_zL_z -
{\textstyle \frac{2\pi}{3}} \, p_z \right) Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \,
f_{m}^\Gamma({\mathbf{k}}) \; ,\end{aligned}$$
where we have put $n_1 = n_x$, $n_2 = n_y$, and $n_3 = 3n_z$, $3n_z + 1$ or $3n_z + 2$ according to (\[G3gamma3\]). We also get $\beta_x =
\frac{p_x}{L}$, $\beta_y = \frac{\sqrt{3}}{3L} \, (2p_y + p_x)$, and $\beta_z
= \frac{p_z}{3L_z}$.
${\mathcal{G}_4}$ {#Ss:G4}
-----------------
The generators for a manifold of class ${\mathcal{G}_4}$ are $\gamma_1 = (I,
{\mathbf{a}})$, $\gamma_2 = (I, {\mathbf{b}})$, and $\gamma_3 = (A, {\mathbf{c}})$, with ${\mathbf{a}} = L \, {\widehat{{\mathbf{e}}}_{x}}$, ${\mathbf{b}} = L \, {\widehat{{\mathbf{e}}}_{y}}$, ${\mathbf{c}} =
\rho \, (\cos \phi + \sin \phi) \, {\widehat{{\mathbf{e}}}_{x}} + \rho \, (\sin \phi - \cos \phi)
\, {\widehat{{\mathbf{e}}}_{y}} + L_z {\widehat{{\mathbf{e}}}_{z}}$, and $A$ given in (\[Rot\]) with $\alpha =
\pi/2$. They satisfy the relations $\gamma_2^{-1} \gamma_3 \gamma_1 =
\gamma_3$ and $\gamma_1 \gamma_3 \gamma_2 = \gamma_3$, which allow us to write any isometry of the covering group by (\[GenIso\]) with $$\label{G4gamma3}
\gamma_3^{n_3} = \left\{ \begin{array}{lcc}
(I, n_3 {\mathbf{c}}_\parallel) & & \mbox{if } n_3 = 0 \mbox{ mod 4} \\
(A, n_3 {\mathbf{c}}_\parallel + {\mathbf{c}}_\perp) & & \mbox{if } n_3 = 1
\mbox{ mod 4} \\
(A^2, n_3 {\mathbf{c}}_\parallel + {\mathcal{O}}_2 {\mathbf{c}}_\perp) & & \mbox{if }
n_3 = 2 \mbox{ mod 4} \\
(A^3, n_3 {\mathbf{c}}_\parallel + {\mathcal{O}}_3 {\mathbf{c}}_\perp) & & \mbox{if }
n_3 = 3 \mbox{ mod 4}
\end{array} \right. \; ,$$ where ${\mathbf{c}}_\perp = \rho \, (\cos \phi + \sin \phi) \, {\widehat{{\mathbf{e}}}_{x}} + \rho \,
(\sin \phi - \cos \phi) \, {\widehat{{\mathbf{e}}}_{y}}$.
Similarly, using (\[GenIso\]) and (\[G4gamma3\]), the expression (\[DefTopSign\]) takes the form
$$\begin{aligned}
\Upsilon^{\Gamma}_{\ell m} ({\mathbf{k}}) & = & \hspace{-0.1cm} \sum_{{\mathbf{n}}
\in \mathbb{Z}^3} \hspace{-0.1cm} e^{-i (n_xk_xL + n_yk_yL + 4n_zk_zL_z)}
\left[ 1 + \sum_{h=1}^3 \omega_4^{-hm} e^{-i {\mathbf{k}} \cdot {\mathcal{O}}_h
{\mathbf{c}}} \right] Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}\!) \\
& = & (2 \pi)^3 \sum_{{\mathbf{p}} \in \mathbb{Z}^3} \, \delta(k_xL - 2 \pi p_x)
\, \delta (k_yL - 2 \pi p_y) \, \delta \left( k_zL_z - {\textstyle
\frac{\pi}{2}} \, p_z \right) \, Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \,
f_{m}^\Gamma({\mathbf{k}}) \; ,\end{aligned}$$
where we have put $n_1 = n_x$, $n_2 = n_y$, and $n_3 = 4n_z$, $4n_z + 1$, $4n_z + 2$ or $4n_z + 3$ according to (\[G4gamma3\]), and $\omega_4$ is the first complex 4th-rooth of unity. We also get $\beta_x = \frac{p_x}{L}$, $\beta_y = \frac{p_y}{L}$, and $\beta_z = \frac{p_z}{4L_z}$.
${\mathcal{G}_5}$ {#Ss:G5}
-----------------
The generators for the rectangular ${\mathcal{G}_5}$ are $\gamma_1 = (I, {\mathbf{a}})$, $\gamma_2 = (I, {\mathbf{b}})$, and $\gamma_3 = (A, {\mathbf{c}})$, with ${\mathbf{a}}
= L \, {\widehat{{\mathbf{e}}}_{x}}$, ${\mathbf{b}} = \frac{L}{2} \, ({\widehat{{\mathbf{e}}}_{x}} + \sqrt{3} \,
{\widehat{{\mathbf{e}}}_{y}})$, ${\mathbf{c}} = \frac{\rho}{2} \, (\cos \phi + \sqrt{3} \sin \phi) \,
{\widehat{{\mathbf{e}}}_{x}} + \frac{\rho}{2} \, (\sin \phi - \sqrt{3} \cos \phi) \, {\widehat{{\mathbf{e}}}_{y}} +
L_z {\widehat{{\mathbf{e}}}_{z}}$, and $A$ given in (\[Rot\]) with $\alpha = \pi/3$. They satisfy the relations $\gamma_2^{-1} \gamma_3 \gamma_1 = \gamma_3$ and $\gamma_1 \gamma_2^{-1} \gamma_3 \gamma_2 = \gamma_3$, which allow us to write any isometry of the covering group by (\[GenIso\]) with $$\label{G5gamma3}
\gamma_3^{n_3} = \left\{ \begin{array}{lcc}
(I, n_3 {\mathbf{c}}_\parallel) & & \mbox{if } n_3 = 0 \mbox{ mod 6} \\
(A, n_3 {\mathbf{c}}_\parallel + {\mathbf{c}}_\perp) & & \mbox{if } n_3 = 1
\mbox{ mod 6} \\
(A^2, n_3 {\mathbf{c}}_\parallel + {\mathcal{O}}_2 {\mathbf{c}}_\perp) & & \mbox{if }
n_3 = 2 \mbox{ mod 6} \\
(A^3, n_3 {\mathbf{c}}_\parallel + {\mathcal{O}}_3 {\mathbf{c}}_\perp) & & \mbox{if }
n_3 = 3 \mbox{ mod 6} \\
(A^4, n_3 {\mathbf{c}}_\parallel + {\mathcal{O}}_4 {\mathbf{c}}_\perp) & & \mbox{if }
n_3 = 4 \mbox{ mod 6} \\
(A^5, n_3 {\mathbf{c}}_\parallel + {\mathcal{O}}_5 {\mathbf{c}}_\perp) & & \mbox{if }
n_3 = 5 \mbox{ mod 6}
\end{array} \right. \; ,$$ where ${\mathbf{c}}_\perp = \frac{\rho}{2} \, (\cos \phi + \sqrt{3} \sin \phi) \,
{\widehat{{\mathbf{e}}}_{x}} + \frac{\rho}{2} \, (\sin \phi - \sqrt{3} \cos \phi) \, {\widehat{{\mathbf{e}}}_{y}}$.
Using (\[GenIso\]) and (\[G5gamma3\]), the expression (\[DefTopSign\]) takes the form
$$\begin{aligned}
\Upsilon^{\Gamma}_{\ell m} ({\mathbf{k}}) & = & \sum_{{\mathbf{n}} \in \mathbb{Z}^3}
e^{-i\left[n_xk_xL + n_y \left(\frac{\sqrt{3}}{2}k_y + \frac{1}{2}k_x\right)L
+ 6n_zk_zL_z\right]} \left[ 1 + \sum_{h=1}^5 \omega_6^{-hm} e^{-i {\mathbf{k}}
\cdot {\mathcal{O}}_h {\mathbf{c}}} \right] Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \\
& = & (2 \pi)^3 \sum_{{\mathbf{p}} \in \mathbb{Z}^3} \, \delta(k_xL - 2 \pi p_x)
\, \delta \left( \left[ {\textstyle \frac{\sqrt{3}}{2}} k_y + {\textstyle
\frac{1}{2}} k_x \right]L - 2 \pi p_y \right) \, \delta \left( k_zL_z -
{\textstyle \frac{\pi}{3}} \, p_z \right) Y_{\ell m}({\mathbf{n}}_{\mathbf{k}}) \,
f_{m}^\Gamma({\mathbf{k}}) \; ,\end{aligned}$$
where we have put $n_1 = n_x$, $n_2 = n_y$, and $n_3 = 6n_z$, $6n_z + 1$, $6n_z + 2$, $\dots$, $6n_z + 5$ according to (\[G5gamma3\]), and $\omega_6$ is the first complex 6th-rooth of unity. We also get $\beta_x =
\frac{p_x}{L}$, $\beta_y = \frac{\sqrt{3}}{3L} \, (2p_y - p_x)$, and $\beta_z
= \frac{p_z}{6L_z}$.
[99]{} M. Tegmark, A. de Oliveira–Costa and A.J.S. Hamilton, Phys. Rev. D [**68**]{}, 123523A (2003); A. de Oliveira–Costa, M. Tegmark, M. Zaldarriaga and A.J.S. Hamilton, [*ibid.*]{} [**69**]{}, 063516 (2004).
C.J. Copi, D. Huterer and G.D. Starkman, Phys. Rev. D [**70**]{}, 043515 (2004); D.J. Schwarz, G.D. Starkman, D. Huterer and C.J. Copi, Phys. Rev. Lett. [**93**]{}, 221301 (2004); G. Katz and J.R. Weeks, Phys. Rev. D [**70**]{}, 063527 (2004); K. Land and J. Magueijo, Mon. Not. R. Astron. Soc. [**357**]{}, 994 (2005); [**362**]{}, 838 (2005); Phys. Rev. Lett. [**95**]{}, 071301 (2005).
L.Z. Fang and M. Houjun, Mod. Phys. Lett. A [**2**]{}, 229 (1987); I.Yu. Sokolov, JETP Lett. [**57**]{}, 617 (1993); A.A. Starobinsky, [*ibid.*]{} [**57**]{}, 622 (1993); D. Stevens, D. Scott and J. Silk, Phys. Rev. Lett. [**71**]{}, 20 (1993); A. de Oliveira–Costa and G.F. Smoot, Astrophys. J. [**448**]{}, 477 (1995); A. de Oliveira–Costa, G.F. Smoot and A.A. Starobinsky, [*ibid.*]{} [**468**]{}, 457 (1996); [astro-ph/9705125]{} (1997); N.J. Cornish, D.N. Spergel and G.D. Starkman, Phys. Rev. D [**57**]{}, 5982 (1998); N.J. Cornish and D.N. Spergel, [*ibid.*]{} [**62**]{}, 087304 (2000); K.T. Inoue, in *AIP Conference Proceedings “3K Cosmology”*, edited by L. Maiani, F. Melchiorri and N. Vittorio (American Institute of Physics, New York, 1999), p.343; Prog. Theor. Phys. [**106**]{}, 39 (2001); Class. Quantum Grav. [**18**]{}, 1967 (2001); K.T. Inoue, K. Tomita and N. Sugiyama, Mon. Not. R. Astron. Soc. [**314**]{}, L21 (2000); J.-P. Luminet, J. Weeks, A. Riazuelo, R. Lehoucq and J.-P. Uzan, Nature (London) [**425**]{}, 593 (2003); J. Weeks, J.-P. Luminet, A. Riazuelo and R. Lehoucq, Mon. Not. R. Astron. Soc. [**352**]{}, 258 (2004); J. Gundermann, [astro-ph/0503014]{}.
J. Levin, E. Scannapieco and J. Silk, Phys. Rev. D [**58**]{}, 103516 (1998); [astro-ph/9811226]{}.
R. Aurich, Astrophys. J. [**524**]{}, 497 (1999); R. Aurich and F. Steiner, Mon. Not. R. Astron. Soc. [**323**]{}, 1016 (2001); R. Aurich, S. Lustig, F. Steiner and H. Then, Class. Quantum Grav. [**21**]{}, 4901 (2004); R. Aurich, S. Lustig and F. Steiner, [*ibid.*]{} [**22**]{}, 2061 (2005); [**22**]{}, 3443 (2005).
K.T. Inoue, [astro-ph/0011539]{}; K.T. Inoue and N. Sugiyama, Phys. Rev. D [**67**]{}, 043003 (2003).
J.R. Weeks, [astro-ph/0412231]{}.
N.J. Cornish, D.N. Spergel and G.D. Starkman, Class. Quantum Grav. [**15**]{}, 2657 (1998); J.R. Weeks, [*ibid.*]{} [**15**]{}, 2599 (1998); B.F. Roukema, [*ibid.*]{} [**17**]{}, 3951 (2000); Mon. Not. R. Astron. Soc. [**312**]{}, 712 (2000); G.I. Gomero, [astro-ph/0310749]{}; B.F. Roukema, B. Lew, M. Cechowska, A. Marecki and S. Bajtlik, Astron. Astrophys. [**423**]{}, 821 (2004); J. Levin, Phys. Rev. D [**70**]{}, 083001 (2004); [astro-ph/0403036]{}; M.O. Calvão, G.I. Gomero, B. Mota and M.J. Rebouças, Class. Quantum Grav. [**22**]{}, 1991 (2005).
N.J. Cornish, D.N. Spergel, G.D. Starkman and E. Komatsu, Phys. Rev. Lett. [**92**]{}, 201302 (2004).
A. Hajian and T. Souradeep, [astro-ph/0301590]{}; Astrophys. J. [**597**]{}, L5 (2003); Pramana [**62**]{}, 793 (2004); [astro-ph/0501001]{}; A. Hajian, T. Souradeep and N. Cornish, Astrophys. J. [**618**]{}, L63 (2004).
T. Souradeep and A. Hajian, [astro-ph/0502248]{}.
J. Levin, J.D. Barrow, E.F. Bunn and J. Silk, Phys. Rev. Lett. [**79**]{}, 974 (1997); J. Levin, E. Scannapieco, G. de Gasperis, J. Silk and J.D. Barrow, Phys. Rev. D [**58**]{}, 123006 (1998).
J. Levin, E. Scannapieco and J. Silk, Class. Quantum Grav. [**15**]{}, 2689 (1998).
A. Riazuelo, J.-P. Uzan, R. Lehoucq and J. Weeks, Phys. Rev. D [**69**]{}, 103514 (2004).
J.-P. Uzan, A. Riazuelo, R. Lehoucq and J. Weeks, Phys. Rev. D [**69**]{}, 043003 (2004).
A. Riazuelo, J. Weeks, J.-P. Uzan, R. Lehoucq and J.-P. Luminet, Phys. Rev. D [**69**]{}, 103518 (2004).
J.R. Bond, D. Pogosyan and T. Souradeep, in *Proceedings of the 18th Texas Symposium on Relativistic Astrophysics*, edited by A. Olinto, J. Frieman and D. Schramm (World Scientific, Singapore, 1997); [astro-ph/9804042]{}; Class. Quantum Grav. [**18**]{}, 2671 (1998); Phys. Rev. D [**62**]{}, 043005 (2000); [**62**]{}, 043006 (2000).
R. Lehoucq, J. Weeks, J.-P. Uzan, E. Gausmann and J.-P. Luminet, Class. Quantum Grav. [**19**]{}, 4683 (2002); R. Lehoucq, J.-P. Uzan and J. Weeks, Kodai Math. Journal [**26**]{}, 119 (2003); J.R. Weeks, [math.SP/0502566]{}.
M. Lachièze-Rey, [math.SP/0304409]{}; Class. Quantum Grav. [**21**]{}, 2455 (2004); [math.SP/0401153]{}; M. Lachièze-Rey and S. Caillerie, [*ibid.*]{} [**22**]{}, 695 (2005).
N.J. Cornish and N.G. Turok, Class. Quantum Grav. [**15**]{}, 2699 (1998); N.J. Cornish and D.N. Spergel, [math.DG/9906017]{}; K.T. Inoue, Class. Quantum Grav. [**16**]{}, 3071 (1999); Phys. Rev. D [**62**]{}, 103001 (2000); Class. Quantum Grav. [**18**]{}, 629 (2001); R. Aurich, F. Steiner and H. Then, [gr-qc/0404020]{}; R. Aurich, S. Lustig, F. Steiner and H. Then, Phys. Rev. Lett. [**94**]{}, 021301 (2005).
G.I. Gomero, A.F.F. Teixeira, M.J. Reboucas and A. Bernui, Int. J. Mod. Phys. D [**11**]{}, 869 (2002); G.I. Gomero, M.J. Reboucas and A.F.F. Teixeira, Phys. Lett. A [**275**]{}, 355 (2000); Class. Quantum Grav. [**18**]{}, 1885 (2001); Int. J. Mod. Phys. D [**9**]{}, 687 (2000).
A.F. Beardon, *The Geometry of Discrete Groups* (Springer, New York, 1983), GTM 91.
G.I. Gomero, M.J. Rebouças and R. Tavakol, Class. Quantum Grav. [**18**]{}, 4461 (2001); [**18**]{}, L145 (2001); Int. J. Mod. Phys. A [**17**]{}, 4261 (2002); G.I. Gomero and M.J. Rebouças, Phys. Lett. A [**311**]{}, 319 (2003); J.R. Weeks, R. Lehoucq and J.-P. Uzan, Class. Quantum Grav. [**20**]{}, 1529 (2003); J.R. Weeks, Mod. Phys. Lett. A [**18**]{}, 2099 (2003); B. Mota, M. Makler and M.J. Rebouças, [astro-ph/0506499]{}.
B. Mota, G.I. Gomero, M.J. Rebouças and R. Tavakol, Class. Quantum Grav. [**21**]{}, 3361 (2004); M. Kunz, N. Aghanim, L. Cayon, O. Forni, A. Riazuelo and J. P. Uzan, [astro-ph/0510164]{}.
D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, *Quantum Theory of Angular Momentum* (World Scientific, 1988).
M.A. Blanco, M. Flórez and M. Bermejo, J. Molec. Struct. (TheoChem), [**419**]{}, 19 (1997).
J.A. Wolf, *Spaces of Constant Curvature* (Publish or Perish Inc., Delaware, 1984).
G.I. Gomero and M.J. Rebouças, Phys. Lett. A [**311**]{}, 319 (2003); G.I. Gomero, Class. Quantum Grav. [**20**]{}, 4775 (2003).
|
---
abstract: 'The [${\mathcal{SW}}(3/2,3/2,2)$]{}superconformal algebra is a $\mathcal{W}$ algebra with two free parameters. It consists of 3 superconformal currents of spins $3/2$, $3/2$ and 2. The algebra is proved to be the symmetry algebra of the coset $\frac{su(2) \oplus su(2) \oplus su(2)}{su(2)}$. At the central charge $c\!=\!10{\frac{1}{2}}$ the algebra coincides with the superconformal algebra associated to manifolds of $G_2$ holonomy. The unitary minimal models of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra and their fusion structure are found. The spectrum of unitary representations of the $G_2$ holonomy algebra is obtained.'
---
[**Unitary minimal models of $\mathcal{SW}(3/2,3/2,2)$\
superconformal algebra and\
manifolds of $G_2$ holonomy\
**]{}
Introduction
============
Recently the manifolds of exceptional holonomy attracted much attention. These are 7–dimensional manifolds of $G_2$ holonomy and 8–dimensional manifolds of $Spin(7)$ holonomy. They are considered in the context of the string theory compactifications.
The supersymmetric nonlinear sigma models on the manifolds of exceptional holonomy are described by conformal field theories, their superconformal chiral algebras were constructed in [@Shatashvili:zw]. We will call them the $G_2$ and $Spin(7)$ superconformal algebras. These are nonlinear ${\mathcal{W}}$–algebras ([@Zamolodchikov:1985wn], for review see [@Bouwknegt:1993wg]) of central charge $10{\frac{1}{2}}$ and $12$ respectively. The conformal field theories were further studied in [@Figueroa-O'Farrill:1996hm; @gn; @Eguchi:2001xa; @Sugiyama:2001qh; @Blumenhagen:2001jb; @Roiban:2001cp; @Eguchi:2001ip; @Blumenhagen:2001qx].
The $Spin(7)$ algebra is identified [@Figueroa-O'Farrill:1996hm] with the [${\mathcal{SW}}(3/2,2)$]{}[superconformal ]{}algebra [@fofs; @Komata:1991cb; @Blumenhagen:1992nm; @Blumenhagen:1992vr; @Eholzer:1992pv; @Mallwitz:1994hh], existing at generic values of the central charge. It consists of the [$N\!=\!1$ ]{}[superconformal ]{}algebra extended by its spin–2 superprimary field. The unitary representation theory of the [${\mathcal{SW}}(3/2,2)$]{}algebra is studied in [@gn], where complete list of unitary representations is determined (including the $c=12$ model, corresponding to the $Spin(7)$ manifolds).
In this paper we identify the $G_2$ algebra with the [${\mathcal{SW}}(3/2,3/2,2)$]{}superconformal algebra (in notations of [@Bouwknegt:1993wg]) at the central charge $c=10{\frac{1}{2}}$ and the coupling constant (see below) ${\lambda}=0$. The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra was first constructed in [@Blumenhagen:1992nm] (see also [@Blumenhagen:1992vr]). It is superconformal ${\mathcal{W}}$–algebra, which besides the energy–momentum supercurrent (the first “$3/2$” in [${\mathcal{SW}}(3/2,3/2,2)$]{}) contains two supercurrents of spins $3/2$ and $2$. The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra has two generic parameters. Along with the central charge there is a free coupling ${\lambda}$ (the self–coupling of the spin–$3/2$ superprimary field), which is not fixed by Jacobi identities.
In [@Mallwitz:1994hh] the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is shown to be the symmetry algebra of the quantized Toda theory corresponding to the $D(2|1;\alpha)$ Lie superalgebra (the only simple Lie superalgebra with free parameter). In the same ref.[@Mallwitz:1994hh] the free field representation of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is constructed.
We study different aspects of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra in the present paper. First we find that the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is the symmetry algebra of the diagonal coset $$\label{coset}
\frac{{su(2)}_{k_1} \oplus {su(2)}_{k_2} \oplus
{su(2)}_2}{{su(2)}_{k_1+k_2 +2}} \, .$$ We define [highest weight representation]{}s of the algebra and study their unitarity. The unitary minimal models are described by the coset (\[coset\]). Their central charge and coupling ${\lambda}$ are given by $$\begin{aligned}
\label{c cpl c}
c_{k_1, k_2}&=
\frac{9}{2} + \frac{6}{ {k_1} + {k_2}+4}
- \frac{6}{ {k_1}+2} - \frac{6}{ {k_2}+2} \, ,\\
\label{c cpl cpl}
{\lambda}_{k_1, k_2}&=
\frac{4\,{\sqrt{2}}\,\left( {k_1} - {k_2} \right) \,
\left( 2\,{k_1} + {k_2}+6 \right) \,
\left( {k_1} + 2\,{k_2} +6 \right) }
{3\,{\left(3\, {k_1} \, {k_2}\, ( {k_1} + {k_2}+6) \right)^{1/2}}
\left( {k_1}+2 \right)
\left( {k_2}+2 \right) \left( {k_1} + {k_2}+4 \right) } \, .\end{aligned}$$ We also obtain all the values of $c$ and ${\lambda}$, where the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra has continuous spectrum of unitary representations. One such model ($c=10{\frac{1}{2}}$, ${\lambda}=0$), which corresponds to the $G_2$ algebra, is discussed in details, the full spectrum of unitary representations is obtained. We also present the complete list of the minimal model representations and their fusion rules.
The diagonal coset constructions of type $\frac{g \oplus g}{g}$ were found very useful in the description of minimal models of different conformal algebras. The minimal models of the Virasoro algebra [@BPZ] ($c_k = 1-\frac{6}{(k+2)(k+3)}$) correspond to the diagonal coset construction [@Goddard:1985vk] $$\frac{{su(2)}_k \oplus {su(2)}_1}{{su(2)}_{k+1}} \, ,
\qquad k \in \mathbb{N} \, .$$ The coset (\[coset N=1\]) is found [@Goddard:1986ee] to form the minimal models of the [$N\!=\!1$ ]{}superconformal algebra ([@Eichenherr:cx; @Bershadsky:dq; @Friedan:1984rv] and appendix \[appN=1\]). The minimal models of the ${\mathcal{W}}_N$ algebra [@Lukyanov:1990tf] are the $su(N)$ diagonal cosets $$\frac{su(N)_k \oplus su(N)_1}{su(N)_{k+1}} \, ,
\qquad k \in \mathbb{N}\, .$$ We present here the first example (to our knowledge) of the conformal chiral algebra, corresponding to the diagonal coset of type $\frac{g \oplus g \oplus g}{g}$. It is nontrivial fact that the coset space (\[coset\]) has the same symmetry algebra for different $k_1$ and $k_2$. It can be explained, probably, by the connection of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra to the Lie superalgebra $D(2|1;\alpha)$, which has a free parameter unlike the other simple Lie algebras.
The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra contains two fields of spin $3/2$ and three fields of spin $2$, making enough room for embedding of different subalgebras, such as the $N\!=\!0$ (Virasoro) and the [$N\!=\!1$ ]{}conformal algebras. Besides the trivial [$N\!=\!1$ ]{}subalgebra (generated by the super energy–momentum tensor) there are 3 different [$N\!=\!1$ ]{}superconformal subalgebras of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra. These embeddings play a crucial role in the understanding of the representation theory of the algebra.
There are four types of [highest weight representation]{}s of the algebra: Neveu–Schwarz (NS), Ramond and two twisted sectors. (The twisted sectors are defined only in the case of vanishing coupling ${\lambda}$.)
The minimal models are labeled by two natural numbers: $k_1$ and $k_2$. The NS and Ramond minimal model representations can be arranged in the form of 3–dimensional table, similarly to the 2–dimensional tables of representations of the $N\!=\!0$ and the [$N\!=\!1$ ]{}conformal algebras. The fusion rules also satisfy the “${su(2)}$ pattern” of the $N\!=\!0$ and [$N\!=\!1$ ]{}minimal model fusions.
The set of the $G_2$ algebra representations consists of 4 sectors: NS, Ramond and two twisted. There are continuous spectrum representations in every sector. We prove, that the $G_2$ conformal algebra is the extended version of the [${\mathcal{SW}}(3/2,2)$]{}algebra at $c=10{\frac{1}{2}}$. Due to this fact we get all the $G_2$ unitary representations from the known spectrum [@gn] of the [${\mathcal{SW}}(3/2,2)$]{}algebra.
The paper is organized as follows. After reviewing the structure of [${\mathcal{SW}}(3/2,3/2,2)$]{} in section \[structure of the algebra\] we prove in section \[coset constr\] that the algebra is the symmetry algebra of the coset space (\[coset\]). In section \[N=1 superconformal subalgebras\] we discuss different embeddings of the [$N\!=\!1$ ]{}[superconformal ]{}algebra into the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra and obtain the unitarity restrictions on the values of $c$ and ${\lambda}$. In section \[Highest weight representations\] the [highest weight representation]{}s of the algebra are introduced, the zero mode algebras in different sectors are discussed. In section \[minimal models\] we concern with the minimal models of the algebra: we explain how the spectrum of unitary representations is obtained, discuss the fusion rules, and give two examples of the [${\mathcal{SW}}(3/2,3/2,2)$]{}minimal models in terms of the $N\!=\!2$ [superconformal ]{}minimal models. Section \[G2 algebra\] is devoted to the $G_2$ algebra, the [superconformal ]{}algebra associated to the manifolds of $G_2$ holonomy. We find it convenient to put some useful (but in some cases lengthy) information in the closed form in the five appendices.
We have to note that substantial part of the calculations was done with a help of [*Mathematica*]{} package [@Thielemans:1991uw] for symbolic computation of operator product expansions.
Structure of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra
=======================================================
\[structure of the algebra\]
Here we review the structure of [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra, which was first constructed in [@Blumenhagen:1992nm].
The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is an extension of [$N\!=\!1$ ]{}superconformal algebra by two superconformal multiplets of dimensions $\frac{3}{2}$ and $2$.
A superconformal multiplet $\widehat \Phi = (\Phi, \Psi)$ of dimension $\Delta$ consists of two Virasoro primary fields of dimensions $\Delta$ and $\Delta + {\frac{1}{2}}$. Under the action of the supersymmetry generator $G$ they transform as $$\begin{aligned}
G(z)\,\Phi(w)&= \frac{\Psi(w)}{z-w} \, , \label{GPhi} \\
G(z)\,\Psi(w)&=\frac{2\, \Delta \, \Phi(w)}{(z-w)^2}+
\frac{{\partial}\Phi(w)}{z-w} \, .\end{aligned}$$
The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra multiplets are denoted by $I=(G, T)$, $\widehat H = (H,M)$ ($\Delta = \frac{3}{2}$), $\widehat W = (W,U)$ ($\Delta = 2$). The structure of the algebra is schematically given by $$\label{algebra_structure}
\begin{aligned}
\widehat H \times \widehat H &=
I + {\lambda}\, \widehat H + {\mu}\, \widehat W \, ,\\
\widehat H \times \widehat W &=
{\mu}\, \widehat H+ {\lambda}\, \widehat W \, ,\\
\widehat W \times \widehat W &=
I + {\lambda}\, \widehat H + {\mu}\, \widehat W
+ {{:} \widehat H \widehat H {:} } \, ,
\end{aligned}$$ where $${\mu}= \sqrt{\frac{9\, c\, (4+{\lambda}^2)}{2 \, (27-2\, c)}} \, ,$$ and the $c$ dependence of the coefficients is omitted.
The explicit operator product expansions ([OPE]{}s) are fixed by the fusions (\[algebra\_structure\]) and by [$N\!=\!1$ ]{}superconformal invariance. We reproduce the [OPE]{}s in appendix \[apA\].
Unitarity is introduced by the standard conjugation relation $\mathcal{O}_n^\dagger = \mathcal{O}_{-n}$ for any generator except $U$. The commutation relation ${\left[G_n,W_m\right]} = U_{n+m}\, $, following from (\[GPhi\]), requires the conjugation $U_n^\dagger = -U_{-n}$.
Coset construction
==================
\[coset constr\]
Preliminary discussion
----------------------
We start from a few supporting arguments, that the coset theory (\[coset\]) indeed possesses the [${\mathcal{SW}}(3/2,3/2,2)$]{}superconformal symmetry.
First, we note that formally ${su(2)}_2 \approx so(3)_1$ and the coset (\[coset\]) can be written in the form of Kazama–Suzuki coset [@Kazama:1989qp] ${\displaystyle}\frac{g \oplus so(\dim{g}-\dim{h})_1}{h}$, where $g= {su(2)}\oplus {su(2)}$ and $h$ is its ${su(2)}$ diagonal subalgebra. It means, that the chiral algebra contains [$N\!=\!1$ ]{}superconformal algebra, obtained from the affine currents by a superconformal generalization of Sugawara construction (see ref. [@Kazama:1989qp] for details).
All other currents, that constitute the chiral algebra should come in pairs of superpartners with difference of scaling dimensions equal to $1/2$.
The central charge (\[c cpl c\]) of the coset models (\[coset\]) has limiting point (when $k_1, k_2 \to \infty $) $c=9/2$. All the known examples of minimal series have limiting central charge $c=n_B+{\frac{1}{2}}n_F$, where $n_B$ and $n_F$ are the number of bosonic and fermionic fields in the correspondent chiral algebra. Adopting the argument to our case we get, that the chiral algebra consists of three supercurrents (including the super–Virasoro operator).
The next argument follows from the simple observation [@Kazama:1989uz], that if there is a sequence of subalgebra inclusions $$g \supset h_1 \supset \ldots \supset h_n$$ then the coset theory can be decomposed to the direct sum $$\frac{g}{h_n} = \frac{g}{h_1} \oplus \frac{h_1}{h_2}
\oplus \ldots \oplus \frac{h_{n-1}}{h_n}\, .$$ In the case of coset (\[coset\]) the inclusion sequence is $${su(2)}_{k_1} \oplus {su(2)}_{k_2} \oplus {su(2)}_2
\supset h_1 \supset {su(2)}_{k_1+k_2+2}$$ with 3 different choices of $h_1$: ${su(2)}_{k_1+2} \oplus {su(2)}_{k_2}\, ,$\
${su(2)}_{k_2+2} \oplus {su(2)}_{k_1}\, ,$ ${su(2)}_{k_1+k_2} \oplus {su(2)}_2\, .$ The correspondent decompositions are: $$\begin{aligned}
\label{decom2}
\frac{{su(2)}_{k_1} \oplus {su(2)}_{2}}
{{su(2)}_{k_1+2}}
&\oplus
\frac{{su(2)}_{k_1+2} \oplus {su(2)}_{k_2}}
{{su(2)}_{k_1+k_2+2}} \, ,\\
\label{decom3}
\frac{{su(2)}_{k_2} \oplus {su(2)}_{2}}
{{su(2)}_{k_2+2}}
&\oplus
\frac{{su(2)}_{k_2+2} \oplus {su(2)}_{k_1}}
{{su(2)}_{k_1+k_2+2}} \, ,\\
\label{decom1}
\frac{{su(2)}_{k_1} \oplus {su(2)}_{k_2}}
{{su(2)}_{k_1+k_2}}
&\oplus
\frac{{su(2)}_{k_1+k_2} \oplus {su(2)}_2}
{{su(2)}_{k_1+k_2+2}} \, .\end{aligned}$$ All three contain coset spaces of type (\[coset N=1\]) with $k=k_1$, $k=k_2$ and $k=k_1+k_2$ respectively. Therefore the chiral algebra contain 3 different [$N\!=\!1$ ]{}superconformal subalgebras (not including the trivial one, generated by the super energy–momentum tensor). This is possible if there are at least 3 operators of scaling dimension 2. They have 3 superpartners of dimensions $\frac{3}{2}$ or $\frac{5}{2}$. One field of dimension $\frac{3}{2}$ can not serve as a superconformal generator for 3 different superconformal subalgebras. The case, when all three are of dimension $\frac{3}{2}$ is also excluded, because then the algebra is trivially a sum of 3 commuting [$N\!=\!1$ ]{}superconformal algebras.
Collecting the arguments we get that the only possibility that the chiral algebra consists of 6 fields of dimensions $\frac{3}{2}$, $\frac{3}{2}$, 2, 2, 2, $\frac{5}{2}$, which can be combined to three supercurrents of dimensions $\frac{3}{2}$, $\frac{3}{2}$, 2, giving the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra.
Explicit construction {#sec:expli_const}
---------------------
In this section we prove by explicit construction that the coset (\[coset\]) contains the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra. The method we use was first proposed in [@Goddard:1986ee] for coset (\[coset N=1\]).
The ${su(2)}$ affine algebra is generated by 3 currents $J_i$, $i=1,2,3$: $$J_i (z) \, J_i (w) = \frac{k/2}{(z-w)^2}+
\frac{{\mathrm{i}}\, \epsilon_{i j k} \, J_k (w)}{z-w} \, .$$ The ${g}$ algebra consists of 3 commuting copies of the $su(2)$ algebra at levels $k_1$, $k_2$ and $2$. The ${su(2)}$ on level $2$ is realized by free fermions in the adjoint representation of ${su(2)}$: $$\psi_i (z) \, \psi_j (w)=\frac{\delta_{i j}}{z-w} \, ,
\qquad i,j = 1,2,3.$$ Then the affine currents are expressed as $$J_i = - \frac{{\mathrm{i}}}{2} \, \epsilon_{i j k} \,
{{:} \psi_j \psi_k {:} } \, .$$ The affine algebra ${h} ={su(2)}_{k_1+k_2+2}$ is diagonally embedded in ${g}={su(2)}_{k_1} \oplus {su(2)}_{k_2} \oplus {su(2)}_2$: $$J_i^{(h)}=J_i^{(1)}+J_i^{(2)}+J_i^{(3)} \, .$$ The coset space ${g}/{h}$ contains operators, constructed from the ${g}$ currents, which commute with the currents of ${h}$. The energy–momentum tensor of the coset ${g}/{h}$ is given by the Sugawara construction: $$T=T^{(g)}-T^{(h)}=T^{(1)}+T^{(2)}+T^{(3)}-T^{(h)}\, ,$$ where $$T^{(n)}=\frac{1}{k_n+2}
\sum_{i=1}^3 {{:} J_i^{(n)} J_i^{(n)} {:} } \, .$$ The general dimension–$3/2$ operator can be written as $$\mathcal{O}_{3/2} =
b_1 \sum_{i=1}^3 {{:} J_i^{(1)} \psi_i {:} } +
b_2 \sum_{i=1}^3 {{:} J_i^{(2)} \psi_i {:} } +
{\mathrm{i}}\, b_3 \, {{:} \psi_1 \psi_2 \psi_3 {:}}\, .$$ It should commute with the $J^{(h)}$ currents. This requirement leads to condition $$b_3 = {\frac{1}{2}}(b_1 \, k_1 +b_2 \, k_2) \, .$$ One has two independent dimension–$3/2$ operators ($G$ and $H$) in the coset theory, since there are two free parameters $b_1$ and $b_2$. In order to close the algebra one needs 3 more operators $M$, $W$, $U$ of scaling dimensions 2, 2, $5/2$. The coset construction of all the 6 operators is given in appendix \[apB\].
We have explicitly checked, that the set of 6 operators $T$, $G$, $H$, $M$, $W$, $U$ satisfy the [OPE]{}s of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra with central charge $c$ and coupling ${\lambda}$ given by (\[c cpl cpl\]).
[$N\!=\!1$ ]{}superconformal subalgebras
========================================
\[N=1 superconformal subalgebras\]
We start this section by observation, that 3 bosonic operators $T, M, W$ do not generate a closed subalgebra because of ${{:} G H {:} }$ term in the [OPE]{} of $M$ with $W$ (\[opeMW\]).
We will construct in this section different [$N\!=\!1$ ]{} superconformal subalgebras of [${\mathcal{SW}}(3/2,3/2,2)$]{}and discuss the unitarity restrictions. The most general dimension–$\frac{3}{2}$ operator is [^1] $${\widetilde G}=\alpha \, G + \beta \, H \, .$$ We calculate its [OPE]{} with itself: $${\widetilde G}(z) \, {\widetilde G}(w) =
\frac{\frac{2}{3} \, c \, (\alpha^2 + \beta^2)}{(z-w)^3}+
\frac{2\, {\widetilde T}}{z-w} \, ,$$ where $${\widetilde T}= (\alpha^2 +\beta^2) \, T +
\beta \, (\alpha + {\frac{1}{2}}\, {\lambda}\, \beta) \, M +
\frac{2}{3} \, {\mu}\, \beta^2 \, W \, ,$$ and take the dimension–2 operator ${\widetilde T}$ as the Virasoro generator of the subalgebra. These two operators, ${\widetilde G}$ and ${\widetilde T}$, generate a closed subalgebra if the following two equations are satisfied: $$\label{eq:alpha_beta}
\begin{aligned}
27 \, \alpha^2 + 27 \, {\lambda}\, \alpha \, \beta +
( 9 \, {\lambda}^2 +4 \, {\mu}^2 + 9) \, \beta^2 -9
&= 0 \, ,\\
\alpha^3 - \alpha +3 \, \alpha \, \beta^2 + {\lambda}\, \beta^3 &= 0 \, .
\end{aligned}$$ Then the central charge of the subalgebra is ${\widetilde c}= c\, (\alpha^2+\beta^2)$. Formally the lefthand side of the first equation should be multiplied by $\beta$. We removed it in order to exclude the trivial solution $\beta=0, \alpha=1$, corresponding to the obvious [$N\!=\!1$ ]{}subalgebra.
The operator $T-{\widetilde T}$ generates another closed subalgebra, namely Virasoro algebra with central charge $c-{\widetilde c}\,$, and it is commutative with the [$N\!=\!1$ ]{}superconformal subalgebra of ${\widetilde G}$ and ${\widetilde T}$.
The equations (\[eq:alpha\_beta\]) are polynomial equations in $\alpha$ and $\beta$ of orders 2 and 3. Generically there are 6 solutions. One should take into account the $\mathbb{Z}_2$ symmetry $\alpha \to - \alpha$ and $\beta \to - \beta$ of the equations, corresponding to the $\mathbb{Z}_2$ transformation of the [$N\!=\!1$ ]{}superconformal algebra ${\widetilde G}\to - {\widetilde G}, \,{\widetilde T}\to {\widetilde T}$. So at any generic $c$ and ${\lambda}$ there are three [$N\!=\!1$ ]{}superconformal subalgebras.
However, we are interested in real subalgebras, i.e. preserving the conjugation relation: the conjugation of the subalgebra ${\widetilde G}_n^\dagger = {\widetilde G}_{-n}, \, {\widetilde T}_n^\dagger = {\widetilde T}_{-n}$ should be consistent with the conjugation relations of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra. This is true, if $\alpha$ and $\beta$ are real. Generically there can be 1 or 3 real solutions of (\[eq:alpha\_beta\]).
Unitary representations of an algebra are necessarily unitary representations of all its real subalgebras. The [$N\!=\!1$ ]{}superconformal algebra has unitary representations only at ${\widetilde c}\ge 3/2$ or at ${\widetilde c}= {c^{\scriptscriptstyle{N\!=\!1}}_{k}}\, , \, k \in \mathbb{N}\,$ (\[C\_k\]). The central charges of all the real [$N\!=\!1$ ]{}superconformal subalgebras should be from this set.
We study the solutions of the set of equations (\[eq:alpha\_beta\]) in the region $0 \le c < \frac{27}{2}\,$. (At $c > \frac{27}{2}$ the coupling ${\mu}$ becomes imaginary.) The results are presented in figure \[figure1\]. The region $0 \le c < \frac{27}{2}$ is divided to two parts by the curve (thick curve in figure \[figure1\]) $$\label{divide curve}
4\, (9-2\, c)^3 = 243\, (2\, c-3) \, {\lambda}^2\, .$$
In the region I ($4\, (9-2\, c)^3 < 243\, (2\, c-3) \, {\lambda}^2$) there is one real solution of (\[eq:alpha\_beta\]), in the region III ($4\, (9-2\, c)^3 > 243\, (2\, c-3) \, {\lambda}^2$) there are 3 real solutions.
The curves in figure \[figure1\] have constant subalgebra central charge along it: ${\widetilde c}={c^{\scriptscriptstyle{N\!=\!1}}_{k}}$. We call them the unitary curves. Taking different [$N\!=\!1$ ]{}subalgebras one gets different curves: $$\begin{aligned}
\label{curveA}
{{\lambda}}^2 &=
\frac{4\,\left( 54 - 4\,c + 9\,k - 2\,c\,k \right) \,
{\left( 4\,c - 9\,k + 2\,c\,k \right) }^2}
{243\,k\,
\left( 16\,c - 18\,k + 12\,c\,k - 3\,k^2 +
2\,c\,k^2 \right)} \\
{{\lambda}}^2 &= \label{curveB}
\frac{4\,\left( 9\,k - 2\,c\,k -8\,c \right)
\,{\left( 8\,c +
2\,c\,k - 9\,k -54 \right) }^2}{243\,
\left( 6 + k \right) \,
\left( 16\,c - 18\,k + 12\,c\,k - 3\,k^2 +
2\,c\,k^2 \right) }\end{aligned}$$ All the region under discussion is spanned by the curves (\[curveA\], \[curveB\]). There are no real solutions of (\[eq:alpha\_beta\]) corresponding to ${\widetilde c}> 3/2\,$.
In the region III the unitarity is restricted to the intersections of the unitary curves (the dots in figure \[figure1\]). There are intersections of exactly 3 curves in every intersection point: two curves of type (\[curveA\]) with $k=k_1$ and $k=k_2$ and the third of type (\[curveB\]) with $k=k_1+k_2$. The intersection points are given by (\[c cpl c\], \[c cpl cpl\]). The formula (\[c cpl c\]) is exactly the formula for central charge of coset theories (\[coset\]). The central charges of three real subalgebras (we will call them the first, the second and the third [$N\!=\!1$ ]{}subalgebras) at $c$ and ${\lambda}$ at the intersection point are ${c^{\scriptscriptstyle{N\!=\!1}}_{k_1}}$, ${c^{\scriptscriptstyle{N\!=\!1}}_{k_2}}$, ${c^{\scriptscriptstyle{N\!=\!1}}_{k_1 + k_2}}$, they also coincide with the central charges of the three [$N\!=\!1$ ]{}subalgebras of the coset (\[coset\]) (see the decompositions (\[decom2\], \[decom3\], \[decom1\])). We conclude that all unitary models of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra in the region III are given by coset models (\[coset\]). One can solve the equations (\[eq:alpha\_beta\]) for the values of $c$ and ${\lambda}$ from (\[c cpl c\], \[c cpl cpl\]) to get the linear connection between the generators $T$, $M$, $W$ and the Virasoro generators of the three subalgebras: [$$\begin{aligned}
\label{h_to_ddd}
T&=\frac{1}{2}\,\left( {\widetilde T}^{(1)}\,
\left({k_1} + 4 \right) +
{\widetilde T}^{(2)}\,\left({k_2} + 4 \right) -
{\widetilde T}^{(3)}\,\left({k_1} +{k_2} + 2 \right)
\right),
\\
\label{m_to_ddd}
M&=
\frac{1}{\left( 6\, k_1 \, k_2 \, (k_1+k_2+6) \right)^{1/2}}
\times \nonumber \\
&\quad \times
\Bigg(
{\widetilde T}^{(3)}\,
\frac{(k_1-k_2)(k_1+k_2+2)(k_1 \, k_2-2\, k_1-2\,k_2-12)}
{(k_1+2)(k_2+2)} \Bigg. \nonumber \\
&\quad - \Bigg.
\left(
{\widetilde T}^{(1)}\,
\frac{(k_1+4)(k_1+2\,k_2+6)(k_2^2+k_1\,k_2-2\,k_1+6\,k_2)}
{(k_2+2)(k_1+k_2+4)}
-\left(1 \leftrightarrow 2 \right)
\right)
\Bigg),
\\
\label{w_to_ddd}
W&=
\left(
\frac{
3\,{{k_1}}^2\,{k_2} +
3\,{k_1}\,{{k_2}}^2 + 2\,{{k_1}}^2 +
2\,{{k_2}}^2 + 20\,{k_1}\,{k_2} +
12\,{k_1} + 12\,{k_2}}
{24\, k_1 \, k_2 \, (k_1+k_2+6)}
\right)^{1/2}
\times \nonumber \\
&\quad \times
\Bigg(
{\widetilde T}^{(3)}\,
\frac{(k_1+k_2+2)(k_1 \, k_2+4\, k_1+4\,k_2+12)}
{(k_1+2)(k_2+2)} \Bigg. \nonumber \\
&\quad - \Bigg.
\left(
{\widetilde T}^{(1)}\,
\frac{(k_1+4)(k_2^2+k_1\,k_2+4\,k_1+6\,k_2+12)}
{(k_2+2)(k_1+k_2+4)}
+\left(1 \leftrightarrow 2 \right)
\right)
\Bigg).\end{aligned}$$ ]{}
In the region I the unitarity is restricted to the unitary curves (\[curveA\]). The region I models with $c$ and ${\lambda}$ satisfying (\[curveA\]) are expected to have continuous spectrum of unitary representations.
On the separating curve (\[divide curve\]) the unitarity is restricted to the limiting points of (\[c cpl c\], \[c cpl cpl\]) at one $k$ fixed and another $k$ taken to infinity. (And $c=9/2, \, {\lambda}=0$, when both $k_1, k_2 \to \infty$.)
The Virasoro subalgebras, generated by $T-{\widetilde T}$, give no new restrictions on unitarity of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra.
The point $c=10{\frac{1}{2}}$, ${\lambda}=0$, which corresponds to the conformal algebra on $G_2$ manifold, is in the region I and lies on the (\[curveA\]) curve with $k=1$. It means that the algebra have one real [$N\!=\!1$ ]{}subalgebra and its central charge is ${c^{\scriptscriptstyle{N\!=\!1}}_{1}}=7/10$ in agreement with results of ref.[@Shatashvili:zw]. The generators of this real subalgebra are $$\begin{aligned}
\label{gsub_cpl_zero}
{\widetilde G}&=\left(\frac{27-2c}{3\,(9-2\,c)}\right)^{1/2} \! H ,\\
\label{tsub_cpl_zero}
{\widetilde T}&=\frac{1}{3\,(9+2\,c)}
\left(\left( 27 - 2\,c \right) \,
T + 2\,{\left( 2\,c\,
\left( 27 - 2\,c \right) \right) }^{1/2}
\,W \right).\end{aligned}$$ (This is true for any ${\lambda}=0$ model.)
The important question for understanding the structure of unitary representations of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is how the algebra is decomposed to the representations of its real [$N\!=\!1$ ]{}subalgebras? The decomposition is $\Phi_{1 1} + \Phi_{3 1}$ under subalgebras corresponding to the (\[curveA\]) curve and $\Phi_{1 1} + \Phi_{1 3}$ under subalgebras corresponding to the (\[curveB\]) curve, where $\Phi_{1 1}$ is the vacuum representation of the [$N\!=\!1$ ]{}[superconformal ]{}algebra, $\Phi_{1 3}$ and $\Phi_{3 1}$ are its degenerate representations, having null vector on level $3/2$. The ${\widetilde T}$, ${\widetilde G}$ and $T$ fields are in the $\Phi_{1 1}$ representation. Three other fields of [${\mathcal{SW}}(3/2,3/2,2)$]{}form $\Phi_{3 1}$ (or $\Phi_{1 3}$) representation, they can be understood in this context as $\Phi_{3 1}$ (spin–$3/2$ field), ${\widetilde G}_{-1/2} \Phi_{3 1}$ (spin–$2$ field) and ${\widetilde T}_{-1} \Phi_{3 1}$ (spin–$5/2$ field). ${\widetilde T}_{-1} {\widetilde G}_{-1/2} \Phi_{3 1}$ is proportional to ${\widetilde G}_{-3/2} \Phi_{3 1}$ ($\approx {{:} {\widetilde G}\Phi_{3 1} {:} }$), since there is a null state on level $3/2$.
Highest weight representations
==============================
\[Highest weight representations\]
The [${\mathcal{SW}}(3/2,3/2,2)$]{}commutation relations admit two consistent choices of generator modes in the general case: NS and Ramond sectors; and two more then the coupling ${\lambda}=0$: first twisted (tw1) and second twisted sectors (tw2).
NS sector.
: The modes of the bosonic operators ($L_n$, $M_n$, $W_n$) are integer ($n \in \mathbb{Z}$) and the modes of the fermionic operators ($G_r$, $H_r$, $U_r$) are half–integer ($r \in \mathbb{Z}+{\frac{1}{2}}$).
Ramond sector.
: The modes of all the operators are integer.
First twisted sector.
: The modes of $L_n$, $W_n$, $H_n$ operators are integer ($n \in \mathbb{Z}$) and the modes of $G_r$, $M_r$, $U_r$ operators are half–integer ($r \in \mathbb{Z}+{\frac{1}{2}}$).
Second twisted sector.
: The modes of $L_n$, $G_n$, $W_n$, $U_n$ operators are integer ($n \in \mathbb{Z}$) and the modes of $H_r$, $M_r$ operators are half–integer ($r \in \mathbb{Z}+{\frac{1}{2}}$).
How can one understand the existence of four different sectors in terms of the coset construction (appendix \[apB\])? In order to get NS or Ramond sectors of the algebra one should take all the three fermions of ${su(2)}_2$ in NS or Ramond sectors respectively. The modes of ${su(2)}_{k_1}$ and ${su(2)}_{k_2}$ currents are integer. The twisted sectors ($k_1=k_2$, since ${\lambda}=0$) are obtained in less obvious way. First twisted sector: one takes one Ramond fermion (say, $\psi_1$) and two NS fermions ($\psi_2$ and $\psi_3$), the modes of $(J_1^{(1)}+J_1^{(2)})$, $(J_2^{(1)}-J_2^{(2)})$, $(J_3^{(1)}-J_3^{(2)})$ are integer and the modes of $(J_1^{(1)}-J_1^{(2)})$, $(J_2^{(1)}+J_2^{(2)})$, $(J_3^{(1)}+J_3^{(2)})$ are half–integer. Second twisted sector: one NS fermion ($\psi_3$) and two Ramond fermions ($\psi_1$ and $\psi_2$), the modes of $(J_1^{(1)}-J_1^{(2)})$, $(J_2^{(1)}-J_2^{(2)})$, $(J_3^{(1)}+J_3^{(2)})$ are integer and the modes of $(J_1^{(1)}+J_1^{(2)})$, $(J_2^{(1)}+J_2^{(2)})$, $(J_3^{(1)}-J_3^{(2)})$ are half–integer. One cannot define the modes of separate bosonic currents (e.g. $J_1^{(1)}$) in the twisted sectors.
The commutation relations of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra include products of the generators. The formula (\[NOepsilon\]) for the mode expansion of composite operators in various sectors is derived in appendix \[NOexp\].
Now are ready to define the [highest weight representation]{}s in all sectors. The [highest weight state]{} is annihilated by positive modes of all generators: $$\mathcal{O}_n { \big|\, \text{hws} \, \big>}=0\, , \qquad n > 0 \, .$$ To deal with the zero modes one should discuss the sectors separately.
NS sector
---------
There are 3 zero modes: $L_0$, $M_0$, $W_0$. It is convenient to choose the [highest weight state]{} to be the eigenstate of these 3 operators and to label the [highest weight representation]{} by the correspondent eigenvalues. This is possible if the set of zero modes is commutative. One finds from (\[opeMW\]) that the commutator $${\left[M_0,W_0\right]} = \frac{9 \, {\mu}}{2\, c}
(M_0+ {{:} G H {:} }_0)$$ is not zero. We rewrite the commutator by expanding ${{:} G H {:} }_0$ in the modes of $G$ and $H$: $$\label{comMW}
{\left[M_0,W_0\right]} = \frac{9 \, {\mu}}{2\, c} \sum_{r=1/2}^\infty
\left( G_{-r} \, H_r - H_{-r} \, G_r \right)$$ The action of righthand side of (\[comMW\]) on [highest weight state]{}vanishes. This is what one effectively needs in order to choose the [highest weight state]{} to be the eigenstate of both $M_0$ and $W_0$. We define the notion of “effective” commutator: the commutation relation, which is true modulo terms, their action on [highest weight state]{} is zero. Concluding: the “effective” commutators of all three zero modes vanish, and one can label the [highest weight representation]{} by three weights, the eigenvalues of the zero modes: $$\begin{aligned}
L_0 \, &{ \big|\, h \, , m \, , w \, \big>} &= h &{ \big|\, h \, , m \, , w \, \big>} \, ,\\
M_0 \, &{ \big|\, h \, , m \, , w \, \big>} &= m &{ \big|\, h \, , m \, , w \, \big>} \, ,\\
W_0 \, &{ \big|\, h \, , m \, , w \, \big>} &= w &{ \big|\, h \, , m \, , w \, \big>} \, .
\end{aligned}$$ One gets all states in the representation acting by negative modes on the [highest weight state]{}.
Ramond sector
-------------
There are 6 zero modes: $L_0$, $M_0$, $W_0$, $G_0$, $H_0$, $U_0$. Since $L_0$ is commutative with all other zero modes, it can be represented by a number $h$. The (anti)commutation relations of other zero modes are $$\begin{aligned}
{\{G_0,G_0\}} \label{zmaGG}
&= 2\, (h - c/24) \, ,\\ {\{G_0,H_0\}}
&=M_0 \, ,\\
{\left[G_0,M_0\right]}
&={\{G_0,U_0\}}=0\, ,\\
{\left[G_0,W_0\right]}
&=U_0 \, ,\\
{\{H_0,H_0\}} \label{zmaHH}
&= 2\, (h - c/24)+
{\lambda}\, M_0 + 4/3 \, {\mu}\, W_0 \, ,\\
{\left[H_0,M_0\right]} \label{zmaHM}
&= 2/3 \, {\mu}\, U_0 \, ,\\
{\left[H_0,W_0\right]} \label{zmaHW}
&= {\lambda}/2 \, U_0 \, ,\\
{\left[M_0,W_0\right]} \label{zmaMW}
&= {\{H_0,U_0\}}=
\frac{9 \, {\mu}}{4\, c} \, (2\, G_0 \, H_0 - M_0) \, ,\\
{\left[M_0,U_0\right]}
&= \frac{9 \, {\mu}}{2\, c} \,
\left( 2\,(h-c/24)\, H_0 -G_0 \, M_0 \right) \, ,\\
{\left[W_0,U_0\right]}
&= \frac{1}{8\,c}\, \Big(
54 \, {\mu}\, U_0
- 54 \, (h - c/24) \, \left( 2\, G_0+ {\lambda}\, H_0
\right)\Big. \nonumber \\
& \quad \Big. -27 \, {\lambda}\, G_0 \, M_0 - 72 \, {\mu}\, G_0 \, W_0
+54 \, H_0 \, M_0 \Big) \, ,\\
{\{U_0,U_0\}} \label{zmaUU}
&= \frac{9}{4\,c} \, \Big(
-12\, (h - c/24)^2-
(h - c/24) \left(6\, {\lambda}\,M_0
+8\,{\mu}\, W_0 \right) \Big. \nonumber \\
&\quad \Big. +4 \, {\mu}\, G_0 \, U_0
+3\, M_0 \, M_0 \Big)\, .\end{aligned}$$ They define finite dimensional $\mathcal{W}$ superalgebra. The commutation relations (\[zmaGG\]–\[zmaHW\]) are exact, and the commutators (\[zmaMW\]–\[zmaUU\]) are “effective”, i.e. modulo terms which annihilate [highest weight state]{}s.
The irreducible representations of the zero mode algebra are one–dimensional or two–dimensional and labeled by three weights: $h$, $w$ and $m$. In one–dimensional representation the zero modes are given by $$\begin{aligned}
W_0&=w, \qquad M_0=m, \qquad U_0=0,\\
G_0&=\sqrt{h-c/24}\, ,\quad
H_0=\frac{m}{2\,\sqrt{h-c/24}}\, .
\end{aligned}$$ Such a representation exists only if the following condition is satisfied: $$\label{eqR1}
12\, (h-c/24)^2 +6\, {\lambda}\, (h-c/24)\, m+
8\, {\mu}\, (h-c/24)\,w-3 \, m^2 = 0\, .$$ Taking $h\!=\!c/24$ one gets $G_0\!=\!0$. Such representation is called Ramond ground state. In the limit $h \to c/24$ the weight $m$ approaches $0$ like $m \sim \left( 8 \, (h- c/24) \, {\mu}\, w /3 \right)^{1/2}$ and then $M_0=0$ and $H_0=\sqrt{2 \, {\mu}\, w /3}\,$.
The definition of the two–dimensional representations of the zero mode algebra (\[zmaGG\]–\[zmaUU\]) is more complicated. The bosonic zero modes $L_0$, $M_0$ and $W_0$ can not be diagonalized simultaneously, since $M_0$ and $W_0$ do not commute, even “effectively”. One can label the [highest weight representation]{}s by $h$, $w={\frac{1}{2}}\text{Trace}(W_0)$ and $m={\frac{1}{2}}\text{Trace}(M_0)$. In the following sections we will use another labels, but they will be always linearly dependent on the $h$, $w$, $m$. The maximal set of commuting operators contains 3 operators, which can be chosen as following: $L_0$, some fermionic operator $F_0$ and its square $F_0^2$. In section \[G2 algebra\] it will be convenient to choose the $F_0$ operator as the zero mode of the [$N\!=\!1$ ]{}subalgebra supersymmetry generator.
First twisted sector
--------------------
There are 3 zero modes: $L_0$, $W_0$ and $H_0$. The zero mode algebra is obtained by setting ${\lambda}=0$ in (\[zmaHH\]) and (\[zmaHW\]): $$\begin{aligned}
{\{H_0,H_0\}}
&= 2\, (h - c/24) + 4/3 \, {\mu}\, W_0 \, ,\\
{\left[H_0,W_0\right]}
&= 0 \, .\end{aligned}$$ Its irreducible representations are one dimensional. The [highest weight state]{} is labeled by two weights: $$\begin{aligned}
L_0 \, { \big|\, h \, , w \, \big>} &= h { \big|\, h \, , w \, \big>} \, ,\\
W_0 \, { \big|\, h \, , w \, \big>} &= w { \big|\, h \, , w \, \big>} \, ,\\
H_0 \, { \big|\, h \, , w \, \big>} &=
\left((h-c/24)+2/3 \, {\mu}\, w \right)^{1/2} { \big|\, h \, , w \, \big>} \, .
\end{aligned}$$
Second twisted sector
---------------------
There are four zero modes: $L_0$, $W_0$, $G_0$ and $U_0$. Again $L_0$ is commutative with other zero modes and represented by its eigenvalue $h$. The commutation relations are $$\begin{aligned}
{\{G_0,G_0\}}
&= 2\, (h - c/24) \, ,\\
{\left[G_0,W_0\right]}
&= U_0 \, ,\\
{\{G_0,U_0\}}
&= 0\, , \\
{\left[W_0,U_0\right]}
&= \frac{1}{2\,c}
\Big( 27\, (9/48-(h-c/24))\, G_0
+9\,{\mu}\, (U_0-2\, G_0 \, W_0) \Big) \, ,\\
{\{U_0,U_0\}}
&=\frac{1}{c}
\Big(
27\, (h-c/24)\,(9/48-(h-c/24)) \Big. \nonumber \\
& \quad \Big.
-18\, {\mu}\,(h-c/24)\, W_0 +9\,{\mu}\,G_0\,U_0 \Big) \, .\end{aligned}$$ The last two commutators are “effective”.
Similarly to the Ramond sector the irreducible representations of the zero mode algebra are one or two–dimensional. They are labeled by two weights. In the one–dimensional representation the zero modes are $$W_0=w\, , \qquad G_0=(h-c/24)^{1/2} \, , \qquad
U_0=0\, .$$ The representation exists only if $$\label{cond_1dim_tw2}
h-c/24=0 \qquad \text{or} \qquad
9 + 2\,c - 48\,h - 32\, {\mu}\, w=0 \, .$$ The two–dimension representation is constructed in the following way. One can choose to diagonalize $L_0$ and $W_0$, and then the zero mode algebra is satisfied by $${\setlength\arraycolsep{2pt} \begin{array}}{c}
W_0=
\left(
{\setlength\arraycolsep{2pt} \begin{array}}{cc}
w_1 & 0\\
0 & w_2\\
{\end{array}}\right),
\qquad
G_0=\left(
{\setlength\arraycolsep{2pt} \begin{array}}{cc}
0&g\\
g&\, 0\\
{\end{array}}\right) ,
\qquad
U_0=\left(
{\setlength\arraycolsep{2pt} \begin{array}}{cc}
0&u\\
-u&\, 0\\
{\end{array}}\right),\\[11pt]
g=(h-c/24)^{1/2}, \qquad u=g\,(w_2-w_1). {\end{array}}$$ There is a connection between $w_1$ and $w_2$: $$\label{connec w1_w2}
2\,c\,(w_1-w_2)^2-9\,{\mu}\,(w_1+w_2)-27\,(h-c/24)+81/16=0\, .$$ The two–dimensional [highest weight state]{} is labeled by two weights: $h$ and $w_1$.
Minimal models
==============
\[minimal models\]
Unitary representations
-----------------------
As we have shown in section \[N=1 superconformal subalgebras\] the existence of [$N\!=\!1$ ]{}superconformal subalgebras restricts the values of $c$ and ${\lambda}$ corresponding to unitary models of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra.
The unitary [highest weight representation]{}s of an algebra are unitary with respect to all its real subalgebras. So there are also restrictions on the weights of unitary [highest weight representation]{}s, coming from the non-unitarity theorem (appendix \[appN=1\]) of the N=1 superconformal algebra.
In the region III of $c$, ${\lambda}$ values (\[c cpl c\], \[c cpl cpl\]) there are 3 different [$N\!=\!1$ ]{}[superconformal ]{}subalgebras. The NS representation is labeled by three weights: $h$, $w$, $m$, which are linear functions of three weights ${d^{(1)}}$, ${d^{(2)}}$, ${d^{(3)}}$ of the three [$N\!=\!1$ ]{}subalgebras of central charge ${c^{\scriptscriptstyle{N\!=\!1}}_{k_1}}$, ${c^{\scriptscriptstyle{N\!=\!1}}_{k_2}}$, ${c^{\scriptscriptstyle{N\!=\!1}}_{k_1+k_2}}$ respectively. The connection between the two sets of weights is taken from (\[h\_to\_ddd\], \[m\_to\_ddd\], \[w\_to\_ddd\]). The necessary condition for NS representation to be unitary is that the weights ${d^{(1)}}$, ${d^{(2)}}$, ${d^{(3)}}$ are included in the correspondent Kac tables (\[eq:N=1\_dimensions\]) of conformal dimensions of the [$N\!=\!1$ ]{}[superconformal ]{}algebra. Therefore there is a finite number of unitary [highest weight representation]{}s in the region III models, and we can call them the minimal models of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra.
The [highest weight representation]{} of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra can be decomposed to the sum of representations of the [$N\!=\!1$ ]{}subalgebra. Let’s take for example the third subalgebra ($c={c^{\scriptscriptstyle{N\!=\!1}}_{k_1+k_2}}$). As we have shown in section \[N=1 superconformal subalgebras\] the generators of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra fall to the $\Phi_{1,1}$ and $\Phi_{1,3}$ representations of the third subalgebra. The fusion rule for $\Phi_{1,3}$ is $$\Phi_{1,3} \times \Phi_{m,n} =
\Phi_{m,n-2}+\Phi_{m,n}+\Phi_{m,n+2}\, ,$$ therefore the [highest weight state]{} of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra with ${d^{(3)}}=d_{m,n}$ has [descendant]{}s lying in the $\Phi_{m,n\pm 2}$, $\Phi_{m,n\pm 4}$, …representation of the third subalgebra, and the [${\mathcal{SW}}(3/2,3/2,2)$]{}[highest weight representation]{} is decomposed to the sum of the third subalgebra representations with dimensions from the same row of the Kac table.
Applying these conclusions to the Ramond sector one gets that ${\widetilde T}^{(3)}_0$ in the two–dimensional representation looks like $\left(
{\setlength\arraycolsep{2pt} \begin{array}}{cc}
d_{m,n} & 0\\
0& d_{m,n+2}
{\end{array}}\right)$ in the basis, where it is diagonal; $d_{m,n}$ is the dimension of Ramond representation of the [$N\!=\!1$ ]{}[superconformal ]{}algebra.
Although the zero modes of the three [$N\!=\!1$ ]{}subalgebras ${\widetilde T}^{(1)}_0$, ${\widetilde T}^{(2)}_0$, ${\widetilde T}^{(3)}_0$, can not be diagonalized simultaneously, one still can label the Ramond representation by 3 pairs of Ramond dimensions (nearest Ramond neighbors in the correspondent Kac tables). Taking trace of zero modes in (\[h\_to\_ddd\]) one obtains the conformal dimension $$\begin{aligned}
h&= \frac{1}{4}\,\Big( -
({d^{(3)}}_{m_3,n_3}+ {d^{(3)}}_{m_3,n_3+2})\left({k_1}
+{k_2} + 2 \right)
\Big. \nonumber \\
& \Big. \quad + ({d^{(1)}}_{m_1,n_1}+ {d^{(1)}}_{m_1+2,n_1})
\left({k_1} + 4 \right) +
({d^{(2)}}_{m_2,n_2}+ {d^{(2)}}_{m_2+2,n_2}) \left({k_2} + 4 \right)
\Big). \label{h ddd R}\end{aligned}$$
For the twisted sectors the situation is different. The twisted representations exist only in the minimal models with $k_1=k_2=k$. Since one cannot mix $G$ and $H$ generators in the twisted sector, the only [$N\!=\!1$ ]{}subalgebra is the third subalgebra, its generators are given by (\[gsub\_cpl\_zero\]) and (\[tsub\_cpl\_zero\]). The two weights of the [highest weight representation]{} in the tw1 sector can be chosen to be the ${\widetilde T}^{(3)}_0$ eigenvalue ${d^{(3)}}$ and the conformal dimension $h$. The tw1 representation is of Ramond type with respect to the third [$N\!=\!1$ ]{}subalgebra. The conditions (\[cond\_1dim\_tw2\]) for existence of the one dimensional representation in the tw2 sector of the $k_1=k_2=k$ minimal model are rewritten as $$h-c/24=0 \qquad \text{or} \qquad {d^{(3)}}=d_{k+2,k+2} \, .$$ The two–dimensional representation of the tw2 type is labeled by two weights: $h$ and a pair $({d^{(3)}}_1, {d^{(3)}}_2)$ of the nearest NS dimensions as the eigenvalues of the ${\widetilde T}^{(3)}_0$ operator. The connection (\[connec w1\_w2\]), being rewritten in terms of ${d^{(3)}}_1$ and ${d^{(3)}}_2$, states exactly that ${d^{(3)}}_1$ and ${d^{(3)}}_2$ are nearest NS neighbors in the row of the correspondent Kac table.
All the conditions of unitarity so discussed are not sufficient. In the case $k_1=1$ one gets additional restrictions on unitary representations by noting that the energy momentum tensor can be decomposed to two commuting parts $$T={\widetilde T}^{(2)} + T^{\text{Vir}}_{k_2+2}
\qquad \text{or} \qquad
T={\widetilde T}^{(3)} + T^{\text{Vir}}_{k_2}\, ,$$ where $T^{\text{Vir}}_{k}$ is the generator of the Virasoro algebra of central charge $$c=c_k^{N\! = \! 0}=1-\frac{6}{(k+2)(k+3)}\, ,$$ corresponding to the unitary minimal models of the Virasoro algebra [@BPZ]. (One could see the decomposition from (\[decom3\]) and (\[decom1\]).) This fact restricts the values of $h-{d^{(2)}}$ and $h-{d^{(3)}}$. For the $k_1=1$ models all the discussed restrictions on the weights are in fact sufficient conditions of unitarity of NS and Ramond representations. (We do not prove it here.) The discussion for $k_1=1$ applies to the $k_2=1$ case as well.
By taking formally $k_1=0$ one should obtain the minimal models of [$N\!=\!1$ ]{}[superconformal ]{}algebra (appendix \[appN=1\]).
In addition we know the examples (see section \[Examples\]) of explicit construction of $k_1=k_2=1$ and $k_1=k_2=2$ minimal models of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra in terms of $N\!=\!2$ models.
Based on these facts and with the help of the coset construction (\[coset\]) we guess the unitary spectrum of the general (arbitrary $k_1$ and $k_2$) [${\mathcal{SW}}(3/2,3/2,2)$]{}minimal model. The full list of minimal model unitary representations is presented in appendix \[list rep\]. The unitarity was also checked by computer calculations of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra Kac determinant on the few first levels.
The list of NS and Ramond representations forms a three–dimensional table with indices $s_1,s_2,s_3$, running in the range (\[s range\]). The twisted sector representations form a two–dimensional table with indices $t_1$ and $t_2$ (\[t range\]). There is a same number of NS and Ramond representations and the same number of tw1 and tw2 representations.
Substantial part of the spectrum could be predicted using the magic relation between the dimensions of any [$N\!=\!1$ ]{}minimal model (\[eq:N=1\_dimensions\]): $$d^k_{m,1}+d^k_{1,n}-d^k_{m,n}=\frac{(m-1)(n-1)}{4}
\qquad \forall k.$$ The relation is to be understood in the context of the fusion rule $$\Phi_{m,1} \times \Phi_{1,n}=\Phi_{m,n} \, .$$ Taking $m=3$ we get, that $\Phi_{1,n}$ fields are local or semilocal with respect to $\Phi_{3,1}\,$. The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is decomposed to $\Phi_{1,1} \oplus \Phi_{3,1}$ representation of the first [$N\!=\!1$ ]{}subalgebra. Therefore the field $\Phi_{1,n}$ (of the first [$N\!=\!1$ ]{}subalgebra) is a valid representation of the whole [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra, since it is local (or semilocal) with respect to all the generators of [${\mathcal{SW}}(3/2,3/2,2)$]{}. This representation is of Ramond or NS type, depending on $n$ is even or odd respectively. The conformal dimension $h$ of such a field coincides with the weight of the first [$N\!=\!1$ ]{}subalgebra. For $n \le 4$ the $\Phi_{1,n}$ field lies in the [highest weight representation]{} of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra, for $n>4$ it is [descendant]{}of some [highest weight representation]{}. We call such a representation the purely internal representation with respect to the first [$N\!=\!1$ ]{}subalgebra. Obviously the set of purely internal representations is closed under fusion rules. (The similar situation is encountered in the case of the [${\mathcal{SW}}(3/2,2)$]{}algebra, which have purely internal representations with respect to its Virasoro subalgebra (section 4 of [@gn]).) Of course, there are also purely internal representations with respect to the second and to the third [$N\!=\!1$ ]{}subalgebras. The representations $(s_1,1,1)$, $(1,s_2,1)$, $(1,1,s_3)$ are purely internal of the first, second and third subalgebras respectively. Such representations have simple fusion rules.
It is interesting to note, that $(1,s_2,s_3)$ fields of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra are local or semilocal with respect to the $(3,1,1)$ field.
Fusion rules {#Fusion rules}
------------
It is well known that the fusion rules of NS and Ramond sector representations have $\mathbb{Z}_2$ grading: (NS $\to 0$, Ramond $\to 1$ under addition modulo 2). The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra has two additional twisted sectors: tw1 and tw2. The full set of fusion rules has $\mathbb{Z}_2 \times \mathbb{Z}_2$ grading: NS $\to (0,0)$, Ramond $\to (1,0)$, tw1 $\to (0,1)$, tw2 $\to (1,1)$. The fusion rules of different sectors are summarized in the table:
NS R tw1 tw2
----- ----- ----- ----- -----
NS NS R tw1 tw2
R R NS tw2 tw1
tw1 tw1 tw2 NS R
tw2 tw2 tw1 R NS
The fusions of the [${\mathcal{SW}}(3/2,3/2,2)$]{}representations have to be consistent with fusion rules of its subalgebras. In the case of minimal models there are three [$N\!=\!1$ ]{}[superconformal ]{}subalgebras and the fusions of the [${\mathcal{SW}}(3/2,3/2,2)$]{}NS and Ramond representations are completely fixed by the fusions of the correspondent [$N\!=\!1$ ]{}minimal models (\[eq:N=1\_fusions\]). The fusion of $({s'_1,s'_2,s'_3})$ and $({s''_1,s''_2,s''_3})$ representation (see appendix \[list rep\]) of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is $$\label{eq:sww_fusions}
({s'_1,s'_2,s'_3}) \times ({s''_1,s''_2,s''_3}) =
\sum_{s_1=|s'_1-s''_1|+1}^
{\min(s'_1+s''_1-1, \atop
2k_1+3-(s'_1+s''_1))}
\,\,
\sum_{s_2=|s'_2-s''_2|+1}^
{\min(s'_2+s''_2-1, \atop
2k_2+3-(s'_2+s''_2))}
\,\,
\sum_{s_3=|s'_3-s''_3|+1}^
{\min(s'_3+s''_3-1, \atop
2(k_1+k_2)+7-(s'_3+s''_3))}
\!\!\!\!\!\!\!\!
({s_1,s_2,s_3}) \, ,$$ where $s_1$, $s_2$ and $s_3$ are raised by steps of 2. The selection of one index is independent on two others and satisfies the “${su(2)}$ pattern”. The fusion rules of the $N\!=\!0$ and the [$N\!=\!1$ ]{}minimal models satisfy the same pattern, the only difference is that in our case the table of representations is three–dimensional.
The proof is based on the first column of formula (\[m n min\_mod\]). The $({s_1,s_2,s_3})$ representation is decomposed to the sum of representations of the first [$N\!=\!1$ ]{}subalgebra from the column number $s_1$ of the correspondent [$N\!=\!1$ ]{}Kac table. Thus the column selection rule of the first [$N\!=\!1$ ]{}subalgebra is preserved and coincides with the $s_1$ selection in (\[eq:sww\_fusions\]). Similarly the $s_2$ and $s_3$ selection rules are adopted from the column and row selection rules of the second and the third [$N\!=\!1$ ]{}subalgebras respectively.
The twisted representations of the minimal models are labeled by two numbers: $t_1$ and $t_2$ (appendix \[list rep\]). But there is only one [$N\!=\!1$ ]{}subalgebra (the third one) in the twisted sector. One can reed from ([\[list tw reps\]]{}) that $t_1$ is the row number in the correspondent Kac table, meaning that $t_1$ has common selection rules with $s_3$. The selection of $t_2$ in the fusion rules can not be fixed by the described methods.
The “corner” entries of the three–dimensional table of NS and Ramond representations are the $(1,1,1)$, $(k_1+1,1,1)$, $(1,k_2+1,1)$, $(1,1,k_1+k_2+3)$ representations. The first one is the vacuum representation. The three others have the following fusion “square”: $$\Phi \times \Phi =I,$$ where $I$ denotes the identity (vacuum) representation. If such a field $\Phi$ is of the Ramond type, then the fusion of it with the other fields defines one-to-one transformation, mapping NS fields to Ramond ones. (This is an analogy of the $U(1)$ flow of the $N\!=\!2$ superconformal algebra [@Schwimmer:mf].) If the field $\Phi$ is of the NS type, then its conformal dimension $h$ is integer or half–integer and the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra can be extended to include this field. In the case, then both $k_1$ and $k_2$ are even, all three “corner” fields are of the NS type; in other cases (at least one $k$ is odd) one “corner” fields is of the NS type and two are of the Ramond type (and then there are two different NS–R isomorphisms).
Examples {#Examples}
--------
The following two examples are the $c=3/2$, ${\lambda}=0$ ($k_1=k_2=1$) and $c=9/4$, ${\lambda}=0$ ($k_1=k_2=2$) minimal models of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra. The former model is realized as $Z_2$ orbifold of a tensor product of free fermion and free boson on radius $\sqrt{3}$, the latter is realized as $Z_2$ orbifold of the sixth ($c=9/4$) minimal model of the $N=2$ superconformal algebra.
### $c=3/2$, ${\lambda}=0$ model {#k1_k2_1}
The boson on radius $\sqrt{3}$ is equivalent to the first minimal model of the $N=2$ superconformal algebra. The generators of [${\mathcal{SW}}(3/2,3/2,2)$]{}are constructed in the following way: $$\label{c32}
\begin{aligned}
T&=T^{N=2}+T^{\mathrm{Ising}} \, , \\
G&=\sqrt{3}\, J \, \psi \, , \\
H&=\sqrt{3/2} \, G_1 \, , \\
\end{aligned}
\qquad
\begin{aligned}
M&=-{\mathrm{i}}\, 3/\sqrt{2} \, G_2 \, \psi \, , \\
W&=1/\sqrt{2}\, T^{N=2}-\sqrt{2}\,T^{\mathrm{Ising}} \, , \\
U&=\sqrt{2/3} \left( 2\,J\,{\partial}\psi-{\partial}J\, \psi \right) , \\
\end{aligned}$$ where $T^{N=2}$, $G_1$, $G_2$ and $J$ are the (real) generators of the $N=2$ superconformal algebra at $c=1$, $\psi$ is the free fermion field and $T^{\mathrm{Ising}}$ is its energy–momentum operator. The expressions in (\[c32\]) are invariant under $\mathbb{Z}_2$ transformation . One can build all the [highest weight representation]{}s in the model (5 in NS, 5 in Ramond, 3 in every twisted sector) by appropriate combinations of representations of the $N=2$ minimal model and the Ising model. The list of representations is presented in table \[table\_k\_1=k\_2=1\] in appendix \[list rep\].
### $c=9/4$, ${\lambda}=0$ model
The NS sector of the sixth ($c=9/4$) $N=2$ minimal model contains [highest weight state]{} $\Psi_6^0$ of conformal dimension $3/2$ and zero $U(1)$ charge. The $N=2$ superconformal algebra can be extended by $N=2$ superprimary field corresponding to this state [@Inami:1989yi]. Then the fields, invariant under $\mathbb{Z}_2$ transformation form the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra: $$\label{c94}
\begin{aligned}
T&= T^{N=2} \, , \\
G&= G_1 \, , \\
\end{aligned}
\qquad
\begin{aligned}
H&=\Psi_6^0 \, , \\
M&=\Phi_6^0 \, , \\
\end{aligned}
\qquad
\begin{aligned}
W&=(2 \, T- 3 \, {{:} J J {:} })/\sqrt{5} \, , \\
U&=2 /\sqrt{5} \, (3\, {\mathrm{i}}\, {{:} J G_2 {:} }-{\partial}G_1) \, ,
\end{aligned}$$ where again $T^{N=2}$, $G_1$, $G_2$ and $J$ are the (real) generators of the $N=2$ superconformal algebra and ${ \big|\, \Phi_6^0 \, \big>}$ is a superpartner of ${ \big|\, \Psi_6^0 \, \big>}$: ${ \big|\, \Phi_6^0 \, \big>}=({G_1})_{-{\frac{1}{2}}} { \big|\, \Psi_6^0 \, \big>}$. The [highest weight representation]{}s of the $N=2$ minimal model can be easily transformed to representations of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra. The list of [highest weight representation]{}s consists of 16 NS, 16 Ramond, 6 tw1 and 6 tw2 representations (too long to be reproduced here explicitly).
Spectrum of the $G_2$ conformal algebra {#G2 algebra}
=======================================
In this section we discuss the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra at $c=10{\frac{1}{2}}, {\lambda}=0$. As we have shown in section \[N=1 superconformal subalgebras\] there is one real [$N\!=\!1$ ]{}subalgebra. It has central charge ${\widetilde c}=7/10$ and thus coincides with the tricritical Ising model. The subalgebra is generated by operators ${\widetilde G}$ (\[gsub\_cpl\_zero\]) and ${\widetilde T}$ (\[tsub\_cpl\_zero\]). The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is decomposed to the $\Phi_{1,1} \oplus \Phi_{3,1}$ representation of the [$N\!=\!1$ ]{}subalgebra. Since at ${\widetilde c}=7/10$ the fields $\Phi_{3,1}=\Phi_{2,2}$ are identical there is a new null state on level $2$: $$\label{null state}
3\,{\widetilde G}_{-3/2}\,{\widetilde G}_{-1/2}-2\,{\widetilde T}_{-2}\,{ \big|\, \Phi_{3,1} \, \big>} .$$ (The null state on level $3/2$ is already encoded in the structure of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra, but the existence of the null state (\[null state\]) is a special feature of the tricritical Ising model.) Translating (\[null state\]) to the language of generators of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra one gets that the null field is $$\label{eq:ideal}
2\,{\sqrt{14}}\,{{:} G W {:} } -
3\,{{:} H M {:} } +
2\,{{:} T G {:} } -
2\,{\sqrt{14}}\,{\partial}U .$$ This is an ideal of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra at $c=10{\frac{1}{2}}, {\lambda}=0$. (The existence of the ideal is known since [@Figueroa-O'Farrill:1996hm].)
In ref. [@Shatashvili:zw] the conformal algebra associated to the manifolds of $G_2$ holonomy is derived. Up to the ideal (\[eq:ideal\]) the $G_2$ conformal algebra coincides with the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra at $c=10{\frac{1}{2}}, {\lambda}=0$. In the free field representation, used by the authors of [@Shatashvili:zw] to obtain the $G_2$ algebra, the ideal (\[eq:ideal\]) vanishes identically. The authors of ref. [@Shatashvili:zw] used different basis of generators of the algebra. Their basis is connected to ours by $$\begin{aligned}
\Phi &={\mathrm{i}}\, H ,\\
K &= {\mathrm{i}}\, M ,
\end{aligned}
\qquad
\begin{aligned}
X &=-(T+\sqrt{14}\,W)/3 ,\\
\tilde M &= -({\partial}G+2\,\sqrt{14}\,U)/6 .
\end{aligned}$$ The $T$ and $G$ generators are the same. We have explicitly checked, that the [OPE]{}s in the first appendix of [@Shatashvili:zw] coincide (up to the ideal (\[eq:ideal\])) with the [OPE]{}s of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra.
Some unitary [highest weight representation]{}s of the $G_2$ algebra are found in [@Shatashvili:zw]. In this section we complete the list of unitary representations. Our calculation is based on the fact, that the $T, G, W, U$ fields of the $G_2$ algebra generate closed subalgebra modulo the same ideal (\[eq:ideal\]) and its [descendant]{}s. This subalgebra is the [${\mathcal{SW}}(3/2,2)$]{}superconformal algebra [@fofs; @Komata:1991cb; @Blumenhagen:1992nm; @Blumenhagen:1992vr; @Eholzer:1992pv; @Mallwitz:1994hh; @gn] of central charge $10{\frac{1}{2}}$. The $G_2$ algebra can be seen as an extended version of the [${\mathcal{SW}}(3/2,2)$]{}algebra. It is interesting to note that the [${\mathcal{SW}}(3/2,2)$]{}algebra at another value of central charge ($c=12$) is the superconformal algebra associated to manifolds of $Spin(7)$ holonomy [@Shatashvili:zw].
The complete spectrum of unitary representations of the [${\mathcal{SW}}(3/2,2)$]{}algebra is found in [@gn]. The $c=10{\frac{1}{2}}$ unitary model spectrum is presented in table \[sw spectrum\], where $x$ stands for real positive number in the continuous spectrum representations.
$$\begin{array}{|c|c||c|c|}
\hline
\multicolumn{2}{|c||}{\text{NS}}
&
\multicolumn{2}{|c|}{\text{Ramond}}\\
\hline
h&a&h&a\\
\hline
0 &0 & 7/16 & 0\\
3/8 & 3/80 & 7/16 & 3/80\\
1/2 & 1/10 & 7/16 & 1/10 \\
7/8 & 7/16 & 7/16 & 7/16 \\
1 &3/5 &15/16 & (1/10,3/5)\\
3/2 & 3/2& 31/16 & (3/5,3/2)\\
\hline
x&0 &7/16+x& (0,1/10)\\
3/8+x & 3/80 &7/16+x& 3/80\\
1/2+x & 1/10 &7/16+x& (3/80,7/16)\\
1+x &3/5 &15/16+x&(1/10,3/5)\\
\hline
\end{array}$$
There are NS and Ramond representations, they are labeled by two numbers: the conformal dimension $h$ and the internal dimension $a$ (the weight of the $c=7/10$ Virasoro subalgebra of the [${\mathcal{SW}}(3/2,2)$]{}algebra).
The $H$ field of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra is identified with the $h=3/2,a=3/2$ NS representation of the [${\mathcal{SW}}(3/2,2)$]{}algebra, $M$ is its superpartner.
The $(H,M)$ supermultiplet is purely internal with respect to the $c=7/10$ Virasoro subalgebra of the [${\mathcal{SW}}(3/2,2)$]{}algebra. Due to this fact we know its fusion rules with all other representations of the [${\mathcal{SW}}(3/2,2)$]{}algebra. In order to get the representations of [${\mathcal{SW}}(3/2,3/2,2)$]{}one have to combine the table \[sw spectrum\] representations in multiplets under the action of the $h=3/2, a=3/2$ field. It can be easily done, however this way one gets only two weights. But we know that the [${\mathcal{SW}}(3/2,3/2,2)$]{}NS and Ramond representations are labeled by three weights. It is convenient to choose them: the conformal dimension $h$; the weight of the [$N\!=\!1$ ]{}subalgebra, which coincides with the [${\mathcal{SW}}(3/2,2)$]{}algebra internal dimension $a$; and the eigenvalue $m$ of the $M$ zero mode. (In the case of two–dimensional Ramond representation $m$ stands for ${\frac{1}{2}}\text{Trace}(M_0)$.) The twisted representations are labeled by two weights: $h$ and $a$.
One obtains the third weight $m$ by using again the null field (\[eq:ideal\]). Acting by it on the [highest weight state]{} one should get zero. Consider, for example, the NS sector. Take the $G_{-1/2}$ [descendant]{} of the ideal: $$2{\sqrt{14}} \left(
2{{:} T W {:} }-{{:} G U {:} }-{\partial}^2 W \right)
-
{{:} G {\partial}G {:} } +
3{{:} H {\partial}H {:} } -
3{{:} M M {:} } +
4{{:} T T {:} } + {\partial}^2 T .$$ The eigenvalue of its zero mode should be set to zero for a consistent representation. This leads to a connection between $m$ and two other weights $h$ and $a$: $$m^2=10\,a\, (2\,h-1) .$$ The similar connection can be found in the Ramond case. Concluding, the unitary representations of the $G_2$ algebra are (again $x>0$):\
NS sector: $$\label{eq:g2 spectrum NS}
\begin{array}{ll}
1) & h=a=m=0 , \\
2) & h=1/2, \quad a=1/10 ,\quad m=0 , \\
3) & h=x ,\quad a=0 ,\quad m=0 , \\
4) & h=1/2+x ,\quad a=1/10 ,\quad m=\sqrt{x} .
\end{array}$$ Ramond sector: $$\label{eq:g2 spectrum R}
\begin{array}{ll}
1) & h=7/16, \quad a=7/16, \quad m=0 , \\
2) & h=7/16, \quad a=3/80, \quad m=0 , \\
3) & h=7/16+x, \quad a=(3/80,7/16), \quad m=0 , \\
4) & h=7/16+x, \quad a=3/80, \quad m=\sqrt{x}/2 .
\end{array}$$ tw1 sector: $$\label{eq:g2 spectrum tw1}
\begin{array}{ll}
1) & h=3/8, \quad a=3/80 , \\
2) & h=7/8, \quad a=7/16 , \\
3) & h=3/8+x, \quad a=3/80 .
\end{array}$$ tw2 sector: $$\label{eq:g2 spectrum tw2}
\begin{array}{ll}
1) & h=7/16, \quad a=1/10 , \\
2) & h=7/16, \quad a=0 , \\
3) & h=7/16+x ,\quad a=(0,1/10) .
\end{array}$$ The NS and Ramond discrete spectrum states (the first two in (\[eq:g2 spectrum NS\]) and (\[eq:g2 spectrum R\])) were found in [@Shatashvili:zw].
The first Ramond representation ($h=7/16, a=7/16, m=0$) is purely internal with respect to the tricritical Ising model. Due to this fact we know its fusions. Since its square is identity, the field serves as an isomorphism mapping, connecting NS and Ramond sectors, and connecting the tw1 and tw2 sectors. The representations of the same line number in (\[eq:g2 spectrum NS\]) and (\[eq:g2 spectrum R\]) and in (\[eq:g2 spectrum tw1\]) and (\[eq:g2 spectrum tw2\]) are isomorphic.
Summary
=======
In this paper we study the [${\mathcal{SW}}(3/2,3/2,2)$]{}superconformal algebra. We show by explicit construction that the coset (\[coset\]) contains the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra. The space of parameters ($c$ and ${\lambda}$) is divided to two regions (figure \[figure1\]).
In the first region there are unitary models at discrete points in the $c, {\lambda}$ space. These are the minimal models of the algebra, they are described by the coset (\[coset\]). In this region the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra has three different nontrivial [$N\!=\!1$ ]{}subalgebras. The conformal dimensions with respect to these subalgebras serve as the weights of the [${\mathcal{SW}}(3/2,3/2,2)$]{}[highest weight representation]{}s. The fusion rules are also dictated by the fusions of the [$N\!=\!1$ ]{}subalgebras. The characters of [highest weight representation]{}s are not discussed in this paper. We suppose that the characters can be easily obtained from the coset construction.
In the second region of parameters the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra has one [$N\!=\!1$ ]{}[superconformal ]{}subalgebra. The unitary models “lie” on unitary curves and have continuous spectrum of unitary representations. One of the continuous unitary models is the $c=10{\frac{1}{2}}$, ${\lambda}=0$ model. The [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra at these values of parameters coincides (up to a null field) with the superconformal algebra, associated to the manifolds of $G_2$ holonomy. From the other point of view it is an extended version of the [${\mathcal{SW}}(3/2,2)$]{}algebra, which at another value of the central charge ($c=12$) corresponds to the manifolds of $Spin(7)$ holonomy. We find the unitary spectrum of the $G_2$ holonomy algebra. The connection of various realizations of the $G_2$ superconformal algebra with the geometric properties of the $G_2$ manifolds is the open problem for study.
Acknowledgment {#acknowledgment .unnumbered}
--------------
It is pleasure to thank Doron Gepner and Alexander Zamolodchikov for useful discussions.
[OPE]{}s of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra {#apA}
------------------------------------------------------
$T$ and $G$ generate [$N\!=\!1$ ]{}superconformal algebra of central charge $c$. $(H,M)$ and $(W,U)$ are its superprimary fields. The nontrivial [OPE]{}s are: [$$\begin{aligned}
H(z)\, H(w)&=
\frac{\frac{2\, c}{3}}{(z-w)^3}+
\frac{{\lambda}\, M + 2\, T + \frac{4\, {{\mu}}}{3}\, W}{z-w} \, , \\
H(z)\, M(w)&=
\frac{3\, G + 3\, {{\lambda}}\, H}{(z-w)^2}+
\frac{-\frac{2\, {{\mu}}}{3}\, U + {\partial}G +
{{\lambda}}\, {\partial}H}{z-w} \, , \\
M(z)\, M(w)&=
\frac{2\, c}{(z-w)^4}+
\frac{4\, {\lambda}\, M +
8\, T + \frac{4\, {{\mu}}}{3}\, W}{(z-w)^2}+
\frac{2\, {\lambda}\, {\partial}M + 4\, {\partial}T
+ \frac{2\, {{\mu}}}{3}\, {\partial}W}{z-w} \, , \\
H(z)\, W(w)&=
\frac{{\mu}\, H}{(z-w)^2}+
\frac{\frac{{\lambda}}{2}\, U +
\frac{{{\mu}}}{3}\, {\partial}H}{z-w} \, , \\
\label{opeMW}
M(z)\, W(w)&=
\frac{\frac{{\mu}}{3} \, M+ 2\, {\lambda}\, W}{(z-w)^2}+
\frac{\frac{9\, {\mu}}{2\,
c}\, {{:} G H {:} } + \frac{{\mu}\, \left( -27 + 2\, c \right) }{12\,
c}\, {\partial}M + {\lambda}\, {\partial}W}{z-w} \, , \\
H(z)\, U(w)&=
\frac{\frac{-2\,{\mu}}{3}\,M + 2\,{\lambda}\,W}{{\left( z - w \right)
}^2} + \frac{\frac{9\,{\mu}}{2\,c}\,{{:} G H {:} } - \frac{{\mu}\,\left( 27 + 2\,c \right) }{12\,c}\,{\partial}M + \frac{{\lambda}}{2}\,{\partial}W}{z - w} \, , \\
M(z)\, U(w)&=
\frac{2\,{\mu}\,H}{{\left( z - w \right) }^3} +
\frac{\frac{5\,{\lambda}}{2}\,U + \frac{2\,{\mu}}{3} \,{\partial}H}{{\left( z
- w \right) }^2} \nonumber\\
&\hspace{15ex} +\frac{-\frac{9\,{\mu}}{2\,c}\,{{:} G M {:} } + \frac{9
\,{\mu}}{c}\,{{:} T H {:} } + {\lambda}\,{\partial}U + \frac{{\mu}\,\left( -27 +
2\,c \right) }{12\,c}\,{\partial}^2 H}{z - w} \, , \\
W(z)\, W(w)&=
\frac{c/2}{(z-w)^4} +
\frac{2\, T+\frac{{\lambda}}{2}\, M
+ \frac{{\mu}(10\, c -27) }{6\,c}\, W}{(z-w)^2}+
\frac{{\partial}T+\frac{{\lambda}}{4}\, {\partial}M
+ \frac{{\mu}(10\, c -27) }{12\,c}\, {\partial}W}{z-w} , \\
W(z)\, U(w)&=
\frac{-3\,G - \frac{3\,{\lambda}}{2}\,H}{{\left( z - w \right) }^3} +
\frac{\frac{{\mu}\,\left( -27 + 10\,c \right) }{12\,c}\,U - {\partial}G -
\frac{{\lambda}}{2}\,{\partial}H}{{\left( z - w \right) }^2} \nonumber\\
&\hspace{-4ex} -\frac{1}{48\, c\, (z - w)}
\Big(
162\,{\lambda}\,{{:} G M {:} } + 432\,{\mu}\,{{:} G W {:} } - 324\,{{:} H M {:} } +
648\,{{:} T G {:} } + 324\,{\lambda}\,{{:} T H {:} } \Big. \nonumber \\
& \hspace{-4ex}\Big. - 8\,{\mu}\,\left( 27 +
2\,c \right) \,{\partial}U + 6\,\left( -27 + 2\,c \right) \,{\partial}^2 G +
3\,\left( -27 + 2\,c \right) \,{\lambda}\,{\partial}^2 H \Big) \, , \\
U(z)\, U(w)&=
-\frac{2\, c}{(z-w)^5}-
\frac{\frac{5\,{\lambda}}{2}\,M + 10\,T +
\frac{{\mu}\,\left( -27 + 10\,c \right) }{3\,c}\,W}
{(z-w)^3}\nonumber \\
& \hspace{-4ex} -
\frac{\frac{5\,{\lambda}}{4}\,{\partial}M + 5\,{\partial}T + \frac{{\mu}\,\left( -27 +
10\,c \right) }{6\,c}\,{\partial}W}{(z-w)^2}
-\frac{1}{16\, c\, (z-w)} \Big(
-144\,{\mu}\,{{:} G U {:} } - 108\,{{:} G {\partial}G {:} }
\Big. \nonumber \\
& \hspace{-4ex} - 54\,{\lambda}\,{{:} G {\partial}H {:} } +
108\,{{:} H {\partial}H {:} } - 108\,{{:} M M {:} } + 216\,
{\lambda}\,{{:} T M {:} } + 432\,{{:} T T {:} } +
288\,{\mu}\,{{:} T W {:} }
\nonumber \\
& \hspace{-4ex} \Big. + 54\,{\lambda}\,{{:} {\partial}G H {:} } - 3\,\left( 9 - 2\,c \right) \,{\lambda}\,{\partial}^2 M + 24\,c\,{\partial}^2 T - 4\,{\mu}\,
\left( 27 - 2\,c \right) \,{\partial}^2 W
\Big) \, .\end{aligned}$$ ]{} where $${\mu}= \sqrt{\frac{9\, c\, (4+{\lambda}^2)}{2 \, (27-2\, c)}}$$ and the fields in the right hand sides of the [OPE]{}s are taken in the point $w$.
Unitary minimal models of the [$N\!=\!1$ ]{}superconformal {#appN=1}
-----------------------------------------------------------
At $c<3/2$ all unitary representations of the [$N\!=\!1$ ]{}superconformal algebra are described by its minimal models. Their central charge is $$\label{C_k}
{c^{\scriptscriptstyle{N\!=\!1}}_{k}} = \frac{3}{2}-\frac{12}{(k+2)(k+4)}\, ,
\qquad k=0,1,2,\ldots$$ The conformal dimensions of the unitary [highest weight representation]{}s $\Phi_{m,n}$ of the $c={c^{\scriptscriptstyle{N\!=\!1}}_{k}}$ minimal model are given in the Kac table $$\begin{aligned}
\label{eq:N=1_dimensions}
d^{k}_{m,n}&=\frac{{\left( \left( k + 2 \right)
\,m - \left( k + 4\right)
\,n \right)}^2 - 4}
{8\,\left( k + 2 \right) \,\left( k + 4 \right) }+\frac{r}{16}\, ,
\qquad
{\setlength\arraycolsep{2pt} \begin{array}}{rcl}
m&=&1,2,\ldots, k+3\, , \\
n&=&1,2,\ldots, k+1\, ,
{\end{array}}\\
r&={(m+n) \bmod 2}=\left\{
{\setlength\arraycolsep{2pt} \begin{array}}{l}
0, \quad \mathrm{NS\ sector,}\\
1, \quad \mathrm{Ramond\ sector.}
{\end{array}}\right.
\end{aligned}$$ The fusion rules are given by ${su(2)}$ like selection rules for every index ($m$ and $n$): $$\label{eq:N=1_fusions}
\Phi_{m_1,n_1} \times \Phi_{m_2,n_2} =
\sum_{m=|m_1-m_2|+1}^
{\min(m_1+m_2-1, \atop
2k+7-(m_1+m_2))} \,\,
\sum_{n=|n_1-n_2|+1}^
{\min(n_1+n_2-1,\atop
2k+3-(n_1+n_2)) } \,\,
\Phi_{m,n} \, ,$$ where the indices $m$ and $n$ in the sums are raised by steps of 2.
The [$N\!=\!1$ ]{}minimal models correspond to the diagonal coset construction [@Goddard:1986ee]: $$\label{coset N=1}
\frac{{su(2)}_k \oplus {su(2)}_2}{{su(2)}_{k+2}} \, .
$$
Coset construction of the [${\mathcal{SW}}(3/2,3/2,2)$]{}algebra {#apB}
----------------------------------------------------------------
We present the explicit construction of the [${\mathcal{SW}}(3/2,3/2,2)$]{}generators in terms of the coset (\[coset\]) currents. For details and notations see section \[sec:expli\_const\]. [$$\begin{aligned}
T &=
\frac{1}{ k_1+k_2+4}
\Bigg(
-{\frac{1}{2}}\, (k_1+k_2) \, {{:} \psi_i {\partial}\psi_i {:} }
- 2 \, {{:} J_i^{(1)} J_i^{(2)} {:} } \Bigg. \nonumber \\
&\Bigg. +
\bigg( \big( \, \frac{k_2+2}{k_1+2} \, {{:} J_i^{(1)} J_i^{(1)} {:} }
- 2 \, {{:} J_i^{(1)} J_i^{(3)} {:} } \big)+
\big(1 \leftrightarrow 2 \big) \bigg)
\Bigg)
, \\
G &=
\frac{\sqrt{2}}{\left(
(k_1+2) (k_2+2) (k_1+k_2+4)\right)^{1/2}} \times \nonumber \\
& \times \Bigg(
{\mathrm{i}}\, (k_1-k_2) \, {{:} \psi_1 \psi_2 \psi_3 {:}}+
\bigg( (k_2+2)\, {{:} J_i^{(1)} \psi_i {:} }
-\big(1 \leftrightarrow 2 \big) \bigg) \Bigg)
, \\
H &=
-\frac{1}{\left(
3\, k_1 \, k_2 \,
(k_1+2) (k_2+2) (k_1+k_2+4) (k_1+k_2+6)\right)^{1/2}} \times \nonumber \\
& \times \Bigg(
3\, {\mathrm{i}}\, k_1 \, k_2 \,(k_1+k_2+4) \,
{{:} \psi_1 \psi_2 \psi_3 {:}}+
\bigg(2\, k_2 \, (2\, k_1+k_2+6)\, {{:} J_i^{(1)} \psi_i {:} }+
\big(1 \leftrightarrow 2 \big) \bigg) \Bigg)
, \\
M &=
\frac{{\lambda}_{k_1,k_2}}{4\,(2\, k_1+k_2+6)(k_1+ 2\, k_2+6)}
\Bigg(
6 \, (k_1 \, k_2 -2 \, k_1-2 \, k_2 -12)
\, {{:} J_i^{(1)} J_i^{(2)} {:} }
\Bigg. \nonumber \\
& -3 \, k_1 \, k_2 \, (k_1+k_2+4) \, {{:} J_i^{(3)} J_i^{(3)} {:} }
+ \bigg( \big(-\frac{6 \, k_2 \, (k_2+2) (2\, k_1 +k_2 +6)}{k_1-k_2}
\, {{:} J_i^{(1)} J_i^{(1)} {:} } \big. \bigg. \nonumber \\
& \Bigg.
\bigg. \big. + \frac{3\, k_2 \, (k_1+k_2+4)
(3 \, k_1 \, k_2 +10 \, k_1+2\, k_2 +12)}
{k_1-k_2}
\, {{:} J_i^{(1)} J_i^{(3)} {:} }
\big)
+\big(1 \leftrightarrow 2 \big) \bigg)
\Bigg)
, \\
W &=
{\left( \frac{c_{k_1, k_2}}
{9 \, k_1 \, k_2 \, (k_1+2)(k_2+2)(k_1+k_2+4) (k_1+k_2+6)}
\right)}^{1/2}
\times \nonumber \\
& \times
\Bigg( {\frac{1}{2}}k_1 \, k_2 (k_1+k_2+4)
\, {{:} J_i^{(3)} J_i^{(3)} {:} }
+2 (k_1 \, k_2 +4\, k_1 +4\, k_2+12)
\, {{:} J_i^{(1)} J_i^{(2)} {:} } \Bigg. \nonumber \\
& \Bigg. + \bigg( \big(
- k_2 (k_2+2) \, {{:} J_i^{(1)} J_i^{(1)} {:} }
-2\, k_2 (k_1+k_2+4) \, {{:} J_i^{(1)} J_i^{(3)} {:} }
\big) +
\big( 1 \leftrightarrow 2 \big) \bigg) \Bigg)
, \\
U &=
{\left(\frac{2 \, c_{k_1, k_2}}
{9 \, k_1 \, k_2 \, (k_1+k_2+6)}
\right)}^{1/2} \times \nonumber \\
& \times
\left(
-6\, {\mathrm{i}}\,
{:} \left|
{\setlength\arraycolsep{2pt} \begin{array}}{ccc}
J_1^{(1)} & J_2^{(1)} & J_3^{(1)} \\
J_1^{(2)} & J_2^{(2)} & J_3^{(2)} \\
\psi_1 & \psi_2 & \psi_3 \\
{\end{array}}\right| {:}+
\bigg( \big(k_2 \, {{:} {\partial}J_i^{(1)} \psi_i {:} }
-2\, k_2 \, {{:} J_i^{(1)} {\partial}\psi_i {:} } \big)
+ \big( 1 \leftrightarrow 2 \big) \bigg)
\right) .\end{aligned}$$ ]{}
[${\mathcal{SW}}(3/2,3/2,2)$]{}minimal models
---------------------------------------------
\[list rep\]
Here we present the complete list of unitary [highest weight representation]{}s of the [${\mathcal{SW}}(3/2,3/2,2)$]{}minimal models. The central charge $c$ and the coupling ${\lambda}$ of the $(k_1, k_2)$ minimal model are given in (\[c cpl c\], \[c cpl cpl\]). The list of NS and Ramond sector representations can be presented in the form of three–dimensional table with indices $s_1$, $s_2$, $s_3$: $$\label{s range}
\begin{aligned}
s_1&= 1,2,\ldots,k_1+1, \\
s_2&= 1,2,\ldots,k_2+1, \\
s_3&= 1,2,\ldots,k_1+k_2+3 .
\end{aligned}$$ The representation ${(s_1,s_2,s_3)}$ is of NS or Ramond type depending on $s_1+s_2+s_3$ is odd or even respectively. The [highest weight representation]{} is labeled by 3 weights ${d^{(1)}}$, ${d^{(2)}}$, ${d^{(3)}}$, the conformal dimensions with respect to the three [$N\!=\!1$ ]{}[superconformal ]{}subalgebras. Their values are taken from the correspondent [$N\!=\!1$ ]{}Kac tables (\[eq:N=1\_dimensions\]): $${d^{(1)}}=d^{k_1}_{m_1,n_1}\, , \qquad
{d^{(2)}}=d^{k_2}_{m_2,n_2}\, , \qquad
{d^{(3)}}=d^{k_1+k_2}_{m_3,n_3}\, ,$$ where the indices are connected to $s_1$, $s_2$, $s_3$ by $$\label{m n min_mod}
\begin{aligned}
n_1&=s_1 \, ,\\ n_2&=s_2 \, ,\\ m_3&=s_3 \, ,
\end{aligned}
\qquad
\begin{aligned}
m_1&=s_1-Y_1 +Y_2+Y_3 \pm r ,\\
m_2&=s_2+Y_1 -Y_2+Y_3 \pm r ,\\
n_3&=s_3+Y_1 +Y_2-Y_3 \pm r .
\end{aligned}$$ $Y_1$, $Y_2$, $Y_3$ are values of the $Y_{a,b}(x)$ function: $$\begin{aligned}
Y_1&=Y_{2,2 k_2+2}\,(s_1-s_2-s_3+1), \\
Y_2&=Y_{2,2 k_1+2}\,(s_2-s_3-s_1+1), \\
Y_3&=Y_{2,2 \min(k_1,k_2)+2}\,(s_3-s_1-s_2+1).
\end{aligned}$$ We define the function $Y_{a,b}(x)$ by its graph:
The number $r$ in (\[m n min\_mod\]) can be 0 or 1. It is 0 in the NS sector ($s_1+s_2+s_3$ odd). In the Ramond sector it is given by $$\begin{aligned}
r&=1-\text{sgn}
\Big(
Y'_{2,2 k_2+2}\,(s_1-s_2-s_3+1) \Big. \nonumber \\
& \quad \Big. +Y'_{2,2 k_1+2}\,(s_2-s_3-s_1+1)
+Y'_{2,2 \min(k_1,k_2)+2}\,(s_3-s_1-s_2+1)
\Big),\end{aligned}$$ where $Y'_{a,b}(x)$ is a derivative of $Y_{a,b}(x)$ with respect to $x$. The $Y'$ function is not continuous, but the values in the points of discontinuity are not important. $r$ distinguishes between one ($r=0$) and two–dimensional ($r=1$) Ramond representations.
The conformal dimension $h$ (the eigenvalue of $L_0$ operator) is calculated from ${d^{(1)}}$, ${d^{(2)}}$, ${d^{(3)}}$ weights by $$\label{h ddd}
h= \frac{1}{2}\,\Big( -
{d^{(3)}}\left({k_1} +{k_2} + 2 \right)
+ {d^{(1)}}
\left({k_1} + 4 \right) +
{d^{(2)}} \left({k_2} + 4 \right)
\Big)$$ in the case of one–dimensional (NS or Ramond) representation. In the case of two–dimensional Ramond representation the ${d^{(1)}}$, ${d^{(2)}}$, ${d^{(3)}}$ in (\[h ddd\]) should be substituted by half of the sum of the correspondent [$N\!=\!1$ ]{}Ramond dimensions (\[h ddd R\]).
The following representations are identical: $$\label{eq:rep ident}
{(s_1, s_2, s_3)}={(k_1+2-s_1, k_2+2-s_2, k_1+k_2+4-s_3)}.$$ There are $[ \frac{(k_1+1)(k_2+1)(k_1+k_2+3)+1}{4} ]$ NS representations and the same number of Ramond representations.
The $k_1=k_2=k$ minimal models contain two additional twisted sectors: tw1 and tw2. The list of tw1 and tw2 representations can be arranged in the two–dimensional table with indices $t_1$ and $t_2$: $$\label{t range}
\begin{aligned}
t_1&= 1,2,\ldots,k+2, \\
t_2&= 1,2,\ldots,k+1.
\end{aligned}$$ The $t_1+t_2$ even entries are of tw2 type and the $t_1+t_2$ odd entries are of tw1 type. The representations in the twisted sectors are labeled by the conformal dimension $h$ and the weight of the third [$N\!=\!1$ ]{}subalgebra ${d^{(3)}}$: $$\label{list tw reps}
{\setlength\arraycolsep{2pt} \begin{array}}{rclcrcl}
h&=& {\displaystyle}\frac{|t_1-t_2|}{4}+
\frac{t_2^2-t_1^2+\delta}{8\,(k+2)} \, , &\, &
\delta&=&\left\{
\begin{array}{ll}
k-1, \phantom{t_2-1} & \quad\text{tw1},\\
3\,k/2, & \quad\text{tw2};
\end{array}
\right.
\\[16pt]
{d^{(3)}}&= & {\displaystyle}{d^{(2\,k)}}_{m,n}\, ,\quad
\begin{aligned}
m&=t_1, \\
n&=t_1+\text{sgn}(t_2-t_1) \pm r,
\end{aligned}
&\quad &
r&=&\left\{
\begin{array}{ll}
0, & \quad\text{tw1},\\
\text{sgn}(t_2-t_1), & \quad\text{tw2}.
\end{array}
\right.
{\end{array}}$$ Again $r$ distinguishes between one ($r=0$) and two–dimensional ($r=1$) tw2 representations. There are $(k+1)(k+2)/2$ tw1 representations and the same number of tw2 representations.
We want to illustrate the formulas of the present appendix by some explicit examples. The simplest model is the $k_1=k_2=1$ ($c=3/2, {\lambda}=0$) model, discussed in section \[k1\_k2\_1\]. Its [highest weight representation]{}s are presented in table \[table\_k\_1=k\_2=1\]. We use the ${
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
h \\[4pt]
{d^{(1)}},{d^{(2)}},{d^{(3)}}
\end{array}
\right)}$ notation for the NS and Ramond representations and the ${
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
h \\[4pt]
{d^{(3)}}
\end{array}
\right)}$ notation for the twisted sectors. One of the four NS/Ramond weights is dependent on other three and is presented for convenience only. The Ramond and tw1 sectors are [*slanted*]{}. The $h=9/16$ Ramond representation is two-dimensional.
$$\!\!\!\!\!
\setlength\arraycolsep{3pt}
\begin{array}{c}
\phantom{s_3 s_3 s_3} s_2 \rightarrow \hfill\\[2pt]
{\begin{array}{c}
s_3\\
\downarrow
\vspace{6cm}
\end{array}
}
{\setlength\arraycolsep{2pt}
\begin{array}{|cc|}
\hline
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
0 \\[4pt]
0,0,0
\end{array}
\right)} & {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{7}{16}}} \\[4pt]
{\it{\frac{3}{80},\frac{7}{16}
,\frac{3}{8}}}
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{1}{16}}} \\[4pt]
{\it{\frac{3}{80},\frac{3}{80},\frac{1}{16}}}
\end{array}
\right)}
& {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{8} \\[4pt]
0,\frac{1}{10},\frac{1}{16}
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{6} \\[4pt]
\frac{1}{10},\frac{1}{10},\frac{1}{6}
\end{array}
\right)} & {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{5}{48
}}} \\[4pt]
{\it{\frac{3}{80},\frac{3}{80},\frac{1}{24}}}
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{9}{16}}} \\[4pt]
{\it{{
\frac{3}{80} \atop \frac{7}{16}} ,{ \frac{3}{80} \atop \frac{7}{16}} ,
{ \frac{1}{16} \atop \frac{9}{16}} }}
\end{array}
\right)} & {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{8} \\[4pt]
\frac{1}{10},0,
\frac{1}{16}
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{2} \\[4pt]
\frac{1}{10},\frac{1}{10},0
\end{array}
\right)} & {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{7}{16}}} \\[4pt]
{\it{\frac{7}{16},\frac{3}{80},\frac{3}{8}}}
\end{array}
\right)}\\[17pt]
\hline
\end{array}
}
\\ \\[-8pt]
\phantom{s_3 s_3} s_1=1
\end{array}
\,\begin{array}{c}
\phantom{s_3 s_3 s_3} s_2 \rightarrow \hfill\\[2pt]
{\begin{array}{c}
s_3\\
\downarrow
\vspace{6cm}
\end{array}
}
{\setlength\arraycolsep{2pt}
\begin{array}{|cc|}
\hline
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{7}{16}}} \\[4pt]
{\it{\frac{7}{16},\frac{3}{80},\frac{3}{8}}}
\end{array}
\right)}
& {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{2} \\[4pt]
\frac{1}{10},\frac{1}{10},0
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{8} \\[4pt]
\frac{1}{10},0,\frac{1}{16}
\end{array}
\right)} &
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{9}{16}}} \\[4pt]
{\it{{
\frac{3}{80} \atop \frac{7}{16}} ,{ \frac{3}{80} \atop \frac{7}{16}} ,
{ \frac{1}{16} \atop \frac{9}{16}} }}
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{5}{48}}} \\[4pt]
{\it{\frac{3}{80},\frac{3}{80},\frac{1}{24}}}
\end{array}
\right)}
& {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{6} \\[4pt]
\frac{1}{10},\frac{1}{10
},\frac{1}{6}
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
\frac{1}{8} \\[4pt]
0,\frac{1}{10},\frac{1}{16}
\end{array}
\right)} & {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{1}{16}}} \\[4pt]
{\it{\frac{3}{80},\frac{3}{80},\frac{1}{16}
}}
\end{array}
\right)} \\[7pt]
{
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
{\it{\frac{7}{16}}} \\[4pt]
{\it{\frac{3}{80},\frac{7}{16},\frac{3}{8}
}}
\end{array}
\right)} & {
\rule{0cm}{28pt}
\left(
\setlength\arraycolsep{0pt}
\begin{array}{c}
0 \\[4pt]
0,0,0
\end{array}
\right)} \\[17pt]
\hline
\end{array}
}
\\ \\[-8pt]
\phantom{s_3 s_3} s_1=2
\end{array}
\quad
$$
[c]{} t\_2\
[c]{} t\_1\
[|cc|]{}
------------------------------------------------------------------------
(
[c]{}\
0
)
&
------------------------------------------------------------------------
(
[c]{} [**]{}\
[*[ ]{}*]{}
)
\
------------------------------------------------------------------------
(
[c]{} [**]{}\
[**]{}
)
&
------------------------------------------------------------------------
(
[c]{}\
)
\
------------------------------------------------------------------------
(
[c]{}\
)
&
------------------------------------------------------------------------
(
[c]{} [**]{}\
[**]{}
)
\
\
\
$$\
NS and Ramond
The second example is the $k_1=2, k_2=3$ ($c=37/15, {\lambda}=-182/(405 \sqrt{11})$) minimal model. Since the list of representations is too long, we reproduce only the conformal dimensions $h$ of the [highest weight representation]{}s (table \[k\_1=2\_k\_2=3\]).
$$\begin{array}{c}
\phantom{s_3 s_3 s_3 s_3} s_2 \rightarrow \hfill\\[2pt]
{\begin{array}{c}
s_3\\
\downarrow
\vspace{4.2cm}
\end{array}
}
\setlength\arraycolsep{2pt}
\begin{array}{|cccc|}
\hline
\rule{0cm}{15pt}
{0} & {\it{\frac{27}{80}}} & {\frac{9}{10}} & {\it{\frac{31}{16}}} \\[7pt] {\it{
\frac{5}{48}}} & {\frac{1}{15}} & {\it{\frac{121}{240}}} & {\frac{7}{6}} \\[7pt] {\frac{
5}{18}} & {\it{\frac{83}{720}}} & {\frac{8}{45}} & {\it{\frac{103}{144}}} \\[7pt] {\it{\frac{
37}{48}}} & {\frac{7}{30}} & {\it{\frac{41}{240}}} & {\frac{1}{3}} \\[7pt] {\frac{5}{6}
} & {\it{\frac{161}{240}}} & {\frac{7}{30}} & {\it{\frac{13}{48}}} \\[7pt] {\it{\frac{175}{14
4}}} & {\frac{61}{90}} & {\it{\frac{443}{720}}} & {\frac{5}{18}} \\[7pt] {\frac{7}{6}
} & {\it{\frac{241}{240}}} & {\frac{17}{30}} & {\it{\frac{29}{48}}} \\[7pt] {\it{\frac{23}{16
}}} & {\frac{9}{10}} & {\it{\frac{67}{80}}} & {1} \\[7pt]
\hline
\end{array}
\\ \\[-8pt]
\phantom{s_3 s_3} s_1=1
\end{array}
\quad
\begin{array}{c}
\phantom{s_3 s_3 s_3 s_3} s_2 \rightarrow \hfill\\[2pt]
{\begin{array}{c}
s_3\\
\downarrow
\vspace{4.2cm}
\end{array}
}
\setlength\arraycolsep{2pt}
\begin{array}{|cccc|}
\hline
\rule{0cm}{15pt}
{\it{\frac{3}{8}}} & {\frac{27}{80}} & {\it{\frac{31}{40}}} & {\frac{23}{16}
} \\[7pt] {\frac{5}{48}} & {\it{\frac{53}{120}}} & {\frac{121}{240}} & {\it{\frac{25}{24}
}} \\[7pt] {\it{\frac{11}{72}}} & {\frac{83}{720}} & {\it{\frac{199}{360}}} & {\frac{103}{1
44}} \\[7pt] {\frac{13}{48}} & {\it{\frac{13}{120}}} & {\frac{41}{240}} & {\it{\frac{17}{
24}}} \\[7pt] {\it{\frac{17}{24}}} & {\frac{41}{240}} & {\it{\frac{13}{120}}} & {\frac{13}{
48}} \\[7pt] {\frac{103}{144}} & {\it{\frac{199}{360}}} & {\frac{83}{720}} & {\it{\frac{1
1}{72}}} \\[7pt] {\it{\frac{25}{24}}} & {\frac{121}{240}} & {\it{\frac{53}{120}}} & {\frac{
5}{48}} \\[7pt] {\frac{23}{16}} & {\it{\frac{31}{40}}} & {\frac{27}{80}} & {\it{\frac{3}{
8}}} \\[7pt]
\hline
\end{array}
\\ \\[-8pt]
\phantom{s_3 s_3} s_1=2
\end{array}
\quad
\begin{array}{c}
\phantom{s_3 s_3 s_3 s_3} s_2 \rightarrow \hfill\\[2pt]
{\begin{array}{c}
s_3\\
\downarrow
\vspace{4.2cm}
\end{array}
}
\setlength\arraycolsep{2pt}
\begin{array}{|cccc|}
\hline
\rule{0cm}{15pt}
{1} & {\it{\frac{67}{80}}} & {\frac{9}{10}} & {\it{\frac{23}{16}
}} \\[7pt] {\it{\frac{29}{48}}} & {\frac{17}{30}} & {\it{\frac{241}{240}}} & {\frac{7}{6}
} \\[7pt] {\frac{5}{18}} & {\it{\frac{443}{720}}} & {\frac{61}{90}} & {\it{\frac{175}{144
}}} \\[7pt] {\it{\frac{13}{48}}} & {\frac{7}{30}} & {\it{\frac{161}{240}}} & {\frac{5}{6}
} \\[7pt] {\frac{1}{3}} & {\it{\frac{41}{240}}} & {\frac{7}{30}} & {\it{\frac{37}{48}
}} \\[7pt] {\it{\frac{103}{144}}} & {\frac{8}{45}} & {\it{\frac{83}{720}}} & {\frac{5}{18}
} \\[7pt] {\frac{7}{6}} & {\it{\frac{121}{240}}} & {\frac{1}{15}} & {\it{\frac{5}{48}
}} \\[7pt] {\it{\frac{31}{16}}} & {\frac{9}{10}} &
{\it{\frac{27}{80}}} & {0} \\[7pt]
\hline
\end{array}
\\ \\[-8pt]
\phantom{s_3 s_3} s_1=3
\end{array}$$
Mode expansions of normal ordered products
------------------------------------------
\[NOexp\]
Here we derive the formula for the mode expansion of normal ordered product of operators in various sectors. The normal ordered product ${{:} P Q {:} }$ is defined as the zero order term in [OPE]{}: $$P(z) \, Q(w)=
\sum_{k=1}^N \frac{R^{(k)}(w)}{(z-w)^k}
+{{:} P Q {:} }(w)+O(z-w) \, .$$ The well known formula for the mode expansion of ${{:} P Q {:} }$ $$\label{NOexpansion}
{{:} P Q {:} }_n =\sum_{m \le -\Delta_P} P_m Q_{n-m}+
(-1)^{PQ} \! \! \! \! \! \sum_{m \ge -\Delta_P +1} Q_{n-m} P_m \, ,$$ is valid only if $m$ has the same modding as $\Delta_P$, i.e. $m$ runs on integer or half integer numbers depending on the spin of $P$ is integer or half integer respectively. So in the NS sector the expansion (\[NOexpansion\]) works. In the case of Ramond or twisted sectors the formula should be modified.
The idea of the following calculation is taken from ref.[@Odake:1988bh] (section 3), where the mode expansion of ${{:} G^+ G^- {:} }$ was obtained using the same method. ($G^+$ and $G^-$ are the supersymmetry generators of the $N=2$ superconformal algebra.) Let’s calculate the integral: $$\label{NOintegral}
\oint_0 \! \mathrm{d} w \, w^{n+\Delta_P+\Delta_Q-1}
\oint_w \! \mathrm{d} z \, \frac{1}{z-w} \, \,
z^{\epsilon} \, P(z) \, Q(w) \, w^{-\epsilon} \, ,$$ where the first integration is around $w$ and the second is around $0$. The integral (\[NOintegral\]) is equal to the $n$–mode $${{:} P Q {:} }_n +
\sum_{k=1}^N { \epsilon \choose k} \, R^{(k)}_n \, .$$
The integration contour in (\[NOintegral\]) can be transformed to $${\oint \! \! \oint \! \!}_{z>w} \mathrm{d} z \, \mathrm{d} w-
{\oint \! \! \oint \! \!}_{w>z} \mathrm{d} w \, \mathrm{d} z \, .$$ The $z^{\epsilon}$ term in the integration function was introduced to compensate the phase change of $P(z)$ around $z=0$.
Expanding $(z-w)^{-1}$ and integrating one gets $$\label{NOepsilon}
{{:} P Q {:} }_n =
- \sum_{k=1}^N { \epsilon \choose k} \, R^{(k)}_n+
\sum_{m \le -\Delta_P + \epsilon} P_m Q_{n-m}+
(-1)^{PQ} \! \! \! \! \! \sum_{m \ge -\Delta_P +1+\epsilon} Q_{n-m} P_m \, ,$$ where $m$ runs on $\mathbb{Z}-\Delta_P + \epsilon$. $\epsilon$ is usually chosen to be $0$ or $1/2$ to produce the correct modding for operator $P$. In the case $\epsilon=0$ we get back the formula (\[NOexpansion\]) as expected. Note that the formula (\[NOepsilon\]) is valid for any $\epsilon$ (not only 0 or $1/2$) consistent with the chosen modding.
Another approach to the calculation of mode expansions of composite operators is presented in [@Eholzer:1992pv] (section 3) and in [@gn] (appendix C) and leads to the same results.
[99]{}
S. L. Shatashvili and C. Vafa, “Superstrings And Manifold Of Exceptional Holonomy,” Selecta Math. [**1**]{} (1995) 347–381 \[arXiv:hep-th/9407025\].\
S. L. Shatashvili and C. Vafa, “Exceptional magic,” Nucl. Phys. Proc. Suppl. [**41**]{} (1995) 345. A. B. Zamolodchikov, “Infinite Additional Symmetries In Two-Dimensional Conformal Quantum Field Theory,” Theor. Math. Phys. [**65**]{} (1985) 1205. P. Bouwknegt and K. Schoutens, “W symmetry in conformal field theory,” Phys. Rept. [**223**]{} (1993) 183 \[arXiv:hep-th/9210010\]. J. M. Figueroa-O’Farrill, “A note on the extended superconformal algebras associated with manifolds of exceptional holonomy,” Phys. Lett. B [**392**]{} (1997) 77 \[arXiv:hep-th/9609113\]. D. Gepner and B. Noyvert, “Unitary representations of SW(3/2,2) superconformal algebra,” Nucl. Phys. B [**610**]{} (2001) 545 \[arXiv:hep-th/0101116\]. T. Eguchi and Y. Sugawara, “CFT description of string theory compactified on non-compact manifolds with G(2) holonomy,” Phys. Lett. B [**519**]{} (2001) 149 \[arXiv:hep-th/0108091\]. K. Sugiyama and S. Yamaguchi, “Cascade of special holonomy manifolds and heterotic string theory,” Nucl. Phys. B [**622**]{} (2002) 3 \[arXiv:hep-th/0108219\]. R. Blumenhagen and V. Braun, “Superconformal field theories for compact G(2) manifolds,” JHEP [**0112**]{} (2001) 006 \[arXiv:hep-th/0110232\]. R. Roiban and J. Walcher, “Rational conformal field theories with G(2) holonomy,” JHEP [**0112**]{} (2001) 008 \[arXiv:hep-th/0110302\]. T. Eguchi and Y. Sugawara, “String theory on G(2) manifolds based on Gepner construction,” arXiv:hep-th/0111012. R. Blumenhagen and V. Braun, “Superconformal field theories for compact manifolds with Spin(7) holonomy,” JHEP [**0112**]{} (2001) 013 \[arXiv:hep-th/0111048\]. J. M. Figueroa-O’Farrill and S. Schrans, “The Conformal bootstrap and super W algebras,” Int. J. Mod. Phys. A [**7**]{} (1992) 591.\
J. M. Figueroa-O’Farrill and S. Schrans, “Extended superconformal algebras,” Phys. Lett. [**B257**]{} (1991) 69. S. Komata, K. Mohri and H. Nohara, “Classical and quantum extended superconformal algebra,” Nucl. Phys. [**B359**]{} (1991) 168. R. Blumenhagen, “Covariant construction of N=1 super-W algebras,” Nucl. Phys. [**B381**]{} (1992) 641. R. Blumenhagen, W. Eholzer, A. Honecker and R. Hubel, “New N=1 extended superconformal algebras with two and three generators,” Int. J. Mod. Phys. A [**7**]{} (1992) 7841 \[arXiv:hep-th/9207072\]. W. Eholzer, A. Honecker and R. Hubel, “Representations of N=1 extended superconformal algebras with two generators,” Mod. Phys. Lett. A [**8**]{} (1993) 725 \[arXiv:hep-th/9209030\]. S. Mallwitz, “On SW minimal models and N=1 supersymmetric quantum Toda field theories,” Int. J. Mod. Phys. A [**10**]{} (1995) 977 \[arXiv:hep-th/9405025\]. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nucl. Phys. [**B241**]{} (1984) 333. P. Goddard, A. Kent and D. Olive, “Virasoro Algebras And Coset Space Models,” Phys. Lett. B [**152**]{}, 88 (1985). P. Goddard, A. Kent and D. Olive, “Unitary Representations Of The Virasoro And Supervirasoro Algebras,” Commun. Math. Phys. [**103**]{}, 105 (1986). H. Eichenherr, “Minimal Operator Algebras In Superconformal Quantum Field Theory,” Phys. Lett. B [**151**]{} (1985) 26. M. A. Bershadsky, V. G. Knizhnik and M. G. Teitelman, “Superconformal Symmetry In Two-Dimensions,” Phys. Lett. B [**151**]{} (1985) 31. D. Friedan, Z. A. Qiu and S. H. Shenker, “Superconformal Invariance In Two-Dimensions And The Tricritical Ising Model,” Phys. Lett. B [**151**]{} (1985) 37. S. L. Lukyanov and V. A. Fateev, “Physics Reviews: Additional Symmetries And Exactly Soluble Models In Two-Dimensional Conformal Field Theory,” [*Chur, Switzerland: Harwood (1990) 117 p. (Soviet Scientific Reviews A, Physics: 15.2)*]{}.
K. Thielemans, “A Mathematica package for computing operator product expansions,” Int. J. Mod. Phys. C [**2**]{} (1991) 787. Y. Kazama and H. Suzuki, “New N=2 Superconformal Field Theories And Superstring Compactification,” Nucl. Phys. [**B321**]{} (1989) 232. Y. Kazama and H. Suzuki, “Characterization Of N=2 Superconformal Models Generated By Coset Space Method,” Phys. Lett. [**B216**]{} (1989) 112. A. Schwimmer and N. Seiberg, “Comments On The N=2, N=3, N=4 Superconformal Algebras In Two-Dimensions,” Phys. Lett. B [**184**]{} (1987) 191. T. Inami, Y. Matsuo and I. Yamanaka, “Extended Conformal Algebra With N=2 Supersymmetry,” Int. J. Mod. Phys. A [**5**]{} (1990) 4441. S. Odake, “Extension Of N=2 Superconformal Algebra And Calabi-Yau Compactification,” Mod. Phys. Lett. A [**4**]{} (1989) 557.
[^1]: The discussion applies to NS and Ramond sectors. One cannot mix $G$ and $H$ in the case of twisted sectors.
|
---
author:
- Dmitri Kharzeev
date: 'Received: date / Revised version: date'
subtitle: |
The First International Conference on Hard and Electromagnetic Probes\
in Relativistic Nuclear Collisions
title: Theoretical Summary
---
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Introduction {#intro}
============
The venue of this Conference – the small town of Ericeira on the Atlantic coast near Lisbon – is both spectacular and symbolic. We are at the western end of Europe, a place which calls to mind the history of how the New World was discovered. At the end of 15th century, nothing was known yet about the new lands hidden by the extensive ocean. Yet, the discoveries were already anticipated by some, and in 1494 the Pope divided the world to be discovered between Portugal and Spain, in the Treaty of Tordesillas. The sharp, straight boundary extended from North to South and divided what was at the time believed to be an empty ocean; less than 10 years later, South America had been discovered. The subsequent exploration of the New World made the shape of the boundary much more complex, and the subsequent developments eventually made it irrelevant altogether. What lessons can be learnt from this story? In my opinion, there are at least three: 0.2cm
i\) the less we know, the sharper are the boundaries;
ii\) sharp boundaries do not last long –
iii\) they disappear with the advance of knowledge.
0.3cm
As Helmut Satz reminded us in his opening talk, this conference grew out of the “Hard Probe café”, which had its first meeting at CERN, in 1994 – five centuries after the Treaty of Tordesillas. The discoveries at the high energy density and small x frontiers were widely anticipated, and the boundaries on the QCD maps were still very sharp. Regarding the statistical properties of QCD, most of us expected to see the weakly interacting quark-gluon gas just above the deconfinement temperature, although the lattice data already at that time indicated large deviations from the ideal gas behavior – see [@fkarsch]. As for the behavior of QCD at high energies (or small Bjorken $x$), it was widely believed that the transition from “soft” to “hard” regimes happens at some typical scale $Q_0 \sim 1\div 2$ GeV, which does not depend on the energy, even though the idea of parton saturation [@GLR; @Mueller:1985wy; @Blaizot:1987nc] was already known and the related classical gluon field approach [@MV] had just been developed.
The experimental heavy ion program at CERN SPS was blooming, and the great potential held by the hard probes had already been made clear by the discovery of $J/\psi$ suppression [@NA50] predicted by Matsui and Satz [@MS] (even though the interpretation of the data was a subject of intense discussions). The low–mass dilepton enhancement [@Agakishiev:1995xb] was observed shortly afterwards and attracted a lot of attention as a potential signature of chiral symmetry restoration, and Drell–Yan pair production proved to be very useful as the baseline. However, high transverse momentum hadrons, let alone jets, were very rare at the SPS energy ($\sqrt{s} \leq 20$ GeV per nucleon pair).
The new millenium brought RHIC – and with it, the era of hard probes in relativistic heavy ion physics has begun. At this Conference, we have heard about the amazing progress made in the experimental study of hard processes in recent years; the excellent overviews of the current situation were made at this conference [@Drees; @Harris:2005vt; @Jacobs:2005pk; @Scomparin; @Specht].
So what have we learnt so far from this wealth of experimental information, and what do we still need to know? In what follows below, I try to address these questions from the theorists’ point of view, based on the talks given at the Conference and on some of my own prejudices. The space limits prevent me from describing all of the reported exciting developments, so instead of presenting a catalogue of the given talks I will concentrate on a few selected topics.
Quark-gluon matter at high temperature {#sec:1}
======================================
Strongly coupled quark-gluon plasma: a surprise? {#subsec:12}
------------------------------------------------
For years, we have been expecting that at “sufficiently high” temperature $T$ the QCD matter will become an “almost” ideal gas of quarks and gluons. Indeed, a typical inter-particle distance in this matter is $\sim 1/T$, and the asymptotic freedom tells us that the interactions at short distances are weak. We still hold this expectation, but the data from RHIC tell us that “sufficiently high” temperatures appear beyond the reach of the current, and perhaps future, experiments: at all accessible temperatures the QCD matter behaves quite differently from an ideal gas, as emphasized at this Conference by E. Shuryak [@shuryak] and others. The dynamics of the quark–gluon plasma is thus much more rich and interesting, and we have to develop new methods to understand it.
In fact, as discussed at the Conference by F. Karsch [@fkarsch], there have been numerous indications from lattice QCD that even above the deconfinement transition the interactions among quarks and gluons remain strong. A particularly telling piece of evidence from the lattice calculations is presented in Fig.\[eff\_as\], which shows the behavior of the QCD running constant as a function of distance for different temperatures. At $T=0$, one observes the celebrated property of asymptotic freedom, or anti–screening of the color charge. Above the deconfinement temperature, the strong force gets screened – in agreement with the qualitative picture in which the range of the interaction is reduced because the exchanged gluons can scatter off the heat bath of deconfined thermal quarks and gluons. However, at experimentally accessible temperatures the screening develops at relatively large distances, at which the coupling constant is quite large. We are thus definitely dealing with a deconfined quark-gluon plasma, in which the long–range confining interactions are screened, but the residual non–perturbative effects are still strong.
This property of the observed quark-gluon plasma makes the traditional quasi–particle description of its excitations questionable, as discussed by J.-P. Blaizot [@Blaizot:2005mj] and K. Rajagopal [@rajagopal], and one has to re–identify the appropriate degrees of freedom. Blaizot pointed out in particular the experimental implications of this problem for the dilepton production rates. Rajagopal also discussed the corresponding problem in the theory of cold quark matter, described as a color super-conductor, and described the applications to the physics of neutron stars. The ways to test the structure of the quark-gluon plasma in lattice simulations and in experiment include the study of fluctuations, as discussed by R. Gavai [@gavai] and various transport coefficients, including viscosity [@shuryak; @Hatsuda].
Quarkonium suppression in a strongly coupled Quark-Gluon Plasma
---------------------------------------------------------------
As pointed out long time ago by Matsui and Satz [@MS], the study of heavy quarkonia in hot QCD matter allows to test the persistence of confining interactions. Indeed, this is probably the closest one can get in experiment to measuring the order parameter of the deconfinement – the large distance limit of the correlation function of the Polyakov loops, which measures the interaction energy of the separated heavy quark and antiquark [@McLerran:1980pk]. Therefore, if some residual non-perturbative interactions are present above $T_c$, they may manifest themselves in the spectra of heavy quarkonia.
Very interesting lattice results on this issue have been presented at the Conference by T. Hatsuda [@Hatsuda], P. Petreczky [@Petreczky:2005bd], K. Petrov [@Petrov:2005sd], S. Digal [@Digal:2005ht], O. Kaczmarek and F. Zantow [@Kaczmarek:2005uv]. All of them point towards the survival of some of the bound charmonium states in the deconfined phase, which is consistent with the large screening radius of Fig. \[eff\_as\].
There are two basic ways of accessing the information about charmonia on the lattice: one is to measure the correlation function of the $\bar{c}c$ current and to reconstruct the corresponding spectral function, another is to compute the effective potential between static sources and to use it in the Schroedinger equation for the bound states.
Each of these methods has advantages and difficulties, so they are complementary to each other: in the spectral function method, one does not have to rely on a potential model, but a reconstruction of the quarkonium spectrum from the data has a limited precision. The effective potential approach provides a precise information on the spectrum, but the validity of the potential model in a heat bath and a treatment of the coupling between the color-singlet and octet components raise some questions.
A representative result for the shape of the quarkonium spectral function as extracted from the lattice vector $\bar{c}c$ correlation functions (the $J/\psi$ channel) with the help of a MEM (Maximal Entropy Method) approach is shown in Fig. \[corrf\]. One can clearly see that up to temperatures of about $T \sim 2\ T_c$ the peak corresponding to the bound $J/\psi$ state still survives in the spectrum. Moreover, in this temperature range little, if any, change in the mass of $J/\psi$ is observed. The effective potential method basing on the lattice results shown in Fig. \[pot\] leads to similar conclusions – the remnants of the confining interaction (“short strings” ?) still exist in the vicinity of the deconfinement phase transition and can support bound states. An interesting analysis aimed at linking the spectral function and potential approaches was presented at the Conference by A. Mocsy [@Mocsy:2004bv].
Do these lattice results imply that no $J/\psi$ suppression from quark-gluon plasma should be seen in experiment? In my opinion, the answer to this question is “no”: even if a quarkonium exists as a bound state, it can still be dissociated by the impact of hard deconfined gluons [@Kharzeev:1994pz], in a process analogous to photo–effect [@Shuryak:1978ij]. The relative importance of the Debye screening and “gluo–effect” processes is governed by the ratio of quarkonium binding energy $\Delta E$ to the temperature of the plasma $T$ [@Kharzeev:1995ju; @Kharzeev:1996se]: $$\Gamma(T) = {\Delta E(T) \over T},$$ where the binding energy depends on the temperature due to Debye screening. In the weakly coupled plasma $\Gamma \ll 1$, and the heavy quark bound state simply falls apart with the rate $$R = {4 \over L} \ \sqrt{{T \over \pi M_Q}},$$ ($L$ is the size of quarkonium, $M_Q$ – the heavy quark mass) which is the classical high temperature, weak coupling limit of the thermal activation rate. On the other hand, in the strongly coupled case of $\Gamma \gg 1$, quarkonium is tightly bound, and the binding energy threshold has to be overcome by the absorption of hard deconfined gluons from the heat bath. In this regime, the heavy quark bound states are quasi–stable, but the dissociation rate is quite large and can lead to a significant quarkonium suppression [@Xu:1995eb].
At the Conference, the fate of heavy quarkonium in the medium was further discussed by D. Blaschke [@Blaschke:2005jg], R. Rapp [@Rapp:2005rr], and R. Thews [@Thews:2005fs]. The latter talks discussed in particular the possibility to create additional quarkonia by recombination of heavy quarks and anti-quarks. In particular, it was shown [@Thews:2005fs] that recombination of heavy quarks leads to a sizable narrowing of the rapidity distribution of $J/\psi$’s in $Au-Au$ collisions at RHIC; a high statistics experimental measurement of this distribution can thus help to extract the contribution of this mechanism, or to put an upper bound on it.
Quarkonium suppression in the percolation approach to deconfinement was discussed by M. Nardi [@nardi]; the signature of the percolation phase transition in this case is a peculiar centrality and mass number dependencies of the $J/\psi$ survival probabilities, which are consistent with the existing NA50, NA60 and PHENIX data. The transverse momentum dependence of the $J/\psi$ suppression in this picture still remains an interesting open problem [@Kharzeev:1997ry].
Percolation of strings as a description of deconfinement was extensively discussed by J. Dias de Deus [@dias] and C. Pajares [@Pajares:2005kk]. It was pointed out that the percolation approach in particular naturally leads to the observed fluctuations in the transverse momentum (see Fig. \[fluct\]) and the universal form of the transverse mass distribution of hadrons in nuclear collisions, similar to the one arising from the color glass condensate [@Schaffner-Bielich:2001qj]. This brings us to the next topic which became one of the focal points of the Conference – the theory of nuclear wave functions on the light cone, at small Bjorken $x$.
High density gluon matter at small $x$
======================================
“Just a change of the reference frame?”
---------------------------------------
Recent years have seen an impressive progress in the understanding of nuclear wave functions at small Bjorken $x$. What makes this problem interesting? After all, nothing changes if we look at the nucleus in a different reference frame, where it is boosted to high momentum – or so it seems at first glance. But we have to remember that in quantum theory the operator of the number of particles does not commute with the operator of Lorentz boost, and so in general a mere change of the reference frame will change the measured number of particles in the system.
This is certainly the situation in QCD, where the boost is accompanied by the evolution of a hadron or nuclear structure function, which leads to a rapid $\sim 1/x^{\lambda}$ growth of the number of gluons and quarks at small $x$. Because the boost also leads to the Lorentz compression of the nucleus, and because the Froissart bound does not allow the area of the nucleus to grow faster than $\sim \ln^2(1/x)$, at sufficiently small $x$ and/or large mass number of the nucleus $A$ the density of partons in the transverse plane becomes large and they can recombine [@GLR; @Mueller:1985wy; @Blaizot:1987nc]; when the occupation number becomes $\sim 1/\alpha_s$, the system can be described as a semi-classical gluon field [@MV]. A broad overview of the semi–classical Color Glass Condensate approach to nuclear wave functions and to the heavy ion collisions has been presented by R. Venugopalan [@raju].
In search of the ultimate evolution equation
--------------------------------------------
Once the density of partons becomes large, the non–linear effects in the parton evolution become important. The quantum processes of parton splitting and recombination in this regime occur in the background of the strong classical field. The general evolution equation in this case still has to be found, and the progress in this direction has been discussed at the Conference by J. Bartels [@bartels], E. Iancu [@iancu] and A.H. Mueller.
A general introduction into the problem of non-linear evolution equations and the underlying physics was given by Mueller, who also discussed the limits of validity of the existing approaches. Iancu in particular discussed the role of rare fluctuations in hadron wave functions which are not captured by the mean–field equation of Balitsky [@Balitsky:1995ub] and Kovchegov [@Kovchegov:1999yj].
One of the important problems of the perturbative QCD approach to high energy scattering emphasized by Bartels is the following: in the impact parameter $b$ space, perturbation theory always predicts the amplitudes which fall off as inverse powers $ (1 / b)^n$ at large $b$. This is because there is no mass gap for the gluon excitations in perturbation theory. On the other hand, in the physical world there are no massless hadronic excitations – pions, as the Goldstone bosons of the spontaneously broken chiral symmetry, are the lightest ones, but their masses $m^2_{\pi} \sim m_q$ do not vanish because of the finite light quark masses $m_q \neq 0$. Therefore, high energy hadronic scattering amplitudes must fall off exponentially at large impact parameters, not slower than $\sim \exp(-2 m_{\pi} b)$ – coupled with the fact that at fixed impact parameter the growth of the amplitude is bounded by a power of energy $s$, this leads to the Froissart bound on the total cross sections. Because of the diffusion to large distances in high energy evolution, one is forced to consider the influence of the mass gap on the scattering amplitudes.
Probing the Color Glass Condensate
----------------------------------
Since the growth of parton distributions in the wave function of a nucleus $A$ at small $x$ is tempered by the non-linear effects, the rescaled by $A$ number of partons in a heavy nucleus is smaller than in a proton. This parton deficit in a heavy nucleus is a quantum effect, which has to manifest itself at sufficiently small $x$, when the longitudinal phase space $\sim \ln(1/x)$ for the emitted gluons is large enough to compensate the smallness of the coupling, $\alpha_s \ln(1/x) \sim 1$. Indeed, at the classical level the total number of partons in a nucleus $A$ is equal to the rescaled number of partons in a nucleons, but they are re-distributed in the transverse momentum which leads to the Cronin effect in nuclear cross sections.
The number of partons in the nuclear wave functions can be measured in hard $p(d)A$ scattering processes at small $x$; at RHIC this corresponds to the forward rapidity region (the deuteron fragmentation region). Therefore one arrives to the prediction that the cross sections of hard $dA$ scattering in the forward rapidity region should be suppressed relative to the $NN$ ones. The physics of this phenomenon has been extensively discussed at the Conference by R. Baier [@baier], B. Gay Ducati [@gay], J. Jalilian-Marian [@jalilian], J. Milhano and C. Salgado [@Albacete:2005ef], D. Triantafyllopoulos [@Triantafyllopoulos:2005eh] and K. Tuchin [@Tuchin:2005ky].
Jalilian-Marian [@jalilian] presented a clear introduction to the problem, and discussed the effects of quantum evolution in the color glass condensate on the production of hadrons, dileptons and photons at forward rapidities. Dilepton and photon production at forward rapidities have also been the topic of talks given by R. Baier [@baier] and Gay Ducati [@gay]. Baier in particular has demonstrated the potential of these probes for understanding the nuclear gluon distributions at small $x$. Salgado [@Albacete:2005ef] has shown that the saturation picture leads to a consistent description of the small $x$ data on deep-inelastic scattering off both protons and nuclei, see Fig. \[geom\]. He argued that this picture also allows to describe the data on hadron multiplicites at RHIC. Triantafyllopoulos [@Triantafyllopoulos:2005eh] discussed the transition from the classical to quantum regimes in $pA$ scattering, and the evolution and disappearance of the Cronin peak with rapidity. Tuchin [@Tuchin:2005ky] presented results on the influence of the color glass condensate on the production of charmed quarks and charmonia. In the latter case, he found an interesting effect of nuclear $J/\psi$ enhancement in a certain window in rapidity, see Fig. \[psi-k\].
Much of the existing theoretical analysis is based on the method of $k_T$ factorization. The limitations of this approach were examined by H. Fujii and F. Gelis [@Fujii:2005rm] using an example of heavy quark and quarkonium production.
Theoretical approaches currently used for the description of $pA$ collisions were discussed by J. Qiu [@qiu]; he analyzed the contributions of higher twist effects resulting from coherent multiple scattering, and their influence on hard nuclear processes. The production of hidden and open charm at RHIC and LHC in the more traditional framework of collinear factorization was discussed by R. Vogt [@Vogt:2004hd]; in particular, she examined the influence of several of the existing approaches to shadowing on the yields of charmed quarks.
Hard probes of hot and dense QCD matter
=======================================
Perturbative QCD – the baseline
-------------------------------
No-one at present doubts the applicability of perturbative QCD to the description of “sufficiently” hard processes. Perturbative methods therefore provide a crucial baseline for the understanding of the attenuation of high momentum partons in hot and dense matter. Of particular interest to the participants was the long–standing puzzle of the apparent discrepancy between the yields of heavy quarks as measured at collider energies and the perturbative calculations. The problem, and possible solutions, was discussed by S. Frixione [@frixione].
Jets and heavy quarks as a probe
--------------------------------
One of the most spectacular successes of the RHIC program is the discovery of the suppression of high transverse momentum particles, predicted as a signature of the quark–gluon plasma. An introduction to the problem, and an overview of the existing and future possibilities with the high momentum probes was given by X.-N. Wang [@wang].
The influence of the quark–gluon plasma on jet shapes and on the propagation of heavy quarks was the topic of U. Wiedemann’s talk [@Wiedemann:2005gm]. The energy loss of heavy quarks was also the discussed by M. Djordjevic [@magdalena]. The results indicate a considerable enhancement in $D/\pi$ ratios (see Fig. \[dead\]) resulting from the interplay between the “dead cone effect” and the coherent multiple scattering, in qualitative agreement with other treatments [@Dokshitzer:2001zm].
A. Accardi [@Accardi:2005fu] investigated the relative importance of Cronin effect and jet quenching at different RHIC energies. An interesting analysis of di–hadron correlations in the fragmentation of the jets was presented by A. Majumder [@Majumder:2005ii], who explored how the dense QCD matter affects the associated hadron distributions (see Fig. \[frag\]).
A novel effect of the influence of the hydrodynamical flow on the jet shape was considered by N. Armesto [@Armesto:2005zn]. He found that the flow can lead to an anisotropic jet shape, as illustrated in Fig. \[flow\].
The influence of the medium on the fragmentation of partons was also the topic of R. Hwa’s talk [@Hwa:2005ay]. He suggested that because of the high density of partons in the quark–gluon plasma, the recombination of partons is a likely mechanism which can affect the composition and the transverse momentum distributions of the produced hadrons. (For a related approach, see also [@Fries:2003kq]).
Electromagnetic probes
----------------------
The production of photons and dileptons from a hot quark-gluon matter remains a subject of vigorous theoretical and experimental studies. The state of the theoretical calculations has been reviewed at the Conference by C. Gale [@Gale:2005ri] and E. Shuryak. Gale emphasized that a variety of phenomena contribute to the photon and dilepton production, and they have to be carefully evaluated to make the extraction of the quark–gluon plasma component possible, see Fig. \[dilept\].
Outlook
=======
This summary clearly does not capture the entirety of the theoretical developments presented at the Conference – it is impossible to fit the entire week of wonderful talks and exciting discussions in a few pages of written text. Nevertheless, I hope that a more complete picture can be reconstructed by looking at the original talks referenced here. This is the picture of the field which is still at the very beginning – prompted by the huge wave of new high quality data, the theorists are still in search of a coherent framework capable of describing the variety of the observed phenomena.
However, in my opinion, the talks at the Conference show that we start to see the essential elements of this unified framework, and enough bright people with enough enthusiasm are working on the problem. Coupled with an impressive progress in experiment, this indicates that the ultimate goal of understanding QCD in the high temperature and strong field regimes may now be within reach.
0.2cm
This work was supported by the U.S. Department of Energy under the contract DE-AC02-98CH10886.
F. Karsch, arXiv:hep-lat/0502014. L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rept. [**100**]{}, 1 (1983). A. H. Mueller and J. w. Qiu, Nucl. Phys. B [**268**]{}, 427 (1986). J. P. Blaizot and A. H. Mueller, Nucl. Phys. B [**289**]{}, 847 (1987). L. D. McLerran and R. Venugopalan, Phys. Rev. D [**49**]{}, 2233 (1994) \[arXiv:hep-ph/9309289\]; Phys. Rev. D [**49**]{}, 3352 (1994) \[arXiv:hep-ph/9311205\]. C. Baglin [*et al.*]{} \[NA38 Collaboration\], Phys. Lett. B [**220**]{}, 471 (1989). T. Matsui and H. Satz, Phys. Lett. B [**178**]{}, 416 (1986). G. Agakishiev [*et al.*]{} \[CERES Collaboration\], Phys. Rev. Lett. [**75**]{}, 1272 (1995). A. Drees, [*in this volume*]{}. J. W. Harris, arXiv:nucl-ex/0503014. P. Jacobs, arXiv:nucl-ex/0503022. E. Scomparin, [*in this volume*]{}. H. Specht, [*in this volume*]{}. E. Shuryak, [*in this volume*]{}.
J. P. Blaizot and F. Gelis, arXiv:hep-ph/0504144. K. Rajagopal, [*in this volume*]{}. R. V. Gavai and S. Gupta, arXiv:hep-ph/0502198. O. Kaczmarek, F. Karsch, F. Zantow and P. Petreczky, Phys. Rev. D [**70**]{}, 074505 (2004) \[arXiv:hep-lat/0406036\]. L. D. McLerran and B. Svetitsky, Phys. Lett. B [**98**]{}, 195 (1981). T. Hatsuda, [*in this volume*]{}.
P. Petreczky, arXiv:hep-lat/0502008. K. Petrov, arXiv:hep-lat/0503002. S. Digal, O. Kaczmarek, F. Karsch and H. Satz, arXiv:hep-ph/0505193. O. Kaczmarek and F. Zantow, arXiv:hep-lat/0502011. A. Mocsy and P. Petreczky, arXiv:hep-ph/0411262. M. Asakawa and T. Hatsuda, Phys. Rev. Lett. [**92**]{}, 012001 (2004) \[arXiv:hep-lat/0308034\]. S. Datta, F. Karsch, P. Petreczky and I. Wetzorke, Phys. Rev. D [**69**]{}, 094507 (2004) \[arXiv:hep-lat/0312037\]. P. Petreczky and K. Petrov, Phys. Rev. D [**70**]{}, 054503 (2004) \[arXiv:hep-lat/0405009\]. D. Kharzeev and H. Satz, Phys. Lett. B [**334**]{}, 155 (1994) \[arXiv:hep-ph/9405414\]. E. V. Shuryak, Phys. Lett. B [**78**]{}, 150 (1978) \[Sov. J. Nucl. Phys. [**28**]{}, 408.1978\]. D. Kharzeev, L. D. McLerran and H. Satz, Phys. Lett. B [**356**]{}, 349 (1995) \[arXiv:hep-ph/9504338\]. D. Kharzeev, Nucl. Phys. A [**610**]{}, 418C (1996) \[arXiv:hep-ph/9609260\]. X. M. Xu, D. Kharzeev, H. Satz and X. N. Wang, Phys. Rev. C [**53**]{}, 3051 (1996) \[arXiv:hep-ph/9511331\]. D. Blaschke, O. Kaczmarek, E. Laermann and V. Yudichev, arXiv:hep-ph/0505053. R. Rapp, arXiv:hep-ph/0502208. R. L. Thews, arXiv:hep-ph/0504226. M. Nardi, [*in this volume*]{}.
D. Kharzeev, M. Nardi and H. Satz, Phys. Lett. B [**405**]{}, 14 (1997) \[arXiv:hep-ph/9702273\]. J. Dias de Deus, [*in this volume*]{}.
C. Pajares, arXiv:hep-ph/0501125. S.S. Adler et al. \[PHENIX Collaboration\], nucl-ex/03100005.
J. Schaffner-Bielich, D. Kharzeev, L. D. McLerran and R. Venugopalan, Nucl. Phys. A [**705**]{}, 494 (2002) \[arXiv:nucl-th/0108048\]. R. Venugopalan, arXiv:hep-ph/0502190. J. Bartels, [*in this volume*]{}.
E. Iancu, arXiv:hep-ph/0503062. I. Balitsky, Nucl. Phys. B [**463**]{}, 99 (1996) \[arXiv:hep-ph/9509348\]. Y. V. Kovchegov, Phys. Rev. D [**60**]{}, 034008 (1999) \[arXiv:hep-ph/9901281\]. R. Baier, [*in this volume*]{}.
B. Gay Ducati, [*in this volume*]{}.
J. Jalilian-Marian, [*in this volume*]{}.
J. L. Albacete, N. Armesto, J. G. Milhano, C. A. Salgado and U. A. Wiedemann, arXiv:hep-ph/0502167. D. N. Triantafyllopoulos, arXiv:hep-ph/0502114. K. Tuchin, arXiv:hep-ph/0504133. R. G. de Cassagnac \[PHENIX Collaboration\], J. Phys. G [**30**]{}, S1341 (2004) \[arXiv:nucl-ex/0403030\]. H. Fujii, F. Gelis and R. Venugopalan, arXiv:hep-ph/0502204. J. Qiu, [*in this volume*]{}.
R. Vogt, arXiv:hep-ph/0412302. S. Frixione, [*in this volume*]{}.
X.-N. Wang, [*in this volume*]{}.
U. A. Wiedemann, arXiv:hep-ph/0503119. N. Armesto, A. Dainese, C. A. Salgado and U. A. Wiedemann, Phys. Rev. D [**71**]{}, 054027 (2005) \[arXiv:hep-ph/0501225\].
M. Djordjevic, [*in this volume*]{}.
Y. L. Dokshitzer and D. E. Kharzeev, Phys. Lett. B [**519**]{}, 199 (2001) \[arXiv:hep-ph/0106202\]. A. Accardi, arXiv:nucl-th/0502033.
A. Majumder, arXiv:nucl-th/0503019. A. Majumder, E. Wang and X. N. Wang, arXiv:nucl-th/0412061. C. Adler et al., \[STAR Collaboration\], Phys. Rev. Lett. [**90**]{}, 082302 (2003).
S.S. Adler et al., \[PHENIX Collaboration\], nucl-ex/0408007.
N. Armesto, arXiv:hep-ph/0501214.
R. C. Hwa, arXiv:nucl-th/0501054. R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, Phys. Rev. C [**68**]{}, 044902 (2003) \[arXiv:nucl-th/0306027\]. C. Gale, arXiv:hep-ph/0504103.
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---
abstract: 'Let $\cal G$ be a family of all $3$-regular $2$-connected plane multigraphs without loops. We prove the following plane version of the Fleischner theorem: Let $G$ be a graph in $\cal G$. For every $2$-factor $X$ of $G$ having $n$-components there exists a plane graph $J$ having a Hamilton cycle omitting all edges of $E(G)\backslash E(X)$ and such that $G \subseteq J \subset G^{2}$, $\Delta(J) \leqslant 5$ and $|E(J)|= |E(G)| + 2n -2$. Moreover, if $G$ is simple, then $J$ is simple too.'
author:
- Jan Florek
date: 'Received: date / Accepted: date'
title: A plane version of the Fleischner theorem
---
Introduction
============
We use [@flobar1] as a reference for undefined terms. In particular, $V(G)$ is the vertex set, and $E(G)$ is the edge set of $G$. A *matching* in $G$ is a set of pairwise non-adjacent links. A *perfect matching* is one which covers every vertex of the graph. A *spanning subgraph* of a multigraph $G$ is a subgraph whose vertex set is the entire vertex set of $G$. Spanning cycles (paths) are called *Hamilton cycles* (*Hamilton paths*) and spanning $k$-regular subgraphs are called *$k$-factors*. A graph is *hamiltonian* if it admits a Hamilton cycle. A graph is *hamiltonian-connected* if for every pair $u$, $v$ of distinct vertices of $G$, there exists a Hamilton $u-v$ path. If $\{V_{1}, V_{2}\}$ is a partition of $V(G)$, the set $E[V_{1}, V_{2}]$ of all edges of $G$ with one end in $V_{1}$ and the other end in $V_{2}$ is called an *edge cut* in $G$. A *bond* of $G$ is a minimal nonempty edge cut of $G$.
Given a connected graph $G$ and a positive integer $k$, we denote by $G^k$ the graph on $V(G)$ in which two vertices are adjacent if and only if they have distance at most $k$ in $G$. The graph $G^2$ and $G^3$ are also referred to as the *square* and *cube*, respectively, of $G$. Karaganis [@flobar7] and Sekanina [@flobar10] proved that if $G$ is a simple connected graph, then the cube of $G$ is hamiltonian-connected, and Fleischner [@flobar5] discovered that the square of $G$ is hamiltonian (see also Říha [@flobar9]). The strengthened result (employing Fleischner’s work) that the square of $G$ is hamiltonian-connected was proved by Chartrand, Kappor and Nash-Williams [@flobar3].
Let ${\cal G}$ be the family of all $3$-regular $2$-connected plane multigraphs without loops. We prove the following plane version of the Fleischner theorem.
\[theorem1.1\] Let $G \in \cal G$. For every $2$-factor $X$ of $G$ having $n$-components there exists a plane graph $J$ having a Hamilton cycle omitting all edges of $E(G)\backslash E(X)$ and such that $G \subseteq J \subset G^{2}$, $|E(J)|= |E(G)| + 2n -2$ and $\Delta(J) \leqslant 5$. Moreover, if $G$ is simple, then $J$ is also simple.
Notice that Petersen [@flobar8] proved that every $3$-regular multigraph without cut edges has a $2$-factor (a perfect matching) (see Bondy and Murty [@flobar1] Theorem 16.14).
A plane version of the Fleischner theorem
=========================================
Let $G \in \cal G$ and suppose that $F(G)$ is the set of all faces of $G$. Each face $f \in F(G)$ is bounded by a cycle $\partial(f)$ called a *facial cycle* of this face (see Bondy and Murty [@flobar1] Theorem 10.7). A *cyclic sequence of faces* is a cyclic sequence $f_{1}f_{2} \ldots f_{k}f_{1}$ of different faces in $G$ such that any two successive faces are adjacent. We say that an edge *$\sim$belongs* to a cyclic sequence of faces (to a sequence of faces) if it is incident with two successive faces of this sequence. Then, we also say that this cyclic sequence of faces *$\sim$contains* this edge. Notice that $B \subseteq E(G)$ is a bond of $G$ if and only if it is the set of all edges which $\sim$belong to a some cyclic sequence of faces. $$\mbox{\bf Proof of Theorem \ref{theorem1.1}}$$
Let $X$ be a $2$-factor of $G$ which has $n$ components. We define a face $2$-colouring ${a: F(G) \rightarrow \{\alpha, \beta\}}$. Fix a face $f$ of $F(G)$. For every $g \in F(G)$, $g \neq f$, there exists a sequence of faces from $f$ to $g$ $\sim$containing only edges of $E(X)$, because $G$ is $3$-regular and $X$ is a $2$-factor of $G$. We set $a(f) = \alpha$ and we colour faces of this sequence with $\alpha$ and $\beta$ alternately. The colouring of $g$ is independent on the choice of the sequence of faces. Indeed, if $D$ is a cyclic sequence of faces and a bond $B \subseteq E(X)$ is the set of all edges $\sim$belonging to $D$, then $|B|$ is even, because $X$ is a $2$-factor of $G$. Notice that
1. any two adjacent faces in $G$ are incident with the same edge belonging to $E(G) \backslash E(X)$ if and only if they are coloured identically by $a$.
Further, every facial cycle of $G$ has an *orientation assigned* by $a$. Namely, we can assume that a facial cycle of an inner (the outer) face has the clockwise-orientation (counter clockwise-orientation, respectively) if and only if this face is coloured $\alpha$.
Let $C_{1}$ be a component of $X$. We can enumerate, by induction, components of $X$ as $C_{1}, C_{2},\ldots, C_{n}$ in such a way that for every $1 \leqslant i < n$ there exist a bond $B_{i}$ of $G$, an edge $b_{i} \in B_i$ and a connected subgraph $S_{i+1}$ of $G$ satisfying the following conditions: $B_i$ is contained in a cut $E[V(S_{i}), V(G) \backslash V(S_{i})]$, $b_{i}$ is connecting $S_{i}$ with $C_{i+1}$ and $S_{i+1} = S_{i} \cup C_{i+1} + b_i $ (we put $S_1 = C_1$). Hence, $S_{i}$ is disjoint with $C_j$, for every $i < j \leqslant n$.
Let $M = \{b_{1}, \ldots, b_{n-1}\}$. Certainly, $E(G) \backslash E(X)$ is a perfect matching of $G$, because $G$ is $3$-regular and $X$ is a $2$-factor of $G$. Notice that $B_i = E[V(S_i), V(C_{i+1})] \subseteq E(G) \backslash E(X)$, $1 \leqslant i < n$. Hence, edges of $B_i$ are not adjacent and $b_i$ is the only edge of $M$ belonging to the bond $B_i$ of $G$. Therefore, the following conditions are satisfied:
1. if a $k$-cycle of $G$ contains an edge of $M$, then $k \geqslant 4$,
2. if $f \in F(G)$ and $|\partial(f)| = 4$, then $\partial(f)$ contains at most one edge of $M$.
Suppose that $\cal K$ is the family of all faces in $G$ each of them is incident with an edge of $M$. Let $f$ be any face of $\cal K$ and suppose that $\partial (f) = c_{1}c_{2} \ldots c_{n}c_{1}$ is the facial cycle of $f$ with the orientation assigned by $a$. Suppose that $c_{i_{1}}c_{i_{1}+1}$, $c_{i_{2}}c_{i_{2}+1}\ldots, c_{i_{p}}c_{i_{p}+1}$ (where $p$ depends on the face $f$) are all successive edges of $\partial (f)$ belonging to $M$. By condition $(2)$, vertices $c_{i_{j}}$ and $c_{i_{j}+2}$ are not adjacent in $G$, for every $j = 1, \ldots, p$. Since $M \subset E(G) \backslash E(X)$, edges of $M$ are not adjacent. Hence, we can draw edges $$e_{1}(f) = c_{i_{1}}c_{i_{1}+2}, e_{2}(f) = c_{i_{2}}c_{i_{2}+2},\ldots, e_{p}(f) = c_{i_{p}}c_{i_{p}+2}$$ in such a way that they are not crossing and their interiors are contained in $f$. By adding to $G$ all edges of $\bigcup_{f \in \cal K} \{e_{1}(f), \ldots, e_{p}(f)\}$ we obtain a plane graph $J$ such that $G \subseteq J \subset G^{2}$ and $|E(J)|= |E(G)| + 2n -2$ . Notice that, by conditions $(2) - (3)$, if $G$ is simple, then $J$ is also simple.
Let $e =ab$ be any edge of $M$ and suppose that $e$ is incident with faces $f_{1}, f_{2} \in F(G)$. Assume that facial cycles $\partial(f_1)$ and $\partial(f_2)$ have orientations assigned by the face $2$-colouring $a$. Since $e \in E(G) \backslash E(X)$, by condition $(1)$, faces $f_1$ and $f_2$ are coloured the same by $a$. Then, by $(2)$, we can assume that $a, b, c$ and $b, a, d$ are successive vertices of $\partial(f_1)$ and $\partial(f_2)$, respectively. Let $D(e)$ be a subgraph of $J$ with $\{a, b, c, d\}$ as the vertex set, and $\{ab, bc, ac, ad, bd\}$ as the edge set. If faces $f_{1}, f_{2}$ are coloured $\alpha$ (or $\beta$), then $D(e)$ is called a *diamond of type $\alpha$* (*diamond of type $\beta$*, respectively). Let $E_{0}(D(e))$ (or $E_{1}(D(e))$) denote the set of all edges of $D(e)$ belonging to $E(G)$ (to $E(J)\backslash E(G)$, respectively). Then, $E_{0}(D(e)) = \{ab, bc, ad\}$ and $E_{1}(D(e)) = \{ac, bd\}$.
Since edges of $M$ are not adjacent, conditions $(2)-(3)$ show the following (see Fig 1 and Fig 2):
1. any different diamonds have no common edge.
Notice that, by condition $(4)$, every vertex of $G$ belongs to at most two diamonds. Hence, $\Delta(J) \leqslant 5$.
\(A) ; (B) at (A.corner 1) ; (C) at (B.corner 5) ; (D) at (A.corner 5) ;
(A.corner 4) – (A.corner 6) – (D.corner 1) – (D.corner 3);
(A.corner 6) – (D.corner 3);
(C.corner 3) – (C.corner 2) – (C.corner 4) – (D.corner 5);
(C.corner 3) – (C.corner 4); (C.corner 3) – (D.corner 5);
(A.corner 3) – (A.corner 2) – (A.corner 1) – (A.corner 6) – (B.corner 5) – (B.corner 6) – (C.corner 1) – (C.corner 6);
(B.corner 2) – (B.corner 1);
(A.corner 4) – (A.corner 5) – (D.corner 4);
(D.corner 5) – (D.corner 6) – (C.corner 5);
at (A.corner 4) [$d$]{}; at (A.corner 5) [$a$]{};
at (B.corner 4) [$b$]{}; at (B.corner 5) [$c$]{}; at (B.corner 6) [$g$]{};
at (D.corner 5) [$f$]{}; at (D.corner 6) [$e$]{};
\(A) ; (B) at (A.corner 1) ; (C) at (B.corner 5) ; (D) at (A.corner 5) ;
(A.corner 4) – (A.corner 6) – (D.corner 1) – (D.corner 3);
(A.corner 6) – (D.corner 3);
(B.corner 5) – (B.corner 1) – (B.corner 6) – (C.corner 4) – cycle;
(C.corner 2) – (C.corner 3);
(A.corner 2) – (A.corner 3) – (A.corner 4) – (A.corner 5) – (D.corner 4) – (D.corner 5) ;
(B.corner 1) – (B.corner 6) – (C.corner 1);
(B.corner 2) – (B.corner 3) – (B.corner 4) –(B.corner 5) – (C.corner 3) – (C.corner 4) – (C.corner 5) – (C.corner 6) ;
at (A.corner 4) [$d$]{}; at (A.corner 5) [$a$]{};
at (B.corner 4) [$b$]{}; at (B.corner 5) [$c$]{}; at (B.corner 6) [$e$]{}; at (B.corner 1) [$g$]{};
at (D.corner 6) [$f$]{};
For every $1 \leqslant i < n$ we define, by induction, a subgraph of $J$ $$H_{i+1} = H_{i} \cup C_{i+1} +E_{1}(D(b_{i})) \backslash E_{0}(D(b_{i}))
\hbox{ (we put } H_1 = C_1).$$ If $H_{i}$ is omitting all edges of $E(G)\backslash E(X)$, then $H_{i+1}$ is also omitting all edges of $E(G)\backslash E(X)$, because $C_{i+1}$ is contained in $X$ and $E_{1}(D(b_i))$ is contained in $E(J) \backslash E(G)$ Assume that $H_i$ is a cycle of $J$. Notice that $H_{i}$ is disjoint with $C_{i+1}$, because $V(H_{i}) = V(S_i)$ and $S_i$ is disjoint with $C_{i+1}$.
By condition $(4)$, diamonds $D(b_i)$ and $D(b_j)$, for $j \neq i$, have no common edge. Hence, one edge of $E_{0}(D(b_i))$ belongs to the cycle $H_{i}$ and another one belongs to the cycle $C_{i+1}$, and any edge of $E_{1}(D(b_i))$ doesn’t belong to $H_{i} \cup C_{i+1}$. Hence, $H_{i+1}$ is a cycle of $J$. Therefore, $H_n$ is a Hamilton cycle of $J$, because $V(H_n) = V(C_{1} \cup\ldots \cup C_{n})= V(G) = V(J)$. Hence, Theorem \[theorem1.1\] holds.
\[exam2.1\] Let $G$ be the Tutte (Bosak and Lederberg, Faulkner and Younger) simple $3$-regular $3$-connected nonhamiltonian plane graph, see Bondy and Murty [@flobar1] (Bosàk [@flobar2] and Lederberg [@flobar7], Faulkner and Younger [@flobar4], respectively). It is not difficult to show a $2$-factor $X$ of $G$ containing only two components. Hence, by Theorem \[theorem1.1\], there exists a simple $3$-connected plane graph $J$ having a Hamilton cycle omitting all edges of $E(G) \backslash E(X)$ and such that $G \subset J \subset G^{2}$ and $|E(J)| =|E(G)| + 2$.
[99]{}
J.A. Bondy and U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics 244, Springer, 2008.
C. Bosàk, Hamiltonian lines in cubic graphs. Theorie des graphes. Dunod, Paris, (1967) 35-46.
G. Chartrand, A. Kappor and C.St.J.A. Nash-Williams, The square of a block is hamiltonian-connected. J. Combin. Theory **16B** (1974) 290-2.
G.B. Faulkner and D.H. Younger, Non-hamiltonian cubic planar maps, Discrete Math. **7** (1974) 67–74. H. Fleischner, The square of every two-connected graph is hamiltonian. J. Combin. Theory **16B** (1974) 399-404.
J.J. Karaganis, On the cube of a graph. Canad. Math. Bull. **11** (1968) 295-6.
J. Lederberg, Hamiltonian circuits of convex trivalent polyhedra (up to 18 vertices). Amer. Math. Monthly **74** (1967) 522-527.
J. Petersen, Die Theorie der regulören Graphen. Acta Math. **15** (1891) 193–220.
S. Říha, A new proof of the theorem by Fleischner, J. Combin. Theory **52B** (1991) 117-123.
M. Sekanina, On an ordering of the set of vertices of a connected graph. Publ. Fac. Sci. Univ. Brno, **412** (1960) 137–42.
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---
abstract: 'In [@BAMU], an ergodic theorem à la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized [@BOYER] to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of [@ROBLI], with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group ${\mathbb{F}_{r}}$ of rank $r$ even if ${\mathbb{F}_{r}}$ is not the fundamental group of a closed manifold, and may have an arithmetic spectrum.'
author:
- Adrien Boyer
- Antoine Pinochet Lobos
bibliography:
- 'bibliobord.bib'
title: 'An ergodic theorem for the quasi-regular representation of the free group'
---
[^1]
[^2]
Introduction
============
In this paper, we consider the action of the free group ${\mathbb{F}_{r}}$ on its boundary ${\mathbf{B}}$, a probability space associated to the Cayley graph of ${\mathbb{F}_{r}}$ relative to its canonical generating set. This action is known to be *ergodic* (see for example [@FIGAT] and [@FIGATPI]), but since the measure is not preserved, no theorem on the convergence of means of the corresponding unitary operators had been proved. Note that a close result is proved in [@FIGATPI Lemma 4, Item (i)].\
We formulate such a convergence theorem in Theorem \[II\]. We prove it following the ideas of [@BAMU] and [@BOYER] replacing [@ROBLI Theorem 4.1.1] by Theorem \[I\].
Geometric setting and notation {#geomsetting}
------------------------------
We will denote ${\mathbb{F}_{r}} = \langle a_1,...,a_r\rangle$ the free group on $r$ generators, for $r \geq 2$. For an element $\gamma \in {\mathbb{F}_{r}}$, there is a unique reduced word in $\{a^{\pm 1}_1,...,a^{\pm 1}_r\}$ which represents it. This word is denoted $\gamma_1 \cdots \gamma_k$ for some integer $k$ which is called the *length* of $\gamma$ and is denoted by $\vert \gamma \vert$. The set of all elements of length $k$ is denoted $S_n$ and is called the *sphere of radius* $k$. If $u \in {\mathbb{F}_{r}}$ and $k \geq \vert u \vert$, let us denote ${Pr}_u(k) := \{\gamma \in {\mathbb{F}_{r}} {\mbox{ } | \mbox{ }}\vert \gamma \vert = k \mbox{, } u \mbox{ is a prefix of } \gamma\}$.\
Let $X$ be the Cayley graph of ${\mathbb{F}_{r}}$ with respect to the set of generators $\{a^{\pm 1}_1,...,a^{\pm 1}_r\}$, which is a $2r$-regular tree. We endow it with the (natural) distance, denoted by $d$, which gives length $1$ to every edge ; for this distance, the natural action of ${\mathbb{F}_{r}}$ on $X$ is isometric and freely transitive on the vertices ; the space $X$ is uniquely geodesic, the geodesics between vertices being finite sequences of successive edges. We denote by $[x,y]$ the unique geodesic joining $x$ to $y$.\
We fix, once and for all, a vertex $ x_0$ in $X$. For $x \in X$, the vertex of $X$ which is the closest to $x$ in $[ x_0,x]$, is denoted by $\lfloor x \rfloor$ ; because the action is free, we can identify $\lfloor x \rfloor$ with the element $\gamma$ that brings $ x_0$ on it, and this identification is an isometry.
### The Cayley tree and its boundary {#the-cayley-tree-and-its-boundary .unnumbered}
As for any other CAT$(-1)$ space, we can construct a boundary of $X$ and endow it with a distance and a measure. For a general construction, see [@BOURD]. The construction we provide here is elementary.
Let us denote by ${\mathbf{B}}$ the set of all right-infinite reduced words on the alphabet $\{a^{\pm 1}_1,...,a^{\pm 1}_r\}$. This set is called the **boundary** of $X$.
We will consider the set $\overline{X} := X \cup {\mathbf{B}}$.
For $u = u_1\cdots u_l \in {\mathbb{F}_{r}}\setminus\{e\}$, we define the sets $$X_u := \left\{x \in X {\mbox{ } | \mbox{ }}u \mbox{ is a prefix of } \lfloor x \rfloor \right\}$$ $${\mathbf{B}}_u := \left\{\xi \in {\mathbf{B}}{\mbox{ } | \mbox{ }}u \mbox{ is a prefix of } \xi \right\}$$ $$C_u := X_u \cup {\mathbf{B}}_u$$
We can now define a natural topology on $\overline{X}$ by choosing as a basis of neighborhoods
1. for $x \in X$, the set of all neighborhoods of $x$ in $X$
2. for $\xi \in {\mathbf{B}}$, the set $\left\{ C_u {\mbox{ } | \mbox{ }}u \mbox{ is a prefix of } \xi\right\}$
For this topology, $\overline{X}$ is a compact space in which the subset $X$ is open and dense. The induced topology on $X$ is the one given by the distance. Every isometry of $X$ continuously extend to a homeomorphism of $\overline{X}$.
### Distance and measure on the boundary {#distance-and-measure-on-the-boundary .unnumbered}
For $\xi_1$ and $\xi_2$ in ${\mathbf{B}}$, we define the **Gromov product** of $\xi_1$ and $\xi_2$ with respect to $x_{0}$ by $$(\xi_1\vert\xi_2) _{x_{0}}:= \sup\left\{k \in {\mathbb{N}}{\mbox{ } | \mbox{ }}\xi_1 \mbox{ and } \xi_2 \mbox{ have a common prefix of length } k\right\}$$ and $$d_{x_{0}}(\xi_1,\xi_2) := e^{-(\xi_1\vert\xi_2)_{x_{0}}}.$$
Then $d$ defines an ultrametric distance on ${\mathbf{B}}$ which induces the same topology ; precisely, if $\xi = u_1u_2u_3 \cdots$, then the ball centered in $\xi$ of radius $e^{-k}$ is just ${\mathbf{B}}_{u_1\dots u_k}$.
On ${\mathbf{B}}$, there is at most one Borel regular probability measure which is invariant under the isometries of $X$ which fix $ x_0$; indeed, such a measure $\mu_{x_{0}}$ must satisfy $$\mu_{x_{0}}({\mathbf{B}}_{u}) = \frac{1}{2r(2r-1)^{\vert u \vert -1}}$$
and it is straightforward to check that the $\ln(2r-1)$-dimensional Hausdorff measure verifies this property.
If $\xi = u_1\cdots u_n \cdots \in {\mathbf{B}}$, and $x,y \in X$, then $\left(d(x,u_1\cdots u_n) - d(y,u_1\cdots u_n)\right)_{n \in {\mathbb{N}}}$ is stationary. We denote this limit $\beta_\xi(x,y)$. The function $\beta_\xi$ is called the **Busemann function** at $\xi$.
Let us denote, for $\xi \in {\mathbf{B}}$ and $\gamma \in {\mathbb{F}_{r}}$ the function $$P(\gamma,\xi) := (2r-1)^{\beta_\xi( x_0,\gamma x_0)}$$
The measure $\mu_{x_{0}}$ is, in addition, quasi-invariant under the action of ${\mathbb{F}_{r}}$. Precisely, the Radon-Nikodym derivative is given for $\gamma\in \Gamma$ and for a.e. $\xi\in \textbf{B}$ by $$\frac{d\gamma_* \mu_{x_{0}}}{d\mu_{x_{0}}} (\xi)= P(\gamma,\xi),$$ where $\gamma_{*}\mu_{x_{0}}(A)=\mu_{x_{0}}(\gamma^{-1}A)$ for any Borel subset $A\subset \textbf{B} $.
### The quasi-regular representation {#the-quasi-regular-representation .unnumbered}
Denote the unitary representation, called the quasi-regular representation of ${\mathbb{F}_{r}}$ on the boundary of $X$ by $$\begin{array}{rcl}
\pi : {\mathbb{F}_{r}} &\rightarrow &\mathcal{U}(L^2({\mathbf{B}}))\\
\gamma &\mapsto &\pi(\gamma)\\
\end{array}$$ defined as $$\big(\pi(\gamma)g\big)(\xi) := P(\gamma,\xi)^{\frac{1}{2}}g(\gamma^{-1}\xi)$$ for $\gamma \in {\mathbb{F}_{r}}$ and for $g \in L^2({\mathbf{B}})$. We define the *Harish-Chandra* function $$\label{HCH}
\Xi(\gamma) :=\langle \pi(\gamma)\textbf{1}_{\textbf{B}},\textbf{1}_{\textbf{B}} \rangle =\int_{{\mathbf{B}}} P(\gamma,\xi)^{\frac{1}{2}} d\mu_{x_{0}}(\xi),$$ where $\textbf{1}_{\textbf{B}}$ denotes the characteristic function on the boundary.
For $f \in C(\overline{X})$, we define the operators $$\label{operators}
M_n(f) : g\in L^2({\mathbf{B}}) \mapsto {\displaystyle\frac{1}{\vert S_n \vert}} \sum\limits_{\gamma \in S_n} f(\gamma x_0){\displaystyle\frac{\pi(\gamma) g}{\Xi(\gamma)}} \in L^2({\mathbf{B}}).$$
We also define the operator $$M(f):=m(f_{|_{{\mathbf{B}}}})P_{\textbf{1}_{\textbf{B}}}$$ where $m(f_{|_{{\mathbf{B}}}})$ is the multiplication operator by $f_{|_{{\mathbf{B}}}}$ on $L^2({\mathbf{B}})$, and $P_{\textbf{1}_{\textbf{B}}}$ is the orthogonal projection on the subspace of constant functions.
Results {#results .unnumbered}
-------
The analog of Roblin’s equidistribution theorem for the free group is the following.
\[I\] We have, in $C(\overline{X}\times\overline{X})^{*}$, the weak-$*$ convergence $$\frac{1}{\vert S_n \vert} \displaystyle\sum_{\gamma \in S_n} D_{\gamma x_0} \otimes D_{\gamma^{-1} x_0} \rightharpoonup \mu_{x_{0}} \otimes \mu_{x_{0}}$$ where $D_x$ denotes the Dirac measure on a point $x$.
It is then straightforward to deduce the weak-$*$ convergence $$\Vert m_{\Gamma} \Vert e^{-\delta n} \displaystyle\sum_{\vert \gamma \vert \leq n} D_{\gamma x_0} \otimes D_{\gamma^{-1} x_0} \rightharpoonup \mu_{x_{0}} \otimes \mu_{x_{0}}$$ $m_{\Gamma}$ denoting the Bowen-Margulis-Sullivan measure on the geodesic flow of $SX/\Gamma$ (where $SX$ is the “unit tangent bundle") and $\delta$ denoting $\ln(2r-1)$, the Hausdorff measure of ${\mathbf{B}}$.
1. Notice that in our case, the spectrum is $\mathbb{Z}$ so the geodesic flow is not topologically mixing, according to [@DALBO] or directly by [@CHARA Ex 1.3].
2. Notice also that our multiplicative term is different of that of [@ROBLI Theorem 4.1.1], which shows that the hypothesis of non-arithmeticity of the spectrum cannot be removed.
We use the above theorem to prove the following convergence of operators.
\[II\] We have, for all $f$ in $C(\overline{X})$, the weak operator convergence $$M_n(f) \underset{n\to+\infty}{\longrightarrow} M(f).$$ In other words, we have, for all $f$ in $C(\overline{X})$ and for all $g$, $h$ in $L^2({\mathbf{B}})$, the convergence $$\frac{1}{\vert S_{n} \vert}\sum_{\gamma \in S_{n}}f(\gamma x_{0})\frac{\langle \pi(\gamma)g,h\rangle}{\Xi(\gamma)} \underset{n\to+\infty}{\longrightarrow} \langle M(f)g,h \rangle.$$
We deduce the irreducibility of $\pi$, and give an alternative proof of this well known result (see [@FIGAT Theorem 5]).
The representation $\pi$ is irreducible.
Applying Theorem \[II\] to $f = \textbf{1}_{\overline{X}}$ shows that the orthogonal projection onto the space of constant functions is in the von Neumann algebra associated with $\pi$. Then applying Theorem \[II\] to $g= \textbf{1}_{{\mathbf{B}}}$ shows that the vector $1_{{\mathbf{B}}}$ is cyclic. Then, the classical argument of [@GARNC Lemma 6.1] concludes the proof.
For $\alpha \in {\mathbb{R}}^*_+$, let us denote by $W_\alpha$ the wedge of two circles, one of length $1$ and the other of length $\alpha$. Let $p : T_\alpha \twoheadrightarrow W_\alpha$ the universal cover, with $T_\alpha$ endowed with the distance making $p$ a local isometry. Then ${\mathbb{F}_{2}} \simeq \pi_1(W_\alpha)$ acts freely properly discontinously and cocompactly on the $4$-regular tree $T_\alpha$ (which is a CAT(-1) space) by isometries. For $\alpha \in {\mathbb{R}}\setminus \mathbb{Q}$, the analog of Theorem \[II\] for the quasi-regular representation $\pi_\alpha$ of ${\mathbb{F}_{2}}$ on $L^2(\partial T_\alpha, \mu_\alpha)$ for a Patterson-Sullivan measure associated to a Bourdon distance is known to hold ([@BOYER]) because [@ROBLI Theorem 4.1.1] is true in this setting. Now if $\alpha_1$ and $\alpha_2$ are such that $\alpha_1 \not = \alpha^{\pm 1}_2$, then the representations $\pi_\alpha$ are not unitarily equivalent (). For $\alpha \in \mathbb{Q}^*_+ \setminus \{1\}$, it would be interesting to formulate and prove an equidistribution result like Theorem \[I\] in order to prove Theorem \[II\] for $\pi_\alpha$.
Proofs
======
Proof of the equidistribution theorem
-------------------------------------
For the proof of Theorem \[I\], let us denote $$E := \left\{f : C(\overline{X}\times\overline{X}) {\mbox{ } | \mbox{ }}{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in S_n} f(\gamma x_0,\gamma^{-1} x_0) \rightarrow \int_{\overline{X}\times\overline{X}} f d(\mu_{x_{0}} \otimes \mu_{x_{0}})\right\}$$
The subspace $E$ is clearly closed in $C(\overline{X} \times \overline{X})$ ; it remains only to show that it contains a dense subspace of it.
Let us define a modified version of certain characteristic functions : for $u \in {\mathbb{F}_{r}}$ we define $$\chi_u(x) :=
\left\{\begin{array}{ccl}
\max\{1 - d_X(x,C_u),0\} &\mbox{ if } &x \in X\\
0 &\mbox{ if } &x \in {\mathbf{B}}\setminus {\mathbf{B}}_u\\
1 &\mbox{ if } &x \in {\mathbf{B}}_u\\
\end{array}\right.$$
It is easy to check that he function $\chi_u$ is a continuous function which coincides with $\chi_{C_u}$ on ${\mathbb{F}_{r}} x_0$ and ${\mathbf{B}}$.
The proof of the following lemma is straightforward.
\[subalgebra\] Let $u \in {\mathbb{F}_{r}}$ and $k \geq \vert u \vert$, then $\chi_u - \displaystyle\sum\limits_{\gamma \in Pr_u(k)} \chi_\gamma$ has compact support included in $X$.
\[algebra\]The set $\chi :=\{ \chi_u {\mbox{ } | \mbox{ }}u \in {\mathbb{F}_{r}} \setminus\{e\}\}$ separates points of ${\mathbf{B}}$, and the product of two such functions of $\chi$ is either in $\chi$, the sum of a function in $\chi$ and of a function with compact support contained in $X$, or zero.
It is clear that $\chi$ separates points. It follows from Lemma \[subalgebra\] that $\chi_u \chi_v = \chi_v$ if $u$ is a proper prefix of $v$, that $\chi_u^2 - \chi_u$ has compact support in $X$, and that $\chi_u \chi_v = 0$ if none of $u$ and $v$ is a proper prefix of the other.
\[combinroblin\] The subspace $E$ contains all functions of the form $\chi_u \otimes \chi_v$.
We make the useful observation that $${\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in S_n} (\chi_u \otimes \chi_v)(\gamma x_0,\gamma^{-1} x_0) = {\displaystyle\frac{\vert S^{u,v}_n \vert}{\vert S_n \vert}}$$ where $S^{u,v}_n$ is the set of reduced words of length $n$ with $u$ as a prefix and $v^{-1}$ as a suffix. We easily see that this set is in bijection with the set of all reduced words of length $n - (\vert u \vert + \vert v \vert)$ that do not begin by the inverse of the last letter of $u$, and that do not end by the inverse of the first letter of $v^{-1}$. So we have to compute, for $s,t \in \{a^{\pm 1}_1,...,a^{\pm 1}_r\}$ and $m \in {\mathbb{N}}$, the cardinal of the set $S_m(s,t)$ of reduced words of length $m$ that do not start by $s$ and do not finish by $t$.
Now we have $$S_m = S_m(s,t) \cup \{ x {\mbox{ } | \mbox{ }}\vert x \vert = m \mbox{ and starts by }s\} \cup \{ x {\mbox{ } | \mbox{ }}\vert x \vert = m \mbox{ and ends by }t\}.$$
Note that the intersection of the two last sets is the set of words both starting by $s$ and ending by $t$, which is in bijection with $S_{m-2}(s^{-1},t^{-1})$.
We have then the recurrence relation :
$\begin{array}{rcl}
\vert S_m(s,t) \vert &= &2r(2r-1)^{m-1} - 2(2r-1)^{m-1} + \vert S_{m-2}(s^{-1},t^{-1}) \vert\\
&= &2(r-1)(2r-1)^{m-1} + 2(r-1)(2r-1)^{m-3} + \vert S_{m-4}(s,t) \vert\\
&= &(2r-1)^{m}{\displaystyle\frac{2(r-1)\left((2r-1)^2 + 1\right)}{(2r-1)^{3}}} + \vert S_{m-4}(s,t) \vert\\
\end{array}$.
We set $C := \frac{2(r-1)\left((2r-1)^2 + 1\right)}{(2r-1)^{3}}$, $n = 4k+j$ with $0\leq j \leq 3$ and we obtain
$\begin{array}{rcl}
\vert S^{s,t}_{4k + j} \vert &= &C(2r-1)^{4k+j} + \vert S^{s,t}_{4(k-1)+j} \vert\\
&= &C(2r-1)^{4k+j} + C(2r-1)^{4(k-1)+j}+ \vert S^{s,t}_{4(k-2) + j} \vert\\
\\
&= &C\displaystyle\sum^{k}_{i=1} (2r-1)^{4i+j} + \vert S^{s,t}_{j} \vert\\
&= &C(2r-1)^{4 + j} {\displaystyle\frac{(2r-1)^{4k} - 1}{(2r-1)^4 - 1}} + \vert S_{j}(s,t) \vert\\
\\
&= &(2r-1)^{1 + j}{\displaystyle\frac{(2r-1)^{4k} - 1}{2r}} + \vert S_{j}(s,t) \vert\\
\end{array}$
Now we can compute
$\begin{array}{rcl}
{\displaystyle\frac{\vert S^{u,v}_{4k+j} \vert}{\vert S_{4k+j} \vert}} &= &{\displaystyle\frac{\left\vert S_{4k+j - (\vert u \vert + \vert v \vert)}(u_{\vert u \vert},v^{-1}_{\vert v \vert}) \right\vert}{\vert S_{4k+j} \vert}}\\
\\
&= &{\displaystyle\frac{(2r-1)^{1 + j}{\displaystyle\frac{(2r-1)^{4k - (\vert u \vert + \vert v \vert)} - 1}{2r}} + \left\vert S_{j}(u_{\vert u \vert},v^{-1}_{\vert v \vert}) \right\vert}{2r(2r-1)^{4k+j - 1}}}\\
\\
&= &{\displaystyle\frac{1}{2r(2r-1)^{\vert u \vert - 1}}}{\displaystyle\frac{1}{2r(2r-1)^{\vert v \vert - 1}}} + o(1)\\
\\
&= &\mu_{x_{0}}({\mathbf{B}}_{u}) \mu_{x_{0}}({\mathbf{B}}_{v}) + o(1)\\
\end{array}$
when $k \to \infty$, and this proves the claim.
The subspace $E$ is dense in $C(\overline{X}\times\overline{X})$.
Let us consider $E'$, the subspace generated by the constant functions, the functions which can be written as $f\otimes g$ where $f,g$ are continuous functions on $\overline{X}$ and such that one of them has compact support included in $X$, and the functions of the form $\chi_u \otimes \chi_v$. By Proposition \[algebra\], it is a subalgebra of $C(\overline{X}\times\overline{X})$ containing the constants and separating points, so by the Stone-Weierstraßtheorem, $E'$ is dense in $C(\overline{X}\times\overline{X})$. Now, by Proposition \[combinroblin\], we have that $E' \subseteq E$, so $E$ is dense as well.
Proof of the ergodic theorem {#sectionergo}
----------------------------
The proof of Theorem \[II\] consists in two steps:
**Step 1**: Prove that the sequence $M_n$ is bounded in $\mathcal{L}(C(\overline{X}),\mathcal{B}(L^2({\mathbf{B}})))$.
**Step 2**: Prove that the sequence converges on a dense subset.
### Boundedness
In the following $\textbf{1}_{\overline{X}}$ denotes the characteristic function of $\overline{X}$. Define $$F_n := \left[M_n(\textbf{1}_{\overline{X}})\right]\textbf{1}_{{\mathbf{B}}}.$$ We denote by $\Xi(n)$ the common value of $\Xi$ on elements of length $n$.
The function $\xi \mapsto \sum\limits_{\gamma \in S_n} \left(P(\gamma,\xi)\right)^{\frac{1}{2}}$ is constant equal to $\vert S_n \vert \times\Xi(n)$.
This function is constant on orbits of the action of the group of automorphisms of $X$ fixing $ x_0$. Since it is transitive on ${\mathbf{B}}$, the function is constant. By integrating, we find
$\begin{array}{rcl}
\displaystyle\sum\limits_{\gamma \in S_n} \left(P(\gamma,\xi)\right)^{\frac{1}{2}} &= &\displaystyle\int_{{\mathbf{B}}} \sum\limits_{\gamma \in S_n} \left(P(\gamma,\xi)\right)^{\frac{1}{2}} d\mu_{x_{0}}(\xi)\\
&= &\displaystyle\sum\limits_{\gamma \in S_n} \displaystyle\int_{{\mathbf{B}}} \left(P(\gamma,\xi)\right)^{\frac{1}{2}} d\mu_{x_{0}}(\xi)\\
&= &\displaystyle\sum\limits_{\gamma \in S_n} \Xi(n)\\
&= &\vert S_n \vert \Xi(n),\\
\end{array}$
The function $F_n$ is constant, equal to $\textbf{1}_{{\mathbf{B}}}$.
Because $\Xi$ depends only on the length, we have that
$\begin{array}{rcl}
F_n(\xi) &:= &{\displaystyle\frac{1}{\vert S_n\vert}} \sum\limits_{\gamma \in S_n} {\displaystyle\frac{\left(P(\gamma,\xi)\right)^{\frac{1}{2}}}{\Xi(\gamma)}}\\
&= &{\displaystyle\frac{1}{\vert S_n \vert \Xi(n)}} \sum\limits_{\gamma \in S_n} \left(P(\gamma,\xi)\right)^{\frac{1}{2}}\\
&=&1,
\end{array}$
and the proof is done.
It is easy to see that $M_n(f)$ induces continuous linear transformations of $L^1$ and $L^\infty$, which we also denote by $M_n(f)$.
\[bornitude\] The operator $M_n(\textbf{1}_{\overline{X}})$, as an element of $\mathcal{L}(L^{\infty}, L^{\infty})$, has norm $1$; as an element of $\mathcal{B}(L^2({\mathbf{B}}))$, it is self-adjoint.
Let $h \in L^{\infty}({\mathbf{B}})$. Since $M_n(\textbf{1}_{\overline{X}})$ is positive, we have that
$\begin{array}{rcl}
\left\Vert \left[M_n(\textbf{1}_{\overline{X}})\right]h \right\Vert_{\infty} &\leq &\left\Vert \left[M_n(\textbf{1}_{\overline{X}})\right]\textbf{1}_{{\mathbf{B}}} \right\Vert_{\infty} \left\Vert h \right\Vert_{\infty}\\
&= &\left\Vert F_n \right\Vert_{\infty} \left\Vert h \right\Vert_{\infty}\\
&= &\Vert h\Vert_\infty\\
\end{array}$
so that $\Vert M_n(\textbf{1}_{\overline{X}})\Vert_{\mathcal{L}(L^{\infty},L^{\infty})} \leq 1$.
The self-adjointness follows from the fact that $\pi(\gamma)^* = \pi(\gamma^{-1})$ and that the set of summation is symmetric.
Let us briefly recall one useful corollary of Riesz-Thorin’s theorem :
Let $(Z,\mu)$ be a probability space.
\[rieszthorin\] Let $T$ be a continuous operator of $L^1(Z)$ to itself such that the restriction $T_2$ to $L^2(Z)$ (resp. $T_\infty$ to $L^\infty(Z)$) induces a continuous operator of $L^2(Z)$ to itself (resp. $L^\infty(Z)$ to itself).
Suppose also that $T_2$ is self-adjoint, and assume that $\Vert T_\infty \Vert_{\mathcal{L}(L^\infty(Z),L^\infty(Z))} \leq 1$.
Then $\Vert T_2 \Vert_{\mathcal{L}(L^2(Z),L^2(Z))} \leq 1$.
Consider the adjoint operator $T^*$ of $(L^1)^* = L^\infty$ to itself. We have that $$\Vert T^* \Vert_{\mathcal{L}(L^\infty,L^\infty)} = \Vert T \Vert_{\mathcal{L}(L^1(Z),L^1(Z))}.$$
Now because $T_2$ is self-adjoint, it is easy to see that $T^* = T_\infty$. This implies $$1 \geq \Vert T^* \Vert_{\mathcal{L}(L^\infty,L^\infty)} = \Vert T \Vert_{\mathcal{L}(L^1(Z),L^1(Z))}.$$
Hence the Riesz-Thorin’s theorem gives us the claim.
\[boundedness\]The sequence $\left(M_n\right)_{n \in {\mathbb{N}}}$ is bounded in $\mathcal{L}(C(\overline{X}),\mathcal{B}(L^2({\mathbf{B}})))$.
Because $M_n(f)$ is positive in $f$, we have, for every positive $g \in L^2({\mathbf{B}})$, the inequality
$- \Vert f \Vert_\infty [M_n(\textbf{1}_{\overline{X}})]g \leq [M_n(f)]g \leq \Vert f \Vert_\infty [M_n(\textbf{1}_{\overline{X}})]g$
from which we deduce, for every $g \in L^2({\mathbf{B}})$
$\begin{array}{rcl}
\Vert [M_n(f)]g \Vert_{L^2} &\leq &\Vert f \Vert_{\infty} \Vert [M_n(\textbf{1}_{\overline{X}})]g \Vert_{L^2}\\
&\leq &\Vert f \Vert_{\infty} \mbox{ } \Vert M_n(\textbf{1}_{\overline{X}}) \Vert_{\mathcal{B}(L^2)} \mbox{ } \Vert g \Vert_{L^2}\\
\end{array}$
which allows us to conclude that
$\Vert M_n(f)\Vert_{\mathcal{B}(L^2)} \leq \Vert M_n(\textbf{1}_{\overline{X}})\Vert_{\mathcal{B}(L^2)} \Vert f \Vert_{\infty}$.
This proves that $\Vert M_n \Vert_{\mathcal{L}(C(\overline{X}),\mathcal{B}(L^2))} \leq \Vert M_n(\textbf{1}_{\overline{X}}) \Vert_{\mathcal{B}(L^2)}$.
Now, it follows from Proposition \[bornitude\] and Proposition \[rieszthorin\] that the sequence $(M_n(\textbf{1}_{\overline{X}}))_{n \in {\mathbb{N}}}$ is bounded by $ 1$ in $\mathcal{B}(L^2)$, so we are done.
### Estimates for the Harish-Chandra function
The values of the Harish-Chandra are known (see for example [@FIGAT Theorem 2, Item (iii)]). We provide here the simple computations we need.
We will calculate the value of
$$\langle \pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_u}\rangle =\displaystyle\int_{{\mathbf{B}}_u} P(\gamma,\xi)^{\frac{1}{2}} d\mu_{x_{0}}(\xi).$$
\[harish1\]Let $\gamma = s_1\cdots s_n \in {\mathbb{F}_{r}}$. Let $l \in \{1,...,\vert \gamma \vert\}$, and $u = s_1\cdots s_{l-1}t_l t_{l+1}\cdots t_{l+k}$[^3], with $t_l \not = s_l$ and $k \geq 0$, be a reduced word. Then $$\langle \pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_u}\rangle = {\displaystyle\frac{1}{2r(2r-1)^{\frac{\vert \gamma \vert}{2}+k}}}$$ and $$\langle \pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_\gamma}\rangle = {\displaystyle\frac{2r-1}{2r(2r-1)^{\frac{\vert \gamma \vert}{2}}}}$$
The function $\xi \mapsto \beta_\xi( x_0,\gamma x_0)$ is constant on ${\mathbf{B}}_u$ equal to $ 2(l-1) - \vert \gamma \vert$.
So $\langle \pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_u}\rangle$ is the integral of a constant function: $$\begin{array}{rcl}
\displaystyle\int_{{\mathbf{B}}_u} P(\gamma,\xi)^{\frac{1}{2}} d\mu_{x_{0}}(\xi) &= &\mu_{x_{0}}({\mathbf{B}}_u) \mbox{ } e^{\log(2r-1)\left((l-1) - \frac{\vert \gamma \vert}{2}\right)}\\
&= &{\displaystyle\frac{1}{2r(2r-1)^{\frac{\vert \gamma \vert}{2}+k}}}\cdot\\
\end{array}$$
The value of $\langle \pi(\gamma) \textbf{1}_{\mathbf{B}}, \textbf{1}_{{\mathbf{B}}_\gamma} \rangle$ is computed in the same way.
*(The Harish-Chandra function)*
Let $\gamma = s_1\cdots s_n$ in $S_{n}$ written as a reduced word. We have that $$\Xi(\gamma) = \left(1+{\displaystyle\frac{r-1}{r}}\vert\gamma\vert\right)(2r-1)^{-\frac{\vert\gamma\vert}{2}}.$$
We decompose ${\mathbf{B}}$ into the following partition: $${\mathbf{B}}= \displaystyle\bigsqcup\limits_{u_1 \not = s_1} {\mathbf{B}}_{u_1} \sqcup \left(\bigsqcup\limits^{\vert \gamma \vert}_{l = 2} \bigsqcup\limits_{\substack{u = s_1\cdots s_{l-1}t_l \\ t_l \not \in \{s_l, (s_{l-1})^{-1}\}}} {\mathbf{B}}_u \right) \sqcup {\mathbf{B}}_\gamma$$
and Lemma \[harish1\] provides us the value of the integral on the subsets forming this partition. A simple calculation yields the announced formula.
The proof of the following lemma is then obvious :
\[harishestimate\] If $\gamma, w \in {\mathbb{F}_{r}}$ are such that $w$ is not a prefix of $\gamma$, then there is a constant $C_w$ not depending on $\gamma$ such that $${\displaystyle\frac{\langle \pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_w}\rangle}{\Xi(\gamma)}} \leq {\displaystyle\frac{C_w}{\vert \gamma \vert}}.$$
### Analysis of matrix coefficients
The goal of this section is to compute the limit of the *matrix coefficients* $\langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle$.
\[matrix1\] Let $u, w \in {\mathbb{F}_{r}}$ such that none of them is a prefix of the other (i.e. ${\mathbf{B}}_u \cap {\mathbf{B}}_w = \emptyset$). Then $$\displaystyle\lim\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{\mathbf{B}}, \textbf{1}_{{\mathbf{B}}_w}\rangle = 0$$
Using Lemma \[harishestimate\], we get $$\begin{array}{rcl}
\langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_w}\rangle &= &{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in S_n} \chi_u(\gamma x_0) {\displaystyle\frac{\langle\pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_w}\rangle}{\Xi(\gamma)}}\\
&= &{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in C_u \cap S_n} {\displaystyle\frac{\langle \pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_w}\rangle}{\Xi(\gamma)}}\\
&\leq &{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in C_u \cap S_n} {\displaystyle\frac{C_w}{\vert \gamma \vert}}\\
&= &O\left({\displaystyle\frac{1}{n}}\right)\\
\end{array}$$
\[matrix2\] Let $u, v \in {\mathbb{F}_{r}}$. Then $$\displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{\mathbf{B}}\rangle \leq \mu_{x_{0}}({\mathbf{B}}_u)\mu_{x_{0}}({\mathbf{B}}_v)$$
$$\begin{array}{rcl}
\langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}}\rangle &= &\langle M_n(\chi_u)^* \textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_v}\rangle\\
&= &{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in S_n} \chi_u(\gamma^{-1} x_0) {\displaystyle\frac{\langle\pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_v}\rangle}{\Xi(\gamma)}}\\
&\leq &{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in S_n} \chi_u(\gamma^{-1} x_0) \chi_v(\gamma x_0)\\
& &+\mbox{ }{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\substack{\gamma \in S_n \\ \gamma \not \in C_v}} \chi_u(\gamma^{-1} x_0) {\displaystyle\frac{\langle \pi(\gamma)\textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}_v}\rangle}{\Xi(\gamma)}}\\
&= &{\displaystyle\frac{1}{\vert S_n \vert}} \displaystyle\sum_{\gamma \in S_n} \chi_u(\gamma^{-1} x_0) \chi_v(\gamma x_0)\\
& &+\mbox{ }O\left({\displaystyle\frac{1}{n}}\right)\\
\end{array}$$
Hence, by taking the $\limsup$ and using Theorem **I**, we obtain the desired inequality.
\[matrix3\] For all $u, v, w \in {\mathbb{F}_{r}}$, we have $$\displaystyle\lim\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle = \mu_{x_{0}}({\mathbf{B}}_u \cap {\mathbf{B}}_w) \mu_{x_{0}}({\mathbf{B}}_v)$$
We first show the inequality $$\displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle \leq \mu_{x_{0}}({\mathbf{B}}_u \cap {\mathbf{B}}_w) \mu_{x_{0}}({\mathbf{B}}_v) .$$ If none of $u$ and $w$ is a prefix of the other, we have nothing to do according to Lemma \[matrix1\]. Let us assume that $u$ is a prefix of $w$ (the other case can be treated analogously). We have, by Lemma \[matrix2\], that
$$\begin{array}{rcl}
\mu_{x_{0}}({\mathbf{B}}_w)\mu_{x_{0}}({\mathbf{B}}_v) &\geq &\displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_w) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{\mathbf{B}}\rangle\\
&\geq &\displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_w) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle\\
&\geq &\displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_w) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle + \sum\limits_{\gamma \in Pr_{u}(\vert w \vert) \setminus \{w\}} \displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_\gamma) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle\\
&= &\displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle
\end{array}$$
We now compute the expected limit. Let us define $$S_{u,v,w} := \{(u',v',w') \in {\mathbb{F}_{r}} {\mbox{ } | \mbox{ }}\vert u \vert = \vert u' \vert, \vert v \vert = \vert v' \vert, \vert w \vert = \vert w' \vert\}.$$ Then $$\begin{array}{rll}
1 &= &\displaystyle\liminf\limits_{n \to \infty} \langle M_n(\textbf{1}_{\overline{X}}) \textbf{1}_{{\mathbf{B}}}, \textbf{1}_{{\mathbf{B}}}\rangle\\
&\leq &\displaystyle\liminf\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle + \displaystyle\sum\limits_{(u',v',w') \in S_{u,v,w} \setminus\{u, v, w\}} \displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_{u'}) \textbf{1}_{{\mathbf{B}}_{v'}}, \textbf{1}_{{\mathbf{B}}_{w'}}\rangle\\
&\leq &\displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle + \displaystyle\sum\limits_{(u',v',w') \in S_{u,v,w} \setminus\{u, v, w\}} \displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_{u'}) \textbf{1}_{{\mathbf{B}}_{v'}}, \textbf{1}_{{\mathbf{B}}_{w'}}\rangle\\
&\leq &\mu_{x_{0}}({\mathbf{B}}_{u} \cap {\mathbf{B}}_{w}) \mu_{x_{0}}({\mathbf{B}}_{v}) + \displaystyle\sum\limits_{(u',v',w') \in S_{u,v,w} \setminus \{u,v,w\}} \mu_{x_{0}}({\mathbf{B}}_{u'} \cap {\mathbf{B}}_{w'}) \mu_{x_{0}}({\mathbf{B}}_{v'})\\
&= &1\\
\end{array}$$
This proves that all the inequalities above are in fact equalities, and moreover proves that the inequalities $$\displaystyle\liminf\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle \leq \displaystyle\limsup\limits_{n \to \infty} \langle M_n(\chi_u) \textbf{1}_{{\mathbf{B}}_v}, \textbf{1}_{{\mathbf{B}}_w}\rangle \leq \mu_{x_{0}}({\mathbf{B}}_{u} \cap {\mathbf{B}}_{w}) \mu_{x_{0}}({\mathbf{B}}_{v})$$ are in fact equalities.
Because of the boundedness of the sequence $(M_n)_{n \in {\mathbb{N}}}$ proved in Proposition \[boundedness\], it is enough to prove the convergence for all $(f,h_1,h_2)$ in a dense subset of $C(\overline{X})\times L^2 \times L^2$, which is what Proposition \[matrix3\] asserts.
[^1]: Weizmann Institute of Science, aadrien.boyer@gmail.com
[^2]: Université d’Aix-Marseille, CNRS UMR7373, a.p.lobos@outlook.com
[^3]: For $l=1$, $s_1\cdots s_{l-1}$ is $e$ by convention.
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abstract: |
We study the two-variable fragments ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ and ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ of dependence logic and independence-friendly logic. We consider the satisfiability and finite satisfiability problems of these logics and show that for ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$, both problems are [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete, whereas for ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$, the problems are [[[$\mathit{\Pi}^0_1$]{}]{}]{}and [[[$\mathit{\Sigma}^0_1$]{}]{}]{}-complete, respectively. We also show that ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ is strictly less expressive than ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ and that already in ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$, equicardinality of two unary predicates and infinity can be expressed (the latter in the presence of a constant symbol).
This is an extended version of a publication in the proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS 2011).
author:
- '[ Juha Kontinen[^1] , Antti Kuusisto[^2] , Peter Lohmann[^3] , Jonni Virtema]{}'
title: '[Complexity of two-variable Dependence Logic and IF-Logic]{}[^4]'
---
[dependence logic, independence-friendly logic, two-variable logic, decidability, complexity, satisfiability, expressivity]{}
[F.4.1 Computability theory, Model theory; F.1.3 Reducibility and completeness]{}
Introduction
============
The satisfiability problem of first-order logic [[[[${\mathsf{FO}}$]{}]{}]{}]{}was shown to be undecidable in [@ch36; @tu36], and ever since, logicians have been searching for decidable fragments of [[[[${\mathsf{FO}}$]{}]{}]{}]{}. Henkin [@he67] was the first to consider the logics ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^k$, i.e., the fragments of first-order logic with $k$ variables. The fragments ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^k$, for $k\ge 3$, were easily seen to be undecidable but the case for $k=2$ remained open. Scott [@sc62] then showed that [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}without equality is decidable. Mortimer [@mo75] extended the result to [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}with equality and showed that every satisfiable [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}formula has a model whose size is doubly exponential in the length of the formula. His result established that the satisfiability and finite satisfiability problems of [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}are contained in [[[${\textsl{2NEXPTIME}}$]{}]{}]{}. Finally, Grädel, Kolaitis and Vardi [@grkova97] improved the result of Mortimer by establishing that every satisfiable [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}formula has a model of exponential size. Furthermore, they showed that the satisfiability problem for [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}is [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete.
The decidability of the satisfiability problem of various extensions of [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}has been studied (e.g. [@grotro97; @grot99; @etvawi02; @kiot05]). One such interesting extension [[[[${\mathsf{FOC^2}}$]{}]{}]{}]{}is acquired by extending [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}with counting quantifiers $\exists^{\geq i}$. The meaning of a formula of the form $\exists^{\geq i} x \phi(x)$ is that $\phi(x)$ is satisfied by at least $i$ distinct elements. The satisfiability problem for the logic [[[[${\mathsf{FOC^2}}$]{}]{}]{}]{}was shown to be decidable by Grädel et al. [@grotro97a], and shown to be in [[[${\textsl{2NEXPTIME}}$]{}]{}]{}by Pacholski et al. [@paszte97]. Finally, Pratt-Hartmann [@Pratt-Hartmann:2005] established that the problem is [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete. We will later use the result of Pratt-Hartmann to determine the complexity of the satisfiability problem of the two-variable fragment of dependence logic.
\[table:results\]
--------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------
*Logic* *Complexity of [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}$]{}]{}]{}]{} / [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}$]{}]{}]{}]{}*
[[[[${\mathsf{FO}}$]{}]{}]{}]{}, ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^3$ [[[$\mathit{\Pi}^0_1$]{}]{}]{} / [[[$\mathit{\Sigma}^0_1$]{}]{}]{} [@ch36; @tu36]
[[[[${\mathsf{ESO}}$]{}]{}]{}]{}, [[[[${\mathsf{D}}$]{}]{}]{}]{}, [[[[${\mathsf{IF}}$]{}]{}]{}]{} [[[$\mathit{\Pi}^0_1$]{}]{}]{} / [[[$\mathit{\Sigma}^0_1$]{}]{}]{} Remark \[eso sat\], [@ch36; @tu36]
[[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{} [[[${\textsl{NEXPTIME}}$]{}]{}]{} [@grkova97]
[[[[${\mathsf{FOC^2}}$]{}]{}]{}]{} [[[${\textsl{NEXPTIME}}$]{}]{}]{} [@Pratt-Hartmann:2005]
[[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{} [[[${\textsl{NEXPTIME}}$]{}]{}]{} Theorem \[dtwo nexptime\]
${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}({{{\protect\ensuremath{\mathrm{I}}}\xspace}})$ [[[$\mathit{\Sigma}^1_1$]{}]{}]{}-hard / [[[$\mathit{\Sigma}^0_1$]{}]{}]{} [@grotro97]
[[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{} [[[$\mathit{\Pi}^0_1$]{}]{}]{} / [[[$\mathit{\Sigma}^0_1$]{}]{}]{} Theorems \[iftwo sat complexity\], \[iftwo finsat complexity\]
--------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------
: Complexity of satisfiability for various logics.The results are completeness results for the full relational vocabulary.
In this article we study the satisfiability of the two-variable fragments of independence-friendly logic ([[[[${\mathsf{IF}}$]{}]{}]{}]{}) and dependence logic ([[[[${\mathsf{D}}$]{}]{}]{}]{}). The logics ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ and ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$ are conservative extensions of ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}$, i.e., they agree with ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}$ on sentences which syntactically are ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}$-sentences. We thereby contribute to the understanding of the satisfiability problems of extensions of ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}$. We briefly recall the history of ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ and ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$. In first-order logic the order in which quantifiers are written determines dependence relations between variables. For example, when using game theoretic semantics to evaluate the formula $$\forall x_0\exists x_1\forall x_2\exists x_3\, \phi,$$ the choice for $x_1$ depends on the value for $x_0$, and the choice for $x_3$ depends on the value of both universally quantified variables $x_0$ and $x_2$. The characteristic feature of [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}is that in these logics it is possible to express dependencies between variables that cannot be expressed in [[[[${\mathsf{FO}}$]{}]{}]{}]{}. The first step in this direction was taken by Henkin [@henkin1961] with his partially ordered quantifiers $$\label{poc}
\left(\begin{array}{cc}\forall x_0& \exists x_1\\ \forall
x_2&\exists x_3\end{array}\right)\phi,$$ where $x_1$ depends only on $x_0$ and $x_3$ depends only on $x_2$. Enderton [@MR44:1546] and Walkoe [@MR43:4646] observed that exactly the properties definable in existential second-order logic ([[[[${\mathsf{ESO}}$]{}]{}]{}]{}) can be expressed with partially ordered quantifiers. The second step was taken by Hintikka and Sandu [@MR1034575; @MR1410063], who introduced independence-friendly logic, which extends [[[[${\mathsf{FO}}$]{}]{}]{}]{}in terms of so-called slashed quantifiers. For example, in $$\forall x_0\exists x_1\forall x_2\exists
x_3/\forall x_0\phi,$$ the quantifier $\exists x_3/\forall x_0$ means that $x_3$ is “independent” of $x_0$ in the sense that a choice for the value of $x_3$ should not depend on what the value of $x_0$ is. The semantics of [[[[${\mathsf{IF}}$]{}]{}]{}]{}was first formulated in game theoretic terms, and [[[[${\mathsf{IF}}$]{}]{}]{}]{}can be regarded as a game theoretically motivated generalization of [[[[${\mathsf{FO}}$]{}]{}]{}]{}. Whereas the semantic game for [[[[${\mathsf{FO}}$]{}]{}]{}]{}is a game of perfect information, the game for [[[[${\mathsf{IF}}$]{}]{}]{}]{}is a game of imperfect information. The so-called team semantics of [[[[${\mathsf{IF}}$]{}]{}]{}]{}, also used in this paper, was introduced by Hodges [@MR1465612].
Dependence logic, introduced by Väänänen [@va07], was inspired by [[[[${\mathsf{IF}}$]{}]{}]{}]{}-logic, but the approach of Väänänen provided a fresh perspective on quantifier dependence. In dependence logic the dependence relations between variables are written in terms of novel atomic dependence formulas. For example, the partially ordered quantifier can be expressed in dependence logic as follows $$\forall x_0\exists x_1\forall x_2\exists
x_3({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x_2,x_3)\wedge\phi).$$ The atomic formula ${{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x_2,x_3)$ has the explicit meaning that $x_3$ is completely determined by $x_2$ and nothing else.
In recent years, research related to [[[[${\mathsf{IF}}$]{}]{}]{}]{}and [[[[${\mathsf{D}}$]{}]{}]{}]{}has been active. A variety of closely related logics have been defined and various applications suggested, see e.g. [@Abramsky:2007; @Bradfield:2005; @Gradel+aananen:2010; @lovo10; @Sevenster:2009; @Vaananen+Hodges:2010]. While both [[[[${\mathsf{IF}}$]{}]{}]{}]{}and [[[[${\mathsf{D}}$]{}]{}]{}]{}are known to be equi-expressive to [[[[${\mathsf{ESO}}$]{}]{}]{}]{}, the relative strengths and weaknesses of the two different logics in relation to applications is not understood well. In this article we take a step towards a better understanding of this matter. After recalling some basic properties in Section \[preliminaries\], we compare the expressivity of the finite variable fragments of [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}in Section \[comparison\]. We show that there is an effective translation from [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}to [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}(Theorem \[dtoif\]) and from [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}to ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3$ (Theorem \[iftod\]). We also show that [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}is strictly more expressive than [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}(Proposition \[d less than if\]). This result is a by-product of our proof in Section \[ifsatsection\] that the satisfiability problem of [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}is undecidable (Theorem \[iftwo sat complexity\] shows [[[$\mathit{\Pi}^0_1$]{}]{}]{}-completeness). The proof can be adapted to the context of finite satisfiability, i.e., the problem of determining for a given formula $\phi$ whether there is a finite structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ such that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi$ (Theorem \[iftwo finsat complexity\] shows [[[$\mathit{\Sigma}^0_1$]{}]{}]{}-completeness). The undecidability proofs are based on tiling arguments. Finally, in Section \[sec:satd2\], we study the decidability of the satisfiability and finite satisfiability problems of [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}. For this purpose we reduce the problems to the (finite) satisfiability problem for [[[[${\mathsf{FOC^2}}$]{}]{}]{}]{}(Theorem \[DtoESO\]) and thereby show that they are [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete (Theorem \[dtwo nexptime\]). Table \[table:results\] gives an overview of previously-known as well as new complexity results.
Preliminaries
=============
In this section we recall the basic concepts and results relevant for this article. The domain of a structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ is denoted by $A$. We assume that the reader is familiar with first-order logic [[[[${\mathsf{FO}}$]{}]{}]{}]{}. The extension of [[[[${\mathsf{FO}}$]{}]{}]{}]{}in terms of counting quantifiers $\exists ^{\ge i}$ is denoted by [[[[${\mathsf{FOC}}$]{}]{}]{}]{}. We also consider the extension ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}({{{\protect\ensuremath{\mathrm{I}}}\xspace}})$ of [[[[${\mathsf{FO}}$]{}]{}]{}]{}by the Härtig quantifier ${{{\protect\ensuremath{\mathrm{I}}}\xspace}}$. The interpretation of the quantifier [[[$\mathrm{I}$]{}]{}]{}is defined by the clause $${{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s\models {{{\protect\ensuremath{\mathrm{I}}}\xspace}}\, xy(\phi(x),\psi(
y))\Leftrightarrow |\phi(x)^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s}|=|\psi(y)^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s}|,$$ where $\phi(x)^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s}:=\{a\in A\ |\ {{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s\models \phi(a)\}$. The $k$-variable fragments ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^k$, ${{{{\protect\ensuremath{{\mathsf{FOC}}}}\xspace}}}^k$, and ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^k({{{\protect\ensuremath{\mathrm{I}}}\xspace}})$ are the fragments of [[[[${\mathsf{FO}}$]{}]{}]{}]{}, [[[[${\mathsf{FOC}}$]{}]{}]{}]{}, and ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}({{{\protect\ensuremath{\mathrm{I}}}\xspace}})$ with formulas in which at most $k$, say $x_1,\ldots,x_k$, distinct variables appear. In the case $k=2$, we denote these variables by $x$ and $y$. The existential fragment of second-order logic is denoted by [[[[${\mathsf{ESO}}$]{}]{}]{}]{}. For logics ${{{\protect\ensuremath{\mathcal{L}}}\xspace}}$ and $\mathcal L'$, we write $\mathcal L \leq \mathcal L'$ if for every sentence $\phi$ of $\mathcal L$ there is a sentence $\phi^*$ of $\mathcal L'$ such that for all structures ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ it holds that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi^*$. We write ${{{\protect\ensuremath{\mathcal{L}}}\xspace}}\mathcal \equiv \mathcal L'$ if $\mathcal L \leq \mathcal L'$ and $\mathcal L' \leq \mathcal L$. We assume that the reader is familiar with the basics of computational complexity theory. In this article we are interested in the complexity of the satisfiability problems of various logics. For any logic [[[$\mathcal{L}$]{}]{}]{}the *satisfiability problem* ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathcal{L}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathcal{L}}}\xspace}}\nobreak)}}}}\xspace}}}$ is defined as $${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathcal{L}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathcal{L}}}\xspace}}\nobreak)}}}}\xspace}}} := \{\phi \in {{{\protect\ensuremath{\mathcal{L}}}\xspace}}\mid \text{there is a structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ such that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi$}\}.$$
The finite satisfiability problem ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathcal{L}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathcal{L}}}\xspace}}\nobreak)}}}}\xspace}}}$ is the analogue of ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathcal{L}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathcal{L}}}\xspace}}\nobreak)}}}}\xspace}}}$ in which we require the structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ to be finite. The following observation will be useful later.
\[eso sat\] If $\phi$ is a formula over the vocabulary $\tau$ and $$\psi := \exists R_1 \dots \exists R_n \exists f_1 \dots \exists f_m \phi$$ with $R_1,\dots,R_n,f_1,\dots,f_m\in \tau$, then $\phi$ is satisfiable iff the second-order formula $\psi$ is satisfiable.
The logics D and IF
-------------------
In this section we define independence-friendly logic and dependence logic and recall some related basic results. For ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ we follow the exposition of [@CaicedoDJ09] and the forthcoming monograph [@ASS].
\[def:iftwo intuitive\] The syntax of ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ extends the syntax of ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}$ defined in terms of $\vee$, $\wedge$, $\neg$, $\exists$ and $\forall$, by adding quantifiers of the form $$\begin{aligned}
&\exists x/W\phi\\
&\forall x/W\phi\end{aligned}$$ called slashed quantifiers, where $x$ is a first-order variable, $W$ a finite set of first-order variables and $\phi$ a formula.
\[def:dtwo intuitive\] The syntax of ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$ extends the syntax of ${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}$, defined in terms of $\vee$, $\wedge$, $\neg$, $\exists$ and $\forall$, by new atomic (dependence) formulas of the form $${{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{t_1,\ldots,t_n}{}}{}{(\nobreakt_1,\ldots,t_n\nobreak)}}}}\xspace}}},$$ where $t_1,\ldots,t_n$ are terms.
The set ${{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{\phi}{}}{}{(\nobreak\phi\nobreak)}}}}\xspace}}}$ of free variables of a formula $\phi\in {{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}\cup {{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ is defined as for first-order logic except that we have the new cases $$\begin{array}{lcl}
{{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t_1,\ldots,t_n))={{{{\protect\ensuremath{\mathord{{\mathrm{Var}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t_1)\cup\cdots \cup {{{{\protect\ensuremath{\mathord{{\mathrm{Var}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t_n)\\
{{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(\exists x /W\psi)=W\cup ({{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(\psi)\setminus \{x\})\\
{{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(\forall x /W\psi)=W\cup ({{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(\psi)\setminus \{x\})\\
\end{array}$$ where ${{{{\protect\ensuremath{\mathord{{\mathrm{Var}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t_i)$ is the set of variables occurring in the term $t_i$. If ${{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(\phi)=\emptyset$, we call $\phi$ a sentence.
\[def:diftwo\] Let $\tau$ be a relational vocabulary, i.e., $\tau$ does not contain function or constant symbols.
a) The *two-variable independence-friendly* logic ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}(\tau)$ is generated from $\tau$ according to the following grammar: $$\begin{aligned}
\phi::= &t_1=t_2\mid R(t_1,\dots,t_n) \mid \neg t_1=t_2\mid \neg R(t_1,\dots,t_n) \mid\\
&(\phi\wedge\phi) \mid (\phi\vee\phi) \mid \forall x \phi \mid \forall y \phi\mid \exists x/W\phi\mid \exists y/W\phi\end{aligned}$$
b) The *two-variable dependence* logic ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}(\tau)$ is generated from $\tau$ according to the following grammar: $$\begin{aligned}
\phi::= &t_1=t_2\mid R(t_1,\dots,t_n) \mid \neg t_1=t_2\mid \neg R(t_1,\dots,t_n) \mid {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{t_1,t_2}{}}{}{(\nobreakt_1,t_2\nobreak)}}}}\xspace}}} \mid \neg {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{t_1,t_2}{}}{}{(\nobreakt_1,t_2\nobreak)}}}}\xspace}}}\mid\\
& {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{t_1}{}}{}{(\nobreakt_1\nobreak)}}}}\xspace}}} \mid \neg {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{t_1}{}}{}{(\nobreakt_1\nobreak)}}}}\xspace}}}\mid(\phi\wedge\phi) \mid (\phi\vee\phi) \mid \forall x \phi \mid \forall y \phi\mid \exists x\phi\mid \exists y\phi\end{aligned}$$
Here $R\in\tau$ is an $n$-ary relation symbol, $W\subseteq\{x,y\}$ and $t_1,\dots,t_n \in \{x,y\}$. We identify existential first-order quantifiers with existential quantifiers with empty slash sets, and therefore if $W=\emptyset$ we simply write $\exists x\phi(x)$ instead of $\exists x /W \phi(x)$. When $\tau$ is clear we often leave it out. To simplify notation, we assume in the following that the relation symbols $R\in\tau$ are at most binary.
Note that in Definition \[def:diftwo\] we have only defined formulas in negation normal form and for that reason we do not need the slashed universal quantifier in [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}[@MR1465612]. Defining syntax in negation normal form is customary in ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ and [[[[${\mathsf{D}}$]{}]{}]{}]{}. A formula $\phi$ with arbitrary negations is considered an abbreviation of the negation normal form formula $\psi$ obtained from $\phi$ by pushing the negations to the atomic level in the same fashion as in first-order logic. It is important to note that the game theoretically motivated negation $\neg$ of [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}does not satisfy the law of excluded middle and is therefore not the classical Boolean negation. This is manifested by the existence of sentences $\phi$ such that for some ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ we have ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\not\models\phi$ and ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\not\models\neg\phi$.
In order to define the semantics of ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ and ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$, we first need to define the concept of a *team*. Let ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ be a model with the domain $A$. [*Assignments*]{} over ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ are finite functions that map variables to elements of $A$. The value of a term $t$ in an assignment $s$ is denoted by $t^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\langle s\rangle$. If $s$ is an assignment, $x$ a variable, and $a\in A$, then $s(a/x)$ denotes the assignment (with the domain ${{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(s)\cup \{x\}$) which agrees with $s$ everywhere except that it maps $x$ to $a$.
Let $A$ be a set and $\{x_1,\ldots,x_k\}$ a finite (possibly empty) set of variables. A [*team*]{} $X$ of $A$ with the domain ${{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)=\{x_1,\ldots,x_k\}$ is any set of assignments from the variables $\{x_1,\ldots,x_k\}$ into the set $A$. We denote by ${{{{\protect\ensuremath{\mathord{{\mathrm{rel}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)$ the $k$-ary relation of $A$ corresponding to $X$ $${{{{\protect\ensuremath{\mathord{{\mathrm{rel}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)=\{(s(x_1),\ldots,s(x_k)) \mid s\in X \}.$$ If $X$ is a team of $A$, and $F\colon X\rightarrow A$, we use $X(F/x)$ to denote the team $\{s(F(s)/x) \mid s\in X \}$ and $X(A/x)$ the team $\{s (a/x)
\mid s\in X\ \textrm{and}\ a\in A \}$. For a set $W\subseteq {{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)$ we call $F$ *$W$-independent* if for all $s,s' \in X$ with $s(x)=s'(x)$ for all $x\in {{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)\setminus W$ we have that $F(s)=F(s')$.
We are now ready to define the semantics of [[[[${\mathsf{IF}}$]{}]{}]{}]{}and [[[[${\mathsf{D}}$]{}]{}]{}]{}.
\[def:semantics\] Let ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ be a model and $X$ a team of $A$. The satisfaction relation ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X \phi$ is defined as follows:
1. If $\phi$ is a first-order literal, then ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \phi$ iff for all $s\in X$: ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s\models_{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}}\phi$.
2. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \psi \wedge \phi$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X \psi$ and ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X \phi$.
3. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \psi \vee \phi$ iff there exist teams $Y$ and $Z$ such that $X=Y\cup Z$, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_Y \psi$ and ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _Z \phi$.
4. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \exists x \psi$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _{X(F/x)} \psi$ for some $F\colon X\to A$.
5. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \forall x\psi$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _{X(A/x)} \psi$.
For [[[[${\mathsf{IF}}$]{}]{}]{}]{}we further have the following rules:
6. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \exists x/W\phi$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(F/x)}\phi$ for some $W$-independent function $F:X\rightarrow A$.
7. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \forall x/W\phi$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(A/x)}\phi$.
And for [[[[${\mathsf{D}}$]{}]{}]{}]{}we have the additional rules:
8. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t_{1},\ldots,t_{n})$ iff for all $s,s'\in
X$ such that $t_1^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\langle s\rangle =t_1^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\langle
s'\rangle ,\ldots, t_{n-1}^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\langle s\rangle
=t_{n-1}^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\langle s'\rangle $, we have $t_n^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\langle
s\rangle =t_n^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\langle s'\rangle $.
9. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \neg {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t_{1},\ldots,t_{n})$ iff $X=\emptyset$.
Above, we assume that the domain of $X$ contains ${{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{\phi}{}}{}{(\nobreak\phi\nobreak)}}}}\xspace}}}$. Finally, a sentence $\phi$ is true in a model ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ (${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi$) if ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _{\{\emptyset\}} \phi$.
From Definition \[def:semantics\] it follows that many familiar propositional equivalences of connectives do not hold in [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}. For example, the idempotence of disjunction fails, which can be used to show that the distributivity laws of disjunction and conjunction do not hold either. We refer to [@va07 Section 3.3] for a detailed exposition on propositional equivalences of connectives in [[[[${\mathsf{D}}$]{}]{}]{}]{}(and also [[[[${\mathsf{IF}}$]{}]{}]{}]{}). Another feature of Definition \[def:semantics\] is that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _{\emptyset}\phi$ for all ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ and all formulas $\phi$ of [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}. This observation is important in noting that, for sentences $\phi$ and $\psi$, the interpretation of $\phi\vee \psi$ coincides with the classical disjunction of $\phi$ and $\psi$.
Basic properties of D and IF
----------------------------
In this section we recall some basic properties of [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}.
Let $X$ be a team with the domain $\{x_1,\ldots,x_k\}$ and $V\subseteq \{x_1,\ldots,x_k\}$. We denote by $X\upharpoonright V$ the team $\{s\upharpoonright V \mid s\in X\}$ with the domain $V$. The following proposition shows that the truth of a [[[[${\mathsf{D}}$]{}]{}]{}]{}-formula depends only on the interpretations of the variables occurring free in the formula.
\[freevar\] Let $\phi\in {{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$ be any formula or $\phi\in {{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ a sentence. If $V\supseteq {{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(\phi)$, then ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X\phi$ if and only if ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _{X\upharpoonright V} \phi$.
The analogue of Proposition \[freevar\] does not hold for open formulas of [[[[${\mathsf{IF}}$]{}]{}]{}]{}. In other words, the truth of an ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$-formula may depend on the interpretations of variables that do not occur in the formula. For example, the truth of the formula $\phi$ $$\phi = \exists x/\{y\}(x=y)$$ in a team $X$ with domain $\{x,y,z\}$ depends on the values of $z$ in $X$, although $z$ does not occur in $\phi$.
The following fact is a fundamental property of all formulas of [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}:
\[Downward closure\] Let $\phi$ be a formula of [[[[${\mathsf{D}}$]{}]{}]{}]{}or [[[[${\mathsf{IF}}$]{}]{}]{}]{}, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ a model, and $Y\subseteq X$ teams. Then ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \phi$ implies ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_Y\phi$.
The expressive power of sentences of ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$ and [[[[${\mathsf{IF}}$]{}]{}]{}]{}coincides with that of existential second-order sentences:
\[d equiv if equiv eso\] ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}\equiv {{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}\equiv {{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}$.
The fact ${{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}\le {{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$ (and ${{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}\le {{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$) is based on the analogous result of [@MR44:1546; @MR43:4646] for partially ordered quantifiers. For the converse inclusions, see [@va07] and [@MR1638352].
\[FO extension\] Let $\phi$ be a formula of [[[[${\mathsf{D}}$]{}]{}]{}]{}or [[[[${\mathsf{IF}}$]{}]{}]{}]{}without dependence atoms and without slashed quantifiers, i.e., $\phi$ is syntactically a first-order formula. Then for all ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$, $X$ and $s$:
1. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _{\{s\}}\phi$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s\models_{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}}\phi$.
2. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X\phi$ iff for all $s\in X:\, {{{\protect\ensuremath{\mathfrak{A}}}\xspace}},s\models_{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}}\phi$.
Comparison of IF and D {#comparison}
======================
In this section we show that $${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}< {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\leq {{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3.$$ We also further discuss the expressive powers and other logical properties of ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ and ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$.
\[dtoif\] For any formula $\phi\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ there is a formula $\phi^*\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ such that for all structures ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ and teams $X$, where ${{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)=\{x,y\}$, it holds that $$\begin{aligned}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X\phi &\Leftrightarrow& {{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X\phi^*.
\end{aligned}$$
The translation $\phi\mapsto \phi^*$ is defined as follows. For first-order literals the translation is the identity, and negations of dependence atoms are translated by $\neg x=x$. The remaining cases are defined as follows: $$\begin{aligned}
{{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x) &\mapsto& \exists y/\{x,y\}(x=y)\\
{{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x,y) &\mapsto& \exists x/\{y\}(x=y)\\
\phi \wedge \psi &\mapsto& \phi^* \wedge \psi^*\\
\phi \vee \psi &\mapsto& \phi^* \vee \psi^*\\
\exists x \phi &\mapsto& \exists x \phi^* \\
\forall x \phi &\mapsto& \forall x \phi^*\end{aligned}$$ The claim of the lemma can now be proved using induction on $\phi$. The only non-trivial cases are the dependence atoms. We consider the case where $\phi$ is of the form ${{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x,y)$.
Let us assume that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \phi$. Then there is a function $F\colon A\to A$ such that $$\label{dtoif6}
\text{for all }s\in X:\ s(y)=F(s(x)).$$ Define now $F'\colon X \to A$ as follows: $$\label{dtoif3}
F'(s):=F(s(x)).$$ $F'$ is $\{y\}$-independent since, if $s(x)=s'(x)$, then $$F'(s)= F(s(x)){=}F(s'(x))=F'(s').$$ It remains to show that $$\label{dtoif2}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(F'/x)}(x=y).$$ Let $s\in X(F'/x)$. Then $$\label{dtoif5}
s= s'(F'(s')/x)\text{ for some }s'\in X.$$ Now $$s(x)\stackrel{\eqref{dtoif5}}{=}F'(s')\stackrel{\eqref{dtoif3}}{=}F(s'(x))\stackrel{\eqref{dtoif6}}{=}s'(y)\stackrel{\eqref{dtoif5}}{=}s(y).$$ Therefore, holds, and hence also $${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \exists x/\{y\}(x=y).$$
Suppose then that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\not \models_X \phi$. Then there must be $s,s'\in X$ such that $s(x)=s'(x)$ and $s(y)\neq s'(y)$. We claim now that $$\label{dtoif1}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\not \models_X \exists x/\{y\}(x=y).$$ Let $F\colon X\rightarrow A$ be an arbitrary $\{y\}$-independent function. Then, by $\{y\}$-independence, $F(s)=F(s')$ and since additionally $s(y)\neq s'(y)$, we have $$s(F(s)/x)(x)=F(s)\neq s(y)=s(F(s)/x)(y)$$ or $$s'(F(s')/x)(x)=F(s')\neq s'(y)=s'(F(s')/x)(y).$$ This implies that $${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\not\models_{X(F/x)}(x=y),$$ since $s(F(s)/x),\,s'(F(s')/x) \in X(F/x)$.
Since $F$ was arbitrary, we may conclude that holds.
Next we show a translation from ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ to ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3$.
\[iftod\] For any formula $\phi\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ there is a formula $\phi^*\in {{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3$ such that for all structures ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ and teams $X$, where ${{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)=\{x,y\}$, it holds that $$\begin{aligned}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X\phi &\Leftrightarrow& {{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X\phi^*.
\end{aligned}$$
The claim follows by the following translation $\phi\mapsto \phi^*$: For atomic and negated atomic formulas the translation is the identity, and for propositional connectives and first-order quantifiers it is defined in the obvious inductive way. The only non-trivial cases are the slashed quantifiers: $$\begin{array}{rcl}
\exists x /\{y\}\psi &\mapsto& \exists z(x=z\wedge \exists x ({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(z,x) \wedge \psi^*)),\\
\exists x /\{x\}\psi &\mapsto& \exists x ({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(y,x) \wedge \psi^*),\\
\exists x /\{x,y\}\psi &\mapsto& \exists x({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x)\wedge \psi^*).\\
\end{array}$$ Again, the claim can be proved using induction on $\phi$. We consider the case where $\phi$ is of the form $\exists x /\{y\}\psi$. Assume ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models _X \phi$. Then there is a $\{y\}$-independent function $F\colon X\rightarrow A$ such that $$\label{iftod0}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(F/x)} \psi.$$ By $\{y\}$-independence, $s(x)=s'(x)$ implies that $F(s)=F(s')$ for all $s,s'\in X$. Our goal is to show that $$\label{iftod1}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \exists z(x=z\wedge \exists x ({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(z,x) \wedge \psi^*)).$$ Now, holds if for $G\colon X \to A$ defined by $G(s)=s(x)$ for all $s\in X$ it holds that $$\label{iftod2}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(G/z)} \exists x ({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(z,x) \wedge \psi^*).$$ Define $F'\colon X(G/z)\rightarrow A$ by $F'(s)=F(s\upharpoonright\{x,y\})$. Now we claim that $${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(G/z)(F'/x)} {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(z,x) \wedge \psi^*,$$ implying and hence .
First we show that $$\label{iftod42}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(G/z)(F'/x)} {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(z,x).$$ At this point it is helpful to note that every $s\in {X(G/z)(F'/x)}$ arises from an $s'\in X$ by first copying the value of $x$ to $z$ and then replacing the value of $x$ by $F(s\upharpoonright\{x,y\})$, i.e., that $s(z)=s'(G(s')/z)(z)=G(s')=s'(x)$ and $s(x)=F(s')$. Now, to show , let $s_1,s_2\in X(G/z)(F'/x)$ with $s_1(z)=s_2(z)$ and let $s'_1,s'_2\in X$ as above, i.e., $s_1$ (resp. $s_2$) arises from $s'_1$ (resp. $s'_2$). Then it follows that $s'_1(x)=s'_2(x)$. Hence, by $\{y\}$-independence, $F(s'_1)=F(s'_2)$, implying that $s_1(x)=F(s'_1)=F(s'_2)=s_2(x)$ which proves . Let us then show that $$\label{iftod3}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(G/z)(F'/x)}\psi^*.$$ Note first that by the definition of the mapping $\phi\mapsto \phi^*$ the variable $z$ cannot appear free in $\psi^*$. By Proposition \[freevar\], the satisfaction of any ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$-formula $\theta$ only depends on those variables in a team that appear free in $\theta$, therefore holds iff $$\label{iftod4}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{X(G/z)(F'/x)\upharpoonright\{x,y\}}\psi^*.$$ We have chosen $G$ and $F'$ in such a way that $$X(G/z)(F'/x)\upharpoonright\{x,y\}=X(F/x),$$ hence now follows from and the induction hypothesis.
We omit the proof of the converse implication which is analogous.
For sentences, Lemmas \[dtoif\] and \[iftod\] now imply the following.
\[dtwo le if2\] ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\le {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\le {{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3$
The claim follows by Lemmas \[dtoif\] and \[iftod\]. First of all, if $\phi$ is a sentence of ${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}$ or ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$, then, by Proposition \[freevar\], for every model ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ and team $X\neq \emptyset$ $$\label{sentences}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \phi\text{ iff } {{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{\{\emptyset\}} \phi.$$ It is important to note that, even if $\phi\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ is a sentence, it may happen that $\phi^*$ has free variables since variables in $W$ are regarded as free in subformulas of $\phi^*$ of the form $\exists x /W\psi$. However, this is not a problem. Let $Y$ be the set of all assigments of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ with the domain $\{x,y\}$. Now $$\begin{aligned}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{\{\emptyset\}}\phi &\textrm{ iff }& {{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_Y\phi \textrm{ iff } {{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_Y \forall x\forall y \phi \\ &\textrm{ iff }& {{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_Y\forall x\forall y \phi^* \text{ iff } {{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{\{\emptyset\}}\forall x\forall y \phi^* ,\end{aligned}$$ where the first and the last equivalence hold by , the second by the semantics of the universal quantifier and the third by Lemma \[dtoif\]. An analogous argument can be used to show that for every sentence $\phi\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ there is an equivalent sentence of the logic ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3$.
Examples of properties definable in D\^2
----------------------------------------
We end this section with examples of definable classes of structures in [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}(and in [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}by Theorem \[dtwo le if2\]).
The following properties can be expressed in [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}:
a) \[example1\] For unary relation symbols $P$ and $Q$, [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}can express $|P|=|Q|$. This shows ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\not \le {{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}$.
b) \[example2\] If the vocabulary of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ contains a constant $c$, then [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}can express that $A$ is infinite.
c) \[example3\] $|A|\le k$ can be expressed already in ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^1$.
Let us first consider part \[example1\]). Clearly, it suffices to express $|P|\le |Q|$. Define $\phi$ by $$\phi:= \forall x\exists y( {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(y,x)\wedge (\neg P(x)\vee Q(y))).$$ Now, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi$iffthere is an injective function $F\colon A\rightarrow A$ such that $F[P^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}]\subseteq Q^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}$ iff$|P^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}|\le |Q^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}|$.
For part \[example2\]), we use the same idea as above. Define $\psi$ by $$\psi:= \forall x\exists y({{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(y,x)\wedge \neg c=y).$$ Now, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models\psi$iffthere is an injective function $F\colon A\rightarrow A$ such that $c^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\notin F[A]$iff$A$ is infinite.
Finally, we show how to express the property from part \[example3\]). Define $\theta$ as $$\forall x( \bigvee_{1\le i\le k}\chi_i),$$ where $\chi_i$ is ${{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x)$. It is now immediate that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \theta$ iff $|A|\le k$.
It is interesting to note that, although part \[example1\]) holds, the difference in [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}$]{}]{}]{}]{}-complexity of ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}({{{\protect\ensuremath{\mathrm{I}}}\xspace}})$ and ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ is a major one. The former is [[[$\mathit{\Sigma}^1_1$]{}]{}]{}-hard [@grotro97] whereas the latter is decidable – as is shown in section \[sec:satd2\]. Part \[example1\]) also implies that [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}does not have a zero-one law, since the property $|P|\le |Q|$ (which can be expressed in [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}) has the limit probability $\frac{1}{2}$.
\[d less than if\] ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}<{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$. This holds already in the finite.
The property of being grid-like (see Definition \[def:gridlike\]) can be expressed in [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}but not in [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}since [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}is decidable by Theorem \[dtwo nexptime\]. In the finite, there exists no [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}sentence equivalent to the [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}sentence $\phi_{\mathrm{torus}}$ (see Section \[iffinsatsection\]), since the finite satisfiability problem of [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}is decidable.
Satisfiability for IF\^2 is undecidable {#ifsatsection}
=======================================
In this section we will use tiling problems, introduced by Hao Wang in [@Wang:1961], to show the undecidability of [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} as well as [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{}. In this paper a *Wang tile* is a square in which each edge is assigned a color. It is a square that has four colors (up, right, down, left). We say that a set of tiles can tile the ${\protect\ensuremath{{\mathbb{N}}}}\times{\protect\ensuremath{{\mathbb{N}}}}$ plane if a tile can be placed on every point $(i,j)\in {\protect\ensuremath{{\mathbb{N}}}}\times{\protect\ensuremath{{\mathbb{N}}}}$ s.t. the right color of the tile in $(i,j)$ is the same as the left color of the tile in $(i+1,j)$ and the up color of the tile in $(i,j)$ is the same as the down color in the tile in $(i,j+1)$. Notice that turning and flipping tiles is not allowed.
We then define some specific structures needed later.
\[def:grid\] The model ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}:=(G,V,H)$ where
- $G={\protect\ensuremath{{\mathbb{N}}}}\times{\protect\ensuremath{{\mathbb{N}}}}$,
- $V=\{((i,j),(i,j+1))\subseteq G\times G\mid i,j\in{\protect\ensuremath{{\mathbb{N}}}}\}$ and
- $H=\{((i,j),(i+1,j))\subseteq G\times G\mid i,j\in{\protect\ensuremath{{\mathbb{N}}}}\}$
is called the *grid*.
A finite model ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}=(D,V,H,V',H')$ where
- $D=\{0,\dots, n\}\times\{0,\dots,m\}$,
- $V=\{((i,j),(i,j+1))\subseteq D\times D\mid i\leq n,j<m\}\}$,
- $H=\{((i,j),(i+1,j))\subseteq D\times D\mid i<n,j\leq m\}$,
- $V'=\{((i,m),(i,0))\subseteq D\times D\mid i\leq n\}$ and
- $H'=\{((n,j),(0,j))\subseteq D\times D\mid j\leq m\}$
is called a *torus*.
\[def:tiling\] A set of *colors* $C$ is defined to be an arbitrary finite subset of the natural numbers. The set of all *(Wang) tiles* over $C$ is $C^4$, i.e., a tile is an ordered list of four colors, interpreted as the colors of the four edges of the tile in the order top, right, bottom and left.
Let $C$ be a set of colors, $T\subseteq C^4$ a finite set of tiles and ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ a first-order structure with binary relations $V$ and $H$ interpreted as vertical and horizontal successor relations. Then a $T$-*tiling* of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ is a total function $t\colon A \to T$ such that for all $x,y\in A$ it holds that
i) $(t(x))_0=(t(y))_2$ if $(x,y)\in V$, i.e., the top color of $x$ matches the bottom color of $y$, and
ii) $(t(x))_1=(t(y))_3$ if $(x,y)\in H$, i.e., the right color of $x$ matches the left color of $y$.
Next we define the tiling problem for a structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$.
A structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ is called $T$-*tilable* iff there is a $T$-tiling of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$.
For any structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ we define the problem $${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\nobreak)}}}}\xspace}}} := \{T\mid \text{${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ is $T$-tilable}\}.$$
We say that a structure ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}=(B,V,H,V',H')$ is $T$-tilable if and only if the structure $(B,V\cup V',H\cup H')$ is $T$-tilable. Hence a torus ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}=(D,V,H,V',H')$ is $T$-tilable if and only if the structure $(D,V\cup V',H\cup H')$ is $T$-tilable. Now we define the problem $${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}\nobreak)}}}}\xspace}}} := \{T\mid \text{there is a torus ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}$ that is $T$-tilable}\}.$$
Note that the set [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathfrak{G}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathfrak{G}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} consists of all $T$ such that there is a $T$-tiling of the infinite grid and [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} consists of all $T$ such that there is a *periodic* $T$-tiling of the grid. Further note that [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} cannot be expressed in the form [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathfrak{D}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathfrak{D}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} for a fixed torus [[[$\mathfrak{D}$]{}]{}]{}since a fixed torus has a fixed size and we want the problem to be the question whether there is a torus of *any* size.
We will later use the following two theorems to show the undecidability of [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} and, resp., [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{}.
\[tiling pizeroone\] [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathfrak{G}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathfrak{G}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} is ${{{\protect\ensuremath{\mathit{\Pi}^0_1}}\xspace}}$-complete.
\[periodic tiling pizeroone\] [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} is ${{{\protect\ensuremath{\mathit{\Sigma}^0_1}}\xspace}}$-complete.
To prove the undecidability of [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} (Theorem \[iftwo sat complexity\]) we will, for every set of tiles $T$, define a formula $\phi_T$ such that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi_T$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ has a $T$-tiling. Then we will define another formula $\phi_\mathrm{grid}$ and show that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi_\mathrm{grid}$ iff ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ contains (an isomorphic copy of) the grid as a substructure. Therefore $\phi_T\wedge \phi_\mathrm{grid}$ is satisfiable if and only if there is a $T$-tiling of the grid. For the undecidability of [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} (Theorem \[iftwo finsat complexity\]) we will define a formula $\phi_\mathrm{torus}$ which is a modification of the formula $\phi_\mathrm{grid}$.
\[def:tiling formula\] Let $T=\{t^0,\dots,t^k\}$ be a set of tiles, and for all $i\leq k$, let ${{{{\protect\ensuremath{\mathord{{\mathrm{right}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t^i)$ (resp. ${{{{\protect\ensuremath{\mathord{{\mathrm{top}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t^i)$) be the set $$\{t^j\in\{0,\dots,k\}\mid t^i_1 = t^j_3\text{ (resp.~}t^i_0 = t^j_2)\},$$ i.e., the set of tiles matching $t^i$ to the right (resp. top).
Then we define the first-order formulas $$\begin{array}{r@{}l}
\psi_T:=\forall x \forall y \bigg(\Big(&H(x,y)\rightarrow \bigwedge\limits_{i\leq k}\big(P_i(x)\rightarrow \bigvee\limits_{t^j\in {{{{\protect\ensuremath{\mathord{{\mathrm{right}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t^i)} P_j(y)\big)\Big)\ \wedge\\
\Big(&V(x,y)\rightarrow \bigwedge\limits_{i\leq k}\big(P_i(x)\rightarrow \bigvee\limits_{t^j\in {{{{\protect\ensuremath{\mathord{{\mathrm{top}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(t^i)} P_j(y)\big)\Big)\bigg),\\[3ex]
\multicolumn{2}{l}{\theta_T:=\forall x \bigvee\limits_{i\leq k}\big(P_i(x)\wedge \bigwedge\limits_{\substack{j\leq k\\j\neq i}} \neg P_j(x)\big)\text{ and}}\\
\multicolumn{2}{l}{\phi_T:=\psi_T \wedge \theta_T,}
\end{array}$$ over the vocabulary $V,H,P_0,\dots,P_k$. In an [[[[${\mathsf{IF}}$]{}]{}]{}]{}or [[[[${\mathsf{D}}$]{}]{}]{}]{}context, $\phi \rightarrow \psi$ is considered to be an abbreviation of $\phi^\neg \vee \psi$, where $\phi^\neg$ is the negation normal form of $\neg \phi$.
\[tiling iff tiling formula\] Let $T=\{t_0,\dots,t_k\}$ be a set of tiles and ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ a structure. Then ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ is $T$-tilable iff there is an expansion ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*=(A,V,H,P_0,\dots,P_k)$ of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ such that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*\models\phi_T$.
\[torustiling iff tiling formula\] Let $T=\{t_0,\dots,t_k\}$ be a set of tiles and ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}=(B,V,H,V',H')$ a structure. There is an [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}sentence $\gamma_T$ of the vocabulary $\{V,H,V',H',P_0,\dots,P_k\}$ such that ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ is $T$-tilable iff there is an expansion ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}^*=(A,V,H,V',H',P_0,\dots,P_k)$ of ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ such that ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}^*\models\gamma_T$.
Notice that $\phi_T$ is an ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}$-sentence. Therefore $T$-tiling is expressible even in ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}$. The difficulty lies in expressing that a structure is (or at least contains) a grid. This is the part of the construction where [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}or even [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}formulas are no longer sufficient and the full expressivity of [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}is needed.
\[def:gridlike\] A structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ is called *grid-like* iff it satisfies the conjunction $\phi_\mathrm{grid}$ of the formulas $$\begin{array}{lcl} \phi_{\mathrm{functional}}(R) & := & \forall x\forall y \big(R(y,x)\,\to\, \exists y/\{x\}\,x=y \big)\\
& & \quad\text{for $R\in\{V,H\}$},\\
\phi_{\mathrm{injective}}(R) & := & \forall x\forall y \big(R(x,y)\, \to\, \exists y/\{x\} \,x=y \big)\\
& & \quad\text{for $R\in\{V,H\}$},\\
\phi_{\mathrm{root}} & := & \exists x\forall y \big(\neg V(y,x)\wedge \neg H(y,x)\big),\\
\phi_{\mathrm{distinct}} & := & \forall x\forall y\, \neg \big(V(x,y)\wedge H(x,y)\big),\\
\phi_{\mathrm{edge}}(R,R') & := & \forall x \Big(\big(\forall y \,\neg R(y,x)\big)\,\to \forall y \big(R'(x,y)\to \forall x\,\neg R(x,y)\big)\Big)\\
& & \quad\text{for $(R,R')\in\{(V,H),(H,V)\}$,}\\
\phi_{\mathrm{join}} & := & \forall x \forall y \Big(\big(V(x,y)\vee H(x,y)\big)\,\to \exists x/\{y\}\, \big(V(y,x)\vee H(y,x)\big)\Big),\\
\phi_{\mathrm{infinite}}(R) & := & \forall x\exists y R(x,y) \text{ for $R\in\{V,H\}$}.
\end{array}$$
The grid-likeness of a structure can alternatively be described in the following more intuitive way.
\[gridlike description\] A structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ is grid-like iff
i) \[degree\] $V$ and $H$ are (graphs of) injective total functions, i.e., the out-degree of every element is exactly one and the in-degree at most one ($\phi_{\mathrm{infinite}}$, $\phi_{\mathrm{functional}}$ and $\phi_{\mathrm{injective}}$),
ii) there is an element, called the *root*, that does not have any predecessors ($\phi_{\mathrm{root}}$),
iii) for every element, its $V$ successor is distinct from its $H$ successor ($\phi_{\mathrm{distinct}}$),
iv) for every element $x$ such that $x$ does not have a $V$ (resp. $H$) predecessor, the $H$ (resp. $V$) successor of $x$ also does not have a $V$ (resp. $H$) predecessor ($\phi_{\mathrm{edge}}$),
v) \[join description\] for every element $x$ there is an element $y$ such that $(x,y) \in (V\circ H)\cap (H\circ V)$ or $(x,y) \in (V\circ V)\cap (H\circ H)$,
We show that a structure ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}\models \phi_{\mathrm{grid}}$ satisfies the above five properties. The only difficult case is property \[join description\]). First note that $\phi_\mathrm{join}$ is equivalent to the first-order formula $$\forall x \exists x' \forall y \Big(\big(V(x,y)\vee H(x,y)\big)\,\to \, \big(V(y,x')\vee H(y,x')\big)\Big).$$ Since $\phi_{\mathrm{functional}}$, $\phi_{\mathrm{distinct}}$ and $\phi_{\mathrm{infinite}}$ hold as well, ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ satisfies $$\forall x \exists x' \exists y_1 \exists y_2 \Big(y_1\neq y_2 \wedge V(x,y_1)\wedge H(x,y_2) \wedge \big(V(y_1,x')\vee H(y_1,x')\big) \wedge \big(V(y_2,x')\vee H(y_2,x')\big)\Big).$$ Due to $\phi_{\mathrm{injective}}$, neither $V(y_1,x') \wedge V(y_2,x')$ nor $H(y_1,x') \wedge H(y_2,x')$ can be true if $y_1\neq y_2$. Hence, ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ satisfies $$\begin{array}{l}
\forall x \exists x' \exists y_1 \exists y_2 \Big(y_1\neq y_2 \wedge \Big(\big(V(x,y_1)\wedge H(x,y_2) \wedge V(y_1,x') \wedge H(y_2,x')\big) \vee\\
\quad \big(V(x,y_1)\wedge H(x,y_2) \wedge H(y_1,x') \wedge V(y_2,x')\big)\Big)\Big).
\end{array}$$ From this formula the property \[join description\]) is immediate (with $x:=x$ and $y:=x'$).
Now we will use Remark \[gridlike description\] to show that a grid-like structure, although it need not be the grid itself, must at least contain an isomorphic copy of the grid as a substructure.
\[gridlike includes grid\] Let ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ be a grid-like structure. Then ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ contains an isomorphic copy of ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}$ as a substructure.
If ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ is a model with two binary relations $R$ and $R'$, $b\in B$ and $i\in {\protect\ensuremath{{\mathbb{N}}}}$ then the *$i$-$b$-generated substructure* of ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ (denoted by ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}^i(b)$) is defined inductively in the following way: $$\begin{array}{lcl}
{{{\protect\ensuremath{\mathfrak{B}}}\xspace}}^0(b) &=& {{{\protect\ensuremath{\mathfrak{B}}}\xspace}}\upharpoonright\{b\},\\
{{{\protect\ensuremath{\mathfrak{B}}}\xspace}}^{i+1}(b) &=& {{{\protect\ensuremath{\mathfrak{B}}}\xspace}}\upharpoonright\big(B^i(b)\,\cup \{x\in B\mid \exists y\in B^i(b):(y,x)\in R\cup R'\}\big).
\end{array}$$
Let $r\in A$ be a root of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ (which exists because ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi_{\mathrm{root}}$). We call a point $a\in{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ a *west border point* (resp. *south border point*) if $(r,a)\in V^n$ (resp. $(r,a)\in H^n$) for some $n\in{\protect\ensuremath{{\mathbb{N}}}}$. Due to Remark \[gridlike description\], every point in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ has $V$- and $H$-in-degree at most one while the west border points have $H$-in-degree zero and the south border points have $V$-in-degree zero. We call a substructure ${{{\protect\ensuremath{\mathfrak{H}}}\xspace}}$ of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ *in-degree complete* if every point in ${{{\protect\ensuremath{\mathfrak{H}}}\xspace}}$ has the same in-degrees in ${{{\protect\ensuremath{\mathfrak{H}}}\xspace}}$ as it has in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$.
We will prove by induction that there exists a family of isomorphisms $\{f_i\mid i\in{\protect\ensuremath{{\mathbb{N}}}}\}$ such that
1. $f_i$ is an isomorphism from ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^i((0,0))$ to ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^i(r)$,
2. ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^i(r)$ is in-degree complete and
3. $f_{i-1}\subseteq f_{i}$
for all $i\in{\protect\ensuremath{{\mathbb{N}}}}$.
The basis of the induction is trivial. Clearly the function $f_0$ defined by $f_0((0,0)):=r$ is an isomorphism from ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^0((0,0))$ to ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^0(r)$. And since $r$ is a root it has no $V$- or $H$-predecessors. Hence, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^0(r)$ is in-degree complete.
![The inductively defined substructures[]{data-label="fig:submodel"}](pic)
Let us then assume that $f_k$ is an isomorphism from ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^k((0,0))$ to ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^k(r)$, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^k(r)$ is in-degree complete and $f_{k-1}\subseteq f_k$. Then the $k$-$r$-generated substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^{k+1}(r)$ (which is ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^k(r)$) is isomorphic to ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^k((0,0))$ and the isomorphism is given by $f_k$.
We will now show how to extend $f_k$ to the isomorphism $f_{k+1}$. This is done by extending $f_k$ element by element along the diagonal (Figure \[fig:submodel\] shows the first extension step). We will abuse notation and denote the extensions of the function $f_k$ by $h$ throughout the proof. We will show by induction on $j$ that we can extend the isomorphism by assigning values for $h(j,(k+1)-j)$ for all $0\leq j\leq k+1$ – still maintaining the isomorphism between ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}\upharpoonright{{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(h)$ and ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright {{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}$, and the in-degree completeness of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright {{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}$.
Due to $\phi_{\mathrm{infinite}}$ and $\phi_{\mathrm{functional}}$ the west border point $f_{k}((0,k))$ has a unique $V$-successor $a$. Since the $k$-$r$-generated substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^{k+1}(r)$ is isomorphic to ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^k((0,0))$ and $(0,k)$ has no $V$ successor in $G^k((0,0))$ we know that $f_k(y)\neq a$ for every $y\in G^k((0,0))$. Note that due to $\phi_{\mathrm{edge}}$ and since $f_k((0,k))$ is a west border point and has no $H$-predecessors in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$, $a$ is also a west border point and has no $H$-predecessor in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$. Thus ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright ({{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}\cup\{a\})$ is in-degree complete. Since ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright {{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}$ is in-degree complete, $a$ has no $V\cup H$-successors in ${{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}$. Due to $\phi_\mathrm{edge}$ and $\phi_\mathrm{injective}$, $a$ has no reflexive loops. We extend $h$ by $h((0,k+1)):=a$. Clearly the extended function $h$ is an isomorphism and ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright {{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}$ in-degree complete. Now let $m\in\{0,\dots,k-1\}$ and assume that $h((j,(k+1)-j))$ is defined for all $j \leq m$, $h$ is an isomorphism extending $f_k$ and $h(G)$ is in-degree complete. We will prove that we can extend $h$ by assigning a value for $h(m+1,(k+1)-(m+1))$, still maintaining the required properties. By the induction hypothesis we have defined a value for $h((m,(k+1)-m))$. Now $h((m,(k+1)-m))$ is the $V^2$-successor of $h((m,(k-1)-m))$. Since $h((m,(k-1)-m))$ has no $H^2$ successor in the structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright {{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}$, the $H^2$- and $V^2$-successors of $h((m,(k-1)-m))$ in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ cannot be the same point. Now by Remark \[gridlike description\]\[join description\], this implies that there is a point $c\in A\setminus {{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}$ such that $c$ is the $H\circ V$- and $V\circ H$-successor of $h((m,(k-1)-m))$ in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$. We extend $h$ by $h((m+1,k-m)):=c$ and observe that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright ({{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}\cup \{c\})$ is still in-degree complete. By $\phi_\mathrm{injective}$ and in-degree completeness of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright({{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h}{}}{}{(\nobreakh\nobreak)}}}}\xspace}}}\setminus\{c\})$, the extended function $h$ is an isomorphism.
Finally we extend the south border. This is possible by reasoning similar to the case where we extended the west border.
Let $f_{k+1}$ be the isomorphism from ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^{k+1}((0,0))$ to ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^{k+1}(r)$ that exists by the inductive proof. Clearly ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^{k+1}(r)$ is in-degree complete and $f_k\subseteq f_{k+1}$. Now since the isomorphisms $f_i$ for $i\in{\protect\ensuremath{{\mathbb{N}}}}$ constitute an ascending chain, $\bigcup_{i\in{\protect\ensuremath{{\mathbb{N}}}}}f_i$ is an isomorphism from ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}$ to a substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$. Therefore ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ has an isomorphic copy of the grid as a substructure.
The last tool needed to prove the main theorem is the following trivial lemma.
\[tiling supergrids\] Let $T$ be a set of tiles and ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}=(B,V,H)$ a structure. Then ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ is $T$-tilable iff there is a structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ which is $T$-tilable and contains a substructure that is isomorphic to ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$.
The following is the main theorem of this section.
\[iftwo sat complexity\] [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} is [[[$\mathit{\Pi}^0_1$]{}]{}]{}-complete.
For the upper bound note that ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}}{}}{}{(\nobreak{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}\nobreak)}}}}\xspace}}}\in{{{\protect\ensuremath{\mathit{\Pi}^0_1}}\xspace}}$ by Gödel’s completeness theorem. By Remark \[eso sat\] it follows that ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}}{}}{}{(\nobreak{{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}\nobreak)}}}}\xspace}}}\in{{{\protect\ensuremath{\mathit{\Pi}^0_1}}\xspace}}$ and by the computable translation from ${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}$ into ${{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}$ from [@va07 Theorem 6.2], it follows that ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3}{}}{}{(\nobreak{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^3\nobreak)}}}}\xspace}}} \in {{{\protect\ensuremath{\mathit{\Pi}^0_1}}\xspace}}$. Finally, the computability of the reductions in Lemma \[iftod\] and Theorem \[dtwo le if2\] implies ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}}}\xspace}}}\in {{{\protect\ensuremath{\mathit{\Pi}^0_1}}\xspace}}$.
The lower bound follows by the reduction $g$ from [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathfrak{G}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathfrak{G}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} to our problem defined by $g(T):=\phi_{\mathrm{grid}} \wedge \phi_T$. To see that $g$ indeed is such a reduction, first let $T$ be a set of tiles such that ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}$ is $T$-tilable. Then, by Lemma \[tiling iff tiling formula\], it follows that there is an expansion ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^*$ of ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}$ such that ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^*\models \phi_T$. Clearly, ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^*\models \phi_{\mathrm{grid}}$ and therefore ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}^* \models \phi_{\mathrm{grid}}\wedge \phi_T$. If, on the other hand, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*$ is a structure such that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*\models \phi_{\mathrm{grid}}\wedge \phi_T$, then by Theorem \[gridlike includes grid\], the $\{V,H\}$-reduct ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*$ contains an isomorphic copy of ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}$ as a substructure. Furthermore, by Lemma \[tiling iff tiling formula\], ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ is $T$-tilable. Hence, by Lemma \[tiling supergrids\], ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}$ is $T$-tilable.
Finite satisfiability for IF\^2 is undecidable {#iffinsatsection}
----------------------------------------------
We will now discuss the problem [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} whose undecidability proof is similar to the above, the main difference being that it uses tilings of tori instead of tilings of the grid.
\[toruslike description\]
A finite structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H,V',H')$ is *torus-like* iff it satisfies the following two conditions
i) there exist unique and distinct points $SW$, $NW$, $NE$, $SE$ such that
1. $SW$ has no $V$- and no $H$-predecessor,
2. $NW$ has no $H$-predecessor and no $V$-successor,
3. $NE$ has no $V$- and no $H$-successor and
4. $SE$ has no $H$-successor and no $V$-predecessor,
ii) there exist $m,n\in{\protect\ensuremath{{\mathbb{N}}}}$ such that
1. $(A,V,H)$ is a model that has an isomorphic copy of the $m\times n$ grid as a component with $SW$, $NW$, $NE$ and $SE$ as corner points,
2. $(A,V',H)$ is a model that has an isomorphic copy of the $m\times 2$ grid as a component with $NW$, $SW$, $SE$ and $NE$ as corner points and $(NW,SW),(NE,SE)\in V'$,
3. $(A,V,H')$ is a model that has an isomorphic copy of the $2\times n$ grid as a component with $SE$, $NE$, $NW$ and $SW$ as corner points and $(SE,SW),(NE,NW)\in H'$.
By a *component* of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ we mean a maximal weakly connected substructure ${{{\protect\ensuremath{\mathfrak{M}}}\xspace}}$, i.e., any two points in $M$ are connected by a path along $R:=V\cup H\cup V^{-1}\cup H^{-1}$, and furthermore, for all $M'$ such that $M\subset M' \subseteq A$, there exist two points in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright M'$ that are not connected by $R$.
In order to define torus-likeness of a structure with an [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}formula we first need to express that a finite structure has a finite grid as a component. This is done in essentially the same way as expressing that a structure has a copy of the infinite grid as a substructure.
\[def:fingridlike\] A finite structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ is called *fingrid-like* iff it satisfies the conjunction $\phi_\mathrm{fingrid}$ of the formulas $$\begin{array}{l@{\ }c@{\ }l}
\phi_{\mathrm{SWroot}} & := &\exists x\forall y \big(\neg V(y,x)\wedge \neg H(y,x) \wedge \exists y V(x,y)\wedge\exists y H(x,y)\big),\\
\phi_{\mathrm{functional}}(R) & := & \forall x\forall y \big(R(y,x)\,\to\, \exists y/\{x\}\,x=y \big)\\
& & \quad\text{for $R\in\{V,H\}$},\\
\phi_{\mathrm{injective}}(R) & := & \forall x\forall y \big(R(x,y)\, \to\, \exists y/\{x\} \,x=y \big)\\
& & \quad\text{for $R\in\{V,H\}$},\\
\phi_{\mathrm{distinct}} & := & \forall x\forall y\, \neg \big(V(x,y)\wedge H(x,y)\big),\\
\phi_{\mathrm{SWedge}} & := & \forall x \Big(\big(\forall y \,\neg R(y,x)\big)\to\forall y \big((R'(x,y)\vee R'(y,x)) \to \forall x\,\neg R(x,y)\big)\Big)\\
& & \quad\text{for $(R,R')\in\{(V,H),(H,V)\}$,}\\
\phi_{\mathrm{NEedge}} & := & \forall x \Big(\big(\forall y \,\neg R(x,y)\big)\to\forall y \big((R'(x,y)\vee R'(y,x))\to \forall x\,\neg R(y,x)\big)\Big)\\
& & \quad\text{for $(R,R')\in\{(V,H),(H,V)\}$,}\\
\phi_{\mathrm{finjoin}} & := & \forall x \Big(\forall y \neg V(x,y)\vee \forall y \neg H(x,y)\vee \forall y \Big(\big(V(x,y)\vee H(x,y)\big)\\
& & \quad\to \exists x/\{y\}\, \big(V(y,x)\vee H(y,x)\big)\Big)\Big),
\end{array}$$
Fingrid-likeness can also be described in the following intuitive way.
\[fingridlike description\] A structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ is fingrid-like iff
i) $V$ and $H$ are (graphs of) injective partial functions, i.e., the in- and out-degree of every element is at most one ($\phi_{\mathrm{functional}}$ and $\phi_{\mathrm{injective}}$),
ii) there exists a point, denoted by $SW$, that has a $V$-successor and an $H$-successor but does not have $V\cup H$-predecessors, ($\phi_{\mathrm{SWroot}}$),
iii) for every element, its $V$-successor is distinct from its $H$-successor ($\phi_{\mathrm{distinct}}$),
iv) for every element $x$ such that $x$ does not have a $V$ (resp. $H$) predecessor, the $H$ (resp. $V$) successor and predecessor of $x$ also do not have a $V$ (resp. $H$) predecessor ($\phi_{\mathrm{SWedge}}$),
v) for every element $x$ such that $x$ does not have a $V$ (resp. $H$) successor, the $H$ (resp. $V$) successor and predecessor of $x$ also do not have a $V$ (resp. $H$) successor ($\phi_{\mathrm{NEedge}}$),
vi) for every element $x$ that has a $V$-successor and an $H$-successor there is an element $y$ such that $(x,y) \in (V\circ H)\cap (H\circ V)$ or $(x,y) \in (V\circ V)\cap (H\circ H)$.
Notice that for a grid ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}$ to be grid-like, it is required that the grid is not of the type $1\times n$ or $n\times 1$ for any $n\in{\protect\ensuremath{{\mathbb{N}}}}$. A grid that is grid-like is called a *proper* grid. Now we can show that a fingrid-like structure contains a proper finite grid as a component.
\[fingridlike includes grid\] Let ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H)$ be a finite fingrid-like structure. Then ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ contains an isomorphic copy of a proper finite grid as a component.
Due to $\phi_\mathrm{SWroot}$ there exists a point denoted by $SW\in A$ that has a $V$-successor and an $H$-successor, but has no $V\cup H$-predecessors. Now since $V$ is an injective partial function and $A$ is finite, there exists $n\in{\protect\ensuremath{{\mathbb{N}}}}$ such that for all $x\in A$ $(SW,x)\notin V^{n+1}$. For similar reasons there exists $m\in{\protect\ensuremath{{\mathbb{N}}}}$ such that for all $x\in A$, $(SW,x)\notin H^{m+1}$. Let $m$ and $n$ be the smallest such numbers. We will show that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ has an isomorphic copy of the $m\times n$ grid as a component.
We will first show by induction on $k\le n$ that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ has an isomorphic copy of the $m\times k$ grid as an in-degree complete substructure with $SW$ as a corner point. By the selection of $m$ the point $SW$ has a $H^i$ successor $v_i$ for each $i\leq m$. Since $H$ is an injective partial function and $SW$ has no $H$-predecessors, the points $v_i$ are all distinct and unique. Due to $\phi_\mathrm{SWedge}$ none of the points $v_i$ has a $V$-predecessor and therefore the $V$-successors of the points $v_i$ are not in the set $\{v_i\mid i\leq m\}$. Therefore ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright\{v_i\mid i\leq m\}$ is an isomorphic copy of the $m\times 1$ grid. Due to $\phi_\mathrm{SWedge}$, $\phi_\mathrm{SWroot}$ and $\phi_\mathrm{injective}$ the structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright\{v_i\mid i\leq m\}$ is in-degree complete.
Let us then assume that ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$, an in-degree complete substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$, is an isomorphic copy of the $m\times k$ grid ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}_{(m,k)}$ with $SW$ as a corner point and $k<n$. Let $h$ be the corresponding isomorphism from ${{{\protect\ensuremath{\mathfrak{G}}}\xspace}}_{(m,k)}$ to ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$. We will now extend $h$ to $h'$ such that $h'$ is an isomorphism from the $m\times (k+1)$ grid to an in-degree complete substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$. Since $k+1\leq n$ there exists a point $a_0\in A$ such that $a_0$ is the $V$-successor of $h((0,k))$. Due to $\phi_\mathrm{NEedge}$ and since $h((0,k))$ has a $V$-successor, each of the points $h((i,k))$, $i\leq m$, has a $V$-successor $a_i$. Since $V$ is a partial injective function and the points $h((i,k))$ are all distinct, the points $a_i$ are also all distinct. The structure ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ is in-degree complete, and hence neither any of the points $a_i$ nor any of their $V\cup H$-successors is in $B$.
We will next show that $(a_i,a_{i+1})\in H^{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ for all $i<m$. For $i\leq m-2$, the point $h((i,k))$ has an $H^2$-successor but has no $V^2$-successor in the structure ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$. Therefore for all $i\leq m-2$, if the $V^2$-successor of $h((i,k))$ exists in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$, it cannot be the same as the $H^2$-successor of $h((i,k))$. Notice that each of the points $h((i,k))$, $i\leq m-2$, has a $V$- and $H$-successor in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$. Therefore due to $\phi_\mathrm{finjoin}$ the $V\circ H$-successor and the $H\circ V$-successor of the point $h((i,k))$, $i\leq m-2$, are the same. Therefore the $H$-successor of $a_i$ is $a_{i+1}$ for all $i\leq m-2$.
It needs still to be shown that $(a_{m-1},a_m)\in H^{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$. The point $h((m-1,k))$ has no $H^2$ successor in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ since $h((m,k))$ is an east border point (due to $\phi_\mathrm{NEedge}$ and the selection of $m$). Therefore there cannot be a point $a$ in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ such that it is both an $H^2$-successor and a $V^2$-successor of $h((m-1,k))$. Now due to $\phi_\mathrm{finjoin}$ and the fact that $h((m-1,k))$ has a $V$- and an $H$-successor in ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$, the $H\circ V$-successor and $V\circ H$-successor of $h((m-1,k))$ have to be the same point. Therefore $(a_{m-1},a_m)\in H^{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$.
We define $h':=h\cup \{((i,k+1),a_i)\mid i\leq m\}$. Each point $a_i$ with the exception of the west border point $a_0$ has an $H$-predecessor $a_{i-1}$. Hence, due to the in-degree completeness of ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$, injectivity of $V$ and $H$, and since each of the points $a_i$ has a $V$-predecessor in the set $B$, we conclude that the structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\upharpoonright{{{{\protect\ensuremath{\mathord{{\mathrm{range}}\ifthenelse{\equal{h'}{}}{}{(\nobreakh'\nobreak)}}}}\xspace}}}$ is an in-degree complete substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$. We also notice that due to injectivity, the points $a_i$ have no reflexive loops. Due to in-degree completeness of $B$, none of the $V\cup H$-successors of the points $a_i$, $i\leq m$, are in the set $B$. Hence it is sraightforward to observe that $h'$ is the desired isomorphism from the $m\times (k+1)$ grid to an in-degree complete substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$.
We have now proven that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ has an isomorphic copy of the $m\times n$ grid as a substructure with $SW$ as a corner point. Let $h$ be the isomorphism from the $m\times n$ grid to a substructure of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ with $SW$ as a corner point. By the selection of $m$ and $n$, the point $h((0,n))$ has no $V$-successors and $h((m,0))$ has no $H$-successors. Therefore, due to $\phi_\mathrm{NEedge}$, none of the points $h((i,n))$, $i\leq m$, have a $V$-successor and none of the points $h((m,j))$, $j\leq n$, have a $H$-successor. This together with functionality and injectivity of $H$ and $V$, and the fact that west border points have no $H$-predecessors and south border points have no $V$-predecessors, imply that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ has an isomorphic copy of the $m\times n$ grid as a component. Since the point $SW$ has a $V$-successor and an $H$-successor, the $m\times n$ grid is a proper grid.
We now define some auxiliary [[[${{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2$]{}]{}]{}-formulas.
$$\begin{array}{lcl} \phi_{\mathrm{NStape}} & := & \exists x(\phi_{\mathrm{SW}}^{(V,H)}(x)\wedge\phi_{\mathrm{NW}}^{(V',H)}(x))\wedge\exists x(\phi_{\mathrm{SE}}^{(V,H)}(x)\wedge\phi_{\mathrm{NE}}^{(V',H)}(x))\\
& & \wedge\exists x(\phi_{\mathrm{NW}}^{(V,H)}(x)\wedge\phi_{\mathrm{SW}}^{(V',H)}(x))\wedge\exists x(\phi_{\mathrm{NE}}^{(V,H)}(x)\wedge\phi_{\mathrm{SE}}^{(V',H)}(x))\\
& & \wedge\exists x\exists y (\phi_{\mathrm{NW}}^{(V,H)}(x)\wedge\phi_{\mathrm{SW}}^{(V,H)}(y)\wedge V'(x,y)),\\\\
\phi_{\mathrm{EWtape}} & := & \exists x(\phi_{\mathrm{SW}}^{(V,H)}(x)\wedge\phi_{\mathrm{SE}}^{(V,H')}(x))\wedge\exists x(\phi_{\mathrm{SE}}^{(V,H)}(x)\wedge\phi_{\mathrm{SW}}^{(V,H')}(x))\\
& & \wedge\exists x(\phi_{\mathrm{NW}}^{(V,H)}(x)\wedge\phi_{\mathrm{NE}}^{(V,H')}(x))\wedge\exists x(\phi_{\mathrm{NE}}^{(V,H)}(x)\wedge\phi_{\mathrm{NW}}^{(V,H')}(x))\\
& & \wedge\exists x\exists y (\phi_{\mathrm{SE}}^{(V,H)}(x)\wedge\phi_{\mathrm{SW}}^{(V,H)}(y)\wedge H'(x,y)),\\\\
\phi_\mathrm{uniquecorners} & := & \bigwedge\limits_{P\in C}\forall x\forall y ((P(x)\wedge P(y))\to x=y),
\end{array}$$
where $C=\{\phi_{\mathrm{T}}^{(R,S)}(x)\mid T\in\{\mathrm{SW},\mathrm{NW},\mathrm{NE},\mathrm{SE}\}, (R,S)\in \{ (V,H), (V',H), (V,H')\} \}$
and $$\begin{array}{lcl}
\phi_{\mathrm{SW}}^{(R,S)}(x) & := & \forall y \big(\neg R(y,x)\wedge \neg S(y,x)\big)\wedge \exists y R(x,y)\wedge\exists y S(x,y),\\
\phi_{\mathrm{NW}}^{(R,S)}(x) & := & \forall y \big(\neg R(x,y)\wedge \neg S(y,x)\big)\wedge \exists y R(y,x)\wedge\exists y S(x,y),\\
\phi_{\mathrm{NE}}^{(R,S)}(x) & := & \forall y \big(\neg R(x,y)\wedge \neg S(x,y)\big)\wedge \exists y R(y,x)\wedge\exists y S(y,x),\\
\phi_{\mathrm{SE}}^{(R,S)}(x) & := & \forall y \big(\neg R(y,x)\wedge \neg S(x,y)\big)\wedge \exists y R(x,y)\wedge\exists y S(y,x),
\end{array}$$ $(R,S)\in \{ (V,H), (V',H), (V,H')\}$.
Let ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H,V',H')$ be a finite structure such that the underlying structures $(A,V,H)$, $(A,V',H)$ and $(A,V,H')$ are fingrid-like. In this context the intuitive meaning of the above three formulas is the following.
- The formula $\phi_\mathrm{uniquecorners}$ expresses that the structures $(A,V,H)$, $(A,V',H)$ and $(A,V,H')$ each have four unique corner points, exactly one of each type, i.e., southwest corner, northwest corner, northeast corner and southeast corner. In each structure the corner points definine a boundary of a proper finite grid.
- The formula $\phi_{\mathrm{NStape}}$ expresses that the proper finite grid in $(A,V',H)$ is of the type $m\times 2$ and connects the north border of the grid in $(A,V,H)$ to the south border of the grid in $(A,V,H)$. (The grids in $(A,V,H)$ and $(A,V',H)$ form a tube.)
- The formula $\phi_{\mathrm{EWtape}}$ expresses that the proper finite grid in $(A,V,H')$ is of the type $2\times n$ and connects the east border of the grid in $(A,V,H)$ to the west border of the grid in $(A,V,H)$. (The grids in $(A,V,H)$ and $(A,V,H')$ form a tube. The three grids together form a torus.)
\[lemma:torus\] Let ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H,V',H')$ be a finite structure such that the underlying structures $(A,V,H)$, $(A,V',H)$ and $(A,V,H')$ are fingrid-like and the structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ satisfies the conjunction of the formulas $\phi_{\mathrm{NStape}}$, $\phi_{\mathrm{EWtape}}$ and $\phi_\mathrm{uniquecorners}$. Then [[[$\mathfrak{A}$]{}]{}]{}is torus-like.
Notice that for a torus ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}$ to be torus-like, it is required that the finite grid $(D,V,H)$ is not of the type $1\times n$ or $n\times 1$ for any $n\in{\protect\ensuremath{{\mathbb{N}}}}$. A torus that is torus-like is called a *proper* torus.
It immediately follows from the previous lemma that there is a sentence $\phi_{\mathrm{torus}}\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ such that for all finite structures ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H,V',H')$, if ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi_{\mathrm{torus}}$ then ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ is torus-like, and furthermore, every proper torus satisfies $\phi_\mathrm{torus}$. We say that a structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,\{R_i^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}\}_{i\le n})$ is a *topping* of a structure ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}=(B,\{R_i^{{{{\protect\ensuremath{\mathfrak{B}}}\xspace}}}\}_{i\le n})$ iff $A=B$ and $R_i^{{{{\protect\ensuremath{\mathfrak{B}}}\xspace}}}\subseteq R_i^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}$ for all $i\le n$.
\[toruslike includes torus\] Let ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}=(A,V,H,V',H')$ be a finite structure with ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models\phi_{\mathrm{torus}}$. Then there is a torus ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}$ such that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ contains an isomorphic copy of a topping of ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}$ as a substructure.
Immediate from Definition \[toruslike description\] and the definition of a torus, i.e., Definition \[def:grid\].
The following theorem is the finite analogue of Theorem \[iftwo sat complexity\].
\[iftwo finsat complexity\] [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} is [[[$\mathit{\Sigma}^0_1$]{}]{}]{}-complete.
For the upper bound, note that since all finite structures can be recursively enumerated and the model checking problem of ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}$ over finite models is clearly decidable, we have ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2}}\xspace}}\nobreak)}}}}\xspace}}}\in{{{\protect\ensuremath{\mathit{\Sigma}^0_1}}\xspace}}$.
The lower bound follows by a reduction $g$ from [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Tiling}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{\mathrm{Torus}}}\xspace}}\nobreak)}}$]{}]{}]{}]{} to our problem defined by $g(T):=\phi_{\mathrm{torus}} \wedge \gamma_T$. To see that $g$ indeed is such a reduction, first let $T$ be a set of tiles such that there is a torus ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}'$ which is $T$-tilable. Therefore there clearly exists a proper torus ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}$ that is $T$-tilable. Then, by Lemma \[torustiling iff tiling formula\], it follows that there is an expansion ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}^*$ of ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}$ such that ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}^*\models \gamma_T$. We have ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}^*\models \phi_{\mathrm{torus}}$ and therefore ${{{\protect\ensuremath{\mathfrak{D}}}\xspace}}^* \models \phi_{\mathrm{torus}}\wedge \gamma_T$. If, on the other hand, ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*$ is a finite structure such that ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*\models \phi_{\mathrm{torus}}\wedge \gamma_T$, then by Lemma \[toruslike includes torus\], ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*$ has a substructure ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}^*_+$, which is an expansion of an isomorphic copy of a topping of a torus ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$. Furthermore, by Lemma \[torustiling iff tiling formula\], the $\{V,V',H,H'\}$-reduct ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ of the structure ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}^*$ is $T$-tilable. Hence, by the obvious analogue of Lemma \[tiling supergrids\], the $\{V,V',H,H'\}$-reduct ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}_+$ of ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}^*_+$ is $T$-tilable. Therefore ${{{\protect\ensuremath{\mathfrak{B}}}\xspace}}$ is clearly $T$-tilable.
Satisfiability for D\^2 is NEXPTIME-complete {#sec:satd2}
============================================
In this section we show that [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} and [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} are [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete. Our proof uses the fact that [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{{\protect\ensuremath{{\mathsf{FOC^2}}}}\xspace}}}}{}}{}{(\nobreak{{{{\protect\ensuremath{{\mathsf{FOC^2}}}}\xspace}}}\nobreak)}}$]{}]{}]{}]{} and [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{{\protect\ensuremath{{\mathsf{FOC^2}}}}\xspace}}}}{}}{}{(\nobreak{{{{\protect\ensuremath{{\mathsf{FOC^2}}}}\xspace}}}\nobreak)}}$]{}]{}]{}]{} are [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete [@Pratt-Hartmann:2005].
\[DtoESO\] Let $\tau$ be a relational vocabulary. For every formula $\phi\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}[\tau]$ there is a sentence $\phi^*\in {{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}[\tau\cup\{R\}]$ (with ${{{{\protect\ensuremath{\mathord{{\mathrm{arity}}\ifthenelse{\equal{R}{}}{}{(\nobreakR\nobreak)}}}}\xspace}}}=|{{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{\phi}{}}{}{(\nobreak\phi\nobreak)}}}}\xspace}}}|$), $$\phi^* :=\exists R_1\ldots\exists R_k\psi,$$ where $R_i$ is of arity at most $2$ and $\psi\in {{{{\protect\ensuremath{{\mathsf{FOC^2}}}}\xspace}}}$, such that for all ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ and teams $X$ with ${{{{\protect\ensuremath{\mathord{{\mathrm{dom}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X)={{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(\phi)$ it holds that $$\label{dtoesoequiv}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_X \phi \text{ iff } ({{{\protect\ensuremath{\mathfrak{A}}}\xspace}},{{{{\protect\ensuremath{\mathord{{\mathrm{rel}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X))\models \phi^*,$$ where $({{{\protect\ensuremath{\mathfrak{A}}}\xspace}},{{{{\protect\ensuremath{\mathord{{\mathrm{rel}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(X))$ is the expansion ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}'$ of ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ into vocabulary $\tau\cup\{R\}$ defined by $R^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}'} := {{{{\protect\ensuremath{\mathord{{\mathrm{rel}}\ifthenelse{\equal{X}{}}{}{(\nobreakX\nobreak)}}}}\xspace}}}$.
Using induction on $\phi$ we will first translate $\phi$ into a sentence $\tau_\phi \in {{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}[\tau\cup\{R\}]$ satisfying . Then we note that $\tau_{\phi}$ can be translated into an equivalent sentence $\phi^*$ that also satisfies the syntactic requirement of the theorem. The proof is a modification of the proof from [@va07 Theorem 6.2]. Below we write $\phi(x,y)$ to indicate that ${{{{\protect\ensuremath{\mathord{{\mathrm{Fr}}\ifthenelse{\equal{\phi}{}}{}{(\nobreak\phi\nobreak)}}}}\xspace}}}=\{x,y\}$. Also, the quantified relations $S$ and $T$ below are assumed not to appear in $\tau_{\psi}$ and $\tau_{\theta}$.
1. Let $\phi(x,y)\in \{x=y,\neg x=y, P(x,y),\neg P(x,y)\}$. Then $\tau_{\phi}$ is defined as $$\forall x\forall y( R(x,y)\rightarrow \phi(x,y)).$$
2. Let $\phi(x,y)$ be of the form ${{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x,y)$. Then $\tau_{\phi}$ is defined as $$\forall x\exists ^{\le 1}yR(x,y).$$
3. Let $\phi(x,y)$ be of the form $\neg {{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{}{}}{}{(\nobreak\nobreak)}}}}\xspace}}}(x,y)$. Then $\tau_{\phi}$ is defined as $$\forall x\forall y\neg R(x,y).$$
4. Let $\phi(x,y)$ be of the form $\psi(x,y)\vee \theta(y)$. Then $\tau_{\phi}$ is defined as $$\exists S\exists T(\tau_{\psi}(R/S)\wedge\tau_{\theta}(R/T) \wedge \forall x\forall y (R(x,y)\rightarrow S(x,y)\vee T(y))).$$
5. \[0-ary\] Let $\phi(x)$ be of the form $\psi(x)\vee \theta$. Then $\tau_{\phi}$ is defined as $$\exists S\exists T(\tau_{\psi}(R/S)\wedge\tau_{\theta}(R/T) \wedge \forall x (R(x)\rightarrow S(x)\vee T)).$$
6. Let $\phi(x)$ be of the form $\psi(x)\wedge \theta(y)$. Then $\tau_{\phi}$ is defined as $$\exists S\exists T(\tau_{\psi}(R/S)\wedge\tau_{\theta}(R/T)\wedge \forall x\forall y (R(x,y)\rightarrow S(x)\wedge T(y))).$$
7. Let $\phi(x)$ be of the form $\exists y\psi(x,y)$. Then $\tau_{\phi}$ is defined as $$\exists S(\tau_{\psi}(R/S)\wedge \forall x\exists y(R(x)\rightarrow S(x,y))).$$
8. Let $\phi(x)$ be of the form $\forall y\psi(x,y)$. Then $\tau_{\phi}$ is defined as $$\exists S(\tau_{\psi}(R/S)\wedge \forall x \forall y(R(x)\rightarrow S(x,y))).$$
It is worth noting that in the translation above we have not displayed all the possible cases, e.g., $\phi$ of the form ${{{{\protect\ensuremath{\mathord{{\mathrm{=}}\ifthenelse{\equal{x}{}}{}{(\nobreakx\nobreak)}}}}\xspace}}}$ or $P(x)$, for which $\tau_{\phi}$ is defined analogously to the above. Note also that, for convenience, we allow $0$-ary relations in the translation. The possible interpretations of a $0$-ary relation $R$ are $\emptyset$ and $\{\emptyset\}$. Furthermore, for a $0$-ary $R$, we define ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models R$ if and only if $R^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}=\{\emptyset\}$. Clause \[0-ary\] exemplifies the use of $0$-ary relations in the translation. It is easy to see that $\tau_{\phi}$ in \[0-ary\] is equivalent to $$\exists S(\tau_{\theta}(R/\top)\vee (\tau_{\psi}(R/S)\wedge \forall x (R(x)\rightarrow S(x)))).$$ Furthermore, the use of $0$-ary relations in the above translation can be easily eliminated with no essential change in the translation.
A straightforward induction on $\phi$ shows that $\tau_{\phi}$ can be transformed into $\phi^*$ of the form $$\exists R_1\ldots\exists R_k (\forall x\forall y\psi\wedge \bigwedge_{i}\forall x \exists y\theta_i\wedge \bigwedge_{j} \forall x\exists y^{\le1}R_{m_j}(x,y)),$$ where $\psi$ and $\theta_i$ are quantifier-free.
Note that if $\phi\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ is a sentence, the relation symbol $R$ is 0-ary and ${{{{\protect\ensuremath{\mathord{{\mathrm{rel}}\ifthenelse{\equal{X}{}}{}{(\nobreakX\nobreak)}}}}\xspace}}}$ (and $R^{{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}}$) is either $\emptyset$ or $\{\emptyset\}$. Hence, Theorem \[DtoESO\] implies that for an arbitrary sentence $\phi \in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}[\tau]$ there is a sentence $\phi^*(R/ {{{\protect\ensuremath{\top}}\xspace}}) \in {{{{\protect\ensuremath{{\mathsf{ESO}}}}\xspace}}}[\tau]$ such that for all ${{{\protect\ensuremath{\mathfrak{A}}}\xspace}}$ it holds that $$\label{d2eso sentences}
{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi \text{ iff }{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models_{\{\emptyset\}} \phi \text{ iff }{{{\protect\ensuremath{\mathfrak{A}}}\xspace}}\models \phi^*(R/{{{\protect\ensuremath{\top}}\xspace}}).$$
It is worth noting that, if $\phi\in{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ does not contain any dependence atoms, i.e., $\phi\in{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}$, the sentence $\phi^*$ is of the form $$\exists R_1\ldots\exists R_k (\forall x\forall y\psi\wedge \bigwedge _{i}\forall x \exists y\theta_i)$$ and the first-order part of this is in Scott normal form. So, in Theorem \[DtoESO\] we essentially translate formulas of ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ into Scott normal form [@sc62].
Theorem \[DtoESO\] now implies the following:
\[dtwo nexptime\] [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} and [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} are [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete.
Let $\phi\in {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ be a sentence. Then, by , $\phi$ is (finitely) satisfiable if and only if $\phi^*$ is. Now $\phi^*$ is of the form $$\exists R_1\ldots\exists R_k\psi,$$ where $\psi\in {{{{\protect\ensuremath{{\mathsf{FOC^2}}}}\xspace}}}$. Clearly, $\phi^*$ is (finitely) satisfiable iff $\psi$ is (finitely) satisfiable as a ${{{{\protect\ensuremath{{\mathsf{FOC^2}}}}\xspace}}}[\tau\cup\{R_1,\dots,R_k\}]$ sentence. Now since the mapping $\phi\mapsto \phi^*$ is clearly computable in polynomial time and (finite) satisfiability of $\psi$ can be checked in [[[${\textsl{NEXPTIME}}$]{}]{}]{}[@Pratt-Hartmann:2005], we get that ${{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}}}\xspace}}}, {{{{\protect\ensuremath{\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}}}\xspace}}}\in {{{\protect\ensuremath{{\textsl{NEXPTIME}}}}\xspace}}$. On the other hand, since ${{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}\le {{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}$ and [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{}, [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{FO}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} are [[[${\textsl{NEXPTIME}}$]{}]{}]{}-hard [@grkova97], it follows that [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{Sat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} and [[[[$\mathord{{\mathrm{{{{\protect\ensuremath{{\textsc{FinSat}}}}\xspace}}}}\ifthenelse{\equal{{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}}{}}{}{(\nobreak{{{\protect\ensuremath{{{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2}}\xspace}}\nobreak)}}$]{}]{}]{}]{} are as well.
Conclusion
==========
We have studied the complexity of the two-variable fragments of dependence logic and independence-friendly logic. We have shown (Theorem \[dtwo nexptime\]) that both the satisfiablity and finite satisfiability problems for [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}are decidable, [[[${\textsl{NEXPTIME}}$]{}]{}]{}-complete to be exact. We have also proved (Theorems \[iftwo sat complexity\] and \[iftwo finsat complexity\]) that both problems are undecidable for [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}; the satisfiability and finite satisfiabity problems for [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}are [[[$\mathit{\Pi}^0_1$]{}]{}]{}-complete and [[[$\mathit{\Sigma}^0_1$]{}]{}]{}-complete, respectively. While the full logics [[[[${\mathsf{D}}$]{}]{}]{}]{}and [[[[${\mathsf{IF}}$]{}]{}]{}]{}are equivalent over sentences, we have shown that the finite variable variants [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}and [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}are not, the latter being more expressive. This was obtained as a by-product of the deeper result concerning the decidability barrier between these two logics.
There are many open questions related to these logics. We conclude with two of them:
1. What is the complexity of the validity problems of [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}and [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}?
2. Is it possible to define NP-complete problems in [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}or in [[[${{{{\protect\ensuremath{{\mathsf{IF}}}}\xspace}}}^2$]{}]{}]{}?
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Phokion G. Kolaitis for suggesting to study the satisfiability of [[[${{{{\protect\ensuremath{{\mathsf{D}}}}\xspace}}}^2$]{}]{}]{}. The authors would also like to thank Johannes Ebbing, Lauri Hella, Allen Mann, Jouko Väänänen and Thomas Zeume for helpful discussions and comments during the preparation of this article.
[GOR97b]{}
S. Abramsky, *A compositional game semantics for multi-agent logics of imperfect information*, J. van Benthem, D. Gabbay and B. Lowe, eds., Texts in Logic and Games **1** (2007), no. 6, 11–48.
R. Berger, *The undecidability of the domino problem*, Memoirs of the American Mathematical Society, no. 66, American Mathematical Society, 1966.
J. C. Bradfield and S. Kreutzer, *The complexity of independence-friendly fixpoint logic*, CSL, 2005, pp. 355–368.
X. Caicedo, F. Dechesne, and T. M. V. Janssen, *Equivalence and quantifier rules for logic with imperfect information*, Logic Journal of the IGPL **17** (2009), no. 1, 91–129.
A. Church, *A note on the entscheidungsproblem*, Journal of Symbolic Logic **1** (1936), no. 1, 40–41.
H. B. Enderton, *Finite partially-ordered quantifiers*, Z. Math. Logik Grundlagen Math. **16** (1970), 393–397.
K. Etessami, M. Y. Vardi, and T. Wilke, *First-order logic with two variables and unary temporal logic*, Inf. Comput. **179** (2002), no. 2, 279–295.
Y. S. Gurevich and I. O. Koryakov, *Remarks on [Berger]{}’s paper on the domino problem*, Siberian Mathematical Journal **13** (1972), 319–321.
E. Gr[ä]{}del, P. G. Kolaitis, and M. Y. Vardi, *On the decision problem for two-variable first-order logic*, The Bulletin of Symbolic Logic **3** (1997), no. 1, 53–69.
E. Gr[ä]{}del and M. Otto, *On logics with two variables*, Theor. Comput. Sci. **224** (1999), 73–113.
E. Gr[ä]{}del, M. Otto, and E. Rosen, *Two-variable logic with counting is decidable*, Logic in Computer Science, 1997. LICS ’97. Proceedings., 12th Annual IEEE Symposium on, jun. 1997, pp. 306 –317.
[to3em]{}, *Undecidability results on two-variable logics*, STACS ’97: Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science (London, UK), Springer-Verlag, 1997, pp. 249–260.
E. Grädel and J. Väänänen, *Dependence and independence*, December 2010, preprint, 9p.
D. Harel, *Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness*, Journal of the ACM **33** (1986), no. 1, 224–248.
L. Henkin, *Some remarks on infinitely long formulas*, Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), Pergamon, Oxford, 1961, pp. 167–183.
L. Henkin, *Logical systems containing only a finite number of symbols*, Presses De l’Université De Montréal, Montreal, 1967.
J. Hintikka, *The principles of mathematics revisited*, Cambridge University Press, Cambridge, 1996.
W. Hodges, *Compositional semantics for a language of imperfect information*, Log. J. IGPL **5** (1997), no. 4, 539–563 (electronic).
[to3em]{}, *Some strange quantifiers*, Structures in logic and computer science, Lecture Notes in Comput. Sci., vol. 1261, Springer, Berlin, 1997, pp. 51–65.
J. Hintikka and G. Sandu, *Informational independence as a semantical phenomenon*, Logic, methodology and philosophy of science, VIII (Moscow, 1987), Stud. Logic Found. Math., vol. 126, North-Holland, Amsterdam, 1989, pp. 571–589.
E. Kieronski and M. Otto, *Small substructures and decidability issues for two-variable first-order logic*, Proceedings of 20th [IEEE]{} Symposium on Logic in Computer Science [LICS]{}’05, 2005, pp. 448–457.
P. Lohmann and H. Vollmer, *Complexity results for modal dependence logic*, Proceedings 19th Conference on Computer Science Logic, Lecture Notes in Computer Science, vol. 6247, Springer Berlin / Heidelberg, 2010, pp. 411–425.
M. Mortimer, *On languages with two variables*, Mathematical Logic Quarterly **21** (1975), no. 1, 135–140.
A. L. Mann, G. Sandu, and M. Sevenster, *Independence-friendly logic: A game-theoretic approach*, London Mathematical Society Lecture Note Series, Cambridge University Press, 2011.
I. Pratt-Hartmann, *Complexity of the two-variable fragment with counting quantifiers*, J. of Logic, Lang. and Inf. **14** (2005), 369–395.
L. Pacholski, W. Szwast, and L. Tendera, *Complexity of two-variable logic with counting*, LICS ’97. Proceedings., 12th Annual IEEE Symposium on Logic in Computer Science., 1997, pp. 318 –327.
D. Scott, *A decision method for validity of sentences in two variables*, J. Symbolic Logic **27** (1962), 377.
M. Sevenster, *Model-theoretic and computational properties of modal dependence logic*, J. Log. Comput. **19** (2009), no. 6, 1157–1173.
A. Turing, *On computable numbers, with an application to the entscheidungsproblem*, Proceedings of the London Mathematical Society, Series 2 **42** (1936), 230–265.
J. V[ä]{}[ä]{}n[ä]{}nen, *Dependence logic: A new approach to independence friendly logic*, London Mathematical Society student texts, no. 70, Cambridge University Press, 2007.
J. V[ä]{}[ä]{}n[ä]{}nen and W. Hodges, *Dependence of variables construed as an atomic formula*, Ann. Pure Appl. Logic **161** (2010), no. 6, 817–828.
W. J. Walkoe, Jr., *Finite partially-ordered quantification*, J. Symbolic Logic **35** (1970), 535–555.
H. Wang, *Proving theorems by pattern recognition ii*, Bell System Technical Journal **40** (1961), 1–41.
[^1]: University of Helsinki, juha.kontinen@helsinki.fi
[^2]: University of Tampere, {antti.j.kuusisto, jonni.virtema}@uta.fi
[^3]: Leibniz University Hannover, lohmann@thi.uni-hannover.de
[^4]: This work was supported by grants 127661, 129208, 129761, 129892 and 138163 of the Academy of Finland and by DAAD grant 50740539
|
---
abstract: 'Combinatorial characterisations are obtained of symmetric and anti-symmetric infinitesimal rigidity for two-dimensional frameworks with reflectional symmetry in the case of norms where the unit ball is a quadrilateral and where the reflection acts freely on the vertex set. At the framework level, these characterisations are given in terms of induced monochrome subgraph decompositions, and at the graph level they are given in terms of sparsity counts and recursive construction sequences for the corresponding signed quotient graphs.'
address: |
Dept. Math. Stats.\
Lancaster University\
Lancaster LA1 4YF\
U.K.
author:
- Derek Kitson and Bernd Schulze
title: 'Motions of grid-like reflection frameworks'
---
\[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Fact]{} \[theorem\][Observation]{} \[theorem\][Claim]{}
\[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Conjecture]{} \[theorem\][Open Problem]{} \[theorem\][Problem]{} \[theorem\][Question]{}
\[theorem\][Remark]{} \[theorem\][Note]{}
[^1] [^2]
Introduction
============
The objects considered in this article are geometric constraint systems where the constraints are determined by a possibly non-Euclidean choice of norm. The main results are new contributions in both geometric and combinatorial rigidity. At the geometric level, characterisations are provided for rigid two-dimensional symmetric frameworks constrained by norms with a quadrilateral unit ball (the $\ell^1$ and $\ell^\infty$ norms for example). At the combinatorial level, the problem of deciding whether a graph can be realized as a forced symmetric or anti-symmetric isostatic reflection framework is considered and complete characterisations are obtained. Overall this article builds on recent work analyzing the rigidity of frameworks in normed linear spaces, with and without symmetry (see for example [@kitson; @kit-pow; @kit-sch; @kit-sch1]).
A bar-joint framework in the plane is referred to as [*grid-like*]{} if the bar-lengths are determined by a norm with a quadrilateral unit ball. The allowable motions of such a framework constrain vertices adjacent to any pinned vertex to move along the boundary of a quadrilateral which is centred at the pinned vertex and obtained from the unit ball by translation and dilation (see Fig. \[fig:gridfw\]). This is an important context from the point of view of applications. For example, the problem of maintaining rigid formations of mobile autonomous agents is a well-known application of geometric rigidity theory and its associated “pebble game" algorithms (see [@EAMWB]). However, the Euclidean metric may not always be the most natural choice for controlling a formation. For instance, it may not be possible to detect Euclidean distances between agents (eg. due to obstacles in the terrain). Moreover, if the agents have restricted mobility (eg. with only vertical and horizontal directions of motion possible) then standard methods from Euclidean rigidity theory will have limited use. In these cases it may be desirable to have a rigidity theory for a non-Euclidean norm (such as the $\ell^1$ or $\ell^\infty$ norm) as an alternative approach to formation control. An accompanying theory for [*symmetric*]{} frameworks may provide more efficient architectures for the control of formations due to the smaller size of the quotient graphs and their associated constraint systems.
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(-1,-1)–(1,-1)–(1,1)–(-1,1)–(-1,-1);
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(p1) at (1,0.2) ; (p2) at (0.6,1) ;
(p0)–(p1); (p0)–(p2); (p1)–(p2);
(p2)–(0,1);
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(p0)–(p1); (p0)–(p2); (p1)–(p2); (p3)–(p0); (p3)–(p1); (p3)–(p2);
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There are three main aims of this article. The first is to formally introduce and develop symmetric and anti-symmetric infinitesimal rigidity for ${{\mathbb{Z}}}_2$-symmetric frameworks in general normed linear spaces. This is achieved in Section \[Sect:orbit\]. Each infinitesimal flex is shown to decompose in a unique way as a sum of a symmetric and an anti-symmetric flex. Moreover, the rigidity operator is shown to admit a block decomposition which leads in a natural way to a consideration of orbit matrices. Sparsity counts, expressed in terms of an associated signed quotient graph, are then derived for symmetrically and anti-symmetrically isostatic frameworks. When applied to Euclidean frameworks, the block decomposition reduces to that studied in [@KG2; @owen; @BS2], while the orbit matrices and sparsity counts coincide with those in [@jkt; @schtan; @BSWW].
The second aim is to characterise symmetric, anti-symmetric and general infinitesimal rigidity for grid-like frameworks with reflectional symmetry, where the reflection acts [*freely*]{} on the vertex set. In Section \[Sect:framework\], characterisations are obtained in terms of edge colourings for the signed quotient graph. These edge colourings are induced from a symmetric edge-colouring of the covering graph which is in turn induced by the positioning of the framework relative to the unit ball. This may be viewed as an extension to symmetric frameworks of methods used in [@kitson; @kit-pow].
The third aim, which is in the spirit of Laman’s theorem (see [@Lamanbib; @tay; @W1]), is to provide combinatorial characterisations for graphs which admit placements as rigid grid-like frameworks with reflectional symmetry. This is achieved in Section \[Sect:graph\] for both symmetric and anti-symmetric infinitesimal rigidity. The characterisations provide the sufficiency direction for the necessary sparsity counts derived in the general theory of Section \[Sect:orbit\]. The proof applies an inductive construction for signed quotient graphs together with the results of Section \[Sect:framework\]. Note that these matroidal counts can be checked in polynomial time using a straightforward adaptation of the algorithm described in [@jkt Sect. 10] (see also [@bhmt]).
The results of Section \[Sect:graph\] are analogous to the corresponding results for Euclidean reflection frameworks in [@jkt; @MT2]. It is important to note, however, that unlike the Euclidean situation (see [@schtan]), the respective characterisations of graphs which admit symmetric or anti-symmetric rigid placements as grid-like reflection frameworks cannot be combined to characterise graphs which admit rigid placements as grid-like reflection frameworks. This is due to the fact that the respective sets of symmetric and anti-symmetric rigid grid-like realisations of a graph may be disjoint (see Fig. \[fig:antisym\] for example). A combinatorial characterisation of graphs which admit a realisation as a grid-like *isostatic* reflection framework was recently given in [@kit-sch1]. However, as shown in [@kit-sch; @kit-sch1], such a framework must have a vertex which is fixed by the reflection.
${{\mathbb{Z}}}_2$-symmetric frameworks in normed spaces {#Sect:orbit}
========================================================
Throughout this article $G=(V,E)$ will denote a finite simple undirected graph with vertex set $V$ and edge set $E$. An edge $e\in E$ which is incident to vertices $v,w\in V$ will be denoted $vw$. An automorphism of $G$ is a bijective map $h:V\to V$ with the property that $vw\in E$ if and only if $h(v)h(w)\in E$. The group (under composition) of graph automorphisms of $G$ is denoted ${\operatorname{Aut}}(G)$. Consider the multiplicative group ${{\mathbb{Z}}}_2$ with elements $\{1,-1\}$. A [*${{\mathbb{Z}}}_2$-symmetric graph*]{} is a pair $(G,\theta)$ consisting of a graph $G$ and a group homomorphism $\theta:{{\mathbb{Z}}}_2\to \textrm{Aut}(G)$. When there is no danger of ambiguity, $\theta(-1)v$ will be denoted by $-v$ for each vertex $v\in V$ and $(-v)(-w)$ will be denoted by $-e$ for each edge $e=vw\in E$. The action $\theta$ is assumed throughout to be [*free*]{} on the vertex set of $G$ which means that $v \neq -v$ for all $v\in V$. It will not be assumed that the action is free on the edge set of $G$ and so there may be edges $e\in E$ such that $e=-e$. Such an edge is said to be [*fixed*]{} by $\theta$. The [*vertex orbit*]{} of a vertex $v\in V$ under the action $\theta$ is the pair $[v]:=\{v,-v\}$. The set of all vertex orbits is denoted $V_0$. Similarly, the [*edge orbit*]{} of an edge $e\in E$ is the pair $[e]:=\{e,-e\}$ and the set of all edge orbits is denoted $E_0$.
Symmetric and anti-symmetric motions
------------------------------------
Let $(X,\|\cdot\|)$ be a finite dimensional normed real linear space. A [*rigid motion*]{} of $(X,\|\cdot\|)$ is a family of continuous paths $\{\alpha_x:[-1,1]\to X\}_{x\in X}$, such that $\alpha_x(t)$ is differentiable at $t=0$ with $\alpha_x(0)=x$ and $\|\alpha_x(t)-\alpha_y(t)\| =\|x-y\|$ for all pairs $x,y\in X$ and all $t\in [-1,1]$.
The [*rigidity map*]{} for $G=(V,E)$ and $(X,\|\cdot\|)$ is defined by, $$f_G:X^{|V|}\to {{\mathbb{R}}}^{|E|},\quad (x_v)_{v\in V}\mapsto (\|x_v-x_w\|)_{vw\in E}.$$ The directional derivative of the rigidity map $f_G$ at a point $p\in X^{|V|}$ and in the direction of a vector $u\in X^{|V|}$ is denoted $D_uf_G(p)$, $$D_uf_G(p) = \lim_{t\to 0}\,\frac{1}{t}\left(f_G(p+tu)-f_G(p)\right).$$
A [*bar-joint framework*]{} in $(X,\|\cdot\|)$ is a pair $(G,p)$ where $p=(p_v)_{v\in V}\in X^{|V|}$ and $p_v\not=p_w$ for all $vw\in E$. A [*subframework*]{} of $(G,p)$ is a bar-joint framework $(H,p_H)$ (or simply $(H,p)$) where $H=(V(H),E(H))$ is a subgraph of $G$ and $p_H=(p_v)_{v\in V(H)}$. A subframework $(H,p)$ is [*spanning*]{} in $(G,p)$ if $H$ is a spanning subgraph of $G$ and [*proper*]{} if $H\not=G$.
An [*infinitesimal flex*]{} for $(G,p)$ is a vector $u\in X^{|V|}$ such that $D_uf_G(p)=0$. The collection of all infinitesimal flexes of $(G,p)$ forms a linear subspace of $X^{|V|}$, denoted ${{\mathcal{F}}}(G,p)$. It can be shown (see [@kit-pow Lemma 2.1]) that if $\{\alpha_x\}_{x\in X}$ is a rigid motion of $(X,\|\cdot\|)$ then $(\alpha_{p_v}'(0))_{v\in V}\in X^{|V|}$ is an infinitesimal flex of $(G,p)$. An infinitesimal flex of this type is said to be [*trivial*]{} and the collection of all trivial infinitesimal flexes forms a linear subspace of ${{\mathcal{F}}}(G,p)$, denoted ${{\mathcal{T}}}(G,p)$. A bar-joint framework is said to be [*infinitesimally rigid*]{} if every infinitesimal flex is trivial and [*isostatic*]{} if, in addition, no proper spanning subframework is infinitesimally rigid.
If the rigidity map $f_G$ is differentiable at $p$ then the differential is denoted $df_G(p)$. In this case, $(G,p)$ is said to be [*well-positioned*]{} in $(X,\|\cdot\|)$ and $df_G(p)$ is referred to as the [*rigidity operator*]{} for $(G,p)$. Note that the rigidity operator $df_G(p)$ satisfies, $$\begin{aligned}
\label{OpFormula}
df_G(p)u = \left(\,\varphi_{v,w}(u_v-u_w)\,\right)_{vw\in E},\end{aligned}$$ for all $u=(u_v)_{v\in V}\in X^{|V|}$ where $\varphi_{v,w}:X\to{{\mathbb{R}}}$ is a linear functional defined by, $$\varphi_{v,w}(x) = \lim_{t\to 0}\,
\frac{1}{t}(\|p_v-p_w+tx\|-\|p_v-p_w\|), \,\,\,\,\,\,\, \forall\,x\in X.$$ In this way the rigidity operator may be represented by a [*rigidity matrix*]{} of linear functionals with rows indexed by $E$ and columns indexed by $V$. (For details see [@kit-sch]).
Let ${\operatorname{Isom}}(X,\|\cdot\|)$ denote the group of linear isometries of $(X,\|\cdot\|)$. A bar-joint framework $(G,p)$ is said to be [**]{} with respect to an action $\theta:{{\mathbb{Z}}}_2\rightarrow {\operatorname{Aut}}(G)$ and a group representation $\tau:{{\mathbb{Z}}}_2\rightarrow {\operatorname{Isom}}(X,\|\cdot\|)$ if $\tau(-1)(p_v)=p_{-v}$ for all $v\in V$.
\[decomp\] \[Decomp\] Let $(G,p)$ be a well-positioned bar-joint framework in $(X,\|\cdot\|)$ which is ${{\mathbb{Z}}}_2$-symmetric with respect to an action $\theta:{{\mathbb{Z}}}_2\rightarrow {\operatorname{Aut}}(G)$ and a representation $\tau:{{\mathbb{Z}}}_2\rightarrow {\operatorname{Isom}}(X,\|\cdot\|)$.
1. $X^{|V|}$ may be expressed as a direct sum $X^{|V|}=X_1\oplus X_2$ where, $$\begin{aligned}
X_1 &=& \{(x_v)_{v\in V}\in X^{|V|}:\, x_{-v}
= \,\tau(-1)x_v,\,\,\,\,\forall\,\,v\in V\},\end{aligned}$$ $$\begin{aligned}
X_2 &=& \{(x_v)_{v\in V}\in X^{|V|}: x_{-v}
= -\tau(-1)x_v,\,\,\forall\,\,v\in V\}.\end{aligned}$$
2. ${{\mathbb{R}}}^{|E|}$ may be expressed as a direct sum ${{\mathbb{R}}}^{|E|} = Y_1\oplus Y_2$ where, $$\begin{aligned}
Y_1 &=& \{(y_e)_{e\in E}\in {{\mathbb{R}}}^{|E|}:\, y_{-e}=\, y_e,\,\,\,\,
\forall\,\,e\in E\},\end{aligned}$$ $$\begin{aligned}
Y_2 &=& \{(y_e)_{e\in E}\in {{\mathbb{R}}}^{|E|}: y_{-e}=- y_e,\,\,
\forall\,\,e\in E\}.\end{aligned}$$
3. With respect to the direct sum decompositions, $$X^{|V|} = X_1\oplus X_2, \,\,\,\,\,\mbox{ and, } \,\,\,\,\,\,
{{\mathbb{R}}}^{|E|} = Y_1\oplus Y_2,$$ the differential $df_G(p)$ may be expressed as a direct sum of linear transformations, $$df_G(p)=R_1\oplus R_2,$$ where $R_1:X_1\to Y_1$ and $R_2:X_2\to Y_2$.
Each $(x_v)_{v\in V}\in X^{|V|}$ may be expressed as a sum $a+b$ where $a=\left(\frac{1}{2}(x_v+\tau(-1)(x_{-v}))\right)_{v\in V}$ and $b=\left(\frac{1}{2}(x_v-\tau(-1)(x_{-v}))\right)_{v\in V}$. Note that $a\in X_1$ and $b\in X_2$. Similarly, each $(y_e)_{e\in E}\in {{\mathbb{R}}}^{|E|}$ may be expressed as a sum $a+b$ where $a=\left(\frac{1}{2}(y_e+y_{-e})\right)_{e\in E}\in Y_1$ and $b=\left(\frac{1}{2}(y_e-y_{-e})\right)_{e\in E}\in Y_2$. To prove $(i)$ and $(ii)$ it only remains to note that $X_1\cap X_2=\{0\}$ and $Y_1\cap Y_2=\{0\}$.
To prove $(iii)$, let $vw\in E$ and note that if $(x_v)_{v\in V}\in X_1$ then, $$\varphi_{v,w}(x_v-x_w) = \varphi_{-v,-w}(\tau(-1)(x_v-x_w))
=\varphi_{-v,-w}(x_{-v}-x_{-w}).$$ Similarly, if $(x_v)_{v\in V}\in X_2$ then, $$\varphi_{v,w}(x_v-x_w) = \varphi_{-v,-w}(\tau(-1)(x_v-x_w))
=\varphi_{-v,-w}(-(x_{-v}-x_{-w}))=-\varphi_{-v,-w}(x_{-v}-x_{-w}).$$By equation \[OpFormula\]), $df_G(p)(X_1)\subset Y_1$ and $df_G(p)(X_2)\subset Y_2$ and so the result follows.
A vector $u=(u_v)_{v\in V}\in X^{|V|}$ will be called [*symmetric*]{} if $u\in X_1$ and [*anti-symmetric*]{} if $u\in X_2$. The vector spaces of symmetric and anti-symmetric infinitesimal flexes of $(G,p)$ are respectively denoted ${{\mathcal{F}}}_1(G,p)$ and ${{\mathcal{F}}}_2(G,p)$. Similarly, the vector spaces of symmetric and anti-symmetric trivial infinitesimal flexes are respectively denoted ${{\mathcal{T}}}_1(G,p)$ and ${{\mathcal{T}}}_2(G,p)$. A straight-forward verification shows that ${{\mathcal{F}}}(G,p)={{\mathcal{F}}}_1(G,p)\oplus {{\mathcal{F}}}_2(G,p)$ and ${{\mathcal{T}}}(G,p) = {{\mathcal{T}}}_1(G,p)\oplus {{\mathcal{T}}}_2(G,p)$.
The following observation will be applied in the next section. The identity operator on $X$ is denoted $I$.
\[lem:dimmotions\] Let $(G,p)$ be a well-positioned and ${{\mathbb{Z}}}_2$-symmetric bar-joint framework in $(X,\|\cdot\|)$. If the group of linear isometries ${\operatorname{Isom}}(X,\|\cdot\|)$ is finite then,
1. $\dim {{\mathcal{T}}}(G,p) = \dim X$.
2. $\dim {{\mathcal{T}}}_1(G,p) = {\operatorname{rank}}(I+\tau(-1))$.
3. $\dim {{\mathcal{T}}}_2(G,p) = {\operatorname{rank}}(I-\tau(-1))$.
It is shown in [@kit-pow] that if ${\operatorname{Isom}}(X,\|\cdot\|)$ is finite then ${{\mathcal{T}}}(G,p) = \{(x,\ldots,x)\in X^{|V|}:x\in X\}$. Part $(i)$ is an immediate consequence of this while $(ii)$ and $(iii)$ follow on considering the definitions of $X_1$ and $X_2$.
A ${{\mathbb{Z}}}_2$-symmetric bar-joint framework $(G,p)$ in $(X,\|\cdot\|)$ is said to be,
1. [*(anti-) symmetrically infinitesimally rigid*]{} if every (anti-) symmetric infinitesimal flex of $(G,p)$ is a trivial infinitesimal flex.
2. [*(anti-) symmetrically isostatic*]{} if it is (anti-) symmetrically infinitesimally rigid and no ${{\mathbb{Z}}}_2$-symmetric proper spanning subframework of $(G,p)$ is (anti-) symmetrically infinitesimally rigid.
Let $G=(V,E)$ be a ${{\mathbb{Z}}}_2$-symmetric graph with $V_0$ the set of vertex orbits and $E_0$ the set of edge orbits. The subset of $E_0$ consisting of edge orbits for edges in $G$ which are not fixed is denoted $E_0'$.
\[Counts\] Let $(G,p)$ be a well-positioned and ${{\mathbb{Z}}}_2$-symmetric bar-joint framework in $(X,\|\cdot\|)$.
1. If $(G,p)$ is symmetrically infinitesimally rigid then, $$|E_0| \geq (\dim X)|V_0| - \dim {{\mathcal{T}}}_1(G,p).$$
2. If $(G,p)$ is anti-symmetrically infinitesimally rigid then, $$|E_0'| \geq (\dim X)|V_0| - \dim {{\mathcal{T}}}_2(G,p).$$
Consider the decompositions constructed in Lemma \[Decomp\]. Note that $\dim X_1 = (\dim X)|V_0|$, $\dim X_2 = (\dim X)|V_0|$, $\dim Y_1 = |E_0|$ and $\dim Y_2 = |E_0'|$. (In the case of $Y_2$ the dimension is determined by the number of edge orbits for edges which are not fixed). If $(G,p)$ is symmetrically infinitesimally rigid then ${{\mathcal{T}}}_1(G,p)={{\mathcal{F}}}_1(G,p)=\ker R_1$ and so, $$|E_0| \geq {\operatorname{rank}}R_1 =
(\dim X)|V_0| - \dim \ker R_1 = (\dim X)|V_0| - \dim {{\mathcal{T}}}_1(G,p).$$ A similar argument applies if $(G,p)$ is anti-symmetrically infinitesimally rigid.
\[AntiSymFixedEdges\] Let $(G,p)$ be a well-positioned and ${{\mathbb{Z}}}_2$-symmetric bar-joint framework in $(X,\|\cdot\|)$. If $(G,p)$ is anti-symmetrically isostatic then $G$ contains no fixed edges.
Suppose $e=v(-v)$ is a fixed edge in $G$ and let $H=G-e$. Then there exists a non-trivial anti-symmetric infinitesimal flex $u\in {{\mathcal{F}}}_2(H,p)$. Note that $u\in \ker df_H(p)$ and the linear functional $\varphi_{v,-v}$ satisfies, $$\varphi_{v,-v}(u_v-u_{-v}) = \varphi_{-v,v}(\tau(-1)(u_v-u_{-v}))
=\varphi_{-v,v}(-(u_{-v}-u_{v}))=-\varphi_{-v,v}(u_{-v}-u_{v})
=-\varphi_{v,-v}(u_{v}-u_{-v}).$$ Thus $\varphi_{v,-v}(u_v-u_{-v})=0$ and so, from equation (\[OpFormula\]), it follows that $u\in \ker df_G(p)$. In particular, $u$ is a non-trivial anti-symmetric infinitesimal flex of $(G,p)$.
Let $Z$ and $W$ be linear subspaces of $X$ such that $X=Z\oplus W$ and suppose $W$ has dimension $1$. A linear isometry $T\in{\operatorname{Isom}}(X,\|\cdot\|)$ is called a [*reflection*]{} in the mirror $Z$ along $W$ if $T=I-2P$, where $P:X\to X$ is the linear projection with range $W$ and kernel $Z$.
\[K2Lemma\] Let $(K_2,p)$ be a placement of $K_2$ in $(X,\|\cdot\|)$ which is ${{\mathbb{Z}}}_2$-symmetric with respect to an action $\theta:{{\mathbb{Z}}}_2\to {\operatorname{Aut}}(G)$ and a representation $\tau:{{\mathbb{Z}}}_2\to {\operatorname{Isom}}(X,\|\cdot\|)$. If $\theta$ acts freely on $V(K_2)$ and $\tau(-1)$ is a reflection then $(K_2,p)$ is symmetrically isostatic.
Let $v$ and $-v$ be the vertices of $K_2$ and let $u\in {{\mathcal{F}}}_1(K_2,p)$ be a symmetric infinitesimal flex of $(K_2,p)$. The isometry $\tau(-1)$ has the form $\tau(-1)=I-2P$ where $P$ is a projection as described above. Note that, $$\varphi_{v,-v}(Pu_v) =\frac{1}{2}\varphi_{v,-v}((I-\tau(-1))u_v)
= \frac{1}{2}\varphi_{v,-v}(u_v-u_{-v}) =
0.$$ Thus $u_v\in Z$ or $W\subset \ker\varphi_{v,-v}$. Note that $p_v-p_{-v} = (I-\tau(-1))p_v =2P(p_v)\in W$. Thus if $W\subset \ker \varphi_{v,-v}$ then, $$\|p_v-p_{-v}\| = \varphi_{v,-v}(p_v-p_{-v})=0,$$ and so $p_v=p_{-v}$ which is a contradiction. We conclude that $u_v\in Z$ and so $u_{-v}=\tau(-1)u_v=u_v$. Thus $u$ is a trivial infinitesimal flex.
Signed quotient graphs {#sec:basicdef}
----------------------
The [*quotient graph*]{} $G_0=G/{{\mathbb{Z}}}_2$ for a ${{\mathbb{Z}}}_2$-symmetric graph $(G,\theta)$ has vertex set $V_0$ consisting of the vertex orbits for $(G,\theta)$ and edge set $E_0$ consisting of the edge orbits. An edge $[e]\in E_0$ is regarded as incident to a vertex $[v]\in V_0$ if $e$ (equivalently, $-e$) is incident to either $v$ or $-v$ in $G$. In general, $G_0$ is not a simple graph as if $e\in E$ is a fixed edge in $G$ then $[e]$ is a loop in $G_0$. Also, if $e=vw$ and $e'=v(-w)$ are distinct edges in $G$ then $[e]$ and $[e']$ are parallel edges in $G_0$.
Let $\tilde{V}_0=\{\tilde{v}_1,\ldots,\tilde{v}_n\}$ be a choice of representatives for the vertex orbits of $(G,\theta)$. A [*signed quotient graph*]{} (or *quotient $\mathbb{Z}_2$-gain graph* [@jkt; @schtan]) is a pair $(G_0,\psi)$ consisting of a quotient graph $G_0$ and an edge-labeling (or *gain*) $\psi:E_0\to {{\mathbb{Z}}}_2$ where $\psi([e])=1$ if either $e$ or $-e$ is incident to two vertices in $\tilde{V}_0$ and $\psi([e])=-1$ otherwise. See Figure \[fig:gaingraph\] for an example.
In the following, $G$ will be referred to as the *covering graph* of $(G_0,\psi)$ and, to simplify notation, $\psi([e])$ will be denoted $\psi_{[e]}$. Note that the covering graph is required to be a simple graph and so signed quotient graphs are characterised by the following two properties.
1. If two edges $[e]$ and $[e']$ in $G_0$ are parallel then $\psi_{[e]}\not=\psi_{[e']}$.
2. If $[e]$ is a loop in $G_0$ then $\psi_{[e]}=-1$.
The [*gain*]{} of a set of edges $F$ in a signed quotient graph $(G_0,\psi)$ is defined as the product $\psi(F) =\Pi_{[e]\in F} \,\psi_{[e]}$. A set of edges $F$ is [*balanced*]{} if it does not contain a cycle of edges, or, has the property that every cycle of edges in $F$ has gain $1$. A subgraph of $G_0$ is [*balanced*]{} in $(G_0,\psi)$ if it is spanned by a balanced set of edges, otherwise, the subgraph is [*unbalanced*]{}. (See also [@jkt; @zas; @zas1]).
=\[circle, draw=black, fill=white, inner sep=0pt, minimum width=5pt\];
(p1) at (0,0) ; (p2) at (4,0) ; (p1o) at (1,0.5) ; (p2o) at (3,0.5) ; (p1l) at (0.5,1.5) ; (p2r) at (3.5,1.5) ; (p1)–(p1o); (p2)–(p2o); (p1)–(p2o); (p2)–(p1o); (p1l)–(p1o); (p1l)–(p2r); (p1l)–(p1); (p2r)–(p2o); (p2r)–(p2); (2,-0.3)–(2,1.9); (a) at (2,-0.9) [(a)]{};
\(a) at (6.8,0.5) [$1$]{}; (a) at (7.3,0.6) [$1$]{};
\(a) at (7.29,0.3) [$1$]{}; (a) at (7.3,1.3) [$-1$]{};
\(a) at (7.6,-0.2) [$-1$]{};
(p11) at (7,0) ; (p22) at (7,1) ; (p33) at (8,0.5) ; (p22)–(p11); (p22)–(p33); (p33)–(p11); (p11) edge \[bend right=40\] (p33); (p22) edge \[loop above, >=stealth,shorten >=1pt,looseness=20\] (p22); (a) at (7.5,-0.9) [(b)]{};
\[switch\] Let $(G_0,\psi)$ be a signed quotient graph for a ${{\mathbb{Z}}}_2$-symmetric graph $(G,\theta)$ and let $H_0$ be a balanced subgraph in $(G_0,\psi)$. Then,
1. $H_0$ is a balanced subgraph in $(G_0,\psi')$ for every gain $\psi'$ induced by a choice of vertex orbit representatives for $(G,\theta)$, and,
2. there exists a choice of vertex orbit representatives $\tilde{V}_0$ for $(G,\theta)$ such that the induced gain $\psi'$ satisfies $\psi'_{[e]}=1$ for all $[e]\in E(H_0)$.
A subgraph of $G_0$ will be referred to as balanced if it is balanced in $(G_0,\psi)$ for some (and hence every) gain $\psi$ induced by a choice of vertex orbit representatives.
A subgraph of $G_0$ for which every connected component contains exactly one cycle, each of which is unbalanced, is called an [*unbalanced map graph*]{} in $G_0$.
If a representative vertex ${\tilde{v}}$ is replaced by the vertex $-{\tilde{v}}$, then a new signed quotient graph $(G_0,\psi')$ is obtained, where $\psi'_{[e]}=-\psi_{[e]}$ if $[e]$ is incident with $[v]$, and $\psi'_{[e]}=\psi_{[e]}$ otherwise. This is referred to as a *switching operation* on $[v]$.
Orbit matrices and sparsity counts
----------------------------------
Let $(G,p)$ be a well-positioned and ${{\mathbb{Z}}}_2$-symmetric bar-joint framework in $(X,\|\cdot\|)$ and let $\tilde{V}_0$ be a choice of vertex orbit representatives.
A [*symmetric orbit matrix*]{} for $(G,p)$ is a matrix of linear functionals on $X$, denoted $O_1(G,p)$ or simply $O_1$, with rows indexed by $E_0$ and columns indexed by $V_0$.
The matrix entry for a pair $([e],[v])\in E_0\times V_0$ is given by, $$O_1([e],[v])=
\left\{
\arraycolsep=2pt\def\arraystretch{1.4}
\begin{array}{ll}
\varphi_{\tilde{v},\psi_{[e]}\tilde{w}}
& \mbox{ if }[e]=[vw] \mbox{ and } [e] \mbox{ is not a loop},\\
2\varphi_{\tilde{v},-\tilde{v}}
& \mbox{ if }[e] \mbox{ is a loop at } [v],\\
0 & \mbox{ otherwise,}
\end{array}\right.$$ where $\tilde{v},\tilde{w}\in \tilde{V}_0$ are the representative vertices for $[v]$ and $[w]$ respectively and $\psi$ is the gain on $G_0$ induced by $\tilde{V}_0$.
Each symmetric orbit matrix determines a linear map $O_1(G,p):X^{|V_0|}\to {{\mathbb{R}}}^{|E_0|}$. Explicitly, the row entries of $O_1(G,p)$ which correspond to an edge orbit $[e]=[vw]$ which is not a loop are, $$\kbordermatrix{
& & & & \lbrack v \rbrack & & & & \lbrack w \rbrack & & & \\
\lbrack e \rbrack & 0 &\cdots\,& 0 & \varphi_{\tilde{v},\psi_{[e]}\tilde{w}} &0 &\cdots
\cdots &0&\varphi_{\tilde{w},\psi_{[e]}\tilde{v}}&0&\cdots &0},$$ while if $[e]$ is a loop at a vertex $[v]$ then the row entries are, $$\kbordermatrix{
&&&&\lbrack v \rbrack &&& \\
\lbrack e \rbrack & 0 &\,\cdots\,& 0& \,\,\,\, 2\varphi_{\tilde{v},-\tilde{v}}
\,\,\,\,&0&\,\cdots\, &0}.$$
\[CommDiagramLemma1\] Let $(G,p)$ be a well-positioned and ${{\mathbb{Z}}}_2$-symmetric bar-joint framework in $(X,\|\cdot\|)$. If $O_1$ is a symmetric orbit matrix for $(G,p)$ then there exist linear isomorphisms, $$S_1:X^{|V_0|}\to X_1, \qquad T_1:{{\mathbb{R}}}^{|E_0|}\to Y_1,$$ such that the following diagram commutes. $$\begin{tikzcd}
X^{|V_0|} \arrow{r}{O_1} \arrow[swap]{d}{S_1} & {{\mathbb{R}}}^{|E_0|} \arrow{d}{T_1} \\
X_1 \arrow{r}{R_1} & Y_1
\end{tikzcd}$$ In particular, $R_1$ and $O_1$ are (isomorphically) equivalent linear transformations.
Let $\tilde{V}_0$ be the choice of vertex orbit representatives from which $O_1(G,p)$ is derived. Each vertex $v\in V$ is expressible in the form $v=\gamma_v \tilde{v}$ for some $\gamma_v\in{{\mathbb{Z}}}_2$ where $\tilde{v}\in\tilde{V}_0$ is the chosen representative for $[v]$. Define, $$S_1:X^{|V_0|}\to X_1, \,\,\,\,\,\,
(x_{[v]})_{[v]\in V_0}\mapsto
(\tau(\gamma_v)x_{[v]})_{v\in V},$$ $$T_1:{{\mathbb{R}}}^{|E_0|}\to Y_1, \,\,\,\,\,\,
(y_{[e]})_{[e]\in E_0}\mapsto
(y_{[e]})_{e\in E}.$$ Let $u=(u_{[v]})_{[v]\in V_0}\in X^{|V_0|}$. It is sufficient to compare the entries of $(T_1\circ O_1)u$ and $(R_1\circ S_1)u$ in $Y_1$ (note that these entries are indexed by $E$).
Suppose $e=vw\in E$ is an edge in $G$ which is not fixed. Then the edge orbit $[e]$ is not a loop in the quotient graph $G_0$ and so the entry of $O_1(u)$ corresponding to $[e]$ is given by, $$\begin{aligned}
\label{RigidityMatrixEqn}
\varphi_{\tilde{v},\psi_{[e]}\tilde{w}}(u_{[v]})-\varphi_{\psi_{[e]}\tilde{v},\tilde{w}}(u_{[w]})
&=& \varphi_{\tilde{v},\psi_{[e]}\tilde{w}}(u_{[v]})-\varphi_{\tilde{v},\psi_{[e]}\tilde{w}}(\tau(\psi_{[e]})u_{[w]}) \\
&=& \varphi_{\tilde{v},\psi_{[e]}\tilde{w}}(u_{[v]} - \tau(\psi_{[e]})u_{[w]}).\end{aligned}$$ This is also the entry of $(T_1\circ O_1)u$ corresponding to $e$. Note that $e=(\gamma_v \tilde{v})(\gamma_w \tilde{w})$ where $\psi_{[e]}=\gamma_v\gamma_w$. Thus, the entry of $(R_1\circ S_1)u$ corresponding to $e$ is, $$\begin{aligned}
\varphi_{v,w}(\tau(\gamma_v)u_{[v]}
-\tau(\gamma_w)u_{[w]})
&=& \varphi_{\gamma_v \tilde{v},\gamma_w \tilde{w}}(\tau(\gamma_v)u_{[v]}
-\tau(\gamma_w)u_{[w]}) \\
&=& \left(\varphi_{\tilde{v},\psi_{[e]}\tilde{w}}\circ\tau(\gamma_v)\right)(\tau(\gamma_v)u_{[v]}
-\tau(\gamma_w)u_{[w]}) \\
&=&\varphi_{\tilde{v},\psi_{[e]}\tilde{w}}(u_{[v]}
-\tau(\psi_{[e]})u_{[w]}).\end{aligned}$$
Now suppose $e=\tilde{v}(-\tilde{v})\in E$ is a fixed edge in $G$. The edge orbit $[e]$ is a loop in the quotient graph and so the entry of $(T_1\circ O_1)u$ corresponding to $e$ is $2\varphi_{\tilde{v},-\tilde{v}}(u_{[\tilde{v}]})$. Likewise, the entry of $(R_1\circ S_1)u$ corresponding to $e$ is, $$\begin{aligned}
\label{RigidityMatrixEqn2}
\varphi_{\tilde{v},-\tilde{v}}(u_{[\tilde{v}]}-\tau(-1)u_{[\tilde{v}]})
&=& \varphi_{\tilde{v},-\tilde{v}}(u_{[\tilde{v}]})+\varphi_{-\tilde{v},\tilde{v}}(\tau(-1)u_{[\tilde{v}]}) \\
&=& \varphi_{\tilde{v},-\tilde{v}}(u_{[\tilde{v}]}) + \varphi_{\tilde{v},-\tilde{v}}(u_{[\tilde{v}]}) \\
&=& 2\varphi_{\tilde{v},-\tilde{v}}(u_{[\tilde{v}]}).\end{aligned}$$
Consider again a ${{\mathbb{Z}}}_2$-symmetric bar-joint framework $(G,p)$ and fix an orientation on the edges of the quotient graph which lie in $E_0'$ (i.e. the edges in $G_0$ which are not loops).
An [*anti-symmetric orbit matrix*]{} for $(G,p)$ is a matrix of linear functionals on $X$, denoted $O_2(G,p)$ or $O_2$, with rows indexed by $E_0'$ and columns indexed by $V_0$.
The matrix entry for a pair $([e],[v])\in E_0'\times V_0$ is given by, $$O_2([e],[v])=
\left\{
\arraycolsep=2pt\def\arraystretch{1.4}
\begin{array}{ll}
\varphi_{\tilde{v},\psi_{[e]}\tilde{w}}
& \mbox{ if }[e]=[vw] \mbox{ and } [e] \mbox{ is oriented from } $[v]$ \mbox{ to }$[w]$,\\
\psi_{[e]}\varphi_{\tilde{v},\psi_{[e]}\tilde{w}}
& \mbox{ if }[e]=[vw] \mbox{ and } [e] \mbox{ is oriented from } $[w]$ \mbox{ to }$[v]$,\\
0 & \mbox{ otherwise,}
\end{array}\right.$$ where $\tilde{v},\tilde{w}\in \tilde{V}_0$ are the representative vertices for $[v]$ and $[w]$ respectively and $\psi$ is the gain on $G_0$ induced by $\tilde{V}_0$.
The row entries of $O_2(G,p)$ corresponding to an edge orbit $[e]$ oriented from $[v]$ to $[w]$ are, $$\kbordermatrix{
& & & & \lbrack v \rbrack & & & & \lbrack w \rbrack & & & \\
\lbrack e\rbrack& 0 &\cdots& 0& \varphi_{\tilde{v},\psi_{[e]}\tilde{w}}&0&\cdots
\cdots &0&\psi_{[e]}\,\varphi_{\tilde{w},\psi_{[e]}\tilde{v}}
&0&\cdots &0 }.$$
Let $(G,p)$ be a well-positioned and ${{\mathbb{Z}}}_2$-symmetric bar-joint framework in $(X,\|\cdot\|)$. If $O_2$ is an anti-symmetric orbit matrix for $(G,p)$ then there exist linear isomorphisms, $$S_2:X^{|V_0|}\to X_2,\qquad \,T_2:{{\mathbb{R}}}^{|E_0'|}\to Y_2$$ such that the following diagram commutes. $$\begin{tikzcd}
X^{|V_0|} \arrow{r}{O_2} \arrow[swap]{d}{S_2} & {{\mathbb{R}}}^{|E_0'|} \arrow{d}{T_2} \\
X_2 \arrow{r}{R_2} & Y_2
\end{tikzcd}$$ In particular, $R_2$ and $O_2$ are (isomorphically) equivalent linear transformations.
Each vertex $v\in V$ is expressible in the form $v=\gamma_v \tilde{v}$ for some $\gamma_v\in{{\mathbb{Z}}}_2$ where $\tilde{v}\in\tilde{V}_0$ is the chosen representative for $[v]$. For each edge $e=vw\in E$ which is not fixed, define $\gamma_e=\gamma_v$ if $[e]$ is oriented from $[v]$ to $[w]$. Also define, $$S_2:X^{|V_0|}\to X_2, \,\,\,\,\,\,
(x_{[v]})_{[v]\in V_0}\mapsto
(\gamma_v\tau(\gamma_v)x_{[v]})_{v\in V},$$ $$T_2:{{\mathbb{R}}}^{|E_0'|}\to Y_2, \,\,\,\,\,\,
(y_{[e]})_{[e]\in E_0'}\mapsto
(\gamma_e y_{[e]})_{e\in E},$$ where, in the definition of $T_2$, we formally set $\gamma_e y_{[e]}=0$ if $e$ is a fixed edge of $G$. The commutativity of the diagram can now be verified in a manner analogous to the proof of Lemma \[CommDiagramLemma1\].
Let $(H,p)$ be a ${{\mathbb{Z}}}_2$-symmetric framework. If $H_0$ is balanced then, by Lemma \[switch\], there exists a choice of vertex orbit representatives $\tilde{V}_0$ such that the induced gain is identically $1$ on the edges of $H_0$. It follows that $H_0$ may be identified with the vertex-induced subgraph on $\tilde{V}_0$ in $H$. With this identification, $(H_0,p)$ is a well-defined subframework of $(H,p)$.
\[lem:symcounts\] Let $(G,p)$ be a well-positioned and ${{\mathbb{Z}}}_2$-symmetric bar-joint framework in $(X,\|\cdot\|)$ and let $(H,p)$ be a ${{\mathbb{Z}}}_2$-symmetric subframework of $G$.
1. If $(G,p)$ is symmetrically isostatic then, $$|E(H_0)| \leq (\dim X)|V(H_0)| - \dim {{\mathcal{T}}}_1(H,p),$$ and if $H_0$ is balanced in $G_0$ then, $$|E(H_0)| \leq (\dim X)|V(H_0)| - \dim {{\mathcal{T}}}(H_0,p).$$
2. If $(G,p)$ is anti-symmetrically isostatic then, $$|E(H_0)| \leq (\dim X)|V(H_0)| - \dim {{\mathcal{T}}}_2(H,p),$$ and if $H_0$ is balanced in $G_0$ then, $$|E(H_0)| \leq (\dim X)|V(H_0)| - \dim {{\mathcal{T}}}(H_0,p).$$
By Lemma \[CommDiagramLemma1\], if $(G,p)$ is symmetrically isostatic then $O_1(H,p)$ is row independent and, $$|E(H_0)| ={\operatorname{rank}}O_1(H,p)
= (\dim X)|V(H_0)| - \dim \ker O_1(H,p)
\leq (\dim X)|V(H_0)|-\dim {{\mathcal{T}}}_1(H,p).$$ If $H_0$ is balanced then for some choice of vertex orbit representatives each edge of $H_0$ has gain $1$. By the remark preceding the lemma, $(H_0,p)$ is a well-positioned framework in $(X,\|\cdot\|)$ and, by equation (\[OpFormula\]), $df_{H_0}(p) = O_1(H,p)$. Thus, $$|E(H_0)|={\operatorname{rank}}O_1(H,p)={\operatorname{rank}}df_{H_0}(p)
\leq (\dim X)|V(H_0)| - \dim {{\mathcal{T}}}(H_0,p).$$ This proves $(i)$ and the proof of $(ii)$ is similar.
Grid-like frameworks with reflectional symmetry {#Sect:Grid}
===============================================
In this section we consider bar-joint frameworks in $({{\mathbb{R}}}^2, \|\cdot\|_{{\mathcal{P}}})$ where the norm $\|\cdot\|_{{\mathcal{P}}}$ has the property that the closed unit ball ${{\mathcal{P}}}=\{x\in {{\mathbb{R}}}^2: \|x\|_{{\mathcal{P}}}\leq 1\}$ is a quadrilateral. (The $\ell^1$ and $\ell^\infty$ norms are familiar examples of such norms. In general, every absolutely convex quadrilateral is the closed unit ball for a unique norm on ${{\mathbb{R}}}^2$ defined by the Minkowski functional for the quadrilateral). The norm is expressed by the formula, $$\|x\|_{{\mathcal{P}}}= \max_{j=1,2}\, |\hat{F}_j\cdot x|, \,\,\,\,\,\,\, \forall x\in {{\mathbb{R}}}^2,$$ where ${{\mathcal{P}}}= \bigcap_{j=1,2} \,\{x\in{{\mathbb{R}}}^2: |x\cdot\hat{F}_j|\leq 1\}$. Note that the boundary of ${{\mathcal{P}}}$ consists of four facets $\pm F_1$, $\pm F_2$ and that for each $j=1,2$, $\hat{F}_j$ is the unique extreme point of the polar set of ${{\mathcal{P}}}$ for which $F_j=\{x\in {{\mathcal{P}}}:\hat{F}_j\cdot x =1\}$. Also note that each facet $F_j$ determines a linear functional, $$\varphi_{F_j}:X\to{{\mathbb{R}}},\quad x\mapsto \hat{F}_{j}\cdot x.$$
Monochrome subgraph decompositions {#Sect:framework}
----------------------------------
Let $(G,p)$ be a bar-joint framework in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$ and let $F$ be a facet of ${{\mathcal{P}}}$. An edge $vw\in E$ is said to have the induced [*framework colour*]{} $[F]$ if $p_v-p_w$ is contained in the cone of $F$ or $-F$. The subgraph of $G$ spanned by edges with framework colour $[F]$ is denoted by $G_{F}$ and referred to as an induced [*monochrome subgraph*]{} of $G$. Note that if $(G,p)$ is well-positioned then each edge $vw$ has exactly one framework colour $[F]$ and the linear functional $\varphi_{v,w}$ is given by either $\varphi_{F}$ or $\varphi_{-F}$. The following result was obtained (for $d$-dimensional frameworks) in [@kit-pow].
Let $(G,p)$ be a well-positioned bar-joint framework in $({{\mathbb{R}}}^2, \|\cdot\|_{{\mathcal{P}}})$. Then $(G,p)$ is isostatic if and only if the monochrome subgraphs $G_{F_1}$ and $G_{F_2}$ are both spanning trees in $G$.
We will now prove symmetric analogues of the above theorem for frameworks with reflectional symmetry. Let $(G,p)$ be ${{\mathbb{Z}}}_2$-symmetric with respect to $\theta:{{\mathbb{Z}}}_2\to {\operatorname{Aut}}(G)$ and $\tau:{{\mathbb{Z}}}_2\to {\operatorname{Isom}}(\mathbb{R}^2,\|\cdot\|_{{\mathcal{P}}})$ where $\tau(-1)$ is a reflection in the mirror $\ker\varphi_{F_1}$ along $\ker \varphi_{F_2}$. Then for each edge $e\in E$, both $e$ and $-e$ have the same induced framework colour and this will be referred to as the framework colour of the edge orbit $[e]$. Define $G_{F,0}$ to be the [*monochrome subgraph*]{} of the quotient graph $G_0$ spanned by edges $[e]$ with framework colour $[F]$.
In the following, the set of vertex orbit representatives for $G$ will be denoted by $\tilde{V}_0=\{{\tilde{v}}_1,\ldots, {\tilde{v}}_n\}$ and $\tilde{V}_1$ will denote the set $\{-{\tilde{v}}_1,\ldots, -{\tilde{v}}_n\}$.
\[thm:forced1\] Let $(G,p)$ be a well-positioned and $\mathbb{Z}_2$-symmetric bar-joint framework in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$ where ${{\mathcal{P}}}$ is a quadrilateral and $G\not=K_2$. Suppose $\theta$ acts freely on $V$ and $\tau(-1)$ is a reflection in the mirror $\ker\varphi_{F_1}$ along $\ker \varphi_{F_2}$. The following are equivalent.
1. $(G,p)$ is symmetrically isostatic.
2. $G_{F_1,0}$ is a spanning unbalanced map graph in $G_0$ and $G_{F_2,0}$ is a spanning tree in $G_0$.
$(i)\Rightarrow (ii)$ Suppose there exists a vertex $[v_0]\in V_0 \setminus V(G_{F_1,0})$. Choose a non-zero vector $x\in \ker\varphi_{F_2}$ and for all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } v={\tilde{v}}_0, \\
-x & \mbox{ if } v=-{\tilde{v}}_0, \\
0 & \mbox{ otherwise. } \\
\end{array}\right.$$ Then $u$ is a non-trivial symmetric infinitesimal flex for $(G,p)$. Similarly, if there exists a vertex $[v_0]\in V_0 \setminus V(G_{F_2,0})$ then choose a non-zero vector $x\in \ker\varphi_{F_1}$. For all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } [v]=[v_0], \\
0 & \mbox{ otherwise. } \\
\end{array}\right.$$ Again, $u$ is a non-trivial symmetric infinitesimal flex for $(G,p)$. In each case we obtained a contradiction and so $G_{F_2,0}$ and $G_{F_2,0}$ are both spanning subgraphs of $G_0$.
Suppose $G_{F_1,0}$ has a connected component $H_0$ which is a balanced subgraph of $G_0$. Then by Lemma \[switch\], by applying switching operations if necessary, we may assume each edge of $H_0$ has trivial gain. Thus, if $H$ is the covering graph for $H_0$, then there is no edge $vw\in E(H)$ with $v\in \tilde{V}_0$ and $w\in \tilde{V}_1$. Choose a non-zero vector $x\in \ker\varphi_{F_2}$ and for all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } [v]\in V(H_0) \mbox{ and } v\in \tilde{V}_0, \\
-x & \mbox{ if } [v]\in V(H_0) \mbox{ and } v\in \tilde{V}_1, \\
0 & \mbox{ otherwise. }
\end{array}\right.$$ Then $u$ is a non-trivial symmetric infinitesimal flex for $(G,p)$ which is a contradiction. Thus each connected component of $G_{F_1,0}$ is an unbalanced subgraph of $G_0$.
Suppose $G_{F_2,0}$ is not connected, and let $H_0$ be a connected component of $G_{F_2,0}$. Choose a non-zero vector $x\in \ker\varphi_{F_1}$ and for all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } [v]\in V(H_0), \\
0 & \mbox{ otherwise. } \\
\end{array}\right.$$ Again $u$ is a non-trivial symmetric infinitesimal flex for $(G,p)$ and this is a contradiction. Thus $G_{F_2,0}$ is a connected spanning subgraph of $G_0$.
By Lemma \[lem:dimmotions\], $\dim{{\mathcal{T}}}_1(G,p)={\operatorname{rank}}(I+\tau(-1))=1$. Thus by Lemmas \[Counts\] and \[lem:symcounts\], $|E_0|=2|V_0|-1$. Note that each connected component of $G_{F_1,0}$ must contain a cycle (since it is unbalanced) and so if $G_{F_1,0}$ has $n$ connected components, $H_1,H_2,\ldots, H_n$ say, then $|E(H_j)|\geq|V(H_j)|$ for each $j$ and, $$|E(G_{F_1,0})|=\sum_{j=1}^n |E(H_j)|\geq\sum_{j=1}^n|V(H_j)|= |V_0|.$$ Since $G_{F_2,0}$ is connected it must contain a spanning tree and so $|E(G_{F_2,0})|\geq |V_0|-1$. It follows that $|E(G_{F_1,0})|= |V_0|$, $|E(G_{F_2,0})|=|V_0|-1$ and $|E(H_j)|=|V(H_j)|$ for each $j$. Thus $G_{F_1,0}$ is an unbalanced spanning map graph and $G_{F_2,0}$ is a spanning tree in $G_0$.
$(ii)\Rightarrow (i)$ Suppose $(ii)$ holds and let $u$ be a symmetric infinitesimal flex of $(G,p)$. Let $v\in V$ and note that since $G_{F_1,0}$ has a unique unbalanced cycle, the covering graph for $H_0$ is a connected subgraph of $G_{F_1}$ which contains both $v$ and $-v$. In particular, there is a path $vv_1,v_1v_2,\ldots,v_n(-v)$ in $G_{F_1}$ from $v$ to $-v$ and so, $$u_{v}-u_{-v} =
(u_v-u_{v_1})+(u_{v_1}-u_{v_2})+\cdots+(u_{v_n}-u_{-v})
\in\ker\varphi_{F_1}.$$ Also note that $u_v-u_{-v}=(I-\tau(-1))u_{v}=2Pu_v\in \ker\varphi_{F_2}$. Thus $u_{v}=u_{-v}$ for all $v\in V$. Since $u_{-v}=\tau(-1)u_{v}$ it also follows that $u_{v}\in\ker\varphi_{F_1}$ for all $v\in V$. Let $e=vw\in E$. It is clear that $u_{v}-u_{w}\in \ker \varphi_{F_1}$. Since $G_{F_2,0}$ is a spanning tree in $G_0$ there exists a path in $G_{F_2,0}$ from $[v]$ to $[w]$ with gain $\gamma'$ say. Thus there exists a path in $G_{F_2}$ from $v$ to $\gamma' w$ and so $u_v-u_w=u_{v}-u_{\gamma' w}\in \ker \varphi_{F_2}$. We conclude that $u_{v}=u_{w}$ for all $vw\in E$ and so $u$ is a trivial infinitesimal flex of $(G,p)$. To see that $(G,p)$ is symmetrically isostatic note that $|E_0|= 2|V_0|-1$ and apply Lemma \[Counts\].
The following theorem characterises anti-symmetric isostatic frameworks and is a counterpart to the previous theorem. While the statement and proof are similar there are some key differences. In particular, the roles of the monochrome subgraphs are reversed.
\[thm:antisym1\] Let $(G,p)$ be a well-positioned and $\mathbb{Z}_2$-symmetric bar-joint framework in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$ where ${{\mathcal{P}}}$ is a quadrilateral. Suppose $\theta$ acts freely on $V$ and $\tau(-1)$ is a reflection in the mirror $\ker\varphi_{F_1}$ along $\ker \varphi_{F_2}$. The following are equivalent.
1. $(G,p)$ is anti-symmetrically isostatic.
2. $G_{F_1,0}$ is a spanning tree in $G_0$ and $G_{F_2,0}$ is a spanning unbalanced map graph in $G_0$.
$(i)\Rightarrow (ii)$ Suppose there exists a vertex $[v_0]\in V_0 \setminus V(G_{F_1,0})$. Choose a non-zero vector $x\in \ker\varphi_{F_2}$. For all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } [v]=[v_0],\\
0 & \mbox{ otherwise. } \\
\end{array}\right.$$ Similarly, suppose there exists a vertex $[v_0]\in V_0 \setminus V(G_{F_2,0})$. Choose a non-zero vector $x\in \ker\varphi_{F_1}$ and for all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } v={\tilde{v}}_0, \\
-x & \mbox{ if } v=-{\tilde{v}}_0, \\
0 & \mbox{ otherwise. } \\
\end{array}\right.$$ In each case $u$ is a non-trivial anti-symmetric infinitesimal flex for $(G,p)$.
Suppose $G_{F_2,0}$ has a connected component $H_0$ which is a balanced subgraph of $G_0$. Then, using some switching operations if necessary, we may assume $H_0$ has trivial gain. Choose a non-zero vector $x\in \ker\varphi_{F_1}$ and for all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } [v]\in V(H_0) \mbox{ and } v\in \tilde{V}_0,\\
-x & \mbox{ if } [v]\in V(H_0) \mbox{ and } v\in \tilde{V}_1, \\
0 & \mbox{ otherwise. }
\end{array}\right.$$ Similarly, suppose $G_{F_1,0}$ is not connected, and let $H_0$ be a connected component of $G_{F_1,0}$. Choose a non-zero vector $x\in \ker\varphi_{F_2}$ and for all $v\in V$ define, $$u_{v} = \left\{ \begin{array}{ll}
x & \mbox{ if } [v]\in V(H_0) ,\\
0 & \mbox{ otherwise. }
\end{array}\right.$$ Again, in each case $u$ is a non-trivial anti-symmetric infinitesimal flex for $(G,p)$. The remainder of the proof is similar to Theorem \[thm:forced1\].
$(ii)\Rightarrow (i)$ Apply an argument as in Theorem \[thm:forced1\] but with the roles of $G_{F_1,0}$ and $G_{F_2,0}$ reversed.
The previous two theorems can be combined to obtain the following characterisation of general infinitesimal rigidity, again expressed in terms of monochrome subgraph decompositions in the quotient graph.
\[cor\] Let $(G,p)$ be a well-positioned and $\mathbb{Z}_2$-symmetric bar-joint framework in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$ where ${{\mathcal{P}}}$ is a quadrilateral. Suppose $\theta$ acts freely on $V$ and $\tau(-1)$ is a reflection in the mirror $\ker\varphi_{F_1}$ along $\ker \varphi_{F_2}$. The following are equivalent.
1. $(G,p)$ is infinitesimally rigid.
2. The monochrome subgraphs of $G_0$ both contain connected spanning unbalanced map graphs.
$(i)\Rightarrow (ii)$ If $(G,p)$ is infinitesimally rigid then it is both symmetrically and anti-symmetrically infinitesimally rigid. By removing edge orbits from $G$ we arrive at a ${{\mathbb{Z}}}_2$-symmetric spanning subgraph $A$ such that $(A,p)$ is symmetrically isostatic. By Theorem \[thm:forced1\], $A_{F_1,0}$ is a spanning unbalanced map graph in $G_0$ and $A_{F_2,0}$ is a spanning tree. Similarly, by removing edge orbits from $G$ we arrive at a ${{\mathbb{Z}}}_2$-symmetric spanning subgraph $B$ such that $(B,p)$ is anti-symmetrically isostatic. By Theorem \[thm:antisym1\], $B_{F_1,0}$ is a spanning tree and $B_{F_2,0}$ is a spanning unbalanced map graph in $G_0$. Let $C_1,\ldots,C_n$ be the connected components of $A_{F_1,0}$. For $k=2,\ldots, n$, let $T_k$ be a spanning tree for $C_k$. Note that there exist edges $[e_1],\ldots,[e_{n-1}]$ in the spanning tree $B_{F_1,0}$ such that $C_1\cup T_2\cup\cdots\cup T_n \cup\{[e_1],\ldots,[e_{n-1}]\}$ is a connected spanning unbalanced map graph in $G_{F_1,0}$. A similar argument shows that $G_{F_2,0}$ contains a connected spanning unbalanced map graph.
$(ii)\Rightarrow (i)$ Suppose $(ii)$ holds. Let $H_{F_1,0}$ and $H_{F_2,0}$ be connected spanning unbalanced map graphs in $G_0$. Note that $H_{F_2,0}$ contains a spanning tree for $G_0$ and so, by Theorem \[thm:forced1\], $(G,p)$ is symmetrically infinitesimally rigid. Similarly, $H_{F_1,0}$ contains a spanning tree for $G_0$ and so, by Theorem \[thm:antisym1\], $(G,p)$ is anti-symmetrically infinitesimally rigid. Hence $(G,p)$ is infinitesimally rigid.
Existence of rigid placements with reflectional symmetry {#Sect:graph}
--------------------------------------------------------
In this section, necessary and sufficient conditions are obtained for a $\mathbb{Z}_2$-symmetric graph to have a well-positioned symmetric or anti-symmetric infinitesimally rigid realisation as a grid-like reflection framework. A signed quotient graph $(G_0,\psi)$ is *$(2,2,1)$-gain-sparse* if it satisfies
- $|F|\leq 2|V(F)|-2$ for every balanced $F\subseteq E_0$;
- $|F|\leq 2|V(F)|-1$ for every $F\subseteq E_0$.
If, in addition, $|E_0|= 2|V_0|-1$, then $(G_0,\psi)$ is said to be *$(2,2,1)$-gain-tight*.
We will now describe a number of recursive operations on a $(2,2,1)$-gain tight signed quotient graph $(G_0,\psi)$. See also [@jkt; @anbs; @schtan] for a description of some of these moves.
A *Henneberg 1 move* is an addition of a new vertex $[v]$ and two new edges $[e_1]$ and $[e_2]$ to $(G_0,\psi)$, where $[e_1]$ and $[e_2]$ are incident with $[v]$ and are not both loops at $[v]$. If $[e_1]$ and $[e_2]$ are parallel edges, then the gain labels are assigned so that $\psi_{[e_1]}\neq \psi_{[e_2]}$.
If $[e_1]$ and $[e_2]$ are non-parallel and neither is a loop then the move is called $H1a$. If these edges are parallel the move is called $H1b$. If one of the edges is a loop, then the move is called $H1c$. See also Figure \[fig:inductive\].
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A *Henneberg 2 move* deletes an edge $[e]$ of $(G_0,\psi)$ and adds a new vertex $[v]$ of degree $3$ to $(G_0,\psi)$ as follows. The edge $[e]$ is subdivided into two new edges $[e_1]$ and $[e_2]$ (both incident with $[v]$) so that the gains of the new edges satisfy $\psi_{[e_1]}\cdot \psi_{[e_2]}=\psi_{[e]}$. Finally, the third new edge, $[e_3]$, joins $[v]$ to a vertex $[z]$ of $(G_0,\psi)$ so that every 2-cycle $[e_i][e_j]$, if it exists, is unbalanced.
Suppose first that the edge $[e]$ is not a loop. If none of the edges $[e_i]$ are parallel, then the move is called H2a. If two of the edges $[e_i]$ are parallel (i.e., $[z]$ is an end-vertex of $[e]$), then the move is called H2b. If the edge $[e]$ is a loop, then the move is called H2c. See Figure \[fig:inductiveH2\].
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A *vertex-to-$K_4$* move removes a vertex $[v]$ (of arbitrary degree) and all the edges incident with $[v]$, and adds in a copy of $K_4$ with only trivial gains. Each removed edge $[x][v]$ is replaced by an edge $[x][y]$ for some $[y]$ in the new $K_4$, where the gain is preserved. If the deleted vertex $[v]$ is incident to a loop, then this loop is replaced by an edge $[y][z]$ with gain $-1$, where $[y]$ and $[z]$ are two (not necessarily distinct) vertices of the new $K_4$.
See Figure \[fig:inductiveH3\](a).
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(p1)–(p2); (p3)–(p2); (p3)–(p4); (p1)–(p4); (p1)–(p3); (p4)–(p2); (p1) edge \[bend left=30\] (p4);
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(p1) edge \[bend left=30\] (p2); (p1) edge \[loop below, >=stealth,shorten >=1pt,looseness=20\] (p1); (1.5,0.1)–(2,0.1); (a) at (1.7,-0.42) [(b)]{};
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An *edge-to-$K_3$* move (also called *vertex splitting* [@NOP; @W2]) on a vertex $[v]$ which is incident to the edge $[v][u]$ with trivial gain and the edges $[v][u_i]$, $i=1,\ldots, t$ (which may include the edges $[v][u]$ and $[v][v]$ with gain $-1$), removes $[v]$ and its incident edges, and adds two new vertices $[v_0]$ and $[v_1]$ as well as the edges $[v_0][v_1]$, $[v_0][u]$ and $[v_1][u]$ with trivial gains. Finally, each edge $[v][u_i]$ (with $[u_i]\neq [v]$), $i=1,\ldots, t$, is replaced by the edge $[v_0][u_i]$ or the edge $[v_1][u_i]$ so that the gain of the new edge $[v_j][u_i]$, $j\in \{0,1\}$, is the same as the gain of the deleted edge $[v][u_i]$. The loop at $[v]$ (if it exists) is replaced by a loop either at $[v_0]$ or $[v_1]$ with gain $-1$.
See Figure \[fig:inductiveH3\](b).
For each of the above moves, an inverse move performed on a $(2,2,1)$-gain-tight signed quotient graph is called *admissible* if it results in another $(2,2,1)$-gain-tight signed quotient graph.
\[Symmetrically isostatic graphs\] \[thm:forced2\] Let $\|\cdot\|_{{\mathcal{P}}}$ be a norm on ${{\mathbb{R}}}^2$ for which ${{\mathcal{P}}}$ is a quadrilateral, and let $G$ be a $\mathbb{Z}_2$-symmetric graph where the action $\theta$ is free on the vertex set of $G$. Let $(G_0,\psi)$ be the signed quotient graph of $G$. The following are equivalent.
1. There exists a representation $\tau:\mathbb{Z}_2\to {\operatorname{Isom}}(\mathbb{R}^2)$, where $\tau(-1)$ is a reflection in the mirror $\ker\varphi_{F_1}$ along $\ker\varphi_{F_2}$, and a realisation $p$ such that the bar-joint framework $(G,p)$ is well-positioned, $\mathbb{Z}_2$-symmetric and symmetrically isostatic in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$;
2. $(G_0,\psi)$ is $(2,2,1)$-gain tight;
3. $(G_0,\psi)$ can be constructed from a single unbalanced loop by a sequence of H1a,b,c-moves, H2a,b,c-moves, vertex-to-$K_4$ moves, and vertex splitting moves.
$(i) \Rightarrow (ii)$. Suppose $(G,p)$ is a well-positioned symmetrically isostatic framework in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$. Then we clearly have $|E_0|= 2|V_0|-1$ since, by Lemma \[lem:dimmotions\], the space of symmetric infinitesimal trivial flexes is of dimension 1 (spanned by the infinitesimal translation along the mirror). Similarly, by Lemma \[lem:symcounts\], there does not exist an edge subset $F$ of $E_0$ with $|F|>2|V(F)|-1$, for otherwise the symmetric orbit matrix of $(G_0,\psi)$ would have a row dependence. So it remains to show that we have $|F|\leq 2|V(F)|-2$ for every balanced edge subset $F$. However, this also follows immediately from Lemma \[lem:symcounts\].
$(ii)\Rightarrow(iii)$. Suppose $(G_0,\psi)$ is $(2,2,1)$-gain tight. If $(G_0,\psi)$ is a single unbalanced loop, then we are done. So suppose $(G_0,\psi)$ has more than two vertices. Then $(G_0,\psi)$ has a vertex $[v]$ of degree $2$ or $3$. If there exists a vertex $[v]$ which is incident to two edges (one of which may be a loop), then there clearly exists an admissible inverse H1a,b- or c-move. If there is no such vertex, then there is a vertex $[v]$ which is incident to three non-loop edges, and $[v]$ has either two or three neighbours. If $[v]$ has two neighbours $[a]$ and $[b]$, and $[v],[a],[b]$ induce a graph with $5$ edges (i.e., a $2K_3-[e]$), then there exists an admissible inverse H2c-move. Otherwise, we may use the argument in [@anbs] for $(2,2,1)$-gain-tight signed graphs to show that there exists an admissible inverse H2b-move. If $[v]$ has three distinct neighbours, then it was again shown in [@anbs] that there exists an admissible inverse H2a-move for $[v]$, unless $[v]$ and its three neighbours $[a],[b]$ and $[c]$ induce a $K_4$ in $(G_0,\psi)$ with gain $1$ on every edge (plus possibly an additional edge with gain $-1$).
In this case there is an admissible inverse vertex-to-$K_4$ move, unless there exists a vertex $[x] \notin V(K_4)$ such that $[x][a]$ and $[x][b]$ are edges in $(G_0,\psi)$ which have the same gain. Let $A_0$ denote the $K_4$ and let $A_1$ be the graph consisting of $A_0$ together with the vertex $[x]$ and the edges $[x][a]$ and $[x][b]$. By switching $[x]$, we may assume that the gains of $[x][a]$ and $[x][b]$ are both $1$. Note that $[x][a]$ and $[x][b]$ cannot both have a parallel edge, and so, without loss of generality, we assume that the edge $[x][a]$ with gain $-1$ is not present.
If there exists a vertex $[y]\notin V(A_1)$ and edges $[y][a]$ and $[y][x]$ with the same gain then let $A_2$ denote the union of $A_1$ with $[y]$ and these two edges (see Fig. \[fig:pf\]). By switching $[y]$ we may assume that all edges in $A_2$ have gain $1$. Again, note that $[y][a]$ and $[y][x]$ cannot both have a parallel edge, and so, without loss of generality, we assume that the edge $[y][a]$ with gain $-1$ is not present. If there exists a vertex $[z]\notin V(A_2)$ and edges $[z][y]$ and $[z][a]$ with the same gain then let $A_3$ denote the union of $A_2$ with $[z]$ and these two edges. Continuing this process we obtain an increasing sequence of subgraphs $A_1,A_2,A_3,\ldots$ of $G_0$ each of which is balanced and satisfies $|E(A_i)|=2|V(A_i)|-2$. This sequence must terminate after finitely many iterations at a subgraph $A_t$ of $G_0$. Let $[w]$ be the vertex in $A_t\backslash A_{t-1}$ and suppose $[w]$ is incident to the vertices $[i]$ and $[j]$ in $A_{t-1}$. By switching $[w]$ we may assume that all edges in $A_t$ have gain $1$. By construction, one of the edges incident to $[w]$ in $A_t$, $[w][i]$ say, does not have a parallel edge and has the property that there is no vertex $[k]\notin V(A_t)$ which is adjacent to both $[w]$ and $[i]$ such that the edges $[k][w]$ and $[k][i]$ both have the same gain.
Clearly, there cannot exist a subgraph $H_0$ of $(G_0,\psi)$ with $|E(H_0)|=2|V(H_0)|-1$ which contains $[w]$ and $[i]$, but not $[j]$, for otherwise $A_{t}\cup H_0$ violates the $(2,2,1)$-gain-sparsity counts. To see this note that $|E(A_{t-1}\cup H_0)|=2|V(A_{t-1}\cup H_0)|-1$ and $A_{t}\cup H_0$ is obtained by adjoining the edge $[w][j]$ to $A_{t-1}\cup H_0$. Similarly, there cannot exist a balanced subgraph $H_0$ of $(G_0,\psi)$ with $|E(H_0)|=2|V(H_0)|-2$ which contains $[w]$ and $[i]$, but not $[j]$. To see this, note that $A_t\cap H_0$ must be connected since otherwise $A_t\cup H_0$ violates the $(2,2,1)$-gain-sparsity counts. By [@jkt Lemma 2.5], $A_t\cup H_0$ is balanced and so, by Lemma \[switch\], we may assume every edge in $A_t\cup H_0$ has gain $1$. Note that $A_{t-1}$ and $H_0$ have a non-empty (balanced) intersection. Therefore, $|E(A_{t-1}\cup H_0)|=2|V(A_{t-1}\cup H_0)|-2$. However, if we add the edge $[w][j]$ to $A_{t-1}\cup H_0$, then this creates a balanced subgraph of $G_0$ which violates the $(2,2,1)$-gain-sparsity counts. It follows that an inverse edge-to-$K_3$ move on the edge $[w][i]$ is admissible.
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\(a) at (3.44,0.75) [$[x]$]{}; (a) at (2.62,1.7) [$[y]$]{};
(p1)–(p2); (p2)–(p3); (p3)–(p4); (p1)–(p4); (p3)–(p1); (p2)–(p4); (p2)–(p5); (p3)–(p5);
(p6)–(p5); (p6)–(p3);
$(iii)\Rightarrow (i)$. We employ induction on the number of vertices of $G_0$. If $G_0$ is a single unbalanced loop with vertex $[v]$, choose $p_v\notin \ker \varphi_{F_1}$ and set $p_{-v}=\tau(-1)p_v$. Then $(G,p)$ is well-positioned and ${{\mathbb{Z}}}_2$-symmetric and so the statement holds by Lemma \[K2Lemma\].
Now, let $n\geq 2$, and suppose $(i)$ holds for all signed quotient graphs satisfying $(iii)$ with at most $n-1$ vertices. Let $(G_0,\psi)$ have $n$ vertices, and let $(G'_0,\psi')$ be the penultimate graph in the construction sequence of $(G_0,\psi)$. If $(G'_0,\psi')$ is a single unbalanced loop, then $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by a H1b-, H1c-, or vertex-to-$K_4$ move. The loop of $G'_0$ belongs to the induced monochrome subgraph $G'_{F_1,0}$ of $G'_0$, and for each of the three moves, it is easy to see how to place the new vertex (vertices) so that the induced monochrome subgraphs $G_{F_1,0}$ and $G_{F_2,0}$ of $G_0$ have the property that $G_{F_1,0}$ is a spanning unbalanced map graph and $G_{F_2,0}$ is a spanning tree of $G_0$ (see also the discussion below). The result then follows from Theorem \[thm:forced1\]. Thus, we may assume that $G'_0$ has at least two vertices.
In this case, it follows from the induction hypothesis and Theorem \[thm:forced1\] that there exists a well-positioned $\mathbb{Z}_2$-symmetric realisation $p'$ of the covering graph $G'$ of $(G'_0,\psi')$ in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$ (where the reflection $\tau(-1)$ is in the mirror $\ker\varphi_{F_1}$) so that the induced monochrome subgraphs $G'_{F_1,0}$ and $G'_{F_2,0}$ of $G'_0$ are both spanning, $G'_{F_1,0}$ is an unbalanced map graph, and $G'_{F_2,0}$ is a tree. By Theorem \[thm:forced1\] it now suffices to show that the vertex (or vertices) of $G\setminus G'$ can be placed in such a way that the corresponding framework $(G,p)$ is $\mathbb{Z}_2$-symmetric and well-positioned, the induced monochrome subgraphs $G_{F_1,0}$ and $G_{F_2,0}$ are both spanning in $G_0$, $G_{F_1,0}$ is an unbalanced map graph, and $G_{F_2,0}$ is a tree.
Choose points $x_1$ and $x_2$ in the relative interiors of $F_1$ and $F_2$ respectively. Suppose first that $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by a H1a-move, where $[v]\in G_0\setminus G'_0$ is adjacent to the vertices $[v_1]$ and $[v_2]$ of $G'_0$ with respective gains $\gamma_1$ and $\gamma_2$. Set $p_w=p_w'$ for all vertices $w$ in $G$ with $[w]\not=[v]$. Let $a\in {{\mathbb{R}}}^2$ be the point of intersection of the lines $L_1=\{\tau(\gamma_1)p_{{\tilde{v}}_1}+tx_1:t\in {{\mathbb{R}}}\}$ and $L_2=\{\tau(\gamma_2)p_{{\tilde{v}}_2}+tx_2:t\in {{\mathbb{R}}}\}$ and let $B(a,r)$ be an open ball with centre $a$ and radius $r>0$. Choose $p_{{\tilde{v}}}$ to be any point in $B(a,r)$ which is distinct from $\{p_w:w\in V(G')\}$ and which is not fixed by $\tau(-1)$. Set $p_{-{\tilde{v}}}=\tau(-1)p_{{\tilde{v}}}$. Then $(G,p)$ is a ${{\mathbb{Z}}}_2$-symmetric bar-joint framework and, by applying a small perturbation to $p_{\tilde{v}}$ if necessary, we may assume that $(G,p)$ is well-positioned. If $r$ is sufficiently small then the induced framework colours for $[v][v_1]$ and $[v][v_2]$ are $[F_1]$ and $[F_2]$ respectively. Thus, the induced monochrome subgraphs of $(G_0,\psi)$ are $G_{F_1,0}=G'_{F_1,0}\cup \{[v][v_1]\}$ and $G_{F_2,0}=G'_{F_2,0}\cup \{[v][v_2]\}$. Clearly, $G_{F_1,0}$ is a spanning unbalanced map graph and $G_{F_2,0}$ is a spanning tree of $G_0$. For an illustration of the monochrome subgraphs of the signed quotient graph see Fig. \[fig:inductive\](a). The edges of $G_{F_1,0}$ are shown in gray and the edges of $G_{F_2,0}$ are shown in black.
If $(G_0,\psi)$ is obtained from $(G'_0,\psi)$ by a H1b-move, then the proof is completely analogous to the proof above. (See Fig. \[fig:inductive\](b)).
Suppose $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by a H1c-move, where $[v]\in G_0\setminus G'_0$ is incident to the unbalanced loop $[e]$ and adjacent to the vertex $[w]$ of $(G'_0,\psi')$ with gain $\gamma$. If we choose $p_{{\tilde{v}}}$ to be any point on the line $L=\{\tau(\gamma)p_{{\tilde{w}}}+tx_2:t\in {{\mathbb{R}}}\}$, then the induced framework colouring for $[v][w]$ is $[F_2]$. Moreover, as we have seen before, the induced framework colouring for the loop $[e]$ is $[F_1]$. It follows that we may place ${\tilde{v}}$ and $-{\tilde{v}}$ in such a way that $(G,p)$ is well-positioned and ${{\mathbb{Z}}}_2$-symmetric, and the induced monochrome subgraphs of $G_0$ are $G_{F_1,0}=G'_{F_1,0}\cup \{[e]\}$ and $G_{F_2,0}=G'_{F_2,0}\cup \{[v][w]\}$. Clearly, $G_{F_1,0}$ is an unbalanced spanning map graph and $G_{F_2,0}$ is a spanning tree of $(G_0,\psi)$. (See Fig. \[fig:inductive\](c)).
Next, we suppose that $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by a H2a-move where $[v]\in G_0\setminus G'_0$ subdivides the edge $[e]$ into the edges $[e_1]$ and $[e_2]$ with respective gains $\gamma_1$ and $\gamma_2$, and $[v]$ is also incident to the edge $[e_3]$ with end-vertex $[z]$ and gain $\gamma_3$. Without loss of generality we may assume that $[e]\in G'_{F_1,0}$. Let $a\in {{\mathbb{R}}}^2$ be the point of intersection of the line $L_1$ which passes through the points $\tau(\gamma_1)p_{{\tilde{v}}_1}$ and $\tau(\gamma_2)p_{{\tilde{v}}_2}$ with $L_2=\{\tau(\gamma_3)p_{{\tilde{z}}}+tx_2:t\in{{\mathbb{R}}}\}$. Let $B(a,r)$ be the open ball with centre $a$ and radius $r>0$ and choose $p_{{\tilde{v}}}$ to be a point in $B(a,r)$ which is distinct from $\{p_w:w\in G'\}$ and which is not fixed by $\tau(-1)$. Set $p_{-{\tilde{v}}}=\tau(-1)p_{{\tilde{v}}}$. As above, $(G,p)$ is ${{\mathbb{Z}}}_2$-symmetric and we may assume it is well-positioned. If $r$ is sufficiently small then $[e_1]$ and $[e_2]$ have induced framework colour $[F_1]$ and $[e_3]$ has framework colour $[F_2]$. The induced monochrome subgraphs of $G_0$ are $G_{F_1,0}=(G'_{F_1,0}\backslash\{[e]\})\cup \{[e_1],[e_2]\}$ and $G_{F_2,0}=G'_{F_2,0}\cup \{[e_3]\}$. Clearly, $G_{F_1,0}$ is a spanning unbalanced map graph and $G_{F_2,0}$ is a spanning tree of $G_0$. (See Fig. \[fig:inductiveH2\](a)).
The cases where $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by a H2b- or a H2c-move can be proved completely analogously to the case above for the H2a-move. Note, however, that for the H2c-move, the edges $[e_1]$ and $[e_2]$ are forced to be in the subgraph $G_{F_1,0}$. (See Fig. \[fig:inductiveH2\](b),(c)).
Next, we suppose that $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by a vertex-to-$K_4$-move, where the vertex $[v]$ of $G'_0$ (which may be incident to an unbalanced loop $[e]$) is replaced by a copy of $K_4$ with a trivial gain labelling (and $[e]$ is replaced by the edge $[f]$). It was shown in [@kit-pow Ex. 4.5] that $K_4$ has a well-positioned and isostatic placement in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$. Moreover, we may scale this realisation so that all of the vertices of the $K_4$ lie in a ball of arbitrarily small radius. For any such realisation, the induced monochrome subgraphs of $K_4$ are both paths of length $3$. Let $B(p_{{\tilde{v}}},r)$ be the open ball with centre $p_{{\tilde{v}}}$ and radius $r>0$. Choose a placement of the representative vertices of the new $K_4$ to lie within $B(p_{{\tilde{v}}},r)$ such that the vertices are distinct from $\{p_w:w\in V(G')\backslash\{{\tilde{v}},-{\tilde{v}}\}\}$, none of the vertex placements are fixed by $\tau(-1)$ and the resulting placement of the new $K_4$ is isostatic. If $r$ is sufficiently small then the edge $[f]$ (if present) has the induced framework colour $[F_1]$. It can be assumed that the corresponding ${{\mathbb{Z}}}_2$-symmetric placement of $G$ is well-positioned. Moreover, the induced monochrome subgraphs $G_{F_1,0}$ and $G_{F_2,0}$ of $G_0$ clearly have the desired properties. (See Fig. \[fig:inductiveH3\](a)).
Finally, we suppose that $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by an edge-to-$K_3$-move, where the vertex $[v]$ of $G'_0$ (which is replaced by the vertices $[v_0]$ and $[v_1]$) is incident to the edge $[v][u]$ with trivial gain and the edges $[v][u_i]$, $i=1,\ldots, t$, in $G'_0$. Without loss of generality we may assume that $[v][u]\in G'_{F_1,0}$. If we choose $p_{{\tilde{v}}_0}=p_{{\tilde{v}}}$ and $p_{{\tilde{v}}_1}$ to be a point on the line $L=\{p_{{\tilde{v}}}+tx_2:t\in {{\mathbb{R}}}\}$ which is sufficiently close to $p_{{\tilde{v}}}$, then the induced framework colour for $[v_0][v_1]$ is $[F_2]$ and the induced framework colour for $[v_0][u]$ and $[v_1][u]$ is $[F_1]$. (Again we may assume the framework is well-positioned). Moreover, all other edges of $G'_0$ which have been replaced by new edges in $G_0$ clearly retain their induced framework colouring if $p_{{\tilde{v}}_1}$ is chosen sufficiently close to $p_{{\tilde{v}}}$. It is now easy to see that for such a placement of ${\tilde{v}}_0$ and ${\tilde{v}}_1$, $(G,p)$ is ${{\mathbb{Z}}}_2$-symmetric and for the induced monochrome subgraphs $G_{F_1,0}$ and $G_{F_2,0}$ of $G_0$ we have that $G_{F_1,0}$ is a spanning unbalanced map graph and $G_{F_2,0}$ is a spanning tree of $(G_0,\psi)$. (See Fig. \[fig:inductiveH3\](b)). This completes the proof.
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\[ex:2k3minusedge\] The smallest signed quotient graph $(G_0,\psi)$ whose covering graph $G$ can be realised as a $\mathbb{Z}_2$-symmetric framework in $(\mathbb{R}^2,\|\cdot\|_{{\mathcal{P}}})$ which is anti-symmetrically isostatic is the graph $2K_3-{\tilde{e}}$ shown in Figure \[fig:antisym\] (b,d). Figure \[fig:antisym\] (c) illustrates such a realisation $(G,p)$ in $(\mathbb{R}^2,\|\cdot\|_\infty)$. To obtain a realisation $(G,\tilde{p})$ in $(\mathbb{R}^2,\|\cdot\|_{{\mathcal{P}}})$ construct a linear isometry $T:(\mathbb{R}^2,\|\cdot\|_\infty) \to
(\mathbb{R}^2,\|\cdot\|_{{\mathcal{P}}})$ and set $\tilde{p}_v=T(p_v)$ for each $v\in V$.
A *$2K_3-[e]$ edge joining* move joins a signed quotient graph $2K_3-[e]$ to $(G_0,\psi)$ via one new edge of arbitrary gain, where $2K_3-[e]$ consists of $3$ vertices and $5$ edges.
\[Anti-symmetrically isostatic graphs\] \[thm:anti2\] Let $\|\cdot\|_{{\mathcal{P}}}$ be a norm on ${{\mathbb{R}}}^2$ for which ${{\mathcal{P}}}$ is a quadrilateral, and let $G$ be a $\mathbb{Z}_2$-symmetric graph with respect to the action $\theta$ which is free on the vertex set of $G$. Let $(G_0,\psi)$ be the signed quotient graph of $G$. The following are equivalent.
1. There exists a representation $\tau:\mathbb{Z}_2\to {\operatorname{Isom}}(\mathbb{R}^2)$, where $\tau(-1)$ is a reflection in the mirror $\ker\varphi_{F_1}$ along $\ker\varphi_{F_2}$, and a realisation $p$ such that the bar-joint framework $(G,p)$ is well-positioned, $\mathbb{Z}_2$-symmetric and anti-symmetrically isostatic in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$;
2. $(G_0,\psi)$ has no loops and is $(2,2,1)$-gain tight;
3. $(G_0,\psi)$ can be constructed from $2K_3-[e]$ by a sequence of H1a,b-moves, H2a,b-moves, vertex-to-$K_4$ moves, vertex splitting moves and $2K_3-[e]$ edge joining moves.
$(i) \Rightarrow (ii)$. Suppose $(G,p)$ is a well-positioned anti-symmetrically isostatic framework in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$. Then, by Lemma \[AntiSymFixedEdges\], $(G_0,\psi)$ cannot contain a loop. The rest of the proof is completely analogous to the proof of Theorem \[thm:forced2\] ((i) $\Rightarrow$ (ii)), since the space of anti-symmetric infinitesimal trivial flexes is also of dimension 1, by Lemma \[lem:dimmotions\].
$(ii) \Rightarrow (iii)$. Suppose $(G_0,\psi)$ is $(2,2,1)$-gain tight with no loops. If $(G_0,\psi)$ is a $2K_3-[e]$, then we are done. So suppose $(G_0,\psi)$ has more than three vertices. Then $(G_0,\psi)$ has a vertex $[v]$ of degree $2$ or $3$. It was shown in [@anbs] that there exists an admissible inverse Henneberg 1a,b- or 2a,b-move for $[v]$, unless $[v]$ either has three distinct neighbours $[a], [b]$ and $[c]$ in $(G_0,\psi)$ and $[v], [a], [b], [c]$ induce a $K_4$ with gain $1$ on every edge (plus possibly an additional edge with gain $-1$) or $[v]$ has two distinct neighbors $[a]$ and $[b]$, and $[v], [a], [b]$ induce a $2K_3-[e]$.
In the first case, there is an admissible inverse vertex-to-$K_4$ move or an admissible inverse vertex splitting move, as shown in the proof of Theorem \[thm:forced2\] ($(ii) \Rightarrow (iii)$). Thus, we may assume that every vertex of degree $3$ is in a copy of $2K_3-[e]$. But now we may use a similar argument as in the proof for the characterisation of $(2,2,1)$-gain-tight signed quotient graphs given in [@anbs] (see also [@nixowen Lemma 4.10]) to show that at least one of the copies of $2K_3-[e]$ has the property that there is exactly one edge which joins a vertex $[x]\notin 2K_3-[e]$ with a vertex in $2K_3-[e]$. For a signed quotient graph $(H,\phi)$ with vertex set $V(H)$ and edge set $E(H)$, we define $f(H)=2|V(H)|-|E(H)|$. Let $Y=\{Y_1,\ldots, Y_k\}$ be the copies of $2K_3-[e]$ in $(G_0,\psi)$. Then the $Y_i$ are pairwise disjoint and satisfy $f(Y_i)=1$ for all $i$. Let $W_0$ and $F_0$ be the sets of vertices and edges of $(G_0,\psi)$ which do not belong to any of the $Y_i$. Then we have $f(G_0)=\sum_{i=1}^k f(Y_i)+2|W_0|-|F_0|$, and since $f(G_0)=1$, $|F_0|=2|V_0|+k-1$. Every vertex in $W_0$ is of degree at least $4$. So if every $Y_i$ is incident to at least two edges in $F_0$, then there are at least $4|W_0|+2k$ edge-vertex incidences for the edges in $F_0$. But then we have $|F_0|\geq 2|W_0|+k$, a contradiction. If there exists a $Y_i$ with the property that none of the vertices of $Y_i$ are incident with an edge in $F_0$, then $G_0=Y_i$, contradicting our assumption that $G_0$ has more than $3$ vertices. It follows that there exists an inverse $2K_3-[e]$ edge joining move.
$(iii) \Rightarrow (i)$. We employ induction on the number of vertices. For the signed graph $2K_3-[e]$, the statement follows from Example \[ex:2k3minusedge\].
Now, let $n\geq 4$, and suppose (i) holds for all signed quotient graphs satisfying (iii) with at most $n-1$ vertices. Let $(G_0,\psi)$ have $n$ vertices, and suppose first that the last move in the construction sequence of $(G_0,\psi)$ is not a $2K_3-[e]$ edge joining move. Then we let $(G'_0,\psi')$ be the penultimate graph in the construction sequence of $(G_0,\psi)$. By the induction hypothesis and Theorem \[thm:antisym1\], there exists a well-positioned $\mathbb{Z}_2$-symmetric realisation of the covering graph of $(G'_0,\psi')$ in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$ (where the reflection $\tau(-1)$ is in the mirror $\ker\varphi_{F_1}$) so that the induced monochrome subgraphs $G'_{F_1,0}$ and $G'_{F_2,0}$ of $G'_0$ are both spanning, $G'_{F_1,0}$ is a tree, and $G'_{F_2,0}$ is an unbalanced map graph. By Theorem \[thm:antisym1\] it suffices to show that the vertex (or vertices) of $G\setminus G'$ can be placed so that $(G,p)$ is well-positioned, ${{\mathbb{Z}}}_2$-symmetric and the induced monochrome subgraphs $G_{F_1,0}$ and $G_{F_2,0}$ of $G_0$ are both spanning, $G_{F_1,0}$ is a tree and $G_{F_2,0}$ is an unbalanced map graph.
If $(G_0,\psi)$ is obtained from $(G'_0,\psi')$ by a H1a-, H1b-, H2a-, H2b-, vertex-to-$K_4$, or edge-to-$K_3$ move, then we may use exactly the same placement for the vertex (or vertices) of $G\setminus G'$ as in the proof of Theorem \[thm:forced2\] to obtain the desired realisation of $G$.
So it remains to consider the case where the last move in the construction sequence of $(G_0,\psi)$ is a $2K_3-[e]$ edge joining move. Suppose $(G_0,\psi)$ is obtained by joining the signed quotient graphs $(G'_0,\psi')$ and $(G''_0,\psi'')$ by an edge $[f]$ with end-vertices $[u]\in G'_0$ and $[v]\in G''_0$, where $G''_0=2K_3-[e]$. By the induction hypothesis, Theorem \[thm:antisym1\], and Example \[ex:2k3minusedge\], the covering graphs of $(G'_0,\psi')$ and $(G''_0,\psi'')$ can be realised as $\mathbb{Z}_2$-symmetric frameworks $(G',p)$ and $(G'',q)$ in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$ (where the reflection $\tau(-1)$ is in the mirror $\ker\varphi_{F_1}$) so that the induced monochrome subgraphs $G'_{F_1,0}$ and $G'_{F_2,0}$ of $G'_0$, and $G''_{F_1,0}$ and $G''_{F_2,0}$ of $G''_0$, are all spanning, $G'_{F_1,0}$ and $G''_{F_1,0}$ are trees, and $G'_{F_2,0}$ and $G''_{F_2,0}$ are unbalanced map graphs. Now, consider the line $L$ which passes through the points $p_{{\tilde{u}}}$ and $\tau(-1)p_{{\tilde{u}}}$, and translate the framework $(G'',q)$ along the mirror line $\ker\varphi_{F_1}$ (thereby preserving the reflection symmetry of $(G'',q)$) so that the points $\hat q_{{\tilde{v}}}$ and $\tau(-1)\hat q_{{\tilde{v}}}$ of the translated framework $(G'',\hat q)$ lie on $L$. If there are vertices of $(G',p)$ and $(G'',\hat q)$ which are now positioned at the same point in $({{\mathbb{R}}}^2,\|\cdot\|_{{\mathcal{P}}})$, then we perturb the vertices of $(G'',\hat q)$ slightly without changing the induced colourings of the edges of $G$ until all of the vertices have different positions. Then $[f]$ has induced framework colour $[F_1]$, the realisation of $G$ is well-positioned, and the induced monochrome subgraphs of $G_0$ are $G_{F_1,0}=G'_{F_1,0}\cup G''_{F_1,0}\cup\{[f]\}$ and $G_{F_2,0}=G'_{F_2,0}\cup G''_{F_2,0}$. Clearly, $G_{F_1,0}$ is a spanning tree and $G_{F_2,0}$ is a spanning unbalanced map graph of $G_0$.
Note that the final argument in the proof of Theorem \[thm:anti2\] can immediately be generalised to show that in the recursive construction sequence in Theorem \[thm:anti2\] $(iii)$, we may replace the $2K_3-[e]$ edge joining move with an edge joining move that joins two *arbitrary* $(2,2,1)$-gain tight signed quotient graphs by an edge of arbitrary gain.
Further remarks
===============
At the graph level, we provided characterisations for symmetric and anti-symmetric infinitesimal rigidity in terms of gain-sparsity counts and recursive constructions (see Theorems \[thm:forced2\] and \[thm:anti2\]). However, a characterisation in terms of monochrome subgraph decompositions (analogous to the results in Section \[Sect:framework\]) was not given, as it is not clear whether for an arbitrary decomposition of a signed quotient graph into a monochrome spanning unbalanced map graph and a monochrome spanning tree, there always exists a grid-like realisation of the covering graph with reflectional symmetry which respects the given edge colourings. These realisation problems are non-trivial [@kit-sch; @kit-sch1] and even arise in the non-symmetric situation [@kitson].
It is easy to see that a necessary count for the existence of a $2$-dimensional infinitesimally rigid grid-like $\mathbb{Z}_2$-symmetric realisation of a graph $G$ is that its signed quotient graph $(G_0,\psi)$ contains a spanning subgraph with $F$ edges which is *$(2,2,0)$-gain-tight*, i.e., $|F|= 2|V(F)|$, $|F'|\leq 2|V(F')|-2$ for every balanced $F'\subseteq F$, and $|F'|\leq 2|V(F')|$ for every $F'\subseteq F$. This is because $(G_0,\psi)$ needs to contain two monochrome connected unbalanced spanning map graphs, by Corollary \[cor\]. However, these conditions are clearly not sufficient.
Finally, it is natural to ask whether the results of this paper can be extended to grid-like frameworks in the plane with half-turn symmetry. A necessary condition for a grid-like half-turn-symmetric framework to be symmetrically isostatic is that the associated signed quotient graph $(G_0,\psi)$ satisfies $|E_0|= 2|V_0|$, as there are no symmetric trivial infinitesimal flexes with respect to the half-turn symmetry group. In fact, $(G_0,\psi)$ must clearly be $(2,2,0)$-gain-tight. A combinatorial characterisation of $(2,2,0)$-gain-tight graphs, however, has not yet been obtained (see also [@anbs]). For anti-symmetric isostaticity, the situation is much easier, as we need $(G_0,\psi)$ to satisfy $|E_0|= 2|V_0|-2$ and $|F|\leq 2|V(F)|-2$ for every $F\subseteq E_0$, and these types of signed quotient graphs have been described in [@anbs].
More generally, it would of course also be of interest to extend the results of this paper to frameworks with larger symmetry groups and to different normed spaces.
M. Berardi, B. Heeringa, J. Malestein and L. Theran, *Rigid components in fixed-lattice and cone frameworks*, CCCG, (2011).
R. Connelly, P.W. Fowler, S.D. Guest, B. Schulze and W. Whiteley, *When is a symmetric pin-jointed framework isostatic?*, International Journal of Solids and Structures **46** (2009), 762–773.
T. Eren, B. Anderson, S. Morse, W. Whiteley and P. Belhumeur, *Operations on rigid formations of autonomous agents*, Commun. Inf. Syst. **3** (2004), no. 4, 223–258.
T. Jordán, V. Kaszanitzky and S. Tanigawa *Gain-sparsity and symmetry-forced rigidity in the plane*, Discrete Comput. Geom. **55** (2016), no. 2, 314–372.
R.D. Kangwai and S.D. Guest, *Symmetry-adapted equilibrium matrices*, International Journal of Solids and Structures **37** (2000), 1525–1548.
D. Kitson, *Finite and infinitesimal rigidity with polyhedral norms*, Discrete Comput. Geom. **54** (2015), no. 2, 390–411.
D. Kitson and S.C. Power, *Infinitesimal rigidity for non-Euclidean bar-joint frameworks*, Bull. Lond. Math. Soc. **46** (2014), no. 4, 685–697.
D. Kitson and B. Schulze, *Maxwell-Laman counts for bar-joint frameworks in normed spaces*, Linear Algebra Appl. **481** (2015), 313–329.
D. Kitson and B. Schulze, *Symmetric isostatic frameworks with $\ell^1$ or $\ell^\infty$ distance constraints*, The Electronic Journal of Combinatorics **23** (2016), No. 4, P4.23.
G. Laman, *On graphs and rigidity of plane skeletal structures*, J. Engrg. Math. **4** (1970), 331–340.
J. Malestein and L. Theran, *Generic rigidity of reflection frameworks*, preprint, arXiv:1203.2276, 2012.
A. Nixon J.C. Owen and S.C. Power, [*A characterisation of generically rigid frameworks on surfaces of revolution*]{}, SIAM Journal on Discrete Mathematics **28** (4) (2014), 2008–2028.
A. Nixon and J. Owen, [*An inductive construction of $(2,1)$-tight graphs*]{}, Contributions to Discrete Mathematics **9** (2) (2014), 1–16.
A. Nixon and B. Schulze, [*Symmetry-forced rigidity of frameworks on surfaces*]{}, Geometriae Dedicata, **182** (2016), no. 1, 163–201.
J.C. Owen and S.C. Power, *Frameworks, symmetry and rigidity*, Int. J. Comput. Geom. Appl. **20** (2010), 723–750.
B. Schulze, *Block-diagonalized rigidity matrices of symmetric frameworks and applications*, Contributions to Algebra and Geometry **51** (2010), No. 2, 427–466.
B. Schulze, *Symmetric Laman theorems for the groups $C_2$ and $C_s$*, The Electronic Journal of Combinatorics **17** (2010), No. 1, R154, 1–61.
B. Schulze and S. Tanigawa, *Infinitesimal rigidity of symmetric bar-joint frameworks*, SIAM J. Discrete Math. **29** (2015), no. 3, 1259–1286.
B. Schulze and W. Whiteley, *The orbit rigidity matrix of a symmetric framework*, Discrete Comp. Geom. **46** (2011), No. 3, 561–598.
T.-S. Tay, *A New Proof of Laman’s Theorem*, Graphs and Combinatorics **9** (1993), 365–370.
W. Whiteley, *Vertex Splitting in Isostatic Frameworks*, Structural Topology, **16** (1991), 23–30.
W. Whiteley, *Some [M]{}atroids from [D]{}iscrete [A]{}pplied [G]{}eometry*, Contemporary Mathematics, AMS **197** (1996), 171–311.
T. Zaslavsky, *Signed graphs*, Discrete Applied Mathematics, **4**(1) (1982), 47–74.
T. Zaslavsky, *Biased graphs “[I]{}”: Bias, balance, and gains*, J. Combin. Theory Ser. B, **47** (1989), 32–52.
[^1]: The first named author is supported by EPSRC grant EP/P01108X/1.
[^2]: The second named author is supported by EPSRC grant EP/M013642/1.
|
---
abstract: |
We show that estimation of error in the iterative solution can reduce uncertainty in convergence by a factor $\sim \kappa(A,x)$ compared to the case of using the relative residue as a stopping criterion. Here $\kappa(A,x)$ is the condition number of the forward problem of computing $Ax$ given $x$, $A$, and $1 \leq \kappa(A,x) \leq\kappa(A)$ where $\kappa(A)$ is the condition number of matrix $A$. This makes error estimation as important as preconditioning for efficient and accurate solution of moderate to high condition problems ($\kappa(A)>10$). An $\mathcal{O}(1)$ estimator (at every iteration) was proposed more than a decade ago, for efficient solving of symmetric positive definite linear systems by the CG algorithm. Later, an $\mathcal{O}(k^2)$ estimator was described for the GMRES algorithm which allows for non-symmetric linear systems as well, and here $k$ is the iteration number. We suggest a minor modification in this GMRES estimation for increased stability. Note that computational cost of the estimator is expected to be significantly less than the $\mathcal{O}(n^2)$ evaluation at every iteration of these methods in solving problems of dimension $n$. In this work, we first propose an $\mathcal{O}(n)$ error estimator for A-norm and $l_{2}$ norm of the error vector in Bi-CG algorithms that can as well solve non-symmetric linear systems. Secondly, we present an analysis of performance of these error estimates proposed for CG, Bi-CG and GMRES methods. The robust performance of these estimates as a stopping criterion results in increased savings and accuracy in computation, as condition number and size of problems increase.\
author:
- 'Puneet Jain, Krishna Manglani and Murugesan Venkatapathi'
bibliography:
- 'references.bib'
title: 'Significance of error estimation in iterative solution of linear systems : estimation algorithms and analysis for CG, Bi-CG and GMRES'
---
: Conjugate Gradients; Bi-CG; GMRES; error; stopping criteria; condition number
Introduction
============
Solving a system of linear equations in the form $Ax = b$ is a ubiquitous requirement in science and engineering (where $A$ is a given matrix, $x$ and $b$ are the unknown and known vectors respectively; $x$ $\in \mathbb{R}^n$ and $b$ $\in \mathbb{R}^n$ if $A$ $\in \mathbb{R}^{n \times n}$). Iterative methods like CG (Conjugate Gradient), Bi-CG (Bi-Conjugate Gradient) and GMRES (Generalized Minimal Residual) are commonly used to solve large linear problems as they require $\mathcal{O}(n^2)$ operations compared to direct solvers which can evaluate $A^{-1}$ explicitly in $\mathcal{O}(n^3)$ operations for a square matrix. Iterations should be stopped when the norm of the error $\epsilon_{k} = x - x_{k}$ is less than a desired tolerance, where $x$ is the final solution to the linear system and $x_k$ is the iterate. Since the the actual error is unknown, relative residue ($\frac{\lVert r_k \rVert}{\lVert b \rVert}$) is considered as stopping criteria where $r_{k} = b - Ax_{k}$ is the residual vector at $k^{th}$ iteration. Such stopping criteria can work when the system is well-conditioned and can be erroneous depending on the condition number of $A$ and the choice of initial approximation, as it can stop the iterations too early when the norm of error is still much larger than tolerance, or not stop early enough and too many floating point operations having done for the required accuracy. Also when condition number (denoted by $\kappa$) of the matrix is large ($\kappa > 10^2$), the residual of a CG/Bi-CG algorithm need not show monotonic behaviour and oscillate while the actual error might still be (however slowly) converging (and vice-versa for the GMRES algorithm). The norm of the relative residue can be as large as $\kappa$ times or as small as $\frac{1}{\kappa}$ times the norm of the relative error.
Even when most iterative algorithms are used with preconditioners, it is not guaranteed that the condition number of the problems will be reduced and this is observed with matrices of larger dimensions. In cases where the condition number of the matrix is indeed reduced, a reduction of condition number of the backward problem (i.e. computing $x$ given $A$ and $b$) might not be guaranteed. Further more, a reduction in the condition number of the backward problem by an ideal preconditioner will accelerate convergence but still leave a large uncertainty in the relationship between the residue and the error, and thus the relative residue remains a poor and inefficient indicator of convergence in general. Moreover, the condition number of matrix $\kappa(A)$ is typically unknown and costly to compute. Thus for even marginally high condition numbers of matrices ($\kappa(A)>10$), either the accuracy or the efficiency of computation is degraded by the above conundrum. The precision in measurements and engineering today renders both the size and condition number of most problems large; making accurate stopping and restarting criteria indispensable in ensuring computational efficiency of solvers. This motivated methods to compute estimates of some norms of the error in iterative solvers. Such estimators (e.g CGQL) are available for CG algorithm [@meurant2006lanczos]. For solving non-symmetric linear systems using FOM (Full Orthogonalization method) and GMRES (Generalized Minimal Residual) methods, formulas for estimation of errors have been suggested [@meurant2011estimates] recently. We suggest a minor modification to this estimator proposed by Meurant, to increase its stability and precision. Our objective is to derive an efficient estimator for solving non-symmetric linear systems using BiCG, and present an analysis highlighting the significance of these estimation algorithms for CG, BiCG and GMRES methods. This analysis shows that these estimators are robust and increase the efficiency/accuracy of computing notably. Note that this gain is expected even when errors in estimation itself may not be negligible, as the factor scaling the residue to the actual error can be more significant, i.e. as large as $\kappa(A)$ or as small as $1/\kappa(A)$.
Section \[section2\] presents related work and discuss CG, Bi-CG and GMRES algorithms and their error estimates. Section \[Section3\] presents the analysis of error estimates and its performance as an efficient stopping criteria for these iterative methods of solving linear systems.
Methods {#section2}
=======
Related work: Algorithms for CG and GMRES
-----------------------------------------
### CGQL Algorithm (Conjugate Gradient and Quadrature Lanczos)
One of the most commonly used methods for solving linear systems with a real Symmetric Positive Definite (SPD) matrix is the Conjugate Gradient (CG) algorithm. It can be derived from several different perspectives, (i) an orthogonalization problem (ii) minimization problem and (iii) Lanczos algorithm (Algorithm \[Lanczosalgorithm\] in appendix).
The idea of CGQL algorithm is to use CG instead of the Lanczos algorithm, to compute explicitly the entries of the corresponding tridiagonal matrices ($T_k$), and then to derive recursive formulas to compute the A-norm of error. The formulas are summarized as CGQL Algorithm \[CGQL algorithm\] in the appendix (QL standing again for Quadrature and Lanczos), whose most recent version is described [@golub1997matrices].\
\
\
The square of the A-norm of error at CG iteration $k$ is given by:
$$\lVert \epsilon_{k} \rVert_{A}^{2} = \lVert r_{0} \rVert ^{2} [(T_{n}^{-1})_{(1,1)} - (T_{k}^{-1})_{(1,1)} ]$$
Here $T_{k}$ is the tridiagonal matrix from Lanczos algorithm whose coefficients can be computed from Equation \[relation\]. Also $\epsilon_{k}$,$r_{0}$ are the error and residual respectively. Let $d$ be a delay integer, the approximation of the A-norm of error at iteration $k-d$ is given by:
$$\lVert \epsilon_{k-d} \rVert_{A}^{2} = \lVert r_{0} \rVert ^{2} [(T_{k}^{-1})_{(1,1)} - (T_{k-d}^{-1})_{(1,1)} ]$$
The main essence of the CGQL lies in computing difference between $(1,1)$ elements of the inverse of two tridiagonal matrices generated from a Lanczos algorithm with the same starting vectors as CG algorithm. If $s_{k-1}$ be the estimate of $\lVert \epsilon_{k} \rVert_{A}$, for sufficiently large $k$ and $d=1$ the estimator is given by:
$$s_{k-1} = \lVert r_{0} \rVert^{2} \frac{\eta_{k-1}^{2}c_{k-1}}{\delta_{k}(\alpha_{k}\delta_{k-1} - \eta_{k}^{2})}$$
All above coefficients are related to CGQL algorithm as shown in the appendix (Algorithm \[CGQL algorithm\]).\
### Estimator for GMRES
Let $V_k$ be a matrix whose columns are orthonormal basis vectors $v_j,\ j=1,...,k$, of Krylov subspace $K_k(A,r^0)$. The iterates of GMRES are defined as $x_k = x_0 + V_k z_k$. We also have $H_k = V_k^T A V_k$ and $AV_n = V_n H_n$ with the assumption that Arnoldi process does not terminate early, that is, $h_{k+1,k} \neq 0$ for $k=1,2,...,n-1$. See appendix for a brief description of GMRES method and the algorithm for estimation of its error.
At $k^{th}$ iteration, we have $H_k$ that can be decomposed blockwise as:
$$H_k =
\begin{pmatrix}
H_{k-d} & W_{k-d} \\
Y_{k-d}^T & \tilde{H}_{k-d}
\end{pmatrix}$$ Let $$\gamma_{k-d} = \dfrac{h_{k-d+1,k-d}\big( e_{k-d}, H_{k-d}^{-1}e_1 \big)}{1 - h_{k-d+1,k-d}\big( e_{k-d}, H_{k-d}^{-1}w_{k-d} \big)}$$ where $w_{k-d} = W_{k-d}\tilde{H}_{k-d}^{-1}e_1 $\
Let the vector $t_k$ be the last column of $\big( H_k^T H_k \big)^{-1} $, $t_{kk}$ its last element and $$\delta_{k+1} = \dfrac{h_{k+1,k}^2}{1+h_{k+1,k}^2 t_{kk}}$$ and $$u_k = \delta_{k+1}t_k$$
Then, error estimates at $(k-d)^{th}$ iteration for GMRES as provided by Meurant[@meurant2011estimates] are given by:
$$\dfrac{\chi_{k-d}^2}{{\left\lVertr_0\right\rVert}^2} = \gamma_{k-d}^2 {\left\lVert\tilde{H}_{k-d}^{-1}e_1\right\rVert}^2 + {\left\lVert\gamma_{k-d} H_{k-d}^{-1}w_{k-d} + (e_{k-d},H_{k-d}^{-1}e_1)u_{k-d}\right\rVert}^2$$
### Proposed modification to the GMRES error estimator
It can be seen that the estimator for GMRES proposed by Meurant[@meurant2011estimates] satisfies the equation \[estanlysis\].
$$\begin{gathered}
\label{estanlysis}
\dfrac{\chi_{k-d}^2}{{\left\lVertr_0\right\rVert}^2} = \dfrac{{\left\lVert\epsilon_{k-d}\right\rVert}^2 - {\left\lVert\epsilon_{k}\right\rVert}^2}{{\left\lVertr_0\right\rVert}^2} + {\left\lVerts_k\right\rVert}^2 + \bigg[ 2h_{k+1,k} \left( e_k,H_k^{-1}e_1 \right) \left((H_n^{-1}e_{k+1})^k,H_k^{-1}e^1 + s_k \right) \\
\qquad \qquad - 2h_{k-d+1,k-d} \left( e_{k-d},H_{k-d}^{-1}e_1 \right) \left( (H_n^{-1}e_{k-d+1})^{k-d} - (H_k^{-1}e_{k-d+1})^{k-d} ,H_{k-d}^{-1}e_1 + s_{k-d} \right) \bigg]\end{gathered}$$
where $s_k = \big( e_k,H_k^{-1}e_1 \big) u_k$.
The terms inside the square bracket are negligible compared to other terms, hence ignoring that, we get, $$\dfrac{\chi_{k-d}^2}{{\left\lVertr_0\right\rVert}^2} \approx \dfrac{{\left\lVert\epsilon_{k-d}\right\rVert}^2 - {\left\lVert\epsilon_{k}\right\rVert}^2}{{\left\lVertr_0\right\rVert}^2} + {\left\lVerts_k\right\rVert}^2$$ Note that for large $d$, term ${\left\lVerts_k\right\rVert}^2 $ also becomes negligible. However, the delay $d$ is kept small in practice ( 10), hence in such cases, term ${\left\lVerts_k\right\rVert}^2 $ significantly contributes to the value of error estimate and needs to be accounted.
For converging problems, ${\left\lVert\epsilon_{k-d}\right\rVert}^2 >> {\left\lVert\epsilon_{k}\right\rVert}^2$, $$\label{offseteq}
\chi_{k-d}^2 \approx {\left\lVert\epsilon_{k-d}\right\rVert}^2 + {\left\lVertr_0\right\rVert}^2 {\left\lVerts_k\right\rVert}^2$$
The equation \[offseteq\] shows that the estimator is offset from exact value of error by term ${\left\lVertr_0\right\rVert}^2 {\left\lVerts_k\right\rVert}^2$ and should be accounted in the formula of error estimator. This offset term can also cause unstable overshoots in estimation as seen in figure \[fig:ust2\]. This figure illustrates that original estimator can have very large errors in estimation due to term ${\left\lVerts_k\right\rVert}^2$ and removal of the term gives better results. Hence, we propose the modification in estimator with new formula given in equation \[newesteq\].
$$\label{newesteq}
\dfrac{\chi_{k-d}^2}{{\left\lVertr_0\right\rVert}^2} = {\left\lvert \gamma_{k-d}^2 {\left\lVert\tilde{H}_{k-d}^{-1}e_1\right\rVert}^2 + {\left\lVert\gamma_{k-d} H_{k-d}^{-1}w_{k-d} + (e_{k-d},H_{k-d}^{-1}e_1)u_{k-d}\right\rVert}^2 - {\left\lVert(e_{k},H_{k}^{-1}e_1)u_{k}\right\rVert}^2 \right\rvert}$$
The absolute operation is necessary as in the non-converging situations, the operand can become negative. The comparison of the Meurant[@meurant2011estimates] estimator and proposed correction is demonstrated in the figure \[fig:unstable\_est\].
It was also seen that GMRES error estimator even after correction may behave erratically when numerical precision of computing system is exhausted as seen in figure \[blowup\]. The exhaustion of numerical precision can lead to near singularity of Hessenberg matrix $H_k$ formed during the Arnoldi iteration which can cause large errors in $H_k^{-1}$ and its functions. Note that ${\left\lVerts_k\right\rVert}$ is also a function of $H_k^{-1}$ and can be used as trigger to predict this exhaustion of numerical precision as seen in figure \[blowup\] making this error estimate a robust stopping criterion.
![Behaviour of relative error estimates in GMRES when the numerical precision is exhausted. The term ${\left\lVerts_k\right\rVert}$ can be used as trigger to improve the estimator and for detection of exhaustion of numerical precision.[]{data-label="blowup"}](images/unstablenspd.png){width="0.98\columnwidth"}
Proposed estimator for A-norm and $l_2$ norm of errors in Bi-Conjugate gradient Algorithm
-----------------------------------------------------------------------------------------
Similar to the CGQL for CG, A-norm of error in this case can be represented in term of residual vector of BiCG algorithm and tri-diagonal matrices of the corresponding Non-symmetric Lanczos algorithm. A-norm of error (which we better denote as A-measure for matrices which are not positive definite) when matrix is Non symmetric, is given by:
$$\begin{aligned}
\label{A-norm}
\lVert \epsilon_{k} \rVert_{A}^{2} = \epsilon_{k}^{T}A\epsilon_k = r_{k}^{T}(A^{T})^{-1}r_{k} = r_{k}^{T}A^{-1}r_{k}
\end{aligned}$$
And, when $A$ $\in \mathbb{R}^{N \times N}$ and $r$ $\in \mathbb{R}^N$ ; $r_{k}^{T}(A^{T})^{-1}r_{k}$ is a scalar quantity whose transpose will be itself and thus $r_{k}^{T}(A^{T})^{-1}r_{k} = r_{k}^{T}A^{-1}r_{k}$. Here, $r$ is the residual vector pertaining to the BiCG method. When $A$ is positive definite, the right side of the above equation is always positive. In case of indefinite matrices, the absolute value of the above equation is considered and we define such an A-measure of the error in these cases. Moreover the $l_{2}$ norm of error is given by:
$$\label{l2 norm}
\lVert \epsilon_{k} \rVert_{2}^{2} = \epsilon_{k}^{T}\epsilon_{k} = r_{k}^{T}(A^{T})^{-1}A^{-1}r_{k}$$
If $A=A^{T}$, A-norm and $l_{2}$ norm of error is given by $r_{k}^{T}A^{-1}r_{k}$ and $r_{k}^{T}A^{-2}r_{k}$ respectively. We are interested in approximating (\[A-norm\]) and (\[l2 norm\]). In the following sections we derive approximation of A-norm and $l_{2}$ norm of error for every iteration of BiCG iteration. The BiCG method is shown as Algorithm \[BiCG algorithm\].
### $\mathcal{O}(n)$ expression to estimate A-norm of error
Writing the consecutive difference between A-norm of error at iteration $k$ and $k+1$ we get the following relation when A is a non-symmetric matrix:
$$\begin{split}
r_{k}^T (A^{T})^{-1} r_{k} - r_{k+1}^T(A^{T})^{-1} r_{k+1} & = r_{k}^T (A^{T})^{-1} r_{k} - (r_{k} - \alpha_{k}Ap_{k}) (A^{T})^{-1} (r_{k} - \alpha_{k}Ap_{k})\\
& = - \alpha_{k} r_{k}^T p_{k} - \alpha_{k}r_{k}^{T}(A^{T})^{-1}Ap_{k} + \alpha_{k}^{2} p_{k}^T A p_{k}
\end{split}$$
In the above equation the first and third terms can be trivially computed using iterates of the Bi-CG Algorithm. Second term involves computation of $A^{-1}$ hence we further reduce the equation. As $ r_{k+1} = r_{k} - \alpha_{k} A p_{k} $ and $Ap_{k} = \dfrac{r_{k} - r_{k+1}}{\alpha_{k}} $ we can derive the following relation:
$$\label{eq:11}
\begin{array}{ccl}
r_{k}^T A^{-1} r_{k} - r_{k+1}^T A^{-1} r_{k+1} = \alpha_{k} r_{k}^T p_{k} - \alpha_{k} r_{k}^T (A^{T})^{-1} (\dfrac{r_{k} - r_{k+1}}{\alpha_{k}}) - \alpha_{k}^{2} p_{k}^T A p_{k}\\
\implies r_{k+1}^T A^{-1} r_{k+1} = - \alpha_{k} r_{k}^T p_{k} + r_{k+1}^T A^{-1} r_{k} + \alpha_{k}^{2} p_{k}^T A p_{k}
\end{array}$$
Error ($\epsilon_{k}$) is given as $A^{-1}r_{k}$. Also, $\epsilon_{k} = x - x_{k}$ and we know $x_{k}$ but we do not know the final $x$ and hence error at each iteration is difficult to compute. Also $\epsilon_k$ is given as weighted sum of search directions from $k$ to $n$ given by Equation \[eq:13\].
$$\label{eq:13}
\epsilon_{k} = \sum_{j=k}^{n} \alpha_{j} p_{j}$$
If $\epsilon_{k+d}$ denotes the error at $k+d$ iteration then $\epsilon_{k+d} \ll \epsilon_{k}$ (assuming error falls as the iteration increases and $d \geq 0$) so we can neglect the further terms of the series sum. We can use the error vector in Equation \[eq:13\] to give an estimation to A-norm of error after inducing a delay of $d$ iterations. However this will require us to store previous $d$ vectors. So Equation \[eq:11\] now becomes:
$$\label{eq:14}
r_{k+1}^T A^{-1} r_{k+1} \approx - \alpha_{k} r_{k}^T p_{k} + r_{k+1}^T(\sum_{j=k}^{k+d}\alpha_{j}p_{j}) + \alpha_{k}^{2} p_{k}^T A p_{k}$$
It should be noted that when A is positive definite, the above expression is always positive and thus provides a lower bound for the square of A-norm of error. Note that since the above derivation is done using Bi-CG which is an extended version of CG, when $A = A^T$, our estimator for $A$-norm becomes equivalent to CGQL estimator for $A$-norm (Section \[BCGtoCG\] from Appendix). Though RHS of Equation \[eq:14\] is not a unique formula of evaluation, it provides a convenient way to use Bi-CG iterates.
### $\mathcal{O}(n)$ expression to estimate $l_{2}$ norm of error
Writing the consecutive difference between $l_{2}$ norm of error at iteration $k$ and $k+1$ we get the following relation when A is a non-symmetric matrix:
$$\begin{gathered}
r_{k}^T (A^{T})^{-1} A^{-1} r_{k} - r_{k+1}^T(A^{T})^{-1}A^{-1} r_{k+1} = r_{k}^T (A^{T})^{-1} A^{-1}r_{k} \\ - (r_{k} - \alpha_{k}Ap_{k}) (A^{T})^{-1} A^{-1}(r_{k} - \alpha_{k}Ap_{k}) \end{gathered}$$
i.e., $$\label{eq:15}
r_{k}^T (A^{T})^{-1} A^{-1} r_{k} - r_{k+1}^T(A^{T})^{-1}A^{-1} r_{k+1} = \alpha_{k}r_{k}^{T}(A^{T})^{-1}p_{k} + \alpha_{k} p_{k}^T A^{-1}r_{k} - \alpha_{k}^{2} p_{k}^T p_{k}$$
Taking transpose of first term of Equation \[eq:15\] and rewriting the equation:
$$\label{eq:20}
r_{k}^T (A^{T})^{-1} A^{-1} r_{k} - r_{k+1}^T(A^{T})^{-1}A^{-1} r_{k+1} = 2\alpha_{k} p_{k}^T A^{-1}r_{k} - \alpha_{k}^{2} \lVert p_{k} \rVert ^{2}$$
Rearranging Equation \[eq:20\] we get:
$$\label{eq:17}
r_{k+1}^T(A^{T})^{-1}A^{-1} r_{k+1} = -\alpha_{k} p_{k}^T A^{-1}r_{k} + \epsilon_{k}^{T}\epsilon_{k} - \alpha_{k}p_{k}^{T}A^{-1}r_{k} + \alpha_{k}^{2} \lVert p_{k} \rVert ^{2}\\$$
Again using \[eq:13\] we can estimate \[eq:18\].
$$\label{eq:18}
\begin{array}{ccl}
r_{k+1}^T(A^{T})^{-1}A^{-1} r_{k+1} \approx -2\alpha_{k} p_{k}^T (\sum_{j=k}^{k+d} \alpha_{j} p_{j}) + \lVert \sum_{j=k}^{k+d} \alpha_{j} p_{j} \rVert ^{2} + \alpha_{k}^{2} \lVert p_{k} \rVert ^{2}\\ \\
\implies r_{k+1}^T(A^{T})^{-1}A^{-1} r_{k+1} \approx -2\alpha_{k} p_{k}^T (\sum_{j=k+1}^{k+d} \alpha_{j} p_{j}) + \lVert \sum_{j=k+1}^{k+d} \alpha_{j} p_{j} \rVert ^{2}
\end{array}$$
Here, $d$ again signifies the delay in approximation. Also, the Bi-CG method shows irregular convergence, in such cases larger values of $d$ can result in less accurate approximations. Hence, values of $d < 10$ is recommended which are anyway much lesser than $N$ for both accurate and efficient estimation. Note that by $\mathcal{O}(n)$ we mean arithmetic complexity. Lemma \[lemma\] further justifies the arithmetic complexity for the above expressions.\
\
\[lemma\] Equation \[eq:14\] and \[eq:18\] involve arithmetic operations of $\mathcal{O}(n)$.
\
Equation \[eq:14\] is an approximation to $A$ norm of error at $k+1$ iteration and it involves three terms.
$$r_{k+1}^T A^{-1} r_{k+1} \approx - \alpha_{k} r_{k}^T p_{k} + r_{k+1}^T(\sum_{j=k}^{k+d}\alpha_{j}p_{j}) + \alpha_{k}^{2} p_{k}^T A p_{k}$$
All the terms individually are inner products of vectors in $\mathbb{R}^{n}$ and require $n$ multiplication operations; referred to as $\mathcal{O}(n)$ arithmetic operations here. Here $d$ denotes the delay in estimation where $\alpha_k$ is a scalar. Note that the third term involves $A p_k$ which is a vector provided by the Bi-CG Algorithm at every iteration with no additional cost to this estimator. Thus estimation of $A$ norm of error in BiCG algorithm requires $\mathcal{O}(3n + nd)$ arithmetic operations in total.\
Similarly for Equation \[eq:17\] the first term requires $\mathcal{O}(n + nd)$ operations in total and result of first term is partially used to compute the second term, hence requiring only $\mathcal{O}(n)$ operations. Thus estimation of $l_2$ norm requires $\mathcal{O}(2n + nd)$ operations.
$$r_{k+1}^T(A^{T})^{-1}A^{-1} r_{k+1} \approx -2\alpha_{k} p_{k}^T (\sum_{j=k+1}^{k+d} \alpha_{j} p_{j}) + \lVert \sum_{j=k+1}^{k+d} \alpha_{j} p_{j} \rVert ^{2}$$
\
But note that in both of the above expressions $d$ vectors are available at each iteration from previous iterations and thus estimation of $A$ norm and $l_2$ norm of error only requires two or three instances of inner-products of vectors in $\in \mathbb{R}^n$, thus making the arithmetic complexity $\mathcal{O}(3n)$ and $\mathcal{O}(2n)$ repectively.
Numerical results and an analysis of estimators {#Section3}
===============================================
A-norm and $l_2$ norm estimators for Bi-CG
------------------------------------------
In Figure \[fig:figure 2\] plot of estimator along with A-norm of the error is shown when A is a Non-symmetric matrix.\
![BiCGQL estimator for a Non-symmetric matrix (indefinite); absolute values are considered for $r^{T}A^{-1}r$ and its approximation; dimension of the matrix = $500 \times 500$; condition number of the matrix is $10^{6}$ and $d = 10$[]{data-label="fig:figure 2"}](images/fig2){width="\linewidth"}
Figure \[fig:figure 3\] shows the comparison between $l_{2}$ norm approximation, actual $l_{2}$ norm of the error and $l_{2}$ norm of the residue.
![Comparison between BiCGQL $l_{2}$ norm of estimator, actual $l_{2}$ norm of the error and $l_{2}$ norm of the residue; dimension of the matrix = $500 \times 500$; condition number of the matrix is $10^{6}$ and $d=10$[]{data-label="fig:figure 3"}](images/fig3){width="\linewidth"}
It is evident that BiCGQL estimators work efficiently both cases. In both the figures we see that the error norm is also more stable than the norm of the residue. Similar behaviour can be seen when the matrix A is Non-symmetric positive definite, and the convergence is faster and more sable as compared to indefinite cases.\
Analysis of estimators
----------------------
### Condition number of the problem
Condition number plays a valuable role in matrix computations as they enable us to estimate the accuracy of computed result. Condition number of a forward problem (that is computing b given A and x) and backward problem (computing x from A and b) respectively are:
$$\label{cn of prob1}
{\kappa}(A,x) = \lVert A \rVert \frac{\lVert x \rVert}{\lVert Ax \rVert}$$
$$\label{cn of prob2}
{\kappa}(A,b) = \lVert A^{-1} \rVert \dfrac{\lVert b \rVert}{\lVert A^{-1}b \rVert}$$
Condition number of the matrix is given by the product of ${\kappa}(A,x)$ and ${\kappa}(A,b)$ and is given by:
$${\kappa} = \lVert A\rVert \lVert A^{-1}\rVert$$
### Data set and Performance metrics
Equation \[cn of prob1\] and \[cn of prob2\] relate to the condition number of the forward and backward problem respectively when one solves for a linear system. In order to test the estimator, we choose the relative error in estimating norm ($l_2$-norm or $A$-norm) of error by the estimator as an uncertainty metric of estimator. This metric for $k^{th}$ iteration can be expressed as follows:
$$\label{eq:1}
\left \lvert \dfrac { \dfrac{ \chi_{k} }{\lVert x \rVert} - \dfrac{\lVert \epsilon_{k} \rVert}{\lVert x \rVert} } { \dfrac{\lVert \epsilon_{k} \rVert}{\lVert x \rVert} } \right \rvert$$
\
However, we are more interested in comparing the estimator and relative residual in order to come up with robust stopping criterion in krylov subspace based algorithms. Also the metric should consider all iterations on which we could measure the uncertainties, Hence, we define uncertainty ratio $U.R.^{(j)}$ of $j^{th}$ order as Performance or Uncertainty metric in estimating the norm of error as follows: $$\label{UR_def}
U.R.^{(j)} = \dfrac{1}{n-d} \sum_{k = 0}^{n-d-1} \left( {\left\lvert \dfrac{ \dfrac{{\left\lVertr_k\right\rVert}^j}{ {\left\lVertb\right\rVert}^j } - \dfrac{{\left\lVert\epsilon_k\right\rVert}^j}{ {\left\lVertx\right\rVert}^j} }{ \dfrac{\chi_k^j}{ {\left\lVertx\right\rVert}^j } - \dfrac{{\left\lVert\epsilon_k\right\rVert}^j}{ {\left\lVertx\right\rVert}^j } } \right\rvert} \right )$$\
where $\chi_k$ is the estimate of norm of error at $k^{th}$ iteration and $n$ is dimension of matrix. We consider j = 1 and 2 for subsequent analysis which are defined as follows:
$$\label{UR1_def}
U.R.^{(1)} = \dfrac{1}{n-d} \sum_{k = 0}^{n-d-1} \left( {\left\lvert \dfrac{ \dfrac{{\left\lVertr_k\right\rVert}}{ {\left\lVertb\right\rVert} } - \dfrac{{\left\lVert\epsilon_k\right\rVert}}{ {\left\lVertx\right\rVert}} }{ \dfrac{\chi_k}{ {\left\lVertx\right\rVert} } - \dfrac{{\left\lVert\epsilon_k\right\rVert}}{ {\left\lVertx\right\rVert} } } \right\rvert} \right )$$
\
$\&$
$$\label{UR2_def}
U.R.^{(2)} = \dfrac{1}{n-d} \sum_{k = 0}^{n-d-1} \left( {\left\lvert \dfrac{ \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVertb\right\rVert}^2 } - \dfrac{{\left\lVert\epsilon_k\right\rVert}^2}{ {\left\lVertx\right\rVert}^2} }{ \dfrac{\chi_k^2}{ {\left\lVertx\right\rVert}^2 } - \dfrac{{\left\lVert\epsilon_k\right\rVert}^2}{ {\left\lVertx\right\rVert}^2 } } \right\rvert} \right )$$
\
According to \[UR1\_def\] and \[UR2\_def\], it can be seen that $U.R.^{(1)}$ and $U.R.^{(2)}$ are functions of matrix $A$ and vectors $b$, $x^0$ and delay parameter $d$ of estimator. Below, we show why the forward condition number of problem \[cn of prob1\] encapsulates the parameters of the problem in this estimation. Calculating $\lVert x \rVert $ is not trivial but we use the norm of $x$ at each iterate (i.e ${\left\lVertx_k\right\rVert}$) in place of $\lVert x \rVert$ since $\lVert x_k \rVert$ converges to $\lVert x \rVert$ in first few iterations and doing so brings marginal changes in the estimation.
The Dataset consists of $O(n^2)$ problems of high condition numbers on which the performance metrics are measured. The convergence of an krylov subspace based iterative methods largely depend on the eigenvalue spectrum of matrix $A$. The uncertainties in the estimation of error by estimator or residual can also depend on the convergence behaviour apart from other parameters. Hence, we consider two kinds of matrices i.e. positive definite and indefinite matrices.
### Theorem on Expectation of $U.R.^{(1)}$ and $ U.R.^{(2)}$
\[theorem2\] For given singular value distribution and $\kappa(A,x)$,
$$E \left( U.R.^{(1)} \right) \approx \sqrt{\dfrac{2}{3}} \left( \dfrac{{\left\lVertA\right\rVert}_F}{\sqrt{n}} \dfrac{{\left\lVertx\right\rVert}}{{\left\lVertb\right\rVert}} \right) \left ( 1+\dfrac{d}{n-d}\log \left ( \dfrac{\sqrt{n} - \sqrt{\dfrac{d}{2}}}{\sqrt{d+1} - \sqrt{\dfrac{d}{2}}} \right ) + \left( \dfrac{\sqrt{2d}}{\sqrt{n}+\sqrt{d}} \right ) \right ) ,$$
$\&$
$$E \left( U.R.^{(2)} \right) \approx \left( \dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{{\left\lVertb\right\rVert}^2} \right) \left ( 1+\dfrac{d}{n-d}\log(n-d) \right )$$
Provided, $\kappa(A,x) >> 1$.
**Derivation for** $E(U.R.^{(2)})$:\
The $U.R.^{(2)}$ can be written as $$U.R.^{(2)} = \dfrac{1}{n-d} \sum_{k=0}^{n-d-1} a_k$$
where $$a_k = {\left\lvert \dfrac{ \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} - 1 }{ \dfrac{\chi_k^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } - 1 } \right\rvert}$$
Now, consider the error at $k^{th}$ iteration in GMRES or other krylov-subspace based methods, $$\epsilon_k = \epsilon_0 - V_kd_k$$
where $V_k$ is the matrix representing basis of subspace of krylov vectors or search directions.
The error in krylov subspace based iterative algorithm generally decreases and convergence is guaranteed in atmost $n$ iterations where $n$ is dimension of matrix $A$. The basis $V_k$ can atmost span k dimensions and if we average across the problems with constant forward condition number and singular values then the error vector $\epsilon_k$ on average will lie in n-k dimensional subspace provided $\epsilon_0$ lies in entire n-dimensional space with any direction being equally probable. Hence, we can write, $$\label{err_k}
\epsilon_k = c_1v_{i_1} + c_2v_{i_2} + ... + c_{n-k}v_{i_{n-k}}$$
$\&$
$$\label{res_k}
r_k = c_1 \sigma_{i_1}u_{i_1} + c_2\sigma_{i_2}u_{i_2} + ... + c_{n-k}\sigma_{i_{n-k}}u_{i_{n-k}}$$
where $u_{i_{k}}$ and $v_{i_{k}}$ are the left and right singular vectors of matrix A respectively.\
Now, lets consider the problems where $\sigma_{i_{k}}$ are fixed. In such scenario, $$E_{ \sigma_{i_1},..,\sigma_{i_{n-k}} } \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) = \left( \dfrac{{\left\lVertA\right\rVert}_F^2 {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \left(\dfrac{1}{{\left\lVertA\right\rVert}_F^2} \right ) E_{ \sigma_{i_1},..,\sigma_{i_{n-k}} } \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \right)$$ This can be done as forward condition number and singular values are constant across problems on which we are averaging. The operator $E_{ \sigma_{i_1},..,\sigma_{i_{n-k}} }$ will be replaced by $P_{n-k}$ for following analysis.\
$$\begin{split}
P_{n-k} \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) &= \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) P_{n-k} \left( \dfrac{{\left\lVertc_1 \sigma_{i_1}u_{i_1} + c_2\sigma_{i_2}u_{i_2} + ... + c_{n-k}\sigma_{i_{n-k}}u_{i_{n-k}}\right\rVert}^2}{ {\left\lVertc_1v_{i_1} + c_2v_{i_2} + ... + c_{n-k}v_{i_{n-k}}\right\rVert}^2 } \right) \\
&= \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) P_{n-k} \left( \dfrac{ (c_1^2 \sigma_{i_1}^2 + c_2^2\sigma_{i_2}^2 + ... + c_{n-k}^2\sigma_{i_{n-k}}^2 )}{ (c_1^2 + c_2^2 + ... + c_{n-k}^2) } \right)
\end{split}$$
As all directions in $n-k$ dimensional space are equally probable for error to point at, hence, the coefficients will follow gaussian distribution for such a vector and hence, $P_{n-k}(c_i^2) = 1$. $$\begin{split}
P_{n-k} \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) & \approx \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \left( \dfrac{ P_{n-k}(c_1^2 \sigma_{i_1}^2 + c_2^2\sigma_{i_2}^2 + ... + c_{n-k}^2\sigma_{i_{n-k}}^2 )}{ P_{n-k} (c_1^2 + c_2^2 + ... + c_{n-k}^2) } \right) \\
& \approx \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \left( \dfrac{ ( \sigma_{i_1}^2 + \sigma_{i_2}^2 + ... + \sigma_{i_{n-k}}^2 )}{ n-k } \right)
\end{split}$$
However, to find total expectation, we should consider all possible combinations of singular values to be equally probable and thus, $$E \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \approx \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \dfrac{1}{\binom{n}{n-k}} \sum_{(i_1,..,i_{n-k}) \in S} \left( \dfrac{ ( \sigma_{i_1}^2 + \sigma_{i_2}^2 + ... + \sigma_{i_{n-k}}^2 )}{ n-k } \right)$$ where $S$ is index set containing all $\binom{n}{n-k}$ combinations.
$$E \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \approx \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \dfrac{1}{\binom{n}{k} (n-k)} \left( \binom{n}{k} {\left\lVertA\right\rVert}_F^2 - \sum_{(i_1,..,i_{k}) \in S'} \left( \sigma_{i_1}^2 + \sigma_{i_2}^2 + ... + \sigma_{i_{k}}^2 \right) \right)$$ where $S'$ is the complementary index set which contains all $\binom{n}{k}$ combinations.
$$E \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \approx \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \dfrac{1}{ (n-k)} \left( {\left\lVertA\right\rVert}_F^2 - \dfrac{\sum_{(i_1,..,i_{k}) \in S'} \left( \sigma_{i_1}^2 + \sigma_{i_2}^2 + ... + \sigma_{i_{k}}^2 \right)}{ \binom{n}{k}} \right)$$
$$E \left( \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \approx \left( \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \dfrac{1}{ (n-k)} \left( {\left\lVertA\right\rVert}_F^2 - \dfrac{{\left\lVertA\right\rVert}_F^2 \binom{n-1}{k-1}}{ \binom{n}{k}} \right)$$
$$E \left( \dfrac{{\left\lVertr^k\right\rVert}^2}{ {\left\lVert\epsilon^k\right\rVert}^2 } \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \approx \left( \dfrac{{\left\lVertA\right\rVert}_F^2 {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \right) \dfrac{1}{ (n-k)} \left( 1 - \dfrac{ k}{ n} \right) \approx \dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}$$
Now, consider $$S = \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{ {\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}$$ It can be seen that the random variable S has well defined bounds as per Eqn. \[boundsS\] which can be derived using Cauchy-Schwartz inequality: $$\label{boundsS}
\dfrac{\kappa(A,x)^2}{\kappa^2} \leq S \leq \kappa(A,x)^2$$
let $a=\dfrac{\kappa(A,x)^2}{\kappa^2}$.\
The lower bound $a$ lies between 0 and 1 where as upperbound is much greater than 1. The random variable S can be seen to follow the relation \[rmv\_onenorm\] (proof in section \[ExpofS\] of Appendix): $$\label{rmv_onenorm}
E({\left\lvertS-1\right\rvert}) = E(S) - 1 + 2\int_a^1 (1-s)f(s)ds$$
Since, $E(S) \approx \dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} >> 1$, $$E({\left\lvertS-1\right\rvert}) \approx \dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}$$
Now, $$E(a_k) \approx \dfrac{ E \left( {\left\lvert \dfrac{{\left\lVertr_k\right\rVert}^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} - 1 \right\rvert} \right) }{ E \left( {\left\lvert \dfrac{\chi_k^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } - 1 \right\rvert} \right) }$$
For delay-based estimators, $\chi_k^2 \approx {\left\lVert\epsilon_k\right\rVert}^2 - {\left\lVert\epsilon_{k+d}\right\rVert}^2$, which means estimator is slightly less than actual error in general and thus the probability mass should be significant for $\dfrac{\chi_k^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \leq 1$. This justification allows us to write the following step.
$$E(a_k) \approx \dfrac{\dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}}{1-E\left ( \dfrac{\chi_k^2}{ {\left\lVert\epsilon_k\right\rVert}^2 } \right )} \approx \dfrac{\dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}}{E \left( \dfrac{{\left\lVert\epsilon_{k+d}\right\rVert}^2}{{\left\lVert\epsilon_k\right\rVert}^2} \right) } \approx \dfrac{\dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}}{ \dfrac{E({\left\lVert\epsilon_{k+d}\right\rVert}^2)}{E({\left\lVert\epsilon_k\right\rVert}^2)} }$$
Thus, According to Eqn. \[err\_k\], $$E(a_k) \approx \dfrac{\dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}}{ \dfrac{E(c_1^2 + c_2^2 + ... + c_{n-k-d}^2)}{E(c_1^2 + c_2^2 + ... + c_{n-k}^2)} }$$ It can be stated that $E(c_i^2) = 1$ as averaging over all combinations of singular values will not change the average since every combination is equally probable.\
Thus, $$E(a_k) \approx \dfrac{\dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2}}{ \dfrac{n-k-d}{n-k} }$$
$$E(a_k) \approx \dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{ {\left\lVertb\right\rVert}^2} \left( 1 + \dfrac{d}{n-k-d} \right)$$ Thus, $$E \left( U.R.^{(2)} \right) \approx \left( \dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{{\left\lVertb\right\rVert}^2} \right) \dfrac{1}{n-d}\sum_{k=0}^{n-d-1} \left ( 1 + \dfrac{d}{n-k-d} \right)$$
$$E \left( U.R.^{(2)} \right) \approx \left( \dfrac{{\left\lVertA\right\rVert}_F^2}{n} \dfrac{{\left\lVertx\right\rVert}^2}{{\left\lVertb\right\rVert}^2} \right) \left ( 1+\dfrac{d}{n-d}\log(n-d) \right )$$
**Derivation for $E(U.R.^{(1)})$:**\
Consider $$Q = \dfrac{{\left\lVertr_k\right\rVert}}{ {\left\lVert\epsilon_k\right\rVert} } \dfrac{{\left\lVertx\right\rVert}}{ {\left\lVertb\right\rVert}}$$ The random variable Q has well defined bounds as follows: $$\dfrac{\kappa(A,x)}{\kappa} \leq Q \leq \kappa(A,x)$$ It can be assumed to follow the triangular distribution with mode at 1. This can be considered as safe assumption as it is in the favor of residual stopping criterion. However, this assumption also gives good intuition that most likely value of relative residual is relative error however due to huge variation, the relative residual on average is too far from relative error. The figure \[Qdist\] depicts the distribution of Q.
![Triangular Distribution of Q[]{data-label="Qdist"}](images/tdistQ.jpg)
For such a skewed triangular distribution, $$E(Q^2) \approx \dfrac{3}{2} E(Q)^2$$ Thus, $$E(Q) \approx \sqrt{\dfrac{2}{3}} \left( \dfrac{{\left\lVertA\right\rVert}_F}{\sqrt{n}} \dfrac{{\left\lVertx\right\rVert}}{{\left\lVertb\right\rVert}} \right)$$
Let $$\psi = \dfrac{\chi_k}{{\left\lVert\epsilon_k\right\rVert}}$$ The distribution of random variable $\psi$ can be approximated well by the exponential distribution as the significant mass of probability lies between 0 and 1 and decreases very fast after 1. For exponential distribution, $$E(\psi^2) = 2 \left( E(\psi) \right)^2$$ Hence, $$E \left( \psi^2 \right) = 1-\dfrac{n-k-d}{n-k} = \dfrac{d}{n-k}$$ $$E \left( \dfrac{\chi_k}{{\left\lVert\epsilon_k\right\rVert}} \right) = \sqrt{ \dfrac{1}{2} \left( \dfrac{d}{n-k} \right) }$$ Now, $$E \left( U.R.^{(1)} \right) \approx \sqrt{\dfrac{2}{3}} \left( \dfrac{{\left\lVertA\right\rVert}_F}{\sqrt{n}} \dfrac{{\left\lVertx\right\rVert}}{{\left\lVertb\right\rVert}} \right) \left( \dfrac{1}{n-d} \sum_{k=0}^{n-d-1} \dfrac{1}{1 - \sqrt{ \dfrac{1}{2} \left( \dfrac{d}{n-k} \right) } } \right)$$
$$E \left( U.R.^{(1)} \right) \approx \sqrt{\dfrac{2}{3}} \left( \dfrac{{\left\lVertA\right\rVert}_F}{\sqrt{n}} \dfrac{{\left\lVertx\right\rVert}}{{\left\lVertb\right\rVert}} \right) \left( \dfrac{1}{n-d} \sum_{k=0}^{n-d-1} \left( 1 + \dfrac{d}{n-k-\dfrac{d}{2}} + \dfrac{\sqrt{\dfrac{d}{2}}}{\sqrt{n-k} +\sqrt{\dfrac{d}{2}}} \right) \right)$$
$$E \left( U.R.^{(1)} \right) \approx \sqrt{\dfrac{2}{3}} \left( \dfrac{{\left\lVertA\right\rVert}_F}{\sqrt{n}} \dfrac{{\left\lVertx\right\rVert}}{{\left\lVertb\right\rVert}} \right) \left ( 1+\dfrac{d}{n-d}\log \left ( \dfrac{\sqrt{n} - \sqrt{\dfrac{d}{2}}}{\sqrt{d+1} - \sqrt{\dfrac{d}{2}}} \right ) + \left( \dfrac{\sqrt{2d}}{\sqrt{n}+\sqrt{d}} \right ) \right )$$
\[lemma1\]
$$U.R.^{(1)} = O\left( \kappa(A,x) \left( \dfrac{1}{E^2} \right)^{\dfrac{d}{n}} \right)$$ $$U.R.^{(2)} = O\left( \kappa(A,x)^2 \left( \dfrac{1}{E^2} \right)^{\dfrac{d}{n}} \right)$$ where $E$ is the stopping tolerance on relative error.
Using Cauchy-Schwartz inequality, the $U.R.^{(1)}$ can be seen to satisfy equation \[ineqUR\]. $$\label{ineqUR}
U.R.^{(1)} \leq \dfrac{\kappa(A,x)}{n-d} \left( \sum_{k=0}^{n-d-1} \dfrac{1}{{\left\lvert1-\dfrac{\chi_k}{{\left\lVert\epsilon_k\right\rVert}}\right\rvert}} \right)$$
For delay-based estimators, $\dfrac{\chi_k}{{\left\lVert\epsilon_k\right\rVert}} \approx \sqrt{1-\dfrac{{\left\lVert\epsilon_{k+d}\right\rVert}^2}{{\left\lVert\epsilon_{k}\right\rVert}^2}}$.\
Since, we are interested in upperbound, term $\dfrac{{\left\lVert\epsilon_{k+d}\right\rVert}^2}{{\left\lVert\epsilon_{k}\right\rVert}^2} << 1$ and error decreases exponentially. The positive definite matrices have exponential convergence rates and are considered as good matrices from convergence point of view. Hence, use of exponential convergence rates is justified for indefinite matrices in order to find upperbound on convergence rate. Therefore, we have, $$\sqrt{1-\dfrac{{\left\lVert\epsilon_{k+d}\right\rVert}^2}{{\left\lVert\epsilon_{k}\right\rVert}^2}} \approx 1 - \dfrac{1}{2} \dfrac{{\left\lVert\epsilon_{k+d}\right\rVert}^2}{{\left\lVert\epsilon_{k}\right\rVert}^2}$$
Let $UB$ be upperbound on $U.R.^{(1)}$. Then, $$UB \approx 2 \left( \dfrac{\kappa(A,x)}{n-d} \right) \left( \sum_{k=0}^{n-d-1} \dfrac{{\left\lVert\epsilon_{k}\right\rVert}^2}{{\left\lVert\epsilon_{k+d}\right\rVert}^2} \right)$$
Now, we know that error can fall upto prescribed tolerance level only and we can assume that relative error at starting iteration is $O(1)$. Therefore, $$\label{ekratio}
\dfrac{{\left\lVert\epsilon_{k}\right\rVert}^2}{{\left\lVert\epsilon_{k+d}\right\rVert}^2} \approx \left( \dfrac{1}{E^2} \right)^{\dfrac{d}{n}}$$
Thus, $$UB \approx 2 \left( \kappa(A,x) \right) \left( \dfrac{1}{E^2} \right)^{\dfrac{d}{n}}$$ Hence, $$\label{UR1O}
U.R.^{(1)} = O\left( \kappa(A,x) \left( \dfrac{1}{E^2} \right)^{\dfrac{d}{n}} \right)$$
Again using Cauchy-Schwartz inequality, the $U.R.^{(2)}$ can be seen to satisfy equation \[ineqUR2\]. $$\label{ineqUR2}
U.R.^{(2)} \leq \dfrac{\kappa(A,x)^2}{n-d} \left( \sum_{k=0}^{n-d-1} \dfrac{1}{{\left\lvert1-\dfrac{\chi_k^2}{{\left\lVert\epsilon_k\right\rVert}^2}\right\rvert}} \right)$$ i.e. $$U.R.^{(2)} \leq \dfrac{\kappa(A,x)^2}{n-d} \left( \sum_{k=0}^{n-d-1} \dfrac{{\left\lVert\epsilon_k\right\rVert}^2 }{{\left\lVert\epsilon_{k+d}\right\rVert}^2} \right)$$
According to equation \[ekratio\], $$\label{UR2O}
U.R.^{(2)} = O\left( \kappa(A,x)^2 \left( \dfrac{1}{E^2} \right)^{\dfrac{d}{n}} \right)$$
Experimental results on performance of error estimates as the stopping criteria
-------------------------------------------------------------------------------
Figures \[A\_norm\_sp\] and \[l\_2norm\_sp\] show the increase in performance ratio (both for$A$-norm and $l_2$ norm estimation) as the condition number of the problem increases when matrices are taken to be Non-symmetric and Indefinite for Bi-CG Algorithm. Here each blue dot signifies the mean average value of uncertainty ratio (Equation \[UR1\_def\]) for a particular problem (i.e for a particular matrix $A$ and a right hand side vector $b$).
![Linear increase in performance ratio for estimation of A-norm of error with increase in condition number of the problem when A is Non-symmetric and Indefinite, green line represents $\kappa(A,x)$ and red line represents the threshold line below which $\frac{{\left\lVertr\right\rVert}}{{\left\lVertb\right\rVert}}$ is a better estimator of error as compared to $\frac{{\left\lVertg\right\rVert}}{{\left\lVertx\right\rVert}}$[]{data-label="A_norm_sp"}](images/A_norm_sp){width="\linewidth"}
![Linear increase in performance ratio for estimation of $l_{2}$ norm of error with increase in condition number of the problem when A is Non-symmetric and Indefinite, green line represents $\kappa(A,x)$ and red line represents the threshold line below which $\frac{{\left\lVertr\right\rVert}}{{\left\lVertb\right\rVert}}$ is a better estimator of error as compared to $\frac{{\left\lVertf\right\rVert}}{{\left\lVertx\right\rVert}}$[]{data-label="l_2norm_sp"}](images/l_2norm_sp){width="\linewidth"}
There is a linear increase in the performance ratio with condition number of the problem, showing the robustness of our estimator. The same trend is expected for Non-symmetric but positive definite matrices as shown in Figure \[l2\_norm\_PD\_Scatterplot\]. The red line in all the above mentioned figures is just a threshold line and a point below the red line only depicts that relative residual was a better estimator of error as compared to our estimator (any measure) for a particular backward problem.\
![Linear increase in performance ratio for estimation of $l_{2}$ norm of error with increase in condition number of the problem when matrix (100 $\times $ 100) is Non-symmetric positive definite[]{data-label="l2_norm_PD_Scatterplot"}](images/l2_norm_PD_Scatterplot){width="\linewidth"}
Note that condition number of the problem for A-norm performance ratio will be defined by A-norm rather than a 2-norm condition number. Moreover the similar trend is expected in case of CGQL and GMRES algorithms where their estimators are compared to relative residual as a stopping criteria as shown in Figures \[BiCGQL\_SPD\] and \[GMRES\].
![Linear increase in performance ratio for estimation of $l_{2}$ norm of error with increase in condition number of the problem when matrix (100 $\times $ 100) is Non-symmetric positive definite in CG algorithm[]{data-label="BiCGQL_SPD"}](images/BiCGQL_SPD){width="\linewidth"}
![Linear increase in performance ratio for estimation of $l_{2}$ norm of error with increase in condition number of the problem when matrix (100 $\times $ 100) is Non-symmetric positive definite in GMRES algorithm[]{data-label="posdefURGMRES"}](images/posdefURGMRES){width="\linewidth"}
Moreover the BiCGQL estimator is equivalent to a CGQL estimator when the matrix is Symmetric and Positive definite which is a requirement for guaranteed convergence of Conjugate Gradient Algorithm. Thus we expect to see a similar pattern of the trend line of performance ratio as the condition number of the problem increases for a CGQL Algorithm in Figure \[BiCGQL\_SPD\]. This confirms the robustness of all three estimators (Bi-CG, CG and GMRES).
![Linear increase in performance ratio for estimation of $l_{2}$ norm of error with increase in condition number of the problem when matrix (100 $\times $ 100) is Non-symmetric in GMRES algorithm[]{data-label="GMRES"}](images/indefURGMRES.png){width="\linewidth"}
### Uncertainty Ratio and the delay parameter
It is evident that increase in delay $d$ will increase the estimator performance and in order to assess that performance experimentally, uncertainty ratio $U.R.^{(1)}$ (nomrlalized by forward condition number based on frobenius norm) i.e. $\kappa_F(A,x) = \dfrac{{\left\lVertA\right\rVert}_F}{\sqrt{n}} \dfrac{{\left\lVertx\right\rVert}}{{\left\lVertb\right\rVert}}$ is averaged over $O(n^2)$ problems for each $d$ with dimension $n = 100$. The same experiment was done for $U.R.^{(2)}$ and results are shown in Fig. \[fig:dplots\]. The experimental results are compared with theoretical results provided in \[theorem2\] and \[lemma1\].
The dataset consists of non-symmetric positive definite matrices of high condition number of $O(10^6)$. The results for both BiCG and GMRES algorithms are above the average line and this could be explained by the increased convergence rates due to positive definiteness of matrices. However, the Upperbound line tightly bounds the average $U.R.$ normalised by $\kappa_F(A,x)$ for tolerance level of $10^{-6}$.
Computation saved with BiCGQL and GMRES estimators
--------------------------------------------------
There can always be an under computation or an over computation involved for desired relative error when dealing with the convergence of an iterative method. Relying on relative residue might stop the iterations too early or might take too many iterations to converge to the final solution. The early stopping leads to loss of accuracy of solution which is proportional to difference of iterations required for relative error and relative residual to reach same stopping tolerance. We will call these iterations as *accuracy loss iterations*. The late stopping of leads to wastage of computation in the form of iterations. we will call these iterations as *computation loss iterations*. Use of error estimators owing to very less uncertainty in estimating relative error as compared to relative residue can reduce these lost iterations to significant amount as shown in figures \[pic:iter\_saved1\] and \[pic:iter\_saved2\].
![Illustration of over computation due to large uncertainty of relative residue leading to computation loss iterations for $1000 \times 1000$ nonsymmetric positive definite matrix with condition number of $O(10^{11})$ with random right hand side. []{data-label="pic:iter_saved1"}](images/itersav1){width="0.9\linewidth"}
![Illustration of under computation due to large uncertainty of relative residue leading to accuracy loss iterations for matrix ’sherman2’ from matrix market provided along with its right hand side. []{data-label="pic:iter_saved2"}](images/itersav2){width="0.9\linewidth"}
The amount of savings should increase with dimensionality ($n$) of matrix as the uncertainty ratio according to equation \[E\_UR1\] increases with size of matrix. It is clear that convergence rate will keep on decreasing as the size of matrix increases leading to more iterations saved by estimator and this could be proportional to dimension of matrix ($n$).
Conclusions
===========
The importance of error estimators for efficient stopping (or restarting) are strongly evident for problems with even moderately high condition number $\kappa > 100$, and is emphasized by numerical examples, and the expected uncertainty in convergence otherwise using the residual. One might choose to use pre-conditioners for a highly conditioned problem but in most cases the structure of the matrix as well as properties of the matrix like condition number, spectra are not known. Even knowing that the condition number of the matrix is very high may not be helpful, as if one were able to reduce the condition number of a matrix it does not always imply reducing the condition number of the backward problem solved. Also one might actually increase the condition number of the forward problem and hence the uncertainty in error.\
For Non-Symmetric matrices Bi-CG shows an irregular convergence which can be improved by using an extended version of the algorithm, BiCGSTAB [@van1992bi] [@sleijpen1994bicgstab] [@sleijpen1993bicgstab]. BiCGSTAB acts as a moving window average of Bi-CG iterates which smoothen the convergence and there are other versions of BiCGSTAB that can be interpreted as the product of BiCG [@fletcher1976conjugate] and repeated application of the Generalized minimal residual method (GMRES) [@saad1986gmres] in which a residual vector is minimized, which leads to a considerably smoother convergence behavior. It should also be noted that relations \[eq:14\] and \[eq:17\] hold valid for BiCGSTAB algorithms. Similarly, results showing the reduction of uncertainty in convergence while using error estimators were presented for GMRES. For HPD systems the Bi-CG delivers the same results as CG, but at twice the cost per iteration. Based on the results discussed in the previous sections, we believe that the estimate for the A-norm or the $l_{2}$ norm of the error should be implemented into software realization of iterative solvers, to use errors as stopping criteria instead of the residual.
Appendix
========
CG and its relation to (symmetric) Lanczos algorithm
----------------------------------------------------
$ \textbf{input: } A,v $ $ \beta_{0} = 0 , v_{0} = 0$ $ v_{1} = \dfrac{v}{\lVert v \rVert} $
$m = Av_{k} - \eta_{k-1}v_{k-1}$ $\omega_{k} = v_{k}^{T}m$ $m = m - \omega_{k}v_{k}$ $\eta_{k} = \lVert m \rVert$ $v_{k+1} = \dfrac{m}{\eta_{k}}$
The Lanczos Algorithm [@lanczos1950iteration] [@parlett1979lanczos]) can be viewed as a simplified Arnoldi’s algorithm in that it applies to Hermitian matrices. The $k^{th}$ step of the algorithm transforms the matrix $A$ into a tri-diagonal matrix $T_{k}$; when k is equal to the dimension of A, $T_{k}$ is similar to $A$.
Given a starting vector $v_1$ and a symmetric matrix $A$, the Lanczos algorithm (Algorithm \[Lanczosalgorithm\]) computes an orthonormal basis $v_{1} , v_{2} .... v_{k+1}$ of the krylov subspace $K_{k+1} (A,v_1)$ and transforms the matrix $A$ to a tri-diagonal matrix $T$.
$$\label{eq:13_A}
K_{k+1} (A,v_1) = span\{v_1,Av_1, \cdots A^{k}v_1\} \\$$
The basis vectors $v_{k}$ satisfy the matrix relation:
$$\label{eq:14_A}
AV_{k} = V_{k}T_{k} + \eta_{k+1}v_{k+1}e_{k}^{T} \\$$
Here, $ e_{k} $ is the $ k^{th} $ canonical vector, where $V_{k} = [v_{1},....v_{k}]$ and $T_{k}$ is the k $\times$ k symmetric tri-diagonal matrix of recurrence coefficients in Algorithm \[Lanczosalgorithm\]:
$$\label{eq:15_A}
T_{n} = \left[
\begin{array}{ccccc}
\omega_{1} & \eta_{1} & 0 & \cdots& 0\\
{\eta_{1}} & \omega_{2} & \eta_{2} & & \vdots\\
0 & \ddots&\ddots&\ddots& 0\\
\vdots& & {\eta_{n-1}} & \omega_{n-1} & \eta_{n-1}\\
0 & \cdots& 0 & {\eta_{n-1}} & \omega_{n}
\end{array}
\right]
\hfill \break
\\$$
\
When solving a system of linear algebraic equations $Ax = b$ with symmetric and positive definite matrix A, the CG method (Algorithm \[CG algorithm\]) computes iterates $x_{k}$ that are optimal since the A-norm of error defined in (\[error\]) is minimized over $x_{0} + \kappa_{k} (A,r_{0})$,
$$\label{eq:16_A}
\lVert x - x_{k} \rVert_{A} = \underset{y \in x_{0} + \kappa_{k}(A,r_{0})} {\mathrm{min}} \lVert x - y \rVert_{A} \\$$
$ \textbf{input: } A,b,x_{0} $ $r_{0} = b-Ax_{0}$ $ p_{0} = r_{0} $
$\alpha_{k} = \dfrac{r_{k}^{T}r_{k}}{p_{k}^{T}Ap_{k}} $ $ x_{k+1} = x_{k} + \alpha_{k}p_{k}$ $r_{k+1} = r_{k} - \alpha_{k}Ap_{k}$ $\beta_{k} = \dfrac{r_{k+1}^{T}r_{k+1}}{r_{k}^{T}r_{k}}$ $p_{k+1} = r_{k+1} + \beta_{k}p_{k}$
CG can be derived from Lanczos Algorithm [@lanczos1950iteration] [@parlett1979lanczos]) and relation between CG and Lanczos coefficients are as follows:
$$\label{relation}
\eta_{k} = \frac{\sqrt{\beta_{k} }}{\alpha_{k}} , \omega_{k} = \frac{1}{\alpha_{k-1}} + \frac{\beta_{k-1}}{\alpha_{k-2}}$$
\
Relating BiCGQL to CGQL {#BCGtoCG}
-------------------------
The difference between two consecutive A-norm of error (A-measure) in case of a Conjugate gradient Algorithm at iteration $'k'$ and $'k+1'$ can be given by:
$$\lVert \epsilon_{k} \rVert_{A}^{2} - \lVert \epsilon_{k+1} \rVert_{A}^{2} = \alpha_{k} r_{k}^{T} r_{k}$$
Hence by inducing a delay of $'d'$ iterations we can easily compute A-norm of error.
$$\begin{array}{ccl}
\lVert \epsilon_{k} \rVert_{A}^{2} - \lVert \epsilon_{k+d} \rVert_{A}^{2} = \sum_{j=k}^{k+d} \alpha_{j} r_{j}^{T} r_{j} \\
\lVert \epsilon_{k} \rVert_{A}^{2} \approx \sum_{j=k}^{k+d} \alpha_{j} r_{j}^{T} r_{j}
\end{array}$$
For A-norm estimation in BiCGQL we derived the following results:
$$r_{k+1}^T A^{-1} r_{k+1} = - \alpha_{k} r_{k}^T p_{k} + r_{k+1}^T A^{-1} r_{k} + \alpha_{k}^{2} p_{k}^T A p_{k}$$
The above result can also be written as:
$$r_{k}^T A^{-1} r_{k} - r_{k+1}^T A^{-1} r_{k+1} = \alpha_{k} r_{k}^T p_{k} + \alpha_{k} r_{k}^T (A^{T})^{-1} A p_{k} - \alpha_{k}^{2} p_{k}^T A p_{k}$$
For a symmetric matrix $A=A^{T}$ the above equation can be further written as:
$$r_{k}^T A^{-1} r_{k} - r_{k+1}^T A^{-1} r_{k+1} = \alpha_{k} r_{k}^T p_{k} + \alpha_{k} r_{k}^Tp_{k} - \alpha_{k}^{2} p_{k}^T A p_{k}$$
For an algorithm like CG $p_{i}^{T}r_{j} = 0$ for $i \neq j $. Also $ r_{k+1} = r_{k} - \alpha_{k} A p_{k} $ and thus substituting $A p_{k} = \dfrac{r_{k} - r_{k+1}}{\alpha_{k}}$ in the last term we get:
$$\label{CGQL_to_BICGQL}
\begin{array}{ccl}
r_{k}^T A^{-1} r_{k} - r_{k+1}^T A^{-1} r_{k+1} = \alpha_{k} r_{k}^T p_{k} + \alpha_{k} r_{k}^Tp_{k} - \alpha_{k}p_{k}^Tr_{k} + \alpha_{k}p_{k}^{T}r_{k+1}\\
\implies r_{k}^T A^{-1} r_{k} - r_{k+1}^T A^{-1} r_{k+1} = \alpha_{k} r_{k}^T p_{k}\\
\implies r_{k}^T A^{-1} r_{k} - r_{k+1}^T A^{-1} r_{k+1} = \alpha_{k} r_{k}^T r_{k} + \beta_{k-1}r_{k}^{T}p_{k-1}\\
\implies r_{k}^T A^{-1} r_{k} - r_{k+1}^T A^{-1} r_{k+1} = \alpha_{k} r_{k}^T r_{k}
\end{array}$$
Equation \[CGQL\_to\_BICGQL\] is equivalent to the result for $A$-norm estimation in CGQL Algorithm. Thus our $A$-norm estimator (BiCGQL) is equivalent to a CGQL estimator when $A=A^T$.
### Relations between Non-Symmetric Lanczos Tridiagonal Matrix ($ T_{k} $) and Residual vectors ($r$ and $\tilde{r}$) of BiCG algorithm
A direct relationship between $T_{k}$ , $r_{k}$ and $\tilde{r_{k}}$ can be given by:
$$(T_{n}^{-1})_{(1,1)} = (T_{k}^{-1})_{(1,1)} + \frac { \tilde{r_{k}}^{T}A^{-1}r_{k} } {\lVert r_{0} \rVert^{2} }
\label{eq: x1}$$
$$(T_{n}^{-1})_{(1,1)} = (T_{k+1}^{-1})_{(1,1)} + \frac { \tilde{r}_{k+1}^{T}A^{-1}r_{k+1} } { \lVert r_{0} \rVert ^{2} }
\label{eq: x2}$$
Subtracting \[eq: x1\] and \[eq: x2\] we get:
$$(T_{k+1}^{-1})_{(1,1)} = (T_{k}^{-1})_{(1,1)} + \frac{ ( \tilde{r_{k}}^{T}A^{-1}r_{k} - \tilde{r}_{k+1}^{T}A^{-1}r_{k+1} ) } { \lVert r_{0} \rVert^{2} }$$
As $\tilde{r_{k}}^{T}A^{-1}r_{k} - \tilde{r}_{k+1}^{T}A^{-1}r_{k+1} = \alpha_{k}\tilde{r}_{k}^{T}r_{k}$ for a Non-Symmetric matrix in BiCG:
$$(T_{k+1}^{-1})_{(1,1)} = (T_{k}^{-1})_{(1,1)} + \frac{ ( \alpha_{k}\tilde{r}_{k}^{T}r_{k} ) } { \lVert r_{0} \rVert^{2} }$$
Thus knowing the relation between two consecutive $T_{k}$ inverse first elements we can relate it with the estimator we developed for A-norm of error and the approach of CGQL Algorithm. However relating Non-Symmetric Lanczos and BiCG through Quadrature based methods will involve Complex Gaussian Quadratures ([@saylor2001gaussian]). Hence we have followed an equivalent but direct approach of estimation using the relations of Bi-CG explicitly.
**input:** A,b,$x_{0}$,$\lambda_{m}$,$\lambda_{M}$ $r_{0} = b-Ax_{0}$, $p_{0} = r_{0}$ $\eta_{0} = 0$, $\alpha_{-1} = 1$, $c_{1} = 1$, $\beta_{0} = 0$, $\delta_{0} = 1$, ${\overline{\omega}}(\mu)^{1} = \lambda_{m}$, ${\underline{\omega}}(\eta)^{1} = \lambda_{M}$
$\omega_{k} = \dfrac{1}{\alpha_{k-1}} + \dfrac{\beta_{k-1}}{\alpha_{k-2}} $ $\eta_{k}^{2} = \dfrac{\beta_{k}}{\alpha_{k-1}^{2}}$ $ \delta_{k} = \omega_{k} - \dfrac{\beta_{k-1}^{2}}{\delta_{k-1}}$ $g_{k} = \lVert r_{0} \rVert \dfrac{c_{k}^{2}}{\delta_{k}}$ $\overline{\delta_{k}} = \omega_{k} - \overline{\omega_{k}}$ , $\overline{\omega_{k+1}} = \lambda_{m} + \dfrac{\beta^{2}}{\overline{\delta_{k}}}$ $\overline{f_{k}} = \lVert r_{0} \rVert^{2} \dfrac{\eta_{k}^{2} c_{k}^{2}}{\delta_{k} (\overline{\omega_{k+1}} \delta_{k} - \eta_{k}^{2})}$ $\underline{\delta_{k}} = \omega_{k} - \underline{\omega}_{k}$ , $\underline{\omega}_{k+1} = \lambda_{M} + \dfrac{\beta^{2}}{\overline{\delta}_{k}}$ $\underline{f_{k}} = \lVert r_{0} \rVert^{2} \dfrac{\eta_{k}^{2} c_{k}^{2}}{\delta_{k} (\underline{\omega}_{k+1} \delta_{k} - \eta_{k}^{2})}$ $\breve{\omega}_{k+1} = \dfrac{\overline{\delta}_{k} \underline{\delta_{k}}}{\overline{\delta{k}} - \underline{\delta_{k}}} ( \dfrac{\lambda_{m}}{\overline{\delta_{k}}} - \dfrac{\lambda_{M}}{\underline{\delta_{k}}} )$ $\breve{\eta}_{k} = \dfrac{\overline{\delta_{k}} \underline{\delta_{k}}}{\overline{\delta{k}} - \underline{\delta_{k}}} ( \lambda_{M} - \lambda_{m})$ $\overline{f_{k}} = \lVert r_{0} \rVert^{2} \dfrac{[\breve{\eta_{k}}]^{2} c_{k}^{2}}{\delta_{k} (\breve{\omega}_{k+1} \delta_{k} - [\breve{\eta_{k}}]^{2})}$ $c_{k+1}^{2} = \dfrac{\eta_{k}^{2} c_{k}^{2}}{\delta_{k}^{2}}$
### Bi-CG and its relation to Non-symmetric Lanczos algorithm
Bi-Conjugate Gradient algorithm is an extension to CG algorithm which is used to solve a system of linear equations and works even for a Non-symmetric (possibly indefinite) matrix.
**input** A, $A^{T}$, b, $x_{0}$, $y_{0}$ $r_{0}$ = b - A$x_{0}$ $\tilde{r_{0}}$ = b - $A^{T}$ $y_{0}$ $p_{0}$ = $r_{0}$ $q_{0}$ = $\tilde{r_{0}}$ ${\alpha}_{k-1}$ = ${\frac {\tilde{r}_{k-1}^T r_{k-1}}{q_{k-1}^T A p_{k-1}}}$ $x_{k}$ = $x_{k-1}$ + $\alpha_{k-1}p_{k-1}$ $y_{k}$ = $y_{k-1}$ + $\alpha_{k-1}q_{k-1}$ $ r_k = r_{k-1} - \alpha_{k-1}Ap_{k-1}$ $ \tilde{r}_k = \tilde{r}_{k-1} - \alpha_{k-1}A^{T}q_{k-1}$ ${\beta}_{k-1}$ = ${\frac {\tilde{r_{k}}^T r_{k}}{\tilde{r}_{k-1}^T r_{k-1}}}$ $p_{k} = r_{k} + \beta_{k-1}p_{k-1}$ $q_{k} = \tilde{r_{k}} + \beta_{k-1}q_{k-1}$
Bi-CG can also be derived from Non-symmetric Lanczos algorithm, for example considering $v_{1}$ and $\tilde{v_{1}}$ be the given starting vectors to Non-symmetric lancozs algorithm (such that $\lVert v_{1} \rVert = 1$ and $(v_{1},\tilde{v_{1}}) = 1)$, the two three term recurrences which help in forming two bi-orthogonal subspaces can be as follows:\
For k=1,2....
$$\begin{array}{ccl}
z_{k} = Av_{k} - w_{k}v_{k} - \eta_{k-1}v_{k-1}\\
\tilde{z_{k}} = A^{T}\tilde{v_{k}} - w_{k}\tilde{v_{k}} - \tilde{\eta}_{k-1}\tilde{v}_{k-1}
\end{array}$$
The coefficient $w_{k}$ being computed as $w_{k} = (\tilde{v_{k}},Av_{k})$. The other coefficients $\eta_{k}$ and $\tilde{\eta_{k}}$ are chosen (provided ($\tilde{z_{k}},v_{k}) = 0$) such that $\eta_{k}\tilde{\eta_{k}} = (\tilde{z_{k}},z_{k})$ and the new vectors at step $k+1$ are given by:
$$\begin{array}{ccl}
v_{k+1} = \dfrac{z_{k}}{{\eta_{k}}}\\ \\
\tilde{v}_{k+1} = \dfrac{\tilde{z_{k}}}{\tilde{\eta}_{k}}
\end{array}$$
These relations can again be written in the form of a non-symmetric tri-diagonal matrix form (under the condition $\tilde{V_k}^T A V_k = T_k$) as:
$$T_{k} = \left[
\begin{array}{ccccc}
\omega_{1} & \eta_{1} & 0 & \cdots& 0\\
\tilde{\eta_{1}} & \omega_{2} & \eta_{2} & & \vdots\\
0 & \ddots&\ddots&\ddots& 0\\
\vdots& & \tilde{\eta}_{k-2} & \omega_{k-1} & \eta_{k-1}\\
0 & \cdots& 0 & \tilde{\eta}_{k-1} & \omega_{k}
\end{array}
\right]
\hfill \break$$
\
Generalized minimal residual method (GMRES)
-------------------------------------------
(GMRES) [@saad1986gmres] is an iterative method for the numerical solution of a non-symmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual.
The krylov vectors in GMRES follow the (n+1)-term recurrence relation as follows:
$$Av_k = h_{1,k}v_1 + ... + h_{k,k}v_k + h_{k+1,k}v_{k+1}$$
where $h_{i,j}$ are the entries of Hessenberg matrix $H_k$.
The Arnoldi iteration in GMRES exploits this relation to compute these entries and vectors using Modified Gram-Schmidt (MGS) technique, as in Algorithm \[GMRESAlgo\].
\[GMRESAlgo\]
set $tol$ set $maxit$ (maximum iterations) set $k = 1$ n = dim(b) $r_0 = b-Ax_0$,$\beta = {\left\lVertr_0\right\rVert}$,$v_1 = r_0/\beta,res = \dfrac{\beta}{{\left\lVertb\right\rVert}} $ $s = Av_k$ $h_{j,k} = (v_j)^Ts$ $s = s - h_{j,k}v_j$ $h_{k+1,k} = {\left\lVerts\right\rVert}$. $v_{k+1} = s/h_{k+1,k}$ Define the $(k+1 \times k)$ Hessenberg matrix $H_k^{e}$ Compute $z_k$, the minimizer of ${\left\lVert\beta e_1 - H_k^{e} z_k\right\rVert}$ $x_k = x_0 + V_k z_k$ $res = \dfrac{{\left\lVertb-Ax_k\right\rVert}}{{\left\lVertb\right\rVert}}$ $k = k+1$
Bin experiment showing the average relative error in estimation of $l_2$ norm of error vector
---------------------------------------------------------------------------------------------
For the purpose of extended tests on the Bi-CG estimator we proposed, six different bins of hundred matrices were generated with varying condition number of matrix A: $10^1 10^{2}....10^{6}$. For each matrix A, 100 different instances of vector $'b'$ were created, each being unique canonical form of order 100. Thus each bin represents the result accumulated from $10,000$ different cases.
In order to verify the relative precision of the estimator for $l_2$ norm of error, we perform the average of equation \[Avg rel error wrt est for l\_2 norm\] over all iterations for each condition number of the matrix ranging from $10^1 - 10^6$.\
$$\label{Avg rel error wrt est for l_2 norm}
\left \lvert \dfrac { \dfrac{\lVert f_{k}\rVert}{\lVert x \rVert} - \dfrac{\lVert \epsilon_{k} \rVert}{\lVert x \rVert} } { \dfrac{\lVert \epsilon_{k} \rVert}{\lVert x \rVert}} \right \rvert$$\
![Average relative error between $l_2$-norm and it’s estimator for $d=10$[]{data-label="fig:figure Avg 2-norm wrt estimator d=10"}](images/avg_rel_error){width="\linewidth"}
Figure \[fig:figure Avg 2-norm wrt estimator d=10\] shows that Equation \[Avg rel error wrt est for l\_2 norm\] is independent of the condition number.
Expectation of ${\left\lvertS-1\right\rvert}$ {#ExpofS}
---------------------------------------------
The random variable $S$ takes values between $a$ and $b$ where $a \leq 1 \leq b$ and let $f(s)$ be the probability density function of S. In such case, $$\begin{split}
E \left({\left\lvert S - 1 \right\rvert} \right) &= \int_{a}^{b} {\left\lverts-1\right\rvert}f(s)ds \\
&= \int_{a}^{1} (1-s)f(s)ds + \int_{1}^{b} (s-1)f(s)ds \\
&= 2\int_{a}^{1} (1-s)f(s)ds + \int_{a}^{b} (s-1)f(s)ds \\
&= E(S) - 1 + 2\int_{a}^{1} (1-s)f(s)ds
\end{split}$$
|
---
abstract: |
We investigate theoretically whether it is feasible to detect $\eta$- and $\omega$-nucleus bound states. As well as the closed shell nuclei, $^{16}$O, $^{40}$Ca, $^{90}$Zr and $^{208}$Pb, we also investigate $^6$He, $^{11}$B and $^{26}$Mg, which are the final nuclei in the proposed experiment involving the (d,$^3$He) reaction at GSI. Potentials for the $\eta$ and $\omega$ mesons in these nuclei are calculated in local density approximation, embedding the mesons in the nucleus described by solving the mean-field equations of motion in the QMC model. Our results suggest that one should expect to find $\eta$- and $\omega$-nucleus bound states in all these nuclei.\
\
[*PACS*]{}: 36.10.G, 14.40, 12.39.B, 21.65, 71.25\
[*Keywords*]{}: Meson-nucleus bound states, Quark-meson coupling model, In-medium meson properties, MIT bag model, Effective mass
author:
- |
K. Tsushima [^1] , D.H. Lu [^2] , A.W. Thomas [^3]\
[Special Research Center for the Subatomic Structure of Matter]{}\
[and Department of Physics and Mathematical Physics]{}\
[The University of Adelaide, SA 5005, Australia]{}\
K. Saito [^4]\
[ Physics Division, Tohoku College of Pharmacy]{}\
[Sendai 981-8558, Japan]{}\
title: 'Are $\eta$- and $\omega$-nuclear states bound ?'
---
= 2.5em
ADP-98-28/T302
The study of the properties of hadrons in a hot and/or dense nuclear medium is one of the most exciting new directions in nuclear physics. In particular, the medium modification of the light vector ($\rho$, $\omega$ and $\phi$) meson masses has been investigated extensively by many authors [@qm97]. It has been suggested that dilepton production in the nuclear medium formed in relativistic heavy ion collisions, can provide a unique tool to measure such modifications as meson mass shifts. For example, the experimental data obtained at the CERN/SPS by the CERES [@ceres] and HELIOS [@hel] collaborations has been interpreted as evidence for a downward shift of the $\rho$ meson mass in dense nuclear matter [@li]. To draw a more definite conclusion, measurements of the dilepton spectrum from vector mesons produced in nuclei are planned at TJNAF [@jlab] and GSI [@gsi] Recently, a new, alternative approach to study meson mass shifts in nuclei was suggested by Hayano [*et al.*]{} [@hayano]. Their suggestion is to use the (d, $^3$He) reaction to produce $\eta$ and $\omega$ mesons with nearly zero recoil. If the meson feels a large enough, attractive (scalar) force inside a nucleus, the meson is expected to form meson-nucleus bound states. Hayano [*et al.*]{} [@hayano2] have estimated the binding energies for various $\eta$-mesic nuclei. They have also calculated some quantities for the $\omega$ meson case. However, they used an $\eta$-nucleus optical potential calculated to first-order in density, taking as input the $\eta$-nucleon scattering length. In this article, we use an alternative, self-consistent method to study whether it is possible to form $\eta$- and $\omega$-nucleus bound states in $^{16}$O, $^{40}$Ca, $^{90}$Zr and $^{208}$Pb, as well as $^{6}$He, $^{11}$B and $^{26}$Mg. The latter three nuclei correspond to the proposed experiments at GSI [@hayano] using the (d,$^3$He) reaction – i.e., the reactions, $^7$Li(d,$^3$He)$^6_{\eta/\omega}$He, $^{12}$C(d,$^3$He)$^{11}_{\eta/\omega}$B and $^{27}$Al(d,$^3$He)$^{26}_{\eta/\omega}$Mg.
In earlier work we addressed the question of whether quarks play an important role in finite nuclei [@finite0; @finite1; @finite2]. The quark-meson coupling (QMC) model [@gui; @finite0], which is based explicitly on quark degrees of freedom, is probably one of the most appropriate models to study whether meson-nucleus bound states are possible. The model has been able to describe successfully the static properties of both nuclear matter and finite nuclei [@finite1; @appl], as well as meson properties in the nuclear medium [@finite2]. Thus, the model is ideally suited to treat a bound meson and the nucleons in a nucleus on the same footing. In this study, we investigate the possible formation of the $\eta$- and $\omega$-nucleus bound states due to downward shifts of the masses. We will use QMC-I [@finite2], where the effective, isoscalar-vector $\omega$ field, which is off mass-shell and mediates the interactions among nucleons, is distinguished from the physical (on mass-shell) $\omega$ meson which is produced inside the nuclei by the above mentioned experiments.
One of the most attractive features of QMC is that, in practice, it is not significantly more complicated than Quantum Hadrodynamics (QHD) [@qhd], although the quark substructure of hadrons is explicitly implemented. Furthermore, it produces a reasonable value for the nuclear incompressibility. A detailed description of the Lagrangian density, and the mean-field equations of motion needed to describe a finite nucleus, is given in Refs. [@finite0; @finite1; @finite2].
At position $\vr$ in a nucleus (the coordinate origin is taken at the center of the nucleus), the Dirac equations for the quarks and antiquarks in the $\eta$ and $\omega$ meson bags are given by [@finite2]: (
[c]{} \_u(x)\
\_(x)\
) = 0, \[diracu\] (
[c]{} \_d(x)\
\_(x)\
) = 0, \[diracd\] \_s (x) ([or]{} \_(x)) = 0. \[diracs\] (Note that we have neglected a possible, very slight variation of the scalar and vector mean-fields inside the meson bag due to its finite size [@finite0].) The mean-field potentials for a bag centered at position $\vr$ in the nucleus, which will be calculated self-consistently, are defined by, $V_\sigma(\vr) = g^q_\sigma \sigma(\vr),
V_\omega(\vr) = g^q_\omega \omega(\vr)$ and $V_\rho(\vr) = g^q_\rho b(\vr)$, with $g^q_\sigma, g^q_\omega$ and $g^q_\rho$ being, respectively, the corresponding quark and meson-field coupling constants. Here we assume that the current masses are given as $m_q \equiv m_u = m_d = m_{\ubar} = m_{\dbar}$. Furthermore, we have assumed that the $\sigma$, $\omega$ and $\rho$ fields only interact directly with the nonstrange quarks and antiquarks [@finite2]. The mean meson fields at position $\vr$ in the nucleus are calculated self-consistently by solving Eqs. (23) – (30) of Ref. [@finite1]. Hereafter we use the notation, $\pomega$, to specify the physical, bound $\omega$ meson, in order to avoid confusion with the isoscalar-vector $\omega$ field appearing in QMC. The static solution for the ground state quarks or antiquarks in the $\eta$ and $\omega$ meson bags may be written as: \_f (x) = N\_f e\^[- i \_f t / R\_j\^\*]{} \_f (), ([for]{} j = , f = u, , d, , s, ), \[wavefunction\] where $N_f$ and $\psi_f (\vx)$ are respectively the normalization factor and corresponding spin and spatial part of the wave function [@finite2]. The bag radius in medium, $R_j^* \,\,(j=\eta,\pomega)$, which depends on the hadron species in which the quarks and antiquarks belong, will be determined self-consistently through the stability condition for the (in-medium) mass of the meson against the variation of the bag radius. (See Eq. (\[equil\]) below.) The eigenenergies for the quarks, in units of $1/R_j^*$, are given by (
[c]{} \_u()\
\_()
) &=& \_q\^\*() R\_j\^\* ( V\_() + V\_() ),\
(
[c]{} \_d()\
\_()
) &=& \_q\^\*() R\_j\^\* ( V\_() - V\_() ),\
\_s() &=& \_() = \_s(), \[quarkenergy\] where $\Omega_q^*(\vr) = \sqrt{x_q^2 + (R_j^* m_q^*)^2}$ and $\Omega_s(\vr) = \sqrt{x_s^2 + (R_j^* m_s)^2}$ with $m_q^* = m_q - g^q_\sigma \sigma(\vr) (q = u, \ubar, d, \dbar)$. The bag eigenfrequencies, $x_q$ and $x_s$, are determined by the usual, linear boundary condition [@finite0].
Next, we consider the $\eta$ and $\pomega$ meson masses in the nucleus. The physical states of the $\eta$ and $\pomega$ mesons are the superpositions of the octet and singlet states: &=& \_8 \_[P,V]{} - \_1 \_[P,V]{},’ = \_8 \_[P,V]{} + \_1 \_[P,V]{}, \[mixing1\]\
[with]{}\
\_1 &=& (u+ d+ s),\_8 = (u+ d- 2 s), \[mixing2\] where ($\xi$, $\xi'$) denotes ($\eta$, $\eta'$) or ($\phi$, $\pomega$), with the mixing angles $\theta_P$ or $\theta_V$, respectively [@pdata]. Then, the masses for the $\eta$ and $\pomega$ mesons in the nucleus at the position $\vr$, are self-consistently calculated by: m\_\^\*() &=& + [43]{}R\_\^[\* 3]{} B, \[meta\]\
m\_\^\*() &=& + [43]{}R\_\^[\* 3]{} B, \[momega\]\
.|\_[R\_j = R\_j\^\*]{} &=& 0, (j = ,), \[equil\]\
[with]{}\
a\_[P,V]{} &=& \_[P,V]{} - \_[P,V]{},b\_[P,V]{} = \_[P,V]{} + \_[P,V]{}. \[abdeff\] In practice, we use $\theta_P = - 10^\circ$ and $\theta_V = 39^\circ$ [@pdata], neglecting any possible mass dependence and imaginary parts. We also assume that the values of the mixing angles do not change in medium, although this is possible and merits further investigation. In Eqs. (\[meta\]) and (\[momega\]), $z_\eta$ and $z_\pomega$ parameterize the sum of the center-of-mass and gluon fluctuation effects, and are assumed to be independent of density [@finite0]. In this study, we chose $m_q = 5$ MeV and $m_s = 250$ MeV, for the current quark masses, and $R_N = 0.8$ fm for the bag radius of the nucleon in free space. Other inputs, parameters, and some of the quantities calculated in the present study, are listed in Table \[bagparam\]. The coupling constants, $g^q_\sigma$, $g^q_\omega$ and $g^q_\rho$, are adjusted to fit the saturation energy and density of symmetric nuclear matter and the bulk symmetry energy. Note that none of the results for nuclear properties depend strongly on the choice of the other parameters – for example, the relatively weak dependence of the final results on the values of the current quark mass and bag radius is shown explicitly in Refs. [@finite0; @finite1]. The parameters at the hadronic level associated with the core nucleus are summarized in Table \[hparamt\]. The value of the $\sigma$ mass for finite nuclei is obtained by fitting the r.m.s. charge radius of $^{40}$Ca to the experimental value, $r_{{\rm ch}}(^{40}$Ca) = 3.48 fm [@finite1]. For more details and explanations of the model parameters, see Refs. [@finite0; @finite1].
mass (MeV) $\Gamma$ (MeV) $z$ $R$ (fm) $m^*$ (MeV) $R^*$ (fm)
----------- --------------- ---------------- ------- --------------- ------------- ------------
$N$ 939.0 (input) — 3.295 0.800 (input) 754.5 0.786
$\eta$ 547.5 (input) 0 (input) 3.131 0.603 483.9 0.600
$\pomega$ 781.9 (input) 8.43 (input) 1.866 0.753 658.7 0.749
: Physical masses fitted in free space, free space full widths, $\Gamma$, the bag parameters, $z$, and the bag radii in free space, $R$. The quantities with an asterisk, are those quantities calculated at normal nuclear matter density, $\rho_0 = 0.15$ fm$^{-3}$. They are obtained with the bag constant, $B = (170$ MeV$)^4$, current quark masses, $m_u = m_d = 5$ MeV and $m_s = 250$ MeV. Note that the free space width of the $\eta$ meson is 1.18 keV [@pdata].[]{data-label="bagparam"}
field mass (MeV) $g^2/4\pi\, (e^2/4\pi)$
---------- ------------ -------------------------
$\sigma$ 418 3.12
$\omega$ 783 5.31
$\rho$ 770 6.93
$A$ 0 1/137.036
: Parameters at the hadronic level (masses and coupling constants of mesons and photon for finite nuclei) [@finite1].[]{data-label="hparamt"}
Through Eqs. (\[diracu\]) – (\[abdeff\]) we self-consistently calculate effective masses, $m^*_\eta(\vr)$ and $m^*_\pomega(\vr)$ at the position $\vr$ in the nucleus. Because the vector potentials for the same flavor of quark and antiquark cancel each other, the potentials for the $\eta$ and $\pomega$ mesons are given respectively by $m^*_\eta(r) - m_\eta$ and $m^*_\pomega(r) - m_\pomega$, where they will depend only on the distance from the center of the nucleus, $r = |\vr|$. Before showing the calculated potentials for the $\eta$ and $\pomega$, we first show in Fig. \[etaomass\] their effective masses and those calculated within an SU(3) quark model basis, $\omega = \frac{1}{\sqrt{2}} (u\ubar + d\dbar)$ (ideal mixing) and $\eta_8 = \xi_8$ in Eq. (\[mixing2\]), in symmetric nuclear matter. One can easily see that the effect of the singlet-octet mixing is negligible for the $\pomega$ mass in matter, whereas it is important for the $\eta$ mass.
As an example, we show the potentials for the mesons in $^{26}$Mg and $^{208}$Pb in Fig. \[ocazrpb\]. Note that the actual calculations for $^{6}$He, $^{11}$B and $^{26}$Mg are performed in the same way as for the closed shell nuclei, $^{16}$O, $^{40}$Ca, $^{90}$Zr and $^{208}$Pb. Although $^{6}$He, $^{11}$B and $^{26}$Mg are not spherical, we have neglected the effect of deformation, which is expected to be small and irrelevant for the present discussion. (We do not expect that deformation should alter the calculated potentials by more than a few MeV near the center of the deformed nucleus, because the baryon (scalar) density there is also expected to be more or less the same as that for a spherical nucleus – close to normal nuclear matter density.) The depth of the potentials are typically 60 and 130 MeV for the $\eta$ and $\pomega$ mesons, respectively, around the center of each nucleus. In addition, we show the calculated potentials using QMC-II [@finite2] in Fig. \[ocazrpb\], for $^{208}$Pb, in order to estimate the ambiguities due to different versions of the QMC model. At the center of $^{208}$Pb, the potential calculated using QMC-II is about 20 MeV shallower than that for QMC-I.
Now we are in a position to calculate single-particle energies for the mesons using the potentials calculated in QMC. Because the typical momentum of the bound $\omega$ is low, it should be a very good approximation to neglect the possible energy difference between the longitudinal and transverse components of the $\omega$ [@saitomega]. Then, after imposing the Lorentz condition, $\partial_\mu \phi^\mu = 0$, solving the Proca equation becomes equivalent to solving the Klein-Gordon equation, \_j() = 0, (j=,), \[kgequation1\] where $E_j$ is the total energy of the meson. An additional complication, which has so far been ignored, is the meson absorption in the nucleus, which requires a complex potential. At the moment, we have not been able to calculate the imaginary part of the potential (equivalently, the in-medium widths of the mesons) self-consistently within the model. In order to make a more realistic estimate for the meson-nucleus bound states, we include the widths of the $\eta$ and $\pomega$ mesons in the nucleus by assuming a specific form: \^\*\_j(r) &=& m\^\*\_j(r) - , (j=,), \[imaginary\]\
&& m\^\*\_j(r) - \^\*\_j (r), \[width\] where, $m_j$ and $\Gamma_j$ are the corresponding masses and widths in free space listed in Table \[bagparam\], and $\gamma_j$ are treated as phenomenological parameters to describe the in-medium meson widths, $\Gamma^*_j(r)$. According to the estimates in Refs. [@hayano; @fri], the widths of the mesons in nuclei and at normal nuclear matter density are $\Gamma^*_\eta \sim 30 - 70$ MeV [@hayano] and $\Gamma^*_\pomega \sim 30 - 40$ MeV [@fri], respectively. Thus, we calculate the single-particle energies for several values of the parameter, $\gamma_j$, which cover the estimated ranges.
From Table \[bagparam\] and the calculated density distributions one can obtain the corresponding widths at normal nuclear matter density, as well as in the finite nuclei. Because of the recoilless condition for meson production in the GSI experiment [@hayano; @hayano2], we may expect that the energy dependence of the potentials would not be strong [@pion]. Thus we actually solve the following, modified Klein-Gordon equations: \_j() = 0, (j=,). \[kgequation2\] This is carried out in momentum space by the method developed in Ref. [@landau]. To confirm the calculated results, we also calculated the single-particle energies by solving the Schrödinger equation. Calculated single-particle energies for the $\eta$ and $\pomega$ mesons, obtained solving the Klein-Gordon equation are respectively listed in Tables \[etaenergy\] and \[omegaenergy\].
$\gamma_\eta$=0 $\gamma_\eta$=0.5 $\gamma_\eta$=1.0
----------------- ---- ----------------- ---------- ------------------- ---------- ------------------- ----------
$E$ $\Gamma$ $E$ $\Gamma$ $E$ $\Gamma$
$^{16}_\eta$O 1s -33.1 0 -32.6 26.7 -31.2 53.9
1p -8.69 0 -7.72 18.3 -5.25 38.2
$^{40}_\eta$Ca 1s -46.5 0 -46.0 31.7 -44.8 63.6
1p -27.4 0 -26.8 26.8 -25.2 54.2
2s -6.09 0 -4.61 17.7 -1.24 38.5
$^{90}_\eta$Zr 1s -53.3 0 -52.9 33.2 -51.8 66.4
1p -40.5 0 -40.0 30.5 -38.8 61.2
2s -22.3 0 -21.7 26.1 -19.9 53.1
$^{208}_\eta$Pb 1s -56.6 0 -56.3 33.2 -55.3 66.2
1p -48.7 0 -48.3 31.8 -47.3 63.5
2s -36.3 0 -35.9 29.6 -34.7 59.5
$^{6}_\eta$He 1s -11.4 0 -10.7 14.5 -8.75 29.9
$^{11}_\eta$B 1s -25.0 0 -24.5 22.8 -22.9 46.1
$^{26}_\eta$Mg 1s -39.2 0 -38.8 28.5 -37.6 57.3
1p -18.5 0 -17.8 23.1 -15.9 47.1
: Calculated $\eta$ meson single-particle energies, $E = Re(E_\eta - m_\eta)$, and full widths, $\Gamma$, (both in MeV), in various nuclei, where the complex eigenenergies are, $E_\eta = E + m_\eta - i \Gamma/2$. See Eq. (\[imaginary\]) for the definition of $\gamma_\eta$. Note that the free space width of the $\eta$ is 1.18 keV, which corresponds to $\gamma_\eta = 0$. []{data-label="etaenergy"}
$\gamma_\omega$=0 $\gamma_\omega$=0.2 $\gamma_\omega$=0.4
------------------- ---- ------------------- ---------- --------------------- ---------- --------------------- ----------
$E$ $\Gamma$ $E$ $\Gamma$ $E$ $\Gamma$
$^{16}_\omega$O 1s -93.5 8.14 -93.4 30.6 -93.4 53.1
1p -64.8 7.94 -64.7 27.8 -64.6 47.7
$^{40}_\omega$Ca 1s -111 8.22 -111 33.1 -111 58.1
1p -90.8 8.07 -90.8 31.0 -90.7 54.0
2s -65.6 7.86 -65.5 28.9 -65.4 49.9
$^{90}_\omega$Zr 1s -117 8.30 -117 33.4 -117 58.6
1p -105 8.19 -105 32.3 -105 56.5
2s -86.4 8.03 -86.4 30.7 -86.4 53.4
$^{208}_\omega$Pb 1s -118 8.35 -118 33.1 -118 57.8
1p -111 8.28 -111 32.5 -111 56.8
2s -100 8.17 -100 31.7 -100 55.3
$^{6}_\omega$He 1s -55.7 8.05 -55.6 24.7 -55.4 41.3
$^{11}_\omega$B 1s -80.8 8.10 -80.8 28.8 -80.6 49.5
$^{26}_\omega$Mg 1s -99.7 8.21 -99.7 31.1 -99.7 54.0
1p -78.5 8.02 -78.5 29.4 -78.4 50.8
2s -42.9 7.87 -42.8 24.8 -42.5 41.9
: As in Tables \[etaenergy\], but for $\omega$ meson single-particle energies. In the light of $\Gamma$ in Refs. [@fri], the results with $\gamma_\omega = 0.2$ are expected to correspond best with the experiment. []{data-label="omegaenergy"}
Our results suggest one should expect to find bound $\eta$- and $\omega$-nuclear states as has been suggested by Hayano [*et al.*]{} [@hayano; @hayano2]. For the $\eta$ single-particle energies, our estimated values lie between the results obtained using two different parameter sets in Ref [@hayano2]. From the point of view of uncertainties arising from differences between QMC-I and QMC-II, the present results for both the single-particle energies and calculated full widths should be no more than 20 % smaller in absolute values according to the estimate from the potential for the $\omega$ in $^{208}$Pb in Fig. \[ocazrpb\]. Nevertheless, for a heavy nucleus and relatively wide range of the in-medium meson widths, it seems inevitable that one should find such $\eta$- and $\omega$-nucleus bound states. Note that the correction to the real part of the single-particle energies from the width, $\Gamma$, can be estimated nonrelativistically, to be of order of $\sim \Gamma^2/8m$ (repulsive), which is a few MeV if we use $\Gamma \simeq 100$ MeV.
In future work we would like to include the effect of $\sigma$-$\omega$ mixing, which (within QHD, at least) becomes especially important at higher densities [@saitomega]. It will also be important for consistency to calculate the in-medium width of the meson within the QMC model and to study the energy dependence of the meson-nucleus potential. While the energy dependence of the potential felt (for example) by the $\omega$ may be quite significant as we move from a virtual $\omega$ ($q^2 \sim 0$) to an almost real $\omega$ ($q^2 \sim m^2_\omega$) [@fri], QHD studies in nuclear matter did not reveal a strong energy dependence for $q^2$ near $m^2_\omega$ [@saitomega] – the region of interest here. Nevertheless, this point merits further study in finite nuclei and within QMC itself.
To summarize, we have calculated the single-particle energies for $\eta$- and $\omega$-mesic nuclei using QMC-I. The potentials for the mesons in the nucleus have been calculated self-consistently in local density approximation, embedding the MIT bag model $\eta$ and $\omega$ mesons in the nucleus described by solving mean-field equations of motion. Although the specific form for the widths of the mesons in medium could not be calculated in this model yet, our results suggest that one should find $\eta$- and $\omega$-nucleus bound states for a relatively wide range of the in-medium meson widths. In the near future, we plan to calculate the in-medium $\omega$ width self-consistently in the QMC model.
[**Acknowledgment**]{}\
We would like to thank S. Hirenzaki, H. Toki and W. Weise for useful discussions. Our thanks also go to R.S. Hayano for discussions at the [*2nd International Symposium on Symmetries in Subatomic Physics*]{} held at University of Washington, Seattle, June 25 – 28, 1997, which triggered the present work, and for providing us the experimental proposal, Ref. [@hayano]. This work was supported by the Australian Research Council.
[99]{} Quark Matter ’97, to be published in Nucl. Phys. A (1998). P. Wurm for the CERES collaboration, Nucl. Phys. A 590 (1995) 103c. M. Masera for the HELIOS collaboration, Nucl. Phys. A 590 (1995) 93c. G.Q. Li, C.M. Ko and G.E. Brown, Nucl. Phys. A 606 (1996) 568; G. Chanfray, R. Rapp and J. Wambach, Phys. Rev. Lett. 76 (1996) 368. M. Kossov [*et al.*]{}, TJNAF proposal PR-94-002 (1994); P.Y. Bertin and P.A.M. Guichon, Phys. Rev. C 42 (1990) 1133. HADES proposal, see HADES home page: http://piggy.physik.uni-giessen.de/hades/; G.J. Lolos [*et al.*]{}. Phys. Rev. Lett. 80 (1998) 241. R.S. Hayano [*et al.*]{}, proposal for GSI/SIS, September, 1997; R.S. Hayano, 2nd Int. Symp. on Symmetries in Subatomic Physics, Seattle (1997); R.S. Hayano and S. Hirenzaki, contributed paper to [*Quark Matter ’97*]{}, Tsukuba (1997); T. Yamazaki [*et al.*]{}, Z. Phys. A 355 (1996) 219. R.S. Hayano, S. Hirenzaki and A. Gillitzer, nucl-th/9806012. P.A.M. Guichon, K. Saito, E. Rodionov and A.W. Thomas, Nucl. Phys. A 601 (1996) 349. K. Saito, K. Tsushima and A.W. Thomas, Nucl. Phys. A 609 (1996) 339. K. Saito, K. Tsushima and A.W. Thomas, Phys. Rev. C 55 (1997) 2637; K. Tsushima, K. Saito, A.W. Thomas and S.W. Wright, Phys. Lett. B 429 (1998) 239. P.A.M. Guichon, Phys. Lett. B 200 (1988) 235. H.Q. Song, R.K. Su, Phys. Lett. B 358 (1995) 179; P.G. Blunden and G.A. Miller, Phys. Rev. C 54 (1996) 359; X. Jin and B.K. Jennings, Phys. Lett. B 374 (1996) 13; H. Müller and B.K. Jennings, Nucl. Phys. A 626 (1997) 966; R. Aguirre and M. Schvellinger, Phys. Lett. B 400 (1997) 245; P. K. Panda, A. Mishra, J. M. Eisenberg and W. Greiner, Phys. Rev. C 56 (1997) 3134; H. Müller, Phys. Rev. C57 (1998) 1974. B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. Review of Particle Physics, Phys. Rev. D 54 (1996) 1. K. Saito, K. Tsushima, A.W. Thomas and A.G. Williams, Phys. Lett. B 433 (1998) 243. B. Friman, nucl-th/9801053; F. Klingl and W. Weise, hep-ph/9802211. T. Waas, R. Brockmann and W. Weise, Phys. Lett. B 405 (1997) 215; T. Yamazaki [*et al.*]{}, Phys. Lett. B 418 (1998) 246. D.H. Lu and R.H. Landau, Phys. Rev. C 49 (1994) 878; Y.R. Kwon and F. Tabakin, Phys. Rev. C 18 (1978) 932; R.H. Landau, [*Quantum Mechanics II*]{} (John Wiley & Sons, New York, 1990).
[^1]: ktsushim@physics.adelaide.edu.au
[^2]: dlu@physics.adelaide.edu.au
[^3]: athomas@physics.adelaide.edu.au
[^4]: ksaito@nucl.phys.tohoku.ac.jp
|
---
abstract: 'We study dwarf galaxy formation at high redshift ($z\ge5$) using a suite of high-resolution, cosmological hydrodynamic simulations and a semi-analytic model (SAM). We focus on gas accretion, cooling and star formation in this work by isolating the relevant process from reionization and supernova feedback, which will be further discussed in a companion paper. We apply the SAM to halo merger trees constructed from a collisionless *N*-body simulation sharing identical initial conditions to the hydrodynamic suite, and calibrate the free parameters against the stellar mass function predicted by the hydrodynamic simulations at $z=5$. By making comparisons of the star formation history and gas components calculated by the two modelling techniques, we find that semi-analytic prescriptions that are commonly adopted in the literature of low-redshift galaxy formation do not accurately represent dwarf galaxy properties in the hydrodynamic simulation at earlier times. We propose 3 modifications to SAMs that will provide more accurate high-redshift simulations. These include 1) the halo mass and baryon fraction which are overestimated by collisionless *N*-body simulations; 2) the star formation efficiency which follows a different cosmic evolutionary path from the hydrodynamic simulation; and 3) the cooling rate which is not well defined for dwarf galaxies at high redshift. [Accurate semi-analytic modelling of dwarf galaxy formation informed by detailed hydrodynamical modelling will facilitate reliable semi-analytic predictions over the large volumes needed for the study of reionization.]{}'
author:
- |
Yuxiang Qin$^{1,2}$[^1], Alan R. Duffy$^{3,2}$, Simon J. Mutch$^{1,2}$, Gregory B. Poole$^{1,3}$, Paul M. Geil$^1$, Andrei Mesinger$^4$ and J. Stuart B. Wyithe$^{1,2}$[^2]\
$^{1}$School of Physics, University of Melbourne, Parkville, VIC 3010, Australia\
$^{2}$ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)\
$^{3}$Centre for Astrophysics and Supercomputing, Swinburne University of Technology, PO Box 218, Hawthorn VIC 3122, Australia\
$^{4}$Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
bibliography:
- 'reference.bib'
date: ' draft - '
title: |
Dark-ages Reionization and Galaxy Formation Simulation - .\
Gas accretion, cooling and star formation in dwarf galaxies at high redshift
---
\[firstpage\]
galaxies: formation – galaxies: dwarf – galaxies: high-redshift – methods: numerical
Introduction
============
About 150 Myr after the Big Bang, the Universe entered the Epoch of Reionization (EoR; ), when the first stars and galaxies were created and started to photoionize the intergalactic neutral hydrogen. Over the past decade, there have been significant advances in the study of galaxies during the EoR and the evolution of the ionized intergalactic medium (IGM). High-redshift galaxies thought to have sourced reionization have been discovered all the way to $z\sim11$ [@Oesch2016], when the Universe was only 400 Myr old. However, the sample of these early galaxies remains small with only ${\sim}1000$ candidates at $z>6$ identified using advanced space-based instruments [@Bouwens2014; @Bouwens2015]. Achieving a physical understanding between galaxies and the progress of reionization is therefore aided by numerical simulations (see e.g. @Bertschinger1998 [@Baugh2006; @Dolag2008; @Somerville2015]).
Two of the main numerical approaches to model galaxy formation are hydrodynamic simulations and semi-analytic models (SAMs) applied to dark matter halo merger trees constructed from cosmological *N*-body simulations. Since they adopt different approximations, their strengths and requirements are diverse. Hydrodynamic simulations include baryons as well as dark matter, and simulate the complex baryonic physics more directly (e.g. @Crain2009 [@Schaye2010; @vogelsberger2014properties; @Schaye2014; @feng2015bluetides; @Pawlik2017]). However, while galaxies form and evolve in the presence of gravity and hydrodynamical forces in a simulation volume, its large computational expense limits the capability of simultaneously resolving small-scale regions and capturing massive systems. For instance, in order to study reionization and galaxy formation during the EoR, simulations are required to have a volume size of at least $10^6 \mathrm{Mpc}^3$ [@Iliev2014MNRAS.439..725I] and a particle resolution that is able to resolve haloes with masses around $5\times10^7\mathrm{M}_\odot$ (i.e. ${>}10$ billion particles; @Barkana1999ApJ...523...54B). While it remains challenging to explore the relevant parameter space in hydrodynamic simulations, SAMs, with their advantages of low expense on computational resources, become the alternative in this case.
Semi-analytical models {#sec:SAMs}
----------------------
*N*-body simulations (e.g. @Navarro1997 [@springel2005simulating; @iliev2008simulating; @Boylan_Kolchin2009; @Klypin2010; @Garrison2017]) solve for the gravitational force on each collisionless dark matter particle but neglect the baryonic physics. In order to implement the galaxy formation physics of baryons into these simulations, halo merger trees are constructed and based on the properties inherited from the merger trees, SAMs parametrize baryonic processes to evolve galaxies, offering an efficient approach to simulate galaxies in a cosmological context [e.g. @Cole2000; @Hatton2003; @Baugh2005; @croton2006many; @DeLucia2007; @Somerville2008; @guo2011dwarf; @Henriques2015].
In general, a SAM first designates haloes as galaxies and endows them with stellar components and several gas reservoirs of varied functionalities. The latter usually include a cold gas disc where star formation occurs and a hot halo of non-star-forming gas. Mass is then manipulated and transferred between these baryonic sectors as a result of varied baryonic processes including accretion, cooling, star formation and feedback from supernovae or active galactic nuclei (AGN). These processes are implemented using scaling functions of astrophysics that are motivated either from first principles (e.g. the thermal cooling rate, see Section \[sec:SAM\_cooling\]), from observations \[e.g. the Kennicutt–Schmidt (KS) star formation law, see Section \[sec:SAM\_SF\]\], or from more complicated simulations such as hydrodynamic and radiative transfer calculations (e.g. reionization feedback, @Sobacchi2013a). Depending on the scientific goal and applicable range, the complexity of SAMs ranges from the treatment of halo profiles to the implementation of baryonic physics. Taking star formation as an example, it can be modelled in a simplified manner by consuming the entire cold gas disc within a given time-scale (e.g. [sage]{}, @croton2006many), or as @Stevens2016 demonstrated (in the updated version, [dark sage]{}), by
1. considering the disc as a combination of a series of gas rings with angular momentum conserved between each annulus and transferred dynamically;
2. using the mid-plane pressure inferred from local observations and theoretical work (e.g. @Blitz2006 [@Krumholz2009]) to split the disc into molecules and atoms, and forming stars directly from the giant ${\mathrm{H}_2}$ clouds; and
3. including disc instabilities and forming additional components representing stellar bulges, which might possess different ages or dynamics from the stellar disc.
Despite these improvements recently implemented into modern SAMs, some prescriptions are still commonly adopted by many of these models. For instance, in order to initialize the baryonic component of haloes from dark matter only *N*-body simulations, the universal baryon fraction is applied to every virialized system with a further suppression due to reionization feedback. These baryons are considered as infalling gas and are usually assumed to share the virial temperature of the host halo as a result of experiencing shock–heating.
However, we note that the degeneracy between parameters of SAMs, as well as their time and mass dependencies introduced by these simplified baryonic prescriptions are not well understood. In addition, lack of available observational data means that semi-analytic predictions have generally been tested against observables that are massive and in the nearby Universe [@croton2006many; @guo2011dwarf; @Stevens2016]. Their consistency and performance in the low-mass regime or at high redshift remain unclear – a model might be able to reproduce a limited number of observed quantities by either careful calibrations or by introducing more free parameters accompanied by *physical motivations*, but incorrectly predict the other properties that cannot be observed at this stage.
This motivates us to 1) test the performance of SAMs at high redshift, which were originally developed from the literature of low-redshift galaxy formation; and 2) inform more accurate semi-analytic modelling of dwarf galaxies, which will be applied to large-volume simulations to study the EoR in the future. In the absence of detailed observational data, constraints on the properties of SAM galaxies can be supplemented through comparisons against physics-rich hydrodynamic simulations, which are expected to predict galaxies that are more representative than simplified calculations using semi-analytic approaches (e.g. @Guo2016 [@mitchell:2017je; @stevens:2017fi; @cote:2017uh]). In this work, we take the [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} SAM [@Mutch2016a] as an example, and apply it to the halo merger trees constructed from a collisionless *N*-body simulation sharing identical initial conditions to the [[[*Smaug*]{}]{}]{} suite of hydrodynamic simulations [@duffy2014low]. We calibrate the SAM against the [[[*Smaug*]{}]{}]{} hydrodynamic simulations and make direct comparisons of the calculated galaxy properties using the two modelling techniques. Since the hydrodynamic simulations were run with different assumptions of galactic or stellar feedback, the free parameters of the SAM are chosen accordingly to reproduce the predicted galaxy properties from the corresponding hydrodynamic simulation.
This paper focuses on modelling of gas accretion, cooling and star formation, the implementation of which is common among many modern SAMs [@croton2006many; @Somerville2008; @guo2011dwarf; @Henriques2015]. The effect of feedback on star formation (which varies between models) is excluded and will be presented in a companion paper (Qin et al. in prep.). We begin with a brief introduction of the [*N*-body]{}/hydrodynamic simulations and SAM utilized in this work in Section \[sec:models\]. Then we discuss the consequence of applying SAMs directly to collisionless *N*-body simulations and propose modifications accordingly in Section \[sec:halo mass\]. In Section \[sec:results\], we proceed with the proposed modifications and calibrate the SAM to reproduce the hydrodynamic result. We investigate galaxy properties in detail and make comparisons between the hydrodynamic simulation and the SAM results. Based on these, we propose additional modifications to the SAM to inform better modelling of dwarf galaxies at high redshift. We conclude in Section \[sec:conclusion\]. In this work, we adopt cosmological parameters from *WMAP7* ($\Omega_{\mathrm{m}}, \Omega_{\mathrm{b}}, \Omega_{\mathrm{\Lambda}}, h, \sigma_8, n_s $ = 0.275, 0.0458, 0.725, 0.702, 0.816, 0.968; @Komatsu2011) in all simulations.
The Dragons Project {#sec:models}
===================
The Dark-ages Reionization And Galaxy Formation Observable from Numerical Simulation (DRAGONS[^3]) project employs *N*-body simulations [@Poole2016], hydrodynamic simulations [@duffy2014low; @Qin2017a] and SAMs [@Mutch2016a; @Qin2017c] to study reionization and galaxy formation at high redshift ($z \ge 5$, @Geil2016 [@Geil2017; @Liu2016; @Liu2017; @Mutch2016b; @Park2017; @Duffy2017; @Qin2017b]). The resulting galaxy catalogues from both the hydrodynamic simulation and SAM with feedback included are in good agreement with observations including the stellar mass and galaxy UV luminosity functions across cosmic time.
In this work, we focus on high-redshift modelling ($z\ge5$) of currently unobservable dwarf galaxies ($M_\mathrm{vir}\lesssim10^{10.5}\mathrm{M}_\odot$) which are believed to be the dominant source of ionizing photons driving reionization [@duffy2014low; @Liu2016]. Using simplified models, we isolate prescriptions of gas accretion, cooling and star formation from reionization and supernova feedback[^4], which will be further discussed in a companion paper. In the following two subsections, we summarize the DRAGONS SAM and hydrodynamic simulation, named [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{}, respectively.
<span style="font-variant:small-caps;">Meraxes</span> {#sec:meraxes}
-----------------------------------------------------
[[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} is based on the *Munich* SAM [@croton2006many; @guo2011dwarf] and optimized to study galaxy formation at high redshift. Based on halo properties read from merger trees, it evolves galaxies using simplified prescriptions, which calculate baryonic infall, cooling, star formation, supernova feedback, metal enrichment, stellar mass recycling, AGN feedback and mergers. In addition, [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} incorporates 21cm[fast]{} [@Mesinger2010], a semi-numerical approach to evolve ionization fields, for the purpose of investigating reionization feedback and exploring the IGM state during the EoR. We refer the interested reader to @Mutch2016a [hereafter ] and @Qin2017c for a detailed description of [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and provide a brief review of the relevant modelling prescriptions in this section.
### Accretion (and stripping) {#sec:SAM_accretion}
To connect *N*-body simulations of dark matter with baryonic physics, the first implementation is the accretion of gas. Following convention, [a universal baryon fraction ($f_\mathrm{b}=\Omega_\mathrm{b}/\Omega_\mathrm{m}$) with suppression due to reionization heating ($f_\mathrm{mod}$) is applied and each halo group (i.e. [fof]{} group, see Appendix \[sec:dark matter halo merger trees\]), with a total mass of $M_\mathrm{vir}$, accretes gas from the IGM]{} $$\label{eq:accretion}
\Delta m_\mathrm{hot} = f_\mathrm{mod}f_\mathrm{b}M_\mathrm{vir}-\sum\limits_{N_\mathrm{halo}} m_\mathrm{baryon},$$ where $m_\mathrm{baryon}$ is the total baryonic mass including[^5] hot gas ($m_\mathrm{hot}$), cold gas ($m_\mathrm{cold}$) and stellar mass ($m_*$), and the summation symbol indicates that baryons of all $N_\mathrm{halo}$ haloes including the central and satellite haloes are considered. $f_\mathrm{mod}\le1$ represents a baryon fraction modifier accounting for the suppression due to the reionization heating background.
In the case of $\Delta m_\mathrm{hot}>0$, all of the mass is assumed to be accreted by the central halo due to its dominant gravitational potential in the entire virialized system, and are stored in its hot gas reservoir. However, when $\Delta m_\mathrm{hot}<0$, baryons will be removed from the hot gas reservoir of the central halo in order to maintain the universal baryon fraction. Note that baryons that have reached the cold gas disc or have formed stars are assumed to be gravitationally bound, and do not get further removed even if $f_\mathrm{mod}f_\mathrm{b}M_\mathrm{vir} - \sum\limits_{N_\mathrm{halo}}\left(m_*+m_\mathrm{cold}\right)<0$.
### Cooling {#sec:SAM_cooling}
Gas falling into a halo is stored in the hot gas reservoir, which is assumed to share the halo virial temperature, $T_\mathrm{vir}$, due to shock heating. It then cools within the following time-scale $$\label{eq:t_cool}
t_{\mathrm{cool}}\left(r\right) = \dfrac{3\bar{\mu}m_\mathrm{p}kT_\mathrm{vir}}{2\rho_\mathrm{hot}\left(r\right)\Lambda\left(T_\mathrm{vir}\right)},$$ where $\bar{\mu}=0.59$, $m_\mathrm{p}$, $k$, $\rho_\mathrm{hot}$ and $\Lambda$ are the mean molecular weight of fully ionized gas, the mass of a proton, the Boltzmann constant, the hot gas density and the cooling function [@Sutherland1993] that is determined by the temperature[^6]. Assuming the hot gas follows a singular isothermal sphere (SIS) profile, we calculate the cooling radius, $r_\mathrm{cool}$, at which the cooling time is equal to the halo dynamical time, $t_\mathrm{dyn}$, through $$r_\mathrm{cool} = \sqrt{\dfrac{m_\mathrm{hot}\Lambda\left(T_\mathrm{vir}\right)}{6\pi \bar{\mu}m_\mathrm{p}kT_\mathrm{vir}V_\mathrm{vir}}},$$ where $V_\mathrm{vir}$ is the virial velocity of the host halo.
Based on the ratio of the cooling radius to the virial radius, $R_\mathrm{vir}$, we calculate the cooling rate and consider the following two scenarios:
1. $r_\mathrm{cool}<R_\mathrm{vir}$ (*static hot halo*): cooling is assumed in thermal equilibrium and the cooling rate is determined by the continuity equation;
2. $r_\mathrm{cool}\geq R_\mathrm{vir}$ (*rapid cooling*): cooling is treated as free-fall collapse and all of the hot gas component becomes cold within the dynamical time-scale, $t_\mathrm{dyn}$.
Therefore, the cooling rate can be calculated by $$\label{eq:mcool}
\dot{m}_{\mathrm{cool}}{=}\dfrac{m_{\mathrm{hot}}}{t_\mathrm{dyn}}\times\min\left(1,\dfrac{r_{\mathrm{cool}}}{R_{\mathrm{vir}}}\right).$$
### Star formation {#sec:SAM_SF}
Cooling leads to the build-up of a cold gas reservoir, which is assumed to form a rotationally supported disc, where stars form, with an exponential surface density profile. When the cold gas mass, $m_\mathrm{cold}$, exceeds a critical value, $m_\mathrm{crit}$, we use a phenomenological relation suggested by observations [@kennicutt1998global] to calculate the star formation rate (SFR)[^7] $$\label{eq:sfr}
\dot{m}_{\star} {=} \alpha_{\mathrm{sf}}\times\dfrac{\mathrm{max}\left(0,m_{\mathrm{cold}}{-}m_{\mathrm{crit}}\right)}{t_{\mathrm{dyn,disc}}},$$ where $t_{\mathrm{dyn,disc}}\equiv1.5\sqrt{2}\lambda t_{\mathrm{dyn}}$ is the dynamical time of the cold gas disc inferred by the halo spin parameter [[@bullock2001apj...555..240b]]{}, $\lambda$, with an additional assumption that the specific angular momentum is conserved between the cold gas disc and the host halo [@Mo1998]. A free parameter, $\alpha_{\mathrm{sf}}$, is introduced for the purpose of adjusting the star formation efficiency.
We note that, in order to be consistent with the hydrodynamic simulations introduced in the next subsection, the newly formed stars are assumed to follow an initial mass function in the mass range of $0.1-120\mathrm{M}_\odot$ with the following form [@chabrier2003galactic] $$\label{eq:imf}
\phi(m) =
\begin{cases}
0.85m^{-1} \mathrm{exp}\left[-\dfrac{\left(\mathrm{log_{10}}m - \mathrm{log_{10}} 0.079\right)^2}{2\times0.69^2}\right],\\
\hfill \text{if}\ 0.1\mathrm{M}_\odot \leq m < 1\mathrm{M}_\odot;\\
\\
0.24 m^{-2.3},\hfill \text{if}\ 1 \leq m \leq 120\mathrm{M}_\odot.
\end{cases}$$
[[[*Smaug*]{}]{}]{} {#sec:smaug}
-------------------
[[[*Smaug*]{}]{}]{} is a series of high-resolution hydrodynamic simulations (@duffy2014low, @Qin2017a) which were run with a modified [gadget]{}-2 *N*-body/hydrodynamics code [@springel2005cosmological]. These models follow the OverWhelmingly Large Simulations project (OWLS; @Schaye2010), in which the full hydrodynamic simulation that involves star formation [@Schaye2008], radiative cooling [@Wiersma2009a], supernova feedback [@DallaVecchia2008; @DallaVecchia2012] and reionization feedback [@Haardt2001], has been shown to successfully reproduce observed galaxy properties at low redshift including the cosmic star formation history [@Schaye2010] and the stellar mass function at $z<2$ [@Haas2013]. In addition, @Katsianis2017 recently showed that the EAGLE simulations, an extension of OWLS, are also in agreement with the observed SFR function across cosmic time ($z\sim 0-8$). These achievements motivate us to make use of the hydrodynamic simulations and to investigate the validity of semi-analytic prescriptions at high redshift where observational constraints are unavailable.
### Galaxy physics {#sec:galaxy physics}
In this work, the simulation from [[[*Smaug*]{}]{}]{} comprises $512^3$ baryon and $512^3$ dark matter particles within a cube of comoving side of $10 h^{-1}\mathrm{Mpc}$. This equates to a mass resolution of $4.7 (0.9){\times} 10^{5} h^{-1}\mathrm{M}_\odot$ per dark matter (gas) particle. The time interval between two outputs is around 11 Myr, and the initial conditions were generated with the [grafic]{} package [@Bertschinger2001] at $z=199$ using the Zel’dovich approximation [@Zeldovich1970]. We summarize the subgrid physics adopted in [[[*Smaug*]{}]{}]{} in terms of cooling and star formation for comparison with the semi-analytic prescriptions (see Section \[sec:meraxes\]), and refer the interested reader to @Schaye2010 and @duffy2014low for more details.
1. *Cooling* [@Wiersma2009a] consists of only primordial elements and is pre-tabulated using the [cloudy]{} package [@ferland1998cloudy]. The cooling rate is calculated in the presence of the cosmic microwave background (CMB), accounting for both free–free scattering between gas particles and Compton scattering between gas particles and CMB photons.
2. *Star formation* [@Schaye2008] occurs by stochastically converting gas particles to star particles in the ISM. In practice, the local density around a gas particle, $\rho_\mathrm{g}$, can be inferred by smoothing the particle mass, $m_\mathrm{g}$, within a certain volume. When $\rho_\mathrm{g}$ exceeds a critical value, the gas particle is assumed to be multiphase ISM state with pressure $P\propto \rho^{\gamma_\mathrm{eff}}$, where $\gamma_\mathrm{eff}=\dfrac{4}{3}$ represents the effective ratio of specific heats. Note that resolving star forming regions in a cosmological hydrodynamic simulation is computationally unrealistic. Therefore, the gas depletion time, $t_\mathrm{g}$, in the hydrodynamic simulation is calculated based on the observed KS (@kennicutt1998global) star formation law, and the SFR is given by $\dot{m}_* = m_\mathrm{g}/t_\mathrm{g}$. We note that although the star formation recipes adopted in the SAM and hydrodynamic simulation are both developed from the KS law, there are discrepancies between the two numerical approaches, which are further discussed in Section \[sec:sfefficiency\].
### Simulations {#sec:simulations}
The two simulations used in this work are
1. *DMONLY*, a collisionless *N*-body simulation performed using the same initial conditions from the full simulation but neglecting hydrodynamic forces from the baryonic component. It consists of $512^3$ collisionless particles with masses of $5.7\times10^5h^{-1}\mathrm{M}_\odot$ within a cube of comoving side of $10h^{-1}\mathrm{Mpc}$. In order to apply the [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} SAM and compare the result with [[[*Smaug*]{}]{}]{}, *DMONLY* is used to construct dark matter halo merger trees (See Appendix \[sec:match\]);
2. *NOSN\_NOZCOOL\_NoRe*, a toy model with cooling in the absence of metal line emissions and reionization heating. It ignores feedback from exploding supernovae and heating due to reionization, and will be used to compare with the SAM for the purpose of investigating gas accretion, cooling and star formation without impacts from any feedback. The top left panel of Fig. \[fig:indicator\] presents an example of the gas density distribution at $z=5$ in the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation.
### Cold gas (star-forming gas) {#sec:cold gas}
In the SAM, cold gas is considered as potentially star-formation gas (see Section \[sec:SAM\_SF\]). In order to facilitate direct comparisons of the gas reservoir with the semi-analytic calculation, we specify, for each galaxy in the hydrodynamic simulation, a star-formation gas component. This reservoir comprises all potential star forming particles of the galaxy, making it comparable to the assumption adopted in the SAM.
As mentioned before, a gas particle located in a cold dense region is considered as multiphase and can potentially form stars. More specifically in [[[*Smaug*]{}]{}]{}, this requires the particle to satisfy *A* AND *B* AND (*C* OR *D*), a combination of the following four criteria [@Schaye2010]:
1. a high physical density, the threshold of which is a fixed value ($0.1\mathrm{cm}^{-3}$) and informs the formation of a multiphase ISM through gravitational instability;
2. a high comoving density, the threshold of which is 57.7 times the cosmic mean and prevents spurious star formation in less dense region at high redshift;
3. a high physical density and a low temperature, the thresholds of which are informed from the equation of state of the unresolved warm dense ISM, and can recover the KS star formation law;
4. a low temperature, the threshold of which is a constant ($10^{5}\mathrm{K}$) and is proposed to capture cold ISM in less dense regions.
The bottom panels of Fig. \[fig:temp\_nosn\_nozcool\] show the gas density–temperature phase diagram of two haloes at $z=5$ in the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation for illustration. Particles located in the lower left region are identified as star-formation gas; otherwise they are non-star-forming gas.
### Hot gas {#sec:hot gas}
On the other hand, hot gas in the SAM represents a reservoir where the temperature is as high as the halo virial temperature as a result of shock–heating during gas accretion (see Section \[sec:SAM\_cooling\]). Gas within the hot gas reservoir cannot form stars in the SAM. Therefore, in order to define a hot gas component for a galaxy in hydrodynamic simulations, which can be compared with the SAM, we first exclude all potential star forming particles from the galaxy. We plot the temperature distribution of non-star-forming gas particles of the two haloes in the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation in the top panels of Fig. \[fig:temp\_nosn\_nozcool\]. In the more massive halo, three populations[^8] can be observed, corresponding to, from right to left, (1) infall hot gas; (2) infall gas that has been through cooling, and the cooling rate decreases significantly when temperature[^9] drops to a few $10^4\mathrm{K}$ [@Wiersma2009a]; and (3) cold gas which is about to reach the multiphase ISM disc and become dense enough to form stars. However, we can only identify the group of less dense (non-star-forming) cold gas in the less massive halo.
![\[fig:temp\_nosn\_nozcool\]*Bottom panels:* gas density–temperature phase diagram of two haloes at $z=5$ in the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation. The stellar mass, gas mass and halo mass are shown on the lower right corner. The piecewise function shown with black solid lines indicates the separation of gas particles with different phases – particles inside (outside) the lower left region are identified as star forming (non-star-forming) gas and are indicated with the blue (red) colour. Using the *KMeans* machine-learning clustering algorithm, the non-star-forming gas particles are further divided into two groups, marked by the dashed (hotter) and dotted (colder) contours, respectively. *Top panels:* number densities of all (red solid line), hotter (black dashed line) and colder (black dotted line) non-star-forming gas of the two haloes as functions of temperature. The vertical red dashed, black dashed and black dotted lines represent the [fof]{} halo virial temperature and the median temperatures of the two non-star-forming gas groups, respectively.](./figures/nosn_nozcool_noret_mass_ratio_baryon_fraction_modifiers/temp_n.pdf){width="\textwidth"}
Based on the density–temperature phase diagram of the non-star-forming gas particles of a galaxy, the hot gas mass and temperature of the galaxy can be determined. In practice, depending on the number of non-star-forming gas particles ($N_\mathrm{nonSF}$) that a galaxy comprises,
1. when $N_\mathrm{nonSF}=0$, the hot gas mass of the galaxy is assigned 0;
2. when $N_\mathrm{nonSF}=1$, the hot gas mass and temperature of the galaxy are assigned the properties of the gas particle ($m_\mathrm{nonSF}$ and $T_\mathrm{nonSF}$) if $T_\mathrm{nonSF}>T_\mathrm{crit}$, otherwise, the hot gas mass of the galaxy is assigned 0;
3. when $N_\mathrm{nonSF}>1$, we use the *scikit-learn* <span style="font-variant:small-caps;">python</span> package [@scikit-learn] – *KMeans*, a clustering algorithm commonly used for machine learning. We consider two features (density and temperature) and separate the non-star-forming gas particles into two groups (see the dashed and dotted contours in the bottom panels of Fig. \[fig:temp\_nosn\_nozcool\]).
1. If the median temperatures of the two groups differ by more than $\Delta_\mathrm{T, crit}$ in logarithm, the hot gas mass and temperature of the galaxy are assigned the median temperature and the total mass of the hotter group (the dashed contour in the left bottom panel of Fig. \[fig:temp\_nosn\_nozcool\]).
2. If the median temperatures of the two groups differ less than $\Delta_\mathrm{T, crit}$ in logarithm, the two groups of the non-star-forming gas particles are considered as one cluster. In this case, the hot gas mass and temperature of the galaxy are assigned the total mass and the median temperature ($\bar{T}_\mathrm{nonSF}$) of all non-star-forming gas particles if $\bar{T}_\mathrm{nonSF}>T_\mathrm{crit}$, otherwise, the hot gas mass of the galaxy is assigned 0.
We adopt $T_\mathrm{crit}=10^5\mathrm{K}$ and $\Delta_\mathrm{T, crit}=0.5$ in this work and we note that, for haloes with efficient cooling[^10], the determination of hot gas is not sensitive to the temperature thresholds (i.e. $T_\mathrm{crit}$ and $\Delta_\mathrm{T, crit}$; see Section \[sec:gas\_infall1\] and Appendix \[app:sec:hot\]).
When the properties of hot gas are defined, we also characterize the remaining particles and identify them as the cold non-star-forming gas component, whose temperature and mass are represented by the median temperature and total mass of the remaining particles.
In the top panels of Fig. \[fig:temp\_nosn\_nozcool\], we show the temperature function of the two groups separately and indicate their median temperatures with black lines with line styles corresponding to the two contours in the bottom panels of Fig. \[fig:temp\_nosn\_nozcool\]. We also indicate the virial temperature[^11] of the system using vertical red dashed lines. We see that the median temperature of the hotter group is in agreement with the virial temperature for the more massive halo, suggesting the infall gas particles of massive haloes have been through virial shocks and are heated to the virial temperature. However, less massive galaxies do not comprise any hot gas component. We will further discuss it in Section \[sec:gas\_infall1\].
Modifications of halo masses and baryon fractions {#sec:halo mass}
=================================================
Before performing any comparisons, there is one modification regarding the suppressed growth of dark matter haloes that needs to be addressed. Hydrostatic pressure keeps gas from collapsing into shallow potential wells of low-mass haloes, which consequently decreases the gravitational potential within these haloes and slows their growths compared to a collisionless universe. This has been illustrated using comparisons between hydrodynamic simulations and *N*-body simulations of dark matter only [@Sawala2013; @Schaller2014; @Velliscig2014], which show that the inclusion of baryons significantly reduces halo masses[^12] (e.g. ${\sim}20$ per cent for haloes around $10^{11} \mathrm{M}_\odot$ at $z\sim0$).
We investigated the same baryonic effect in dwarf galaxies at high redshift ($z>5$) in @Qin2017a [hereafter ]. We found that the reduction of mass can be up to a factor of 2 and that the fraction of baryons in haloes with masses between $10^7$ and $10^9 \mathrm{M}_\odot$, which host dwarf galaxies, never exceeds 90 per cent of the cosmic mean ($\Omega_\mathrm{b}/\Omega_\mathrm{m}$) during reionization. Thus, applying SAMs to halo merger trees generated from a collisionless *N*-body simulation and assuming the baryon fraction of every virialized system is $\Omega_\mathrm{b}/\Omega_\mathrm{m}$ overestimates halo and baryon masses of dwarf galaxies at high redshift, and suggests the necessity of incorporating modifications to SAMs. We proposed two modifiers using a simplified hydrodynamic simulation in – *ADIAB*, where gas only cools adiabatically with no stellar or galactic physics involved. [Note that the halo mass modifier was calculated as the mass ratio of matched haloes (see matching between galaxies in Appendix \[sec:match\]) between *ADIAB* and *DMONLY* while the baryon fraction modifier is the baryon fraction of these haloes in units of $\Omega_{\mathrm{b}}/\Omega_{\mathrm{m}}$ calculated from the *ADIAB* simulation.]{} In this work, which aims to inform a better semi-analytic description of high-redshift dwarf galaxies, we make use of these two modifiers, which are shown in the top panels of Fig. \[fig:modifiers\] as functions of halo mass at $z=13-5$. In practice, the halo mass modifier is included when halo properties are read from the merger trees and the baryon fraction modifier is embedded in $f_\mathrm{mod}$ (see equation \[eq:accretion\]).
However, we also point out the fact that these modifiers do not offer a consistent modification[^13] accounting for all aspects of astrophysical processes. In , these have been shown to provide additional ${\sim}10$ per cent alterations to halo mass due to radiative cooling, star formation and supernova feedback. Reionization, on the other hand, provides a more dramatic suppression, but can be incorporated through more efficient reionization feedback, which will be further discussed in the companion paper.
{width="1.\textwidth"}\
{width="\textwidth"}
In order to demonstrate the effect of incorporating the modifiers within semi-analytic galaxy formation modelling, we apply [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} with different implementations. We first apply the SAM with both the halo mass and baryon fraction modifiers implemented and calibrate the free parameters in the *NOSN\_NOZCOOL\_NoRe* regime where feedback is not included (*SAM\_HB*). The detailed calibration strategy is addressed in the next section. Next with all free parameters remaining the same, we then apply the SAM with only the halo mass modifier included (*SAM\_H*) and without any modifiers (*SAM*). We show the halo mass, gas mass and stellar mass functions at $z=5$ and 13 of the three SAM results in the bottom panels of Fig. \[fig:modifiers\]. The halo mass functions predicted by the [[[*Smaug*]{}]{}]{} simulations are also included in the left subpanels. Note that matching of individual systems (see Appendix \[sec:match between meraxes and smaug\]) is not performed in this section. We see that compared to the hydrodynamic simulations (black solid lines), the collisionless *N*-body simulation (black dotted lines) as well as the uncorrected SAM overestimate the virial mass function, especially at higher redshifts. However, the halo mass modifier results in a SAM virial mass function that is in better agreement with the hydrodynamic calculation. This leads to a reduction as much as a factor of 2 in of the halo number density at $z\sim12$ for a given mass, and since the growth of the system becomes suppressed, it harbours fewer baryons (see Section \[sec:SAM\_accretion\]) and results in suppressed mass functions of gas and stars. In addition, the baryon fraction modifier due to hydrostatic pressure causes a further reduction of baryons for a given halo mass, which subsequently causes less massive stellar masses.
To summarize, the mass ratio modifier can be interpreted as providing slower evolution of haloes when considering the hydrostatic pressure from baryons, while the baryon fraction modification accounts for the fact that the cosmic mean, $\Omega_\mathrm{b}/\Omega_\mathrm{m}$, cannot be achieved for haloes hosting dwarf galaxies – gas accretion is less efficient.
Comparison between dwarf galaxies in SAM and hydrodynamic simulations {#sec:results}
======================================================================
In this section, we present the comparison of accretion and cooling between [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{}. We note that in this section, if not otherwise stated,
1. for the purpose of performing fair comparisons of the gas reservoir, we define star-formation gas, non star-formation gas and hot gas in hydrodynamic simulations following the method introduced in Sections \[sec:cold gas\] and \[sec:hot gas\];
2. for the purpose of facilitating direct comparisons, we match each individual galaxy in the [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{} outputs using the method presented in Appendix \[sec:match between meraxes and smaug\]. The free parameters in the SAM, which are summarized in Table \[tab:SAMs\], are calibrated to reproduce the $z=5$ stellar mass function predicted by the *NOSN\_NOZCOOL\_NoRe* hydrodynamic simulation;
3. for the purpose of minimizing the impact from mismatch and central-satellites switching (see Appendix \[sec:gbptrees\]), in the SAM, central galaxies with $M_\mathrm{vir}<10^{7.5}\mathrm{M}_\odot$ as well as all satellites[^14] are excluded from the final matched galaxy sample;
4. for the purpose of providing more consistent halo properties between [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{}, the two modifiers (i.e. halo mass and baryon fraction; see *SAM\_HB* in Section \[sec:halo mass\]) have been implemented in the SAM.
[lccll]{}
------------------------------------------------------------------------
& & & &\
*SAM*&&&0.05&1\
*SAM\_H*&&&0.05&1\
*SAM\_HB*&&&0.05&1\
*SAM\_HBS*&&&$0.05\times\left(\dfrac{1+z}{6}\right)^{-1.3}$&1\
*SAM\_HBSC*&&&$0.05\times\left(\dfrac{1+z}{6}\right)^{-1.3}$&$\min\left(5, \dfrac{1+z}{6}\right)$\
{width="\textwidth"}
The first row of Fig. \[fig:nosn\_nozcool\_nore\] presents the stellar mass function of matched galaxies between *SAM\_HB* and [[[*Smaug*]{}]{}]{} at $z=13-5$. We see that the two $z=5$ mass functions are in agreement at $M_*>10^7\mathrm{M}_\odot$, below which the models are limited by resolution and cooling mechanism. This indicates that the two modelled galaxy catalogues comprise similar stellar components at $z=5$ in a cosmological context. However, the results diverge towards higher redshifts – *SAM\_HB* produces more massive galaxies than [[[*Smaug*]{}]{}]{}. This highlights the issue that stellar build-up proceeds faster in the SAM. As a consequence of the high-redshift modelling, the SAM produces more ionizing photons[^15] at earlier times compared to the hydrodynamic calculation, overestimating the contribution of high-redshift dwarf galaxies to reionization [@Liu2016].
We investigate the properties of individual galaxies in the two models, and show the comparisons of the total baryon mass, stellar mass and SFR predicted by the hydrodynamic simulation and SAM in the bottom three rows of Fig. \[fig:nosn\_nozcool\_nore\]. The grey 2D histograms and black solid lines represent the distribution and mean of all matched galaxies, respectively. We see that the overall baryon mass is underestimated by the SAM, with the offset becoming larger at the low-mass end. This is because the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation predicts larger halo masses and baryon fractions than *ADIAB* (where only adiabatic cooling of gas is included; see Section \[sec:halo mass\]) due to cooling and star formation (). Therefore, modifying the halo mass and baryon fraction using the *ADIAB* [[[*Smaug*]{}]{}]{} simulation overestimates the baryonic effect when feedback is not included. In addition, we see that both stellar mass and SFR are similar between the two models at $z=5$, which are, however, underestimated in the SAM result when the stellar mass approaches $10^6\mathrm{M}_\odot$ or SFR reaches $10^{-2}\mathrm{M}_\odot\mathrm{yr}^{-1}$. These objects are close to or below the atomic cooling threshold, where cooling becomes inefficient. For these objects, while a fraction of gas particles in the hydrodynamic simulation are still dense enough to form stars, star formation is quenched in the SAM due to the lack of replenishment of cold gas (see Section \[sec:SAM\_SF\]). This causes an underestimation of the number of ionizing photons at the high redshifts when the first galaxies start to ionize the IGM.
We next discuss the involved prescriptions that might induce such discrepancies in the history of star formation. We note that in the *NOSN\_NOZCOOL\_NoRe* case when feedback is not implemented, there are only two processes that might cause the different stellar growth histories between SAMs and hydrodynamic simulations: 1) star formation efficiency; and 2) gas fuelling and cooling efficiencies[^16].
Star formation efficiency {#sec:sfefficiency}
-------------------------
We note again that since star formation is not resolved in cosmological simulations, both the hydrodynamic simulation and SAM start from an empirical relation between SFR and density – the KS law [@kennicutt1998global], which proposes that a galaxy is able to form stars when its surface density ($\Sigma_\mathrm{g}$) exceeds a critical value ($\Sigma_\mathrm{crit}{\sim} \mathrm{a\ few\ M_\odot pc^{-2}}$), and the surface SFR density ($\dot{\Sigma}_\star$) can be estimated by $$\label{eq:ks}
\dfrac{\dot{\Sigma}_\star}{\left(2.5\pm0.7\right)\times10^{-4}\mathrm{M_\odot {yr}^{-1} {pc}^{-2}}} =\left(\dfrac{\Sigma_\mathrm{g}}{\mathrm{1M_\odot kpc^{-2}}}\right)^n,$$ where $n=1.4\pm0.15$ is suggested by observations [@kennicutt1998global]. We have shown that cooling is not well modelled in the SAM when the virial temperature is lower than the atomic cooling threshold, which leads to significant underestimations of the number of low stellar mass and SFR objects in Fig. \[fig:nosn\_nozcool\_nore\]. However, this cannot explain the discrepancy of star formation history for more massive objects. In this subsection, we further probe the star formation laws utilized in [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{}.
### Star formation in hydrodynamic simulations
In the hydrodynamic simulation, local 3D densities, $\rho_\mathrm{g}$ can be calculated through smoothing particle mass, $m_\mathrm{g}$ within a certain volume. [[[*Smaug*]{}]{}]{} converts the critical surface density to three dimensions ($\rho_\mathrm{crit}$) assuming a self–gravitating disc. The gas on the disc is determined as multiphase ISM with an effective ratio of specific heats, $\gamma_\mathrm{eff} {=} 4/3$. Then the pressure of the disc can be derived from the equation of state. When the local pressure of a gas particle with a relatively low temperature (${<}10^5\mathrm{K}$) exceeds the critical value (see Section \[sec:cold gas\]), the particle is considered as a potential star forming region. This infers a SFR of (see more details in @Schaye2008) $$\label{eq:sfr_smaug_tmp}
\dot{m}_{\star,\mathrm{hydro}} {\equiv} \dfrac{m_\mathrm{g}}{t_\mathrm{g}} {\equiv} m_\mathrm{g} \dfrac{\dot{\Sigma}_\star}{\Sigma_\mathrm{g}}\propto m_\mathrm{g} \Sigma_\mathrm{g}^{\left(n{-}1\right)} \propto m_\mathrm{g} \rho_\mathrm{g}^{0.5\left(n{-}1\right)\gamma_\mathrm{eff}},$$ where $t_\mathrm{g} \equiv \Sigma_\mathrm{g}/\dot{\Sigma}_\star$ is the gas depletion time-scale, the third step makes use of equation (\[eq:ks\]) and the last step assumes that $\Sigma_\mathrm{g}$ is of the order of the Jeans column density in the case of self gravitating disc. With a further assumption that the disc follows an isothermal exponential surface density profile [@Schaye2004] $$\rho_\mathrm{g} = \dfrac{G M_\mathrm{disc}^2}{12\pi c_\mathrm{s}^2R_\mathrm{disc}^4},$$ where $M_\mathrm{disc}\propto M_\mathrm{vir}$, $R_\mathrm{disc}\propto R_\mathrm{vir}$ and $c_\mathrm{s}$ are the mass, radius and sound speed of the disc, respectively, we find that for a given halo mass $M_\mathrm{vir}$, $\rho_\mathrm{g} \propto \left(1+z\right)^4$, and hence $$\label{eq:sfr_smaug}
\dot{m}_{\star,\mathrm{hydro}} \propto \left(1+z\right)^{2\gamma_\mathrm{eff}\left(n-1\right)} {\mathrel{\vcenter{
\offinterlineskip\halign{\hfil$##$\cr
\propto\cr\noalign{\kern2pt}\sim\cr\noalign{\kern-2pt}}}}}\left(1+z\right)^{1.1}.$$
### Star formation in SAMs {#sec:sfr_meraxes}
On the other hand, the average SFR of a galaxy in the SAM is calculated using equation (\[eq:sfr\]). For a given halo mass and with $m_\mathrm{cold}\propto M_\mathrm{vir}$, the redshift dependency of $\dot{m}_{\star,\mathrm{SAM}}$ is approximately $$\label{eq:sfr_meraxes}
\dot{m}_{\star,\mathrm{SAM}} = \alpha_{\mathrm{sf}}\times\dfrac{\mathrm{max}\left(0,m_{\mathrm{cold}}{-}m_{\mathrm{crit}}\right)}{t_{\mathrm{dyn,disc}}} {\mathrel{\vcenter{
\offinterlineskip\halign{\hfil$##$\cr
\propto\cr\noalign{\kern2pt}\sim\cr\noalign{\kern-2pt}}}}}\left(1+z\right)^{1.5}.$$ Note that without supernova feedback in the SAM, the fraction of galaxies that comprise a massive cold gas reservoir (i.e. $m_\mathrm{cold}>m_\mathrm{crit}$) increases towards higher redshifts and in more massive objects. This suggests that the index can be larger than 1.5 on average, especially for less massive haloes.
{width="49.50000%"} {width="49.50000%"}
### Redshift-dependent star formation efficiency
Comparing equations (\[eq:sfr\_smaug\]) and (\[eq:sfr\_meraxes\]), we find that although both numerical calculations of star formation are derived from the KS law, they possess different evolutionary histories for a given halo mass. This is essentially due to the different assumption of gas depletion time-scale adopted in the two modelling approaches, which is chosen to be the disc dynamical time in the SAM while in the hydrodynamic simulation is inferred from observations. In order to be consistent with the hydrodynamic simulation, a suppressed star formation efficiency of $\alpha_\mathrm{sf} \propto (1{+}z)^{{-}m}$ towards higher redshifts is required in the SAM.
We next further discuss the value of $m$.
1. The scaling indices in equations (\[eq:sfr\_smaug\]) and (\[eq:sfr\_meraxes\]) assume a disc profile for star-forming gas and a negligible cold gas threshold of star formation (i.e. $m_\mathrm{cold}\gg m_\mathrm{crit}$) in the SAM, respectively. These infer $m\sim0.4$.
2. However, at higher redshifts, galaxies have larger velocity dispersions, indicating increased turbulence and thickened discs [@Newman2012; @Price2015]. Meanwhile, simulations suggest that mergers which happen frequently at high redshift [@Poole2016] can also thicken discs [@Moster2010; @Moster2012]. Therefore, the aforementioned assumption of a self-gravitating disc might not be valid at high redshift. If we assume an SIS profile for star-forming gas, $\rho_\mathrm{g}$ scales $\left(1+z\right)^3$, which leads to a larger value of $m\sim0.7$.
3. Furthermore, during the experiment we find that, in the SAM, galaxies are more likely to have an insufficient cold gas reservoir (i.e. $m_\mathrm{cold}<m_\mathrm{crit}$) at lower redshifts. This requires a more suppressed star formation efficiency at higher redshift, and in practice, $m=1.3$ is adopted in this work.
We note that since the star-forming gas profile varies with different implementations of physics including feedback, $m$ is introduced as a free parameter modulating the global star formation history.
In Fig. \[fig:nosn\_nozcool\_nore\], the semi-analytic result using the redshift-dependent star formation efficiency (*SAM\_HBS*) is presented using dashed lines. We see that suppressing star formation at higher redshifts results in a better agreement of the stellar mass function, stellar mass and SFR with the hydrodynamic calculations. Note that in the absence of feedback, star formation is not limited by the gas reservoir of galaxies in the SAM (see equation \[eq:sfr\]). Therefore, updating the star formation efficiency does not have a significant impact on the gas component discussed in the next subsections.
Gas fuelling {#sec:gas_infall1}
------------
### Hot gas in hydrodynamic simulations
We define hot gas fraction as the mass ratio of the hot gas (see Section \[sec:hot gas\]) to the non-star-forming gas (see Section \[sec:cold gas\]; $f_\mathrm{hot}\equiv M_\mathrm{hot}/M_\mathrm{nonSF}$), and show its correlation with halo mass in [[[*Smaug*]{}]{}]{} at $z=13-5$ in the left-hand panel of Fig. \[fig:temp\_nosn\_nozcool\_nore\]. We highlight galaxies above the atomic cooling threshold with red circles (circle size representing the stellar mass) and we see that these galaxies in general comprise little or no hot gas. We show the numbers of all galaxies, galaxies with $f_\mathrm{hot}=0$, $0<f_\mathrm{hot}<1$ and $f_\mathrm{hot}=1$ in each subpanel, and indicate galaxies with $f_\mathrm{hot}\ne0$ with red points. We see that the majority of galaxies at high redshift do not possess any hot gas, and their non-star-forming gas particles are identified as one group – the cold non-star-forming gas. However, due to the lack of molecular cooling, galaxies with $M_\mathrm{vir}\lesssim10^8\mathrm{M}_\odot$ do comprise two components. In the right-hand panel of Fig. \[fig:temp\_nosn\_nozcool\_nore\], we show the 2D histograms of the hot (red) and cold (blue) non-star-forming gas temperatures as functions of halo mass in [[[*Smaug*]{}]{}]{}. We see that for galaxies with $M_\mathrm{vir}\lesssim10^8\mathrm{M}_\odot$, their hot and cold non-star-forming gas particles[^17] possess median temperatures $>10^3\mathrm{K}$ and $<10^3\mathrm{K}$, respectively. This broad distribution on the density–temperature phase diagram is from adiabatic cooling of the gas as the universe expands, which is also indicated by the decreasing median temperature at lower redshifts of these less massive objects.
### Hot gas in SAMs
Gas in the IGM falls into the gravitational potential of a halo and fuels star formation in the host galaxy (see Section \[sec:SAM\_accretion\]). [Although the hydrodynamic simulation and SAM both assume ionization equilibrium for the accreted gas, the temperature profile varies significantly. While gas particles cover a broad range of temperatures from ${\sim}10^3-10^8\mathrm{K}$ [@Wiersma2009b] in the hydrodynamic simulation, (hot) gas initially accreted by a galaxy in the SAM is assumed to be shock–heated to the virial temperature (see Section \[sec:SAM\_cooling\]).]{} Hot gas then cools through thermal radiation and builds a cold gas disc where stars can form (see Section \[sec:SAM\_SF\]). In the *NOSN\_NOZCOOL\_NoRe* regime where metals cooling is not included, temperature alone (assuming the helium fraction is a constant) determines the cooling rate (see equation \[eq:t\_cool\]) and therefore alters star formation in each galaxy. In order to test the validity of this simplified gas infall prescription for modelling of dwarf galaxies, we compare the temperature of non-star-forming gas in the hydrodynamic simulation to the halo virial temperature. In the right-hand panel of Fig. \[fig:temp\_nosn\_nozcool\_nore\], we highlight galaxies with $M_\mathrm{vir}>10^9\mathrm{M}_\odot$ in [[[*Smaug*]{}]{}]{} with circles and show the correlation between virial temperature and halo mass with green solid lines. We see that ${\lesssim}10$ galaxies with $M_\mathrm{vir}>10^9\mathrm{M}_\odot$ in the [[[*Smaug*]{}]{}]{} simulation comprise hot gas at $z=5$, and their hot gas temperatures are close to the virial temperatures, suggesting that theses galaxies have been heated through shocks. However, the non-star-forming gas of most galaxies in [[[*Smaug*]{}]{}]{} is considered as cold with $T\lesssim10^4K$, which is much lower than the virial temperature and becomes more common at higher redshifts. [This is consistent with the prediction of EAGLE simulations [@Schaye2014], where the mass ratio of hot[^18] to all baryons is less than 1 per cent on average for haloes around $10^{10}\mathrm{M}_\odot$ [@Correa2017], with the fraction decreasing towards higher redshifts and in less massive haloes.]{}
### Gas accretion in cold mode
{width="49.50000%"} {width="49.50000%"}
We note that in the SAM, cold and hot gas, where stars can and cannot form, are defined geometrically. They represent a star forming disc and the outer region, following an exponential and an SIS profile, respectively. However, we show that full virialization of infall gas might be problematic for high-redshift modelling of dwarf galaxy formation. In these less massive systems, virial shocks might not form and the infall gas could then only reach the virial temperature when it arrives at the star forming disc [@Birnboim2003; @Cattaneo2017]. Instead, these cold and less dense non-star-forming gas particles of dwarf galaxies in [[[*Smaug*]{}]{}]{} represent a cold accretion mode [@Keres2005; @Keres2009; @benson2011; @Correa2017]. Therefore, in the SAM, instead of being fully virialized, the infall gas of dwarf galaxies should be separated into hot and cold components, which reach the disc and contribute to star formation on different time-scales (i.e. thermal cooling time, $t_\mathrm{cool}$ in equation \[eq:t\_cool\], and dynamical time, $t_\mathrm{dyn}$ in equation \[eq:mcool\]).
In addition, since the non-star-forming gas particles of the majority of high-redshift dwarf galaxies are *not* hot in the hydrodynamic simulation, we advocate that the terminology of hot gas becomes misleading in the redshift and mass ranges discussed in this work.
Rapid cooling {#sec:rapid_cooling}
-------------
The gas infall and cooling prescription adopted for the SAM distinguishes two regimes – a static hot halo and rapid cooling gas (see Section \[sec:SAM\_cooling\]). In the second regime, gas cools onto the disc at the dynamical time-scale, which has the properties of cold accretion if one combines the process of gas infall and rapid cooling. Therefore, altering the rapid cooling rate is an alternative to incorporating cold accretion[^19] directly [@Cattaneo2017].
The relative importance of the two cooling modes is determined by the ratio of the cooling radius to the virial radius ($R_\mathrm{cool}/R_\mathrm{vir}$), which is shown in the left panel of Fig. \[fig:sf\_nosn\_nozcool\_nore\]. We see that galaxies with $M_* < 10^9\mathrm{M}_\odot$ are mostly in the rapid cooling regime with $R_\mathrm{cool}/R_\mathrm{vir}>1$. The determination of static hot halo and rapid cooling regimes has been discussed in @croton2006many. Comparing with hydrodynamic simulations, they found $M_\mathrm{vir}\sim10^{11}\mathrm{M}_\odot$ separates the two regimes, which is approximately independent with redshift up to $z\sim6$ (see also @Correa2017). This is in agreement with our finding[^20] of $M_* \sim 10^9\mathrm{M}_\odot$.
The balance of instantaneous cooling rate between the two numerical approaches can be inferred from the evolution of the mass ratio of star forming to non-star-forming gas (see Appendix \[app:sam\_properties\] for more information of galaxy properties in the SAM). This is shown in the right panel of Fig. \[fig:sf\_nosn\_nozcool\_nore\]. We see that while the *SAM\_HBS* result is in agreement with the hydrodynamic calculation at low redshift, $M_\mathrm{SF}/M_\mathrm{nonSF}$ in the SAM is underestimated at higher redshifts. We have shown that in the SAM, dwarf galaxies at high redshift are identified as being in the rapid cooling regime. Therefore, in order to understand the different transition rates between non-star-forming reservoir and star-formation gas in [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{}, we next discuss the assumed SIS profile of hot gas reservoir in the SAM.
### Gas profile
{width="49.50000%"} {width="49.50000%"}\
{width="49.50000%"} {width="49.50000%"}
We present the median radial profile of the non-star-forming gas of galaxies with $M_*=10^{7\pm0.5}\mathrm{M}_\odot$ in [[[*Smaug*]{}]{}]{} at $z=13-5$ in Fig. \[fig:1dprofile\]. We see that the density of non-star-forming gas drops at the inner region (where particles become star forming) when compared to an SIS mass profile that follows [@croton2006many] $$\mathscr{M}_\mathrm{nonSF}\left(r\right) = M_\mathrm{nonSF}\times \min\left(\dfrac{r}{R_\mathrm{vir}},1\right),$$ the non-star-forming gas particles 1) have a large dispersion up to ${\gtrsim}2R_\mathrm{vir}$; and 2) are less concentrated at higher redshifts. The SAM assumes the SIS profile for hot gas. Therefore, in the rapid cooling regime (see equation \[eq:mcool\]), the cooling mass during one time step ($\Delta t$) can be calculated through $$M_\mathrm{cool} = M_\mathrm{nonSF}\times\dfrac{\Delta t}{t_\mathrm{dyn}}\equiv \mathscr{M}_\mathrm{nonSF}\left(r=R_\mathrm{vir}\times\dfrac{\Delta t}{t_\mathrm{dyn}}\right).$$ The last step implies there is a radius, within which gas is able to reach the *centre* through free-fall. This is indicated with vertical dashed lines in Fig. \[fig:1dprofile\] at different redshifts. For comparison, we also present the density profile of all gas of the galaxies with $M_*=10^{7\pm0.5}\mathrm{M}_\odot$ in Fig. \[fig:1dprofile\] (coloured dash-dotted lines), and indicate the radii, at which the star-formation gas is as dense as the non-star-forming gas (i.e. $\rho_\mathrm{SF}=\rho_\mathrm{nonSF}$) or becomes deficient (i.e. $\rho_\mathrm{SF}\sim0$), using vertical solid lines. We see that the star-formation gas also possesses a large dispersion. At high redshift ($z>9$), $r(\rho_\mathrm{SF}=0)$ can be larger than the virial radius.
### Transition radius of gas reservoir
We see that the SIS profile significantly overestimates the non-star-forming gas density in the inner region compared to the simulation result. Therefore, with the assumption that gas can only transform from non-star-forming to star forming (or hot to cold in the SAM) gas when collapsing into the centre, the SIS profile leads to an overestimation of the inward collapse rate (see the horizontal dashed line in Fig. \[fig:1dprofile\]). However, due to the large radial region of star-formation gas observed in Fig. \[fig:1dprofile\], this assumption (i.e. transition radius is ${\sim}0$) might not be accurate. Gas particles in dwarf galaxies at $z>9$ can be triggered as star forming regions as far as the virial radius. The combination of corresponding effects between the overestimated collapse rate from the assumed SIS profile and the underestimated transition radius of gas reservoir is the primary cause of the discrepancy in $M_\mathrm{SF}/M_\mathrm{nonSF}$ between [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{}.
### Redshift-dependent maximum cooling factor
Whilst adopting the correct gas profile and finding the accurate transition radius of gas reservoir can help explain the underestimation of $M_\mathrm{SF}/M_\mathrm{nonSF}$ in the SAM at high redshift, an analytic solution is not well defined. The large dispersion in the gas profile is mainly due to frequent mergers at these redshifts, which can be affected by numerical configurations, such as particle resolution, as well as by physics implementations including feedback. In order to take the underestimated cooling rate of dwarf galaxies at high redshift into account, we propose an alteration to equation \[eq:mcool\] as follows $$\label{eq:mcool2}
\dot{m}_{\mathrm{cool}}{=}\dfrac{m_{\mathrm{hot}}}{t_\mathrm{dyn}}\times\min\left(\kappa_\mathrm{cool},\dfrac{r_{\mathrm{cool}}}{R_{\mathrm{vir}}}\right),$$ and allow the free parameter, $\kappa_\mathrm{cool}$ representing the maximum cooling factor to exceed unity at high redshift[^21]. We show the result of $M_\mathrm{SF}/M_\mathrm{nonSF}$ calculated with $\kappa_\mathrm{cool}=\min\left(5,
\dfrac{1+z}{6}\right)$ (*SAM\_HBSC*) in Fig. \[fig:sf\_nosn\_nozcool\_nore\], and we see that with higher cooling rates, the semi-analytic calculation becomes more consistent with the hydrodynamic result at high redshift. However, the stellar mass function becomes higher at $z=13$ compared to the [[[*Smaug*]{}]{}]{} result (see Fig. \[fig:nosn\_nozcool\_nore\]), suggesting the necessity of stronger suppression of the star formation efficiency (i.e. $m>1.3$) due to the relatively larger disc mass at higher redshifts in the SAM (see Section \[sec:sfr\_meraxes\]).
Lastly, we point out that without a self-consistent radiative transfer calculation, atomic cooling and gas temperature might not be properly simulated in the hydrodynamic simulation either, which alters the importance of thermal radiation cooling. For instance, with local sources (of each gas particle) of ionizing radiation neglected, the cooling rate is potentially overestimated [@Schaye2014], and since self-shielding cannot be properly captured in our hydrodynamic simulations, the cooling rate might be underestimated in dense regions [@McQuinn2011]. Moreover, supernova feedback and reionization are expected to regulate galaxy formation through altering the gas component, which will be the topic of a forthcoming paper.
Conclusions {#sec:conclusion}
===========
While most assumptions adopted for semi-analytic galaxy formation models, including the gas reservoirs (i.e. the density profile, temperature and transition), are in good agreement with hydrodynamic simulations for Milky Way size objects [@Guo2016; @Stevens2016b], they become less accurate for less massive galaxies at high redshift. In this work, we propose modifications to SAMs based on the comparison of dwarf galaxy properties calculated by the [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} SAM and the [[[*Smaug*]{}]{}]{} high-resolution hydrodynamic simulation. We focus on gas accretion, cooling and star formation at $z\ge5$, and consider scenarios in the absence of reionization and supernova feedback. We summarize the modifications below:
1. The parent cosmological simulation of a SAM usually includes only collisionless particles, where baryonic physics is neglected. In making comparisons between *N*-body and hydrodynamic simulations starting from identical initial conditions, we previously showed () that dwarf galaxy host halo masses are significantly overestimated when hydrostatic pressure is not considered, and that the fraction of baryons accreted by dwarf galaxies cannot reach the level of cosmic mean. While inclusion of halo masses directly from *N*-body dark matter only simulation and assumption of a universal baryon fraction are standard features of SAMs, the impact of these assumptions becomes significant at high redshift and small scale. We have considered implementations to modify the relevant properties (i.e. the halo mass and baryon fraction) in SAMs (see Section \[sec:halo mass\]).
2. We find that the star formation prescription in the SAM that is based on the consumption of a cold gas reservoir does not represent the evolutionary path followed by gas in the hydrodynamic simulation, while both of the two modelling approaches start from the observational relation between surface density and SFR [@kennicutt1998global]. We find that this results from variation in the calculated depletion time-scale of the gas reservoir. We address this by modifying the efficiency that modulates star formation in the SAM with a redshift dependency (i.e. replacing the constant $\alpha_{\mathrm{sf}}$ to $\alpha_{\mathrm{sf}}\left(z\right)$) $$\dot{m}_* = \alpha_{\mathrm{sf}}\left(z\right)\times \dfrac{m_\mathrm{cold}-m_\mathrm{crit}}{t_{\mathrm{dyn,disc}}}.$$ In this work, we adopt $\alpha_{\mathrm{sf}}\left(z\right)=0.05\times\left[\left(1+z\right)/6\right]^{-1.3}$, which allows the model to follow the *NOSN\_NOZCOOL\_NoRe* hydrodynamic simulation in the [[[*Smaug*]{}]{}]{} suite (see Section \[sec:sfefficiency\]).
3. The majority of dwarf galaxies at high redshift are in the rapid cooling regime, where the infalling gas cannot form stable shocks or remain in hydrostatic equilibrium. This represents a cold accretion mode, which is not well modelled with the assumption that the hot gas reservoir follows the SIS profile and falls into the centre within the dynamical time. We find, in the hydrodynamic simulation, that gas in high-redshift dwarf galaxies can form stars as far as the virial radius and that the SIS profile overestimates the density in the inner regions of these low-mass objects. In order to take this into account, we propose of modulation of the cooling prescription with a redshift-dependent collapse rate (i.e. changing the cooling rate upper limit from $m_\mathrm{hot}t_\mathrm{dyn}^{-1}$ to $\kappa_\mathrm{cool}\left(z\right)m_\mathrm{hot}t_\mathrm{dyn}^{-1}$) $$\dot{m}_{\mathrm{cool}}{=}\dfrac{m_{\mathrm{hot}}}{t_\mathrm{dyn}}\times\min\left[\kappa_\mathrm{cool}\left(z\right),\dfrac{r_{\mathrm{cool}}}{R_{\mathrm{vir}}}\right],$$ where $\kappa_\mathrm{cool}\left(z\right)=\min\left[5, \left(1+z\right)/6\right]$ leads to an agreement with the hydrodynamic calculation on the cold gas mass evolution (see Section \[sec:rapid\_cooling\]).
Furthermore, we point out that the terminology of *hot gas* and *cold gas* becomes misleading when applying SAMs to the formation of high-redshift dwarf galaxies. The hot gas representing non-star-forming gas within a galaxy does not experience full virialization and possess a median temperature that is much lower than the virial temperature of the host halo (Section \[sec:gas\_infall1\]). In a future paper (Qin et al. in prep.), we will discuss star formation in the presence of feedback from reionization and supernovae. We will compare the SAM calculation with the most complete hydrodynamic simulation in the [[[*Smaug*]{}]{}]{} suite that considers reionization as an instantaneous heating background [@Haardt2001] and distributes supernova energy in thermal form [@DallaVecchia2012]. We will also consider an additional star formation prescription to the one shown in this work, in which molecular gas directly drives the star formation history [@Lagos2011]. The goal of these two papers will be more complete and faithful recreation of hydrodynamic simulations at high redshift, which will further leverage the observations that we can make in this early history of our Universe.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by the Victorian Life Sciences Computation Initiative, grant ref. UOM0005, on its Peak Computing Facility hosted at the University of Melbourne, an initiative of the Victorian Government, Australia. Part of this work was performed on the gSTAR national facility at Swinburne University of Technology. gSTAR is funded by Swinburne and the Australian Governments Education Investment Fund. This research was conducted by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. This work was supported by the Flagship Allocation Scheme of the NCI National Facility at the ANU, generous allocations of time through the iVEC Partner Share and Australian Supercomputer Time Allocation Committee. AM acknowledges support from the European Research Council under the European Union’s Horizon 2020 research and innovation program (Grant No. 638809 – AIDA).
Matched Galaxies between <span style="font-variant:small-caps;">Meraxes</span> and Smaug {#sec:match}
========================================================================================
\
In this work, we make direct comparisons of the semi-analytic and hydrodynamic galaxy properties calculated by [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{}, respectively. We connect galaxies produced by [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} to their host haloes in the *N*-body simulation, and then match galaxies in [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} with the corresponding hydrodynamic simulation. We describe the method in detail for matching galaxies between [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{} in this section.
Running <span style="font-variant:small-caps;">Meraxes</span> on dark matter halo merger trees from [[[*Smaug*]{}]{}]{} {#sec:gbptrees}
-----------------------------------------------------------------------------------------------------------------------
The same algorithm of matching haloes between snapshots is adopted to match between simulations. Therefore, we first discuss the construction of halo merger trees in the DRAGONS framework (@Poole2016 [@poole2017mnras.472.3659p]).
### Dark matter halo merger trees {#sec:dark matter halo merger trees}
Dark matter (and baryon) particles evolve (co-evolve) in [[[*Smaug*]{}]{}]{}, and friend-of-friend (or [fof]{}) groups and subgroups (or haloes), to which particles belong are determined using a standard [fof]{} halo finder and subgroup finder [[subfind]{}, @springel2008aquarius] in post-processing. More specifically, the halo finder identifies [fof]{} groups using a standard linking length of 0.2 and then treats each group as a sphere with centre located at the position of its most bounded particle (*MBP*). Then by tracking halo particles in consecutive snapshots, each halo can be linked to its progenitors and form a merger tree. Each merger tree can be traced towards higher redshifts until no progenitors can be identified. This process is applied to all haloes at one snapshot, in order to horizontally construct the merger trees, which allows the SAM to evolve all galaxies at each snapshot and calculate reionization feedback self-consistently. We note that the [[[*Smaug*]{}]{}]{} trees provide 103 snapshots between $z=50$ and 5 with a time interval of ${\sim}11.3$ Myr. This high temporal resolution is considered a major improvement of the DRAGONS project (), which is able to resolve the dynamical time of the galactic disc at high redshift (see equation \[eq:sfr\]).
### Matching haloes {#sec:matching}
We next give a brief overview of the [gbptrees]{} algorithm[^22] used to build dark matter halo merger trees, which is essential for understanding the matching process. We refer the interested reader to @poole2017mnras.472.3659p for more detailed descriptions and numerical tests.
For each halo, [subfind]{} sorts particles according to their kinetic and potential energies, with, in general, more bounded particles ranking higher. More specifically, we consider a halo, $\mathscr{H}^{\left(i\right)}$ at snapshot $i$, which consists of $n^{\left(i\right)}$ particles radially sorted in order of decreasing potential energies with ranks of $r^{\left(i\right)} = 1, 2, ..., n^{\left(i\right)}$. At snapshot $j\ne i$, the particles of $\mathscr{H}^{\left(i\right)}$ can be located within several haloes $\{\mathscr{H}^{\left(j\right)}_{k}; k=1,2,...,n_k\}$, each of which is considered as a matching candidate (i.e. progenitor or descendant) and has $n^{\left(ij\right)}_k$ shared particles with $\mathscr{H}^{\left(i\right)}$. For each candidate, a pseudo-radial moment is defined as $$S_k^{\left(ij\right)}\left(m\right) = \sum_{l=1}^{n_k^{\left(ij\right)}}\left[{r^{\left(i\right)}_l}\right]^m,$$ where $r^{\left(i\right)}_l$ is the rank of particle $l$ of halo $\mathscr{H}^{\left(j\right)}_k$ in halo $\mathscr{H}^{\left(i\right)}$. Therefore[^23], $\sum_{k}S_k^{\left(ij\right)}\leq S_\mathrm{max}^{\left(ij\right)}\equiv1^m+2^m+...+{n^{\left(i\right)}}^m$. We define the goodness of a matching candidate as $$\begin{split}
&\Delta f_k^{\left(ij\right)} {\equiv} \left.f_k^{\left(ij\right)}\right\vert_{m{=}{-}1}{-}\left.f_k^{\left(ij\right)}\right\vert_{m{=}0},\\
&\mathrm{where}\\
&f_k^{\left(ij\right)}\left(m\right) = \dfrac{S_k^{\left(ij\right)}\left(m\right)}{S_\mathrm{max}^{\left(ij\right)}\left(m\right)}.
\end{split}$$ $\mathscr{H}^{\left(i\right)}$ and $\mathscr{H}^{\left(j\right)}_k$ are considered as a good match when $\Delta f_k^{\left(ij\right)}>-0.2$ [@poole2017mnras.472.3659p] Amongst all good matches, the best match is identified by maximizing the statistic of $S_k^{\left(ij\right)}\left(m{=}{-}1\right)$ during the process of scanning for good matches forwards and backwards over 16 snapshot[^24], $|j-i|=1,2,...,16$.
A consequence of this algorithm is that matching is performed by tracking halo cores instead of the majority of halo particles. This central weighting algorithm allows us to build more realistic halo merger trees, distributed on which galaxies form and evolve within the central regions of their host haloes. However, pathologies can also arise in this algorithm, which require special treatments. We briefly review two consequences of central weighting while more dedicated tests and analysis are presented in @poole2017mnras.472.3659p.
1. When a halo orbits around a more massive object for a period of time without a subsequent merger event, it might be identified as a fraction of the larger system by [subfind]{}, leading to a temporary merger event and causing numerical confusions. Under this circumstance and when the two haloes split, the descendant of the temporarily merged system is determined by the dominant object in the central regions before the separation. However, a side effect arises as well that the emerging halo is identified as a newly formed system without any progenitors. Note that losing progenitors in the merger trees leads to a reset of the hosted galaxy and thus discards its entire evolutionary history, which is catastrophic for semi-analytic galaxy formation models. The same impact also arises when the descendant cannot be identified due to tidal disruptions or numerical noise. Therefore,[gbpTrees]{} is optimized to capture these and link the progenitor or locate the descendant with matching dedicated over consecutive and multiple snapshots.
2. When a central halo, which is usually considered as the most massive halo in an [fof]{} group and dominates the gravitational potential of the entire system, is interacting with its satellites, particles are frequently exchanged between central and satellites. This can potentially lead to numerical confusion that the dominant halo switches between different substructures of the [fof]{} group while their cores remain nearly intact. In this case and because matching performed by following the core restricts galaxies in the centres of their host haloes, pathologies might arise that satellite galaxies receive unrealistic increments of mass on their hosts, leading to massive infall from the IGM and dramatic star formation when cooling is efficient.
Match between <span style="font-variant:small-caps;">Meraxes</span> and [[[*Smaug*]{}]{}]{} {#sec:match between meraxes and smaug}
-------------------------------------------------------------------------------------------
[[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} reports the *MBP* of the host halo for each galaxy, which bridges SAM galaxies and dark matter haloes in *DMONLY*. In order to further connect haloes in the collisionless *N*-body simulation with simulations including baryonic physics (i.e. *NOSN\_NOZCOOL\_NoRe*), we match haloes between simulations using the same matching strategy described in Appendix \[sec:gbptrees\]. Instead of matching 33 snapshots in one simulation, we match haloes from two simulations (e.g. *NOSN\_NOZCOOL\_NoRe* and *DMONLY*) using their dark matter particles at each snapshot. This is achievable because all the [[[*Smaug*]{}]{}]{} simulations including *DMONLY* start from the identical cosmological initial conditions and their dark matter particles possess the same IDs over different simulations. However, with information from only two snapshots, mismatches and pathologies occur in particular when reaching the resolution limit. Therefore, final matched sample is limited to haloes which are matched bidirectionally. Fig. \[fig:match\] illustrates how galaxies are finally matched between [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{} with this two-step manner that *MBP*s connect galaxies from [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} to their host haloes which are matched with [[[*Smaug*]{}]{}]{} galaxies through dark matter particles. In this work, only central galaxies with virial masses exceeding the resolution threshold of $10^{7.5}\mathrm{M}_\odot$ in the SAM results are considered, and in order to minimize the impact from central-satellites switching mentioned in Appendix \[sec:gbptrees\], satellite galaxies identified in the SAM are excluded from the final sample as well.
![\[fig:match\] Illustration of matching between galaxies in [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and full-hydrodynamic simulation [[[*Smaug*]{}]{}]{} outputs. The red and black circles indicate galaxies and their host haloes, respectively. We link galaxies in [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} with their host haloes in the *N*-body simulation through the most bonded particle, *MBP*. Then haloes between *N*-body and full-hydrodynamic simulations are matched using dark matter particle IDs.](./figures/match-eps-converted-to.pdf){width="\columnwidth"}
The right-hand panels of Fig. \[fig:indicator\] project the spatial distributions of dark matter particles in the *NOSN\_NOZCOOL\_NoRe* and *N*-body simulations from [[[*Smaug*]{}]{}]{} on to the xy-plane, which illustrates how well the matching procedure works. In Fig. \[fig:indicator\], the bottom left panel shows the distribution of the first matched 1000 galaxies[^25] in [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} (also without supernova feedback and metal cooling) with circle size representing stellar mass and different colours for better visualization. The bottom right panel shows the distribution of dark matter particles of the corresponding host haloes in the *N*-body simulation while the top right panel shows how these particles are distributed in the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation. The bottom right of each panel zooms into four objects for better illustration. In the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation, there are 195562 galaxies, in which 129043 galaxies ($\sim 66$ per cent) are matched with [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{}.
Hot and cold non-star-forming gas {#app:sec:hot}
=================================
In this work, we use a machine learning algorithm to spilt non-star-forming gas particles into hot and cold components. The two groups are considered as one when the offset between their median temperatures is small (i.e. less than $\Delta_\mathrm{T, cirt}$ in logarithm). Fig. \[app:fig:fhot\_DeltaTcrit\] shows the fraction of hot gas in the non-star-forming gas ($f_\mathrm{hot}\equiv M_\mathrm{hot}/M_\mathrm{nonSF}$) at $z=5$ with different choices of $\Delta_\mathrm{T, cirt}$ and we see that a smaller $\Delta_\mathrm{T, cirt}$ introduces more galaxies with $f_\mathrm{hot}>0$. However, this only affects galaxies with inefficient cooling (i.e. below the atomic cooling thresholds) while $f_\mathrm{hot}$ of more massive galaxies remain zero or small. On the other hand, when non-star-forming gas particles of a galaxy are considered as one group, which is common for galaxies with efficient cooling, whether they are determined as hot or cold (i.e. $f_\mathrm{hot}=1$ or 0) is based on their median temperature. We vary the minimum temperature of particles being identified as hot ($T_\mathrm{crit}$) from $10^5\mathrm{K}$ to $10^4\mathrm{K}$ and we find that $f_\mathrm{hot}$ is insensitive to $T_\mathrm{crit}$. Based on these, we note that the choice of $\Delta_\mathrm{T, cirt}$ and $T_\mathrm{crit}$ does not have a significant impact to our conclusion that high-redshift dwarf galaxies accrete gas in cold mode (see Section \[sec:gas\_infall1\]).
![\[app:fig:fhot\_DeltaTcrit\]The fraction of hot gas in the non-star-forming gas ($f_\mathrm{hot}\equiv M_\mathrm{hot}/M_\mathrm{nonSF}$) from the *NOSN\_NOZCOOL\_NoRe* [[[*Smaug*]{}]{}]{} simulation at $z=5$ as a function of the halo mass ($M_\mathrm{vir}$). From top to bottom, $\Delta_\mathrm{T, cirt}$ is 0.5, 0.2 and 0.1. Galaxies with $f_\mathrm{hot}\ne0$ are indicated with red contours (logarithm scale) and dots at low-dense regions. Galaxies with $M_\mathrm{vir}>10^9\mathrm{M}_\odot$ are emphasized with red circles with circle size representing the stellar mass ($M_*$). The fraction of galaxies with $f_\mathrm{hot}=0$, $0<f_\mathrm{hot}<1$ and $f_\mathrm{hot}=1$ are shown in each subpanel. ](./figures/nosn_nozcool_noret_mass_ratio_baryon_fraction_modifiers/fhot_mvir_DeltaTcrit.pdf){width="\textwidth"}
Semi-analytic properties {#app:sam_properties}
========================
A numerical experiment evolves a galaxy by calculating physics processes in a predefined order. For instance, [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} in sequence calculates reionization, baryonic infall, cooling, star formation and the impact of supernova feedback. Thanks to the exquisite time step of calculation (not output) of hydrodynamic simulations, their outputs can be considered as instantaneous galaxy properties at a given redshift. However, this is not the case for [the SAM used in this work]{}, which records properties that either have (e.g. stellar mass and gas mass) or have not been processed (e.g. host halo properties) or are average values (e.g. SFR) during one time step. For instance, in the right panel of Fig. \[fig:sf\_nosn\_nozcool\_nore\], we show $M_\mathrm{SF}/M_\mathrm{nonSF}$, which is the mass ratio of cold to hot gas in the SAM. Three moments of calculation can be adopted:
1. after star formation, when both hot gas and cold gas have been consumed. In this case, gas mass is underestimated due to the lack of replenishment;
2. before star formation and after cooling, when hot gas has transitioned to cold gas. In this case, stellar mass and hot gas are underestimated while cold gas is overestimated;
3. before cooling and after baryon infall, when hot gas has been refurbished but cold gas still represents the remaining gas reservoir from the previous snapshot or ${\sim}11$Myr ago. In this case, stellar mass and cold gas are underestimated while hot gas is overestimated.
We note that case (1) is adopted in the main context, and we show the difference in Fig. \[app:fig:sf\_nosn\_nozcool\_nore\]. We see that because of the high cadence (${\sim}11$Myr) of the SAM, the choice of calculation does not have a significant impact to our results.
![\[app:fig:sf\_nosn\_nozcool\_nore\]The mass ratio of star-formation gas to non-star-forming gas as a function of stellar mass at $z=13-5$ from [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} and [[[*Smaug*]{}]{}]{} *NOSN\_NOZCOOL\_NoRe* results. Three [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} results with redshift-dependent star formation efficiency and cooling rate (*SAM\_HBSC*) are shown with green dashed for calculations performed at different moments.](./figures/nosn_nozcool_noret_mass_ratio_baryon_fraction_modifiers/SF2nonSF_star_appendix.pdf){width="\textwidth"}
\[lastpage\]
[^1]: E-mail: Yuxiang.L.Qin@Gmail.com
[^2]: E-mail: swyithe@unimelb.edu.au
[^3]: [http://dragons.ph.unimelb.edu.au](http://dragons.ph.unimelb.edu.au/)
[^4]: AGN feedback, which is expected to have an insignificant impact to the formation of dwarf galaxies at high redshift, is not considered here.
[^5]: Gas that has been ejected from galaxies due to supernova feedback is not included in this work. We refer the interested readers to for more details about feedback and gas stripping prescriptions.
[^6]: [Metals are predominately created by massive stars that have reached type supernovae. With supernova feedback included, metals are released into the cold gas disc, which are further transferred between various gas reservoirs through heating, cooling and reincorporation. However, these are not considered in this work and $\Lambda$ only accounts for primordial elements.]{}
[^7]: The second channel of star formation is through mergers, which does not have a significant impact in our model.
[^8]: Supernova feedback and reionization are not included in *NOSN\_NOZCOOL\_NoRe*, which heat the ISM and alter the gas temperature-phase diagram as demonstrated in the companion paper.
[^9]: In collisional ionization equilibrium, hydrogen is ionized at this temperature (${\sim} 5\times10^4\mathrm{K}$), which, however, is low for helium ionization and appears as a trough on the cooling function curve of primordial gas.
[^10]: Molecular cooling is not implemented in this work. Therefore only haloes that are more massive than the atomic cooling limit possess efficient cooling.
[^11]: The virial temperature is calculated using the [fof]{} group properties (see Section \[sec:SAM\_accretion\]).
[^12]: [We refer to the halo mass as the total mass of dark matter and baryons.]{}
[^13]: Modifications on subgroup properties are not considered.
[^14]: Satellite haloes in the SAM do not receive any fresh gas, due to the assumption that their central haloes dominate the gravitational potential, which might not be the case in hydrodynamic simulations.
[^15]: Assuming other relevant factors including the escape fraction remain the same.
[^16]: [In this work, we keep the original cooling functions of primordial elements adopted in and @duffy2010impact. The difference in the two cooling curves (i.e. [mapping ii]{} for the SAM, @Sutherland1993 and [cloudy]{} for the hydrodynamic simulations, @ferland1998cloudy) is minor and does not have a significant impact on our results.]{}
[^17]: Note that with a smaller threshold of the temperature offset between hot and cold non-star-forming gas ($T_\mathrm{crit}$; see Section \[sec:hot gas\]), more particles that belong to haloes with $M_\mathrm{vir}\lesssim10^8\mathrm{M}_\odot$ are identified as hot. However, the temperature thresholds (i.e. $T_\mathrm{crit}$ and $\Delta_\mathrm{T,crit}$) have less impact on more massive haloes, and their $f_\mathrm{hot}$ remains zero or small (see Appendix \[app:sec:hot\]).
[^18]: In @Correa2017, hot gas is defined as particles with cooling time longer than the dynamical time.
[^19]: We leave the task of implementing cold accretion in a future project when molecular cooling is included.
[^20]: Although we are now focusing on a simplified scenario where supernova and metals are not included, it is also true when the feedback is implemented as we will show in the companion paper.
[^21]: The other interpretation of $\kappa_\mathrm{cool}$ is to modulate the time-scale (i.e. $\mathrm{t}_\mathrm{inflow}\equiv\kappa_\mathrm{cool}^{-1}t_\mathrm{dyn}$) of gas inflow from the circumgalactic medium to the ISM. See more in the companion paper.
[^22]: <https://github.com/gbpoole/gbpCode>
[^23]: $\sum_{k}S_k^{\left(ij\right)}$ is smaller than $S_\mathrm{max}^{\left(ij\right)}$ if some of the particles are in unresolved haloes.
[^24]: When $i$ is close to 0 or the total number of snapshots, $|j-i|<16$, leading to a worse matching result. This usually affects the performance of the halo merger trees when the simulation reaches the end.
[^25]: The position of galaxy in [[<span style="font-variant:small-caps;">Meraxes</span>]{}]{} is inherited from its host halo in the *N*-body simulation.
|
---
abstract: 'We study the dynamical rheology of spring networks with a percolation model constructed by bond dilution in a two-dimensional triangular lattice. Hydrodynamic interactions are implemented by a Stokesian viscous coupling between the network nodes and a uniformly deforming liquid. Our simulations show that in a critical connectivity regime, these systems display weak power law rheology in which the complex shear modulus scales with frequency as $G^*\sim (i \omega)^{\Delta}$ where $\Delta = 0.41$, in discord with a mean field prediction of $\Delta = 1/2$. The weak power law rheology in the critical regime can be understood from a simple scaling relation between the macroscopic rheology and the nonaffine strain fluctuations, which diverge with vanishing frequency for isostatic networks. We expand on a dynamic effective medium theory, showing that it quantitatively describes the rheology of a diluted triangular lattice far from isostaticity; although the EMT correctly predicts the scaling form for the rheology of near-isostatic networks, there remains a quantitative disparity due to the mean-field nature of the EMT. Surprisingly, by connecting this critical scaling of the rheology with that of the strain fluctuations, we find that the dynamical behavior of disordered spring networks is fully determined by the critical exponents that govern the behavior of elastic network in the absence of viscous interactions.'
author:
- 'M. G. Yucht'
- 'M. Sheinman'
- 'C. P. Broedersz'
title: Dynamical behavior of disordered spring networks
---
Disordered mechanical networks are used to model a variety of systems including network glasses [@Thorpe1983; @Phillips1981; @He1985; @Schwartz1985; @Feng1984; @Arbabi1993; @Feng1985], jammed packings [@LiuNag1998; @Ohern2003; @Wyart2008a; @Ellenbroek2008] and semiflexible biopolymer networks [@Head2003; @Wilhelm2003; @Heussinger2006; @Das2007; @Broedersz2011; @Mao2011]. Though the specific responses of different classes of networks may vary, the general character of the mechanical behavior depends on the network’s connectivity. Above the so-called isostatic connectivity, networks are mechanically stable and resist static shear deformations [@Maxwell1864]. The lack of mechanical rigidty below this connectivity is due to zero-energy, floppy deformation modes; however, the mechanical response can be stabilized by additional weak interactions [@Wyart2008a; @Schwartz1985; @He1985; @Broedersz2011; @Das2012] or internal stresses [@Alexander; @Sheinman2012]. Recently, focus has shifted towards the dynamic behavior of networks at the verge of mechanical stability with additional viscous interactions [@Heussinger2009; @Tighe2011; @Tighe2012; @Tighe2012b; @Andreotti2012; @Lerner2012a; @Lerner2012b; @Wyart2010; @During2013]. Marginally stable spring networks in jammed configurations exhibit a rich dynamic mechanical response in the presence of damping forces [@Tighe2011; @Tighe2012; @Tighe2012b]; in the vicinity of the isostatic connectivity, the dynamic shear modulus was found numerically to scale with frequency as $\sqrt{\omega}$, in accord with a mean field prediction. In contrast, bond-diluted lattice-based networks—a major class of mechanical systems distinguishable from jammed packings [@Ellenbroek2008]—do not exhibit mean-field behavior [@Feng1984; @Schwartz1985; @Arbabi1993; @Broedersz2011]. This class of models has provided insight in both the linear and nonlinear elastic response of disordered spring and fiber networks. However, little is understood about the dynamical response of such systems in the presence of viscous coupling between the network and a liquid.
![Exemplary samples of networks undergoing dynamic deformation, where the coloring of the plots is a qualitative representation of the nonaffinity of the two nodes connected by each bond (redder is more nonaffine). Networks with $p \gg p_c$ (ad), $p \approx p_c$ (be), and $p \ll p_c$ (cf) are illustrated; the top row depicts high-frequency oscillation, $\omega \gg \omega^*$, and the bottom row depicts low-frequency oscillation, $\omega \ll \omega^*$. In the high frequency limit, all networks deform affinely because the spring network couples strongly with the affine deformation of the viscous network (abc). In the low frequency limit, the spring network deformation is nonaffine, and the degree of nonaffinity increases as $p$ approaches the isostatic bond probabilty $p_c$ (def). []{data-label="fig:networks"}](figures/networks.pdf){width="\columnwidth"}
Here we study the linear rheological behavior of percolating disordered spring networks immersed in a Newtonian liquid. Disordered spring networks are constructed on a triangular lattice in 2D, diluting bonds to control the network’s connectivity, ranging from well below to well above the isostatic connectivity. Analysis of the simulated complex shear modulus of such networks reveals three dynamical regimes. For connectivities well above isostaticity, the networks behave as solids with weak viscous coupling to the fluid. The deformation of these networks is nonaffine at low frequencies, but becomes increasingly affine at high frequencies, as illustrated in Fig. \[fig:networks\]. The second regime is in the vicinity of the isostatic connectivity, where we observe weak power law rheology over a frequency range extending to zero-frequency at the isostatic connectivity; this critical slowing down indicates diverging relaxation timescales. In this critical regime the complex shear modulus scales with frequency as $G^*\sim(i \omega
)^\Delta$, where $\Delta \approx 0.41$. In the third regime, with connectivities well below isostaticity, the networks behave as Maxwell fluids, crossing over from a fluid-like to a solid-like response at a connectivity-independent characteristic frequency. Our results are qualitatively consistent with results by Tighe [@Tighe2011; @Tighe2012; @Tighe2012b] on disordered spring networks based on jammed configurations, although in those systems the dynamic exponent $\Delta =1/2$ was found at near-isostatic connectivities. In addition, we construct a framework that builds on a dynamic effective medium theory (EMT) [@Wyart2010; @During2013] for the rheological response of bond-diluted lattice-based networks and compare this directly with numerical results over a broad range of network connectivities. This dynamic EMT serves as a framework that can be expanded to bond-bending and fiber networks [@Arbabi1993; @He1985; @Das2007; @Broedersz2011; @Mao2011; @Das2012]. Our EMT calculation for a bond-diluted triangular lattice, taken together with scaling arguments, indicates that the dynamical properties of the networks are directly implied from both the scaling of the strain fluctuations and the mechanical behavior of purely elastic spring networks.
The mechanical response of spring networks depends sensitively on the network’s coordination number $z$, the average number of springs attached at a node in the network, not including dangling springs. Maxwell’s constraint counting argument indicates a critical condition of $z_c = 2d$ for the onset of mechanical rigidity in a central force network in $d$ dimensions [@Maxwell1864]. To create a network with variable $z$ in a range spanning from well below $z_c$ to well above, springs are arranged on a triangular lattice and are removed with a probability $1 - p$ such that the network connectivity is roughly $z\simeq 6p$, resulting in a network architecture distinguishable from jammed networks [@Ellenbroek2008]. Using units in which the spring rest length $\ell_0$ and stiffness $\mu$ are both $1$, the energy can be written for small relative deformations ${\bf u}_{ij} = {\bf u}_j - {\bf u}_i$ between neighboring nodes $i$ and $j$ as $$\begin{aligned}
\mathcal{H} &= \frac12 \sum\limits_{\langle ij \rangle}
g_{ij}({\bf u}_{ij} \cdot {\bf \hat{r}}_{ij})^2\end{aligned}$$ where $g_{ij}=\mu = 1$ for a present bond or $0$ for an absent bond and ${\bf \hat{r}}_{ij}$ is a unit vector directed along the $ij$-bond in the undeformed reference lattice. This network is embedded in a viscous fluid in the low-Reynolds number limit. As the network is deformed, hydrodynamic interactions between nodes are ignored, and the fluid deforms affinely. Consequently, the net force on a node $i$ is given by the viscous Stokes drag and the elastic central forces due to the springs to which it is connected, $$\begin{aligned}
\label{eq:force}
{\bf f}_i &=4 \pi \eta
a (\dot{{\bf u}}_{i} - \dot{{\bf u}}_{\rm fluid}) + \sum\limits_{\langle j \rangle}
g_{ij} ({\bf u}_{ij} \cdot {\bf \hat{r}}_{ij}) {\bf \hat{r}}_{ij} \end{aligned}$$ where $\dot{{\bf u}}_{\rm fluid}$ is the velocity field of the underlying fluid, $\eta$ is the fluid’s viscosity, and the summation of $\langle j \rangle$ is over nearest neighbors of node $i$. For the 2D drag coefficient in the Stokes term, we associated a hydrodynamic radius $a$ to a network node and chose natural units such that $\eta a=1$.
![Plot of a typical strain (blue) applied to the spring network and the resultant stress (red) of the network for $p = 0.6$ (a). We observe a phase shift $\delta$ between the applied strain and the response, as well as a decrease in amplitude $\sigma_{0}/\gamma_{0}$, from which we can calculate the complex shear modulus given in Eq. . The effect of the dynamical shear strain is shown for a sample network on the right; (b) shows the initial network state at $t_0 = 0$, and (c) shows the network state after a quarter oscillation at $t = \pi/2\omega$. []{data-label="fig:stress-strain"}](figures/stress_strain.pdf){width="\columnwidth"}
{width="2\columnwidth"}
To study the rheology of these diluted networks numerically, we impose a time-dependent oscillatory shear strain along the two parallel sides of the network with frequency $\omega$ (Fig. \[fig:stress-strain\] bc), which is applied by using periodic, Lees-Edwards boundary conditions [@Lees1972]. The resulting macroscopic shear stress is calculated by $$\label{stresscalc} \sigma_{xy} = \eta \dot{\gamma}(t) + \frac{1}{2 A} \sum\limits_{\langle ij \rangle}
{f}_{ij,x} u_{ij,y}$$ where $A$ is the surface area of the network and the x-component of the force between two nodes $f_{ij,x} = g_{ij}({\bf u}_{ij} \cdot {\bf \hat{r}}_{ij}) { \hat{r}}_{ij,x}$. A typical stress-strain relationship for an oscillating network is depicted in Fig. \[fig:stress-strain\] a. From this, we determine the complex shear modulus $$\label{complexshearmodulus} G^*(\omega)= G'( \omega ) + iG''( \omega
) = \frac{\sigma_{0}}{\gamma_{0}} \left[ \cos( \delta (\omega)) + i\sin( \delta (\omega)
)\right],$$ where $\sigma_0$ is the magnitude of the observed shear stress, $\gamma_0$ that of the imposed shear strain, and $\delta (\omega)$ the phase lag between the stress and the strain at frequency $\omega$. Solid-like systems are dominated by $G'$, the storage modulus, whereas liquid-like systems are dominated by $G''$, the loss modulus.
In the quasistatic limit $\omega \to 0$, the behavior of the network is determined by the relation between $z$ and $z_c$ or, equivalently, between $p$ and $p_c$; the elastic shear modulus vanishes continuously at the isostatic bond probability $p_c$ as $G'\sim \Delta p^f$, where the rigidity exponent $f \approx
1.4$ [@Arbabi1993; @Broedersz2011] and $\Delta p=p-p_c$. By contrast, in the high frequency limit $\omega \rightarrow \infty$, the network’s response becomes affine, $G \to G_{\rm affine} = p\sqrt{3}/4$, for all bond probabilities. (Fig. \[fig:rheodata\] a). At high frequencies, nonaffine deformations are suppressed by large drag forces between the network and the affinely deforming fluid; thus, at high strain rates the fluid effectively dictates the behavior of the network.
Networks with connectivities well below the isostatic connectivity behave as Maxwell fluids—the storage modulus vanishes as $\omega^2$ and the loss modulus as $\omega$ at low frequencies, crossing over at a frequency $\omega^* = 1/4\pi$, set by comparing the stretch modulus of a spring to the drag coefficient of a node, to high-frequency affine elastic behavior. By contrast, hyperstatic networks cross over at this characteristic frequency from a nonaffine to an affine solid-like gel at $\omega^*$. This transition is accompanied by a maximum in the loss modulus, $G''$ (Fig. \[fig:rheodata\] b). At connectivities near isostaticity, the shear modulus appears to exhibit a power law regime $G^*\sim (i \omega)^\Delta$, where $\Delta \approx
0.41$, extending to the zero-frequency limit as $z \to z_c$. Spring networks in jammed configurations display a similar rheological behavior but with mean field exponent $\Delta =1/2$ [@Tighe2011; @Tighe2012; @Tighe2012b]. The distinct regimes observed here over a connectivity range $3\le z \le 6$ are visualized by plotting the inverse loss tangent $G'/G''$ for a range of network connectives and frequencies, as shown in Fig. \[fig:heatmap\] a.
To supplement the complex shear modulus as a description of the macroscopic behavior of spring networks, we study the fluctuations in the microscopic network deformations using a simple one-point nonaffinity measure based on the nonaffine component of a node’s displacement, $\delta
{\bf u}$: $$\Gamma \equiv \left\langle \frac{\delta {\bf u}^2 }{\gamma^2}
\right\rangle,$$ where the brackets indicate an average over network nodes and time. Studies on elastic networks have shown that the nonaffine fluctuations diverge at $p = p_c$ like $\Gamma = \Gamma_\pm |\Delta p |^{-\lambda}$ [@Wyart2008a; @Broedersz2011; @Sheinman2012a], where *$\lambda \approx 2.2$* for bond-diluted 2D triangular networks [@Broedersz2011]. In dynamical networks, nonaffine fluctuations will be suppressed by the affinely deforming viscous fluid. Consistent with prior work [@Tighe2012], our simulations show that the nonaffine fluctuations exhibit a frequency dependence in certain regimes, as shown in Figs. \[fig:NA\] and \[fig:heatmap\] b: far away from $p_c$, the nonaffine fluctuations are frequency-independent at low $\omega$. By contrast, near isostaticity, the nonaffinity measure depends on frequency as a power law $\Gamma\sim \omega^{-\delta}$, with $\delta= 0.59$. For all values of $p$, the nonaffinity vanishes as $\Gamma \sim \omega^{-2}$ beyond the crossover frequency $\omega^*$. The frequency and connectivity dependence of the nonaffinity in dynamic networks can be captured by the scaling ansatz [@Tighe2012]: $$\label{eq:naansatz}
\Gamma=\left|\Delta p\right|^{-\lambda} \Psi_\pm \left(\omega / \omega_c \right),$$ where the critical relaxation frequency, $\omega_c = |\Delta p|^{\phi_\omega}$, describes the slowest relaxation rate in the system. Indeed, we find a good collapse with this scaling form using $\lambda=2.2$ as determined previously in elastic networks [@Broedersz2011] and $\phi_\omega=3.6$, as shown in the lower inset of Fig. \[fig:NA\]. Near isostaticity, the nonaffine fluctuations are finite, and $\Gamma \sim \omega^{-\delta}$; therefore, it becomes clear that $\Psi_\pm(x) \sim x^{-\delta}$ with $\delta = \lambda/\phi_\omega$ to eliminate the $\Delta p$ dependence in Eq. .
The nonaffine fluctuations can be related to the shear modulus by estimating the dissipated power in the system in the critical regime in two different ways [@Tighe2012; @Tighe2012b]. The viscous forces scale as $ \sqrt{\Gamma} \gamma_0 \omega$, giving rise to a dissipated power $W\sim \Gamma (\gamma_0 \omega)^2$, while on a macroscopic level the dissipated power is given by $W= \frac{1}{2} G'' \omega \gamma_0^2$. It follows that $G'' \sim \Gamma \omega$. To test this relation over a broad range of connectivities and frequencies, we plot $G''$ against $\Gamma \omega$ and find that all data collapse onto a curve with a slope of $1$, affirming this correlation (Fig. \[fig:NA\] upper inset). Furthermore, since near isostaticity $G'' \sim \omega^\Delta$ with $\Delta \approx 0.41$ (Fig. \[fig:rheodata\] ab), this connection between $G''$ and $\Gamma$ implies that the dynamical exponents $\delta$ and $\Delta$ are related as $$\label{eq:delta}
\delta=1-\Delta,$$ which is consistent with our observation of $\delta = 0.59$ (Fig. \[fig:NA\] inset).
Dynamic effective medium theory
===============================
To provide insight into the simulated dynamic rheology, we use an Effective Medium approach [@Feng1985; @Schwartz1985; @Das2007; @Das2012; @Sheinman2012a; @Mao2011; @Mao2013], a technique dating back to Bruggeman’s model for the AC conductivity of disordered composite media [@Bruggeman1935; @Clerc1990]. More recently, dynamic effective medium theories have been developed for mechanical networks [@Wyart2010; @During2013]. This approach is based on the construction of a mapping from a lattice network where the spring between nodes $i$ and $j$ has a spring constant $g_{ij}$, drawn from a probability distribution $P(g_{ij})$, onto a lattice with uniform, frequency-dependent bond stiffness $\widetilde{g}(\omega)$; this effective lattice mimics the mechanical response of the disordered network at the same global strain, $\epsilon (\omega)$. To derive an expression for $\widetilde{g}\left(\omega\right)$, we follow the procedure in refs [@Wyart2010; @During2013], extending the approach by Feng et al [@Feng1985] by determining the dynamic, effective bond stiffness from a self-consistency requirement, as detailed below.
The effective medium network is subjected to a macroscopic infinitesimal oscillating strain $\epsilon(\omega) = \epsilon_0 e^{i\omega t}$, deforming bond $nm$ by $ {\bf \hat{r}}_{nm} \epsilon(\omega)$. Subsequently, replacing this effective medium bond with one sampled from the distribution $P(g)$ gives rise to an additional, nonaffine deformation $\delta {\bf u}(\omega)$. The original, uniform deformation can be restored by applying a force $${\bf f}(\omega) = {\bf \hat{r}}_{nm} \epsilon(\omega)(\widetilde{g}(\omega) - g)$$ Thus, the nonaffine deformation which arose from the bond replacement can be expressed as $$\delta {\bf u}(\omega) = \frac{{\bf f}(\omega)}{g_{EM}(\omega) - \widetilde{g}(\omega) + g}$$ where $g_{EM}\left(\omega\right) $ is the force on a bond in the effective medium network in response to a unit displacement. This allows us to express the nonaffine displacement as $$\delta\mathbf{u}\left(\omega\right)=\frac{\mathbf{\hat{r}}_{nm}\epsilon(\omega)\left(\widetilde{g}\left(\omega\right)-g\right)}{g_{EM}\left(\omega\right)-\widetilde{g}\left(\omega\right)+g},\label{eq:DisplacementMainText}$$
The self-consistency condition requires that, when averaging over all possible bond replacements, the local fluctuations in the deformation field must vanish, $\left\langle \delta\mathbf{u}\left(\omega\right)\right\rangle =0$, leading to the following equation for $\widetilde{g}(\omega)$, $$\int_{0}^{\infty}\frac{g-\widetilde{g}\left(\omega\right)}{g_{EM}(\omega)+g-\widetilde{g}\left(\omega\right)}P\left(g\right)dg=0.\label{eq:mEff_Integral-1}$$ We solve this equation by first determining $g_{EM}^{-1}(\omega)$ as the displacement in response to a unit force between nodes $ n $ and $ m $, $\mathbf{f}\left(\mathbf{k}\right)=\mathbf{\hat{r}}_{nm}\left(1-e^{i\mathbf{k}\cdot\mathbf{\hat{r}}_{nm}}\right)$, by solving the network’s equation of motion $$\mathbf{u}\left(\mathbf{k}\right)=-D^{-1}\left(\mathbf{k}\right)\cdot\mathbf{f}\left(\mathbf{k}\right),$$ where the dynamical matrix of the effective medium is given by $$D_{nm}=\begin{cases}
-\widetilde{g}\left(\omega\right)\mathbf{r}_{nm}\otimes\mathbf{r}_{nm} & n\neq m\\
\underset{m\neq n}{\sum}\widetilde{g}\left(\omega\right)\mathbf{r}_{nm}\otimes\mathbf{r}_{nm}+4\pi\eta a i \omega\mathbb{I} & n=m
\end{cases},$$ where $\mathbb{I}$ is the unit tensor and $\otimes$ is the external product. As before, we set $\eta a=1$, and the spatial Fourier transform of $D$ is given by $$\begin{aligned}
D\left(\mathbf{k}\right) & = & \underset{ij}{\sum}D_{ij}e^{i\mathbf{k}\cdot\mathbf{r}_{ij}}\nonumber \\
& = & \underset{\mathbf{r}}{\sum}\widetilde{g}(\omega)\mathbf{r}_{ij}\otimes\mathbf{r}_{ij}\left(1-e^{i\mathbf{k}\cdot\mathbf{r}}\right)+i4\pi\omega\mathbb{I}\end{aligned}$$ Thus, the displacement of the $nm$ bond due to a unit force follows $$\begin{aligned}
\label{eq:GEM}
&g&_{EM}^{-1} (\omega) = \frac{1}{N}\mathbf{r}_{nm}\cdot\underset{\mathbf{k}}{\sum}\mathbf{u}\left(\mathbf{k}\right)\left(e^{-i\mathbf{k}\cdot\mathbf{r}_{nm}}-1\right) = \\
&=&-\frac{1}{N}\underset{\mathbf{k}}{\sum}\mathbf{r}_{nm}\cdot\mathbf{f}\left(\mathbf{k}\right)D^{-1}\left(\mathbf{k}\right)\left(e^{-i\mathbf{k}\cdot\mathbf{r}_{nm}}-1\right) =\nonumber \\
&=&\frac{2\widetilde{g}^{-1}(\omega)}{\mathcal{Z}N} \underset{\mathbf{k}}{\sum}Tr\left[\frac{\underset{\mathbf{r}}{\sum}\mathbf{r}_{ij}\otimes\mathbf{r}_{ij}\left(1-e^{i\mathbf{k}\cdot\mathbf{r}}\right)}{\underset{\mathbf{r}}{\sum}\mathbf{r}_{ij}\otimes\mathbf{r}_{ij}\left(1-e^{i\mathbf{k}\cdot\mathbf{r}}\right)+4\pi\frac{i\omega}{\widetilde{g}(\omega)}\mathbb{I}}\right] =\nonumber\\
&=&\frac{2d}{\mathcal{Z}\widetilde{g}}\left\{1-\frac{i4\pi\omega}{dN\widetilde{g}}\underset{\mathbf{k}}{\sum}Tr\left[ \frac{1}{\underset{\mathbf{r}}{\sum}\mathbf{r}_{ij}\otimes\mathbf{r}_{ij}\left(1-e^{i\mathbf{k}\cdot\mathbf{r}}\right)+4\pi\frac{i\omega}{\widetilde{g}}\mathbb{I}}\right] \right\} \nonumber\end{aligned}$$ where $\mathcal{Z}$ is the maximum coordination of the underlying lattice, $ d $ is the dimension of the system and $N$ is the total number of nodes in the network.
For a random bond-diluted lattice, the self-consistency condition (Eq. ) can be written as $$\label{eq:mEff_diluted}
p\frac{\mu-\widetilde{g}\left(\omega\right)}{g_{EM}(\omega)+\mu-\widetilde{g}\left(\omega\right)}-\left(1-p\right)\frac{\widetilde{g}\left(\omega\right)}{g_{EM}(\omega)-\widetilde{g}\left(\omega\right)}=0,$$ where $\mu$ will be set to $1$. By solving this equation for a triangular lattice configuration, we obtain the macroscopic shear modulus, $G^* (\omega)= \widetilde{g}(\omega) \sqrt{3}/4$. Remarkably, this EMT prediction for the rheology captures the main features of the simulation results with reasonable quantitative agreement, as shown in Figs. \[fig:rheodata\]cd. Slow convergence of the numerical integration scheme precludes a high-precision solution of the EMT in the critical regime. However, we can obtain various interesting analytical results by considering the large or small limits of the quantity $\left|\widetilde{g}(\omega)\right|/\omega$.
#### High-frequency limit
When $ \omega \gg \left|\widetilde{g}(\omega)\right|$, Eq. can be written as $g_{EM} \approx 2\pi \omega i$. Using this in the self-consistency equation, we find the shear modulus to be $$G^* (\omega)\approx \frac{\sqrt{3} p}{4}\left(1+i\frac{1-p}{2 \pi \omega}\right)
\label{GemtLow}$$ This high frequency limit corresponds quantitatively with the numerical results, as shown in the insets of Figs. \[fig:rheodata\]cd.
#### Low-frequency limit
This limit is solvable only when $ p $ is not much less than $ p_c $. In this case $\left|\widetilde{g}(\omega)\right|\gg \omega$ and Eq. reduces to $$g_{EM}^{-1}(\omega) \approx \frac{2\widetilde{g}^{-1}(\omega)}{3} \left[1-\frac{2\pi i \omega}{\widetilde{g}(\omega)}\mathcal{A}\right]$$ where $ \mathcal{A} $ is a numerical constant, $$\mathcal{A} = \frac{1}{N}\underset{\mathbf{k}}{\sum}Tr\frac{1}{\underset{\mathbf{r}}{\sum}\mathbf{r}_{ij}\otimes\mathbf{r}_{ij}\left(1-e^{i\mathbf{k}\cdot\mathbf{r}}\right)} \simeq 5.17.$$ By solving the self-consistency equation for a bond-diluted lattice (Eq. ) in this limit, we find the shear modulus, $$\begin{aligned}
\label{Gemt}
G^*(\omega)\approx & \frac{\sqrt{3}}{16} \bigg[&6p-4-8 i \mathcal{A} \pi \omega \nonumber \\
& & \left. +\sqrt{64 i \mathcal{A} \pi \omega+\left(6p-4-8 i \mathcal{A} \pi \omega\right)^2}\right]\end{aligned}$$ This is consistent with results found by Düring et al. [@During2013]. This expression for the dynamic shear modulus captures the low-frequency rheology for $p\gtrsim p_c$, as shown in the insets of Figs. \[fig:rheodata\] cd.
The EMT indicates a critical bond probability at $p_c=2/3$. Close to the critical point, $\left| \Delta p \right| \ll 1 $, and in the limit of small frequencies ($ \omega \ll \left(\mathcal{A} \pi\right)^{-1}\approx 10^{-1}$ for $G'$ and $ \omega \ll \left(\mathcal{A} \pi\right)^{-2}\approx 10^{-2}$ for $G''$), the shear modulus in Eq. can be written in the following scaling form [@Tighe2011] $$\label{eq:widom}
G^*=\left| \Delta p \right|^{f} \mathcal{G}^*_{\pm}\left(\frac{\omega}{\left| \Delta p \right|^{\phi_\omega}}\right)$$ where the EMT predicts $f = 1$ and $\phi_\omega=2$, consistent with the mean field predictions in ref. [@Tighe2011; @Tighe2012b]. The scaling function $\mathcal{G}^*_{\pm}\left(x\right)=\mathcal{G}'_{\pm}\left(x\right)+i\mathcal{G}''_{\pm}\left(x\right)$ is given by $$\begin{aligned}
\label{eq:scalingfunc}
\mathcal{G}'_{\pm}\left(x\right)&=& \frac{3\sqrt{3}}{8}\left\{ \cos\left[\frac{1}{2} \tan^{-1}\left( \alpha x\right) \right]\left[1+\left( \alpha x\right)^2 \right]^{1/4} \pm 1\right\} \nonumber \\
\mathcal{G}''_{\pm}\left(x\right)&=& \frac{3\sqrt{3}}{8} \sin\left[\frac{1}{2} \tan^{-1}\left( \alpha x\right) \right]\left[1+\left( \alpha x\right)^2 \right]^{1/4} \end{aligned}$$ with $\alpha=\frac{16 \mathcal{A} \pi}{9}$. The scaling function $ \mathcal{G}'_{\pm} $ has a $+$ branch above $p_c$ and a $-$ branch below $p_c$. When $x\rightarrow 0$ and $p>p_c$ $\mathcal{G'_+}(x)$ must be constant such that $G'$ scales as $|\Delta p|^f$, while $\mathcal{G''_\pm}(x) \sim x$ so that $G''$ scales as $\omega |\Delta p|^{f - \phi_\omega}$. Furthermore, when $x\rightarrow 0$ and $p<p_c$ $\mathcal{G'_-}(x) \sim x^2$ such that $G'$ scales as $\omega |\Delta p|^{f-2\phi_\omega}$, and $\mathcal{G''_-}(x) \sim x$ such that $G''$ scales as $\omega |\Delta p|^{f - \phi_\omega}$. In the critical connectivity regime, the shear modulus is finite and, thus, $\mathcal{G^*_\pm}\left( x\right) \sim x^{\frac{f}{\phi_\omega}}$ such that $G^*$ is $ \Delta p $-independent. Consequently, $G^*\sim(i \omega)^\Delta$ with $ \Delta=f / \phi_\omega $.
The EMT and numerical results are collapsed according to this scaling form, as shown in Fig. \[fig:rheocollapse\]. We find an excellent collapse for the numerical data with $f=1.4$, determined in prior work on elastic networks [@Arbabi1993; @Broedersz2011], and $\phi_\omega=3.6$, determined from collapsing the nonaffinity data (Fig. \[fig:NA\]). Within the EMT, $\Delta=1/2$ [@Wyart2010; @During2013], consistent with the mean field calculations in refs. [@Tighe2012; @Tighe2012b]. By contrast, from the collapse of the simulation data, we find $\Delta\approx 0.41$. This difference between the exponents predicted by the EMT and our numerical results are due to the mean-field nature of the effective medium approximation. Specifically, the effective medium theory assumes small nonaffine fluctuations [@During2013]. This assumption appears to be justified for most network connectivities and frequencies, as shown by the good comparison between the EMT and numerical results shown in Fig. \[fig:rheodata\]. However, the nonaffine fluctuations become large as the network approaches criticality; in the quasistatic limit, such fluctuations are expected to diverge for networks with dimension greater than or equal to $2$ [@During2013]. Therefore, any approximation that does not take these diverging fluctuations into account cannot be expected to predict the correct exponents.
The scaling behavior of $G^*$ is clearly related to that of the nonaffinity parameter $\Gamma$: in both cases, the critical relaxation frequency is controlled by the exponent $\phi_\omega$. In the first case, we inferred that $\phi_\omega = f/\Delta$, while in the second case, we found that $\phi_\omega = \lambda / \delta$, and thus, $$\frac{f}{\Delta} = \frac{\lambda}{\delta}$$ Solving for $\Delta$ and recalling that $\delta = 1 - \Delta$ (Eq. \[eq:delta\]), we obtain $$\label{eq:epic}
\Delta = \frac{f}{\lambda + f}$$ Strikingly, we find that the dynamical behavior of the network, captured by the exponent $\Delta$, can in fact be inferred from the rigidity exponent $f$ and the nonaffinity exponent $\lambda$ of the elastic network in the absence of viscous interactions. Using previously determined $f = 1.4 \pm 0.1$ and $\lambda = 2.2 \pm 0.4$ [@Broedersz2011], we expect $\Delta = 0.39 \pm 0.08$, in agreement with our numerical observations. Furthermore, using Eq. , we can also recover the mean-field prediction for the nonaffinity exponent $\lambda = 1$, using $\Delta = 1/2$ and $f = 1$ from the EMT calculation [@Wyart2008a]. This argument should be valid for broader classes of disordered networks, e.g. fiber networks for which $f_b = 3.2\pm0.4$ and $\lambda_b = 1.8 \pm 0.3$ in 2D, implying a dynamical scaling of $\Delta_b = 0.64 \pm 0.13$. These results show that viscous interactions act like a field taking the system away from criticality. Furthermore, similar scaling arguments have been constructed, relating the dynamic conductivity of disordered resistor networks to the exponents that govern the DC response [@Efros76]. Thus, the dynamic EMT, combined with the scaling arguments, provide an avenue for exploring dynamical behavior of a wide range of disordered networks.
We thank Cliff Brangwynne for helpful discussions and his hospitality and Brian Tighe, Fred MacKintosh and Boris Shklovskii for insightful discussions. This work was supported by the Lewis-Sigler fellowship (CPB) and FOM/NWO (MS).
[50]{}
M. F. Thorpe, J. Non-Cryst. Solids [**57**]{}, 355 (1983).
J. C. Phillips, J. Non-Cryst. Solids [**43**]{}, 37-77 (1981).
H. He and M. F. Thorpe, Phys. Rev. Lett. [**54**]{}, 2107-2110 (1985).
L. M. Schwartz, S. Feng, M. F. Thorpe and P. N. Sen, Phys. Rev. B [**32**]{}, 4607-4617 (1985).
S. Feng and P. N. Sen, Phys. Rev. Lett., [**52**]{}, 216-219 (1984).
S. Arbabi and M. Sahimi, Phys. Rev. B [**47**]{}, 695-702 (1993).
S. Feng, M. F. Thorpe and E. J. Garboczi, Phys. Rev. B [**31**]{}, 276 (1985).
A. J. Liu and S. R. Nagel, Nature [**396**]{}, 21 (1998).
C. S. O’Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Phys. Rev. E [**68**]{}, 011306 (2003).
M. Wyart, H. Liang, A. Kabla and L. Mahadevan, Phys. Rev. Lett. [**101**]{}, 215501 (2008).
W. G. Ellenbroek, Z. Zeravcic, W. van Saarloos and M. van Hecke, Eur. Phys. Lett. [**87**]{}, 34004 (2009).
D. A. Head, A. J. Levine and F. C. MacKintosh, Phys. Rev. Lett. [**91**]{}, 108102 (2003).
J. Wilhelm and E. Frey, Phys. Rev. Lett. [**91**]{}, 108103 (2003).
C. Heussinger and E. Frey, Phys. Rev. Lett. [**97**]{}, 105501 (2006).
M. Das, F. C. MacKintosh and A. J. Levine, Phys. Rev. Lett. [**99**]{}, 038101 (2007).
C. P. Broedersz, X. Mao, T. C. Lubensky and F.C. MacKintosh, Nat. Phys. [**7**]{}, 983 (2011).
X. Mao, O. Stenull, and T. C. Lubensky, arXiv:1111.1751v1 (2011).
J. C. Maxwell, Philos. Mag. [**27**]{}, 294 (1864).
M. Das, D. A. Quint, J. M. Schwarz, PLoS ONE 7(5): e35939 (2012).
S. Alexander, Phys. Rep. [**296**]{}, 65 (1998).
M. Sheinman, C. P. Broedersz and F. C. MacKintosh, Phys. Rev. Lett. [**109**]{}, 238101 (2012).
C. Heussinger and J-L. Barrat, Phys. Rev. Lett. [**102**]{}, 218303 (2009).
E. Lerner, G. Düring and M. Wyart, Euro. Phys. Lett., [**99**]{}, 58003, (2012).
E. Lerner, G. Düring and M. Wyart, PNAS, [**109**]{} (13), 4798-4803 (2012).
B. P. Tighe, Phys. Rev. Lett. [**107**]{}, 158303 (2011).
B. P. Tighe, Phys. Rev. Lett. [**109**]{}, 168303 (2012).
B. P. Tighe, arXiv:1205.2960 (2012).
B. Andreotti, J-L. Barrat, and C. Heussinger , Phys. Rev. Lett. [**109**]{}, 105901 (2012).
G. Düring , E. Lerner and M. Wyart, Soft Matter, [**9**]{}, 146 (2013).
M. Wyart, Euro. Phys. Lett., [**89**]{}, 64001 (2010).
A. W. Lees and S. F. Edwards, J. Phys. C: Solid State Phys. **5**, 1921 (1972).
M. Sheinman, C. P. Broedersz and F. C. MacKintosh, Phys Rev E, [**85**]{}: 021801 (2012).
X. Mao, O. Stenull, and T. C. Lubensky, arXiv:1301.0870 (2013).
D. A. G. Bruggeman, Annln Phys. [**24**]{}, 636 (1935).
J.P. Clerc, G. Giraud, J.M. Laugier, J.M. Luck, Advances in Physics [**39**]{}, 191 (1990).
Efros and Shklovskii, Phys. Stat. Sol. B [**76**]{}, 476 (1976).
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abstract: 'We investigate the host galaxy and environment properties of a sample of 400 low z ($<$0.5) quasars that were imaged in the SDSS Stripe82. We can detect and study the properties of the host galaxy for more than 75% of the data sample. We discover that quasar are mainly hosted in luminous galaxies of absolute magnitude $M^* -3 < M(R) < M^*$[**[^1]**]{} and that in the quasar environments the galaxy number density is comparable to that of inactive galaxies of similar luminosities. For these quasars we undertake also a study in u,g,r,i and z SDSS bands and again we discover that the mean colours of the quasar host galaxy it is not very different with respect to the values of the sample of inactive galaxies. For a subsample of low z sources the imaging study is complemented by spectroscopy of quasar hosts and of close companion galaxies. This study suggests that the supply and cause of the nuclear activity depends only weakly on the local environment of quasars. Contrary to past suggestions, for low redshift quasar there is a very modest connection between recent star formation and the nuclear activity.'
author:
-
bibliography:
- 'ref.bib'
title: '[ **On the role of the environments and star formation for quasar activity**]{}'
---
Introduction
============
Quasar phenomenon assumes that the nuclear activity can be due to the major merger of two gas-rich galaxies that feed the central engine and enable the growth of a spheroidal stellar component. However, important details on the mechanisms that triggers the gas fueling and how nuclear activity influence the successive evolution of the host galaxies remain poorly understood. The study of correlations among black hole masses, properties of the host galaxies and their environments may provide relevant clues to investigate the fundamental issues on quasar activity and its role on the evolution of galaxies. Simulations [e.g. @Menci; @Hop] suggest that minor and major merging events may have a key role for triggering and fueling the nuclear/quasar activity. These effects strictly depend on the global properties of the galaxy environment [@KH; @DiM].
The quasars, at low-redshift, follow the large-scale structure traced by galaxy clusters but they eschew the very centre of clusters [@SCC1; @SCC2]. On the other hand the quasar environment, on small scales (projected distance $<$ 0.5 Mpc), appear overpopulated by blue disc galaxies having a significant star formation rate [@CL1; @CL2]. At higher redshifts, quasars are in some cases associated with richer environments [@HG; @Djorg] but also examples of modest galaxy environments are observed. The comparison of the environments of quasars, at Mpc scales, to those of galaxies has given contradictory results partially due to small samples and non homogeneous datasets. Early studies suggested that the galaxy environment of quasars is more strongly clustered than that of inactive galaxies [e.g. @SBP], while later studies based on surveys such as the Two Degree Field (2dF) and the Sloan Digital Sky Survey (SDSS) [@abazajian09] found galaxy densities around quasars and inactive galaxies to be comparable to each other [e.g. @SBM; @Wake].
More recent studies using the SDSS archives such as [@Serber] and [@Strand] have taken advantage of the large datasets provided by the survey to study quasars at z $<$ 0.4. Both studies found that quasars are on average located in higher local over-density regions than typical L\* galaxies, and that within 100 kpc from the quasar the density enhancement is strongest. @Serber also claimed that: “high luminosity quasars have denser small-scale environments than those of lower luminosity”. These studies are, however, limited by the deepness of the surveys that do not enable to study the galaxy population much fainter that L\*. Meanwhile, a study by @Bennert found that many low-z quasars show signs of recent or on-going interactions with nearby galaxies, suggesting a connection between the environment and the quasar activation. In spite of these imaging studies our knowledge of the physical association of the galaxies around quasars and their dynamical properties remains uncertain.
To overcome these issues we recently carried out a systematic study of the the host galaxy and environment properties of a large ($\sim$ 400) sample of low redshift (z $<$ 0.5) quasars based on the deep images available from the SDSS Stripe82 survey. These co-added images are $\sim$2 magnitudes more profound with respect to the standard Sloan data and give the possibility to study both the quasar hosts and the immediate environments. We are able to resolve the host galaxy for $\sim$ 300 targets [@F14] and to investigate the the large scale galaxy environment of all quasars [@K14]. For this work we adopt the concordance cosmology with H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$, $\Omega_m$ = 0.3 and $\Omega_\Lambda$ = 0.7.
The sample and analysis
=======================
The sample of quasar we used, described in detail in @F14, is derived from the fifth release of the SDSS Quasar Catalog [@schneider2010] that is based on the SDSS-DR7 data release [@abazajian09]. Our analysis is done only in the region of sky covered by the stripe82 data, these images go deeper of about $\sim$2 magnitudes with respect to the usual Sloan data and make possible the study of the quasar hosts, with these images we reach $m_{i(limit)}$=24.1 [@Annis]. The final sample is composed by 416 quasars in the range of redshift 0.1$<z<$0.5. In this sample we are dominated by radio quiet quasars only 24 are radio loud (about 5%). The mean redshift of the sample is $<z>$ = $0.39\pm0.08$ (median $0.41\pm0.06$ ) and the average absolute magnitude is :$<M_i>$ = $-22.68\pm0.61$ (median $-22.52\pm0.35$). To perform the analysis of the images we used the tool AIDA (Astronomical Image Decomposition & Analysis) [@Uslenghi]. The tool is designed to perform 2D model fitting of quasar images including Adaptive Optics data and with detailed modeling of the PSF and its variations. In figure \[fig:aida\] we show an example of our analysis and of the AIDA fit.
To perform the analysis of the large scale environment we used SExtractor [@BA] to create our own galaxy catalogs for each image, measuring the magnitudes of the galaxies in each of the five filters separately. A comparison of the catalog generated with SExtractor to those of SDSS, and a visual inspection on a number of frames to further to study the validity of our classification is fully described in [@K14]. We found a good match between the SDSS and SExtractor catalogs at apparent magnitudes $m_i <$23, but at fainter magnitudes the number of objects detected by SExtractor dropped dramatically compared to those in the SDSS catalog. A visual inspection of these faint objects showed that they are mainly background noise which is either undetected by SExtractor or classified as an unknown object. We chose a conservative limit of classification value $\leq$0.20 for our galaxies, to make sure we avoid all the stars and majority of the unknown objects.
Host Galaxies
=============
For our sample we found that the morphology of the quasars host galaxy ranges from pure ellipticals to complex/composite morphologies that combine disks, spheroids, lens and halo and are dominated by luminous galaxies with absolute magnitude in the range M\*-3 $<$ M(R) $<$ M\* and the average absolute magnitude of this sample is $<M_i> = -22.68 \pm 0.61$. The galaxy sizes (defined here as the half-light radius) range from very compact (few kpc) up to more extended galaxies (10-15 kpc) and, in our redshift range, not significant trend of change of the galaxy size with z is found. However, note that the sampled redshift range is small thus possibly hiding systematic changes of size with z. It is interesting to note that the distribution of sizes of the quasars hosts is very similar to that of the sample of inactive galaxies of similar luminosity and redshift distribution as shown in figure \[fig:re\]. The black hole mass of the quasar, estimated from the spectral properties of the nuclei, are poorly correlated with the total luminosity of the host galaxies but the relation between M$_{BH}$ and the luminosity of the spheroidal component is consistent with that of local inactive galaxies [@F14]. We can resolve the host galaxy in $\sim$73% of the quasar sample but in u-band only for the 48%.
Only the ($g-i$) colour is slighter bluer (1.06 $\pm$ 0.11) than that of inactive galaxies (1.19$\pm$ 0.25) of similar luminosity and redshift [see @B15]. The average absolute magnitude ($<M_i>_{QSOHOST}$ = -22.52 $<M_i>_{GAL}$ = -22.26) are very similar and also ($u-g$) color of quasar hosts are similar (1.40$\pm$0.30) to that of inactive galaxies (1.54$\pm$0.62). Quasar and control sample have the same distribution of close ($<$ 50 kpc projected) companion galaxies [@B15]. This result is in contrast with that obtained by @M14 who find similar (g-i) color for the quasar hosts but claim they are systematically bluer than the ensemble of normal galaxies in SDSS. Because of the different selection of active and non active galaxies of our and @M14 samples it is not clear what is the cause of this discrepancy.
Galaxy environment
==================
The large scale galaxy environment ($<$1 Mpc) of the 302 resolved quasars compared with those of a sample of inactive galaxies at same redshift [@K14] shows that the galaxy number density is comparable to that of inactive galaxies with similar luminosities, as seen in figure \[fig:env\]. For distances $<$400 kpc both quasars and inactive galaxies environments shows a significant excess compared to the background galaxy density. In particular the quasars, on average, have the tendency to be associated with small group of galaxies. No statistically significant difference is found between the over densities of low z quasar around the quasars and the inactive galaxies [@K14]. No dependence of the over density on redshift, quasar luminosity, the galaxies at same luminosity of the host galaxy, the radio luminosity or BH mass was found.
The lack of a notable difference between the quasars and non-active galaxies [@K14] environments suggests that the connection between the quasar activity and its environment is less important than believed for fueling and activate the quasar. This may also indicate that secular evolution may play an important role in triggering the quasar activity, and that mergers are less important than expected.
As a second step in this environment study we used all five SDSS filters to analyze the colour properties of the quasar environment (Karhunen et al. in preparation). This was done for a subsample of objects for which the SDSS-Stripe82 u-band image was deep enough to allow the detection of nearby galaxies. This implies that we have to discard $\sim$20% of our original sample (typically u-band images with exposures $<$40 min). We find that the colours of the environments of the quasars are slightly bluer than those of inactive galaxies at all distances, with the effect being most noticeable in g-i. The colour of the environment depends on the density of the environment, with denser areas being redder. We also find that quasars at higher redshifts have redder environments.
Close Environments
------------------
To better investigate, in all 5 Sloan bands, the very close environment of quasars in our sample, we selected both from the full sample of 416 quasar and from the comparison sample only objects at z $<$ 0.3 in fact, beyond this limit, the characterization of the quasar host galaxies becomes difficult at bluer filters due to the reduced contrast between the host galaxy and the nuclear emission. The detailed description of the final sample, composed by 52 quasars, is described in @B15. To compare the colours of the host galaxy we need to evaluate the K-correction for each filter, to this purpose we used the KCORRECT package [@BR]. This allow us, using the luminosity+SED fitting, to derive also an estimate of the stellar mass for this subsample of quasar hosts and we used the stellar masses to derive the relation $LogM_*$-Log$M_{BH}$ shown in fig \[fig:BH\]. Our data are well fitted by the relation for nearby normal elliptical and S0 galaxies as derived by @RV, confirming our findings that quasars are hosted in luminous early-type galaxies.
The two samples show similar properties in particular, for our sample of resolved objects, we find an average mass of the quasar host galaxy of $<M_*> = 4.28\pm2.76\times10^{10} M_{\odot}$ and $<M_*> = 5.27\pm3.88\times10^{10} M_{\odot}$ for the comparison sample of normal galaxies. The overall mean colours of the quasar host galaxy are indistinguishable from those of inactive galaxies of similar luminosity and redshift. There is a suggestion that the most massive quasar hosts have bluer colours and show a higher star formation rate than those in the sample of inactive galaxies.
Star formation
--------------
As a last step in our study we obtained with NOT+ALFOSC instrumentation long slit spectra of a subsample of the 52 quasar in @B15 and of the close companions . We found that in 8 out of 12 ($\sim$67%) quasar the closest companion galaxy is associated to the quasar (same redshift). However the average level of star formation of companion galaxies that are associated with the quasar appears similar to that of the companion galaxies that are not associated with the quasar. These results suggest a modest role of the quasar emission for the SF in nearby companion galaxies. Finally for 3 targets we observed also the spectrum of the host galaxy which turned out to be typical of an old stellar population.
Conclusions
===========
From our multi-band study of a large sample of nearby quasar we found that:
Quasar host galaxy luminosity is mainly in the range $M^*-3 < M_R < M^*$.\
Galaxy environments of quasar are similar to those of a sample of inactive galaxies at similar redshift and luminosity. The colours of the galaxies in the immediate environments of the quasars are slightly bluer than those of inactive galaxies at all distances; the effect being most noticeable in ($g-i$) colour.\
Overall the mean colours ($g-i$ = 0.82$\pm$0.26; $r-i$ = 0.26$\pm$0.16 and $u-g$ = 1.32$\pm$0.25) of the quasar host galaxy are indistinguishable from those of inactive galaxies of similar luminosity and redshift.\
In $\sim$50% of the quasar in the z$<0.3$ sample, we found companion galaxies at projected distance $<$50 kpc that could be associated to the quasar.\
Optical spectroscopy of a subsample of 12 quasar shows that these associated companions exhibit \[OII\] $\lambda$ 3727 Å emission lines suggestive of episodes of (recent) star formation possibly induced by past interactions. The rate of star formation is, however, similar to that of companions of inactive galaxies. This may indicate that the presence of the quasar do not change the star formation in the associated close companions. Contrary to past suggestions [e.g. @M14] the role of these associated companion galaxies for triggering and fueling the nuclear activity appears thus marginal as similar conditions are observed in inactive galaxies of mass similar to that of quasar hosts. Contrary to past suggestions \cite{} A significant time delay between the phase of nuclear activity and the star formation in the immediate environments could smear the link between them.
Conflict of Interest Statement {#conflict-of-interest-statement .unnumbered}
==============================
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Author Contributions {#author-contributions .unnumbered}
====================
DB and RF wrote the paper and analyzed the data JK helped in the selection of the sample and the discussion of the results and KK made the NOT observations. All authors participated in the discussion of the final results.
Acknowledgments {#acknowledgments .unnumbered}
===============
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences ( LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
Figure captions {#figure-captions .unnumbered}
===============
![Upper left panel:SDSS-DR7 i-band image, middle left panel: corresponding data from Stripe 82, bottom left panel: contour plot zoomed on the central object of the Stripe 82 i-band image (north at top, east left). Upper right panel: i-band radial profile together with the AIDA decomposition and fit. Middle right panel: distribution of residual for the best fit. Lower right panel: distribution of $\chi^2$.[]{data-label="fig:aida"}](fig_AIDA1.jpg){width="13cm"}
![Distribution of effective radius $R_e$ for quasar hosts (red line) and that of a sample of inactive galaxies (blue line) that have similar luminosity and redshift of quasar hosts.[]{data-label="fig:re"}](Fig_re_onlyres.jpg){width="9cm"}
![Galaxy over-density in the environment of quasar compared with that of inactive galaxies at same redshift and luminosity.[]{data-label="fig:env"}](env_QSO1.jpg){width="9cm"}
![Correlation between quasar virial black hole mass end the total stellar mass of their host galaxies as derived from SED fitting. [**Despite the large scatter the relation is similar**]{} to that obtained by @RV for inactive galaxies and bulges (red solid line). The 1$\sigma$ uncertainties are indicated by blue dashed lines.[]{data-label="fig:BH"}](Mbh_Mstar_fit.jpg){width="10cm"}
[^1]: $M^*$ and $L^*$ are the characteristic absolute magnitude and luminosity of the Schechter luminosity function of galaxies [@SH]
|
---
author:
- Moshe Adrian
title: Characters of some simple supercuspidal representations on split tori
---
Introduction
============
Let $F$ be a nonarchimedean local field of characteristic zero, $\mathfrak{o}$ its ring of integers, $\mathfrak{p}$ the maximal ideal, $p$ the residual characteristic, $q$ the order of the residue field, and $\varpi$ a fixed uniformizer of $F$. Let $G$ be a connected reductive group defined over $F$. If $\pi$ is an irreducible admissible representation of $G$, we denote by $\theta_{\pi}$ its distribution character, which is a linear functional on $C_c^{\infty}(G)$, the locally constant, compactly supported functions on $G$. Harish-Chandra showed that $\theta_{\pi}$ can be represented by a locally constant function on the regular semisimple set of $G$, which we will also denote $\theta_{\pi}$.
Suppose that $\pi$ is a supercuspidal representation. Much is known about $\theta_{\pi}$. The first supercuspidal characters were computed by Sally and Shalika in [@sallyshalika], where they investigated the supercuspidal representations of $SL(2,F)$ when $p \neq 2$. Shimizu calculated the supercuspidal characters of $GL(2,F)$ in [@shimizu], for $p \neq 2$. Kutzko began a study of the supercuspidal characters of $GL(\ell,F)$, $\ell$ a prime (see [@kutzko]), when $\ell \neq p$, and DeBacker computed these characters on elliptic tori (see [@debacker]). Later, Spice calculated the supercuspidal characters of $SL(\ell,F)$, $\ell$ an odd prime (see [@spice]), for $\ell \neq p$ and together with Adler, they computed a large class of supercuspidal characters for very general connected reductive groups (see [@adlerspice]).
Many times one wants to determine character values of a representation on a particular torus, as this can carry much of the information of the representation. For example, discrete series representations of real groups are determined by their character values on the compact (mod center) torus. On the $p$-adic side, it is known (see [@adrian]) that the supercuspidal representations of $GL(n,F)$, $n$ prime, are determined by their character values on a specific elliptic torus, for $p > 2n$.
In this paper, we compute the character values of the simple supercuspidal representations (recently discovered by Gross and Reeder) of $SL(2,F)$ and $SL(3,F)$ on the maximal split torus of each group when $p$ is arbitrary. The character values for $SL(2,F)$ are especially elegant, being a fixed constant times a sum of $q^{\ell(w)}$ over appropriate affine Weyl group elements $w$, where $\ell(w)$ is the length of $w$. We had hoped that the character values of simple supercuspidals for $SL(3,F)$ and more general reductive groups would be as elegant, but this is unfortunately not the case. As one might expect, various Gauss-type sums appear for $SL(3,F)$.
Let $T$ denote the maximal split torus of $SL(2)$, $Z$ the center, and $W^a$ the affine Weyl group. Let $val$ denote valuation. Using the Frobenius character formula for supercuspidal representations (see [@sally]), we will first prove the following theorem.
\[maintheorem\] Let $g = {\left( \begin{array}{cc} {a} & {0} \\ {0} & {a^{-1}}
\end{array} \right)} \in SL(2,F)$ where $a \in 1 + \mathfrak{p}$, and set $r := val(a - a^{-1})$. Let $\pi$ be a simple supercuspidal representation of $SL(2,F)$. Then $$\theta_{\pi}(g) = c_q \left( \displaystyle\sum_{w \in W^{a} : \ell(w) < r} q^{\ell(w)} \right)$$ where $$c_q := \left\{
\begin{array}{rl}
\frac{q-1}{2} & \text{if } p \neq 2 \\
q-1 & \text{if } p = 2
\end{array} \right.$$
It will be clear from our calculations in later sections that $\theta_{\pi}$ vanishes on $T(F) \setminus Z(F) T(1 + \mathfrak{p})$, so up to the central character, we have computed $\theta_{\pi}$ on all of the split torus. Moreover, since the central character is given by the data forming the simple supercuspidal representation, we have therefore computed all of $\theta_{\pi}$ on the split torus.
We will also compute $\theta_{\pi}(g)$ for simple supercuspidal representations $\pi$ of $SL(3,F)$, where $g$ is in the maximal split torus of $SL(3,F)$. Specifically, we will compute the character values on $T(1 + \mathfrak{p})$, where here $T$ is the maximal split torus of $SL(3)$. Then by the same reasoning as for $SL(2,F)$, this will be enough to give the character values on the entire maximal split torus of $SL(3,F)$. However, since the theorem for $SL(3,F)$ is much more complicated, we will defer its statement to a later section. Moreover, as there are nontrivial Gauss sums in this formula, we will compute them in complete generality in the last section of this paper.
We note that the term $a - a^{-1}$ in Theorem \[maintheorem\] is, up to a sign, a canonical square root of the Weyl denominator. In particular, if $D(g)$ is the standard Weyl denominator, then $-D(g) = (a-a^{-1})^2$.
We wish to make another note. In [@sallyshalika], Sally/Shalika have character values on the split torus of $SL(2,F)$ for an arbitrary supercuspidal representation of $SL(2,F)$ when $p \neq 2$, which we briefly recall. For any quadratic extension $V = F(\sqrt{\theta})$ of $F$, let $C_{\theta}$ denote the kernel of the norm $N_{V/F}$, and $\mathfrak{p}_{\theta}$ the prime ideal in $V$. If $V$ is ramified, set $C_{\theta}^{(h)} = (1 + \mathfrak{p}_{\theta}^{2h+1}) \cap C_{\theta}, h \geq 0$. If $\psi \in \hat{C}_{\theta}$, denote the conductor of $\psi$ by cond $\psi$ (this is the largest subgroup in the filtration $\{ C_{\theta}^{(h)} \}$ on which $\psi$ is trivial). The ramified discrete series are indexed by a nontrivial additive character $\eta$ of $F$ and a nontrivial character $\psi \in \hat{C}_{\theta}$, where $V$ is ramified. The corresponding representation is denoted $\Pi(\eta, \psi, V)$. If $g = {\left( \begin{array}{cc} {a} & {0} \\ {0} & {a^{-1}}
\end{array} \right)} \in SL(2,F)$, $a \in 1 + \mathfrak{p}$, and $\Pi(\eta, \psi, V)$ is a ramified discrete series, then $$\theta_{\Pi(\eta, \psi, V)}(g) = \frac{1}{|a-a^{-1}|} - \frac{1}{2} q^h \left( \frac{q+1}{q} \right)$$ where $V = F(\sqrt{\theta})$ and cond $\psi = C_{\theta}^{(h)}, h \geq 1$.
Since these character values are in particular valid for the simple supercuspidal representations when $p \neq 2$ (since simple supercuspidal representations are, in particular, ramified discrete series), we may compare their character values to ours. After some calculation, one can see that their character values agree with ours in the case of simple supercuspidal representations on the split torus.
We now briefly present an outline of the paper. In section \[background\], we define simple supercuspidal representation and we present the relevant background theory that we use to compute the character values. In section \[sl2section\], we compute the character formula for $SL(2,F)$. In section \[sl3section\], we compute the character formula for $SL(3,F)$. In particular, the formula contains various Gauss-type sums. In section \[morecalculations\], we compute these Gauss sums.
Acknowledgements: This paper has benefited from conversations with Gordan Savin, Aaron Wood, Chris Kocs, and Loren Spice.
Background
==========
Let us recall some basic definitions. Let $G$ be a split, simply connected, almost simple, connected reductive group, and $T$ a maximal $F$-split torus in $G$. Associated to $T$ we have the set of roots $\Phi$ of $T$ in $G$, an apartment, together with a set of affine roots $\Psi$, and an affine Weyl group $W^a$. We also have a canonical length function $\ell(w)$ on $W^a$. Fix a Chevalley basis in the Lie algebra of $G$. To each $\psi \in \Psi$ we have an associated affine root group $U_{\psi}$. Fix an alcove $C$ in the apartment with corresponding simple and positive affine roots $\Pi \subset \Psi^+$. Let $T(\mathfrak{o})$ be the maximal compact subgroup of $T(F)$. Let $T(1+\mathfrak{p}) := < t \in T(\mathfrak{o}) : \lambda(t) \in 1 + \mathfrak{p} \ \forall \lambda \in X^*(T)>$, where $X^*(T)$ is the character lattice of $T$. Let $I = <T(\mathfrak{o}), U_{\psi} : \psi \in \Psi^+>$ denote the corresponding Iwahori subgroup, and $I_+ = <T(1 + \mathfrak{p}), U_{\psi} : \psi \in \Psi^+>$ its pro-unipotent radical. Set $I_{++} := <T(1 + \mathfrak{p}), U_{\psi} : \psi \in \Psi^+ \setminus \Pi>$. We set $H := Z(F) I_+$, where $Z$ is the center of $G$. Let $N$ denote the normalizer of $T(F)$ in $G(F)$.
(see [@grossreeder]) The subgroup $I_{++}$ is normal in $I_+$, with quotient $$I_+ / I_{++} \cong \displaystyle\bigoplus_{\psi \in \Pi} U_{\psi} / U_{\psi+1}$$ as $T(\mathfrak{o})$-modules.
(see [@grossreeder]) A character $\chi : H \rightarrow \mathbb{C}^*$ is called *affine generic* if
\(i) $\chi$ is trivial on $I_{++}$ and
\(ii) $\chi$ is nontrivial on $U_{\psi}$ for every $\psi \in \Pi$.
(see [@grossreeder]) Let $\chi : H \rightarrow \mathbb{C}^*$ be an affine generic character. Then $cInd_{H}^{G(F)} \chi$ is an irreducible supercuspidal representation, called a *simple supercuspidal representation*, where $cInd$ denotes compact induction.
Now suppose that $\pi$ is an irreducible smooth supercuspidal representation of $G(F)$. Let $K$ be an open, compact subgroup of $G(F)$, and suppose that $\sigma$ is an irreducible representation of $K$ such that $$\pi = cInd_K^{G(F)} \sigma.$$ Let $\chi_{\sigma}$ denote the distribution character of $\sigma$. The following is the Frobenius formula for the induced character $\theta_{\pi}$.
\[sally\] (see [@sally]) Let $g$ be a regular element of $G(F)$. Then $$\theta_{\pi}(g) = \displaystyle\sum_{x \in K \backslash G(F) / K} \displaystyle\sum_{y \in K \backslash K x K} \dot{\chi}_{\sigma}(ygy^{-1})$$ where
$$\dot{\chi}_{\sigma}(k) = \left\{
\begin{array}{rl}
\chi_{\sigma}(k) & \text{if } k \in K \\
0 & \text{if } k \in G(F) \setminus K
\end{array} \right.$$
There is an integral version of this formula as well (see [@sally]), and these formulas are a main tool in computing characters of supercuspidal representations. We will use this theorem to compute the character values of the simple supercuspidal representations of $SL(2,F)$ and $SL(3,F)$ on their maximal split tori.
We first show that the formula in Theorem \[sally\] simplifies considerably in our situation. Let us first recall the following basic theory about double coset decompositions. If $G$ is a connected reductive group and $K$ is a compact open subgroup of $G$, let us choose a set of representatives $\{t_{\alpha} \}$ for the double cosets of $K \backslash G / K$. Then $K t_{\alpha} K$ is the disjoint union of the cosets $K t_{\alpha} s_1, K t_{\alpha} s_2, ..., K t_{\alpha} s_m$, where $s_1, s_2, ..., s_m$ is a set of representatives of $K / (K \cap t_{\alpha}^{-1} K t_{\alpha})$. We will use this fact repeatedly in this paper. We can now state our reduction formula.
\[reduction\] Let $G$ be simply connected, and $\chi$ an affine generic character of $H$. Set $\pi := cInd_{H}^{G(F)} \chi$. Then $$\theta_{\pi}(g) = |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})| \displaystyle\sum_{x \in W^a} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1}),$$ where the outer sum is meant to be taken over any set of representatives $x$ in $W^a$.
Recall the affine Bruhat decomposition $I \backslash G / I \leftrightarrow W^a \cong N / T(\mathfrak{o})$. As $I / I_+ \cong T(\mathfrak{o}) / T(1 + \mathfrak{p})$, the affine Bruhat decomposition descends to $H \backslash G / H \leftrightarrow N / Z(F) T(1 + \mathfrak{p})$ (we are using here that $G$ is simply connected, so that $Z(F) = Z(\mathfrak{o})$ and therefore $Z(\mathfrak{o}) I_+ = H \subset I$ and $I / H = T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})$), and we have the short exact sequence $$1 \rightarrow T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p}) \rightarrow N / Z(F) T(1 + \mathfrak{p}) \rightarrow W^a = N / T(\mathfrak{o}) \rightarrow 1$$ Therefore, $$\theta_{\pi}(g) = \displaystyle\sum_{x \in N / Z(F) T(1 + \mathfrak{p})} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1})$$ Write $$\sigma(x) := \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1})$$ for $x \in N$. We claim that $\sigma(x) = \sigma(xt) \ \forall t \in T(\mathfrak{o})$. Write $HxH = \displaystyle\cup_{i=1}^m Hx x_i$. Recall that $x_i$ are representatives of $H / (H \cap x^{-1} Hx)$. As $x \in N$, $x^{-1} H x = Z(F) I'_+$, where $I'_+$ is the pro-unipotent radical of another Iwahori subgroup $I'$. It is then easy to see that $H / (H \cap x^{-1} Hx)$ is a direct sum of spaces of the form $U_{\gamma} / U_{\gamma + n}$, where $\gamma \in \Phi^+$ or $\gamma = \gamma' + 1$ where $\gamma' \in \Phi^-$, and where $n$ is a non-negative integer. Now, $$\sigma(xt) = \displaystyle\sum_{y \in H \backslash H xt H} \dot{\chi}_{\sigma}(ygy^{-1}).$$ Write $HxtH = \displaystyle\cup_{j} Hxt y_j$. $y_j$ are representatives of $H / (H \cap (xt)^{-1} H xt)$. Since $t \in T(\mathfrak{o})$, we have that $t^{-1} x^{-1} Hxt = x^{-1} Hx$. Therefore, in particular, $HxtH = \displaystyle\cup_{i=1}^m Hxt x_i$. Then $$\sigma(xt) = \displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xtx_i gx_i^{-1} t^{-1} x^{-1}) =$$ $$\displaystyle\sum_{i = 1}^m \dot{\chi}_{\sigma}(xtx_i t^{-1} t g t^{-1} tx_i^{-1} t^{-1} x^{-1}) = \displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xtx_i t^{-1} g tx_i^{-1} t^{-1} x^{-1}).$$ Since this sum is over spaces of the form $U_{\gamma} / U_{\gamma+n}$ as noted above, and since conjugation by an element $t \in T(\mathfrak{o})$ preserves $U_{\gamma} / U_{\gamma+n}$, we get $$\displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xtx_i t^{-1} g tx_i^{-1} t^{-1} x^{-1}) = \displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xx_i g x_i^{-1} x^{-1}) = \sigma(x).$$
Therefore, since $\sigma$ is constant along fibers of the above exact sequence, we have that $$\theta_{\pi}(g) = |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})| \displaystyle\sum_{x \in W^a} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1})$$
Therefore, we only need to compute $\sigma(x)$ as $x$ varies over a set of representatives of $W^a$. Moreover, since we are only computing $\theta_{\pi}$ on $T(1+\mathfrak{p})$, we only need the data of the affine generic character $\chi$ on $I_+$ (and not on all of $ZI_+$), since if $g \in T(1+\mathfrak{p})$, then the terms $ygy^{-1}$ in $\theta_{\pi}(g)$ will always live in $I_+$.
The character formula for $SL(2,F)$ {#sl2section}
===================================
In this section we prove Theorem \[maintheorem\]. We prove this theorem with a case by case investigation. We compute the inner sum $\sigma(x)$ in Proposition \[reduction\] for any set of representatives of $W^a$ by decomposing $HxH$ into a union of left cosets, as in the paragraph that immediately precedes proposition \[reduction\]. Afterwards, we sum everything up to get $\theta_{\pi}$.
We fix a Haar measure on $F$ such that $\mathfrak{o}$ has volume $1$, and we use the abbreviation $vol$ to denote volume. Fix an element $g = {\left( \begin{array}{cc} {a} & {0} \\ {0} & {a^{-1}}
\end{array} \right)} \in T(1+\mathfrak{p})$, and set $r := val(a - a^{-1})$
\[innersum1\] Let $x = {\left( \begin{array}{cc} {b} & {0} \\ {0} & {b^{-1}}
\end{array} \right)}$. Then
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{
\begin{array}{rl}
vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{-2n+r}) & \text{if } val(b) = n \geq 0 \ \mathrm{and} \ 2n < r \\
vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{2n+r}) & \text{if } val(b) = n < 0 \ \mathrm{and} \ -2n < r \\
0 & \text{otherwise}
\end{array} \right.$$
We first rewrite the double coset $H x H$ as a finite union of single right cosets. Suppose $val(b) = n > 0.$ Since $$H \cap x^{-1} H x = Z(F) {\left( \begin{array}{cc} {1 + \mathfrak{p}} & {\mathfrak{o}} \\ {\mathfrak{p}^{2n+1}} & {1 + \mathfrak{p}}
\end{array} \right)},$$ we can obtain an explicit disjoint union $$H = \displaystyle\bigcup_{z \in \mathfrak{p} / \mathfrak{p}^{2n+1}} Z(F) {\left( \begin{array}{cc} {1 + \mathfrak{p}} & {\mathfrak{o}} \\ {\mathfrak{p}^{2n+1}} & {1 + \mathfrak{p}}
\end{array} \right)} {\left( \begin{array}{cc} {1} & {0} \\ {z} & {1}
\end{array} \right)},$$ where $${\left( \begin{array}{cc} {1 + \mathfrak{p}} & {\mathfrak{o}} \\ {\mathfrak{p}^{2n+1}} & {1 + \mathfrak{p}}
\end{array} \right)} := \left\{ {\left( \begin{array}{cc} {x_1} & {x_2} \\ {x_3} & {x_4}
\end{array} \right)} \in SL(2,F): x_1,x_4 \in 1 + \mathfrak{p}, x_2 \in \mathfrak{o}, x_3 \in \mathfrak{p}^{2n+1} \right\}$$ (we will use this last type of notation throughout the paper). Therefore, we have a disjoint union $$H x H = \displaystyle\bigcup_{z \in \mathfrak{p} / \mathfrak{p}^{2n+1}} H x {\left( \begin{array}{cc} {1} & {0} \\ {z} & {1}
\end{array} \right)}$$
Now suppose that $val(b) = n < 0.$ Then similarly, we get $$H \cap x^{-1} H x = Z {\left( \begin{array}{cc} {1 + \mathfrak{p}} & {\mathfrak{p}^{-2n}} \\ {\mathfrak{p}} & {1 + \mathfrak{p}}
\end{array} \right)},$$ $$H = \displaystyle\bigcup_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} Z {\left( \begin{array}{cc} {1 + \mathfrak{p}} & {\mathfrak{p}^{-2n}} \\ {\mathfrak{p}} & {1 + \mathfrak{p}}
\end{array} \right)} {\left( \begin{array}{cc} {1} & {z} \\ {0} & {1}
\end{array} \right)}.$$ Therefore, we have a disjoint union $$H x H = \displaystyle\bigcup_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} H x {\left( \begin{array}{cc} {1} & {z} \\ {0} & {1}
\end{array} \right)}$$
Finally, if $val(b) = 0$, then we get $H \cap x H x^{-1} = H$. Therefore, $H x H = H x$.
Let us return to the case $val(b) = n > 0$. Suppose $y \in H x H$. We need to check when $y g y^{-1} \in H$ since $\dot{\chi}$ vanishes outside $H$. Using our above double coset decomposition, write $y = i x \tilde{z}$, where $\tilde{z}$ is of the form ${\left( \begin{array}{cc} {1} & {0} \\ {z} & {1}
\end{array} \right)}$ for some $z \in \mathfrak{p} / \mathfrak{p}^{2n+1}$ and for some $i \in H$. Then $y g y^{-1} \in H \Leftrightarrow x \tilde{z} g \tilde{z}^{-1} x^{-1} \in H$. Moreover, $$x \tilde{z} g \tilde{z}^{-1} x^{-1} = {\left( \begin{array}{cc} {a} & {0} \\ {b^{-2} z(a-a^{-1})} & {a^{-1}}
\end{array} \right)}.$$ Notice that $a \in \pm (1 + \mathfrak{p})$ is forced upon us here in order to have $x \tilde{z} g \tilde{z}^{-1} x^{-1} \in H$. (We note that the condition $a \in \pm (1 + \mathfrak{p})$ continues to be forced upon us, for the same reason, when you compute the terms $ygy^{-1}$ that appear in $\theta_{\pi}(g)$ for any other representative $x$ of any element of the affine Weyl group,as simple computations will show. This shows, therefore, that $\theta_{\pi}$ vanishes on $T(F) \setminus Z(F) T(1 + \mathfrak{p})$).
Write $a - a^{-1} = \varpi^r u$ for some unit $u$. Absorbing all units into the $z$ term, we may write $b^{-2} z(a - a^{-1}) = \varpi^{-2n} \varpi^r z'$ for some $z' \in \mathfrak{p} / \mathfrak{p}^{2n+1}$.
Recall that we are only interested in $\chi|_{I_+}$. We will abuse notation and write $\chi$ for $\chi|_{I_+}$. Now, write $\chi$ on $I_+$ as $$\chi : I_+ \rightarrow \mathbb{C}^*$$ $${\left( \begin{array}{cc} {d_{11}} & {d_{12}} \\ {d_{21}} & {d_{22}}
\end{array} \right)} \mapsto \chi_1(d_{12}) \chi_2(d_{21})$$ where $\chi_1$ is a level $1$ character of $\mathfrak{o}$ and where $\chi_2(d_{21}) = \chi_2'(\frac{1}{\varpi} d_{21})$, where $\chi_2'$ is a level $1$ character of $\mathfrak{o}$ (a character of $\mathfrak{o}$ is said to be level $1$ if it is trivial on $\mathfrak{p}$, but nontrivial on $\mathfrak{o}$). Set $\dot{\chi}_1(z) := \chi_1(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi}_1(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$. Moreover, set $\dot{\chi}_2(z) := \chi_2'(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi}_2(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$.
Therefore, $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{z' \in \mathfrak{p} / \mathfrak{p}^{2n+1}} \dot{\chi}_2(\varpi^{-2n+r-1} z')$$ Making a change of variables, we get $$\displaystyle\sum_{z' \in \mathfrak{p} / \mathfrak{p}^{2n+1}} \dot{\chi}_2(\varpi^{-2n+r-1} z') = \displaystyle\sum_{z'' \in \mathfrak{p}^{-2n+r} / \mathfrak{p}^{r}} \dot{\chi}_2(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{-2n+r} \cap \mathfrak{o}} \dot{\chi}_2(z'') d(z'')$$ since $\dot{\chi}$ vanishes outside $H$. If $\mathfrak{p}^{-2n+r} \supseteq \mathfrak{o}$, then this integral vanishes since the integral of a nontrivial character over a group vanishes. However, if $\mathfrak{p}^{-2n+r} \varsubsetneq \mathfrak{o}$, which is precisely the condition that $2n < r$, then $$vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{-2n+r} \cap \mathfrak{o}} \dot{\chi}_2(z'') d(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{-2n+r}} d(z'') = vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{-2n+r})$$ since $\dot{\chi}_2$ is trivial on $\mathfrak{p}$.
Now consider the case $val(b) = n < 0$. Suppose $y \in H x H$. By our above double coset decomposition, write $y = i x \tilde{z}$, where $\tilde{z}$ is of the form ${\left( \begin{array}{cc} {1} & {z} \\ {0} & {1}
\end{array} \right)}$ for some $z \in \mathfrak{o} / \mathfrak{p}^{-2n}$ and for some $i \in H$. Moreover, $x \tilde{z} g \tilde{z}^{-1} x^{-1} = {\left( \begin{array}{cc} {a} & {b^2 z(a^{-1} - a)} \\ {0} & {a^{-1}}
\end{array} \right)}.$ Therefore, $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} \dot{\chi}_1( b^2 z (a^{-1} - a))$$ We rewrite $b^{2} z(a^{-1} - a) = \varpi^{2n} p^r z'$, where $z' \in \mathfrak{o} / \mathfrak{p}^{-2n}$. Again, after a change of variables, we get $$\displaystyle\sum_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} \dot{\chi}_1(b^2 z (a^{-1} - a)) = \displaystyle\sum_{z' \in \mathfrak{o} / \mathfrak{p}^{-2n}} \dot{\chi}_1(\varpi^{2n} p^r z' ) =$$ $$\displaystyle\sum_{z'' \in \mathfrak{p}^{2n+r} / \mathfrak{p}^{r}} \dot{\chi}_1(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{2n+r} \cap \mathfrak{o}} \dot{\chi}_1(z'') d(z'')$$ If $\mathfrak{p}^{2n+r} \supseteq \mathfrak{o}$, then again this integral vanishes. However, if $\mathfrak{p}^{2n+r} \varsubsetneq \mathfrak{o}$, which is precisely the condition that $-2n < r$ then $$vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{2n+r} \cap \mathfrak{o}} \dot{\chi}_1(z'') d(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{2n+r}} d(z'') = vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{2n+r})$$ since $\dot{\chi}_1$ is trivial on $\mathfrak{p}$.
Now consider the case $val(b) = 0$. Suppose $y \in H x H$. Recall that in this case, $H x H = H x$. Moreover, $x g x^{-1} = g.$ Therefore, $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \dot{\chi}(g) = 1$$
Let $x = {\left( \begin{array}{cc} {0} & {c} \\ {-c^{-1}} & {0}
\end{array} \right)}$. Then
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{
\begin{array}{rl}
vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{-2n+r-1}) & \text{if } val(c) = n \geq 0 \ \mathrm{and} \ 2n +1 < r \\
vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{2n+r+1}) & \text{if } val(c) = n < 0 \ \mathrm{and} \ -2n -1 < r \\
0 & \text{otherwise}
\end{array} \right.$$
The proof is completely analogous to that of Proposition \[innersum1\].
Now note that
$$|T(\mathfrak{o}) / Z(F)T(1+\mathfrak{p})| = |\mathfrak{o}^* / (\pm (1 + \mathfrak{p}))| = \left\{
\begin{array}{rl}
\frac{q-1}{2} & \text{if } p \neq 2 \\
q-1 & \text{if } p = 2
\end{array} \right.$$
(note that if $p = 2$, $1 + \mathfrak{p} = -1 + \mathfrak{p}$). Thus, our character formula is
$$\theta_{\pi_{\chi}}(g) = c_q vol(\mathfrak{p}^r)^{-1} \bigg[ \displaystyle\sum_{n \in \mathbb{N}, 0 \leq 2n < r} vol(\mathfrak{p}^{-2n+r}) + \displaystyle\sum_{n \in \mathbb{N}, 0 < -2n < r} vol(\mathfrak{p}^{2n+r})$$ $$+ \displaystyle\sum_{n \in \mathbb{N}, 0 \leq 2n+1 < r} \left( vol(\mathfrak{p}^{-2n+r-1}) \right) + \displaystyle\sum_{n \in \mathbb{N}, 0 < -2n-1 < r} \left( vol(\mathfrak{p}^{2n+r+1}) \right) \bigg]$$
It is a straightforward calculation to show that if $x = {\left( \begin{array}{cc} {b} & {0} \\ {0} & {b^{-1}}
\end{array} \right)}$ and $val(b) = k$, then $\ell(x) = |2k|$. Moreover, if $x = {\left( \begin{array}{cc} {0} & {c} \\ {-c^{-1}} & {0}
\end{array} \right)}$ and $val(c) = k$, then $\ell(x) = |2k+1|$.
Making the relevant substitutions, and noting that $vol(\mathfrak{p}^d) = q^{-d}$ for $d > 0$ by our choice of measure, one can see that we have proven Theorem \[maintheorem\].
The character formula for $SL(3,F)$ {#sl3section}
===================================
In this section we compute the character formula for $SL(3,F)$ on the split maximal torus. As in the case of $SL(2,F)$, we compute the formula via a case by case investigation. We compute the inner sum $\sigma(x)$ in Proposition \[reduction\] for any set of representatives of $W^a$. Afterwards, we sum everything up to get $\theta_{\pi}$.
Let $g = {\left( \begin{array}{ccc} {\alpha} & {0} & {0} \\ {0} & {\beta} & {0} \\ {0} & {0} & {\gamma}
\end{array} \right)} \in T(1+\mathfrak{p})$. Suppose $\alpha - \beta = \varpi^r u$, $\beta - \gamma = \varpi^s u'$, $\alpha - \gamma = \varpi^t u''$ for some units $u, u', u''$. Again, we fix a Haar measure on $F$ such that $\mathfrak{o}$ has volume $1$.
Before we state the main theorem, we need to make a few simplifications. First, notice that $val(\alpha - \gamma) = val( (\alpha - \beta) + (\beta - \gamma) ) \geq inf \{val(\alpha - \beta), val(\beta - \gamma) \}$. Similarly, $val(\alpha - \beta) \geq inf \{val(\alpha - \gamma), val(\beta - \gamma) \}$ and $val(\beta - \gamma) \geq inf \{val(\alpha - \gamma), val(\alpha - \beta) \}$. One can conclude therefore that either $t \geq r = s$, $s \geq r = t$, or $r \geq s = t$. Since everything in sight is symmetric, we will assume without loss of generality that $t \geq r = s$, and we will state the main theorem in this case. One can easily state the analogous results in the two other cases.
So assume $t \geq r = s$. A complicated impediment is the character values are different in the cases $t = r = s$, $t = r+ 1 = s + 1$, and $t > r + 1 = s + 1$. We therefore have to state the character formula separately for these three cases.
We need some notation first. Let $\mathcal{A}_{n_i}$ denote the set of affine Weyl group elements who representatives in $N_G(T)$ are of the form $${\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}, {\left( \begin{array}{ccc} {0} & {a} & {0} \\ {0} & {0} & {b} \\ {c} & {0} & {0}
\end{array} \right)}, {\left( \begin{array}{ccc} {0} & {0} & {a} \\ {b} & {0} & {0} \\ {0} & {c} & {0}
\end{array} \right)},$$ $${\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {0} & {b} \\ {0} & {c} & {0}
\end{array} \right)}, {\left( \begin{array}{ccc} {0} & {0} & {a} \\ {0} & {b} & {0} \\ {c} & {0} & {0}
\end{array} \right)}, \ \mathrm{and} \ {\left( \begin{array}{ccc} {0} & {a} & {0} \\ {b} & {0} & {0} \\ {0} & {0} & {c}
\end{array} \right)},$$ respectively. For any of the above types of matrices, let $n_{12} = val(a/b), n_{21} = val(b/a), n_{23} = val(b/c), n_{32} = val(c/b), n_{13} = val(a/c), n_{31} = val(c/a)$. We now define a long list of notation that we need in order to state the main theorem of this section.
Let $\mathcal{B}_1, \mathcal{B}_2, \mathcal{B}_3, \mathcal{B}_4, \mathcal{B}_5, \mathcal{B}_6$ be the inequality conditions $\{n_{21} < r, -n_{31} < t, n_{32} < s \}, \{n_{13} < r-1, n_{21} < s, -n_{23} < t + 1 \}, \{-n_{12} < t + 1, n_{13} < s - 1, n_{32} < r \}, \{n_{21} < t, -n_{31} < r, -n_{23} < s + 1 \}, \{n_{13} < t - 1, -n_{12} < s + 1, -n_{23} < r + 1 \},$ and $\{-n_{12} < r + 1, n_{32} < t, -n_{31} < s \}$, respectively. For example, if $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ and $val(b/a) < r , -val(c/a) < t, val(c/b) < s$, then we say that “$x$ satisfies $\mathcal{B}_1$”.
Let $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3, \mathcal{C}_4, \mathcal{C}_5, \mathcal{C}_6$ be the inequality conditions $\{n_{31} \leq t, -n_{32} \leq s, -n_{21} \leq r \}, \{n_{23} \leq t - 1, -n_{13} \leq r + 1, -n_{21} \leq s \}, \{-n_{32} \leq r, -n_{13} \leq s + 1, n_{12} \leq t - 1 \}, \{n_{23} \leq s - 1, -n_{21} \leq t, n_{31} \leq r \}, \{n_{23} \leq r - 1, n_{12} \leq s - 1, -n_{13} \leq t + 1 \},$ and $\{n_{31} \leq s, n_{12} \leq r -1, -n_{32} \leq t \}$, respectively.
Let $\mathcal{D}_1, \mathcal{D}_2, \mathcal{D}_3, \mathcal{D}_4, \mathcal{D}_5, \mathcal{D}_6$ be the exact same inequality conditions as $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3, \mathcal{C}_4, \mathcal{C}_5, \mathcal{C}_6$, respectively, except that we replace every $\leq$ sign with a $>$ sign, and moreover, within each $\mathcal{C}_i$, replace every comma by the word “or”. For example, if $x = {\left( \begin{array}{ccc} {0} & {a} & {0} \\ {0} & {0} & {b} \\ {c} & {0} & {0}
\end{array} \right)}$ satisfies at least one of the inequalities $val(b/c) > t - 1, -val(a/c) > r + 1$, or $-val(b/a) > s$, then we say that “$x$ satisfies $\mathcal{D}_2$”.
Let $\mathcal{E}_1^0, \mathcal{E}_2^0, \mathcal{E}_3^0, \mathcal{E}_4^0, \mathcal{E}_5^0, \mathcal{E}_6^0$ be the inequality conditions $\{n_{21} \geq 0, n_{31} \geq 0, n_{32} \geq 0 \} \cup \{n_{21} \geq 0, n_{31} < 0, n_{32} < 0 \} \cup \{n_{21} < 0, n_{31} < 0, n_{32} \geq 0 \}, \{n_{13} \geq 0, n_{23} \geq 0, n_{21} \geq 0 \} \cup \{n_{13} < 0, n_{23} < 0, n_{21} \geq 0 \} \cup \{n_{13} \geq 0, n_{23} < 0, n_{21} < 0 \}, \{n_{32} < 0, n_{12} < 0, n_{13} \geq 0 \} \cup \{n_{32} \geq 0, n_{12} < 0, n_{13} < 0 \} \cup \{n_{32} \geq 0, n_{12} \geq 0, n_{13} \geq 0 \}, \{n_{31} < 0, n_{21} \geq 0, n_{23} \geq 0 \} \cup \{n_{31} < 0, n_{21} < 0, n_{23} < 0 \} \cup \{n_{31} \geq 0, n_{21} \geq 0, n_{23} < 0 \}, \{n_{23} \geq 0, n_{13} \geq 0, n_{12} < 0 \} \cup \{n_{23} < 0, n_{13} \geq 0, n_{12} \geq 0 \} \cup \{n_{23} < 0, n_{13} < 0, n_{12} < 0 \},$ and $\{n_{12} < 0, n_{32} \geq 0, n_{31} \geq 0 \} \cup \{n_{12} \geq 0, n_{32} \geq 0, n_{31} < 0 \} \cup \{n_{12} < 0, n_{32} < 0, n_{31} < 0 \}$, respectively. For example, if $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ satisfies $val(b/a) \geq 0, val(c/a) \geq 0$, and $val(c/b) \geq 0$, then we say that “$x$ satisfies $\mathcal{E}_1^0$. If $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ satisfies $val(b/a) \geq 0, val(c/a) < 0$, and $val(c/b) < 0$, then we say that “$x$ satisfies $\mathcal{E}_1^0$. If $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ satisfies $val(b/a) < 0, val(c/a) < 0$, and $val(c/b) \geq 0$, then we say that “$x$ satisfies $\mathcal{E}_1^0$.
Let $\mathcal{E}_1^2, \mathcal{E}_2^2, \mathcal{E}_3^2, \mathcal{E}_4^2, \mathcal{E}_5^2, \mathcal{E}_6^2$ be the inequality conditions $\mathcal{E}_1^0 \setminus \{n_{21} \geq 0, n_{31} \geq 0, n_{32} \geq 0 \}$, $\mathcal{E}_2^0 \setminus \{n_{13} \geq 0, n_{23} \geq 0, n_{21} \geq 0 \}$, $\mathcal{E}_3^0 \setminus \{n_{32} \geq 0, n_{12} \geq 0, n_{13} \geq 0 \}$, $\mathcal{E}_4^0 \setminus \{n_{31} < 0, n_{21} < 0, n_{23} < 0 \}$, $\mathcal{E}_5^0 \setminus \{n_{23} < 0, n_{13} < 0, n_{12} < 0 \}$, and $\mathcal{E}_6^0 \setminus \{n_{12} < 0, n_{32} < 0, n_{31} < 0 \}$, respectively. For example, if $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ satisfies $val(b/a) \geq 0, val(c/a) \geq 0$, and $val(c/b) \geq 0$, then $x$ does not satisfy $\mathcal{E}_1^2$. If $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ satisfies $val(b/a) \geq 0, val(c/a) < 0$, and $val(c/b) < 0$, then “$x$ satisfies $\mathcal{E}_1^2$. If $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ satisfies $val(b/a) < 0, val(c/a) < 0$, and $val(c/b) \geq 0$, then “$x$ satisfies $\mathcal{E}_1^2$.
Set $\mathcal{E}_1^1 = \mathcal{E}_1^2, \mathcal{E}_2^1 = \mathcal{E}_2^2, \mathcal{E}_3^1 = \mathcal{E}_3^2, \mathcal{E}_4^1 = \mathcal{E}_4^0, \mathcal{E}_5^1 = \mathcal{E}_5^0, \mathcal{E}_6^1 = \mathcal{E}_6^0$. For example, if $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$ satisfies $val(b/a) \geq 0, val(c/a) \geq 0$, and $val(c/b) \geq 0$, then $x$ does not satisfy $\mathcal{E}_1^1$, but it does satisfy $\mathcal{E}_1^0$.
Finally, let $\mathcal{F}_i^j$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ satisfies $\mathcal{B}_i, \mathcal{C}_i,$ and $\mathcal{E}_i^j$. Moreover, let $\mathcal{G}_i^j$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ satisfies $\mathcal{B}_i, \mathcal{D}_i,$ and $\mathcal{E}_i^j$. For example, if $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {0} & {b} \\ {0} & {c} & {0}
\end{array} \right)}$, and if $x$ satisfies $val(b/a) < t, -val(c/a) < r, -val(b/c) < s + 1$, and if $x$ satisfies $val(b/c) > s - 1$, and if $x$ satisfies $val(c/a) < 0, val(b/a) \geq 0, val(b/c) \geq 0$, then $x$ satisfies $\mathcal{G}_4^j$, for $j = 0,1,$ and $2$.
We need a few more notations. As in the case of $SL(2,F)$, the character values will be sums of powers of $q$. Some of these powers will be lengths of certain affine Weyl group elements, as before, but some will not. Some powers will be “truncated lengths” of affine Weyl group elements. We will not define “truncated length”. However, we will do an example in the next section, and the computation is analogous in every other case. If an affine Weyl group element $x$ is of type $\mathcal{G}_i^0, \mathcal{G}_i^1$, or $\mathcal{G}_i^2$, then $x$ will contribute the term $q^{\ell'
(x)}$ to $\theta_{\pi}(g)$, where $\ell'(x)$ denotes the “truncated length” of $x$. There will also be two types of Gauss sums that appear in the character formula. If an affine Weyl group element $x$ is of a certain type to be discussed later, then a Gauss sum corresponding to this element will contribute to the character, and we will denote this Gauss sum by either $\Gamma(x)$ or $\Xi(x)$. We will do examples of how these Gauss sums arise in the next section.
Let $\mathcal{H}_i$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ does not satisfy $\mathcal{E}_i^0$. Let $\mathcal{E}_i^3$ be the inequality condition $\mathcal{E}_i^0 \setminus \mathcal{E}_i^2$. Let $\mathcal{J}_i$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ satisfies $\mathcal{E}_i^3$. Write $\Upsilon := |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})|$. Finally, in the statement of the main theorem, when we write a summation over $x \in W^a$, we mean that we are summing over any set of representatives of the elements in $W^a$. Our main theorem for $SL(3,F)$ is the following.
$$\frac{\theta_{\pi}(g)}{\Upsilon} = \left\{
\begin{array}{ll}
\displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{F}_i^0 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{G}_i^0 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell'(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that} \\ x \ \mathrm{satisfies} \ \mathcal{H}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Gamma(x) & \text{if } t = r \\
\displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{F}_i^1 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{G}_i^1 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell'(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that} \\ x \ \mathrm{satisfies} \ \mathcal{H}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Gamma(x) + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{J}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 3}} \Xi(x) & \text{if } t = r + 1 \\
\displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{F}_i^2 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{G}_i^2 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell'(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that} \\ x \ \mathrm{satisfies} \ \mathcal{H}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Gamma(x) + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{J}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Xi(x) & \text{if } t > r + 1
\end{array} \right.$$
We note that the value of term $\Upsilon = |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})|$ depends on whether or not there are cube roots of unity in $F$, which is why we leave this term as is in the above theorem.
The case of $x \in \mathcal{A}_{n_1}$
-------------------------------------
In this section we compute $\sigma(x)$ when $x \in \mathcal{A}_{n_1}$. When $x \in \mathcal{A}_{n_2}$ or $x \in \mathcal{A}_{n_3}$, the calculations are similar. There is a very slight difference, however, when $x \in \mathcal{A}_{n_4}$, $x \in \mathcal{A}_{n_5}$, and $\mathcal{A}_{n_6}$, although these three latter cases are almost completely analogous. We will address them in the next section.
The main result of this section is the following proposition. Some of the notation in the proposition will be explained in the proof.
\[firsttheorem\] Let $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$. Suppose $n_{21} \geq 0, n_{31} \geq 0, n_{32} \geq 0$. If $r \geq t$, we have
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{
\begin{array}{rl}
vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} & \text{if } -n_{21} + r > 0 \ \mathrm{and} \ -n_{32} + s > 0 \ \\
0 & \text{otherwise}
\end{array} \right.$$
If $r < t$, then $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Xi(x)$$
We first rewrite the double coset $H x H$ as a finite union of single right cosets, as in the case of $SL(2,F)$. Since $$H \cap x^{-1} H x = Z(F) {\left( \begin{array}{ccc} {1 + \mathfrak{p}} & {\mathfrak{p}^{n_{21}}} & {\mathfrak{p}^{n_{31}}} \\ {\mathfrak{p}} & {1 + \mathfrak{p}} & {\mathfrak{p}^{n_{32}}} \\ {\mathfrak{p}} & {\mathfrak{p}} & {1 + \mathfrak{p}}
\end{array} \right)},$$ we have a disjoint union $$H x H = \displaystyle\bigcup_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in \mathfrak{o} / \mathfrak{p}^{n_{31}}}} H x {\left( \begin{array}{ccc} {1} & {z_{21}} & {z_{31}} \\ {0} & {1} & {z_{32}} \\ {0} & {0} & {1}
\end{array} \right)}$$
Now, if $z = {\left( \begin{array}{ccc} {1} & {z_{21}} & {z_{31}} \\ {0} & {1} & {z_{32}} \\ {0} & {0} & {1}
\end{array} \right)}$, then $$xzgz^{-1}x^{-1} = {\left( \begin{array}{ccc} {\alpha} & {\frac{a}{b} z_{21} (\beta - \alpha)} & {\frac{a}{c}[z_{21} z_{32}(\alpha - \beta) + z_{31}(\gamma - \alpha)]} \\ {0} & {\beta} & {\frac{b}{c} z_{32} (\gamma - \beta)} \\ {0} & {0} & {\gamma}
\end{array} \right)}$$ Notice that $g \in Z(F) T(1+\mathfrak{p})$ is forced upon us here in order to have $x z g z^{-1} x^{-1} \in H$ (so that $\dot{\chi}$ doesn’t vanish on $x z g z^{-1} x^{-1}$). We note that the condition $g \in Z(F) T(1+\mathfrak{p})$ continues to be forced upon us, for the same reason, when you compute the terms $ygy^{-1}$ that appear in $\theta_{\pi}(g)$ for any other representative $x$ of any element of the affine Weyl group, but we won’t include these calculations. This shows, therefore, that $\theta_{\pi}$ vanishes on $T(F) \setminus Z(F) T(1+\mathfrak{p})$.
Now, write $\chi$ on $I_+$ as $$\chi : I_+ \rightarrow \mathbb{C}^*$$ $${\left( \begin{array}{ccc} {d_{11}} & {d_{12}} & {d_{13}} \\ {d_{21}} & {d_{22}} & {d_{23}} \\ {d_{31}} & {d_{32}} & {d_{33}}
\end{array} \right)} \mapsto \chi_1(d_{12}) \chi_2(d_{23}) \chi_3(d_{31})$$ where $\chi_1, \chi_2$ are level $1$ characters of $\mathfrak{o}$ and where $\chi_3(d_{31}) = \chi_3'(\frac{1}{\varpi} d_{31})$, where $\chi_3'$ is a level $1$ character of $\mathfrak{o}$.
We would like to say that we therefore have $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in \mathfrak{o} / \mathfrak{p}^{n_{31}}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$ where $\dot{\chi_i}(z) = \chi_i(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi_i}(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$, for $i = 1,2$, and where $\dot{\chi}_3(z) = \chi_3'(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi}_3(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$.
However, we need to take into account the fact that we have the term $\frac{a}{c}[z_{21} z_{32}(\alpha - \beta) + z_{31}(\gamma - \alpha)]$. $\dot{\chi}$ is zero outside of $Z(F) I_+$, and so we have to take into account the condition that $\frac{a}{c}[z_{21} z_{32}(\alpha - \beta) + z_{31}(\gamma - \alpha)] \in \mathfrak{o}$.
Absorbing all units into the $z_{21}, z_{32}, z_{31}$ terms, and recalling that $\alpha - \beta = \varpi^r u$, $\beta - \gamma = \varpi^s u'$, $\alpha - \gamma = \varpi^t u''$, we therefore wish to understand the condition $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31} \in \mathfrak{o}$. We separate this into two cases.
Case 1) Suppose $r \geq t$. Notice that there might be negative powers of $\varpi$ in $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ because of the $-n_{31}$ terms. Fix any $z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}}$ and any $z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}}$. Then, $\varpi^{-n_{31}+r} z_{21} z_{32}$ can certainly contain negative powers of $\varpi$. However, we can use $z_{31}$ to cancel out these negative powers of $\varpi$, since $r \geq t$. In order to force $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31} \in \mathfrak{o}$, it is evident that all of the negative $\varpi$ power terms in $\varpi^{-n_{31} + t} z_{31}$ are uniquely determined by the negative $\varpi$ power terms in $\varpi^{-n_{31} + r} z_{21} z_{32}$. Moreover, since $r \geq t$, no matter what $z_{21}$ or $z_{32}$ are, we can always find a $z_{31}$ to force $\varpi^{-n_{31} + r} z_{21} z_{32} + \varpi^{-n_{31}+t} z_{31} \in \mathfrak{o}$. Once we determine the negative $\varpi$ power terms of $\varpi^{-n_{31} + t} z_{31}$ so that the total sum $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ is in $\mathfrak{o}$, we have complete leeway in the non-negative $\varpi$ power terms in $\varpi^{-n_{31} + t} z_{31}$. Therefore, it appears that we have obtained
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in \mathfrak{p}^{n_{31}-t} / \mathfrak{p}^{n_{31}}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$
However, the last thing we need to notice is that $z_{31}$ still has to be in $\mathfrak{o}$, by a condition from earlier. Therefore, the condition $z_{31} \in \mathfrak{p}^{n_{31}-t} / \mathfrak{p}^{n_{31}}$ is not quite correct since it could be the case that $n_{31} < t$. Therefore, what we really get in then end is
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in (\mathfrak{o} \cap \mathfrak{p}^{n_{31}-t}) / \mathfrak{p}^{n_{31}}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$
Absorbing all units into the $z_{21}, z_{32}$ terms, we get $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}}}} \dot{\chi}_1\left(\varpi^{-n_{21}} z_{21} \varpi^r \right) \dot{\chi}_2 \left(\varpi^{-n_{32}} z_{32} \varpi^{s} \right)$$ Making a change of variables, we get $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21}' \in \mathfrak{p}^{-n_{21} + r} / \mathfrak{p}^{r} \\ z_{32}' \in \mathfrak{p}^{-n_{32} + s} / \mathfrak{p}^{s}}} \dot{\chi}_1\left(z_{21}' \right) \dot{\chi}_2 \left(z_{32}' \right) =$$ $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \left( vol(\mathfrak{p}^r)^{-1} \displaystyle\int_{\mathfrak{p}^{-n_{21}+r} \cap \mathfrak{o}} \dot{\chi}_1(z_{21}') dz_{21}' \right) \left( vol(\mathfrak{p}^s)^{-1} \displaystyle\int_{\mathfrak{p}^{-n_{32}+s} \cap \mathfrak{o}} \dot{\chi}_2(z_{32}') dz_{32}' \right)$$ Therefore, since the integral of a nontrivial character over a group vanishes, we get
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{
\begin{array}{rl}
vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} & \text{if } -n_{21} + r > 0 \ \mathrm{and} \ -n_{32} + s > 0 \ \\
0 & \text{otherwise}
\end{array} \right.$$
which concludes Case 1).
Case 2): Suppose $r < t$. In this case, if one picks any random $z_{21}$ and $z_{32}$, one might not be able to find a $z_{31}$ that makes $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ land in $\mathfrak{o}$. However, if one chooses $z_{21}, z_{32}$ such that $z_{21} z_{32} \in \mathfrak{p}^{t-r}$, then one can find a $z_{31}$ such that $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ will be in $\mathfrak{o}$. One can see that $z_{21} z_{32} \in \mathfrak{p}^{t-r}$ is the only additional condition that we need to add to the conditions in Case 1, so we get
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) =$$ $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}}, z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ \mathrm{such \ that \ } z_{21} z_{32} \in \mathfrak{p}^{t-r}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$
Absorbing all units into the $z_{21}, z_{32}$ terms, we get $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}}, z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ \mathrm{such \ that \ } z_{21} z_{32} \in \mathfrak{p}^{t-r}}} \dot{\chi}_1\left(\varpi^{-n_{21}} z_{21} \varpi^r \right) \dot{\chi}_2 \left(\varpi^{-n_{32}} z_{32} \varpi^{s} \right) \ \ \label{xi}$$ Since this sum is quite complicated, we compute this type of sum in full generality in a later section. We will instead denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Xi(x)$ for the $x$ in this proposition, in the case that $r < t$. Moreover, since we will encounter the analogous type of sum in equation (\[xi\]) for many other elements $x$, we will merely denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Xi(x)$ for those $x$ as well.
We note that in Case 1) above, if $n_{31} \leq t$, we get $vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} = q^{n_{31} + n_{21} + n_{32}}$, which equals $q^{\ell(x)}$, as a simple calculation will show. This is how the terms of the form $q^{\ell(x)}$ appear in the character formula. If, on the other hand, $n_{31} > t$, we get $vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} = q^{t + n_{21} + n_{32}}$, which is what we have called $q^{\ell'(x)}$, as $t + n_{21} + n_{32}$ is a “truncated length” of $x$.
We now illustrate one more case in the $\mathcal{A}_{n_1}$ setting, which will show how the second type of Gauss sum in the character formula arises. Some of the notation in the next proposition will be explained in the proof.
Let $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}
\end{array} \right)}$. Suppose $n_{21} < 0, n_{31} < 0, n_{32} < 0$. Then
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Gamma(x)$$
We have a disjoint union $$H x H = \displaystyle\bigcup_{\substack{z_{21} \in \mathfrak{p} / \mathfrak{p}^{-n_{21}+1} \\ z_{31} \in \mathfrak{p} / \mathfrak{p}^{-n_{31}+1} \\ z_{32} \in \mathfrak{p} / \mathfrak{p}^{n_{32}+1}}} H x {\left( \begin{array}{ccc} {1} & {0} & {0} \\ {z_{21}} & {1} & {0} \\ {z_{31}} & {z_{32}} & {1}
\end{array} \right)}$$
Now, if $z = {\left( \begin{array}{ccc} {1} & {0} & {0} \\ {z_{21}} & {1} & {0} \\ {z_{31}} & {z_{32}} & {1}
\end{array} \right)}$, then $$xzgz^{-1}x^{-1} = {\left( \begin{array}{ccc} {\alpha} & {0} & {0} \\ {\frac{b}{a} z_{21} (\alpha - \beta)} & {\beta} & {0} \\ {\frac{c}{a} [ z_{32} z_{21} (\gamma - \beta) + z_{31} ( \alpha - \gamma) ]} & {\frac{c}{b} z_{32} ( \beta - \gamma)} & {\gamma}
\end{array} \right)}$$
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum \dot{\chi}(\frac{c}{a} [ z_{32} z_{21} (\gamma - \beta) + z_{31} ( \alpha - \gamma) ] \ \ \label{gamma}$$
where the sum is over $z_{21} \in \mathfrak{p} / \mathfrak{p}^{-n_{21}+1}, z_{31} \in \mathfrak{p} / \mathfrak{p}^{-n_{31}+1}, z_{32} \in \mathfrak{p} / \mathfrak{p}^{n_{32}+1} \ \mathrm{such \ that} \ \frac{b}{a} z_{21} (\alpha - \beta) \in \mathfrak{p} \ \mathrm{and} \ \frac{c}{b} z_{32} ( \beta - \gamma) \in \mathfrak{p}$. Since this type of sum is quite complicated, we compute this type of sum in full generality in section \[morecalculations\]. We will instead denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Gamma(x)$ for the $x$ in this proposition. Moreover, since we will encounter the type of sum on the right hand side of the above equation (\[gamma\]) for many other elements $x$, we will merely denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Gamma(x)$ for those $x$ as well.
The case of $x \in \mathcal{A}_{n_4}$
-------------------------------------
In this section we will consider the inner sums $\sigma(x)$ when $x \in \mathcal{A}_{n_4}$. This case is slightly different from that of $\mathcal{A}_{n_1}, \mathcal{A}_{n_2}$, and $\mathcal{A}_{n_3}$. In particular, the inequalities between $r,s,$ and $t$ that distinguished between Case 1)’s and Case 2)’s in the previous section are now shifted, as we shall show. This is why we need separate cases in the statement of the main theorem of the distribution character for $SL(3,F)$. We will show why the shifts occur in this section. The cases of $x \in \mathcal{A}_{n_5}$ and $x \in \mathcal{A}_{n_6}$ are similar to the case of $x \in \mathcal{A}_{n_4}$.
Let $x = {\left( \begin{array}{ccc} {a} & {0} & {0} \\ {0} & {0} & {b} \\ {0} & {c} & {0}
\end{array} \right)}$. Suppose $n_{31} < 0, n_{21} < 0, n_{23} < 0$. Then if $s \geq t -1$, we have $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) =$$
$$\left\{
\begin{array}{rl}
vol((\mathfrak{p} \cap \mathfrak{p}^{-n_{21} - t + 1}) / \mathfrak{p}^{-n_{21}+1}) q^{-n_{31} - n_{23}-1} & \text{if } n_{23}+s+1 > 0 \ \mathrm{and} \ n_{31}+r > 0 \ \\
0 & \text{otherwise}
\end{array} \right.$$
If $s < t-1$, then $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Xi(x)$$
We have a disjoint union $$H x H = \displaystyle\bigcup_{\substack{z_{21} \in \mathfrak{p} / \mathfrak{p}^{-n_{21}+1} \\ z_{23} \in \mathfrak{p} / \mathfrak{p}^{-n_{23}} \\ z_{31} \in \mathfrak{p} / \mathfrak{p}^{-n_{31}+1}}} H x {\left( \begin{array}{ccc} {1} & {0} & {0} \\ {z_{31}} & {1} & {0} \\ {z_{21}} & {z_{23}} & {1}
\end{array} \right)}$$
Now, if $z = {\left( \begin{array}{ccc} {1} & {0} & {0} \\ {z_{31}} & {1} & {0} \\ {z_{21}} & {z_{23}} & {1}
\end{array} \right)}$, then $$xzgz^{-1}x^{-1} = {\left( \begin{array}{ccc} {\alpha} & {0} & {0} \\ {\frac{b}{a}[z_{21}(\alpha - \gamma) + z_{31} z_{23} (\gamma - \beta)]} & {\gamma} & {\frac{b}{c} z_{23} (\beta - \gamma)} \\ {\frac{c}{a} z_{31} (\alpha - \beta)} & {0} & {\beta}
\end{array} \right)}$$
We have to take into account the condition that $\frac{b}{a}[z_{21}(\alpha - \gamma) + z_{31} z_{23} (\gamma - \beta)] \in \mathfrak{p}$. We therefore wish to understand the condition $\varpi^{n_{21}+s} z_{31} z_{23} + \varpi^{n_{21} + t} z_{21} \in \mathfrak{p}$. We separate this into two cases, exactly in the way we did in Proposition \[firsttheorem\]. Notice here that $z_{21} \in \mathfrak{p}, z_{23} \in \mathfrak{p}, z_{31} \in \mathfrak{p}$. Therefore, our first case is going to be
Case 1) Suppose $s \geq t -1$. Then an analogous computation as we have done before shows that $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) =$$
$$\left\{
\begin{array}{rl}
vol((\mathfrak{p} \cap \mathfrak{p}^{-n_{21} - t + 1}) / \mathfrak{p}^{-n_{21}+1}) q^{-n_{31} - n_{23}-1} & \text{if } n_{23}+s+1 > 0 \ \mathrm{and} \ n_{31}+r > 0 \ \\
0 & \text{otherwise}
\end{array} \right.$$
Case 2) Suppose $s < t - 1$. Then, using notation from the previous section, we have $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Xi(x)$$
Calculation of the Gauss sums {#morecalculations}
=============================
The calculation of $\Gamma(x)$ {#gammasections}
------------------------------
In this section we calculate the terms of the form $\Gamma(x)$ in a much more general context. In particular, we shall calculate $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z),$$ where $\chi$ is a character of $F$ that is zero on $F \setminus \mathfrak{o}$, and is $1$ on $\mathfrak{p}$, except when
i\) $m + k < a, i < b$, and $n + \ell \leq a$
ii\) $m + k < a, i < b$, and $j \leq b$
The reason that we may omit these cases is that the conditions $j \leq b$ and $n + \ell \leq a$ never occur in the $SL(3,F)$ calculations (as one can check, but we do not include these details here), so we can ignore them. We won’t ignore every possible case where $j \leq b$ or $n + \ell \leq a$, since some of these cases are easy to write down. We will just ignore the above two special cases. We split up the calculation of the above sum into various cases.
Case 1) Suppose $m+k \geq a, i > b$. We then have that $\varpi^{-a} xy, \varpi^{-b} z \in \mathfrak{o}$. Therefore, $\chi(\varpi^{-a} xy + \varpi^{-b} z) = \chi(\varpi^{-a} xy) \chi( \varpi^{-b} z)$, so $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) \displaystyle\sum_{z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi( \varpi^{-b} z)$$ After making a change of variables, we get $$\displaystyle\sum_{z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi( \varpi^{-b} z) = \displaystyle\sum_{z' \in \mathfrak{p}^{i-b} / \mathfrak{p}^{j-b} } \chi( z')$$ $$=vol(\mathfrak{p}^{j-b})^{-1} \int_{\mathfrak{p}^{i-b} \cap \mathfrak{o}} \chi(z') dz' = vol(\mathfrak{p}^{j-b})^{-1} vol(\mathfrak{p}^{i-b}) = q^{j-i}$$ and so $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = q^{j-i} \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy)$$ We now consider two subcases : $m + k > a$ and $m + k = a$. Suppose that $m + k > a$. Then $\varpi^{-a} xy \in \mathfrak{p}$, and therefore $\chi(\varpi^{-a} xy) = 1 \ \forall x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}$ since $\chi$ is trivial on $\mathfrak{p}$. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} 1 = vol(\mathfrak{p}^m / \mathfrak{p}^n) vol(\mathfrak{p}^k / \mathfrak{p}^{\ell}) = q^{n-m} q^{\ell-k}$$ Now suppose $m + k = a$. We argue by fixing values of $x$. Suppose $val(x) = m$. Then $$\displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(\varpi^{-a} xy) = 0$$ since $\varpi^{-a} xy$ ranges over all elements of $\mathfrak{o} / \mathfrak{p}^{\ell-a+m}$. Therefore, the contributions to $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy)$$ from the elements $x$ that have valuation $m$ are all zeroes. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy)$$ But now in this sum, notice that since we have $x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}$, we conclude that $\varpi^{-a} xy$ is always in $\mathfrak{p}$. Therefore, as before, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} 1 = q^{n - (m+1)} q^{\ell-k}$$
Case 2) Suppose $m+k \geq a, i = b$. This case is mostly analagous to Case 1), except we now have $$\displaystyle\sum_{z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi( \varpi^{-b} z) = 0$$ since $i = b$, and therefore $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = 0$$
Case 3) Suppose $m+k \geq a, i < b$. We need to understand the condition $\varpi^{-a} xy + \varpi^{-b} z \in \mathfrak{o}$. We first assume that $j > b$. Thus, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^b / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z)$$ (note that $b < j$, so we can talk about $\mathfrak{p}^b / \mathfrak{p}^j$). But now note that since $z \in \mathfrak{p}^b$, we have that $\varpi^{-a} xy, \varpi^{-b} z \in \mathfrak{o}$. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^b / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) =
\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) \displaystyle\sum_{ z \in \mathfrak{p}^b / \mathfrak{p}^j } \chi(\varpi^{-b} z) = 0$$ since the integral over a group of a nontrivial character vanishes.
If $j \leq b$, then in order for $\varpi^{-a} xy + \varpi^{-b} z$ to be in $\mathfrak{o}$, we require that $z = 0$, since $z$ is assumed to be in $\mathfrak{p}^i / \mathfrak{p}^j$. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy),$$ which may be rewritten as $$\displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(x'y)$$ after a change of variables. This type of sum will be handled in Case 4), which we now present.
Case 4) Suppose $m+k < a, i > b$. We consider two subcases. We therefore get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a }} \chi(\varpi^{-a} xy + \varpi^{-b} z)$$ since if $xy \notin \mathfrak{p}^a$, then $\varpi^{-a} xy + \varpi^{-b} z \notin \mathfrak{o}$. Therefore, we get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a }} \chi(\varpi^{-a} xy + \varpi^{-b} z) =$$ $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a}} \chi(\varpi^{-a} xy) \displaystyle\sum_{ z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi(\varpi^{-b} z) = q^{j-i} \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a}} \chi(\varpi^{-a} xy)$$ We make a change of variables $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a}, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } x'y \in \mathfrak{o}}} \chi(x'y)$$
Since $\chi$ vanishes outside $\mathfrak{o}$ $$\displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a}, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } x'y \in \mathfrak{o}}} \chi(x'y) = \displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(x'y)$$
After reindexing and relabeling, we are now interested in computing the following type of sum $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ We split this into two cases:
Case i) Suppose $n + \ell \geq -2$. Recall that $\chi$ vanishes outside $\mathfrak{o}$, so we are interested in when $xy \in \mathfrak{o}$. We will separate out the $x$ terms from the sum that have no chance of multiplying with an element of $y$ to land in $\mathfrak{o}$ unless $y$ is zero. We consider two subcases. Suppose $1 - \ell > m$. We write $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ (note that we can write $x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n$ since we assumed $1 - \ell \geq m$). Let the first sum be denoted $A$ and the second sum be denoted $B$. For the first sum $A$, $xy$ can never be in $\mathfrak{o}$ unless $y = 0$. Therefore, we get $$A = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n \\ y = 0}} \chi(xy) = \displaystyle\sum_{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n} 1 = q^{n-m} - q^{n - (1 - \ell)}.$$ For $B$, we split up the sum as
$$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1 - \ell} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + 1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1 - \ell+1} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + 2} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ $$+ \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1 - \ell+2} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + 3} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + ... + \displaystyle\sum_{\substack{x \in \mathfrak{p}^n / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$
Note that if $val(x) = i$, we require $y \in \mathfrak{p}^{-i} / \mathfrak{p}^{\ell}$ in order to force $xy \in \mathfrak{o}$. Moreover, note that if $val(x) = i - \ell$, and $y$ ranges over $\mathfrak{p}^{\ell-i} / \mathfrak{p}^{\ell}$, then $xy$ ranges over $\mathfrak{o} / \mathfrak{p}^i$. Coupling this with the fact that the sum of a nontrivial character over a group vanishes, we may compute the above sums easily. We note that the evaluation of these sums can vary depending on $k, \ell, $ and $n$, as one can check.
We now assume $1 - \ell \leq m$. Then there is no analogous term $A$ as above that we need to evaluate, and so the evaluation of the sum $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ is analogous to the sum $B$ above.
Case ii) Assume $n + \ell < -2$. In this case, it’s never possible that $xy \in \mathfrak{o}$ unless $x$ or $y$ is zero. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \displaystyle\sum_{\substack{x = 0 \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y = 0}} \chi(xy) - \chi(0*0)$$ We subtracted $\chi(0*0)$ since we have double counted the term $\chi(0*0)$ in the right hand side of the equality. Thus, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \left(\displaystyle\sum_{ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} 1 \right) + \left(\displaystyle\sum_{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n}} 1 \right) - 1 = q^{n-m} + q^{\ell-k} - 1$$ This finishes the case $n + \ell < -2$.
Case 5) Suppose $m + k < a, i = b$. Then $$\displaystyle\sum_{ z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi(\varpi^{-b} z) = 0$$ since $i = b$ (as in a previous case). Moreover, in order to get $\varpi^{-a} xy + \varpi^{-b} z$ to be in $\mathfrak{o}$ we need the negative valuation terms of $\varpi^{-a} xy$ to be zero, since $\varpi^{-b} z$ is always in $\mathfrak{o}$. Therefore, we are forced to take values of $x,y$ such that $\varpi^{-a} xy \in \mathfrak{o}$. Over these values of $x,y$, we get $\chi(\varpi^{-a} xy + \varpi^{-b} z) = \chi(\varpi^{-a} xy) \chi(\varpi^{-b} z)$. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = 0$$
Case 6) Suppose $m+ k < a, i < b$. Recall from assumptions i) and ii) at the beginning of section \[gammasections\], we may assume that $n + \ell > a$ and $j > b$. Since $m + k < a, i < b$, we will have negative $\varpi$ powers in both $\varpi^{-a} xy$ and $\varpi^{-b} z$. Consider the negative valuation part of a term of the form $\varpi^{-a} xy + \varpi^{-b} z$. We need this negative valuation part to be zero. But once the negative valuation part of this is zero, we are free to let the rest of $x,y,z$ vary. Fix possible negative valuation parts, denoted $(xy)_{-}$ and $z_{-}$, of $\varpi^{-a} xy$ and $\varpi^{-b} z$, respectively. We compute the contribution to $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z)$$ of all terms $\chi(\varpi^{-a} xy + \varpi^{-b} z)$ such that $\varpi^{-a} xy$ has negative valuation $(xy)_{-}$ and $\varpi^{-b} z$ has negative valuation $(z)_{-}$, where $(xy)_{-} = -(z)_{-}$ (this last equality is forced upon us since otherwise $\chi$ will vanish). Since $(xy)_{-} = -(z)_{-}$, this contribution is $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^a, z \in \mathfrak{p}^b }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^a, z \in \mathfrak{p}^b }} \chi(\varpi^{-a} xy) \chi(\varpi^{-b} z) = 0.$$ This argument holds regardless of $(xy)_{-}$ and $(z)_{-}$. Thus, in the end, we are summing up a bunch of zeroes, so we finally get that $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = 0$$
The calculation of $\Xi(x)$
---------------------------
In this section we calculate the terms of the form $\Xi(x)$ in a much more general context. In particular, we shall calculate $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ We will calculate this sum in complete generality. We split up the calculation of the above sum into various cases.
Case 1): Suppose $n + \ell \leq c + 1$. Then it’s never the case that $xy \in \mathfrak{p}^c$ unless at least one of $x,y$ are zero. Therefore, if $n + \ell \leq c + 1$, we get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \ } \chi(y) + \displaystyle\sum_{y \in \mathfrak{p}^m / \mathfrak{p}^n \ } \chi(x) - 1$$ which is easily calculatable, each sum being $0$ or a power of $q$, depending on $k,\ell,m,n$. Note that we subtracted $\chi(0)\chi(0) = 1$ since this is $\chi(x) \chi(y)$ when $x = y = 0$ and we have double counted this term when we added the $x =0$ and $y = 0$ sums above.
Case 2): Now assume that $n + \ell > c+1$. Assume furthermore that $1 - \ell + c > m$. We will separate the sum into two parts. We write $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) =
\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) +$$ $$\displaystyle\sum_{\substack{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n ), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y).$$ (note that we can write $x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n )$ since we assumed that $1 - \ell + c > m$). We first compute the second sum on the right hand side. In this sum, we will never have $xy \in \mathfrak{p}^c$ unless at least one of $x,y$ is zero. But $x$ can’t be zero, since $x \in \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n $. Therefore, we must have $y = 0$. Therefore, the second sum is $$\displaystyle\sum_{\substack{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n ), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n )} \chi(x) \chi(0) =$$ $$\displaystyle\sum_{x \in ( \mathfrak{p}^m / \mathfrak{p}^n)} \chi(x) - \displaystyle\sum_{x \in ( \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n) } \chi(x)$$ which is easy to calculate, each sum being $0$ or a power of $q$, depending on $m,n,\ell,c$.
So we now need to calculate the first sum $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ We separate this into cases:
Case A: $1 - \ell + c > 0$. We will calculate the sum by fixing the valuation of $x$. Namely, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in (\mathfrak{p}^{1 - \ell + c} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + c + 1} / \mathfrak{p}^n), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) +$$ $$\displaystyle\sum_{\substack{x \in (\mathfrak{p}^{1 - \ell + c+1} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + c + 2} / \mathfrak{p}^n) \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) +$$ $$\displaystyle\sum_{\substack{x \in (\mathfrak{p}^{1 - \ell + c+2} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + c + 3} / \mathfrak{p}^n) \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) + ... + \displaystyle\sum_{\substack{x \in \mathfrak{p}^n / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$
Note that if $val(x) = 1 - \ell + c + i$, then in order for $xy$ to be in $\mathfrak{p}^c$, we must have $y \in \mathfrak{p}^{\ell-i-1}$. Coupling this with the fact that $\chi(x) = 1$ in every summation since we have assumed that $1 - \ell + c > 0$, we may compute the above sums easily. We note that the evaluation of these sums can vary depending on $c, k, \ell, $ and $n$, as one can check.
Case B: Suppose $1 - \ell + c \leq 0$ and $\ell > 0$. Suppose first that $n \leq 0$. Then $\chi(x) \chi(y) \neq 0$ iff $x = 0$ since $\chi$ is zero on elements of negative valuation. Therefore, if $n \leq 0$, $$\displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(0) \chi(y) = \displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(y)$$ which is easy to calculate. If $n > 0$, then again, since $\chi$ is zero on elements of negative valuation, and since $1 - \ell + c \leq 0$, so we get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ In the case that $k \geq c$, $xy \in \mathfrak{p}^c$ always holds, so $$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x) \displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(y)$$ which is easy to calculate, depending on whether or not $n$ is zero.
Suppose that $k < c$. We break the sum $$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ into two sums :
$$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^c / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) + \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in (\mathfrak{p}^k / \mathfrak{p}^{\ell} - \mathfrak{p}^c / \mathfrak{p}^{\ell}) \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ Note that we assumed that $1 - \ell + c \leq 0$, so we can talk about $\mathfrak{p}^c / \mathfrak{p}^{\ell}$. Moreover, since we assumed that $k < c$, we can talk about $y \in (\mathfrak{p}^k / \mathfrak{p}^{\ell} - \mathfrak{p}^c / \mathfrak{p}^{\ell})$. Call the first sum D and the second sum E. We first analyze E. Notice that since are subtracting all elements $y$ of valuation $\geq c$, if we take an element $x$ of valuation zero, $xy$ can never be in $\mathfrak{p}^c$. Therefore, $$E = \displaystyle\sum_{\substack{x \in \mathfrak{p} / \mathfrak{p}^n, y \in (\mathfrak{p}^k / \mathfrak{p}^{\ell} - \mathfrak{p}^c / \mathfrak{p}^{\ell}) \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ which equals $$\displaystyle\sum_{\substack{x \in \mathfrak{p} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) - \displaystyle\sum_{\substack{x \in \mathfrak{p} / \mathfrak{p}^n, y \in \mathfrak{p}^c / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y).$$ The first of these two sums can be handled by Case A, and the second sum equals $$\displaystyle\sum_{y \in \mathfrak{p}^c / \mathfrak{p}^{\ell}} \chi(y),$$ which is easy to calculate, and depends on $c$ and $\ell$.
We now handle the first sum D. It’s always the case that $xy \in \mathfrak{p}^c$ in this sum, so we get $$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^c / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n \\ y \in \mathfrak{p}^c / \mathfrak{p}^{\ell}}} \chi(x) \chi(y) = \displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x) \displaystyle\sum_{y \in \mathfrak{p}^c / \mathfrak{p}^{\ell}} \chi(y) = 0$$ since $$\displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x) = 0$$ (recall that we are in the case that $n > 0$). Thus, $D = 0$.
Case C: Suppose $1 - \ell + c \leq 0$ and $\ell \leq 0$. Then since $\chi$ is zero on negative valuation terms, the $y$ terms don’t contribute unless $y = 0$. Then $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n} \chi(x) = \displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x)$$ which is easy to calculate, depending on whether or not $n$ is zero.
Finally, the above analysis for Case 2) assumed that $1 - \ell + c > m$. The case that $1 - \ell + c \leq m$ is simpler and similar to the case of $1 - \ell + c \leq m$, as one can check. We note that there is no sum of the form $$\displaystyle\sum_{\substack{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n ), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ that we need to evaluate, in the case that $1 - \ell + c \leq m$.
[9]{}
*J. Adler, L. Spice*, Supercuspidal characters of reductive, $p$-adic groups, Amer. J. Math. 131 no. 4, 1137–1210
*M. Adrian*, A New Realization of the Local Langlands Correspondence for $GL(n,F)$, $n$ a prime, preprint.
Pierre Cartier, *Representations of $\mathfrak{p}$-adic groups: A survey* Proceedings of Symposia in Pure Mathematics, vol. 33 (1979) part 1, pp. 111-155
*S. DeBacker*, On Supercuspidal Characters of $GL_{\ell}$, $\ell$ a prime, Ph.D. thesis, University of Chicago, 1997.
Benedict Gross and Mark Reeder, *Arithmetic invariants of discrete Langlands parameters.* Duke Math. Journal, 154, (2010), 431-508.
*P. Kutzko*, Character Formulas for supercuspidal representations of $GL_{\ell}$, $\ell$ a prime., Amer. J. Math. 109 (1987), no. 2, 201–221.
Paul J. Sally, Jr., *Some remarks on discrete series characters for reductive p-adic groups.* Representations of Lie groups, Kyoto, Hiroshima, 1986, 1988, pp. 337–348. MR1039842 (91g:22026)
*P. Sally, J. Shalika*, Characters of the discrete series of representations of ${\rm SL}(2)$ over a local field., Proc. Nat. Acad. Sci. U.S.A. 61 1968 1231–1237.
*H. Shimizu*, Some examples of new forms, J. Fac. Sci. Univ. Tokyo 24 (1977), no. 1, 97-113.
*L. Spice*, Supercuspidal Characters of $SL_{\ell}$ over a $p$-adic field, $\ell$ a prime, Amer. J. Math. 121 no. 1, 51–100
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abstract: 'A partial geometry $S$ admitting an abelian Singer group $G$ is called of rigid type if all lines of $S$ have a trivial stabilizer in $G$. In this paper, we show that if a partial geometry of rigid type has fewer than $1000000$ points it must be the Van Lint-Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very rare. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint-Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). We end the paper with a conjecture on this type of geometries.'
author:
- 'Stefaan De Winter[^1]'
- 'Ellen Kamischke[^2]'
- 'Erik Neubert[^3]'
- 'Zeying Wang[^4]'
title: Results on partial geometries with an abelian Singer group of rigid type
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Partial geometry, Partial difference set, Singer group
Introduction
============
Ever since James Singer in his seminal paper [@Singer] showed that every finite Desarguesian projective space admits a cyclic sharply transitive group of automorphisms, finite geometries admitting a (not necessarily cyclic) sharply transitive group of automorphisms have attracted significant interest. Such automorphism groups are now commonly called [*Singer groups*]{} in honor of J. Singer. A lot of research in this realm has focused on projective planes, where the major conjecture is that “[*if a finite projective plane admits a cyclic Singer group it must be Desarguesian*]{}”. Despite much research and some progress the conjecture is still wide open, see for example [@Schmidt]. Generalized quadrangles are another type of geometries that have been studied in this context. We refer to [@Bamberg; @SDWpgst1; @Tao; @Ghinelli; @Yoshiara] among others. Partial geometries can in some sense be thought of as sitting between projective planes and generalized quadrangles. They are first mentioned in the context of abelian Singer groups in [@MA94b] and a more detailed study appeared in [@SDWpgst2] and [@Leung2]. Partial geometries with nonabelian Singer groups are discussed in [@Swartz]. In this paper we obtain further evidence that a certain type of partial geometries with an abelian Singer group is very rare, and we formulate a conjecture on this type of geometries.
Definitions
-----------
We start by providing the necessary definitions and some previous results.
\[pgdef\] A proper partial geometry, denoted $pg(s, t, \alpha)$, is a finite point-line geometry such that each line contains $s + 1$ points, each point is contained in $t+1$ lines, each pair of lines intersects in at most one point, and such that
- given any point $x$ and any line $L$ not containing $x$, there are exactly $\alpha$ lines through $x$ that intersect $L$, with $0<\alpha<min(s,t)$.
Partial geometries were introduced by Bose [@BosePG] in order to provide a setting and generalization for known characterization theorems for strongly regular graphs (it is readily checked that the point graph of a partial geometry is a strongly regular graph). Partial geometries with $\alpha=1$ are the so-called generalized quadrangles. Generalized quadrangles with an abelian Singer group are well-understood, they are the ones arising from (generalized) hyperovals; see [@SDWpgst1]. The natural language to study partial geometries admitting a sharply transitive abelian group of automorphisms is the language of partial difference sets. Partial difference sets were introduced by Ma in [@MA84].
Let $G$ be a finite group of order $v$, and let $D\subseteq G$ be a subset of size $k$. We say $D$ is a $(v,k,\lambda,\mu)$-[*partial difference set*]{} (PDS) in $G$ if the expressions $gh^{-1}$, $g$, $h\in D$, $g\neq h$, represent each non-identity element in $D$ exactly $\lambda$ times, and each non-identity element of $G$ not in $D$ exactly $\mu$ times. If we further assume that $D^{(-1)}=D$ (where $D^{(s)}=\{g^s:g\in D\}$) and $e \notin D$ (where $e$ is the identity element of $G$), then $D$ is called a [*regular*]{} partial difference set. A regular PDS is called [*trivial*]{} if $D\cup\{e\}$ or $G\setminus D$ is a subgroup of $G$. The condition that $D$ be regular is not a very restrictive one, as $D^{(-1)}=D$ is automatically fulfilled whenever $\lambda\neq\mu$, and $D$ is a PDS if and only if $D\cup\{e\}$ is a PDS. In this paper we will also assume all groups are abelian. Throughout this paper we will use the following standard notations: $\beta=\lambda-\mu$ and $\Delta=\beta^2+4(k-\mu)$. For more information on partial difference sets, we refer the reader to a survey of Ma [@MA94b].\
The [*Cayley graph over $G$ with connection set $D$*]{}, denoted by Cay($G$, $D$), is the graph with the elements of $G$ as vertices, and in which two vertices $g$ and $h$ are adjacent if and only if $gh^{-1}$ belongs to $D$. When the connection set $D$ is a regular partial difference set in $G$, Cay($G$, $D$) is a strongly regular graph. Conversely, given a strongly regular graph admitting a sharply transitive abelian group of automorphisms $G$, it is easy to check that one obtains a regular partial difference in $G$ as follows: identify one of the vertices $v$ with the identity of $G$ and any other vertex $w$ with the unique element $g$ of $G$ such that $v^g=w$. The neighborhood of $v$ now provides a regular partial difference set in $G$.
Suppose we have a $pg(s,t,\alpha)$ admitting an abelian automorphism group $G$ acting sharply transitively on the points and we identify the points with the elements of $G$ as above. Let $L_0$, $L_1$, $\cdots$, $L_t$ (seen as sets of element of $G$) be the $t+1$ lines through $e$, the identity element in $G$. Then the following result follows immediately:
\[[@MA94b]\] Suppose we have a $pg(s,t,\alpha)$ with an automorphism group $G$ acting sharply transitively on the points, with the notation above, $D=(\bigcup_{i=0}^t L_i)\setminus\{e\}$ is a regular PDS with parameters $$(v,k,\lambda,\mu)=((s+1)(\alpha+st))/{\alpha}, \,s(t+1),\,s+t(\alpha-1)-1,\,\alpha(t-1)).$$
In [@SDWpgst2] the following observation was made for partial geometries admitting an abelian Singer group $G$: the stabilizer of any line $L$ in $G$ is either trivial or has size $|L|=s+1$. Based on this pairs $(\mathcal{S},G)$ of partial geometries $\mathcal{S}$ admitting an abelian Singer group $G$ were divided into three disjoint classes:
- [*spread type*]{}: all lines have a stabilizer in $G$ of size $s+1$;
- [*mixed type*]{}: some but not all lines have a stabilizer in $G$ of size $s+1$;
- [*rigid type*]{}: all lines have a trivial stabilizer in $G$.
In [@SDWpgst2] it was shown that partial geometries of spread type are in some sense well-understood: they arise through linear representation of a so-called PG-regulus. The known examples are the partial geometries arising from maximal arcs and one other example constructed in [@FDCperp]. In [@SDWpgst2] it was also conjectured that partial geometries of mixed type with $\alpha=2$ do not exist and this conjecture was largely proven in [@Leung2]. The case $\alpha\geq3$ is wide open, but no examples are known. Finally, in [@SDWpgst2] it was shown that a partial geometry of rigid type with $\alpha=2$ must be the pg$(5,5,2)$ of Van Lint and Schrijver (see [@VLS] for the original construction of this geometry using cyclotomy). In this paper we want to provide evidence that partial geometries of rigid type are very rare by showing that for $\alpha\leq1000$ at most five hypothetical parameter sets can give rise to such geometry, including the above mentioned Van Lint - Schrijver geometry. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint - Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). As a consequence it follows that if a partial geometry of rigid type has fewer than $1,000,000$ points it must be the Van Lint-Schrijver geometry or have $1024$ or $4096$ or $194,481$ points. Partial results in this direction were previously obtained in the theses of the second and third author [@ElThesis; @Neubert].
Throughout the rest of this paper $\mathcal{S}$ will be a proper pg$(s,t,\alpha)$, $\alpha\geq2$, admitting the abelian group $G$ as a Singer group and such that the pair $(\mathcal{S},G)$ is of rigid type.
Some necessary conditions
=========================
In this section we derive some basic necessary conditions on the parameters of partial geometries of rigid type. First, we cite a theorem from [@SDWpgst2].
\[Bensontype\] Let $\mathcal{S}$ be a partial geometry $pg(s, t, \alpha)$ and let $\theta$ be any automorphism of $\mathcal{S}$. Let $f$ be the number of fixed points of $\mathcal{S}$ under $\theta$ and $g$ be the number of points $p$ of $\mathcal{S}$ for which $p$ is collinear with $p^\theta$ , where $p^\theta$ denotes the image of $p$ under $\theta$. Then, $(1+t) f +g \equiv (1+s)(1+t) \pmod{s+t - \alpha +1}.$
We are now ready to prove the following:
\[necessary\] Let $\mathcal{S}$ be a pg$(s,t,\alpha)$ such that the pair $(\mathcal{S},G)$ is of rigid type. Then
- $(s+1)(t+1) \equiv 0 \pmod{s+t-\alpha+1}$;
- $v=(s+1)(st+\alpha)/\alpha \equiv 0 \pmod{s+t-\alpha+1}$;
- $t+1=x(s+1)$ for some positive integer $x$;
- $x=\frac{c(s-\alpha)}{(\alpha+1)^2-c(s+1)}$ for some positive integer $c$;
- $s < (\alpha+1)^2 -1$.
[**Proof.** ]{}Clearly no non-identity element of $G$ has a fixed point in its action on $\mathcal{S}$. Let $g$ be an element of $G$ which maps some point $p_0$ to a non-collinear point (such $g$ obviously exists). Then we claim that $g$ maps any other point $p_1$ to a non-collinear point. Let $h$ be the unique element of $G$ that maps $p_0$ to $p_1$. As $h$ is an automorphism it preserves (non)-collinearity. Hence $p_1=p_0^h$ is not collinear with $p_0^{gh}=p_0^{hg}=p_1^g$ (note that we need $G$ to be abelian here). Hence $g$ indeed maps every point to a non-collinear point. So we obtain $f=g=0$ in Theorem \[Bensontype\]. This implies that $(s+1)(t+1) \equiv 0 \pmod{s+t-\alpha+1}$.
In a similar way there exists an element of $G$ that maps every point of $\mathcal{S}$ to a collinear point, so we obtain $f=0$ and $g=v=(s+1)(st+\alpha)/\alpha$ in Theorem \[Bensontype\]. Combining with part a) it follows that $v\equiv 0 \pmod{s+t-\alpha+1}$.
Pick any point $p_0$ and any line $L$ containing $p_0$ from $\mathcal{S}$. Assume that $p_0$, $p_1$, $\cdots$, $p_s$ are the $s+1$ points on $L$. Since $G$ acts sharply transitively on $S$, for any two collinear points $p_0$ and $p_i$, there exists a unique $g_i\in G$ such that $p_i^{g_i}=p_0$. Since the stabilizer in $G$ of $L$ is trivial, $\{L, L^{g_1}, L^{g_2}, \cdots,L^{g_s}\}$ is a set of $s+1$ distinct lines through the point $p_0$. If these are all the lines through $p_0$ we are done, if not, simply repeat the argument for one of the remaining lines through $p_0$. As there are $t+1$ lines passing through the point $p_0$, it follows that $t+1=x(s+1)$ for some positive integer $x$.
Combining $(s+1)(t+1) \equiv 0 \pmod{s+t-\alpha+1}$ and $t+1 = x(s+1)$ we obtain $x(s+1)^2\equiv 0 \pmod{s+t-\alpha+1}$. Replacing $t$ by $t=x(s+1)-1$ gives:
$x(s+1)^2\equiv 0 $ (mod $(x+1)(s+1)-(\alpha+1)).$
Multiplying by $(x+1)^2$ and subtracting $x(\alpha + 1)^2$:
$x\big((x+1)^2(s+1)^2 - (\alpha+1)^2\big) \equiv - x(\alpha+1)^2 \pmod{(x+1)(s+1)-(\alpha+1)}$, so
$x \big((x+1)(s+1)-(\alpha+1)\big)\big((x+1)(s+1) + (\alpha+1)\big) \equiv - x (\alpha+1)^2 \pmod{(x+1)(s+1)-(\alpha+1)}$.
But now the left hand side is equivalent to 0, and we have $x(\alpha+1)^2 \equiv 0 \pmod{(x+1)(s+1) - (\alpha+1)}$. Since $x(\alpha+1)^2$ is positive, there is a positive integer $c$ such that $x(\alpha+1)^2 = c\left((x+1)(s+1) - (\alpha+1)\right)$. Solving for $x$ yields d).
Using the fact that $\alpha<s$ it follows that the denominator in the expression from d) must be strictly positive. Hence e) follows. [$\Box$]{}
We now cite three further necessary conditions. The first two were obtained by Ma:
\[Ma2\] No non-trivial PDS exists in
- an abelian group $G$ with a cyclic Sylow-$p$-subgroup and $o(G)\neq p$;
- an abelian group $G$ with a Sylow-$p$-subgroup isomorphic to $\mathbb{Z}_{p^s}\times\mathbb{Z}_{p^t}$ where $s\neq t$.
\[Ma3\] If $D$ is a non-trivial regular PDS in the abelian group $G$, then $v$, $\Delta$ and $v^2/\Delta$ all have the same prime divisors.
The third appeared in [@Leung2]:
\[[@Leung2]\] \[MLS\] Suppose there exists a proper partial geometry $\mathcal{S}$=pg[(]{}$s$,$t$, $\alpha$[)]{} with an abelian Singer group $G$. Suppose that $\mathcal{S}$ is of rigid type with $s > 2\alpha - 1$. Then every prime divisor of $\Delta$ must divide $s + 1$.
Hypothetical cases with $\alpha\leq1000$ {#table}
========================================
The Mathematica code below searches for all parameter sets $(s,t,\alpha)$, $2\leq\alpha\leq1000$, that survive the restrictions from Theorem \[necessary\], Proposition \[Ma2\] (part a), Proposition \[Ma3\] (the part stating that $v$ and $\Delta$ have the same prime divisors) and Proposition \[MLS\]. Note that part a) from Proposition \[Ma2\] is equivalent to the fact that if $v$ has multiple prime factors none of these can appear with exponent $1$. Despite the simplicity these conditions imply a number of other conditions that should be satisfied, such as the integrality of eigenvalues for strongly regular graphs. We are not aware of other necessary conditions that could speed up the search. Given that for a partial geometry of rigid type we have that $\alpha\leq s\leq t$ it follows that $v\geq (\alpha+1)^2$. Hence the exhaustive search for $2\leq\alpha\leq1000$ implies an exhaustive search for hypothetical partial geometries of rigid type on fewer that $1,000,000$ points. Note however that also many larger values of $v$ are partially covered by the search.
This code produces the list of cases shown in Table \[detabel\].
The first case in the table corresponds to the only known existing partial geometry of rigid type: the Van Lint-Schrijver partial geometry.
Looking for possible patterns we observed that the following parameter family satisfies the necessary conditions used to produce the table: $$pg(\alpha^2+\alpha-1, \alpha^3-\alpha-1, \alpha) \text{ for } \alpha = 2^n + 1 \text{ and } n=0,1,2,\hdots$$
This family seemed particularly promising given that for $n=0$ it yields the Van Lint-Schrijver parameters. For $n=1,2,3,4,5,6,7,8, 9$ the family yields case 2, 3, 4, 6, 7, 8, 13, 15 and 18 from the table respectively. However, in Section \[NER\] we will show that unless $n=0$ or $n=1$ no member of this family can give rise to a partial geometry of rigid type.
Proving nonexistence
====================
We describe below three general techniques for proving nonexistence of regular PDS, labeled T. 1, T. 2 and T. 3. The second and third were pioneered in a slightly less elegant form by three of the authors in [@SDWEKZW].
Assume that $D$ is a nontrivial regular $(v,\,k,\,\lambda,\,\mu)$ partial difference set with $\Delta$ a perfect square in an abelian group $G$, where $G$ has at least two distinct prime divisors. Invoke the following result on “sub-partial difference sets” from S.L. Ma (quoted here in the form given in [@MA94b] (Theorem 7.1)):
\[Ma1\] Let D be a nontrivial regular $(v,\,k,\,\lambda,\,\mu)$ partial difference set in an abelian group G. Suppose $\Delta=\delta^2$ is a perfect square. Let $N$ be a subgroup of $G$ such that $\gcd(\left|N\right|, \left|G\right|/\left|N\right|)=1$ and $\left|G\right|/\left|N\right|$ is odd. Let $$\pi:=\gcd(|N|,\;\sqrt{\Delta}) \quad {\mbox and} \quad \theta:=\lfloor\frac{\beta+\pi}{2\pi}\rfloor.$$
Then $D_1=N\,\cap D$ is a (not necessarily non-trivial) regular $(v_1,\,k_1,\,\lambda_1,\,\mu_1)$ partial difference set in $N$ with $$v_1=|N|, \; \beta_1=\lambda_1-\mu_1=\beta-2\theta \pi, \; \Delta_1=\beta_1^2+4(k_1-\mu_1)=\pi^2,$$ and $$k_1=|N\cap D|=\frac{1}{2}\left[ |N| +\beta_1\pm \sqrt{(|N|+\beta_1)^2-(\Delta_1-\beta_1^2)(|N|-1)} \right].$$
As we assumed that $G$ has at least two distinct prime divisors we can always find a subgroup $N$ of $G$ containing a $(v_1,\,k_1,\,\lambda_1,\,\mu_1)$ partial difference set $D_1$ as in the proposition. The first non-existence result follows immediately:
[**T. 1**]{} If $(|N|+\beta_1)^2-(\Delta_1-\beta_1^2)(|N|-1)$ is not a perfect square $D$ cannot exist.
The second non-existence argument is slightly more elaborate and depends on the so-called Local Multiplier Theorem for partial difference sets cited below:
\[lmt\] Let $D$ be a regular $(v,k,\lambda,\mu)$ PDS in the abelian group $G$. Furthermore assume $\Delta$ is a perfect square. Let $g\in G$ be an element of order $r$. Assume $s$ is a positive integer such that $\gcd(s,r)=1$. Then $g\in D$ if and only if $g^s\in D$.
This yields:
[**T. 2**]{} If $|N|=p^m$ for some prime $p$, and $p-1$ does not divide $k_1=|N\cap D|$, then $D$ cannot exist. If $|G/N|=p^m$ for some prime $p$, and $p-1$ does not divide $k-k_1$, then $D$ cannot exist.
Now note that for the difference of two elements of $D$ to belong to $N$ it is necessary and sufficient for these two elements to belong to the same coset of $N$ in $G$. Denote the distinct cosets of $N$ by $N, Nh_1,\hdots, Nh_{\frac{|G|}{|N|}-1}$ for certain elements $h_i\in G$, and set $B_i=|Nh_i\cap D|$, that is, the number of elements of the partial difference set $D$ in the $i$th coset of $N$. Double counting using the basic properties of a partial difference set yields
$$\sum_{i=1}^{\frac{|G|}{|N|}-1} B_i = k-k_1$$
and
$$\sum_{i=1}^{\frac{|G|}{|N|}-1} B_i(B_i - 1) + k_1(k_1 - 1) = k_1\cdot \lambda + (|N|-1-k_1)\cdot \mu .$$
From these two equations we can now compute the variance of the $B_i$ (up to scaling) in terms of the parameters of the PDS:
$$Var=\left( \frac{|G|}{|N|}-1\right) \sum_{i=1}^{\frac{|G|}{|N|}-1} B_i^2 - \left(\sum_{i=1}^{\frac{|G|}{|N|}-1} B_i\right)^2,$$
which has to be non-negative.
[**T. 3**]{} Whenever the above $Var$ is strictly negative we conclude $D$ cannot exists.
Non-existence results {#NER}
=====================
In this section we show that most of the hypothetical parameter sets obtained in Section \[table\] can be excluded using the techniques T.1, T.2 and T.3 described in the previous section.
Cases excluded by T.1
---------------------
In all of the following cases one finds that the value $d=(|N|+\beta_1)^2-(\Delta_1-\beta_1^2)(|N|-1)$ from T.1 is either negative or not a perfect square. In the table below we provide the case number as well as all parameters needed to compute this value. We do not provide the value of $d$ explicitly as this is in most case a number with many digits. However we indicate with “$<0$” if it is strictly negative and with $\not\!\square$ if it is strictly positive but not a perfect square. Note that the value $|N|$ is sufficient to know what subgroup $N$ we consider. Also note that in some cases multiple choices for $N$ are possible but not all necessary give the desired result (this makes it complicated to efficiently incorporate T. 1 in or exhaustive search).
Case $v$ $k$ $\lambda$ $\mu$ $|N|$ $\pi$ $\theta$ $\beta_1$ $d$
------ ------------------------------- ----------- ----------- ---------- ------------------- ------------------ ---------- ----------- ----------------- --
9 $2^4 \cdot 5^5 \cdot 7^3$ 1409520 115010 115920 $2^4 \cdot 5^5$ 50 $-9$ $-10$ $\not\!\square$
12 $3^4 \cdot 5^3 \cdot 7^4$ 2676240 292845 294840 $5^3$ $5^2$ -40 5 $<0$
14 $2^9 \cdot 11^3$ 279004 113916 114444 $ 2^9$ $2^3$ -33 0 $\not\!\square$
16 $2^{11} \cdot 5^5$ 2320560 840080 842160 $2^{11}$ $2^7$ -8 -32 $<0$
17 $2^6 \cdot 3^{10} \cdot 11^3$ 250873920 12489444 12513600 $2^6 \cdot 11^3 $ $2^2 \cdot 11^2$ -25 44 $<0$
21 $2^5 \cdot 5^3 \cdot 31^3$ 66117420 36664010 36710820 $2^5 \cdot 31^3 $ $2 \cdot 31^2$ -12 -682 $<0$
Cases excluded by T.2
---------------------
We have one case in this category. The table follows the notation introduced so far with $k_1=|N\cap D|$.
Case $v$ $k$ $\lambda$ $\mu$ $|N|$ $\pi$ $\theta$ $\beta_1$ $\Delta_1$ $k_1$
------ ------------------ --------- ----------- --------- -------- -------- ---------- ----------- ------------ --------------
20 $5^4 \cdot 11^4$ 3495030 1334465 1335180 $11^4$ $11^2$ -3 11 $11^4$ 8052 or 6600
Since $10$ doesn’t divide $8052$, it follows that necessarily $k_1=6600$, that is, there are $6600$ elements from $N$ in $D$. On the other hand, it then follows that $k-k_1=3,488,430$, which is not divisible by $4$, so this case cannot occur by T. 2.
Cases excluded by T.3
---------------------
We first consider the hypothetical infinite family described at the end of Section \[table\]. We show that a $pg(\alpha^2+\alpha-1, \alpha^3-\alpha-1, \alpha) \text{ for } \alpha = 2^n + 1 \text{ and } n\geq2$ of rigid type cannot exist by proving the following stronger statement:
\[genRigidPDS\_1\] Let $\alpha = 2^n + 1$ and $n \geq 2$. Then there does not exist an [($m^2$, $r(m+1)$, $-m+r^2+3r$, $r^2+r$)]{}-PDS in an abelian group $G$, where $m=(\alpha+1)^2(\alpha-1)$, $r=\alpha^2-1$.
[**Proof.** ]{}With the same notation as before we obtain the following table:
$v$ $k$ $\lambda$ $\mu$ $|N|$ $\pi$ $\theta$ $\beta_1$ $\Delta_1$ $k_1$
------- ---------- ------------- --------- ------------ ----------- --------------------- ----------- ------------ ------------------------
$m^2$ $r(m+1)$ $-m+r^2+3r$ $r^2+r$ $2^{2n+4}$ $2^{n+2}$ $-2^{n-2}-2^{2n-3}$ 0 $2^{2n+4}$ $2^{2n+3} \pm 2^{n+1}$
In the case where $k_1=2^{2n+3} + 2^{n+1}$ we obtain $$Var= -2^{4n-4}(2^n + 2)(7 \cdot 2^n - 2)(2^{3n} +15 \cdot 2^{2n} +9 \cdot 2^{n+2} + 28)<0,$$
and in the case where $k_1=2^{2n+3} - 2^{n+1}$ we obtain $$Var= -2^{2n-4}(2^n + 2)(7 \cdot 2^{2n}-3 \cdot 2^{n+1}-8)(2^{4n} +15 \cdot 2^{3n} +2^{2n+5} + 3 \cdot 2^{n+2} -16)<0.$$ This concludes the proof. [$\Box$]{}
Note that $n\geq 2$ is necessary in the proof of the above theorem as for $n=0$ or $1$ the number of vertices is a prime power. The above theorem excludes existence of a rigid type partial geometry in cases 3, 4, 6, 7, 8, 13, 15 and 18 from the table in Section \[table\].
One other case from the table in Section \[table\] can be excluded using T.3. The table below provides the necessary parameters to carry out the computations:
Case $v$ $k$ $\lambda$ $\mu$ $|N|$ $\pi$ $\theta$ $\beta_1$ $\Delta_1$ $k_1$ $Var$
------ ------------------- -------- ----------- -------- ------------ ------- ---------- ----------- ------------ ----------- -------
10 $2^{12}\cdot 3^6$ 622440 128952 129960 $ 2^{12} $ $2^6$ -8 16 $2^{12}$ 2600 1512 $<0$
Some comments on the open cases
===============================
The four cases from Table \[detabel\] that we could not deal with are:
Case $s$ $t$ $\alpha$ $x$ $v$ $k$ $\lambda$ $\mu$
------ ------ ------ ---------- ----- -------------------- ---------- ----------- ---------
2 11 23 3 2 $2^{10}$ 264 56 72
5 39 39 15 1 $2^{12}$ 1560 584 600
11 272 272 104 1 $3^4 \cdot7^4$ 74256 28287 28392
19 2295 4591 615 2 $2^{14}\cdot 7^4$ 10538640 2821168 2824080
: Remaining open cases.
We first make a simple observation that also appeared in [@SDWpgst2] and [@Leung2]: [*If $D$ is the PDS in the group $G$ arising from a partial geometry of rigid type then $D$ cannot contain an element of order two.*]{} This is easily seen as any element of order two in $D$ would have to fix a line, contradicting the definition of rigid type.
As an immediate consequence of this observation we see that if $(\mathcal{S},G)$ would be a pair of rigid type in any of the three open cases where $v$ is divisible by $2$ then the Sylow-2-group of $G$ cannot be elementary abelian as Proposition \[Ma1\] shows that the Sylow-2-subgroup in these three cases must contain $k_1>0$ elements of $D$.
We also want to note that J. Polhill in [@Polhill] constructed PDS with parameters $(2^{10}, 264, 56, 72)$ in an abelian group which is not elementary abelian. However, closer inspection of these PDS reveals that all of them contain an involution. Hence, these PDS cannot arise from a partial geometry of rigid type. We do want to point out a subtlety here. Although the PDS constructed by Polhill cannot arise from a partial geometry of rigid type, it is theoretically not impossible for the strongly regular graph arising from these PDS to be geometrical, or even hold a partial geometry of rigid type as there could possibly be non-isomorphic PDS yielding the same strongly regular graph.
Using the techniques used to prove non-existence in the previous section we can derive some information on the hypothetical structure of the corresponding partial difference set in the two cases where $v$ is not a prime power. In particular one can derive that in Case 11, the Sylow-7 subgroup of $G$ must contain $1050$ elements of $D$, whereas the Sylow-3 subgroup must contain either $24$ or $60$ elements of $D$. In the same way one can show that the Sylow-2 subgroup in Case 19 would have to contain $2064$ elements of $D$.
Conclusions
===========
Combining the results of our search and the non-existence proofs in Section \[NER\] we obtain:
If $\mathcal{S}$ is a pg$(s,t,\alpha)$ of rigid type with $\alpha\leq1000$ then $\mathcal{S}$ is either the Van Lint - Schrijver partial geometry or $\mathcal{S}$ is a hypothetical geometry with parameters pg$(11,23,3)$ or pg$(39,39,15)$ or pg$(272,272,104)$ or pg$(2295,4591,615)$.
Based on our observations and attempts to construct partial geometries of rigid type in the open cases we propose the following conjecture:
If $\mathcal{S}$ is a pg$(s,t,\alpha)$ of rigid type then the number of points of $\mathcal{S}$ is either a power of $2$ or a power of $3$.
Based on our observations we do not feel confident to make the stronger conjecture that the Van Lint - Schrijver geometry is the unique partial geometry of rigid type.
Acknowledgement {#acknowledgement .unnumbered}
===============
This material is based upon work done while the first author was supported by and serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
J. Bamberg and M. Giudici, Point regular groups of automorphisms of generalised quadrangles, [*J. Combin. Theory Ser. A*]{} [**118**]{}, 1114-1128 (2011)
R. C. Bose, Strongly regular graphs, partial geometries, and partially balanced designs. [*Pacific J. Math*]{}, [**13**]{}, 389-419, (1963)
F. De Clerck, M. Delanote and R. Mathon, Perp-systems and partial geometries, [*Adv. Geom*]{} [**2**]{}, 1-12 (2002)
S. De Winter, Partial geometries pg(s,t,2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry. [*J. Alg. Combin*]{} [**24**]{}, 285–297, (2006)
S. De Winter, K. Thas, Generalized quadrangles with an Abelian Singer group. [*Designs, Codes, Cryptogr.*]{} [**39**]{}, 81–87 (2006)
S. De Winter, E. Kamischke and Z. Wang, Automorphisms of strongly regular graphs with applications to partial difference sets, [*Designs, Codes, Cryptogr.*]{} [**79**]{}, 471–485 (2016)
T. Feng and W.Li, The point regular automorphism groups of the Payne derived quadrangle of $W(q)$, [*arXiv:1908.10659*]{}
D. Ghinelli, Regular groups on generalized quadrangles and nonabelian difference sets with multiplier $-1$. [*Geom. Dedicata*]{} [**41**]{}, 165-174 (1992)
Y. Huang and B. Schmidt, Uniqueness of Some Cyclic Projective Planes, [*Designs, Codes, Cryptogr.*]{} [**50**]{}, 253-266 (2009)
E. Kamischke, [*Benson’s Theorem for Partial Geometries*]{}, Master’s Thesis, Michigan Technological University (2013)
K. Leung, S.L. Ma and B. Schmidt, Proper partial geometries with Singer groups and pseudogeometric partial difference sets, [*J. Combin. Theory Ser. A*]{} [**115**]{}, 147–177 (2008)
S. L. Ma, Partial difference sets, [*Discrete Mathematics*]{} [**52**]{}, 75-89 (1984)
S.L. Ma, A survey of partial difference sets, [*Designs, Codes, Cryptogr.*]{} [**4**]{}, 221-261 (1994)
E. J. Neubert, [*Some results on partial difference sets and partial geometries*]{}, Master’s Thesis, Michigan Technological University(2019)
J. Polhill, Negative Latin square type partial difference sets and amorphic association schemes with Galois rings, [*Journal of Comb. Designs*]{} [**17**]{} 266-282 (2009)
J. Singer, A theorem in finite projective geometry and some applications to number theory, [*Transactions of the American Mathematical Society*]{}, [**43**]{} No. 3, 377–385 (1938)
E. Swartz and G. Tauscheck, Restrictions on parameters of partial difference sets in nonabelian groups, [*arXiv:2006.00353*]{}
J.H. van Lint, A. Schrijver, Construction of strongly regular graphs, two-weight codes, and partial geometries by finite fields, [*Combinatorica*]{} [**1**]{} 63-73 (1981)
S. Yoshiara, A generalized quadrangle with an automorphism group acting regularly on the points, [*European J.Combin.*]{} [**28**]{}, 653-664 (2007)
[^1]: Department of Mathematical Sciences, Michigan Technological University
[^2]: Department of Mathematical Sciences, Michigan Technological University
[^3]: Department of Mathematical Sciences, Michigan Technological University
[^4]: Department of Mathematical Sciences, Michigan Technological University
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abstract: 'We present data for four ultra-Li-deficient, warm, halo stars. The Li deficiency of two of these is a new discovery. Three of the four stars have effective temperatures $T_{\rm eff}~\sim~6300$ K, in contrast to previously known Li-deficient halo stars which spanned the temperature range of the Spite Plateau. In this paper we propose that these, and previously known ultra-Li-deficient halo stars, may have had their surface lithium abundances reduced by the same mechanism as produces halo field blue stragglers. Even though these stars have yet to reveal themselves as blue stragglers, they might be regarded as “blue-stragglers-to-be.” In our proposed scenario, the surface abundance of Li in these stars could be destroyed (a) during the normal pre-main-sequence single star evolution of their low mass precursors, (b) during the post-main-sequence evolution of a evolved mass donor, and/or (c) via mixing during a mass-transfer event or stellar merger. The warmest Li-deficient stars at the turnoff would be regarded as emerging “canonical” blue stragglers, whereas cooler ones represent sub-turnoff-mass “blue-stragglers-to-be.” The latter are presently hidden on the main sequence, Li depletion being possibly the clearest signature of their past history and future significance. Eventually, the main sequence turnoff will reach down to their mass, exposing those Li-depleted stars as canonical blue stragglers when normal stars of that mass evolve away. Arguing [*against*]{} this unified view is the observation that the three Li-depleted stars at $T_{\rm eff}~\simeq~6300$ K are [*all*]{} binaries, whereas very few of the cooler systems show evidence for binarity; it is thus possible that two separate mechanisms are responsible for the production of Li-deficient main-sequence halo stars.'
author:
- 'Sean G. Ryan'
- 'Timothy C. Beers'
- Toshitaka Kajino
- Katarina Rosolankova
title: 'Ultra-Lithium-Deficient Halo Stars and Blue Stragglers: A Common Origin?[^1]'
---
Introduction
============
$^7$Li is destroyed in stellar interiors where temperatures exceed $2.5\times 10^6$ K, and Li-depleted material can in principle reach the stellar surfaces where it can be observed. Thus, if one is to infer pre-stellar $^7$Li abundances from current-epoch observations, it is important to understand the stellar processing of this species. It has widely, though not universally, been supposed that warm ($T_{\rm
eff}~>~5700$ K), metal-poor (\[Fe/H\] $<~-1$) stars retain their pre-stellar abundances (Spite & Spite 1982; Bonifacio & Molaro et al. 1997; but see also Deliyannis 1995; Ryan et al. 1996). Although claims had been made of an intrinsic spread in the Li abundances by 0.04 – 0.1 dex (Deliyannis, Pinsonneault, & Duncan 1993; Thorburn 1994), Ryan, Norris & Beers (1999) attributed these to an embedded $A$(Li) vs \[Fe/H\] dependence , and underestimated errors, respectively. Ryan et al. (1999) set tight limits on the intrinsic spread of $^7$Li in metal-poor field stars as essentially zero, stated conservatively as $\sigma_{\rm int} < 0.02$ dex. However, the subset of ultra-Li-deficient stars identified by Spite, Maillard, & Spite (1984), Hobbs & Mathieu (1991), Hobbs, Welty & Thorburn (1991), Thorburn (1992), and Spite et al. (1993) stands out as a particular exceptional counter-example to the general result. These stars have only upper limits on their $^7$Li abundances, typically 0.5 dex or more below otherwise similar stars of the same $T_{\rm
eff}$ and metallicity. Detailed studies of other elements in these objects have revealed some chemical anomalies, but none common to all, or which might explain [*why*]{} their Li abundances differ so clearly from those of otherwise similar stars (Norris et al. 1997a; Ryan, Norris & Beers 1998).
In contrast to the situation for Population II stars, a wider range of Li behaviors is seen in Population I. In addition to a stronger increase with metallicity, thought to be due to the greater production of Li in later phases of Galactic chemical evolution (Ryan et al. 2001), there is also substantial evidence of Li depletion in certain temperature ranges. Open cluster observations, for example, show steep dependences on temperature for $T_{\rm
eff} \la 6000$ K (e.g., Hobbs & Pilachowski 1988) and in the region of the F-star Li gap (6400 K $<~T_{\rm eff}~<$ 7000 K; Boesgaard & Tripicco 1986). More problematic, for the young cluster $\alpha$ Per (age 50 Myr) and the Pleiades (age 100 Myr), is the presence of a large apparent Li spread even at a given mass. Various explanations have been proposed involving mixing in addition to that due to convection. Extra mixing processes include rotationally-induced mixing (e.g., Chaboyer, Demarque & Pinsonneault 1995), structural changes associated with rapid rotation (Martín & Claret 1996), and different degrees of suppression of mixing by dynamo-induced magnetic fields (Ventura et al. 1998). Gravity waves have been proposed as yet another different mixing mechanism (Schatzman 1993; Montalbán & Schatzman 1996). Consensus has not yet emerged concerning the range of possible mechanisms, or the relative importance of each. Jeffries (1999) even questions the reality of a Li abundance spread in low mass Pleiades stars, due to a similar spread being seen in the resonance line. Amongst older open clusters, the spread at a given effective temperature is generally much less, though M67 (Jones, Fischer, & Soderblom 1999) is an exception. A class of stars with higher lithium abundances than otherwise similar stars is short-period tidally-locked binaries (Deliyannis et al. 1994; Ryan & Deliyannis 1995) which give credence to the view that physics related to stellar rotation can and does influence the evolution of Li in approximately solar-mass stars.
The fraction of warm, metal-poor stars that fall in the ultra-Li-deficient category has previously been estimated at approximately 5% (Thorburn 1994). However, recent measurements of Li in a sample of 18 warm ($T_{\rm
eff}~\ga~6000$ K), metal-poor ($-2~\la~$\[Fe/H\] $\la~-1$) stars yielded four ultra-Li-deficient objects, i.e. more than 20% of the sample (Ryan et al. 2001). The Poisson probability of a 5% population yielding 4 or more objects in a sample of this size is just 0.013. Clearly, the selection criteria for this sample have opened up a regime rich in ultra-Li-poor stars. We now examine those criteria, and discuss the implications for the origin of such systems and for our understanding of Li-poor and Li-normal stars.
We note some similarities between Li-deficient halo stars and blue stragglers. Although these two groups have traditionally been separated due to the different circumstances of their [*discovery*]{}, we question whether there is a reliable [*astrophysical*]{} basis for this separation. One must ask whether the process(es) that gives rise to blue stragglers is capable only of producing stars whose mass is greater than that of the main sequence turnoff of a $\sim$13 Gyr old population. If, as we think is reasonable, the answer is “no”, then one may ask what the sub-turnoff mass products of this process(es) would be. Our proposal is that they would be Li-deficient, but otherwise difficult to distinguish from the general population, and in this regard very similar to the ultra-Li-deficient halo stars.
Observations of the Ultra-Li-Poor Halo Stars
============================================
The ultra-Li-poor halo stars we consider were identified serendipitously in a study of predominantly high proper-motion halo stars having $T_{\rm
eff}~^>_\sim~6000$ K and $-2~^<_\sim$ \[Fe/H\] $^<_\sim~-1$, and are listed in Table 1(a). Details of the sample selection and abundance analysis are given by Ryan et al. (2001); the key points are that high resolving power ($\lambda/\Delta\lambda~\simeq~50000$) échelle spectra were obtained, equivalent widths were measured, and abundances were computed using a model atmosphere spectrum-synthesis approach. Two of the Li-poor stars were subsequently found to have previous Li measurements; Wolf 550 was identified as G66-30, and G202-65 had been observed by Hobbs & Mathieu (1991) in a study targeted at blue stragglers. The new spectra of the four stars, plus one with normal Li for comparison, are shown in Figure 1. The comparison star, CD$-31^\circ$305, has $T_{\rm eff}~=~5970$ K, \[Fe/H\] = $-1.0$, and $A$(Li) = 2.24 (Ryan et al. 2001). For convenience, previously known Li-depleted halo stars are listed in Table 1(b). The full sample of Ryan et al. (2001) is plotted in Figure 2, along with additional stars from the literature.
It is immediately apparent that three of the four ultra-Li-deficient stars are amongst the hottest in our sample, though not [*the*]{} hottest in the figure. It seems likely that high temperature is one biasing characteristic of these objects. The stars with $T_{\rm eff}~>~6300$ K and [*normal*]{} Li abundances are listed in Table 1(c). These have had comparatively high values of de-reddening applied, and it is possible that they are in reality cooler than Table 1 shows. An indication that high temperature is not the [*only*]{} biasing characteristic of ultra-Li-poor stars is that the Ryan et al. (1999) study of 23 very metal-poor ($-3.5~^<_\sim$ \[Fe/H\] $^<_\sim~-2.5$) stars in the same temperature range included only one ultra-Li-deficient star, G186-26. This rate, 1 in 23, is consistent with previous estimates for Population II stars as a whole. However, very few relatively metal-rich ($-2~^<_\sim $ \[Fe/H\] $^<_\sim~-1$) halo stars in this temperature range had been studied previously, so earlier works may have been biased against discovering ultra-Li-poor objects. It appears, then, that the fraction of ultra-Li-deficient stars is higher as metallicity increases. This may explain why our study, which targeted stars in the higher metallicity range [*and*]{} with $T_{\rm eff}~>~6000$ K, was so successful at yielding ultra-Li-deficient stars. Figure 3 shows the distribution of objects in the $T_{\rm eff}$, \[Fe/H\] plane. Whereas previously no region of parameter space stood out as “preferred” by Li-deficient stars, the objects are now more conspicuous as a result of their high temperatures and relatively high metallicities.
Also shown in Figure 3 are the $T_{\rm eff}$ of the main-sequence turnoff as a function of metallicity, for 14 and 18 Gyr isochrones. The isochrones shown are the oxygen-enhanced curves of Bergbusch & VandenBerg (1992; solid curves; Y=0.235), and, for comparison, the Revised Yale Isochrones of Green, Demarque & King (1987; dotted curves; Y=0.24). Clearly there is disagreement of $\simeq$4 Gyr between the two sets as to the ages that would be assigned to these stars, and there are uncertainties in the color-$T_{\rm eff}$ transformations that have been applied to the observed data. However, these difficulties are not the issue here. Rather, we use the isochrones to indicate the [*shape*]{} of the turnoff locus in the $T_{\rm eff}$ vs \[Fe/H\] plane, and on that point the four isochrones are in overall agreement. They emphasise that even though HD 97916 is cooler than five other Li-depleted stars in the study, it is nevertheless close to the turnoff. That is, a star with $T_{\rm eff}$ = 6100 K would appear below the turnoff if \[Fe/H\] = $-3$, but will be close to the turnoff if \[Fe/H\] = $-1$. Even excluding the definite blue straggler BD+25$^\circ$1981, there are four Li-depleted stars amongst the eight whose symbols lie above or touch the 14 Gyr Revised Yale Isochrone. Clearly, all of these are very close to the turnoff once their metallicities are taken into account.[^2] Besides these Li-depleted stars close to the turnoff, four are 100–200 K cooler than the turnoff. We discuss later in this paper whether the these two groupings might have different origins.
Traditional blue stragglers
===========================
Blue stragglers are recognised observationally as stars that are considerably bluer than the main-sequence turnoff of the population to which they belong, but having a luminosity consistent with main-sequence membership. Such objects were originally identified in globular clusters (e.g., M3; Sandage 1953), but are also known in the field (e.g., Carney & Peterson 1981), and in Population I as well as Population II (e.g., Leonard 1989; Stryker 1993). Their origin is not known with certainty, and it is possible that more than one mechanism is responsible for their presence. A range of explanations was examined by Leonard (1989), but the discovery of Li destruction in blue stragglers in the halo field and the open cluster M67 led Hobbs & Mathieu (1991) and Pritchet & Glaspey (1991) to conclude that “virtually all mechanisms for the production of blue stragglers [*other than*]{} mixing, binary mass transfer, or binary coalescence appear to be ruled out ... .” As Hobbs & Mathieu emphasized, internal mixing alone is also ruled out; mixing out to the surface is required.
Recent advances in high-resolution imaging have verified that the blue straggler fractions in at least some globular clusters are higher in their cores, strongly supporting the view that some blue stragglers are formed through stellar collisions, probably involving the coalescence of binary stars formed and/or hardened through exchanges, in these dense stellar environments (e.g., Ferraro et al. 1999). However, it is neither established nor required that a single mechanism will explain all blue stragglers, and it is unclear how the field examples and those in the tenuous dwarf galaxy Ursa Minor (Feltzing 2000, priv. comm.) relate to those in the dense cores of globular clusters. Probably even the halo field and dwarf spheroidal stars formed in clusters of some description (since the formation of stars in isolation is unlikely), but one should not be too quick to link the properties of surviving globular clusters to diffuse populations. This view is supported by Preston & Sneden’s (2000) conclusion that at more than half (62% – 100%) of their field blue metal-poor binaries are blue stragglers formed by mass transfer rather than mergers, due to the long orbital periods and low eccentricities of the field systems they observed. Their conclusion is entirely consistent with the views of Ferraro et al. (1995), who ascribed blue straggler formation to interactions [*between*]{} systems in high-density environments, but [*within*]{} systems (primordial binaries) in lower-density clusters. In contrast to but not contradicting Preston & Sneden’s result for field systems, Mateo et al. (1990) argue that all of the blue stragglers in the globular cluster NGC 5466 are the result of close binary mergers.
The mechanism for Li destruction in field blue stragglers is not known. It is unclear what degree of mixing will occur as a result of coalescence. Early work by Webbink (1976) suggested substantial mixing would occur, whereas more recent simulations of head-on collisions by Sills et al. (1997), and grazing collisions and binary mergers by Sandquist, Bolte, & Hernquist (1997), have suggested otherwise. Sills, Bailyn & Demarque (1995) argue, however, that to account for the blue stragglers observed in NGC 6397, mixing is nevertheless required (unless the collision products have more than twice the turnoff mass), and may occur after the initial coalescence. This is perhaps consistent with the result of Lombardi, Rasio, & Shapiro (1996) that some mixing could occur as a merger remnant re-contracts to the main sequence. Due to the fragility of Li, if some mixing of surface material does occur during the coalescence it will at least dilute, and possibly also destroy, any lithium remaining in the stars’ thin convective surface zones up to that time. One might suppose that mass transfer in a detached system also destroys Li, though one could also imagine gentle mass-transfer processes where the rate is slow enough that the original envelope is not subjected to additional mixing, and where the transferred matter itself does not undergo additional Li-destruction. Of course, mass transfer via Roche lobe overflow in a detached system, or wind accretion from a more distant companion, involve mass from an evolved star which may have [*already*]{} depleted its surface Li due to single-star evolutionary processes. Consequently, the mass transferred may be already devoid of Li, as in the scenario quantified by Norris et al. (1997a).
We also note the possibility that the accretor in a mass-transfer system, or the progenitors of a coalescence, was (were) devoid of Li prior to that event. Li is (normally) preserved in halo stars only over the temperature range from the turnoff ($T_{\rm eff}~\simeq~6300$ K) to about $T_{\rm eff}~\simeq~5600$ K, corresponding to a mass range from 0.80 down to 0.70 $M_\odot$. Therefore it is likely that any mass accretor, and certain that any merger remnant, now seen in this mass range began life as one (or two) stars with initial mass(es) less than 0.70 $M_\odot$ and had already destroyed Li normally, as lower-mass stars are known to do, prior to mass exchange. In such a scenario, it is not [*necessary*]{} for any Li to have been destroyed as a result of the blue-straggler formation process itself, though this could occur as well.
Discussion
==========
In view of the distributions of the ultra-Li-deficient stars in the $T_{\rm eff}$, \[Fe/H\] plane, with four at the turnoff and four 100–200 K cooler, we consider whether all represent the same phenomenon, or the possibility that two distinct processes have been in operation. It is not a trivial matter to answer this question, because we do not know with certainty what mechanism(s) has affected any of the stars. However, we explore a number of possibilities in the discussion that follows. Ignoring again the obvious blue straggler BD+25$^\circ$1981, of the 111 stars shown in Figure 3, 8 are ultra-Li-deficient. If all ultra-Li-poor stars have the same origin, then we should begin by restating the frequency of such Li-weak objects as $\simeq$7% of plateau stars rather than $\simeq$5% as estimated previously when the parameter space was incompletely sampled, and with strong metallicity and temperature dependences in that fraction.
Do Ultra-Li-Deficient Stars and Field Blue Stragglers Share a Common Origin?
----------------------------------------------------------------------------
Historically, blue stragglers and ultra-Li-deficient stars have been regarded as separate phenomena. However, we have been driven to consider whether there is any astrophysical basis for this separation. One must ask whether the process(es) that gives rise to blue stragglers is capable only of producing stars whose mass is greater than that of the main sequence turnoff of a $\sim$13 Gyr old population. If, as we think is reasonable, the answer is “no”, then one may ask what the sub-turnoff mass products of this process(es) would be. Our proposal is that they would be Li-deficient, but otherwise difficult to distinguish from the general population.[^3]
For ultra-Li-poor stars redder than the main sequence turnoff, Hipparcos parallaxes have established that G186-26 is on the main sequence rather than on the subgiant branch. Of those [*at*]{} the turnoff, Wolf 550, G202-65, and BD+51$^\circ$1817 also have Hipparcos parallaxes; two are almost certainly dwarfs, while G202-65 is subject to larger uncertainties and may be more evolved (see Ryan et al. 2001, Table 2). The argument that the evolutionary rate of subgiants is too rapid to explain the high frequency of observed Li-deficient objects, which persuaded Norris et al. (1997a) to reject the proposition that they might be the [*redward*]{}-evolving (post-turnoff) progeny of blue-stragglers, is therefore redundant. However, the detection of several Li-weak stars at the bluest edge of the colour distribution has prompted us to re-examine their possible association with blue stragglers.
We would describe G202-65 as “at” the turnoff rather than classify it as a blue straggler in the conventional sense, as it is only marginally hotter (bluer) than the main sequence turnoff for its metallicity (see Figure 3). Hobbs & Mathieu, on the other hand, classified it as a blue straggler, based presumably on the photometry of Laird, Carney & Latham (1988) which they referenced. (Indeed, Carney et al (1994) declare it as a “blue straggler candidate”, and Carney et al. (2000) treat it as one, though acknowledging at the same time that some normal stars may be included in this classification.) Our purpose is [*not*]{} to debate how this star should be classified, but rather to underline the main suggestion of our work, that the blue straggler and halo ultra-Li-deficient stars may have a common origin. Although blue stragglers have historically been recognised because they are bluer than the main-sequence turnoff, it is essential to remember that stars that have accreted mass from a companion, or that result from a coalescence can have a mass less than the current turnoff. Such stars would be expected to share many of the properties of blue stragglers, but would not [*yet*]{} appear bluer than the turnoff. However, at some future time, once the main-sequence turnoff reaches lower masses, these non-standard objects would lag the evolution of normal stars and hence appear bluer, showing canonical straggling behaviour. Therefore, such stars might, for the present, be regarded as “blue-stragglers-to-be,”[^4] and our speculation is that the ultra-Li-deficient halo stars in are in fact members of such a population. Note that this proposition is distinct from that of [*redward*]{}-evolving systems considered and rejected by Norris et al. (1997a).
If ultra-Li-deficient stars and blue stragglers are manifestations of the same process, then Li deficiency may be the only way of distinguishing sub-turnoff-mass blue-stragglers-to-be from normal main-sequence stars, prior to their becoming classical blue stragglers. Mass transfer during their formation may also help clarify some of the unusual element abundances found by Norris et al. (1997a; see also Ryan et al. 1998). Whereas an appeal to extra mixing (in a single-star framework) to explain the Li depletion would not necessarily affect other elements, mass transfer in a binary with an AGB donor may be capable of altering s-process abundances as well. In this regard, we recall that two of the ultra-Li-deficient stars studied by Norris et al. (1997a; also Ryan et al. 1998) had non-standard Sr and Ba abundances. Mass transfer from an RGB donor would presumably leave a different chemical signature.[^5]
Some constraints on the progenitors of the Li-deficient stars may be obtained from their rotation rates and radial velocity variations. Webbink’s (1976) calculations of a coalesced star ($M_{\rm total}~=~1.85~M_\odot$) show that a high rotation rate is maintained at least until it reaches the giant branch. In contrast, previously known blue stragglers appear not to have uncommonly high rotation rates (e.g., Carney & Peterson 1981; Pritchet & Glaspey 1991). This tends to argue against the blue stragglers as having originated from coalesced main-sequence contact binaries, and points towards one of the other binary mass-transfer scenarios, unless mass loss (e.g., via Webbink’s excretion disk) and magnetic breaking can dissipate envelope angular momentum during the main sequence lifetime of a coalesced star. To spin down, stars must have a way of losing surface angular momentum. In single stars, most of this is believed to occur during the pre- and early-main-sequence phase when magnetic coupling of the stellar surface to surrounding dust creates a decelerating torque on the star. It is not clear that two mature stars which merge will still have this coupling, because of the much lower mass loss rates beyond the early stages of evolution (unless they produce an excretion disk) and lower magnetic field strengths. (See also discussion by Sills et al. 1997, §5.5.) Leonard & Livio (1995) have proposed that the merger product acquires the distended form of a pre-main-sequence-like star which then spins down as it again approaches the main sequence, losing angular momentum in much the same way as conventional pre-main-sequence stars. [^6]
For the four stars observed in this work, three had previous radial velocity measurements accurate to $\simeq$ 1 km s$^{-1}$ (Carney et al. 1994). The new measurements (Ryan et al. 2001; Table 2) showed residuals of +1.0 (BD+51$^\circ$1817), $-$3.3 (G202-65), and $-$6.9 km s$^{-1}$ (Wolf 550); compared with the expected radial velocity accuracy of $\sigma_v$ = 0.3–0.7 km s$^{-1}$, these are consistent with significant motion. Carney et al. (2000) indicate periods of 168 to 694 days for these systems, and low eccentricities, except for Wolf 550 ($e$ = 0.3). Similarly, the metal-poor field blue straggler CS 22966-043 has an orbital period of 319 days (Preston & Landolt 1999). If the brighter component has a mass of 0.8 M$_\odot$ and its companion has a mass between 0.4 and 1.4 M$_\odot$ (appropriate to a white dwarf) then the [*current*]{} semi-major axis of the system will be in the range $a$ = 200–260 R$_\odot$ (from Kepler’s Third Law).[^7] Their second system, CS 29499-057, may have an even longer period of 2750 days, implying $a$ = 900–1100 R$_\odot$. The periods of these and Carney et al’s systems, and hence their large current separations, are more compatible with mass loss from an evolved companion rather than being short-period systems in contact on the main sequence.
The evidence presented to date has argued against internal mixing alone as an adequate explanation for the ultra-Li-deficient stars whose neutron-capture elements show abundance anomalies. Note, though, that certainly not all ultra-Li-deficient stars and blue stragglers exhibit neutron-capture element anomalies (Carney & Peterson 1981; Norris et al. 1997a; Ryan et al. 1998). If mass transfer has occurred, systems in which s-process elements are abnormal would presumably indicate material originating with an AGB companion, whereas s-process-normal remnants would indicate mass transfer during an earlier stage of evolution (RGB) or from a pre-thermal-pulsing AGB mass donor. (We have no data on the N abundance, and the CH band in these stars is too weak to hope to measure the $^{12}$C/$^{13}$C ratio.) Likewise, the rotation rates of both blue stragglers and ultra-Li-deficient stars are apparently normal, arguing against coalescences having already occurred on the main sequence. Of the three mechanisms found to be viable by Pritchet & Glaspey (1991) and Hobbs & Mathieu (1991), this leaves mass transfer from a companion as the only one remaining, [*if*]{} we are correct in speculating that the ultra-Li-deficient and blue straggler phenomena are manifestations of the same process.
The Hot Stars in Isolation
--------------------------
In the absence of an adequate theory for why eight otherwise-normal halo stars (excluding the traditional blue straggler BD+21$^\circ$1981) should have low (zero?) Li abundances, it may be useful to consider the hot subsample (6200 K $^<_\sim$ $T_{\rm eff}$ $^<_\sim$ 6300 K) as a distinct group. Several possibilities then arise that might account for the observed Li deficiency, including diffusion (the sinking of Li to below the photosphere), the F-star Li dip, and an unknown process that may be responsible for depletion in some (but not all) disk stars. We consider each of these in turn. We note that the three Li-deficient stars with $T_{\rm eff}~\simeq~6300$ K are confirmed binaries, whereas most cooler ones show no evidence of binary motion. The binary/single distinction between warmer/cooler Li-depleted stars is pronounced; see Table 1, where the binary status (Carney et al. 1994, 2000; Latham 2000, priv.comm.) is given in the final column. If such a dichotomy is maintained as more Li-poor systems are discovered, it may indicate a genuine difference in the origin of the turnoff and sub-turnoff systems.
### Diffusion
Deliyannis, Demarque & Kawaler (1990) and Proffitt & Michaud (1991) have computed the predicted effects of diffusion on the surface Li abundances of warm halo stars. Diffusion is more significant in hotter stars because their surface convective zone is thinner. The degree of depletion expected at $T_{\rm eff}~\sim~6300$ K is a function of effective temperature, changing by $\simeq$ 0.2 dex per 100 K in the former (for $\alpha$ = 1.1), and $\simeq$ 0.2 and $>~0.2$ dex per 100 K in the latter (for $\alpha$ = 1.7 and 1.5 respectively). This does not match the behavior observed (see Figure 2). For comparison, our ultra-Li-poor stars are depleted by $^>_\sim0.8$ dex. This alone appears to rule out diffusion as the explanation, except possibly for the lower-$\alpha$ model of Proffitt & Michaud. However, Li diffusion appears to have been inhibited in all other metal-poor samples (e.g., Ryan et al. 1996), so it would be unusual to see it suddenly present and with such effect only in isolated stars in our new sample.
### The F-Star Li Dip
Boesgaard & Tripicco (1986) and Hobbs & Pilachowski (1988) showed that Li is severely depleted in Population I open cluster stars over the interval 6400 K $<~T_{\rm eff}~<~7000$ K. Various explanations have been proposed, including mass loss (e.g., Schramm, Steigman, & Dearborn 1990), diffusion (e.g., Turcotte, Richer & Michaud 1998), and slow mixing of various forms (e.g., Deliyannis & Pinsonneault 1997), but none has been convincingly established as responsible, and several mechanisms may be acting in concert (e.g., Turcotte et al.). Whatever the correct explanation(s), is it possible that the hottest ultra-Li-deficient stars are encroaching on this regime and are affected by this phenomenon? Although this cannot be ruled out completely for the hot subset, especially since we have questioned the reliability of the $E(B-V)$ (and hence $T_{\rm eff}$) values of the hottest Li-preserving stars in Figure 2, the onset of destruction in the F-star dip seems too gradual with $T_{\rm eff}$ to explain the new data. The Hyades observations (Boesgaard & Tripicco 1986) show a decrease of only 0.3 dex from 6200 to 6400 K, substantially less than the $^>_\sim 0.8$ dex deficit in the ultra-metal-poor objects around 6300 K.[^8] As noted above, Hipparcos parallaxes are available for five of the eight known ultra-Li-deficient stars and, with the possible exception of G202-65, rule out the possibility that these stars are redward-evolving [*descendants*]{} of the Li-dip.
Anomalously-Li-Depleted Disk Stars
----------------------------------
Lambert, Heath & Edvardsson (1991) found that, in almost all cases, the low Li abundances in their Population I sample could be ascribed to their being evolved descendants of Li-dip stars, or else being dwarfs exhibiting the Li-depletion that increases towards [*lower*]{} temperature, as is normally associated with pre-main-sequence and/or main-sequence burning. Anomalously high Li depletions were found in only 1–3 cases out of some 26 old-disk stars, and for a similar fraction of young-disk stars. Based on this fraction, Lambert et al. proposed that a new class of highly Li-depleted stars, comprising less than about 10% of the population, might exist. It is interesting to note that this proposal pre-dated the discovery of ultra-Li-deficient halo dwarfs.
The uncertain number of cases stated above arises because Lambert et al. recognised that uncertainties in the stellar luminosities, and hence mass, could drive stars into or out of the region of importance. We now have the benefit of accurate Hipparcos parallaxes. These indicate that two of the seven stars highlighted by their study, HD 219476 and HR 4285, are indeed considerably more massive than reported in Lambert et al.’s tables and hence are probably descendants of the Li gap, thus reducing the number of [*genuine*]{} cases to 2 out of 26 old-disk stars, and 3 out of a similar number of young-disk stars. That is, the fraction of anomalously Li-depleted stars appears to be around 8-10%, albeit sensitive to small-number statistics. [^9] Ultra-Li-depleted Population I stars are also seen in young open clusters. They can be recognised, for example, in Fig. 1 of Ryan & Deliyannis (1995), where $\simeq$6% of the Hyades stars cooler than the F-star dip appear to be ultra-Li-deficient.
Is it possible that the Li-depleted halo stars are of the same type? The lack of examples in the two Pop I and Pop II classes to compare with precludes a detailed analysis, but we note that we see Li deficiency in about 7% of halo objects, which is comparable to the ratio for the Pop I objects. That is, the Pop I and Pop II examples could arise due to the same process, even though it remains unclear what that process is. We note, for completeness, that Ryan et al. (2001) showed that the kinematics of the new ultra-Li-depleted stars are clearly those of halo objects, and thus they genuinely belong to the halo Population, despite their metallicities being close to those of the most metal-poor thick-disk stars.
The stars remaining on Lambert et al’s list of unusually Li-deficient objects are: HR 3648, HR 4657, HR 5968, HR 6541, and HD 30649. Upon searching the literature for evidence of binarity or abundance anomalies in these systems, we found that not only was HR 4657 a 850 day period binary, but Fuhrmann & Bernkopf (1999) had also been driven to consider this star as a blue straggler. It has an unexpectedly high rotational velocity (in contrast to the blue stragglers studied by Carney & Peterson 1981). There is no evidence of s-process anomalies, but other unusual characteristics of the system include an observable soft X-ray flux and the very likely association of this object with GRB 930131. HR 3648 (= 16 UMa = HD 79028) is a 16.2 day period chromospherically-active single-lined spectroscopic binary (Basri, Laurent, & Walter 1985). HD 30649 (= G81-38) and HR 6541 (=HD 159332), in contrast, show no significant evidence of binarity (Carney et al. 1994). HR 5968 (= $\rho$ CrB) does not appear to have a stellar companion, though it has a planetary companion (Noyes et al. 1997), but Ryan (2000) argues that Li in this star is [*not*]{} anomalous. HR 3648 and HR 4657 have Ba abundance measurements from the study by Chen et al. (2000). The latter also has been observed by Fuhrmann & Bernkopf (1999), but neither star appears abnormal in this element.
Implications and Summary
========================
Ryan et al. (1999) have argued that the ultra-Li-deficient halo stars are distinct from the majority of halo stars that occupy the Spite plateau, and, in particular, that they do [*not*]{} merely represent the most extreme examples of a [*continuum*]{} of Li depletion. If the association with blue stragglers (or, for that matter, any distinct evolutionary phenomenon) is correct, then the mechanism for their unusual abundances will at last be understood and they will be able to be neglected with certainty from future discussion of the Spite plateau.
In the present work, we have proposed and discussed the possibility that ultra-Li-depleted halo stars and blue stragglers are manifestations of the same phenomenon, and described the former as “blue-stragglers-to-be.” We proposed that their Li was destroyed either during the formation process of blue stragglers or during the [*normal*]{} single-star evolutionary processes of their precursors, namely during pre-main-sequence and/or main-sequence phases of low-mass stars, or during post-main-sequence evolution of mass donors, as in the scenario quantified by Norris et al. (1997a). We note that in a study carried out separately but over the same time period as ours, Carney et al. (2000) have examined the orbital characteristics of blue stragglers, and have been driven towards similar considerations as we have. There are clearly still details to be clarified, but our two groups appear to be converging on a view unifying blue stragglers and ultra-Li-deficient systems.
Because there are numerous observational and theoretical issues surrounding this unified view, we seek to clarify the main arguments and possibilities using an itemised summary.
Observations: $\bullet$ In a study of 18 halo stars with $-2~^<_\sim$ \[Fe/H\] $^<_\sim~-1$ and 6000 K $^<_\sim~T_{\rm eff}~^<_\sim$ 6400 K, we have found four ultra-Li-deficient objects, i.e. a 22% detection rate. $\bullet$ The fraction of ultra-Li-deficient stars is very much higher amongst the hottest and most metal-rich halo main-sequence stars ($\simeq$20%) than amongst cooler and more metal-poor ones ($\simeq$5%). $\bullet$ Ultra Li-deficient stars exist both at the turnoff, and cooler than the turnoff, and with well-determined main-sequence luminosities from Hipparcos. $\bullet$ All of the turnoff ultra-Li-deficient halo stars, but none of the sub-turnoff ultra-Li-deficient halo stars, appear to be binaries. This may indicate that two different mechanisms are causing the halo ultra-Li-deficient phenomenon.
Theoretical framework: $\bullet$ Blue stragglers may form from [*several*]{} mechanisms, but seem to require at least one of either complete mixing, binary mass transfer, or coalescence[^10] (Hobbs & Mathieu 1991; Pritchet & Glaspey 1991).
Origins: $\bullet$ We speculate that ultra-Li-deficient stars and blue stragglers are manifestations of the same process, and that sub-turnoff-mass ultra-Li-deficient stars may be regarded as “blue-stragglers-to-be.” $\bullet$ Li could be destroyed at several stages: (i) in a mass-transfer event which induces extensive mixing; (ii) by single-star evolutionary processes (convective mixing) in a post-main-sequence mass donor; (iii) by single-star evolutionary processes (mixing) in pre-main-sequence (or possibly main-sequence) low-mass stars prior to their gaining mass. $\bullet$ Mass-transfer scenarios from an AGB star seem better able to explain the unusual neutron-capture element ratios [*sometimes*]{} seen in ultra-Li-depleted stars (Norris et al. 1997a) than internal mixing, since $\simeq$ 0.8 M$_\odot$ core-hydrogen-burning stars are not expected to process neutron-capture elements. This argues against internal mixing as the sole explanation for the existence of ultra-Li-depleted stars with unusual neutron-capture abundances. (Mass transfer from pre-AGB (most likely RGB) donors would produce the stars with normal neutron-capture abundances.) $\bullet$ Coalesced binaries are expected to maintain high rotation rates until they reach the giant branch, but neither blue stragglers nor ultra-Li-depleted halo stars have high rotation rates. This argues against coalescence of a binary as the explanation for these objects unless they have spun down. $\bullet$ The orbital periods of metal-poor field blue stragglers (Preston & Landolt 1999; Carney et al. 2000) suggest current semi-major axes in the range 200–1100 R$_\odot$, arguing against these being coalescing stars (unless they began their lives as triple systems). $\bullet$ The arguments against solely internal mixing, and against coalescence of main-sequence contact binaries, leaves mass transfer as the most viable mechanism for field binaries. This is [*not*]{} to say that Li was destroyed during the transfer; it may have been destroyed by single-star mechanisms already. $\bullet$ The observed d$A$(Li)/d$T_{\rm eff}$ is too steep compared with models of diffusion to be due to that process. $\bullet$ The observed d$A$(Li)/d$T_{\rm eff}$ is too steep compared with the Hyades data to be due to the F-star Li dip. $\bullet$ The halo ultra-Li-deficient stars could be related to the Pop I anomalously-Li-depleted stars identified in the field by Lambert et al. (1991) and also seen in open clusters. $\bullet$ Hipparcos parallaxes rule out the possibility that the ultra-Li-deficient stars are redward-evolving post-turnoff stars. They have not descended from the F-star Li dip.
Implications: $\bullet$ Severe Li depletion may be the (only?) signature of sub-turnoff-mass blue stragglers. The halo population fraction comprising ultra-Li-poor stars is 7%. $\bullet$ Understanding the ultra-Li-depleted stars as resulting from a distinct process (not normally affecting single stars) would eliminate the need to include them in discussions of processes affecting the evolution of normal Spite plateau stars, and would explain why they appear so radically different from the vast majority of halo stars (Ryan et al. 1999).
Acknowledgements
================
The authors gratefully acknowledge the support for this project given by the Australian Time Assignment Committee (ATAC) and Panel for the Allocation of Telescope Time (PATT) of the AAT and WHT respectively, and for practical support given by the staff of these facilities. They also express gratitude to D. A. Latham and B. W. Carney for conveying the results of their program in advance of publication, and to an anonymous referee for his/her comments that helped us clarify our arguments. S.G.R. sends a special thanks to colleagues at the University of Victoria: to C. J. Pritchet for a most memorable snow-shoeing expedition on 1991 February 10 during which Li deficiency in blue stragglers was discussed, to D. A. VandenBerg for discussing and supplying isochrones, and to F. D. A. Hartwick for once asking whether there were blue stragglers in the halo field. T.C.B acknowledges partial support from grant AST 95-29454 from the National Science Foundation.
Basri, G., Laurent, R., & Walter, F. M. 1985, ApJ, 298, 761 Bergbusch, P. A., & VandenBerg, D. A. 1992, ApJS, 81, 163 Boesgaard, A. M. & Tripicco, M. J. 1986, ApJ, 302, L49 Bonifacio, P. & Molaro, P. 1997, MNRAS, 285, 847 Carney, B. W., Latham, D. W., Laird, J. B., Grant, C. E, & Morse, J. A. 2000, preprint Carney, B. W., Latham, D. W., Laird, J. B., & Aguilar, L. A. 1994, AJ, 107, 2240 Carney, B. W., & Peterson, R. C. 1981, ApJ, 251, 190 Chaboyer, B. 2000, “The Galactic Halo; from Globular Clusters to Field Stars” A. Noels, P. Magain, D. Caro, E. Jehin, G. Parmentier, & A. Thoul (eds) (U. Liège, Liège) in press Chaboyer, B., Demarque, P., & Pinsonneault, M. H. 1995, ApJ, 441, 876 Chen, Y. Q., Nissen, P. E., Zhao, G., Zhang, H.W., & Benoni, T. 2000, A&AS, 141, 491 Deliyannis, C. P. 1995, The Light Element Abundances, ed. P. Crane, (Berlin: Springer-Verlag), 395 Deliyannis, C. P., Demarque, P., & Kawaler, S. D. 1990, ApJS, 73, 21 Deliyannis, C. P., King, J. R., Boesgaard, A. M., & Ryan, S. G. 1994, ApJ, 434, L71 Deliyannis, C. P. & Pinsonneault, M. H. 1997, ApJ, 488, 836 Deliyannis, C. P., Pinsonneault, M. H. & Duncan, D. K. 1993, ApJ, 414, 740 Ferraro, F. R., Fusi Pecci, F., & Bellazini, M. 1995, A&A, 294, 80 Ferraro, F. R., Palterinieri, B., Rood, R. T., & Dorman, B. 1999, ApJ, 522, 983 Fuhrmann, K., & Bernkopf, J. 1999, A&A, 247, 897 to Field Stars, ed. A Noels & P. Magain, 35$^{\mbox{\rm th}}$ Liège Int. Astroph. Coll., in press Green, E. M., Demarque, P., & King, C. R. 1987, The Revised Yale Isochrones and Luminosity Functions (New Haven : Yale Univ. Press) Hobbs, L. M., & Mathieu, R. D. 1991, PASP, 103, 431 Hobbs, L. M. & Pilachowski, C. 1988, ApJ, 334, 734 Hobbs, L. M., Welty, D. E., & Thorburn, J. A. 1991, ApJ, 373, L47 Jeffries, R. D. 1999, MNRAS, 309, 189 Jones, B. F., Fischer, D., & Soderblom, D. R. 1999, AJ, 117, 330 Laird, J. B., Carney, B. W., & Latham, D. W. 1988, AJ, 95, 1843 Lambert, D. L., Heath, J. E., and Edvardsson, B. 1991, MNRAS, 253, 610 Leonard, P. J. T. 1989, AJ, 98, 217 Leonard, P. J. T., & Livio, M. 1995, ApJ, 447, L121 Lombardi, J. C. Jr, Rasio, F. A., & Shapiro, S. L. 1996, ApJ, 468, 797 Martín, E. L., & Claret, A. 1996, A&A, 306, 408 Mateo, M., Harris, H. C., Nemec, J., & Olszewski, E. W. 1990, AJ, 100, 469 Montalbán, J., & Schatzman, E. 1996, A&A, 305, 513 Norris, J. E., Ryan, S. G., & Beers, T. C. 1997b, ApJ, 488, 350 Norris, J. E., Ryan, S. G., & Beers, T. C. 1997c, ApJ, 489, L169 Norris, J. E., Ryan, S. G., Beers, T. C., & Deliyannis, C. P. 1997a, ApJ, 485, 370 Noyes, R. W., Jha, S., Korzennik, S. G., Krockenberger, M., Nisenson, P., Brown, T. M., Kennelly, E. J., & Horner, S. D. 1997, ApJ, 483, L111 Piotto, G. 2000, “The Galactic Halo; from Globular Clusters to Field Stars” A. Noels, P. Magain, D. Caro, E. Jehin, G. Parmentier, & A. Thoul (eds) (U. Liège, Liège) in press Portegies Zwart, S. 2000, Proc. Conf. The Influence of Binaries on Stellar Population Studies, Brussels 2000, D. Vanbeveren (ed), in press Preston, G. W., & Landolt, A. U. 1999, AJ, 118, 3006 Preston, G. P. & Sneden, C. 2000, in press Pritchet, C. J., & Glaspey, J. W. 1991, ApJ, 373, 105 Proffitt, C. R., & Michaud, G. 1991, ApJ, 371, 584 Rebolo, R., Molaro, P. & Beckman, J. E. 1988, A&A, 192, 192 Ryan, S. G. 2000, MNRAS, 316, L35 Ryan, S. G., Beers, T. C., Deliyannis, C. P., & Thorburn, J. A. 1996, ApJ, 458, 543 Ryan, S. G. & Deliyannis, C. P. 1995, ApJ, 453, 819 Ryan, S. G., Kajino, T., Beers, T. C., Suzuki, T., Romano, D., Matteucci, F., & Rosolankova, K. 2001, ApJ, 548 (20 Feb) Ryan, S. G., Norris, J. E., & Beers, T. C. 1998, ApJ, 506, 892 Ryan, S. G., Norris, J. E., & Beers, T. C. 1999, ApJ, 523, 654 Sandage, A. R. 1953, AJ, 58, 61 Sandquist, E. L., Bolte, M., & Hernquist, L. 1997, ApJ, 477, 335 Schatzman, E., 1993, A&A, 279, 431 Schramm, D. N., Steigman, G., and Dearborn, D. S. P. 1990, ApJ, 359, L55 Sills, A. P., Bailyn, C. D., & Demarque, P. 1995, ApJ, 455, L163 Sills, A., Lombardi, J. C. Jr, Bailyn, C. D., Demarque, P., Rasio, F. A., & Shapiro, S. L. 1997, ApJ, 487, 290 Spite, F. & Spite, M. 1982, A&A, 115, 357 Spite, M., Maillard, J. P., & Spite, F. 1984, A&A, 141, 56 Spite, M., Molaro, P., François, P., & Spite, F. 1993, A&A, 271, L1 Spite, M., Spite, F., Cayrel, R., Hill, V., Depagne, E., Nordstrom, B., Beers, T., & Nissen, P. E. 2000, “The Evolution of the Light Elements: IAU Symp. 198” L. da Silva, R. de Medeiros, & M. Spite (eds), (ASP Conf Ser.), in press Stryker, L. L. 1993, PASP, 105, 1081 Thorburn, J. A. 1992, ApJ, 399, L83 Thorburn, J. A. 1994, ApJ, 421, 318 Thorburn, J. A. & Beers, T. C. 1993, ApJ, 404, L13 Turcotte, S., Richer, J., & Michaud, G. 1998, ApJ, 504, 559 Ventura, P., Zeppieri, Mazzitelli, I., & D’Antona, F. 1998, A&A, 331, 1011 Webbink, R. F. 1976, ApJ, 209, 829
[^1]: Based on observations obtained with the University College London échelle spectrograph (UCLES) on the Anglo-Australian Telescope (AAT) and the Utrecht échelle spectrograph (UES) on the William Herschel Telescope (WHT).
[^2]: We resist the temptation to speak of a [*single*]{} locus for the turnoff because of the possibility that an age spread exists at a given metallicity. That issue has not yet been settled for the globular clusters (see Piotto et al. 2000 and Chaboyer 2000), despite those systems being better constrained. For the same reason, and because of random errors in the effective temperature estimates, we refrain from debating whether a particular star lying close to the turnoff is definitely above or below the turnoff.
[^3]: The likelihood of sub-turnoff mass objects being produced by the blue-straggler forming process is independently addressed in the model by Preston & Sneden (2000, §5.3), which came to our attention during finalisation of this manuscript.
[^4]: Independently, Carney et al. (2000) have noted this possibility, and models by Portegies Zwart (2000) predict the existence of such objects.
[^5]: Amongst very metal-poor stars with \[Fe/H\] $<$ $-2.5$, as many as 25% have C overabundances (e.g. Norris, Ryan, & Beers 1997b). At least some but not all of these (Norris, Ryan, & Beers 1997c) have s-process anomalies. Detailed studies have yet to be completed, so it is unclear what fraction of stars are formed from anomalous material and what fraction became modified later in their life. Whilst we cannot presently rule out the possibility that the s-process anomalies seen in some ultra-Li-deficient stars were inherited at birth, our expectation is that mass transfer from a companion star will be a more common mechanism.
[^6]: Although stellar collisions will be rare for stars in the field, we should recall that most stars are probably born in clusters, and prior to cluster dissolution, collisions would have greater probability.
[^7]: Carney et al. (2000) argue that all of their blue-straggler observations are consistent with 0.55 M$_\odot$ companions having a canonical white-dwarf mass.
[^8]: The critic could object that there are deficiencies in comparing metal-rich and metal-poor objects in this fashion. We would agree, but would also note that such a comparison is justifiable if only to show that the two behaviors are dissimilar.
[^9]: Errors in temperature could reduce these cases further.
[^10]: Coalescence may be between the components of an existing binary, possibly having been hardened via interactions with a third star, or through direct collisions (which may also be moderated by binary interactions).
|
----------------
NORDITA–98–1 P
hep-ph/9801222
----------------
[**Light-Cone Sum Rules**]{}
V.M. Braun[^1]\
[*NORDITA, Blegdamsvej 17, DK–2100 Copenhagen Ø, Denmark*]{}
[**Abstract**]{}\
[*Plenary talk given at the\
IVth International Workshop on Progress in Heavy Quark Physics\
Rostock, Germany, 20–22 September 1997\
To appear in the Proceedings*]{}
[**Light-Cone Sum Rules**]{}\
[*V.M. Braun*]{}\
[NORDITA, Blegdamsvej 17, DK–2100 Copenhagen Ø, Denmark]{}\
[**Introduction**]{}
Already a long time ago it was realized that large momentum transfer to an extended object (hadron) requires a specific configuration of its constituents. One possibility is to pick up a configuration in which almost all momentum is carried by one parton. The large momentum can be transferred to this fast parton which eventually recombines with the soft cloud. The second possibility is to pick up the Fock state with a minimum number of constituents (quark and antiquark for a meson) at small transverse separations, and exchange a hard gluon. In the first case the contributing transverse distances are not restricted which makes this mechanism difficult for theory; in the second case a factorizaton formula can be derived[@BLreport] and the relevant nonperturbative information can be parametrized by hadron distribution amplitudes given by vacuum-to-meson matrix elements of light-cone operators.
Which mechanism actually dominates the cross section — this is a nontrivial question which has to be studied case by case. For pion electromagnetic form factor it has been proven[@exclusive] that in the theoretical limit $Q^2\to\infty$ the “soft” (or “end-point”) contribution is suppressed compared to the “hard” contribution by one power of $1/Q^2$. For heavy-to-light B decay form factors at large recoil (e.g. $B\to \pi e\bar\nu$) both soft and hard contributions are of the same order in the $1/m_b$ expansion. For practical values of $Q^2$ and $m_b$ the soft contribution is always numerically important and often dominates. Taking it into account is difficult and presents a notorious problem in the theory of hard exclusive processes, which is not solved until now.
An important theoretical progress which has allowed for quantitative estimates of soft contributions was made with the arrival of QCD sum rules[@SVZ]. Within this approach, matrix elements of a certain operator $J$ sandwiched between two hadron states $h_1$ and $h_2$ can be evaluated by studying correlation functions of the type $$\!\int \!\! dx\, dy\, e^{-ip_1x+i p_2 y}
\langle 0| T\{H_2(y) J(0) H_1(x)\}|0\rangle \sim
\langle 0|H_2|h_2\rangle \frac{1}{m_2^2-p_2^2}
\langle h_2|J| h_1\rangle \frac{1}{m_1^2-p_1^2}
\langle h_1 |H_1 |0\rangle
\label{SVZ}$$ where $H_1$ and $H_2$ are suitable interpolation currents. The idea is to make a matching between the short-distance expansion in Euclidian space and the expansion in hadron states in the two variables $p_1^2$ and $p_2^2$ with fixed value of $q^2=(p_2-p_1)^2$. The detailed procedures have been worked out in Ref.[@3ptSR] and involve double dispersion relations, double Borel transformation to suppress contributions of higher states, and using vacuum condensates[@SVZ] to take into account nonperturbative effects.
The case $q^2=(p_2-p_1)^2 =0$ is special and requires a certain modification of the operator product expansion (OPE) to include so-called bilocal power corrections corresponding to contributions of large distances in the “t-channel” (the region of large $x$ and $y$ in (\[SVZ\]) such that $|x-y| \sim 1/|p_{1,2}| \to 0$)[@bilocal]. The structure of such modified OPE is well understood if one keeps $p_1=p_2 \equiv p$ identically. Because of this restriction, the procedure is somewhat different compared to the form factor case: One uses an ordinary dispersion relation in the single remaining variable and finds contribution of interest as the one which multiplies a double-pole term $\sim 1/(m^2-p^2)^2$.
This extension of the original SVZ sum rules to three-point functions has proved to be quite successful and has a lot of applications, for example to pion and nucleon form factors at intermediate momentum transfers, to semileptonic form factors of D decays, to baryon magnetic moments and axial constants, to $g_{\pi NN}$ and $g_{\pi B B^*}$ couplings, and to many other physical observables.
The increase in sophistication has its price, however. The three-point sum rules have specific problems which severely restrict their potential accuracy and region of applicability. These subtleties are well known to experts, but very often escape due attention of the majority of sum rules “users” and the physics community in general.
[**Problems of Three-Point Sum Rules**]{}
The first major problem is that
- OPE (short-distance expansion in condensates) upsets power counting in the large momentum/mass.
This problem is known since already the first sum rules for the pion electromagnetic form factor[@3ptSR] which have the following (schematic) structure: $$F_\pi(Q^2) \sim \#\cdot \frac{1}{Q^2}
+\#\cdot \frac{\langle g^2G^2\rangle}{M^4}
+\#\cdot Q^2 \frac{\langle \bar q q\rangle^2}{M^8}+\ldots$$ Here $M^2$ is the Borel parameter which is of order 1 GeV$^2$. The first contribution is due to perturbation theory and it has the expected $1/Q^2$ behavior[^2]. Contribution of the gluon condensate is independent of $Q^2$ and it is easy to convince oneself that condensates of higher dimension are accompanied by increasing powers of $Q^2$. If plotted as a function of $Q^2$, the sum rule result for $F(Q^2)$ starts to [*rise*]{} at $Q^2\geq 3-5$ GeV$^2$. Such behavior is clearly unphysical and indicates that at high momentum transfers the OPE breaks down. Requiring that contributions of higher dimension constitute a moderate fraction of the perturbative result, one obtains that the sum rule is only legitimate in a narrow interval of $0.5 \leq Q^2 \leq 1.5$ GeV$^2$. (The lower limit is due to the neglect of bilocal power corrections in this approximation.)
The three-point sum rules for heavy-to-light decays have the similar problem at large recoil. For example, the sum rule for the form factor $A_1$ in $B\to\rho e\bar\nu$ at the maximum recoil $q^2=0$ has the following structure: $$A_1(q^2=0) \sim \# \cdot\frac{1}{m_b^{3/2}} +
\#\cdot m_b^{1/2} \langle \bar q q\rangle +
\#\cdot m_b^{3/2} \langle \bar qg\sigma G q\rangle + \ldots$$ Here $q^2$ is the invariant mass of the lepton pair. For generic values of $q^2$ the relevant large parameter is the energy of the outgoing $\rho$ meson $(m^2_b-q^2)/(2m_b)$ in the B rest frame, which plays the role of the hard momentum transfer $Q$ in the above example. The similarity is clear. The rise of the form factor $A_1$ observed in calculations using three-point sum rules is entirely due to this principal problem: expansion in slowly varying (vacuum) fields is inadequate if a short-distance subprocess is involved. For decays of D mesons[@BBD] the recoil energy is not large - comparable to the region of applicability of sum rules for the pion form factor, and the traditional approach works well. For B decays it does not work apart from the specific case of $B\to\pi e\bar\nu$ transition[@BBD91; @Ball93] where (accidentally) the quark condensate contribution is only $\sim m_b^{-1/2}$ and the problem is numerically less important.
The second general problem of three-point sum rules is
- Contamination of the sum rule by “nondiagonal” transitions of the ground state to excited states.
This is a notorious problem in calculations of hadron matrix elements at zero momentum transfer. The contribution of interest corresponds in this case to the double-pole term in the correlation function (\[SVZ\]) at $p^2=m_h^2$ while transitions from the ground state to excited states generically produce single-pole terms which are not suppressed by the Borel transformation: $$\frac{1}{(m^2_h-p^2)^2}\cdot \langle h|J|h\rangle +
\frac{1}{(m^2_h-p^2)}\frac{1}{(m^2_h-m^2_{h'})}
\cdot \langle h|J|h'\rangle +\ldots$$ In order to get rid of “parasitic” single-pole contributions one is forced to introduce additional parameters or take the derivative of the sum rule in respect to the Borel parameter, resulting in a considerable loss of accuracy. In addition, it becomes not possible to take into account mass difference of the initial and final hadrons since one can rewrite $$\frac{1}{(m^2_1-p^2)}\frac{1}{(m^2_2-p^2)}
= \frac{1}{(m^2_1-m^2_2)}\left[
\frac{1}{(m^2_2-p^2)}-\frac{1}{(m^2_1-p^2)}\right]$$ and there is no double-pole term at all. This does not allow for calculations of transition matrix elements of the type $\Sigma\to p\gamma$, $\Delta\to N\gamma$ etc., where mass differences are large.
For form factors at sufficient values of $q^2$ it was proposed[@3ptSR] to get rid of nondiagonal transitions by using the double dispersion relation and taking Borel transform in both variables. In practical applications there are several caveats, however: First, the results depend on the shape of the duality region in plane of the two dispersion variables and this dependence can be significant. Second, there are formal problems with double dispersion relations in the decay kinematics in presence of Landau singularities[@BBD]. Third, it is becoming increasingly clear that suppresion of nondiagonal transitions by the double Borel transform is more formal than real[^3].
Light-cone sum rules (LCSR)[@BBK; @BF1; @CZ90] were developed in late 80-th in an attempt to solve or at least moderate the problems of three-point sum rules by making a partial resummation of the OPE to all orders and reorganizing the expansion in terms of twist of relevant operators rather than their dimension[^4]. In physical terms, the difference is that the expansion at short distances is substituted by the expansion in the [*transverse*]{} distance between partons in the infinite momentum frame. In this way one incorporates certain additional information on QCD correlation functions related to approximate conformal symmetry of the theory. Technically, the LCSR approach presents a marriage of QCD sum rules with the theory of hard exclusive processes. As a bonus, SVZ vacuum condensates are substituted by light-cone hadron distribution functions of increasing twist which have a direct physical significance.
10.0cm
[**A Heuristic Discussion**]{}
Consider the semileptonic decay $B\to\pi e\bar\nu$ at zero invariant mass of the lepton pair $q^2=0$, pictured schematically in Fig. 1. The $u$ quark in the final state has large energy of order $E_u\sim m_b/2$ in the B meson rest frame and has to recombine with the soft spectator antiquark with $E_{\bar d}
\sim \Lambda_{\rm QCD}$ to form a pion. If no hard gluons are exchanged as in Figs. 1a,b (we will discuss this “hard”[@hard] contribution later), the form factor is proportional to the overlap integral of such an asymmetric configuration — fast quark and slow antiquark — with the pion state, see Fig. 1c. Schematically, we can write $$f_+^{B\to\pi}(q^2=0) \sim m_b^{1/2}\cdot
\!\!\!\!\!\!\!\!\int\limits_{1-O(1/m_b)}^1
\hspace{-0.5cm} dx\,\phi_\pi(x,b)
\label{ffac}$$ where $x=2E_u/m_B$ is the $u$ quark energy fraction and $b$ is the separation between the quark and the antiquark in the plane transverse to the (large) pion momentum[^5]. The extra factor $m_b^{1/2}$ is due to the normalization of the B meson coupling to the $b\bar d$ pair; it is not very important for what follows.
For sufficiently small $b$ one can derive the asymptotic behavior of the pion distribution amplitude $\phi_\pi(x,b)$ at large x $$\phi_\pi(x,b\ll 1/\Lambda_{QCD}) \stackrel{x\to 1}{=} - N(b)\cdot (1-x)
\label{largex}$$ where the $b$-dependent normalization factor $N(b)$ $$N(b) = 6\left[1+
6\, a_2(b_0) \left(\frac{\alpha_s(b)}{\alpha_s(b_0)}\right)^{50/81}
+ 15\, a_4(b_0) \left(\frac{\alpha_s(b)}{\alpha_s(b_0)}\right)^{364/405}
+\ldots\right]$$ can be calculated in terms of (nonperturbative) coefficients $a_n$ at a certain reference scale. Provided that this expansion is convergent — which certainly is the case at very small transverse separations — the behaviour $\phi(x)\sim (1-x)$ is maintained by the renormalization group evolution. Assuming (\[largex\]), the contribution of small transverse separations to the form factor scales as[@CZ90; @ABS] $$f_+^{B\to\pi}(q^2=0) \sim m_b^{1/2}\cdot \frac{1}{m_b^2} =
\frac{1}{m_b^{3/2}}\,.
\label{3/2}$$ My discussion was purely heuristic. It can be made more rigorous with the result that (\[3/2\]) is indeed the correct behavior in the theoretical limit $m_b\to\infty$, by observing[@ASY94] that contributions of large transverse separations are suppressed by Sudakov effects. Problem is, however, that the Sudakov suppression is very weak. With the b quark mass of order 5 GeV it becomes effective at $b\sim 1$ fm only, which is of order or even larger than the B meson radius determined by nonperturbative effects. Taking them into account is mandatory for a quantitative analysis.
It is here that ideas of the QCD sum rules enter the stage: I will try to make a [*matching*]{} between the QCD calculation at small $b$ with the expansion in hadron states at large transverse separations. If the behaviour in (\[largex\]) is correct at relatively low scales of order 1 GeV where the matching is made to hadronic states — which is supported by the existing evidence, see below — then the power counting in the quark mass (\[3/2\]) is correct for realistic values of the b quark mass at which the perturbative (Sudakov) dominance of small impact parameters does not hold yet.
It is instructive to explain in this language why traditional three-point sum rules fail to describe B meson decays. The leading nonperturbative effect is then given by the diagram in Fig. 1(d), where the light quark is soft and interacts with the nonperturbative QCD vacuum, forming the quark condensate. Since quarks in a condensate have zero momentum, this diagram yields a contribution to the distribution amplitude that is naively proportional to $\delta(1-x)$. The corresponding contribution to the decay form factor (\[ffac\]) remains unsuppressed for $m_b\to\infty$ and obviously violates the power counting discussed above. The contradiction must be resolved by including the contributions of higher-order condensates to the sum rules and subtracting the contribution of excited states. The suppression of the end-point region $x\to 1$, which is expected in QCD, can only hold as a [*numerical*]{} cancellation between different contributions, which becomes the more delicate (and requires more fine-tuning) the more $m_b$ increases. For $m_b\approx 5\,$GeV a suppression of the quark condensate contribution by a factor $\sim 1\,$GeV$^2/m_b^2\sim 1/25$ is required. This explains why the three-point sum rules become unreliable.
[**A Simple Light-Cone Sum Rule**]{}
After these preliminary remarks, I will now derive the simplest LCSR for the $B\to\pi e\bar\nu$ form factor[@CZ90; @BBD; @BKR93; @Bel95]. To this end, consider the correlation function $$\Pi_\mu(p_B^2,q^2) = i\int\!d^4z\,e^{-ip_Bz}
\langle \pi(p_\pi)|T\{\bar u(0)\gamma_\mu b(0) b(z)i\gamma_5 d(z)\}|0\rangle.
\label{B-Pi}$$ At large negative $m_b^2-p_B^2$ and fixed (small and positive) $q^2$ this correlation function can systematically be calculated in QCD. The leading contribution is expressed in terms of the pion distribution amplitude: $$\Pi_\mu(p_B^2,q^2) = (p_B-q)_\mu f_\pi m_b
\int\limits_0^1 \!dx\, \frac{\phi_\pi(x,\mu)}{m^2_b-x p_B^2-(1-x)q^2}
+\ldots
\label{lt}$$ where $\mu^2\sim m^2_b-p_B^2$ and the corrections are suppressed either by powers of $\alpha_s(\mu)$ (radiative corrections) or by powers of $1/(m_b^2-p_B^2)$ (higher twist corrections)[^6]. On the other hand, $\Pi_\mu(p_B^2,q^2)$ has a pole at $p_B^2=m_B^2$ corresponding to the B meson intermediate state: $$\Pi_\mu^{\rm B meson}(p_B^2,q^2) = \frac{f_B m_B^2}{m_b}\cdot
\frac{1}{m_B^2-p_B^2}\cdot
\left[(2p_B+q)_\mu \,f_+^{B\to\pi}(q^2)+q_\mu\, f_-^{B\to\pi}(q^2)\right] .
\label{pole}$$ We can relate the two above representations, observing that $\Pi(p_B^2,q^2)$ is analytic in the cut $p_B^2$ plane, and assuming that the B meson contribution is given by integral of the QCD spectral density over the [*interval of duality*]{} $m_b^2 < s < s_0$: $$\Pi_\mu^{\rm B meson}(p_B^2,q^2) = (p_B-q)_\mu
\int\limits_{m_b^2}^{s_0}\!\frac{ds}{s-p_B^2}\, \rho(s,q^2).
\label{duality}$$ The explicit expression for $\rho(s,q^2)$ can easily be read off (\[lt\]), making a change of variables $x\to s = (m^2_b -q^2)/x+q^2$. Equating Eqs. (\[pole\]) and (\[duality\]) and making the Borel transformation $(s-p_B^2)^{-1}\to \exp(-s/M^2)$, $(m_B^2-p_B^2)^{-1}\to \exp(-m_B^2/M^2)$, we obtain (after some rewriting) the [*light-cone sum rule*]{} $$\begin{aligned}
\frac{f_B m_B^2}{f_\pi m_b}f_+^{B\to\pi}(q^2) e^{-(m_B^2-m_b^2)/M^2}&=&
\frac{1}{2}\int\limits_{x_0}^1\!\frac{dx}{x}\,\phi_\pi(x,\mu)
\,e^{-\frac{(1-x)(m_b^2-q^2)}{xM^2}},
\nonumber\\
x_0 &\equiv& \frac{m_b^2-q^2}{s_0-q^2}.
\label{LCSR1}\end{aligned}$$ Note that the restriction in the maximum invariant mass of the heavy-light quark pair $s<s_0$ translates to the lower limit in the momentum fraction carried by the b quark $x>x_0$. In the heavy quark limit $s_0\simeq (m_b+ 1$ GeV$)^2$ and $x_0\simeq 1- O(1/m_b)$ in agreement with the heuristic discussion above.
Compared to the traditional three-point QCD sum rules, note: (i) a single variable dispersion relation; (ii) no condensates; (iii) resummation of contributions of operators of leading twist. The last statement follows from the definition of the pion distribution amplitude: Moments of $\phi_\pi$ equal vacuum-to-pion matrix elements of twist two operators $$\langle \pi(p)|\bar u\gamma_\nu \gamma_5
\stackrel{\leftrightarrow}{D}_{\mu_1}\ldots
\stackrel{\leftrightarrow}{D}_{\mu_{n-1}} d|0\rangle =
-i f_\pi p_\nu p_{\mu_1}\ldots p_{\mu_{n-1}}
\int\limits_0^1\! dx\,(2x-1)^n\phi_\pi(x,\mu)$$ Naively, each such matrix element presents an independent nonperturbative parameter $M_n \equiv \int_0^1\! dx\,(2x-1)^n\phi_\pi(x)$. It is easy to check that expansion of the r.h.s. of the sum rule (\[LCSR1\]) in moments $M_n$ would correspond to expansion of the distribution function in derivatives of $\delta(1-x)$, which is the origin of problems with the traditional sum rules. The crucial idea of the LCSR approach is that the expansion in moments, alias in operators of increasing dimension, is replaced by the expansion in conformal partial waves, each of which takes into account a subset of operators to all dimensions.
The trick is analogous to the partial wave expansion of the wave function in usual quantum mechanics. The rotational symmetry of the potential allows one (in quantum mechanics) to separate angular and radial degrees of freedom. The dependence on the angular coordinates is included in spherical harmonics which form an irreducible representation of the $O(3)$ group, and the dependence on the single remaining radial coordinate is governed by a one-dimensional Schrödinger equation. Similar, the conformal expansion of distribution amplitudes in QCD aims to separate longitudinal degrees of freedom from transverse ones. For the pion distribution amplitude it has a simple form[@exclusive] $$\begin{aligned}
\phi_\pi(x,\mu) &=& 6x(1-x) \left\{1+
a_2(\mu_0)\left(\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\right)^{50/81}
C_2^{3/2}(2x-1)\right.
\nonumber\\&&\hspace*{2.3cm}{}\left.
+a_4(\mu_0) \left(\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\right)^{364/405}
C_4^{3/2}(2x-1)
+\ldots\right\}.\end{aligned}$$ All dependence on the longitudinal momentum fraction is included in Gegenbauer polynomials $C_n^{3/2}$ which form an irreducible representation of the so-called collinear subgroup $SL(2,R)$ of the conformal group corresponding to Möbius transformations on the light-cone, and the transverse-momentum dependence (the scale-dependence) is governed by simple renormalization group equations: The different partial waves, labeled by different “conformal spins” $j=n+2$, do not mix with each other. Since conformal invariance is broken in QCD by quantum corrections, mixing of different conformal partial waves is absent to leading logarithmic accuracy only. Still, conformal spin is a good quantum number in hard processes, up to small corrections of order $\alpha_{s}^{2}$, and it is natural to expect that the hierarchy of contributions of different conformal partial waves is preserved at sufficiently low scales, meaning that only a few first “harmonics” are numerically important in B decays. This assumption is supported by the recent CLEO measurement of the $\pi\gamma^\ast\gamma $ form factor[@CLEO] which indicates that at scales of order 1 GeV the pion distribution amplitude is already close to its asymtotic form $6 x(1-x)$.
Since the Gegenbauer polynomials oscillate rapidly in high orders, their convolution with smooth functions like in the r.h.s. of the sum rule (\[LCSR1\]) is strongly suppressed. For realistic values of the b quark mass it turns out that contributions of all polynomials with $n=4,6,\ldots$ are not important (unless the coefficients $a_n$ are abnormally large). The only significant potential correction to the “S-wave” contribution $6x(1-x)$ is with $n=2$. The parameter $a_2(1$ GeV) can be estimated from the CLEO data[@CLEO] or from additional sum rules[@CZreport]. The (conservative) range is $0<a_2(1$ GeV$)<0.5$. This uncertainty will eventually be eliminated when more high-precision data on exclusive processes involving pions become available.
[**$B\to \pi e\bar\nu$: State of the Art**]{}
The LCSR considered above has to be complemented by higher twist and radiative corrections. Higher twist effects were calculated in Ref.[@Bel95] using the complete set of twist 3 and twist 4 pion distribution amplitudes available from[@BF2]. The radiative correction was calculated very recently in Ref.[@KRWY; @BBB].
The structure of the radiative correction in the heavy quark limit is instructive and deserves to be mentioned here. The full expression is rather complicated, so I quote the answer[@BBB] in the so-called local duality approximation corresponding to the limiting case $M^2\to\infty$[^7] in the sum rule: $$\begin{aligned}
\lefteqn{
\frac{f^{\rm stat}(m_b)}{f_\pi}[m_b^{3/2}f_+(0)]=}
\nonumber\\
&=&-\omega_0^2\phi_\pi'(1,\mu)\Bigg[ 1+ \frac{\alpha_s}{\pi}C_f
\Bigg(\frac{1+\pi^2}{4} +\ln\frac{m_b}{2\omega_0}
-\frac{1}{2}\ln^2\frac{m_b}{2\omega_0}
+\frac{1}{2}\ln \frac{2\omega_0}{\mu}
\Bigg)\Bigg]
\nonumber\\
&&{} -\omega_0^2 \frac{\alpha_s}{\pi}C_f
\left[
\left(1-\ln\frac{2\omega_0}{\mu}\right)
\int_0^1 dx \left(\frac{\phi_\pi(x)}{\bar x^2}+
\frac{\phi_\pi'(1)}{\bar x}\right)
- \ln\frac{2\omega_0}{\mu}\int_0^1 dx \,\frac{\phi_\pi(x)}{\bar x}
\right]
\label{SR:hqlrad}\end{aligned}$$ where $\phi_\pi'(x) = (d/dx)\phi_\pi(x)$, $\bar x\equiv 1-x$ and $\omega_0$ is the nonrelativistic continuum threshold $s_0 \simeq (m_b+\omega_0)^2$. Local duality means that we identify the B meson with a b quark accompanied by an arbitrary number of light quarks and gluons with total energy less than $\omega_0$ (in the b quark rest frame).
9.0cm
Let us interpret the two pieces: the first term on the right-hand side can be identified with the soft (end-point) contribution including the Born-term and its radiative correction, while the second term corresponds to the usual mechanism[@hard] of hard gluon exchange.
The dependence on the collinear factorization scale $\mu$ must cancel the scale dependence of the pion distribution amplitude. This implies that the structure of terms in $\ln \mu$ in the hard contribution is fixed by the structure of the leading order soft term which is proportional to $\phi_\pi'(1,\mu)$. Indeed, we find $$\begin{aligned}
\frac{d}{d \ln \mu} \phi_\pi'(1,\mu) &=& \frac{\alpha_s}{\pi}C_f
\frac{d}{dx}\left[\int_0^1 dy\, V_0(x,y)\,\phi_\pi(y,\mu)\right]_{x\to 1}
\nonumber\\
&=&{}-\frac{\alpha_s}{\pi}C_f\left\{
\int_0^1 dx\,\left[\frac{\phi_\pi(x)+\bar x \phi_\pi'(1)}{\bar x^2} +
\frac{\phi_\pi(x)}{\bar x}\right]-\frac{1}{2}\phi_\pi'(1)
\right\},
\label{scaledepend}\end{aligned}$$ where $V_0(x,y)$ is the usual Brodsky-Lepage kernel, so that the structure of $\ln \mu$ terms in (\[SR:hqlrad\]) is reproduced. Note the subtraction term accompanying the naively divergent expression $\int dx\, \phi_\pi(x,\mu)/\bar x^2$ [@hard], which is similar to the usual “plus” prescription in the evolution kernel. The lesson to be learnt is that LCSRs are fully consistent with QCD and in fact can be used to study the factorization of hard and soft (end-point) contributions.
Some numerical results are shown in Fig. 2 and Fig. 3 [@BBB]. It attracts attention that the radiative correction is small, at most 7% in the whole $q^2$ range, and the higher twist effects appear to be under control[^8]. Possible deviation of the pion distribution amplitude from its asymptotic form mainly affects the slope of the form factor and has little impact on the normalization. The corresponding spectrum $dB/d q^2$ has to increase somewhat from $q^2=0$ to $q^2 \leq 15$ GeV$^2$ if the pion distribution amplitude is close to its asymptotic expression, and it decreases with $q^2$ if the distribution amplitude has large corrections[@CZreport], see Fig. 3 in Ref.[@BBB]. This behavior can be checked experimentally in the near future.
9.0cm
The analysis of theoretical uncertainties in the sum rule method is a difficult issue in general. Using state-of-the-art LCSRs including radiative corrections and higher twist effects up to twist 4, and with some better knowledge of $m_b$ and $f_B$, one can expect a theoretical accuracy up to 10% in form factors which translates to 20% uncertainty in the decay rates. Yet higher accuracy is not feasible within the sum rule method.
[**Other Heavy-to-Light Decays**]{}
Apart from the simplest process $B\to\pi e\bar\nu$ which historically attracted most of the attention, LCSRs have been derived for semileptonic $B\to\rho e\bar\nu$ decays[@ABS; @BB97], see Fig. 4, and for rare radiative decays induced by flavor-changing neutral currents, most notably $B\to K^*\gamma$ [@ABS]. Other decays studied are $B_s\to K^*\gamma$, $B_u\to\rho(\omega)\gamma $ and $B_s\to\phi\gamma$ [@ABS]. In addition, the $B\to K^* l^+l^-$, $B\to K l^+l^-$ decay form factors have been calculated using the light-cone approach in Ref.[@ALIEV96; @ALIEV97]. The relevant form factors are too numerous to be presented here.
8.5cm
The results of[@ABS; @ALIEV96] have eventually to be updated to include radiative corrections and using the revised distribution amplitudes of the vector meson[@BB96] and surface terms for the continuum subtraction[@BB97]. In the analysis of [@ALIEV96] one should also take into account $SU(3)$ violating asymmetry in the K meson distribution amplitude. I do not expect significant numerical changes, however.
The LCSR method can be used to estimate the long-distance contributions of four-fermion operators to the decay $B^+\to\rho^+\gamma$ [@KHO95; @ALI95] which appear to be of order 20% of the short-distance contribution to the decay rate. The same approach was applied to the decay $B\to \mu \bar\nu_\mu\gamma$ in Ref.[@KHO95; @EIL95].
[**Other Applications**]{}
My discussion so far was concentrated on the B decays which are topical for this conference. The LCSR approach is, however, quite general and is equally useful for form factors of light hadrons, where it has the similar advantages of being applicable in a wide range of momentum transfers and using simpler dispersion relations. Sample applications include the (electromagnetic) pion form factor [@BH94], $\pi A_1\gamma$ form factor[@Bel95a], $\gamma^*\rho\to\pi$ [@BH94; @Kho97] and $\gamma^*\gamma\to \pi^0$ [@Kho97] transitions.
[**Light-Cone Sum Rules for Hadron Matrix Elements**]{}
A conceptually similar but technically somewhat more complicated modification of the LCSR approach is useful for calculations of matrix elements of local operators between hadron states (like baryon magnetic moments) or involving two heavy and one light hadron, like $D^*\to D\pi$ decays. In this type of problems there is no large scale involved (except, possibly, a heavy quark mass) and the light-cone approach has to be compared with the method of Ref.[@bilocal] with explicit separation of local and bilocal power corrections. The work Ref.[@Bel95] contains a rather detailed introduction to the LCSR technique in this context, which is more readable than the original papers[@BBK; @BF1].
To give an example, I will consider calculation of the $g_{BB^*\pi}$ coupling. The starting point in both approaches is the same correlation function (\[B-Pi\]) where the vector current now serves as an interpolating field for the $B^*$ meson and it is therefore convenient to change the notation for the corresponding momentum $q\to p_{B^*}$: $$\Pi_\mu(p_B^2,p_{B^*}^2) = i\int\!d^4z\,e^{-ip_Bz}
\langle \pi(p_\pi)|T\{\bar u(0)\gamma_\mu b(0) b(z)i\gamma_5 d(z)\}|0\rangle.
\label{CFgBBpi}$$ As explained in detail in Ref.[@Bel95], in order to apply the short-distance expansion to this correlation function one has to take the soft pion limit $p_\pi\to 0$ so that $p_B=p_{B^*}=p$. Therefore, a double Borel transformation cannot be applied and nondiagonal transitions from ground to excited states produce a single-pole contribution to (\[CFgBBpi\]): $$\frac{g_{BB^*\pi}}{(p^2-m_B^2)^2} + A\cdot \frac{1}{(p^2-m_B^2)}.$$ The constant $A$ creeps into the sum rule, which has the following schematic structure[@EK85]: $$\#\cdot g_{BB^*\pi} + M^2 A = m_b f_\pi M^2 \exp\left[
\frac{m_{B^*}^2-m_b^2}{2M^2}+\frac{m_{B}^2-m_b^2}{2M^2}\right]
+\ldots$$ Thus, one sum rule has to be used to determine two unknown constants — $g_{BB^*\pi}$ and $A$ — which reduces the accuracy. In addition, it is in principle not possible to keep the B and B$^*$ masses different from each other, since, as I already mentioned in the introduction, in this case the double-pole term is not present in the correlation function. This deficiency is marginal for the case in question, but it can be crucial in other applications. Historically, the need to take into account the mass difference of the proton and $\Sigma$-hyperon in the weak decay $\Sigma\to p\gamma$ [@BBK] has been the prime motivation for the development of the LCSR approach.
As I emphasized already, the main characteristic feature of LCSRs is that short-distance expansion is replaced by expansion in powers of the deviation from the light-cone (or transverse distance in light-cone coordinates). The light-cone expansion corresponds to a more general kinematics, with the pion being on-shell $p_\pi^2=m_\pi^2\simeq 0$ but with nonzero momentum, so that $2(p_\pi\cdot p_B) = p_{B}^2-p_{B^*}^2$ can be arbitrary large and one can take $p_B^2$ and $p_{B^*}^2$ as two independent variables. Taking the Borel transform in both of them one obtains the sum rule[@Bel95] $$\#\cdot g_{BB^*\pi} = m_b f_\pi \phi_\pi(x) \exp\left[
\frac{m_{B^*}^2-m_b^2}{2M^2_1}+\frac{m_{B}^2-m_b^2}{2M^2_2}\right]
+\ldots$$ where the argument of the pion distribution amplitude is fixed by the ratio of the two Borel parameters $$x = \frac{M_1^2}{M_1^2+M_2^2}\sim \frac{m_{B^*}^2}{m_{B^*}^2+m_{B}^2}
\simeq \frac{1}{2}.$$ The premium is that single-pole terms are absent and one can keep $m_{B^*}\neq m_B$, while the price to pay is that one has a nontrivial new input: The pion distribution amplitude in approximately the middle point. Lacking direct experimental measurement of $\phi_\pi(1/2)$ one can consider this quantity as a nonperturbative parameter to be found from one suitable sum rule and used elsewhere[@BF1], similar to the usual way how the gluon condensate is determined and used in the sum rules. Note that this quantity is not related to a matrix element of any local operator, which illustrates that the sum rule is not related to a short distance expansion. The dedicated study in Ref.[@BF1] resulted in the estimate $\phi_\pi(1/2) = 1.2\pm 0.2$ which is only slightly below the asymptotic value $\phi_\pi(1/2) = 3/2$.
The same approach is applicable to the calculation of amplitudes involving emission of a real photon with the advantage that the photon distribution amplitudes are expected to be very close to their asymptotic form[@BBK]. As the result, for photon radiation there are no free parameters and it was checked that the LCSR approach works very well for the proton and neutron magnetic moments[@BF1]. Other applications include: $\Sigma\to p\gamma$ decay[@BBK], $g_{\pi NN}$ and $g_{\rho\omega\pi}$ couplings[@BF1] and the radiative decays $D^*\to D\gamma$, $B^*\to B\gamma$ [@BBgamma].
The LCSR result is $g_{B^*B\pi}= 29\pm 3 $ [@Bel95] with an error corresponding to the estimated theoretical uncertainty. In the same framework, the strong coupling constants of the scalar and axial B mesons with the pion have been estimated yielding the following predictions for the observable strong decay widths[@COL95]: $\Gamma( B(0^{++}) \to B \pi)\simeq \Gamma( B(1^{++})
\to B^* \pi)\simeq 360 ~\mbox{MeV}$. An analogous method was used in Ref.[@AlievBrho] to obtain the $BB^*\rho$ coupling.
[**Summary and Further Prospects**]{}
I have given a short introduction to the technique of light-cone sum rules, their theoretical background and main modifications. This approach is a derivative of the QCD sum rule method[@SVZ] and combines characteristic features of sum rules with the theory of hard exclusive processes. Main idea and the defining feature of a generic LCSR is that the short-distance Wilson operator product expansion is substituted by the light-cone expansion in operators of increasing twist; for given twist the expansion in local operators is replaced by the expansion in conformal partial waves. Each term in the partial wave expansion is well defined and has the expected asymptotic behavior at the end-points of the phase-space. The approach involves an implicit physical assumption that the conformal spin presents a “good” approximate quantum number in hard exclusive processes in QCD, and it is this physics issue that will eventually be decided by the success (or failure) of the LCSR programm.
Although the approach is already 10 years old, full understanding of its advantages and potential is rather recent. LCSRs can be used for a broad range of processes from which I mainly discussed applications to heavy quark decays. There is room for further improvement: The existing LCSRs are in most cases derived to leading twist accuracy only and do not include radiative corrections. A few methodical questions need to be clarified as well.
Main input in the sum rules is provided by hadron light-cone distribution amplitudes. They have a direct physical interpretation and in this sense are as basic as conventional parton distributions. For a further progress in LCSR calculations a systematic study of distribution amplitudes is mandatory. The present situation is not satisfactory and requires both theoretical and experimental efforts. The leading twist distributions can be studied experimentally and there is increasing evidence that they are not far from their asymptotic form (the ’S-wave’ contribution to the conformal partial wave expansion). Several works on higher-twist meson distribution amplitudes are in progress. Results on higher-twist baryon distributions are so far absent and would be most welcome.
[**Acknowledgements**]{}
It is a pleasure for me to thank A. Ali, I. Balitsky, P. Ball, V.M. Belyaev, H.G. Dosch, I. Filyanov (Halperin), A. Khodjamirian, R. Rückl and H. Simma for a very rewarding collaboration on subjects related to this report. Special thanks are due to the organizers of this conference for the invitation and hospitality.
[**References**]{}
[99]{}
S.J. Brodsky and G.P. Lepage, in: [*Perturbative Quantum Chromodynamics*]{}, ed. by A.H. Mueller, p. 93, World Scientific (Singapore) 1989.
V.L. Chernyak and A.R. Zhitnitsky, JETP Lett. [**[25]{}**]{} (1977) 510; Yad. Fiz. [**31**]{} (1980) 1053; A.V. Efremov and A.V. Radyushkin, Phys. Lett. B [**94**]{} (1980) 245; Teor. Mat. Fiz. [**[42]{}**]{} (1980) 147; G.P. Lepage and S.J. Brodsky, Phys. Lett. B [**87**]{} (1979) 359; Phys. Rev. D [**22**]{} (1980) 2157; V.L. Chernyak, V.G. Serbo and A.R. Zhitnitsky, JETP Lett. [ **26**]{} (1977) 594; Sov. J. Nucl. Phys. [**31**]{} (1980) 552.
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl.Phys. [**B147**]{} (1979) 385, 448, 519.
V. Nesterenko and A.V. Radyushkin, Phys. Lett. B [**115**]{} (1982) 410; JETP Lett. [**35**]{} (1982) 488; B.L. Ioffe and A.V. Smilga, Phys. Lett. B [**114**]{} (1992) 353; Nucl. Phys. [**B216**]{} (1983) 373.
I.I. Balitsky and A.V. Yung, Phys. Lett. B [**129**]{} (1983) 328; B.L. Ioffe and A.V. Smilga, Nucl. Phys. [**B232**]{} (1984) 109.
P. Ball, V.M. Braun and H.G. Dosch, Phys. Rev. D [**44**]{} (1991) 3567.
P. Ball, V.M. Braun and H.G. Dosch, Phys. Lett. B [**273**]{} (1991) 316. P. Ball, Phys. Rev. D [**48**]{} (1993) 3190.
B. Blok and M. Lublinsky, hep-ph/9706484.
I.I. Balitsky, V.M. Braun and A.V. Kolesnichenko, Nucl. Phys. [**B312**]{} (1989) 509; Sov. J. Nucl. Phys. [**44**]{} (1986) 1028; [*ibid.*]{} [**48**]{} (1988) 348, 546;
V.M. Braun and I.E. Filyanov, Z. Phys. C [**[44]{}**]{} (1989) 157.
V.L. Chernyak and I.R. Zhitnitskii, Nucl. Phys. [**B345**]{} (1990) 137.
A. Szczepaniak, E.M. Henley and S.J. Brodsky, Phys. Lett. B [**243**]{} (1990) 287; G. Burdman and J.P. Donoghue, Phys. Lett. B [**270**]{} (1991) 55.
A. Ali, V.M. Braun and H. Simma, Z. Phys. C [**63**]{} (1994) 437.
R. Akhoury, G. Sterman and Y.P. Yao, Phys. Rev. D [**50**]{} (1994) 358; R. Akhoury and I.Z. Rothstein, Phys. Lett. B [**337**]{} (1994) 176.
V.M. Belyaev, A. Khodjamirian and R. Rückl, Z. Phys. C [**60**]{} (1993) 349; A. Khodjamirian and R. Rückl, Talk presented at 28th International Conference on High-energy Physics (ICHEP 96), Warsaw, Poland, July 1996; published in ICHEP 96, 902 (hep–ph/9610367).
V.M. Belyaev et al., Phys. Rev. D [**51**]{} (1995) 6177.
J. Gronberg et al. (CLEO coll.), Preprint CLNS-97-1477 (hep-ex/9707031).
V.L. Chernyak and A.R. Zhitnitsky, Phys. Rept. (1984) 173.
V.M. Braun and I.E. Filyanov, Z. Phys. C [**[48]{}**]{} (1990) 239.
A. Khodjamirian et al., Phys. Lett. B [**410**]{} (1997) 275.
E. Bagan, P. Ball and V.M. Braun, Preprint NORDITA-97-59, hep-ph/9709243.
P. Ball and V.M. Braun, Phys. Rev. D [**55**]{} (1997) 5561.
J. Flynn, Talk given at Lattice 96, 14th International Symposium on Lattice Field Theory, St. Louis, USA, June 1996; published in Nucl. Phys. Proc. Suppl. [**53**]{} (1997) 168 (hep–lat/9610010).
T. M. Aliev, A. Ozpineci and M. Savci, Phys. Rev. D [**56**]{} (1997) 4260.
T. M. Aliev et al., Phys. Lett. [**B400**]{} (1997) 209.
P. Ball and V.M. Braun, Phys. Rev. D [**54**]{} (1996) 2182.
A. Khodjamirian, G. Stoll and D. Wyler, Phys. Lett. [**B358**]{} (1995) 129.
A. Ali and V.M. Braun, Phys. Lett. [**B359**]{} (1995) 223.
G. Eilam, I. Halperin and R.R. Mendel, Phys. Lett. [**B361**]{} (1995) 137.
V.M. Braun and I. Halperin, Phys. Lett. [**B328**]{} (1994) 457.
V.M. Belyaev, Z. Phys. C [**65**]{} (1995) 93.
A. Khodjamirian, Preprint WUE-ITP-97-051, hep-ph/9712451
V.L. Eletskii and Ya.I. Kogan, Z. Phys. C [**28**]{} (1985) 155.
T.M. Aliev et al., Phys. Rev. [**D54**]{} (1996) 857.
P. Colangelo et al., Phys. Rev. [**D52**]{} (1995) 6422.
T.M. Aliev et al., Phys. Rev. [**D53**]{} (1996) 355.
[^1]: On leave of absence from St. Petersburg Nuclear Physics Institute, 188350 Gatchina, Russia.
[^2]: The leading-order term is only $\sim 1/Q^4$ and the $1/Q^2$ behaviour starts with the radiative correction.
[^3]: For example, nondiagonal transitions upset QCD sum rules for the b quark kinetic energy in B meson, see Ref.[@BL97] and references therein.
[^4]: The term “light-cone sum rules” first appears in Ref.[@BBD].
[^5]: It can be argued that contributions of other than valence states (with additional gluons and/or $q\bar q$ pairs) are suppressed by extra powers of $1/m_b$.
[^6]: From now on I will set the scale in the distribution amplitude by the cutoff in the transverse momentum rather than position space, which is more convenient in practical calculations.
[^7]: The local duality limit has to be taken consistently with the heavy quark expansion, in particular the order of limits $m_b\to\infty$ and $M^2 \to \infty$ is important, see Ref.[@BBB] for the details.
[^8]: The large twist 3 correction is exactly calculable in terms of the quark condensate.
|
---
abstract: 'Predictions of the existence of well-defined deeply bound pionic atom states for heavy nuclei and the eventual observation of such states by the $(d,^3He)$ reaction have revived interest in the pion-nucleus interaction at threshold and in its relation to the corresponding pion-nucleon interaction. Explanation of the ‘anomalous’ $s$-wave repulsion in terms of partial restoration of chiral symmetry and/or in terms of energy-dependence effects have been tested in [*global*]{} fits to pionic atom data and in a recent [*dedicated*]{} elastic scattering experiment. The role of neutron density distributions in this context is discussed in detail for the first time.'
author:
- 'E. Friedman'
title: 'Overview of pion-nucleus interaction at low energies'
---
Motivation and Background {#sec:mot}
=========================
The motivation for presenting this overview is the revival in recent years of interest in the pion-nucleus interaction at threshold and in particular in its relation to the corresponding pion-nucleon interaction. For low energy pions one may relate the two interactions by the Ericson Ericson (EE) potential [@EEr66], which is inserted into the Klein-Gordon (KG) equation [@BFG97] to calculate shifts and widths of pionic atom levels. Traditionally, experiments on pionic atoms involved the measurement of X-rays from atomic transitions of $\pi ^-$ which terminate when the nuclear absorption of pions becomes dominant. It was believed that in any case the deepest atomic levels for heavy nuclei will be too broad to be well-defined. Friedman and Soff [@FSo85] predicted in 1985 that such states, owing to the repulsive $s$-wave $\pi N$ interaction at threshold, are sufficiently narrow so as to be well defined, and three years later Toki and Yamazaki discussed ways to populate such states [@TYa88]. The first observation of these states was by Yamazaki et al. [@YHI96]. Figure \[fig1\] illustrates how nuclear absorption becomes dominant when the angular momentum $l$ goes down and how its initial increase is greatly reduced due to the repulsion of the wavefunction out of the nucleus, which is the mechanism responsible for making the level widths sufficiently small to be observed.
![Radiation (dashed) and total widths of pionic atom levels in Pb.[]{data-label="fig1"}](PbwidthsK.eps){height="8cm" width="10cm"}
![$\chi ^2$ [*vs.*]{} the neutron-excess radius parameter $\gamma$ for different shapes of the neutron density (lower part); derived values of the isovector parameter $b_1$ (upper).[]{data-label="fig2"}](piZRCK.eps){height="11cm" width="10cm"}
Interest in the pion-nucleus interaction at low energies has been focused on the $s$-wave part of the optical potential [@BFG97]
$$\begin{aligned}
\label{equ:EE1s}
q(r) & = & -4\pi(1+\frac{\mu}{M})\{{\bar b_0}(r)
[\rho_n(r)+\rho_p(r)]
+b_1[\rho_n(r)-\rho_p(r)] \} \nonumber \\
& & -4\pi(1+\frac{\mu}{2M})4B_0\rho_n(r) \rho_p(r) ,\end{aligned}$$
where $M$ is the mass of the nucleon and $\rho_n$ and $\rho_p$ are the neutron and proton density distributions normalized to the number of neutrons $N$ and number of protons $Z$, respectively. The function ${\bar b_0}(r)$ is given in terms of the [*local*]{} Fermi momentum $k_{\rm F}(r)$ corresponding to the isoscalar nucleon density distribution: $$\label{equ:b0b}
{\bar b_0}(r) = b_0 - \frac{3}{2\pi}(b_0^2+2b_1^2)k_{\rm F}(r).$$ This term shows enhanced repulsion compared to the free pion-nucleon interaction, resulting mostly from the in-medium isovector $s$-wave $\pi N$ amplitude $b_1$ which plays a dominant role due to the nearly vanishing of the corresponding isoscalar amplitude $b_0$. Some extra repulsion comes also from the empirical dispersive component of the two-nucleon absorption term $B_0$. Recent interest in this topic included attempts to explain the enhancement of the $s$-wave repulsion in terms of chiral-motivated density dependence of the pion decay constant [@Wei01] or by energy-dependent effects [@ETa82; @FGa03].
In Section \[sec:dens\] we discuss the role of the neutron density distributions which, together with the proton distributions, form an essential ingredient of the EE potential [@EEr66; @BFG97]. In Section \[sec:atoms\] we present results from global fits to pionic atom data, with emphasis placed on dependence on the neutron distribution used in the analysis. Recent extensions to the scattering regime are presented in Section \[sec:scatt\]. Section \[sec:sum\] is a summary.
The Role of Neutron Densities {#sec:dens}
=============================
With nine parameters in the EE pion-nucleus optical potential [@BFG97] the only way of gaining significant information from fits to pionic atom data is to perform large scale fits to as wide a data base as possible. The proton density distributions $\rho _p$ are known quite well from electron scattering and muonic X-ray experiments, and can be obtained from the nuclear charge distributions by numerical unfolding of the finite size of the proton. In contrast, the neutron densities $\rho _n$ are not known to sufficient accuracy and we have therefore performed fits while scanning over these densities. This procedure is based on the expectation that for a large data set over the whole of the periodic table some local variations will cancel out and that an [*average*]{} behavior may be established.
Experience showed [@FGa03] that the feature of neutron density distributions which is most relevant in determining strong interaction effects in pionic atoms is the radial extent, as represented for example by $r_n$, the neutron density rms radius. Other features such as the detailed shape of the distribution have only minor effect. For that reason we chose the rms radius as the prime parameter in the present study and adopted the two-parameter Fermi distribution both for the proton and for the neutron density distributions as follows:
$$\label{equ:2pF}
\rho_{n,p}(r) = \frac{\rho_{0n,0p}}{1+{\rm exp}((r-R_{n,p})/a_{n,p})}~~ .$$
The use of single-particle densities is not expected to be more appropriate when there are many nuclei far removed from closed shells and when in any case parameters of $\rho _n$ are being varied. In order to allow for possible differences in the shape of the neutron distribution, the ‘skin’ and the ‘halo’ forms of Ref. 9 were used, as well as an average of the two. In this way we have used three shapes of the neutron distribution for each value of its rms radius all along the periodic table. The rms radius $r_n$ for the various nuclei was parameterised using the following expression for the neutron-proton differences
$$\label{equ:RMF}
r_n-r_p = \gamma \frac{N-Z}{A} + \delta \; ,$$
and scanning over the values of the parameter $\gamma$. This approach has been made in analyses of antiprotonic atoms data [@TJL01; @FGM05].
Results for Pionic Atoms {#sec:atoms}
========================
Figure \[fig2\] shows values of $\chi ^2$ for 100 data points in the lower part and the derived values of the $s$-wave parameter $b_1$ in the upper part, as functions of the neutron radius parameter $\gamma$. It is seen that the best fit is obtained for the ‘skin’ shape and that the $b_1$ parameter is then in good agreement with its free-space value. We note, however, that this best fit is obtained for a value of $\gamma$ which leads to unacceptably large neutron rms radii in heavy nuclei [@Jas04]. Attempting to introduce finite-range folding into the potential, as was successfully done with antiprotons [@FGM05], the fits deteriorate monotonically with increasing range. However, when finite range is introduced separately into the $s$-wave and the $p$-wave parts of the potential, it is found that a finite range with an rms radius of 0.9$\pm$0.1 fm applied only to the $p$-wave part, leads to the best fit and Fig. \[fig3\] shows that it is obtained for $\gamma$ close to 1.0 fm, which is acceptable when comparing with other sources of information on neutron radii. The discrepancy between the value obtained for $b_1$ and its free-space value is clearly observed. It is also found that Re$B_0$/Im$B_0$=$-$2, which is unacceptable [@BFG97]. Both results represent the well known ‘anomaly’, or enhanced repulsion in nuclei.
![Same as Fig. \[fig2\] but for finite range in the $p$-wave term.[]{data-label="fig3"}](piYCK.eps){height="11cm" width="10cm"}
![Same as Fig. \[fig3\] but with the chiral model and energy dependence applied to the $s$-wave term.[]{data-label="fig4"}](piYEWK.eps){height="11cm" width="10cm"}
In the chiral-approach of Weise [@Wei01] the in-medium $b_{1}$ is related to possible partial restoration of chiral symmetry in dense matter, as follows. Since $b_{1}$ in free-space is well approximated in lowest chiral-expansion order by the Tomozawa-Weinberg expression $b_{1}=-\mu_{\pi N}/8 \pi f^{2}_{\pi}=-0.08~m^{-1}_{\pi}$, then it can be argued that $b_{1}$ will be modified in pionic atoms if the pion decay constant $f_\pi$ is modified in the medium. The square of this decay constant is given in leading order, as a linear function of the nuclear density, $$\label{eq:fpi2}
f_\pi ^{*2} = f_\pi ^2 - \frac{\sigma }{m_\pi ^2} \rho$$ with $\sigma$ the pion-nucleon sigma term. This leads to a density-dependent isovector amplitude such that $b_1$ becomes $$\label{eq:ddb1}
b_1(\rho) = \frac{b_1(0)}{1-2.3\rho}$$ for $\sigma $=50 MeV and with $\rho$ in units of fm$^{-3}$. With this [*ansatz*]{} the best fit is obtained for $\gamma$ close to 1.0 fm and with $b_1$ almost in agreement with the free value. When the energy dependence of the $b_0$ parameter is also included [@ETa82; @FGa03], Fig. \[fig4\] shows a perfect agreement between the derived $b_1$ and its free value, for acceptable neutron rms radii. For this case the dispersive term Re$B_0$ (not shown) turns out to be consistent with zero, in contrast to the unacceptably large repulsive values obtained for the conventional potential. It is worth mentioning that although the values of $r_n$ obtained here agree with the values found from analyses of antiprotonic atoms [@TJL01; @FGM05], the latter favor the ‘halo’ shape for the neutron density distribution. This could be the result of using the over-simplified Fermi distribution and the fact that whereas pionic atoms are sensitive to densities around 50% of the central value, antiprotonic atoms sample regions where the density is only 5% of that of nuclear matter.
Finally, it should be noted that the ‘deeply bound’ pionic atom states fit precisely into the picture emerging from global fits to conventional pionic atom data in the sense that predictions made with potentials obtained from fits to the latter are in full agreement with experimental results for the former. That is a natural consequence of the repulsion of the atomic wavefunction out of the nucleus which prevent really deep penetration of the deeply-bound atomic wavefunctions.
Elastic Scattering at 21.5 MeV {#sec:scatt}
==============================
With the picture emerging from global analyses of pionic atoms it is interesting to extend the study of the $s$-wave part of the pion-nucleus potential into the scattering regime, where below 30 MeV the pions penetrate well into nuclei. In the scattering scenario, unlike in the atomic case, one can study both charge states of the pion, thus increasing sensitivities to isovector effects and to the energy dependence of the isoscalar amplitude due to the Coulomb interaction. Note that both were found to play a role in the atomic case. For that reason precision measurements of elastic scattering of 21.5 MeV $\pi ^+$ and $\pi ^-$ by several nuclei were performed very recently at PSI [@FBB04; @FBB05] and analyzed in terms of the same effects as in pionic atoms. The experiment was dedicated to studying the elastic scattering of both pion charge states and special emphasis was placed on the absolute normalization of the cross sections, which was based on the parallel measurements of Coulomb scattering of muons. The potentials tested were the conventional (C) one, the chiral motivated potential of Weise [@Wei01] (W), the energy-dependent potential [@ETa82; @FGa03] (E) and a potential with both effects included (EW).
[ccccc]{} model & C & W & E & EW\
$b_1 (m_\pi ^{-1}$)& $-0.114\pm0.006$ & $-0.081\pm0.005$ & $-0.119\pm0.006$ &$-0.083\pm0.005$\
$\chi ^2$ for 72 points & 134 & 88 & 80 & 88\
Table \[tab:scatt\] summarizes the results, showing that clearly the better fits to the data require that at least one of these effects is included, and that the derived $b_1$ agrees with the free $\pi N$ interaction only when the chiral-motivated density dependence is included.
Summary {#sec:sum}
=======
Global analyses of strong interaction effects in pionic atoms, fitting parameters to the EE potential across the periodic table, consistently led to ‘anomalous’ repulsion in the $s$-wave part of the potential when compared to the free $\pi N$ interaction at threshold. This is particularly clear when the dependence on rms radii of neutron density distributions is considered. Introducing into the $s$-wave part of the potential a chiral-motivated dependence on density of the isovector interaction and the energy dependence of the isoscalar interaction fully explain the enhanced repulsion and the best fit is then obtained with neutron densities that are in agreement with other sources of information. Dedicated experiment at 21.5 MeV measuring the elastic scattering of $\pi ^{\pm}$ by several targets show that the pion-nucleus potential changes smoothly from threshold into the scattering regime. Enhanced repulsion is observed also at 21.5 MeV, and is accounted for by the same mechanisms as for pionic atoms. However, in contrast with pionic atoms, the quality of fits to the data clearly require the inclusion of at least the chiral-motivated density dependence.
Acknowledgments {#acknowledgments .unnumbered}
===============
I wish to thank A. Gal for many years of collaboration on topics of the present manuscript. This work was supported in part by the Israel Science Foundation grant 757/05.
[00]{}
M. Ericson, T.E.O. Ericson, [*Ann. Phys. (NY)*]{} [**36**]{}, 323 (1966).
C.J. Batty, E. Friedman and A. Gal, [*Phys. Rep.*]{} [**287**]{}, 385 (1997).
E. Friedman and G. Soff, [*J. Phys. G: Nucl. Phys.*]{} [**11**]{}, L37 (1985).
H. Toki and T. Yamazaki, [*Phys. Lett. B*]{} [**213**]{}, 129 (1988).
T. Yamazaki, et al., [*Z. Phys. A*]{} [**355**]{}, 219 (1996).
W. Weise, [*Nucl. Phys. A*]{} [**690**]{}, 98c (2001).
T.E.O. Ericson and L. Tauscher, [*Phys. Lett. B*]{} [**112**]{}, 425 (1982).
E. Friedman and A. Gal, [*Nucl. Phys. A*]{} [**724**]{}, 143 (2003).
A. Trzcińska et al., [*Phys. Rev. Lett.*]{} [**87**]{} 082501, (2001).
E. Friedman, A. Gal and J. Mareš, [*Nucl. Phys. A*]{} [**761**]{}, 283 (2005).
J. Jastrzȩbski et al., [*Int. J. Mod. Phys. E*]{} [**13**]{}, 343 (2004).
E. Friedman et al., [*Phys. Rev. Lett.*]{} [**93**]{}, 122302 (2004).
E. Friedman et al., [*Phys. Rev. C*]{} [**72**]{}, 034609 (2005).
|
---
abstract: |
The atomic and magnetic structures of (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl and Br) are investigated using the density-functional calculations. Among several dozens of examined structures, an orthorhombic distorted $2\times 2$ structure, in which the displacement pattern of $X$ halogens resembles the model conjectured previously based on the empirical information is identified as the most stable one. The displacements of $X$ halogens, together with those of Cu ions, result in the formation of $X$-Cu-$X$-Cu-$X$ zigzag chains in the two materials. Detailed analyses of the atomic structures predict that (Cu$X$)LaNb$_2$O$_7$ crystallizes in the space group $Pbam$. The nearest-neighbor interactions within the zigzag chains are determined to be antiferromagnetic (AFM) for (CuCl)LaNb$_2$O$_7$ but ferromagnetic (FM) for (CuBr)LaNb$_2$O$_7$. On the other hand, the first two neighboring interactions between the Cu cations from adjacent chains are found to be AFM and FM respectively for both compounds. The magnitudes of all these in-plane exchange couplings in (CuBr)LaNb$_2$O$_7$ are evaluated to be about three times those in (CuCl)LaNb$_2$O$_7$. In addition, a sizable AFM inter-plane interaction is found between the Cu ions separated by two NbO$_6$ octahedra. The fourth-neighbor interactions are also discussed. The present study strongly suggests the necessity to go beyond the square $J_1-J_2$ model in order to correctly account for the magnetic property of (Cu$X)$LaNb$_2$O$_7$.\
PACS: 71.15.Mb, 75.45.+j\
author:
- 'Chung-Yuan Ren$^{a,\dagger}$ and Ching Cheng$^{b}$'
title: 'Atomic and magnetic structures of (CuCl)LaNb$_2$O$_7$ and (CuBr)LaNb$_2$O$_7$: Density functional calculations '
---
INTRODUCTION
============
Low-dimensional quantum spin systems with frustrated interactions have drawn considerable attention for several decades [@book1]. In particular, the square-lattice $S=1/2$ frustrated Heisenberg magnets with first-neighbor exchange constant $J_1$ and second-neighbor constant $J_2$ are increasingly interesting due to their unusual ground states and quantum phenomena [@DM]-[@SMS]. Based on the $J_1-J_2$ model studies, there exist several phases as a function of $J_2/J_1$. When $J_1$ dominates or $J_2$ is ferromagnetic (FM), the system is either Néel antiferromagnetic (NAFM) or FM depending on the sign of $J_1$ (Refs. [@CCL; @SOW; @SSPT; @SMS]). When $J_2$ is antiferromagnetic (AFM) and dominates, there appears the so-called columnar AFM (CAFM) order [@CCL; @SOW] with antiferromagnetically coupled FM chains. The CAFM and FM, or CAFM and NAFM ordered phases are separated by the intermediate quantum-disordered phases, the nature of which is not yet fully resolved [@DM; @BFP; @SWHO; @CBPS; @SOW; @SSPT; @SMS]. The recent discoveries of quasi-two-dimensional materials are realizations to test the validity of the $J_1-J_2$ model. Prominent among them are Li$_2$VO(Si,Ge)O$_4$ (Refs. [@MS; @MCLMTMM]), $AB$(VO)(PO$_4$)$_2$ ($A,B$=Pb, Zr, Sr, Ba) (Refs. [@KPhD; @KKG; @KRSSG]), (CuBr)$A^{'}_2B^{'}_3$O$_{10}$ ($A^{'}$=Ca, Sr, Ba, Pb; $B^{'}$=Nb, Ta) (Ref. [@TKBKYNK]), and (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl, Br) (Refs. [@YOTYIKKAY; @YOTKKAY]). (Cu$X$)LaNb$_2$O$_7$ compounds are of particular interest because they allow systematic tuning and understanding of the structural and magnetic properties, which are plausibly connected with the phenomenon of high-$T_c$ superconducting cuprates.
Although divalent copper with the electronic configuration $d^{9}$ should be Jahn-Teller active and lead to the cooperative lattice distortion (e.g., perovskite KCuF$_{3}$ (Ref. [@LAZ])), the precise crystal structure of the layered copper oxyhalides (Cu$X$)LaNb$_2$O$_7$ is still under debate. Earlier structural studies on (Cu$X$)LaNb$_2$O$_7$ were carried out with the tetragonal space group $P_4/mmm$, where the Cu and $X$ sites possess the $C_4$ symmetry [@KKZW; @KLZCSFSOW]. While the Rietveld refinement gave satisfactory results, the thermal parameter for halogens remained large. Besides, in this structure copper is in a significantly squeezed octahedral coordination with two short Cu-O bonds (about 1.9 Å) and four rather long Cu-$X$ bonds (2.7 Å), which are also quite unusual. Subsequently, the neutron diffraction experiment [@CKW] proposed that the Cl ions in (CuCl)LaNb$_2$O$_7$ ((CuCl)LNO) shifted away from the ideal Wyckoff $1b$ position [@KLZCSFSOW]. The transmission electron microscopy measurement on (CuCl)LNO (Ref. [@YOTYIKKAY]) revealed superlattice reflections corresponding to an enlarged $2\times 2$ unit cell. The nuclear magnetic resonance and the nuclear quadrupole resonance experiments for (CuCl)LNO and (CuBr)LaNb$_2$O$_7$ ((CuBr)LNO) further demonstrated the lack of the tetragonal symmetry at both Cu and Cl/Br sites [@YOTKKAY; @YOTYIKKAY].
The magnetic properties of (CuCl)LNO and (CuBr)LNO are also unusual and lack a clear microscopic interpretation. The former exhibits a spin liquid phase with a spin gap [@KKONNHVWYB; @YOTYIKKAY; @KYKTNHKBOAY] that are incompatible [@WD] with the square $J_1-J_2$ model. On the other hand, it has been reported [@OKKYBNHNHKSAY] that the replacement of Cl by Br leads to a CAFM order in (CuBr)LNO at low temperatures. However, it is unclear whether the Cu ions connected with the dominant exchange interaction couple ferromagnetically or antiferromagnetically [@YOTKKAY]. Moreover, both (CuCl)LNO and (CuBr)LNO are claimed to be FM $J_1$ compounds [@KKONNHVWYB; @OKKYBNHNHKSAY] whose justifications largely rely on the $J_1-J_2$ model. Yet, the structural study [@YOTKKAY] raised doubts over the validity of the model. Therefore, unambiguous determination of the crystal structure is crucial for understanding these complex systems.
At present, there are several structural models proposed for the Cu$X$ plane. Whangbo and Dai [@WD] suggested a model that consists of different ring clusters to explore the exchange couplings. However, the existence of inequivalent Cu and Cl sites in such a model is in contradiction to the experimental results that both Cu and $X$ occupy a unique crystallographic site with no substantial disorder [@YOTYIKKAY; @YOTKKAY]. Yoshida [*et al.*]{} [@YOTYIKKAY], based on the empirical evidence, proposed an orthorhombic distorted $2\times 2$ structure (hereafter referred to as the YY model). In this model, the displacement of Cl ions generates different exchange couplings among the nearest neighboring Cu pairs. A Cu dimer formed by the dominant exchange interaction was considered [@YOTYIKKAY] to study the spin-gap behavior. The same structural model was shown [@YOTKKAY] to consistently account for (CuBr)LNO. The third model, suggested by Tsirlin and Rosner (TR) [@TR] is also characterized by an ordering pattern but with a $2\times 1$ periodicity, where the local environment of copper is distorted to form the CuO$_2$Cl$_2$ plaquette.
First-principles calculations have proven to be an appealing method to deal with complex systems [@Er; @MV; @RSZODP]. Such a method can efficiently and reliably calculate the total energy, which is crucial in determining the most stable structure in order to study all relevant physical properties. In this work, we will investigate the atomic structure and resultant magnetic property of (CuCl)LNO and (CuBr)LNO based on the density functional theory. Our results show that, among several dozens of examined structures, the distortion pattern of the most stable one is similar to that of the YY model. The displacement of the $X$ ions changes the environment of copper to form the CuO$_2$$X_2$ plaquette. In addition, these two materials crystallize in the space group $Pbam$. The FM chains in CAFM (CuBr)LNO are found to be along the direction which is contrary to the previous conjecture [@YOTKKAY]. It will be shown that (CuCl)LNO still belongs to the AFM $J_1$ compound. The first- and second-neighbor exchange couplings of (CuCl)LNO and (CuBr)LNO are also discussed in detail.
CRYSTAL STRUCTURE AND COMPUTATIONAL DETAILS
===========================================
Figure \[fig1\] illustrates the basic crystal structure of the copper oxyhalides (Cu$X$)LaNb$_2$O$_7$. It is made up of copper-halogen planes and nonmagnetic double-perovskite LaNb$_2$O$_7$ slabs. The La ions are located at the 12-coordinate sites of the double-perovskite slabs. The Cu$X$ planes and the LaNb$_2$O$_7$ slabs alternate along the $c$ direction such that the copper is six-fold coordinated, bridging between the apical O ions of NbO$_6$ octahedra and surrounded by four $X$ halogens. Because of the short Cu-O bond length ($\sim$1.9 Å), the Cu$X$ plane is more appropriately considered as a Cu$X$O$_2$ layer. The initial structural study on (Cu$X$)LaNb$_2$O$_7$ was carried out with the space group $P_4/mmm$, where both Cu and $X$ have the $C_4$ symmetry [@KKZW; @KLZCSFSOW]. In this model (hereafter referred to as C4), the Cu and $X$ ions are located at the Wyckoff $1d$ and $1b$ positions, respectively (Fig. \[fig2\](a)). Later studies [@CKW; @YOTYIKKAY] proposed that Cl ions are displaced from the $C_4$-symmetry positions. The YY $2 \times 2$ model is represented in Fig. \[fig2\](b). The displacement of $X$ ions on the Cu$X$ plane leads to the formation of the $X$-Cu-$X$-Cu-$X$ zigzag chains, as indicated in Fig. \[fig2\](b). The original equivalent and perpendicular Cu chains are now distinguishable. Here, the direction extending along the zigzag chains is defined as the $b$ axis.
The present calculations were based on the generalized gradient approximation (GGA) [@PBE] to the exchange-correlation energy functional of the density functional theory. It is known [@ZR; @CS; @MG] that Cu-derived oxide compounds are usually strongly correlated systems. The correlation effect is important for the present systems to understand their ground state. Therefore, the on-site Coulomb interaction $U$ for Cu $3d$ electrons was also included [@LAZ] (GGA+$U$) in this work. Since the on-site exchange interaction $J$ is expected to be less influenced by the solid state effects [@TR], the relation $J=0.1U$ was used [@SHT] for different choices of $U$. The projector-augmented-wave potentials, as implemented in VASP [@KJ; @KF], were employed for the interactions between the ions and valence electrons. The plane-wave basis set with an energy cut-off of 500 eV was used. To minimize numerical uncertainties, structural optimizations were performed using a $2\times2$ supercell for all the test structures unless specified otherwise. The $6\times6\times4$ Monkhorst-Pack grids were taken to sample the corresponding Brillouin zone. The lattice parameters and atomic positions were relaxed until the total energy changed by less than $10^{-6}$ eV per conventional cell and the residual force was smaller than 0.01 eV/Å.
RESULTS AND DISCUSSION
======================
Energetics
----------
We first calculated the total energies of the C4 structure and several $2\times1$ and $2\times2$ distorted structures with different displacement patterns of the $X$ halogens. The YY $2\times2$ model is found to be the most stable one. As compared to the C4 structure, the YY model has a significant 0.3 and 0.2 eV/fu lowering in the energy of (CuCl)LNO and (CuBr)LNO, respectively. This directly rules out the possibility that the two compounds crystallize in the $C_4$ symmetry. Particularly, over the full $U$ range from 0 eV to 8 eV, the YY model is 0.1 eV/fu lower than the TR $2\times1$ model (see Fig. 4 in Ref. [@TR]) for both materials. The nonmagnetic calculations with and without $U$ lead to the same conclusion. Therefore, [*the structural distortion outweights the magnetism and on-site correlation effects*]{} in determining the atomic structure. Note that, besides Refs. [@YOTYIKKAY] and [@YOTKKAY], very recent experimental evidences [@KTKANNKIUUY] also confirm that the original unit cell should be double along both the $a$ and $b$ axes for the family of these compounds. Our study therefore provided theoretical support for the stabilization of the YY $2\times2$ model.
To examine whether there exists other more stable structure with the $X$ ions restricted to the Cu plane, we perform the calculations for (CuCl)LNO with twenty sets of random displacements of all four Cl ions from the positions in the YY model. However, no such structure was found. The resultant configurations of the trial structures are either relaxed back to the YY model or trapped into a nearby higher energy minimum.
Next, we allow the halogens in the YY model to move off the plane. It is found that the $X$ ions in the relaxed structures are $0.02-0.04$ Å away from the Cu plane. However, the change in the total energy is rather small. At $U$=0 and 8 eV, the results of both materials show that the energy differences are only within 2 meV per $2\times2$ supercell while the energies for the structures with the $X$ ions fixed in the plane remain lower. We also examine the two structures in Figs. 16(e) and 16(f) of Ref. [@YOTYIKKAY], which are based on another $2\times 2$ configuration with the Cl ions displaced away from the Cu plane in a different way. The calculations indicate that, after relaxation, both are energetically about 0.2 eV/fu higher than the YY model. The increase in the total energy mostly comes from the different in-plane structural distortions. Again, the contribution from the $z$-component shift of Cl ions is rather minor. Hence, the distortion on the Cu$X$ plane is predominantly crucial to stabilize the atomic structure. In the following discussion, we shall focus on the YY model with $X$ ions kept in the Cu plane [@com1].
Now, we analyze the total energies influenced by the on-site Coulomb interaction and the different magnetic configurations shown in Fig. \[fig3\]. Here, SC1 and SC3 are FM and NAFM. SC2 and SC4 are both CAFM, with the FM chain along the $b$ and $a$ directions, respectively. The results are displayed in Fig. \[fig4\], where the energy of SC2 was chosen as a reference. For (CuCl)LNO, the energies of SC1, SC2, and SC4 are very competing. The differences among them are within 1 meV/fu when $U\geq 6$. The first two are even almost identical around $U$=4 eV. Clearly, Fig. \[fig4\](a) shows that the SC3 is the lowest energy state and its energy is well separated from those of the other three magnetic structures. In the (CuBr)LNO case, similar tiny energy differences but between SC1, SC2, and SC3 are also found. Interestingly, when the FM chain in the CAFM state is set parallel along the $a$ axis, as in SC4, the total energy over the examined $U$ range is much higher than those of the other three configurations, indicating that the Cu ions along the $a$-axis should not couple ferromagnetically. This finding is contrary to the previous conjecture [@YOTKKAY]. The different energy ordering for the four magnetic configurations between the two compounds are conceivable since the magnetic interactions through the path Cu-$X$-Cu depend subtly on the small structural variation via the $X$-ion size effect. We will return to this issue in Sec. IIID when considering the various exchange couplings.
Atomic structure
----------------
Tables \[tab1\] lists the fully optimized structural parameters of both materials. For comparison, those obtained by the C4 model are also included. As can be seen in this table, the evaluated lattice constants are in good agreement with the experimental data [@CKW; @KKZW]. The discrepancies between them are only within $ 1\%$, the typical errors in the density-functional calculation. The $a$ and $b$ lattice constants of (CuCl)LNO are smaller than those of (CuBr)LNO, which is due primarily to the size effect of Br in the layered structure.
To discuss the structural distortion, we take the (CuCl)LNO case as an example. In the C4 model, copper is in the squeezed octahedral coordination with four long Cu-Cl bonds \[$d$(Cu-Cl)=2.77 Å\] and two short Cu-O bonds \[$d$(Cu-O)=1.85 Å\]. The displacements of the Cl and Cu ions in the YY model lead to two shorter Cu-Cl bonds of 2.38 and 2.39 Å, forming the Cl-Cu-Cl-Cu-Cl zigzag chain to stabilize the structure. The rest two Cu-Cl interatomic distances are increased to $3.27-3.29$ Å. In particular, the Cu-O bond length remains short after the structural distortion (from 1.85 Å to 1.88 Å), indicating the strong bonding character between Cu and O ions. The calculated interatomic distances are comparable to those reported previously [@CKW]. As a result, the distortion yields the nearly planar CuO$_2$Cl$_2$ rather than the octahedral CuO$_2$Cl$_4$ environment around the Cu ion (Fig. \[fig5\](a)). The resultant CuO$_2$Cl$_2$ planar structure is reminiscent of the conventional CuO$_4$ plaquette, which is commonly observed in copper oxides, e.g., La$_2$CuO$_4$ (Ref. [@CS]) and Sr$_2$CuO$_3$ (Ref. [@MG]). It should be noted that the CuO$_2$Cl$_2$-plaquette zigzag chains was also reported in the TR model [@TR]. Additionally, the basic electronic structure is similar to that of the CuO$_4$ planar unit, which will be demonstrated in the next section. Combined with the energetic advantage mentioned above, the YY model provides a realistic description for the atomic structures of (CuCl)LNO and (CuBr)LNO .
From a closer analysis of the positions of all ions in (CuCl)LNO, we found that the distorted atomic structure in the YY model belongs to the space group [*Pbam*]{} (No. 55) [@Book0]. The atomic positions are summarized in Table \[tab2\]. Clearly, the deviations of the Cl ions from the $C_4$-symmetry positions are as large as 0.66 Å, and these values are four times larger than those of Cu ions. Note that the displacements along the $a$ axis are more significant than those along the $b$ axis for both ions to form the zigzag chains.
As expected, the structural distortion on the CuCl plane leads La, Nb and O ions to shift from the the $C_4$-symmetry positions. Figure \[fig5\](b) and Table \[tab2\] show the significant tilting and distortion of NbO$_6$ octahedra. Such a tilting distortion is typical for perovskite oxides structures [@Book1]. Particularly, La ions shift along the $b$ axis by an amount of 0.10 Å. This displacement of La from the $C_4$-symmetry positions well agrees with the experimental nonzero value of the EFG tensor at La sites [@YOTYIKKAY], a strong evidence for the structural distortion in (CuCl)LNO. We also found that Nb ions shift along the $a$ axis by a relatively smaller amount of 0.02 Å. Note that the $a$ ($b$) component of La (Nb) displacement is almost negligible.
Taking into account the tilting of the NbO$_6$ octahedra in the (CuCl)LNO is important for providing a realistic description of the distortion on the CuCl plane. Figure \[fig5\](c), the top view of the atomic structure, clearly demonstrates that the cooperative tilting of the NbO$_6$ octahedra in the space group $Pbam$ results in a $2\times 2$ periodicity and leads to the zigzag chains with the same periodicity. It is worth pointing out that the higher symmetric $2\times 1$ zigzag chains in the TR model were investigated without consideration of the effect due to the tilting of the NbO$_6$ octahedra, where these octahedra were still kept at the C4 tetragonal sites [@TR]. Allowing the tilting distortion of NbO$_6$ octahedra in (CuCl)LNO lowers the symmetry of the atomic structure and correspondingly that of the zigzag chains and therefore leads to a lower total energy. In the YY model, the zigzag chains have the glide symmetry about $u=1/4$ and $v=1/4$. Specifically, the Cu-Cl bond of 2.39 Å in the zigzag chain is next to the Cl-Cu bond of 2.38 Å in the adjacent chain and vice versa. As compared to those in the TR model, such a [*complementary*]{} arrangement between adjacent chains in the YY model allows a further lowering in energy. Now, it is evident that the YY $2\times 2$ model in the present study is energetically more stable than the TR $2\times 1$ one. Similar conclusion holds for (CuBr)LNO.
electronic structure
--------------------
Figure \[fig6\] depicts the orbital- and site-projected densities of states (DOS) of (CuCl)LNO with $U$=6 eV, where the valence-band maximum ($E_v$) is set to zero. The orbitals are projected in the local coordinates with the $x$ and $y$ axes directed to the neighboring Cl ions and the $z$ axis coinciding with the crystal $c$ axis (Fig. \[fig2\](b)). Among the major valence state region of 6.3 eV, the higher-energy part consists almost exclusively of O and Cl $p$ states. There is larger contribution from the Cl $p$ state just below $E_v$. The lower-energy part, dominated by the Cu $d$ states, is splitted into the doubly occupied $dxy$, $dyz$, $dzx$, $d(x^2-y^2)$, and singly occupied $d(3z^2-r^2)$ states. As compared to the GGA DOS (not shown here), the GGA+$U$ shows an essential redistribution of the Cu $3d$ DOS, i.e., from being above to below the O and Cl $p$ states. That the energy gap ($E_g$) lies between the occupied anion $p$ states and the unoccupied Cu $d$ states is similar to those in the charge-transfer insulators, e.g., La$_2$CuO$_4$ (Ref. [@CS]) and Sr$_2$CuO$_3$ (Ref. [@MG]). The sharp peak of the low-lying Cu $d(3z^2-r^2)$ state is a result of the strong bonding between the Cu and O ions with a considerably short Cu-O bond length of 1.88 Å (see Table \[tab1\]). Note that the $d(3z^2-r^2)$ orbital was hybridized with little Cl $p$ component. The on-site correlation $U$ leads to the half-filling of the $d(3z^2-r^2)$ orbital, of which the lobes point to the O ions. These results imply a single orbital ground state. Figure \[fig6\] shows that, due to the hybridization with the O $p$ state [@CS], the $d(3z^2-r^2)$ bonding-antibonding separation (8.3 eV) is larger than the value of $U$. Therefore, the electronic structure due to the CuO$_2$Cl$_2$ plaquette in the YY model is very similar to those of other copper oxides [@CS] with planar CuO$_4$ units.
We found that the structural distortion and magnetism together already open up the gap. The $E_g$ of (CuCl)LNO obtained by the GGA is 0.3 eV. However, this result is insufficient to account for the green color appearance [@KKZW] of this compound. At $U=6$ eV, the $E_g$ is increased to 1.8 eV. Further increase of $U$ makes no significant widening for the band gap. The main structures in the DOS of (CuCl)LNO are also found in that of (CuBr)LNO, except for the smaller $E_g$ of 1.5 eV. At this choice of $U$, the local magnetic moment at the Cu site of (CuBr)LNO is evaluated to be 0.6 Bohr magneton, which agrees with the experiments [@YOTKKAY; @OKKYBNHNHKSAY]. An amount of 0.1 Bohr magneton at Br sites is also observed. Hence, we choose the optimal $U$ = 6 eV case to discuss the corresponding atomic and electronic properties.
exchange interaction
--------------------
Finally, we discuss the exchange couplings for both (CuCl)LNO and (CuBr)LNO. In the undistorted C4 structure, the interactions between the Cu ions can be approximately modeled by the Heisenberg Hamiltonian $\hat{H}=J_1\sum_{NN}S_i\cdot S_j +J_2\sum_{2NN}S_i\cdot S_j$, where the sums run over the first and second nearest-neighbor pairs, respectively, and $S_i$ corresponds to the spin moment at site $i$. The relevant exchange couplings can be then determined by applying the model to the calculated energies of different spin configurations. For the YY model, the formation of the $X$-Cu-$X$-Cu-$X$ zigzag chains along the $b$ axis (Fig. \[fig2\](b)) lifts the tetragonal symmetry and leads to inequivalent superexchange pathways, as indicated in Fig. \[fig2\]. As a result, the $J_1$ in the C4 structure is split into $J_{11}$, $J_{12}$, and $J_{13}$, with the former two now being the first neighboring inter-chain interactions and the latter the first neighboring intra-chain interaction. The original $J_2$ coupling is split into two inequivalent $J_{21}$ and $J_{22}$, which are correspondingly the second neighboring inter-chain interactions. We investigate these interactions via the various spin configurations in Fig. \[fig3\]. The results are summarized in Table \[tab3\].
We first discuss the results from the C4 model. Table \[tab3\] shows that $J_1$ is almost negligible as compared to $J_2$. This is reasonable because, as illustrated in Fig. \[fig7\](a), there is no overlap between the Cu2 $d(x^2-y^2)$ and $X$4 $p$ orbitals. Therefore, even with the obvious overlapping of the Cu1 $d$ and $X$4 $p$ orbitals, Cu1 and Cu2 could hardly interact with each other. On the other hand, Cu1 can interact with Cu3 via the $X$4 $p$ orbital.
Based on the $J_1-J_2$ model, both (CuCl)LNO and (CuBr)LNO were previously claimed [@KKONNHVWYB; @OKKYBNHNHKSAY] to be FM $J_1$ magnets with competing AFM $J_2$ interactions, as in the case of Pb$_2$VO(PO$_4$)$_2$ (Ref. [@KRSSG]). Table \[tab3\] indeed shows that $J_1<0$ and $J_2>0$ for both materials in the C4 model, a direct consequence of the Hund’s coupling and virtual electron hopping. However, the recent structural study [@YOTKKAY] has raised serious doubt over the validity of the $J_1-J_2$ model in such materials. Our calculations also indicate that consideration of the structural distortion leads to the opposite results. For (CuCl)LNO, the effective interactions $(J_{11}+J_{12}+2J_{13})/4$ and $(J_{21}+J_{22})/2$ in the YY model are found to be AFM and FM, respectively. And they both become FM for (CuBr)LNO at large $U$s. These results come from the complicated interplay between the Hund’s coupling, virtual electron hopping, the distorted structure and $X$-ion size effect. It should be noted that the TR model [@TR] also results in a leading AFM coupling in (CuCl)LNO. This implies that the simple $J_1-J_2$ model is unable to describe the present systems. Moreover, the first neighboring interactions become more significant as compared to the second neighboring ones. Figure \[fig7\](b) clearly shows that, unlike the C4 case, $\angle$ Cu1-$X4$-Cu2 is no longer 90$^o$ (Table \[tab1\]) due to the structural distortion. This will lead to the overlapping of Cu2 (Cu4) $d$ and $X$4 $p$ orbitals, and enhance the interaction between Cu1 and Cu2 (Cu4).
In fact, Fig. \[fig8\](a) shows that for (CuCl)LNO, $(J_{11}+J_{12})/2>0$, $J_{13}>0$, and $(J_{21}+J_{22})/2<0$ for all the $U$s considered. It is now clear that, since the interactions due to all the corresponding spin pairs in SC3 satisfy these conditions, (CuCl)LNO in SC3 is much more stable than in the rest three configurations of Fig. \[fig3\]. Actually, SC3 is the most stable structure among all the spin configurations with the interactions up to second-nearest neighbors. However, by comparing Fig. \[fig8\](b) with Fig. \[fig8\](a), we found that the $J_{13}$ in (CuBr)LNO becomes FM. This could be ascribed to the fact that the interaction $J_{13}$ depends sensitively on the angle of the Cu1-$X$4-Cu2 superexchange path in Fig. \[fig7\], and the replacement of Cl by Br will change this angle and modify the interaction. Therefore, for (CuBr)LNO, the first neighboring couplings within the chains are FM, and the first two neighboring couplings between adjacent chains are AFM and FM, respectively. None of the four structures in Fig. \[fig3\] satisfies these conditions. Specifically, the interactions due to the corresponding spin pairs in SC4 are all opposite to these couplings, giving rise to the result of the SC4 being the highest-energy structure for (CuBr)LNO.
The above analysis seems to indicate that, contrary to previous expectations [@KKONNHVWYB; @YOTYIKKAY; @KYKTNHKBOAY; @OKKYBNHNHKSAY], (CuCl)LNO rather than (CuBr)LNO is less frustrated [@OKKYBNHNHKSAY]. To check the reliability of our calculations, we perform the total-energy calculation for an additional structure SC5 with three of the four spins being the same but opposite to the fourth one. For all the possible choices of four magnetic structures containing SC5 in solving the Heisenberg Hamiltonian, the deviations of the relevant couplings (dashed lines in Fig. \[fig8\]) from those obtained by SC1$-$SC4 are less than 1.0 meV. More importantly, the signs of these interactions remain unaltered for both materials.
One plausible explanation for the above puzzling discrepancy is that the third neighboring couplings between different zigzag chains [@com3] may not be completely negligible [@TR; @KKONNHVWYB]. Actually, in the YY model, the two CuO$_2$$X_2$ plaquettes with the Cu1 and Cu5 ions in Fig. \[fig7\](b) could be considered approximately coplanar. Kageyama [*et al.*]{} [@KKLW] argue that this kind of coplanarity provides an opportunity for the interaction between Cu1 and Cu5 through the overlap of the Cu1 $d(x^2-y^2)-$X4 $p-X$5 $p-$Cu5 $d(x^2-y^2)$ orbitals. So, from the structural geometry point of view, such a long-range coupling (8.5 Å) could be possible. However, to examine and identify these couplings, one has to take into account additional eight inequivalent couplings and use a larger supercell whose corresponding calculations are very time-consuming and yet, likely, not accurate enough for the present purposes (Fig. \[fig4\]). Therefore, we will not address this issue presently. Further theoretical and experimental work is required to clarify this point.
It is worth pointing out that the first neighboring interactions within the chains and between the adjacent chains have the opposite signs for (CuBr)LNO. The spatial asymmetry of these results again demonstrates the inappropriateness of the square $J_1-J_2$ model for the Br compound. Furthermore, for both materials, the couplings between the adjacent chains are very competitive to the intra-chain interactions, sharply contrary to the previous conclusion [@TR]. Tsirlin and Rosner have argued [@TR] that in the TR model, where the basic structure element is also the CuO$_2$Cl$_2$-plaquette zigzag chain, the large hopping runs along the chain and leads to the strongest interaction. According to their discussion, the inter-chain interaction is rather weak due to the long “nonbonding” Cu-Cl distance and the lack of the proper superexchange path. In the present study, despite the similar backbone in the YY model, the couplings between the adjacent chains are shown to be still substantial. As mentioned before, the strength of exchange interactions between two spin sites should be determined by the overlap of orbitals rather than the distance between them. The interactions between the Cu ions from adjacent chains could be significant through the path mediated by the extended $3p$ orbital of Cl ions (vs O$^{2-}$) and would be enhanced in the Br case with the further extended $4p$ orbital. Indeed, our calculations show that all the in-plane exchange couplings in (CuBr)LNO are three times larger than those in (CuCl)LNO.
For the inter-layer interaction $J_{\perp}$, Table \[tab3\] shows that the $J_{\perp}$ is AFM, in agreement with the experiment [@OKKYBNHNHKSAY]. When compared to the in-plane interaction, the $J_{\perp}$ in (CuCl)LNO is non-negligible, implying that some long-path (12 Å) interaction between the Cu ions is still cooperative. The origin of this long-range coupling could be associated with the interaction through the Cu $d(3z^2-r^2)$ orbital. As discussed in Sec. IIIC, this orbital strongly overlaps with the O $p_z$ orbital. The O $p_z$ orbitals further couple with Nb $4d$ orbitals. Therefore, the inter-plane coupling $J_{\perp}$ shall involve the Cu-O-Nb-O-Nb-O-Cu path. In (CuBr)LNO, however, the coupling is found to be relatively less significant. The strength of all the interactions interested here is decreased with increasing $U$. The evolution is expected since adding $U$ makes the wavefunctions more localized and the virtual electrons hopping less favorable.
CONCLUSIONS {#con}
===========
In conclusion, we have investigated the atomic structure and magnetic property of the copper oxyhalides (Cu$X$)LaNb$_2$O$_7$($X$=Cl and Br) based on the density functional theory. The calculations show that, among the examined structures, the YY $2\times 2$ model proposed by Yoshida [*et al.*]{} [@YOTYIKKAY] has the lowest energy. This model is significantly more stable than both the undistorted C4 structure and the TR $2\times 1$ model suggested recently by Tsirlin and Rosner [@TR]. The $X$ and Cu ions in the YY model are displaced to form the $X$-Cu-$X$-Cu-$X$ zigzag chains and the local environment of copper is distorted to form a nearly CuO$_2$$X_2$ plaquette. We found that (Cu$X$)LaNb$_2$O$_7$ crystallizes in the space group $Pbam$. The cooperative tilting of the NbO$_6$ octahedra leads to the lower symmetry of the zigzag chains with a $2\times 2$ periodicity. With consideration of the on-site Coulomb interaction, the YY model shows the single-orbital scenario typical for copper oxides and oxyhalides.
We concluded that (CuCl)LNO is still the AFM $J_1$ magnet with mixing FM $J_2$ interactions. For (CuCl)LNO, the first neighboring interactions within the zigzag chains are AFM, and the first two neighboring couplings between adjacent chains are AFM and FM, respectively. However, the replacement of Cl by Br modifies the first neighboring intra-chain interaction to be FM for (CuBr)LNO. Despite the “well”-separated zigzag chains in the YY model, the couplings between adjacent chains are comparable to those within the chain. The opposite signs of the inter- and intra-chain interactions in (CuBr)LNO reflect the spatial asymmetry and therefore the failure of the simple $J_1-J_2$ model for such material. All the in-plane exchange couplings in (CuBr)LNO are shown to be three times those in the Cl counterpart. It is found that the inter-plane interaction $J_{\perp}$ is AFM, in agreement with the experiment [@OKKYBNHNHKSAY]. The present study strongly suggests that the simple square $J_1-J_2$ model should be modified to explore the magnetic property of Cu$X$LaNb$_2$O$_7$. We hope the present calculations will shed light on the precise crystallographic determination and the magnetic properties of Cu$X$LaNb$_2$O$_7$.
We are grateful to P. Sindzingre for bringing these systems to our attention. Computer resources provided by the National Center for High-performance Computing are gratefully acknowledged. This work was supported by the National Science Council and National Center for Theoretical Sciences of Taiwan.
[99]{} , edited by H.T. Diep (World Scientific, Singapore, 1994); [*Frustrated spin systems*]{}, edited by H.T. Diep (World Scientific, Singapore, 2004). E. Dagotto and A. Moreo, Phys. Rev. Lett. [**63**]{}, 2148 (1989). P. Chandra, P. Coleman, and A. I. Larkin, Phys. Rev. Lett. [**64**]{}, 88 (1990). R. F. Bishop, D. J. J. Farnell, and J. B. Parkinson, Phys. Rev. B [**58**]{}, 6394 (1998). R. R. P. Singh, Z. Weihong, C. J. Hamer, and J. Oitmaa, Phys. Rev. B [**60**]{}, 7278 (1999). L. Capriotti, F. Becca, A. Parola, and S. Sorella, Phys. Rev. Lett. [**87**]{}, 097201 (2001). O. P. Sushkov, J. Oitmaa, and Z. Weihong, Phys. Rev. B [**63**]{}, 104420 (2001). J. Sirker, Z. Weihong, O. P. Sushkov, and J. Oitmaa, Phys. Rev. B [**73**]{}, 184420 (2006). D. Schmalfuß, R. Darradi, J. Richter, J. Schulenburg, and D. Ihle, Phys. Rev. Lett. [**97**]{}, 157201 (2006). J. R. Viana and J. R. de Sousa, Phys. Rev. B [**75**]{}, 052403 (2007). A. O’Hare, F. V. Kusmartsev, and K. I. Kugel, Phys. Rev. B [**79**]{}, 014439 (2009). N. Shannon, B. Schmidt, K. Penc, and P. Thalmeier, Eur. Phys. J. B [**38**]{}, 599 (2004). N. Shannon, T. Momoi, and P. Sindzingre, Phys. Rev. Lett. [**96**]{}, 027213 (2006). P. Millet and C. Satto, Mater. Res. Bull. [**33**]{}, 1339 (1998). R. Melzi, P. Carretta, A. Lascialfari, M. Mambrini, M. Troyer, P. Millet, and F. Mila, Phys. Rev. Lett. [**85**]{}, 1318 (2000). E. Kaul, Ph.D. thesis, Technische Universit$\ddot{a}$t Dresden, Dresden, 2005. N. S. Kini, E. E. Kaul, and C. Geibel, J. Phys.: Condens. Matter [**18**]{}, 1303 (2006). E. E. Kaul, H. Rosner, N. Shannon, R. V. Shpanchenko and C. Geibel, J. Magn. Magn. Mater. [**272-276(II)**]{}, 922 (2004). Y. Tsujimoto, H. Kageyama, Y. Baba, A. Kitada, T. Yamamoto, Y. Narumi, K. Kindo, M. Nishi, J. P. Carlo, A. A. Aczel, T. J. Williams, T. Goko, G. M. Luke, Y. J. Uemura, Y. Ueda, Y. Ajiro, and K. Yoshimura, Phys. Rev. B [**78**]{}, 214410 (2008). M. Yoshida, N. Ogata, M. Takigawa, J. Yamaura, M. Ichihara, T. Kitano, H. Kageyama, Y. Ajiro, K. Yoshimura, J. Phys. Soc. Jpn. [**76**]{}, 104703 (2007). M. Yoshida, N. Ogata, M. Takigawa, T. Kitano, H. Kageyama, Y. Ajiro, K. Yoshimura, J. Phys. Soc. Jpn. [**77**]{}, 104705 (2008). A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B [**52**]{}, R5467 (1995). T. A. Kodenkandath, J. N. Lalena, W. L. Zhou, E. E. Carpenter, C. Sangregorio, A. U. Falster, W. B. Simmons, C. J. O’Connor, J. B. Wiley, J. Am. Chem. Soc. [**121**]{}, 10743 (1999). T. A. Kodenkandath, A. S. Kumbhar, W. L. Zhou, and J. B. Wiley, Inorg. Chem. [**40**]{}, 710 (2001). G. Caruntu, T. A. Kodenkandath, J. B. Wiley, Mater. Res. Bull. [**37**]{}, 593 (2002). H. Kageyama, J. Yasuda, T. Kitano, K. Totsuka, Y. Narumi, M. Hagiwara, K. Kindo, Y. Baba, N. Oba, Y. Ajiro, and K. Yoshimura, J. Phys. Soc. Jpn. [**74**]{}, 3155 (2005). H. Kageyama, T. Kitano, N. Oba, M. Nishi, S. Nagai, K. Hirota, L. Viciu, J. B. Wiley, J. Yasuda, Y. Baba, Y. Ajiro, and K. Yoshimura, J. Phys. Soc. Jpn. [**74**]{}, 1702 (2005). M. H. Whangbo and D. Dai, Inorg. Chem. [**45**]{}, 6227 (2006). N. Oba, H. Kageyama, T. Kitano, J. Yasuda, Y. Baba, M. Nishi, K. Hirota, Y. Narumi, M. Hagiwara, K. Kindo, T. Saito, Y. Ajiro, and K. Yoshimura, J. Phys. Soc. Jpn. [**75**]{}, 113601 (2006). A. A. Tsirlin and H. Rosner, Phys. Rev. B [**79**]{},214416 (2009). S. C. Erwin, Phys. Rev. Lett. [**91**]{}, 206101 (2003). A. Malashevich and D. Vanderbilt, Phys. Rev. Lett. [**101**]{}, 037210 (2008). H. Rosner, R. R. P. Singh, W. H. Zheng, J. Oitmaa, S. -L. Drechsler, and W. E. Pickett, Phys. Rev. Lett. [**88**]{}, 186405 (2002). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996). F. C. Zhang and T. M. Rice, Phys. Rev. B [**37**]{}, 3759 (1988). M. T. Czyżyk and G. A. Sawatzky, Phys. Rev. B [**49**]{}, 14211 (1994). W. C. Mackrodt and H. J. Gotsis, Phys. Rev. B [**62**]{}, 10728 (2000). I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev. B [**53**]{}, 7158 (1996). G. Kresse and D. Joubert, Phys. Rev. B [**59**]{}, 1758 (1999). G. Kresse and J. Furthmüller, Comput. Mater. Sci. [**6**]{}, 15 (1996). A. Kitada, Y. Tsujimoto, H. Kageyama, Y. Ajiro, M. Nishi, Y. Narumi, K. Kindo, M. Ichihara, Y. Ueda, Y. J. Uemura, and K. Yoshimura, Phys. Rev. B [**80**]{}, 174409 (2009). The model in Ref. [@OKSAPKAY] originally proposed for (FeCl)LaNb$_2$O$_7$ (the same shown in Fig. 16(a) of Ref. [@YOTYIKKAY]) is energetically less favorable than the YY model. N. Oba, H. Kageyamaa, T. Saito, M. Azuma, W. Paulus, T. Kitano, Y. Ajiro, and K. Yoshimur, J. Magn. Magn. Mater. [**310**]{}, 1337 (2007). , edited by T. Hahn (Kluwer, Dordrecht, 1989), Vol. A. , edited by C. N. R. Rao and B. Raveau (World Scientific, Singapore, 1998). From the viewpoint of the square $J_1-J_2$ model, these interactions should be regarded as the fourth-nearest-neighbor interactions. H. Kageyama, J. Kang, C. Lee, and M.-H. Whangbo, (unpublished).
Fig. 1: (Color online) Crystal structure of (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl, Br) in the tetragonal space group $P_4/mmm$.\
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Fig. 2: (Color online) (a) The undistorted tetragonal model and (b) the distorted model proposed by Yoshida [*et al.*]{} [@YOTYIKKAY] for (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl, Br). Large and small spheres denote $X$ and Cu ions, respectively. The relevant exchange couplings are also indicated.\
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Fig. 3: (Color online) The four different spin configurations of Cu ions considered in the present study. Large and small spheres denote Cl/Br and Cu ions, respectively.\
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Fig. 4: (Color online) Total energies of (a) (CuCl)LaNb$_2$O$_7$ and (b) (CuBr)LaNb$_2$O$_7$ in the spin configurations shown in Fig. \[fig3\]. The energies are relative to that of SC2.\
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Fig. 5: (Color online) Perspective view of (a) the CuO$_2X_2$-plaquette zigzag chains and (b) tilted NbO$_6$ octahedra of (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl, Br) in the space group $Pbam$. (c) Top view of (b). Here, the symbols for the various atomic species are the same as those in Fig. \[fig1\].\
\
Fig. 6: (Color online) Orbital- and site-projected density of states (DOS) of (CuCl)LaNb$_2$O$_7$, obtained by $U=6$ eV and SC2 in Fig. \[fig3\]. The energy is relative to the valence-band maximum.\
\
Fig. 7: (Color online) Schematic plot of Cu $d(x^2-y^2)$ and $X$ $p$ orbitals of (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl, Br) in the space group (a) $P_4/mmm$ and (b) $Pbam$.\
\
Fig. 8: (Color online) The $U$-dependence of the in-plane exchange couplings of (a) (CuCl)LaNb$_2$O$_7$ and (b) (CuBr)LaNb$_2$O$_7$. The solid lines are calculated from SC1$-$SC4 in Fig. \[fig3\]. The dashed lines show the uncertainty of the calculations. See the text for details. Note that the scale of exchange coupling in (a) is only half of that in (b).
(CuCl)LaNb$_2$O$_7$ (CuBr)LaNb$_2$O$_7$
------------------------ ----------------------------- ------------------------------------
$a$ 7.868 7.889
$b$ 7.883 7.914
$c$ 11.878 11.853
Cu1-$X1$ 2.39 (2.40$^{\dagger}$) 2.54
Cu1-$X2$ 3.27 (3.14$^{\dagger}$) 3.11
Cu1-$X3$ 3.29 3.11
Cu1-$X4$ 2.38 2.52
Cu1-O 1.88 (1.84$^{\dagger}$) 1.88
$\angle$ $X$1-Cu1-$X2$ 83.6 81.5
$\angle$ $X2$-Cu1-$X4$ 87.1 88.8
$\angle$ $X4$-Cu1-$X3$ 102.7 101.3
$\angle$ $X3$-Cu1-$X1$ 86.6 88.3
$\angle$ Cu1-$X4$-Cu2 112.2 103.4
$a^{\star}$ 3.914$~(3.884^{\dagger})$ 3.942 $(3.899^{\dagger\dagger})$
$c^{\star}$ 11.892 (11.736$^{\dagger}$) 11.853 (11.706$^{\dagger\dagger}$)
Cu-$X$$^{\star}$ 2.77 2.79
Cu-O$^{\star}$ 1.85 1.87
: The calculated lattice constants (Å), relevant interatomic distances (Å), and bond angles ($^{\bf o}$) of (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl, Br), obtained by $U=6$ eV and SC2 in Fig. \[fig3\]. The latter two refer to those in Fig. \[fig2\](b). In the bottom part, the results in the undistorted tetragonal model are listed for comparison. The values in parentheses are the corresponding experimental data. []{data-label="tab1"}
$\dagger$: Ref. [@CKW]\
$\dagger\dagger$: Ref. [@KKZW]
----- ------ -------- -------- -------- -------- -------- --------
ion site $u$ $v$ $w$ $u$ $v$ $w$
Cu $4h$ 0.2706 0.0077 0.5 0.2720 0.0057 0.5
$X$ $4h$ 0.4185 0.2711 0.5 0.4481 0.2713 0.5
La $4g$ 0.5000 0.2622 0 0.5000 0.2624 0
Nb $8i$ 0.2522 0.0000 0.1911 0.2520 0.9998 0.1903
O1 $4f$ 0.5 0 0.1336 0.5 0 0.1334
O2 $4e$ 0 0 0.1841 0 0 0.1834
O3 $8i$ 0.2498 0.2501 0.1522 0.2498 0.2499 0.1520
O4 $4g$ 0.2004 0.0001 0 0.2004 0.0000 0
O5 $8i$ 0.2830 0.0008 0.3417 0.2818 0.0000 0.3413
----- ------ -------- -------- -------- -------- -------- --------
: The calculated atomic structural parameters of (Cu$X$)LaNb$_2$O$_7$ ($X$=Cl and Br) in the space group [*Pbam*]{}, obtained by $U=6$ eV and SC2 in Fig. \[fig3\]. $u$, $v$, and $w$ denote fractional coordinates based on the $a$, $b$, and $c$ lattice constants, respectively.[]{data-label="tab2"}
----- --------------------- ---------- --------------------- ------------- --------- --------- --------------------- ---------- --------------------- ------------- --------- ---------
$U$ $(J_{11}+J_{12})/2$ $J_{13}$ $(J_{21}+J_{22})/2$ $J_{\perp}$ $J_{1}$ $J_{2}$ $(J_{11}+J_{12})/2$ $J_{13}$ $(J_{21}+J_{22})/2$ $J_{\perp}$ $J_{1}$ $J_{2}$
0 8.9 21.3 $-3.7$ 5.1 $-1.8$ 18.4 28.2 $-16.8$ $-10.6$ 5.4 $-2.9$ $-1.5$
4 3.6 5.4 $-1.8$ 2.0 $-3.6$ 26.9 11.9 $-11.9$ $-5.9$ 2.2 $-1.8$ 23.3
6 2.2 3.0 $-1.0$ 1.3 $-2.5$ 19.4 7.9 $-8.6$ $-3.5$ 1.5 $-0.1$ 25.5
8 1.7 2.1 $-0.7$ 0.7 $-1.7$ 13.5 5.3 $-6.2$ $-2.0$ 0.9 $-0.3$ 18.6
----- --------------------- ---------- --------------------- ------------- --------- --------- --------------------- ---------- --------------------- ------------- --------- ---------
: The exchange couplings (meV) of (CuCl)LaNb$_2$O$_7$ and (CuBr)LaNb$_2$O$_7$. The notation is explained in the text. $U$ (eV) is the on-site Coulomb correlation interaction.[]{data-label="tab3"}
|
---
author:
- 'Philip Greengard, Kirill Serkh'
title: 'Zernike Polynomials: Evaluation, Quadrature, and Interpolation'
---
Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we provide a self-contained reference on Zernike polynomials, algorithms for evaluating them, and what appear to be new numerical schemes for quadrature and interpolation. We also introduce new properties of Zernike polynomials in higher dimensions. The quadrature rule and interpolation scheme use a tensor product of equispaced nodes in the angular direction and roots of certain Jacobi polynomials in the radial direction. An algorithm for finding the roots of these Jacobi polynomials is also described. The performance of the interpolation and quadrature schemes is illustrated through numerical experiments. Discussions of higher dimensional Zernike polynomials are included in appendices.
[**Zernike Polynomials: Evaluation, Quadrature, and Interpolation**]{}\
Philip Greengard$^{\dagger\, \star}$ and Kirill Serkh$\mbox{}^{\ddagger \, \diamond}$
$^{\star}$ This author’s work was supported in part under ONR N00014-14-1-0797, AFOSR FA9550-16-0175, and NIH 1R01HG008383.\
$\mbox{}^{\diamond}$ This author’s work was supported in part by the NSF Mathematical Sciences Postdoctoral Research Fellowship (award no. 1606262) and AFOSR FA9550-16-1-0175.\
$\mbox{}^{\dagger}$ Dept. of Mathematics, Yale University, New Haven, CT 06511\
$\mbox{}^{\ddagger}$ Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
[**Keywords:**]{} [*Zernike polynomials, Orthogonal polynomials, Radial basis functions, Approximation on a disc*]{}
Introduction
============
Zernike polynomials are a family of orthogonal polynomials that are a natural basis for the approximation of smooth functions on the unit disk. Among other applications, they are widely used in optics and atmospheric sciences and are the natural basis for representing Generalized Prolate Spheroidal Functions (see [@slepian]).
In this report, we provide a self-contained reference on Zernike polynomials, including tables of properties, an algorithm for their evaluation, and what appear to be new numerical schemes for quadrature and interpolation. We also introduce properties of Zernike polynomials in higher dimensions and several classes of numerical algorithms for Zernike polynomial discretization in $\R^n$. The quadrature and interpolation schemes provided use a tensor product of equispaced nodes in the angular direction and roots of certain Jacobi polynomials in the radial direction. An algorithm for the evaluation of these roots is also introduced.
The structure of this paper is as follows. In Section \[secmprem\] we introduce several technical lemmas and provide basic mathematical background that will be used in subsequent sections. In Section \[secnumev\] we provide a recurrence relation for the evaluation of Zernike polynomials. Section \[seczernquad\] describes a scheme for integrating Zernike polynomials over the unit disk. Section \[secapprox\] contains an algorithm for the interpolation of Zernike polynomials. In Section \[secnumres\] we give results of numerical experiments with the quadrature and interpolation schemes introduced in the preceding sections. In Appendix A, we describe properies of Zernike polynomials in $\R^n$. Appendix B contains a description of an algorithm for the evaluation of Zernike polynomials in $\R^n$. Appendix C includes an description of Spherical Harmonics in higher dimensions. In Appendix D, an overview is provided of the family of Jacobi polynomials whose roots are used in numerical algorithms for high-dimensional Zernike polynomial discretization. Appendix D also includes a description of an algorithm for computing their roots. Appendix E contains notational conventions for Zernike polynomials.
Mathematical Preliminaries {#secmprem}
==========================
In this section, we introduce notation and several technical lemmas that will be used in subsequent sections.
For notational convenience and ease of generalizing to higher dimensions, we will be denoting by $S_N^\ell(\theta):
\R \rightarrow \R$, the function defined by the formula $$\label{20}
S_N^\ell(\theta) = \left\{
\begin{array}{ll}
(2\pi)^{-1/2} & \mbox{if $N = 0$}, \\
\sin(N\theta)/\sqrt{\pi} & \mbox{if $\ell = 0$, $N>0$}, \\
\cos(N\theta)/\sqrt{\pi} & \mbox{if $\ell = 1$, $N>0$}.
\end{array}
\right.$$ where $\ell \in\{0,1\}$, and $N$ is a non-negative integer. In accordance with standard practice, we will denoting by $\delta_{i,j}$ the function defined by the formula $$\begin{aligned}
\label{40}
\delta_{i,j} = \left\{
\begin{array}{ll}
1 & \mbox{if $i = j$}, \\
0 & \mbox{if $i \ne j$}.
\end{array}
\right.
\end{aligned}$$ The following lemma is a classical fact from elementary calculus.
\[60\] For all $n \in \{1,2,...\}$ and for any integer $k\geq n+1$, $$\label{80}
\frac{1}{k}\sum_{i=1}^k \sin(n\theta_i)=\int_0^{2\pi}
\sin(n\theta)d\theta=0$$ and $$\label{100}
\frac{1}{k}\sum_{i=1}^k \cos(n\theta_i)=\int_0^{2\pi}
\cos(n\theta)d\theta=0$$ where $$\label{120}
\theta_i=i\frac{2\pi}{k}$$ for $i=1,2,...,k$.
The following technical lemma will be used in Section \[seczernquad\].
\[140\] For all $m\in \{0,1,2,...\}$, the set of all points $(N,n,\ell) \in \R^3$ such that $\ell\in \{0,1\}$, $N,n$ are non-negative integers, and $N+2n \leq 2m-1$ contains exactly $2m^2+2m$ elements.
Lemma \[140\] follows immediately from the fact that the set of all pairs of non-negative integers $(N,n)$ satisfying $N+2n \leq 2m-1$ has $m^2+m$ elements where $m$ is a non-negative integer.
The following is a classical fact from elementary functional analysis. A proof can be found in, for example, [@stoer].
\[130\] Let $f_1,...,f_{2n-1}:[a,b]\rightarrow \R$ be a set of orthonormal functions such that for all $k\in \{1,2,...,2n-1\}$, $$\int_a^b f_k(x) dx = \sum_{i=1}^n f_k(x_i) \omega_i dx$$ where $x_i \in [a,b]$ and $\omega_i \in \R$. Let $\phi:[a,b]\rightarrow \R$ be defined by the formula $$\phi(x)=a_1f_1(x)+...+a_{n-1}f_{n-1}(x).$$ Then, $$a_k=\int_a^b \phi(x) f_k(x) dx=\sum_{i=1}^n \phi(x_i)f_k(x_i)\omega_i.$$ for all $k\in \{1,2,...,n-1\}$.
Jacobi Polynomials {#secjacpol}
------------------
In this section, we define Jacobi polynomials and summarize some of their properties.\
Jacobi Polynomials, denoted $P_n^{(\alpha,\beta)}$, are orthogonal polynomials on the interval $(-1,1)$ with respect to weight function $$w(x)=(1-x)^{\alpha} (1+x)^{\beta}.$$ Specifically, for all non-negative integers $n,m$ with $n\neq m$ and real numbers $\alpha,\beta>-1$, $$\label{192}
\int_{-1}^{1}P_n^{(\alpha,\beta)}(x)P_m^{(\alpha,\beta)}(x)
(1-x)^{\alpha} (1+x)^{\beta}dx=0$$ The following lemma, provides a stable recurrence relation that can be used to evaluate a particular class of Jacobi Polynomials (see, for example, [@abramowitz]).
For any integer $n\geq1$ and $N\geq0$, $$\begin{aligned}
\label{195}
&\hspace*{-5em} P_{n+1}^{(N,0)}(x)=
\frac{(2n+N+1)N^2+(2n+N)(2n+N+1)(2n+N+2)x}
{2(n+1)(n+N+1)(2n+N)}
P_n^{(N,0)}(x) \notag \\
&-\frac{2(n+N)(n)(2n+N+2)}
{2(n+1)(n+N+1)(2n+N)}
P_{n-1}^{(N,0)}(x),\end{aligned}$$ where $$P_{0}^{(N,0)}(x)=1$$ and $$P_{1}^{(N,0)}(x)=\frac{N+(N+2)x}{2}.$$ The Jacobi Polynomial $P_n^{(N,0)}$ is defined in (\[192\]).
The following lemma provides a stable recurrence relation that can be used to evaluate derivatives of a certain class of Jacobi Polynomials. It is readily obtained by differentiating (\[195\]) with respect to $x$,
\[196\] For any integer $n\geq1$ and $N\geq0$, $$\begin{aligned}
\label{197}
&\hspace*{-4em} P_{n+1}^{(N,0)\prime}(x)=
\frac{(2n+N+1)N^2+(2n+N)(2n+N+1)(2n+N+2)x}
{2(n+1)(n+N+1)(2n+N)}
P_n^{(N,0)\prime}(x) \notag \\
&-\frac{2(n+N)(n)(2n+N+2)}
{2(n+1)(n+N+1)(2n+N)}
P_{n-1}^{(N,0)\prime}(x) \notag \\
&+\frac{(2n+N)(2n+N+1)(2n+N+2)}
{2(n+1)(n+N+1)(2n+N)}
P_n^{(N,0)}(x),\end{aligned}$$ where $$P_{0}^{(N,0)\prime}(x)=0$$ and $$P_{1}^{(N,0)\prime}(x)=\frac{(N+2)}{2}.$$ The Jacobi Polynomial $P_n^{(N,0)}$ is defined in (\[192\]) and $P_n^{(N,0)\prime}(x)$ denotes the derivative of $P_n^{(N,0)}(x)$ with respect to $x$.
The following lemma, which provides a differential equation for Jacobi polynomials, can be found in [@abramowitz]
\[205\] For any integer $n$, $$\label{210}
(1-x^2)P_n^{(k,0)\prime\prime}(x)+(-k-(k+2)x)P_n^{(k,0)\prime}(x)
+n(n+k+1)P_n^{(k,0)}(x)=0$$ for all $x\in [0,1]$ where $P_n^{(N,0)}$ is defined in (\[192\]).
We will be denoting by $\widetilde{P}_n: [0,1] \rightarrow \R$ the shifted Jacobi polynomial defined for any non-negative integer $n$ by the formula $$\label{215}
\widetilde{P}_n(x)=\sqrt{2n+2}P_n^{(1,0)}(1-2x)$$ where $P_n^{(1,0)}$ is defined in (\[192\]). The roots of $\widetilde{P}_n$ will be used in Section \[seczernquad\] and Section \[secapprox\] in the design of quadrature and interpolation schemes for Zernike polynomials.
It follows immediately from the combination of (\[192\]) and (\[215\]) that the polynomials $\widetilde{P}_n$ are orthogonal on $[0,1]$ with respect to weight function $$w(x)=x.$$ That is, for any non-negative integers $i,j$, $$\label{740}
\int_0^1\widetilde{P}_i(r)\widetilde{P}_j(r)rdr=\delta_{i,j}.$$
Gaussian Quadratures {#secgengauss}
--------------------
In this section, we introduce Gaussian Quadratures.
\[220\] A Gaussian Quadrature with respect to a set of functions $f_1,...,f_{2n-1}:[a,b]\rightarrow \mathbb{R}$ and non-negative weight function $w:[a,b]\rightarrow \mathbb{R}$ is a set of $n$ nodes, $x_1,...,x_n\in[a,b]$, and $n$ weights, $\omega_1,...,\omega_n \in \R$, such that, for any integer $j\leq 2n-1$, $$\label{240}
\int_a^b f_j(x)w(x)dx=\sum_{i=0}^n\omega_i f_j(x_i).$$
The following is a well-known lemma from numerical analysis. A proof can be found in, for example, [@stoer].
\[260\] Suppose that $p_0,p_1,...:[a,b]\rightarrow \R$ is a set of orthonormal polynomials with respect to some non-negative weight function $w:[a,b]
\rightarrow \mathbb{R}$ such that polynomial $p_i$ is of degree $i$. Then,\
\
i) Polynomial $p_i$ has exactly $i$ roots on $[a,b]$.\
\
ii) For any non-negative integer $n$ and for $i=0,1,...,2n-1$, we have $$\label{280}
\int_a^b p_i(x)w(x)dx=\sum_{k=1}^n\omega_k p_i(x_k)$$ where $x_1,...,x_n \in [a,b]$ are the $n$ roots of $p_n$ and where weights $\omega_1,...,\omega_n \in \R$ solve the $n \times n$ system of linear equations $$\label{300}
\sum_{k=1}^n \omega_k p_j(x_k) = \int_a^b w(x)p_j(x)dx$$ with $j=0,1,...,n-1$.\
\
iii) The weights, $\omega_i$, satisfy the identity, $$\label{320}
\omega_i=\left(\sum_{k=0}^{n-1} p_k(x_i)^2 \right)^{-1}$$ for $i=1,2,...,n$.
Zernike Polynomials {#seczern}
-------------------
In this section, we define Zernike Polynomials and describe some of their basic properties.
Zernike polynomials are a family of orthogonal polynomials defined on the unit ball in $\R^n$. In this paper, we primarily discuss Zernike polynomials in $\R^2$, however nearly all of the theory and numerical machinery in two dimensions generalizes naturally to higher dimensions. The mathematical properties of Zernike polynomials in $\R^n$ are included in Appendix A.
Zernike Polynomials are defined via the formula $$\label{340}
Z_{N,n}^\ell(x) = R_{N,n}(r)S_N^{\ell}(\theta)$$ for all $x\in \R^2$ such that $\|x\|\leq1$, $(r,\theta)$ is the representation of $x$ in polar coordinates, $N,n$ are non- negative integers, $S_N^\ell$ is defined in (\[20\]), and $R_{N,n}$ are polynomials of degree $N+2n$ defined by the formula $$\begin{aligned}
\label{360}
R_{N,n}(x) = x^N \sum_{k=0}^n (-1)^k {n+N+\frac{p}{2} \choose k}
{n\choose k} (x^2)^{n-k} (1-x^2)^k,
\end{aligned}$$ for all $0\le x\le 1$. Furthermore, for any non-negative integers $N,n,m$, $$\begin{aligned}
\label{1.45}
\int_0^1 R_{N,n}(x)R_{N,m}(x) x\, dx = \frac{\delta_{n,m}}
{2(2n+N+1)}
\end{aligned}$$ and $$\begin{aligned}
\label{1.35}
R_{N,n}(1) = 1.
\end{aligned}$$ We define the normalized polynomials $\overline{R}_{N,n}$ via the formula $$\begin{aligned}
\label{1.65}
\overline{R}_{N,n}(x) = \sqrt{2(2n+N+1)} R_{N,n}(x),
\end{aligned}$$ so that $$\begin{aligned}
\label{1.75}
\int_0^1 \bigl(\overline{R}_{N,n}(x)\bigr)^2 x\, dx = 1,
\end{aligned}$$ where $N$ and $n$ are non-negative integers. We define the normalized Zernike polynomial, $\overline{Z}_{N,n}^\ell$, by the formula $$\label{10.85}
\overline{Z}_{N,n}(x)=\overline{R}_{N,n}(r)S_{N}^\ell(\theta)$$ where $x\in\R^2$ satisfies $\|x\|\leq 1$, and $N,n$ are non-negative integers. We observe that $\overline{Z}_{N,n}^\ell$ has $L^2$ norm of $1$ on the unit disk.
In an abuse of notation, we use $Z_{N,n}^\ell(x)$ and $Z_{N,n}^\ell(r,\theta)$ interchangeably where $(r,\theta)$ is the polar coordinate representation of $x\in \R^2$.
Numerical Evaluation of Zernike Polynomials {#secnumev}
===========================================
In this section, we provide a stable recurrence relation (see Lemma \[lem755\]) that can be used to evaluate Zernike Polynomials.
\[lem755\] The polynomials $R_{N,n}$, defined in (\[360\]) satisfy the recurrence relation $$\begin{aligned}
\label{7550}
&\hspace*{-4em}
R_{N,n+1}(x)= \notag \\
&\hspace*{-4em}-\frac{((2n+N+1)N^2+(2n+N)(2n+N+1)(2n+N+2)(1-2x^2))}
{2(n+1)(n+N+1)(2n+N)}
R_{N,n}(x) \notag \\
&\hspace*{-4em}-\frac{2(n+N)(n)(2n+N+2)}
{2(n+1)(n+N+1)(2n+N)}
R_{N,n-1}(x)\end{aligned}$$ where $0\le x\le 1$, $N$ is a non-negative integer, $n$ is a positive integer, and $$R_{N,0}(x)=x^N$$ and $$R_{N,1}(x)=-x^N\frac{N+(N+2)(1-2x^2)}{2}.$$
According to [@abramowitz], for any non-negative integers $n$ and $N$, $$\label{6.245}
R_{N,n}(x) = (-1)^n x^N P_n^{(N,0)}(1-2x^2),$$ where $0\le x\le 1$, $N$ and $n$ are nonnegative integers, and $P^{(N,0)}_n$ denotes a Jacobi polynomial (see (\[192\])).
Identity (\[7550\]) follows immediately from the combination of (\[6.245\]) and (\[195\]).
The algorithm for evaluating Zernike polynomials using the recurrence relation in Lemma \[lem755\] is known as Kintner’s method (see [@kintner] and, for example, [@chong]).
Quadrature for Zernike Polynomials {#seczernquad}
==================================
In this section, we provide a quadrature rule for Zernike Polynomials.
The following lemma follows immediately from applying Lemma \[260\] to the polynomials $\widetilde{P}_n$ defined in (\[215\]).
\[lem4.30\] Let $\{r_1,...,r_{m}\}$ be the $m$ roots of $\widetilde{P}_m$ (see (\[215\])) and $\{\omega_1,...,\omega_m\}$ the $m$ weights of the Gaussian quadrature (see (\[240\])) for the polynomials $\widetilde{P}_0,\widetilde{P}_1,...,
\widetilde{P}_{2m-1}$ (see (\[215\])). Then, for any polynomial $q$ of degree at most $2m-1$, $$\label{10.60}
\int_0^1 q(x)xdx=\sum_{i=1}^{m}
q(r_i)\omega_i.$$
The following theorem provides a quadrature rule for Zernike Polynomials.
\[lem4.70\] Let $\{r_1,...,r_{m}\}$ be the $m$ roots of $\widetilde{P}_m$ (see (\[215\])) and $\{\omega_1,...,\omega_m\}$ the $m$ weights of the Gaussian quadrature (see (\[240\])) for the polynomials $\widetilde{P}_0,\widetilde{P}_1,...,
\widetilde{P}_{2m-2}$ (see (\[215\])). Then, for all $\ell
\in \{0,1\}$ and for all $N,n\in \{0,1,...\}$ such that $N+2n
\leq 2m-1$, $$\label{10.7}
\int_{D}Z_{N,n}^\ell(x)dx=\sum_{i=1}^{m}
R_{N,n}(r_i)\omega_i
\sum_{j=1}^{2m}
\frac{2\pi}{2m}S_N^\ell(\theta_j)$$ where $R_{N,n}$ is defined in (\[360\]), $\theta_j$ is defined by the formula $$\theta_j=j\frac{2\pi}{2m}$$ for $j\in \{1,2,...,2m\}$, and $D\subseteq \R^2$ denotes the unit disk. Furthermore, there are exactly $2m^2+m$ Zernike Polynomials of degree at most $2m-1$.
Applying a change of variables, $$\label{10.10}
\int_{D}Z_{N,n}^\ell(x)dx=\int_0^1\int_0^{2\pi}
R_{N,n}(r)S_N^\ell(\theta)rdrd\theta,$$ where $Z_{N,n}^\ell$ is a Zernike polynomial (see (\[340\])) and where $R_{N,n}$ is defined in (\[1.45\]). Changing the order of integration of (\[10.10\]), we obtain $$\label{10.20}
\int_{D}Z_{N,n}^\ell(x)dx
=\int_0^1rR_{N,n}(r)dr\int_0^{2\pi}S_N^\ell(\theta)d\theta.$$ Applying Lemma \[60\] and Lemma \[lem4.30\] to (\[10.20\]), we obtain $$\label{10.30}
\int_{D}Z_{N,n}^\ell(x)dx
=\sum_{i=1}^{m} R_{N,n}(r_i)\omega_i
\sum_{j=1}^{2m} \frac{2\pi}{2m}S_N^\ell(\theta_j)$$ for $N+2n\leq 2m-1$. The fact that there are exactly $2m^2+m$ Zernike polynomials of degree at most $2m-1$ follows immediately from the combination of Lemma \[140\] with the fact that there are exactly $m$ Zernike polynomials of degree at most $2m-1$ that are of the form $Z_{0,n}^{\ell}$.
It follows immediately from Lemma \[lem4.70\] that for all $m\in \{1,2,...\}$, placing $m$ nodes in the radial direction and $2m$ nodes in the angular direction (as described in Lemma \[lem4.70\]), integrates exactly the $2m^2+m$ Zernike polynomials on the disk of degree at most $2m-1$.
The $n$ roots of $\widetilde{P}_n$ (see \[740\]) can be found by using, for example, the algorithm described in Section \[secalg\].
For Zernike polynomial discretization in $\R^{k+1}$, roots of the polynomials $\widetilde{P}_n^k$ are used, where $\widetilde{P}_n^k$ is defined by the formula $$\label{210.60}
\widetilde{P}_n^k(x)=\sqrt{k+2n+1}P_n^{(k,0)}(1-2x).$$ Properties of this class of Jacobi polynomials are provided in Appendix D in addition to an algorithm for finding their roots.
The following remark illustrates that the advantage of quadrature rule (\[10.7\]) is especially noticeable in higher dimensions.
Quadrature rule (\[10.7\]) integrates all Zernike polynomials up to order $2m-1$ using the $m$ roots of $\widetilde{P}_m$ (see (\[740\])) as nodes in the radial direction. Using Guass-Legendre nodes instead of roots of $\widetilde{P}_m$ would require using $m+1$ nodes in the radial direction.
The high-dimensional equivalent of quadrature rule (\[10.7\]) uses the roots of $\widetilde{P}_m^{p+1}$ (see (\[760.2\])) as nodes in the radial direction. Using Gauss-Legendre nodes instead of these nodes would require using an extra $p+1$ nodes in the radial direction or approximately $(p+1)m^{p+1}$ extra nodes total.
\[scale=4.0\]
(0,0) circle (1cm);
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( -0.30371, -0.93472) circle \[radius=0.0075\]; ( -0.15375, -0.97072) circle \[radius=0.0075\]; ( 0.00000, -0.98282) circle \[radius=0.0075\]; ( 0.15375, -0.97072) circle \[radius=0.0075\]; ( 0.30371, -0.93472) circle \[radius=0.0075\]; ( 0.44619, -0.87570) circle \[radius=0.0075\]; ( 0.57769, -0.79512) circle \[radius=0.0075\]; ( 0.69496, -0.69496) circle \[radius=0.0075\]; ( 0.79512, -0.57769) circle \[radius=0.0075\]; ( 0.87570, -0.44619) circle \[radius=0.0075\]; ( 0.93472, -0.30371) circle \[radius=0.0075\]; ( 0.97072, -0.15375) circle \[radius=0.0075\]; ( 0.99672, 0.00000) circle \[radius=0.0075\]; ( 0.98445, 0.15592) circle \[radius=0.0075\]; ( 0.94794, 0.30800) circle \[radius=0.0075\]; ( 0.88809, 0.45250) circle \[radius=0.0075\]; ( 0.80637, 0.58586) circle \[radius=0.0075\]; ( 0.70479, 0.70479) circle \[radius=0.0075\]; ( 0.58586, 0.80637) circle \[radius=0.0075\]; ( 0.45250, 0.88809) circle \[radius=0.0075\]; ( 0.30800, 0.94794) circle \[radius=0.0075\]; ( 0.15592, 0.98445) circle \[radius=0.0075\]; ( 0.00000, 0.99672) circle \[radius=0.0075\]; ( -0.15592, 0.98445) circle \[radius=0.0075\]; ( -0.30800, 0.94794) circle \[radius=0.0075\]; ( -0.45250, 0.88809) circle \[radius=0.0075\]; ( -0.58586, 0.80637) circle \[radius=0.0075\]; ( -0.70479, 0.70479) circle \[radius=0.0075\]; ( -0.80637, 0.58586) circle \[radius=0.0075\]; ( -0.88809, 0.45250) circle \[radius=0.0075\]; ( -0.94794, 0.30800) circle \[radius=0.0075\]; ( -0.98445, 0.15592) circle \[radius=0.0075\]; ( -0.99672, 0.00000) circle \[radius=0.0075\]; ( -0.98445, -0.15592) circle \[radius=0.0075\]; ( -0.94794, -0.30800) circle \[radius=0.0075\]; ( -0.88809, -0.45250) circle \[radius=0.0075\]; ( -0.80637, -0.58586) circle \[radius=0.0075\]; ( -0.70479, -0.70479) circle \[radius=0.0075\]; ( -0.58586, -0.80637) circle \[radius=0.0075\]; ( -0.45250, -0.88809) circle \[radius=0.0075\]; ( -0.30800, -0.94794) circle \[radius=0.0075\]; ( -0.15592, -0.98445) circle \[radius=0.0075\]; ( 0.00000, -0.99672) circle \[radius=0.0075\]; ( 0.15592, -0.98445) circle \[radius=0.0075\]; ( 0.30800, -0.94794) circle \[radius=0.0075\]; ( 0.45250, -0.88809) circle \[radius=0.0075\]; ( 0.58586, -0.80637) circle \[radius=0.0075\]; ( 0.70479, -0.70479) circle \[radius=0.0075\]; ( 0.80637, -0.58586) circle \[radius=0.0075\]; ( 0.88809, -0.45250) circle \[radius=0.0075\]; ( 0.94794, -0.30800) circle \[radius=0.0075\]; ( 0.98445, -0.15592) circle \[radius=0.0075\];
The following remark shows that we can reduce the total number of nodes in quadrature rule (\[10.7\]) while still integrating the same number of functions.
Quadrature rule (\[10.7\]) integrates all Zernike polynomials of order up to $2m-1$ using a tensor product of $2m$ equispaced nodes in the angular direction and the $m$ roots of $\widetilde{P}_m$ (see \[215\]) in the radial direction. However, for large enough $N$ and small enough $j$, $Z_{N,n}(r_j)$ is of magnitude smaller than machine precision, where $r_j$ denotes the $j^{\text{th}}$ smallest root of $\widetilde{P}_m$. As a result, in order to integrate exactly $Z_{N,n}$ for large $N$, we can use fewer equispaced nodes in the angular direction at radius $r_j$.
Approximation of Zernike Polynomials {#secapprox}
====================================
In this section, we describe an interpolation scheme for Zernike Polynomials.
We will denote by $r_1,...,r_M$ the $M$ roots of $\widetilde{P}_M$ (see \[215\]).
\[thminterp\] Let $M$ be a positive integer and $f:D \rightarrow \R$ be a linear combination of Zernike polynomials of degree at most $M-1$. That is, $$\label{1100}
f(r,\theta)=\sum_{i,j} \alpha_{i,j}^\ell \overline{Z}_{i,j}^\ell(r,\theta)$$ where $i,j$ are non-negative integers satisfying $$\label{1120}
i+2j\leq M-1$$ and where $\overline{Z}_{i,j}^\ell(r,\theta)$ is defined by (\[10.85\]) and $S_{i}^\ell$ is defined by (\[20\]). Then, $$\label{1140}
\alpha_{i,j}^\ell=\sum_{k=1}^M \left[ \overline{R}_{i,j}(r_k)\omega_k
\sum_{l=1}^{2M-1} \frac{2\pi}{2M-1} f(r_k,\theta_l) S_{i}^{\ell}
(\theta_l)\right]$$ where $r_1,...,r_M$ denote the $M$ roots of $\widetilde{P}_M$ (see \[215\]) and $\theta_l$ is defined by the formula $$\label{1160}
\theta_l=l\frac{2\pi}{2M-1}$$ for $l=1,2,...,2M-1$.
Clearly, $$\label{1180}
\alpha_{i,j}^\ell=\int_D f(r,\theta)\overline{Z}_{i,j}^\ell
=\int_{0} ^{2\pi} \int_0^1 f(r,\theta)\overline{R}_{i,j}(r)
S_{i}^\ell(\theta)rdrd\theta.$$ Changing the order of integration of (\[1180\]) and applying Lemma \[60\] and Lemma \[130\], we obtain $$\label{1200}
\begin{split}
\alpha_{i,j}^\ell&=\int_{0} ^{1} \overline{R}_{i,j}(r)
r \int_{0}^{2\pi} f(r,\theta)S_{i}^\ell(\theta)d\theta dr \\
&=\int_{0} ^{1} \overline{R}_{i,j}(r) r \sum_{l=1}^{2M-1}
\frac{2\pi}{2M-1} f(r,\theta_l) S_{i}^\ell(\theta_l)dr.
\end{split}$$ Applying Lemma \[130\] to (\[1200\]), we obtain $$\label{1210}
\alpha_{i,j}^\ell=\sum_{k=1}^{M} \left[ \overline{R}_{i,j}(r_k)
\omega_k \sum_{l=1}^{2M-1}
\frac{2\pi}{2M-1} f(r_k,\theta_l) S_{i,j}^\ell(\theta_l)\right].$$
Suppose that $f:D\rightarrow \R$ is a linear combination of Zernike polynomials of degree at most $M-1$. It follows immediately from Theorem \[thminterp\] and Theorem \[lem4.70\] that we can recover exactly the $M^2/2+M/2$ coefficients of the Zernike polynomial expanison of $f$ by evaluation of $f$ at $2M^2-M$ points via (\[1140\]).
Recovering the $M^2/2+M/2$ coefficients of a Zernike expansion of degree at most $M-1$ via (\[1210\]) requires $O(M^3)$ operations by using a FFT to compute the sum $$\sum_{l=1}^{2M-1} \frac{2\pi}{2M-1}
f(r,\theta_l) S_{i,j}^\ell(\theta_l)$$ and then naively computing the sum $$\label{1240}
\begin{split}
\alpha_{i,j}^\ell=\sum_{k=1}^{M} \overline{R}_{i,j}(r_k)
\omega_k \sum_{l=1}^{2M-1}
\frac{2\pi}{2M-1} f(r_k,\theta_l) S_{i,j}^\ell(\theta_l).
\end{split}$$
Sum (\[1240\]) can be computed using an FMM (see, for example, [@alpert]) which would reduce the evaluation of sum (\[1210\]) to a computational cost of $O(M^2 \log(M))$.
Standard interpolation schemes on the unit disk often involve representing smooth functions as expansions in non-smooth functions such as $$\label{1720}
T_n(r)S_N^\ell(\theta)$$ where $n$ and $N$ are non-negative integers, $T_n$ is a Chebyshev polynomial, and $S_N^\ell$ is defined in (\[20\]). Such interpolation schemes are amenable to the use of an FFT in both the angular and radial directions and thus have a computational cost of only $O(M^2\log(M))$ for the interpolation of an $M$-degree Zernike expansion.
However, interpolation scheme (\[1140\]) has three main advantages over such a scheme:\
i) In order to represent a smooth function on the unit disk to full precision, a Zernike expansion requires approximately half as many terms as an expansion into functions of the form (\[1720\]) (see Figure \[8400\]).\
ii) Each function in the interpolated expansion is smooth on the disk.\
iii) The expansion is amenable to filtering.
Numerical Experiments {#secnumres}
=====================
The quadrature and interpolation formulas described in Sections \[seczernquad\] and \[secapprox\] were implemented in Fortran 77. We used the Lahey/Fujitsu compiler on a 2.9 GHz Intel i7-3520M Lenovo laptop. All examples in this section were run in double precision arithmetic.
In each table in this section, the column labeled “nodes” denotes the number of nodes in both the radial and angular direction using quadrature rule (\[10.7\]). The column labeled “exact integral” denotes the true value of the integral being tested. This number is computed using adaptive gaussian quadrature in extended precision. The column labeled “integral via quadrature” denotes the integral approximation using quadrature rule (\[10.7\]).
We tested the performance of quadrature rule (\[10.7\]) in integrating three different functions over the unit disk. In Table \[7600\] we approximated the integral over the unit disk of the function $f_1$ defined by the formula $$\label{2100.10}
f_1(x,y)=\frac{1}{1+25(x^2+y^2)}.$$ In Table \[7700\] we use quadrature rule (\[10.7\]) to approximate the integral over the unit disk of the function $f_2$ defined by the formula $$\label{2200}
f_2(r,\theta)=J_{100}(150r)\cos(100\theta)).$$ In Table \[7900\], we use quadrature rule (\[10.7\]) to approximate the integral over the unit disk of the function $f_3$ defined by the formula $$\label{2400}
f_3(r,\theta)=P_8(x)P_{12}(y).$$ We tested the performance of interpolation scheme (\[1100\]) on two functions defined on the unit disk.
In Figure \[8000\] we plot the magnitude of the coefficients of the Zernike polynomials $R_{0,n}$ for $n=0,1,...,10$ using interpolation scheme (\[1100\]) with $21$ nodes in the radial direction and $41$ in the angular direction on the function $f_1$ defined in (\[2100.10\]). All coeficients of other terms were of magnitude smaller than $10^{-14}$. In Table \[7800\] we list the interpolated coefficients of the Zernike polynomial expansion of the function $f_4$ defined by the formula $$f_4(x,y)=P_2(x)P_4(y)$$ where $P_i$ is the $i$th degree Legendre polynomial. Listed are the coefficients using interpolation scheme (\[1100\]) with $5$ points in the radial direction and $9$ points in the angular direction of Zernike polynomials $$R_{N,n}\cos(N\theta)$$ where $N=0,1,...,8$ and $n=0,1,2,3,4$. All other coefficients were of magnitude smaller than $10^{-14}$. We interpolated the Bessel function $$\label{9800}
J_{10}(10r)cos(10\theta)$$ using interpolation scheme (\[1100\]) and plot the resulting coefficients of the Zernike polynomials $$\label{9850}
R_{10,n}\cos(10\theta)$$ for $n=0,...,16$ in Figure \[8400\]. All other coefficients were approximately $0$ to machine precision. In Figure \[8400\], we plot the coefficients of the Chebyshev expansion obtained via Chebyshev interpolation of the radial component of (\[9800\]).
radial nodes angular nodes exact integral integral via quadrature
-------------- --------------- ---------------- ------------------------------------------------
$ 5$ $ 10$ $0$ $\hphantom- 0.2670074163846569\times 10^{-1}$
$ 10$ $ 20$ $0$ $\hphantom- 0.2606355680939063\times 10^{-2}$
$ 15$ $ 30$ $0$ $\hphantom- 0.3119143925398078\times 10^{-15}$
$ 20$ $ 40$ $0$ $\hphantom- 0.0000000000000000\times 10^{0}$
$ 25$ $ 50$ $0$ $\hphantom- 0.3228321977714574\times 10^{-1}$
$ 30$ $ 60$ $0$ $\hphantom- 0.4945592102178045\times 10^{-16}$
$ 35$ $ 70$ $0$ $\hphantom- 0.1147861841710902\times 10^{-16}$
$ 40$ $ 80$ $0$ $\hphantom- 0.8148891073315595\times 10^{-16}$
$ 45$ $ 90$ $0$ $-0.7432759692263743\times 10^{-16}$
$ 50$ $100$ $0$ $\hphantom- 0.3207999037057322\times 10^{-1}$
$ 55$ $110$ $0$ $-0.1399753743762347\times 10^{-15}$
$ 60$ $120$ $0$ $\hphantom- 0.3075136040459932\times 10^{-16}$
$ 65$ $130$ $0$ $-0.9458788981593222\times 10^{-16}$
$ 70$ $140$ $0$ $\hphantom- 0.2045957446273746\times 10^{-17}$
$ 75$ $150$ $0$ $\hphantom- 0.2416178317504225\times 10^{-16}$
: Quadratures for $f_2(r,\theta)=J_{100}(150r)\cos(100\theta)$ using several different numbers of nodes[]{data-label="7700"}
coordinates [ (0,0.23) (1,0.18) (2,0.13) (3,0.09) (4,0.06) (5,0.04) (6,0.028) (7,0.019) (8,0.012) (9,0.0087) (10,0.006) ]{};
coordinates [ ( 1, -0.9898973637462893) ( 2, -1.5819586550661100) ( 3, -2.3588831162234810) ( 4, -3.2818384538878600) ( 5, -4.3265899327402020) ( 6, -5.4763255881423960) ( 7, -6.7185889712528420) ( 8, -8.0437256952401060) ( 9, -9.4440037263722940) ( 10, -10.9130714107455100) ( 11, -12.4457655714899900) ( 12, -14.0303912755872500) ( 13, -15.8930077195246900) ( 14, -16.3457963962040700) ( 15, -15.7586036824183800) ( 16, -15.6123213406515100) ]{};
coordinates [ ( 1, -1.0595713478383200) ( 2, -0.8238192166464383) ( 3, -1.0293346132799840) ( 4, -1.4184210516931150) ( 5, -2.2236928015527780) ( 6, -2.4320890160470980) ( 7, -2.5458791787004040) ( 8, -3.1529353540202170) ( 9, -4.0684262705269870) ( 10, -4.0164562320349430) ( 11, -4.8033818867489570) ( 12, -5.3788871564204650) ( 13, -5.7771701061737610) ( 14, -7.5330217317784680) ( 15, -7.0809635324045650) ( 16, -8.3203906346855500) ( 17, -8.5720716277244970) ( 18, -9.5869646653661940) ( 19, -10.2017710131164700) ( 20, -11.1192420938158200) ( 21, -11.9441235125173900) ( 22, -12.8058618265459900) ( 23, -13.7926138070249200) ( 24, -14.6726336554743400) ( 25, -15.3358881075438200) ( 26, -15.6272308358046700) ( 27, -15.8313508184606000) ( 28, -15.9754399091641200) ( 29, -15.9973417506119500) ( 30, -15.9478563875320300) ( 31, -15.8915380244439100) ( 32, -15.8549738638070200) ( 33, -15.8416916517988200) ( 34, -15.8469558917911700) ( 35, -15.8212504915562700) ( 36, -15.7458293943819500) ( 37, -15.6907318775671300) ( 38, -15.6753007742882200) ( 39, -15.6761934662405600) ( 40, -15.6916568927877700) ( 41, -15.6802334440423000) ]{};
Appendix A: Mathematical Properties of Zernike Polynomials
==========================================================
In this appendix, we define Zernike polynomials in $\R^{p+2}$ and describe some of their basic properties. Zernike polynomials, denoted $Z_{N,n}^\ell$, are a sequence of orthogonal polynomials defined via the formula $$\begin{aligned}
Z_{N,n}^\ell(x) = R_{N,n}(\|x\|)
S_N^\ell(x/\|x\|),
\label{6.10}
\end{aligned}$$ for all $x\in \R^{p+2}$ such that $\|x\| \le 1$, where $N$ and $n$ are nonnegative integers, $S_N^\ell$ are the orthonormal surface harmonics of degree $N$ (see Appendix C), and $R_{N,n}$ are polynomials of degree $2n+N$ defined via the formula $$\begin{aligned}
R_{N,n}(x) = x^N \sum_{m=0}^n (-1)^m {n+N+\frac{p}{2} \choose m}
{n\choose m} (x^2)^{n-m} (1-x^2)^m,
\label{6.20}
\end{aligned}$$ for all $0\le x\le 1$. The polynomials $R_{N,n}$ satisfy the relation $$\begin{aligned}
R_{N,n}(1) = 1,
\label{6.30}
\end{aligned}$$ and are orthogonal with respect to the weight function $w(x) = x^{p+1}$, so that $$\begin{aligned}
\int_0^1 R_{N,n}(x)R_{N,m}(x) x^{p+1}\, dx = \frac{\delta_{n,m}}
{2(2n+N+\frac{p}{2}+1)},
\label{6.40}
\end{aligned}$$ where $$\begin{aligned}
\delta_{n,m} = \left\{
\begin{array}{ll}
1 & \mbox{if $n = m$}, \\
0 & \mbox{if $n \ne m$}.
\end{array}
\right.
\label{6.50}
\end{aligned}$$ We define the polynomials $\overline{R}_{N,n}$ via the formula $$\begin{aligned}
\overline{R}_{N,n}(x) = \sqrt{2(2n+N+p/2+1)} R_{N,n}(x),
\label{6.60}
\end{aligned}$$ so that $$\begin{aligned}
\int_0^1 \bigl(\overline{R}_{N,n}(x)\bigr)^2 x^{p+1}\, dx = 1,
\label{6.70}
\end{aligned}$$ where $N$ and $n$ are nonnegative integers. We define the normalized Zernike polynomial, $\overline{Z}_{N,n}^\ell$, by the formula $$\label{10.85.2}
\overline{Z}_{N,n}(x)=\overline{R}_{N,n}(\|x\|)S_{N}^\ell(x/\|x\|)$$ for all $x\in \R^{p+2}$ such that $\|x\| \le 1$, where $N$ and $n$ are nonnegative integers, $S_N^\ell$ are the orthonormal surface harmonics of degree $N$ (see Appendix C), and $R_{N,n}$ is defined in (\[6.20\]). We observe that $\overline{Z}_{N,n}^\ell$ has $L^2$ norm of $1$ on the unit ball in $\R^{p+2}$.
In an abuse of notation, we refer to both the polynomials $Z^\ell_{N,n}$ and the radial polynomials $R_{N,n}$ as Zernike polynomials where the meaning is obvious.
When $p=-1$, the Zernike polynomials take the form $$\begin{aligned}
&Z_{0,n}^1(x) = R_{0,n}(|x|) = P_{2n}(x), \label{6.90} \\
&Z_{1,n}^2(x) = {\mathop{\mathrm{sgn}}}(x)\cdot R_{1,n}(|x|) = P_{2n+1}(x), \label{6.100}
\end{aligned}$$ for $-1\le x\le 1$ and nonnegative integer $n$, where $P_n$ denotes the Legendre polynomial of degree $n$ and $$\begin{aligned}
{\mathop{\mathrm{sgn}}}(x) = \left\{
\begin{array}{ll}
1 & \mbox{if $x > 0$}, \\
0 & \mbox{if $x = 0$}, \\
-1 & \mbox{if $x < 0$},
\end{array}
\right.
\label{6.110}
\end{aligned}$$ for all real $x$.
\[rem6.1\] When $p=0$, the Zernike polynomials take the form $$\begin{aligned}
&Z_{N,n}^1(x_1,x_2)
= R_{N,n}(r) \cos(N\theta)/\sqrt{\pi}, \label{6.71} \\
&Z_{N,n}^2(x_1,x_2)
= R_{N,n}(r) \sin(N\theta)/\sqrt{\pi}, \label{6.80}
\end{aligned}$$ where $x_1=r\cos(\theta)$, $x_2=r\sin(\theta)$, and $N$ and $n$ are nonnegative integers.
Special Values {#sect6.2}
--------------
The following formulas are valid for all nonnegative integers $N$ and $n$, and for all $0\le x\le 1$. $$\begin{aligned}
& R_{N,0}(x) = x^N, \label{6.180} \\
& R_{N,1}(x) = x^N \bigl((N+p/2+2)x^2 - (N+p/2+1)\bigr), \label{6.190} \\
& R_{N,n}(1) = 1, \label{6.200} \\
& R_{N,n}^{(k)}(0) =0 \quad \mbox{for $k=0,1,\ldots,N-1$}, \label{6.210} \\
& R_{N,n}^{(N)}(0) = (-1)^n N! {n+N+\frac{p}{2} \choose n}. \label{6.220}
\end{aligned}$$
Hypergeometric Function {#sect6.3}
-----------------------
The polynomials $R_{N,n}$ are related to the hypergeometric function $_2 F_1$ (see [@abramowitz]) by the formula $$\begin{aligned}
&\hspace*{-3em} R_{N,n}(x) = (-1)^n {n+N+\frac{p}{2} \choose n} x^N
{_2 F_1}\Bigl(-n, n+N+\frac{p}{2}+1; N+\frac{p}{2}+1; x^2\Bigr),
\label{6.230}
\end{aligned}$$ where $0\le x\le 1$, and $N$ and $n$ are nonnegative integers.
Interrelations {#sect6.4}
--------------
The polynomials $R_{N,n}$ are related to the Jacobi polynomials via the formula $$R_{N,n}(x) = (-1)^n x^N P_n^{(N+\frac{p}{2},0)}(1-2x^2),
\label{6.240.2}$$ where $0\le x\le 1$, $N$ and $n$ are nonnegative integers, and $P^{(\alpha,\beta)}_n$, $\alpha, \beta > -1$, denotes the Jacobi polynomials of degree $n$ (see [@abramowitz]).
When $p=-1$, the polynomials $R_{N,n}$ are related to the Legendre polynomials via the formulas $$\begin{aligned}
R_{0,n}(x) &= P_{2n}(x), \label{6.250} \\
R_{1,n}(x) &= P_{2n+1}(x), \label{6.260}
\end{aligned}$$ where $0\le x\le 1$, $n$ is a nonnegative integer, and $P_n$ denotes the Legendre polynomial of degree $n$ (see [@abramowitz]).
Limit Relations {#sect6.5}
---------------
The asymptotic behavior of the Zernike polynomials near $0$ as the index $n$ tends to infinity is described by the formula $$\lim_{n\to\infty} \frac{(-1)^n R_{N,n}\bigl(\frac{x}{2n}\bigr)}
{(2n)^{p/2}}
= \frac{J_{N+p/2}(x)}{x^{p/2}},
\label{6.270}$$ where $0\le x \le 1$, $N$ is a nonnegative integer, and $J_\nu$ denotes the Bessel functions of the first kind (see [@abramowitz]).
Zeros {#sect6.6}
-----
The asymptotic behavior of the zeros of the polynomials $R_{N,n}$ as $n$ tends to infinity is described by the following relation. Let $x_{N,m}^{(n)}$ be the $m$th positive zero of $R_{N,n}$, so that $0 <
x_{N,1}^{(n)} < x_{N,2}^{(n)} < \ldots$. Likewise, let $j_{\nu,m}$ be the $m$th positive zero of $J_\nu$, so that $0 < j_{\nu,1} < j_{\nu,2} < \ldots$, where $J_\nu$ denotes the Bessel functions of the first kind (see [@abramowitz]). Then $$\lim_{n\to\infty} 2n x_{N,m}^{(n)} = j_{N+p/2,m},
\label{6.280}$$ for any nonnegative integer $N$.
Inequalities {#sect6.7}
------------
The inequality $$|R_{N,n}(x)| \le {n+N+\frac{p}{2} \choose n}
\label{6.290}$$ holds for $0\le x\le 1$ and nonnegative integer $N$ and $n$.
Integrals {#sect6.8}
---------
The polynomials $R_{N,n}$ satify the relation $$\int_0^1 \frac{J_{N+p/2}(xy)}{(xy)^{p/2}} R_{N,n}(y) y^{p+1}\, dy
= \frac{(-1)^n J_{N+p/2+2n+1}(x)}{x^{p/2+1}},
\label{6.300}$$ where $x\ge 0$, $N$ and $n$ are nonnegative integers, and $J_\nu$ denotes the Bessel functions of the first kind.
Generating Function {#sect6.9}
-------------------
The generating function associated with the polynomials $R_{N,n}$ is given by the formula $$\frac{\bigl(1+z - \sqrt{1+2z(1-2x^2)+z^2}\bigr)^{N+p/2}}
{(2zx)^{N+p/2} x^{p/2} \sqrt{1+2z(1-2x^2)+z^2}}
= \sum_{n=0}^\infty R_{N,n}(x) z^n,
\label{6.310}$$ where $0\le x\le 1$ is real, $z$ is a complex number such that $|z|\le 1$, and $N$ is a nonnegative integer.
Differential Equation {#sect6.10}
---------------------
The polynomials $R_{N,n}$ satisfy the differential equation $$(1-x^2)y''(x) - 2xy'(x) + \biggl(\chi_{N,n}
+ \frac{\frac{1}{4} - (N+\frac{p}{2})^2}{x^2}\biggr)y(x) = 0,
\label{6.320}$$ where $$\begin{aligned}
\chi_{N,n} = (N+\tfrac{p}{2}+2n+\tfrac{1}{2})
(N+\tfrac{p}{2}+2n+\tfrac{3}{2}),
\label{6.330}
\end{aligned}$$ and $$\begin{aligned}
y(x) = x^{p/2+1} R_{N,n}(x),
\label{6.340}
\end{aligned}$$ for all $0 < x < 1$ and nonnegative integers $N$ and $n$.
Recurrence Relations {#sect6.11}
--------------------
The polynomials $R_{N,n}$ satisfy the recurrence relation $$\begin{aligned}
&\hspace*{-4em} 2(n+1)(n+N+\tfrac{p}{2}+1)(2n+N+\tfrac{p}{2})R_{N,n+1}(x)
\notag \\
&= -\bigl( (2n+N+\tfrac{p}{2}+1)(N+\tfrac{p}{2})^2 + (2n+N+\tfrac{p}{2})_3
(1-2x^2)\bigr) R_{N,n}(x) \notag \\
&\hspace*{2em} - 2n(n+N+\tfrac{p}{2})(2n+N+\tfrac{p}{2}+2) R_{N,n-1}(x),
\label{6.350.2}
\end{aligned}$$ where $0\le x\le 1$, $N$ is a nonnegative integer, $n$ is a positive integer, and $(\cdot)_n$ is defined via the formula $$\begin{aligned}
(x)_n = x(x+1)(x+2)\ldots (x+n-1),
\label{6.360}
\end{aligned}$$ for real $x$ and nonnegative integer $n$. The polynomials $R_{N,n}$ also satisfy the recurrence relations $$\begin{aligned}
&\hspace*{-4em} (2n+N+\tfrac{p}{2}+2)x R_{N+1,n}(x)
= (n+N+\tfrac{p}{2}+1) R_{N,n}(x)
+ (n+1) R_{N,n+1}(x),
\label{6.370}
\end{aligned}$$ for nonnegative integers $N$ and $n$, and $$\begin{aligned}
(2n+N+\tfrac{p}{2})x R_{N-1,n}(x) = (n+N+\tfrac{p}{2}) R_{N,n}(x)
+ n R_{N,n-1}(x),
\label{6.380}
\end{aligned}$$ for integers $N\ge 1$ and $n\ge 0$, where $0\le x\le 1$.
Differential Relations {#sect6.12}
----------------------
The Zernike polynomials satisfy the differential relation given by the formula $$\begin{aligned}
&\hspace*{-4em} (2n+N+\tfrac{p}{2})x(1-x^2)\frac{d}{dx} R_{N,n}(x)
\notag \\
&= \bigl( N(2n+N+\tfrac{p}{2})+2n^2 - (2n+N)(2n+N+\tfrac{p}{2})x^2 \bigr)
R_{N,n}(x) \notag \\
&\hspace*{4em} + 2n(n+N+\tfrac{p}{2})R_{N,n-1}(x),
\label{6.390}
\end{aligned}$$ where $0< x< 1$, $N$ is a nonnegative integer, and $n$ is a positive integer.
Appendix B: Numerical Evaluation of Zernike Polynomials in $\R^{p+2}$
=====================================================================
The main analytical tool of this section is Lemma \[lem750\] which provides a recurrence relation that can be used for the evaluation of radial Zernike Polynomials, $R_{N,n}$.
According to [@abramowitz], radial Zernike polynomials, $R_{N,n}$, are related to Jacobi polynomials via the formula $$\label{6.240}
R_{N,n}(x) = (-1)^n x^N P_n^{(N+\frac{p}{2},0)}(1-2x^2),$$ where $0\le x\le 1$, $N$ and $n$ are nonnegative integers, and $P^{(\alpha,0)}_n$ is defined in (\[192\]).
The following lemma provides a relation that can be used to evaluate the polynomial $R_{N,n}$.
\[lem750\] The polynomials $R_{N,n}$ satisfy the recurrence relation $$\begin{aligned}
&\hspace*{-4em} 2(n+1)(n+N+\tfrac{p}{2}+1)(2n+N+\tfrac{p}{2})R_{N,n+1}(x)
\notag \\
&= -\bigl( (2n+N+\tfrac{p}{2}+1)(N+\tfrac{p}{2})^2 + (2n+N+\tfrac{p}{2})_3
(1-2x^2)\bigr) R_{N,n}(x) \notag \\
&\hspace*{2em} - 2n(n+N+\tfrac{p}{2})(2n+N+\tfrac{p}{2}+2) R_{N,n-1}(x),
\label{6.350}
\end{aligned}$$ where $0\le x\le 1$, $N$ is a nonnegative integer, $n$ is a positive integer, and $(\cdot)_n$ is defined via the formula $$\begin{aligned}
(x)_n = x(x+1)(x+2)\ldots (x+n-1),
\label{6.360.2}
\end{aligned}$$ for real $x$ and nonnegative integer $n$.
It is well known that the Jacobi polynomial $P_n^{(\alpha,0)}(x)$ satisfies the recurrence relation $$\label{230.2}
a_{1n}P_{n+1}^{(\alpha,0)}=(a_{2n}+a_{3n}x)P_n^{(\alpha,0)}(x)
-a_{4n}P_{n-1}^{(\alpha,0)}(x)$$ where $$\label{240.2}
\begin{aligned}
a_{1n}&=2(n+1)(n+\alpha+1)(2n+\alpha)\\
a_{2n}&=(2n+\alpha+1)\alpha^2\\
a_{3n}&=(2n+\alpha)(2n+\alpha+1)(2n+\alpha+2)\\
a_{4n}&=2(n+\alpha)(n)(2n+\alpha+2)
\end{aligned}$$ Identity (\[6.350\]) follows immediately from the combination of (\[230.2\]) and (\[240.2\]).
Appendix C: Spherical Harmonics in $\R^{p+2}$
=============================================
Suppose that $S^{p+1}$ denotes the unit sphere in $\R^{p+2}$. The spherical harmonics are a set of real-valued continuous functions on $S^{p+1}$, which are orthonormal and complete in $L^2(S^{p+1})$. The spherical harmonics of degree $N\ge 0$ are denoted by $S_N^1, S_N^2,
\ldots, \allowbreak S_N^\ell, \ldots, S_N^{h(N)}\colon S^{p+1} \to \R$, where $$\begin{aligned}
h(N)=(2N+p) \frac{(N+p-1)!} {p!\,N!},
\end{aligned}$$ for all nonnegative integers $N$.
The following theorem defines the spherical harmonics as the values of certain harmonic, homogeneous polynomials on the sphere (see, for example, [@batemanII]).
For each spherical harmonic $S_N^\ell$, where $N\ge 0$ and $1\le \ell
\le h(N)$ are integers, there exists a polynomial $K_N^\ell \colon \R^{p+2} \to \R$ which is harmonic, i.e. $$\begin{aligned}
\nabla^2 K_N^\ell(x) = 0,
\end{aligned}$$ for all $x\in \R^{p+2}$, and homogenous of degree $N$, i.e. $$\begin{aligned}
K_N^\ell(\lambda x) = \lambda^N K_N^\ell(x),
\end{aligned}$$ for all $x \in \R^{p+2}$ and $\lambda\in \R$, such that $$\begin{aligned}
S_N^\ell(\xi) = K_N^\ell(\xi),
\end{aligned}$$ for all $\xi \in S^{p+1}$.
The following theorem is proved in, for example, [@batemanII].
Suppose that $N$ is a nonnegative integer. Then there are exactly $$\begin{aligned}
(2N+p) \frac{(N+p-1)!} {p!\,N!}
\end{aligned}$$ linearly independent, harmonic, homogenous polynomials of degree $N$ in $\R^{p+2}$.
The following theorem states that for any orthogonal matrix $U$, the function $S_N^\ell(U\xi)$ is expressible as a linear combination of $S_N^1(\xi), S_N^2(\xi), \ldots, S_N^{h(N)}(\xi)$ (see, for example, [@batemanII]).
\[spher.rot\] Suppose that $N$ is a nonnegative integer, and that $S_N^1,S_N^2,\ldots,S_N^{h(N)} \colon S^{p+1} \to \R$ are a complete set of orthonormal spherical harmonics of degree $N$. Suppose further that $U$ is a real orthogonal matrix of dimension $p+2 \times p+2$. Then, for each integer $1\le \ell \le h(N)$, there exists real numbers $v_{\ell,1},v_{\ell,2},\ldots,v_{\ell,h(N)}$ such that $$\begin{aligned}
S_N^\ell(U\xi) = \sum_{k=1}^{h(N)} v_{\ell,k} S_N^k(\xi),
\end{aligned}$$ for all $\xi \in S^{p+1}$. Furthermore, if $V$ is the $h(N)
\times h(N)$ real matrix with elements $v_{i,j}$ for all $1\le i,j \le
h(N)$, then $V$ is also orthogonal.
From Theorem (\[spher.rot\]), we observe that the space of linear combinations of functions $S_N^\ell$ is invariant under all rotations and reflections of $S^{p+1}$.
The following theorem states that if an integral operator acting on the space of functions $S^{p+1}\to \R$ has a kernel depending only on the inner product, then the spherical harmonics are eigenfunctions of that operator (see, for example, [@batemanII]).
Suppose that $F\colon [-1,1] \to \R$ is a continuous function, and that $S_N\colon S^{p+1}\to \R$ is any spherical harmonic of degree $N$. Then $$\begin{aligned}
\int_\Omega F({\ensuremath{\langle \xi,\eta \rangle}}) S_N(\xi) \, d\Omega(\xi) =
\lambda_N S_N(\eta),
\end{aligned}$$ for all $\eta\in S^{p+1}$, where ${\ensuremath{\langle \cdot,\cdot \rangle}}$ denotes the inner product in $\R^{p+2}$, the integral is taken over the whole area of the hypersphere $\Omega$, and $\lambda_N$ depends only on the function $F$.
Appendix D: The Shifted Jacobi Polynomials $P_n^{(k,0)}(2x-1)$ {#secxkpol}
==============================================================
In this section, we introduce a class of Jacobi polynomials that can be used as quadrature and interpolation nodes for Zernike polynomials in $\R^{p+2}$.
We define $\widetilde{P}_n^k(x)$ to be the shifted Jacobi polynomials on the interval $[0,1]$ defined by the formula $$\label{760.2}
\widetilde{P}_n^k(x)=\sqrt{k+2n+1}P_n^{(k,0)}(1-2x)$$ where $k>-1$ is a real number and where $P_n^{(k,0)}$ is defined in (\[192\]). It follows immediately from (\[760.2\]) that $\widetilde{P}_n^k(x)$ are orthogonal with respect to weight function $x^k$. That is, for all non-negative integers $n$, the Jacobi polynomial $\widetilde{P}_n^k$ is a polynomial of degree $n$ such that $$\label{740.2}
\int_0^1\widetilde{P}_i^k(x)\widetilde{P}_j^k(x)x^kdx=\delta_{i,j}$$ for all non-negative integers $i,j$ where $k>-1$.\
The following lemma, which follows immediately from the combination of Lemma \[205\] and (\[760.2\]), provides a differential equation satisfied by $\widetilde{P}_n^k$.
\[780\] Let $k>-1$ be a real number and let $n$ be a non-negative integer. Then, $\widetilde{P}_n^k$ satisfies the differential equation, $$\label{800}
r-r^2\widetilde{P}_n^{k\prime\prime}(r)
+(k-rk+1-2r)\widetilde{P}_n^{k\prime}(r)+
n(n+k+1)\widetilde{P}_n^k(r)=0.$$ for all $r\in (0,1)$.
The following recurrence for $\widetilde{P}_n^k$ follows readily from the combination of Lemma \[760.2\] and (\[195\]).
\[900.2\] For all non-negative integers $n$ and for all real numbers $k>-1$, $$\label{920}
\begin{split}
\hspace*{-6em}
\widetilde{P}_{n+1}^{k}(r)&=
\frac{(2n+N+1)N^2+(2n+N)(2n+N+1)(2n+N+2)(1-2r)}{2(n+1)(n+N+1)(2n+N)}\\
&\cdot \frac{\sqrt{2n+k+1}}{\sqrt{2(n+1)+k+1}}
\widetilde{P}_n^{k}(r)\\
&-\frac{2(n+N)(n)(2n+N+2)}{2(n+1)(n+N+1)(2n+N)}
\frac{\sqrt{2(n-1)+k+1}}{\sqrt{2(n+1)+k+1}}
\widetilde{P}_{n-1}^{k}(r)
\end{split}$$
Numerical Evaluation of the Shifted Jacobi Polynomials {#secxkeval}
------------------------------------------------------
The following observations provide a way to evaluate $\widetilde{P}_n^k$ and its derivatives.
\[765\] Combining (\[195\]) with (\[760.2\]), we observe that $\widetilde{P}_n^k(x)$ can be evaluated by first evaluating $P_n^{(k,0)}(1-2x)$ via recurrence relation (\[195\]) and then multiplying the resulting number by $$\sqrt{k+2n+1}.$$
\[767\] Combining (\[197\]) with (\[760.2\]), we observe that the polynomial $\widetilde{P}_n^{k\prime}(x)$ (see (\[760.2\])), can be evaluated by first evaluating $P_n^{(k,0)\prime}(1-2x)$ via recurrence relation (\[197\]) and then multiplying the resulting number by $$-2\sqrt{k+2n+1}.$$
Pr[ü]{}fer Transform {#secprufer}
--------------------
In this section, we describe the Pr[ü]{}fer Transform, which will be used in Section \[secxkquad\]. A more detailed description of the Pr[ü]{}fer Transform can be found in [@20].
\[380\] Suppose that the function $\phi: [a,b] \rightarrow \R$ satisfies the differential equation $$\label{400}
\phi^{\prime\prime}(x)+\alpha(x)\phi^{\prime}(x)
+\beta(x)\phi(x)=0,$$ where $\alpha,\beta:(a,b) \rightarrow \R$ are differential functions. Then, $$\label{440}
\frac{d\theta}{dx}=-\sqrt{\beta(x)}-\left(\frac{\beta^{\prime}(x)}
{4\beta(x)}+\frac{\alpha(x)}{2}\right)sin(2\theta),$$ where the function $\theta :[a,b]\rightarrow \R$ is defined by the formula, $$\label{420}
\frac{\phi^\prime(x)}{\phi(x)}=\sqrt{\beta(x)}\tan(\theta(x)).$$
Introducing the notation $$\label{460}
z(x)=\frac{\phi^{\prime}(x)}{\phi(x)}$$ for all $x\in [a,b]$, and differentiating (\[460\]) with respect to $x$, we obtain the identity $$\label{480}
\frac{\phi^{\prime\prime}}{\phi}=\frac{dz}{dx}+z^2(x).$$ Substituting (\[480\]) and (\[460\]) into (\[400\]), we obtain, $$\label{500}
\frac{dz}{dx}=-(z^2(x)+\alpha(x)z(x)+\beta(x)).$$ Introducing the notation, $$\label{520}
z(x)=\gamma(x)\tan(\theta(x)),$$ with $\theta,\gamma$ two unknown functions, we differentiate (\[520\]) and observe that, $$\label{540}
\frac{dz}{dx}=\gamma(x)\frac{\theta^{\prime}}{\cos^2(\theta)}
+\gamma^{\prime}(x)\tan(\theta(x))$$ and squaring both sides of (\[520\]), we obtain $$\label{560}
z(x)^2=\tan^2(\theta(x))\gamma(x)^2.$$ Substituting (\[540\]) and (\[560\]) into (\[500\]) and choosing $$\label{580}
\gamma(x)=\sqrt{\beta(x)}$$ we obtain $$\label{600}
\frac{d\theta}{dx}=-\sqrt{\beta(x)}-\left(\frac{\beta^{\prime}(x)}
{4\beta(x)}+\frac{\alpha(x)}{2}\right)sin(2\theta).$$
\[620\] The Pr[ü]{}fer Transform is often used in algorithms for finding the roots of oscillatory special functions. Suppose that $\phi: [a,b] \rightarrow \R$ is a special function satisfying differential equation (\[400\]). It turns out that in most cases, coefficient $$\label{640}
\beta(x)$$ in (\[400\]) is significantly larger than $$\label{660}
\frac{\beta^{\prime}(x)}{4\beta(x)}+\frac{\alpha(x)}{2}$$ on the interval $[a,b]$, where $\alpha$ and $\beta$ are defined in (\[400\]).
Under these conditions, the function $\theta$ (see (\[420\])), is monotone and its derivative neither approaches infinity nor $0$. Furthermore, finding the roots of $\phi$ is equivalent to finding $x \in [a,b]$ such that $$\label{670}
\theta(x)=\pi/2+k\pi$$ for some integer $k$. Consequently, we can find the roots of $\phi$ by solving well-behaved differential equation (\[600\]).
\[680\] If for all $x\in [a,b]$, the function $\sqrt{\beta(x)}$ satisfies $$\label{700}
\sqrt{\beta(x)}>
\frac{\beta^{\prime}(x)}
{4\beta(x)}+\frac{\alpha(x)}{2},$$ then, for all $x\in [a,b]$, we have $\frac{d\theta}{dx} < 0$ (see (\[440\])) and we can view $x:[-\pi,\pi]\rightarrow \R$ as a function of $\theta$ where $x$ satisfies the first order differential equation $$\label{720}
\frac{dx}{d\theta}=\left(-\sqrt{\beta(x)}-\left(\frac{\beta^{\prime}(x)}
{4\beta(x)}+\frac{\alpha(x)}{2}\right)sin(2\theta)\right)^{-1}.$$
Roots of the Shifted Jacobi Polynomials {#secxkquad}
---------------------------------------
The primary purpose of this section is to describe an algorithm for finding the roots of the Jacobi polynomials $\widetilde{P}_n^k$. These roots will be used in Section \[seczernquad\] for the design of quadratures for Zernike Polynomials.\
The following lemma follows immediately from applying the Prufer Transform (see Lemma \[380\]) to (\[800\]).
\[900\] For all non-negative integers $n$, real $k>-1$, and $r\in(0,1)$, $$\label{eq1140}
\frac{d\theta}{dr}=
-\left(\frac{n(n+k+1)}{r-r^2}\right)^{1/2}
-\left(
\frac{1-2r+2k-2kr}{4(r-r^2)}
\right)
\sin(2\theta(r)).$$ where the function $\theta:(0,1) \rightarrow \R$ is defined by the formula $$\label{eq1200}
\frac{\widetilde{P}_n^k(r)}{\widetilde{P}_n^{k\prime}(r)}=
\left(\frac{n(n+k+1)}{r-r^2}\right)^{1/2}\tan(\theta(r)),$$ where $\widetilde{P}_n^k$ is defined in (\[740.2\]).
For any non-negative integer $n$, $$\frac{d\theta}{dr}<0$$ for all $r\in(0,1)$. Therefore, applying Remark \[680\] to (\[eq1140\]), we can view $r$ as a function of $\theta$ where $r$ satisfies the differential equation $$\label{eq1150}
\frac{dr}{d\theta}=
\left(
-\left(\frac{n(n+k+1)}{r-r^2}\right)^{1/2}
-\left(
\frac{1-2r+2k-2kr}{4(r-r^2)}
\right)
\sin(2\theta(r))
\right)^{-1}.$$
### Algorithm {#secalg}
In this section, we describe an algorithm for the evaluation of the $n$ roots of $\widetilde{P}_n^k$. We denote the $n$ roots of $\widetilde{P}_n^k$ by $r_1<r_2<...<r_n$.\
\
Step 1. Choose a point, $x_0 \in (0,1)$, that is greater than the largest root of $\widetilde{P}_n^k$. For example, for all $k \geq 1$, the following choice of $x_0$ will be sufficient: $$\begin{aligned}
x_0 = \left\{
\begin{array}{ll}
1-10^{-6} & \mbox{if $n <10^3$}, \\
1-10^{-8} & \mbox{if $10^3 \leq n <10^4$}, \\
1-10^{-10} & \mbox{if $10^4 \leq n <10^5$}.
\end{array}
\right.
\end{aligned}$$\
\
Step 2. Defining $\theta_0$ by the formula $$\theta_0=\theta(x_0),$$ where $\theta$ is defined in (\[eq1200\]), solve the ordinary differential equation $\frac{dr}{d\theta}$ (see (\[eq1150\])) on the interval $(\pi/2,\theta_0)$, with the initial condition $r(\theta_0)=x_0$. To solve the differential equation, it is sufficient to use, for example, second order Runge Kutta with $100$ steps (independent of $n$). We denote by $\tilde{r}_n$ the approximation to $r(\pi/2)$ obtained by this process. It follows immediately from $(\ref{670})$ that $\tilde{r}_n$ is an approximation to $r_n$, the largest root of $\widetilde{P}_n^k$.\
\
Step 3. Use Newton’s method with $\tilde{r}_n$ as an initial guess to find $r_n$ to high precision. The polynomials $\widetilde{P}_n^k$ and $\widetilde{P}_n^{k\prime}$ can be evaluated via Observation \[765\] and Observation \[767\].\
\
Step 4. With initial condition $$x(\pi/2)=r_n,$$ solve differential equation $\frac{dr}{d\theta}$ (see (\[eq1150\])) on the interval $$(-\pi/2,\pi/2)$$ using, for example, second order Runge Kuta with $100$ steps. We denote by $\tilde{r}_{n-1}$ the approximation to $$r(-\pi/2)$$ obtained by this process.\
\
Step 5. Use Newton’s method, with initial guess $\tilde{r}_{n-1}$, to find to high precision the second largest root, $r_{n-1}$.\
\
Step 6. For $k=\{1,2,...,n-1\}$, repeat Step 4 on the interval $$(-\pi/2-k\pi,-\pi/2-(k-1)\pi)$$ with intial condition $$x(-\pi/2-(k-1)\pi)=r_{n-k+1}$$ and repeat Step 5 on $\tilde{r}_{n-k}$.
Appendix E: Notational Conventions for Zernike Polynomials {#not:1}
==========================================================
In two dimensions, the Zernike polynomials are usually indexed by their azimuthal order and radial order. In this report, we use a slightly different indexing scheme, which leads to simpler formulas and generalizes easily to higher dimensions (see Section \[seczern\] for our definition of the Zernike polynomials $Z_{N,n}^\ell$ and the radial polynomials $R_{N,n}$). However, for the sake of completeness, we describe in this section the standard two dimensional indexing scheme, as well as other widely used notational conventions.
If $|m|$ denotes the azimuthal order and $n$ the radial order, then the Zernike polynomials in standard two index notation (using asterisks to differentiate them from the polynomials $Z_{N,n}^\ell$ and $R_{N,n}$) are [ $$\begin{aligned}
\accentset{*}{Z}_n^m(\rho,\theta) = \accentset{*}{R}_n^{|m|}(\rho) \cdot
\left\{
\begin{array}{cc}
\sin(|m|\theta) & \text{if $m<0$}, \\
\cos(|m|\theta) & \text{if $m>0$}, \\
1 & \text{if $m=0$},
\end{array}
\right.
\label{not:2}
\end{aligned}$$ ]{} where $$\begin{aligned}
\accentset{*}{R}_n^{|m|}(\rho) =
\sum_{k=0}^{\frac{n-|m|}{2}}
\frac{ (-1)^k (n-k)! }{ k! \bigl(\frac{n+|m|}{2} - k\bigr)!
\bigl(\frac{n-|m|}{2} - k\bigr)! } \rho^{n-2k},
\label{not:3}
\end{aligned}$$ for all $m=0,\pm 1,\pm 2, \ldots$ and $n=|m|,|m|+2,|m|+4,\ldots$ (see Figure \[not:2\]); they are normalized so that $$\begin{aligned}
\accentset{*}{R}_n^{|m|}(1) = 1,
\end{aligned}$$ for all $m=0,\pm 1,\pm 2, \ldots$ and $n=|m|,|m|+2,|m|+4,\ldots$ . We note that $$\begin{aligned}
\accentset{*}{R}_n^{|m|}(\rho) = R_{|m|,\frac{n-|m|}{2}}(\rho),
\label{not:4}
\end{aligned}$$ for all $m=0,\pm 1,\pm 2, \ldots$ and $n=|m|,|m|+2,|m|+4,\ldots$, where $R$ is defined by (\[360\]) (see Figure \[fig:rnn\_notation\]); equivalently, $$\begin{aligned}
R_{N,n}(\rho) = \accentset{*}{R}^N_{N+2n}(\rho),
\label{not:5}
\end{aligned}$$ for all nonnegative integers $N$ and $n$.
\[not:6\] The quantity $n+|m|$ is sometimes referred to as the “spacial frequency” of the Zernike polynomial $\accentset{*}{Z}_n^m(\rho,\theta)$. It roughly corresponds to the frequency of the polynomial on the disc, as opposed to the azimuthal frequency $|m|$ or the order of the polynomial $n$.
Zernike Fringe Polynomials
--------------------------
The Zernike Fringe Polynomials are the standard Zernike polynomials, normalized to have $L^2$ norm equal to $\pi$ on the unit disc and ordered by their spacial frequency $n+|m|$ (see Table \[tab:fringe\] and Figure \[fig:fringe\_ord\]). This ordering is sometimes called the “Air Force” or “University of Arizona” ordering.
ANSI Standard Zernike Polynomials
---------------------------------
The ANSI Standard Zernike polynomials, also referred to as OSA Standard Zernike polynomials or Noll Zernike polynomials, are the standard Zernike polynomials, normalized to have $L^2$ norm $\pi$ on the unit disc and ordered by $n$ (the order of the polynomial on the disc; see Table \[tab:ansi\] and Figure \[fig:ansi\_ord\]).
Wyant and Creath Notation
-------------------------
[ In [@wyant], James Wyant and Katherine Creath observe that it is sometimes convienient to factor the radial polynomial $\accentset{*}{R}_{2n-|m|}^{|m|}$ into $$\begin{aligned}
\accentset{*}{R}_{2n-|m|}^{|m|}(\rho) = Q_n^{|m|}(\rho) \rho^{|m|},
\end{aligned}$$ for all $m=0,\pm 1,\pm 2, \ldots$ and $n=|m|,|m|+1,|m|+2,\ldots$, where the polynomial $Q_n^{|m|}$ is of order $2(n-|m|)$ (see Figure \[fig:wyant\]). Equivalently, the factorization can be written as $$\begin{aligned}
\accentset{*}{R}_{n}^{|m|}(\rho) = Q_{\frac{n+|m|}{2}}^{|m|}(\rho) \rho^{|m|},
\end{aligned}$$ for all $m=0,\pm 1,\pm 2, \ldots$ and $n=|m|,|m|+2,|m|+4,\ldots$ . ]{}
(-1,-) – (-1,); (0,--1) – (,--1);
in [-,...,]{} (-0.1-1,) – (0.1-1,);
in [-,...,-1]{} at (-1.67,) [$\y$]{}; in [0,...,]{} at (-1.45,) [$\y$]{};
in [0,...,]{}[ (,--1-0.1) – (,--1+0.1); at (,--1-0.5) [$\x$]{}; ]{}
at (-2.2,0.1) [$m$]{}; at (2.7,--2.1) [$n$]{};
(-1,-) – (-1,); (0,--1) – (,--1);
in [-,...,]{} (-0.1-1,) – (0.1-1,);
in [-,...,-1]{} at (-1.67,) [$\y$]{}; in [0,...,]{} at (-1.45,) [$\y$]{};
in [0,...,]{}[ (,--1-0.1) – (,--1+0.1); at (,--1-0.5) [$\x$]{}; ]{}
at (-2.2,0.1) [$m$]{}; at (2.7,--2.1) [$n$]{};
(-1,-) – (-1,); (0,--1) – (,--1);
in [-,...,]{} (-0.1-1,) – (0.1-1,);
in [-,...,-1]{} at (-1.67,) [$\y$]{}; in [0,...,]{} at (-1.45,) [$\y$]{};
in [0,...,]{}[ (,--1-0.1) – (,--1+0.1); at (,--1-0.5) [$\x$]{}; ]{}
at (-2.2,0.1) [$m$]{}; at (2.7,--2.1) [$n$]{};
(-1,-) – (-1,); (0,--1) – (,--1);
in [-,...,]{} (-0.1-1,) – (0.1-1,);
in [-,...,-1]{} at (-1.67,) [$\y$]{}; in [0,...,]{} at (-1.45,) [$\y$]{};
in [0,...,]{}[ (,--1-0.1) – (,--1+0.1); at (,--1-0.5) [$\x$]{}; ]{}
at (-2.2,0.1) [$m$]{}; at (2.7,--2.1) [$n$]{};
(-1,-) – (-1,); (0,--1) – (,--1);
in [-,...,]{} (-0.1-1,) – (0.1-1,);
in [-,...,-1]{} at (-1.67,) [$\y$]{}; in [0,...,]{} at (-1.45,) [$\y$]{};
in [0,...,]{}[ (,--1-0.1) – (,--1+0.1); at (,--1-0.5) [$\x$]{}; ]{}
at (-2.2,0.1) [$m$]{}; at (2.7,--2.1) [$n$]{};
[llllll]{} & & & & &\
& & & & &\
0 & 0 & 0 & 0 & $1$ & piston\
1 & 1 & 1 & 2 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,0}(\rho) \cos(\theta)$ & tilt in $x$-direction\
2 & 1 & -1 & 2 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,0}(\rho) \sin(\theta)$ & tilt in $y$-direction\
3 & 2 & 0 & 2 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{0,1}(\rho)$ & defocus (power)\
4 & 2 & 2 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,0}(\rho) \cos(2\theta)$ & defocus + astigmatism 45$^\circ$/135$^\circ$\
5 & 2 & -2 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,0}(\rho) \sin(2\theta)$ & defocus + astigmatism 90$^\circ$/180$^\circ$\
6 & 3 & 1 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,1}(\rho) \cos(\theta)$ & tilt + horiz. coma along $x$-axis\
7 & 3 & -1 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,1}(\rho) \sin(\theta)$ & tilt + vert. coma along $y$-axis\
8 & 4 & 0 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{0,2}(\rho)$ & defocus + spherical aberration\
9 & 3 & 3 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,0}(\rho) \cos(3\theta)$ & trefoil in $x$-direction\
10 & 3 & -3 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,0}(\rho) \sin(3\theta)$ & trefoil in $y$-direction\
11 & 4 & 2 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,1}(\rho) \cos(2\theta)$ &\
12 & 4 & -2 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,1}(\rho) \sin(2\theta)$ &\
13 & 5 & 1 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,2}(\rho) \cos(\theta)$ &\
14 & 5 & -1 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,2}(\rho) \sin(\theta)$ &\
15 & 6 & 0 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{0,3}(\rho)$ &\
16 & 4 & 4 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{4,0}(\rho) \cos(4\theta)$ &\
17 & 4 & -4 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{4,0}(\rho) \sin(4\theta)$ &\
18 & 5 & 3 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,1}(\rho) \cos(3\theta)$ &\
19 & 5 & -3 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,1}(\rho) \sin(3\theta)$ &\
20 & 6 & 2 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,2}(\rho) \cos(2\theta)$ &\
21 & 6 & -2 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,2}(\rho) \sin(2\theta)$ &\
22 & 7 & 1 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,3}(\rho) \cos(\theta)$ &\
23 & 7 & -1 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,3}(\rho) \sin(\theta)$ &\
24 & 8 & 0 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{0,4}(\rho)$ &\
[llllll]{} & & & & &\
& & & & &\
0 & 0 & 0 & 0 & $1$ & piston\
1 & 1 & -1 & 2 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,0}(\rho) \sin(\theta)$ & tilt in $y$-direction\
2 & 1 & 1 & 2 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,0}(\rho) \cos(\theta)$ & tilt in $x$-direction\
3 & 2 & -2 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,0}(\rho) \sin(2\theta)$ & defocus + astigmatism 90$^\circ$/180$^\circ$\
4 & 2 & 0 & 2 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{0,1}(\rho)$ & defocus (power)\
5 & 2 & 2 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,0}(\rho) \cos(2\theta)$ & defocus + astigmatism 45$^\circ$/135$^\circ$\
6 & 3 & -3 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,0}(\rho) \sin(3\theta)$ & trefoil in $y$-direction\
7 & 3 & -1 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,1}(\rho) \sin(\theta)$ & tilt + vert. coma along $y$-axis\
8 & 3 & 1 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,1}(\rho) \cos(\theta)$ & tilt + horiz. coma along $x$-axis\
9 & 3 & 3 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,0}(\rho) \cos(3\theta)$ & trefoil in $x$-direction\
10 & 4 & -4 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{4,0}(\rho) \sin(4\theta)$ &\
11 & 4 & -2 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,1}(\rho) \sin(2\theta)$ &\
12 & 4 & 0 & 4 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{0,2}(\rho)$ &\
13 & 4 & 2 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,1}(\rho) \cos(2\theta)$ &\
14 & 4 & 4 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{4,0}(\rho) \cos(4\theta)$ &\
15 & 5 & -5 & 10 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{5,0}(\rho) \sin(5\theta)$ &\
16 & 5 & -3 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,1}(\rho) \sin(3\theta)$ &\
17 & 5 & -1 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,2}(\rho) \sin(\theta)$ &\
18 & 5 & 1 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{1,2}(\rho) \cos(\theta)$ &\
19 & 5 & 3 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{3,1}(\rho) \cos(3\theta)$ &\
20 & 5 & 5 & 10 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{5,0}(\rho) \cos(5\theta)$ &\
21 & 6 & -6 & 12 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{6,0}(\rho) \sin(6\theta)$ &\
22 & 6 & -4 & 10 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{4,1}(\rho) \sin(4\theta)$ &\
23 & 6 & -2 & 8 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{2,2}(\rho) \sin(2\theta)$ &\
24 & 6 & 0 & 6 & ${\mkern 1.5mu\overline{\mkern-1.5mu R\mkern-1.5mu}\mkern 1.5mu}_{0,3}(\rho)$ &\
[99]{}
Abramowitz, Milton, and Irene A. Stegun, eds. [*Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables.*]{} Washington: U.S. Govt. Print. Off., 1964.
Alpert, Bradley K. and Vladimir Rokhlin. “A fast algorithm for the evaluation of Legendre expansions.” [*SIAM J. Sci. Stat. Comput.*]{} 12.1 (1991): 158–179.
Bateman, Harry. [*Higher Transcendental Functions, Vol. 2*]{}. Malabar, Florida: Robert E. Krieger Publishing Company, Inc., 1985.
Born, Max, and Emil Wolf. [*Principles of Optics.*]{} 6th ed. (with corrections). New York: Pergamon Press Inc., 1980.
Boyd, John P. and Fu Yu. “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike Polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier Series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions”[*J. Comput. Phys.*]{} 230.4 (2011): 1408–1438.
Chong, Chee-Way, P. Raveendran, and R. Mukundan. “A comparative analysis of algorithms for fast computation of Zernike moments.” [Pattern Recognition]{} 36 (2003): 731–742.
Glaser, Andreas, Xiangtao Liu, and Vladimir Rokhlin. “A Fast Algorithm for the Calculation of the Roots of Special Functions.”[*SIAM J. Sci. Comput.*]{} 29.4 (2007): 1420–1438.
Jagerman, Louis S. [*Ophthalmologists, meet Zernike and Fourier!*]{}. Victoria, BC, Canada: Trafford Publishing, 2007.
Kintner, Eric C. “On the mathematical properties of the Zernike polynomials.” [*Optica Acta*]{} 23.8 (1976): 679–680.
. ANSI Standard Z80.23-2008 (R2013).
Radiant ZEMAX LLC. [*ZEMAX: Optical Design Program User’s Manual.*]{} Redmond, WA: Author, 2011.
Slepian, David. “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions.” [*Bell Labs Technical Journal*]{} 43.6 (1964): 3009-3057.
Stoer, Josef and Roland Bulirsch. [*Introduction to Numerical Analysis*]{}, 2nd ed., Springer-Verlag, 1992.
Wyant, James C., and Katherine Creath. “Basic wavefront aberration theory for optical metrology.” Applied optics and optical engineering II (1992): 28–39.
|
---
abstract: 'Calculations of high multiplicity Higgs amplitudes exhibit a rapid growth that may signal an end of perturbative behavior or even the need for new physics phenomena. As a step towards this problem we consider the quantum mechanical equivalent of $1 \to n$ scattering amplitudes in a spontaneously broken $\phi^4$-theory by extending our previous results on the quartic oscillator with a single minimum [@Jaeckel:2018ipq] to transitions $\langle n \lvert \hat{x} \rvert 0 \rangle$ in the symmetric double-well potential with quartic coupling $\lambda$. Using recursive techniques to high order in perturbation theory, we argue that these transitions are of exponential form $\langle n \lvert \hat{x} \rvert 0 \rangle \sim \exp \left( F (\lambda n) / \lambda \right)$ in the limit of large $n$ and $\lambda n$ fixed. We apply the methods of “exact perturbation theory" put forward by Serone *et al.* in [@Serone:2016qog; @Serone:2017nmd] to obtain the exponent $F$ and investigate its structure in the regime where tree-level perturbation theory violates unitarity constraints. We find that the resummed exponent is in agreement with unitarity and rigorous bounds derived by Bachas [@Bachas:1991fd].'
author:
- |
Joerg Jaeckel and Sebastian Schenk\
\
\
title:
---
Introduction {#sec:introduction}
============
Perturbative[@Cornwall:1990hh; @Goldberg:1990qk; @Brown:1992ay; @Voloshin:1992mz; @Argyres:1992np; @Smith:1992kz; @Smith:1992rq] as well as semiclassical[^1]$^{,}$[^2] [@Son:1995wz; @Khoze:2017ifq; @Khoze:2018kkz] calculations of high multiplicity $1 \to n$ scattering amplitudes in scalar field theories exhibit an extremely rapid growth with the number of final state particles. The experimentally observed [@Aad:2012tfa; @Chatrchyan:2012xdj] existence of a scalar Higgs boson with a mass of $125$ GeV has taken this problem out of the realm of purely field theoretic interest and has provided us with an explicit upper scale $\lesssim 1600$ TeV [@Voloshin:1992rr; @Jaeckel:2014lya] (possibly significantly smaller), where we may need better calculational techniques or even new physics.
A particularly interesting form of a new phenomenon could be the “Higgsplosion" and “Higgspersion" effect recently proposed in [@Khoze:2017tjt], that could potentially even address the hierarchy problem and provide for interesting phenomenology [@Khoze:2017lft; @Khoze:2017uga; @Khoze:2018bwa][^3].
For this work our aim is, to some extent, more conservative. We want to shed light on what are the relevant features that give rise to the growth in high multiplicity amplitudes and whether it can be cured by improved calculational techniques. To this end we consider a very simplistic toy model: quantum mechanics with a quartic potential. Here, the vacuum transitions ${\langle n \lvert \hat{x} \rvert 0 \rangle}$ correspond to $1 \to n$ scattering amplitudes.
In studying this toy model we should be keenly aware that, due to its $(3+1)$- instead of $(0+1)$-dimensional nature, quantum field theory is subject to additional features and complications that have to be taken into account. Important examples are the presence of a non-trivial phase space and the possibility of having weakly coupled, spatially separated final states (see [@Khoze:2018mey] for details). Nevertheless, we think that it can give important insights into those features, such as the quartic potential, or the existence of a single or multiple minima, that are shared between the two theories. As we find that advanced calculational techniques probably stop the growth of quantum mechanical amplitudes, it in turn focusses efforts to establish the onset of new phenomena on those aspects of quantum field theory that are different from quantum mechanics.
In a recent paper [@Jaeckel:2018ipq] we have provided significant evidence that vacuum transition amplitudes in the anharmonic oscillator with a single-well potential with quartic coupling $\lambda$ take on the exponential form (conjectured in [@Voloshin:1992nu; @Khlebnikov:1992af; @Libanov:1994ug; @Libanov:1995gh; @Bezrukov:1995qh; @Libanov:1996vq]), $${\langle n \lvert \hat{x} \rvert 0 \rangle} = \exp \left( \frac{1}{\lambda} F \right) \, .$$ At tree-level $F$ turns positive beyond a critical value of $\lambda n$. This indicates a rapid growth of ${\langle n \lvert \hat{x} \rvert 0 \rangle}$ in the double scaling limit $n\to\infty$ with $\lambda n$ fixed. However, suitably resummed perturbation theory results in the exponent – often called *holy grail function* – being negative, $F < 0$, preventing a rapid growth of the amplitude at high energies [@Jaeckel:2018ipq].
In this work we extend these results to the symmetric double-well potential, $$V(x) = -x^2 + \lambda x^4 \, ,$$ which is the quantum mechanical analogue of spontaneously broken $\phi^4$-theory. We therefore realize an important feature that is essential in the case of the Standard Model Higgs. In particular, our aim is to establish that also in case of the double-well potential the amplitude takes on an exponential form and to find the sign of $F$. To do so we use recursive relations to compute $F$ to high orders in a perturbative expansion. We then apply the method of *exact perturbation theory* (EPT) put forward in [@Serone:2016qog; @Serone:2017nmd]. Using this we investigate the behavior of $F$ at values of $\lambda n$ beyond the point where tree-level perturbation theory violates unitarity constraints and find strong indications for a restoration of unitarity.
This work is arranged as follows. In Section \[sec:wavefunctions\] we start with a brief review of how to reconstruct wave functions and energy levels of a Schroedinger problem using recursive methods to high order in perturbation theory. These are the main building blocks for computing vacuum transition amplitudes to highly excited states. Furthermore, we argue that these amplitudes are of exponential form. Section \[sec:ept\] is devoted to introducing the concept of exact perturbation theory and applying it to the holy grail function computed before. In particular, we present a specific example of EPT to obtain the holy grail function associated to the symmetric double-well potential. Finally, we conclude in Section \[sec:conclusions\] by giving a brief summary of our results and future perspectives.
Reconstructing Vacuum Transitions {#sec:wavefunctions}
=================================
The main building blocks for computing transition amplitudes from the vacuum to an excited state, ${\langle n \lvert \hat{x} \rvert 0 \rangle}$, are the wave functions and their corresponding energy eigenvalues. That is, we need to find the eigenfunctions of the Schroedinger operator $$\left( - \frac{d}{dx^2} + V(x) - E \right) \psi = 0
\label{eq:schroedingerop}$$ in the anharmonic oscillator with a symmetric double-well potential, $$V(x) = m^2 x^2 + \lambda x^4 \quad \mathrm{with} \quad m^2 <0 \, , \, \lambda > 0 \, .$$ An efficient way to find those solutions is to use recursive relations to high order in perturbation theory, first introduced by Bender and Wu in [@Bender:1969si; @Bender:1990pd]. A detailed review and application of this approach to transition amplitudes in the anharmonic oscillator with a single-well potential is given in [@Jaeckel:2018ipq].
Yet, there is an issue in applying perturbative techniques to double-well potentials. In fact, these methods rely on perturbations around the harmonic oscillator solution. If we naively tried to use them for the double-well potential, we would have to do perturbation theory in an inverted harmonic oscillator background – with the obvious problems arising from the instability of the potential. However, we can choose another reference point of the perturbative expansion which is locally harmonic. In our example a suitable point is one of the two minima of the double-well potential[^4]. Expanding around $x_+$, shifting the coordinate $\tilde{x} = x - x_+$ and subtracting the zero-point energy yields the asymmetric double-well potential $$\tilde{V} (\tilde{x}) = \tilde{m}^2 \tilde{x}^2 + 2 \sqrt{\tilde{m}^2} \sqrt{\lambda} \tilde{x}^3 + \lambda \tilde{x}^4 \, ,
\label{eq:doublewell_asymmetric}$$ where we defined $\tilde{m}^2 \equiv -2 m^2$. Due to its positive mass term the potential $\tilde{V} (x)$ is well suited for the perturbative approach we want to pursue. Calculationally the cost is the introduction of an additional cubic term $\sqrt{\lambda} \tilde{x}^3$. For convenience we will later set $m^2=-1$, such that the excitations in the minima are of mass $\tilde{m}^2=2$.
Before we continue let us remark that a constant shift in the ground state energy or in the definition of the position operator does not alter the vacuum transition amplitude ${\langle n \lvert \hat{x} \rvert 0 \rangle}$. In the case of the ground state energy this is obvious as, even in quantum mechanics, only energy differences are relevant, and an additive constant in the Hamiltonian has no effect on those. Furthermore, ${\langle n \lvert \hat{x} \rvert 0 \rangle}$ involves different energy eigenstates, which are orthogonal to each other. Therefore, a constant shift in the position operator does not affect the transition amplitude between those states.
Let us now continue, closely following Bender and Wu’s approach [@Bender:1969si; @Bender:1990pd]. We can reconstruct the wave functions and energy levels order by order[^5] in the coupling $\lambda$ of the Schroedinger operator associated to the potential $\tilde{V}(x)$. This standard perturbative ansatz allows us to compute the normalized vacuum transition amplitudes as described in detail in [@Jaeckel:2018ipq], $$\mathcal{A}_n \equiv \frac{{\langle n \lvert \hat{x} \rvert 0 \rangle}}{\sqrt{\bra n \vert n \ket \bra 0 \vert 0 \ket}} = \int_{\mathbb{R}} dx \, x \bar{\psi}_n \psi_0 \, .$$ Similar to the single-well ($m^2 > 0$) these can be factorized into a tree-level and higher order contributions, $$\mathcal{A}_n = \mathcal{A}_n^{\mathrm{tree}} \mathcal{A}_\Sigma \, ,$$ where the former is given by (cf. [@Brown:1992ay]) $$\mathcal{A}_n^{\mathrm{tree}} = \sqrt{\frac{n!}{2 \tilde{m}}} \left( \frac{\lambda}{2 \tilde{m}^3} \right)^{\frac{n-1}{2}} \, .
\label{eq:antree}$$ In particular, it turns out that the perturbative series of the amplitude is reproduced *exactly* by $$\label{eq:AnSymbolic}
\mathcal{A}_n = \mathcal{A}_n^{\mathrm{tree}} \exp \left( \frac{1}{\lambda} F_\Sigma \right) \, ,$$ where $F_\Sigma$ can be constructed systematically as a series expansion in $1/n$ (cf. [@Jaeckel:2018ipq]), $$F_\Sigma \left( \lambda, n \right) = F_0 (\lambda n) + \frac{1}{n} F_1 (\lambda n) + \frac{1}{n^2} F_2 (\lambda n) + \ldots \, .
\label{eq:FSigmaSymbolic}$$ Note that, in the coefficient functions $F_i$, the coupling and energy level only appear as the product $\lambda n$, so for convenience we will define the abbreviation $$\epsilon = \lambda n \, .$$ As we describe in more detail in [@Jaeckel:2018ipq], the coefficient functions $F_i$ can be obtained by making a polynomial ansatz in $\epsilon$ for each of the $F_{i}$. We can then expand the exponential in and compare the corresponding coefficients to the power series $\mathcal{A}_\Sigma$. Remarkably, a finite number of coefficients of $F_\Sigma$ reproduce infinitely many terms of the perturbative series of $\mathcal{A}_\Sigma$ *exactly*, rendering this resummation very powerful. We have checked this exact replication to order $\mathcal{O} \left( \lambda^{16} \right)$ of $\mathcal{A}_\Sigma$.
In order to study the behavior of the vacuum transition as $n \to \infty$, it is useful to also rewrite the tree-level factor in exponential form, $$\mathcal{A}_n = \mathcal{A}_n^{\mathrm{tree}} \mathcal{A}_\Sigma \sim \exp \left\{ \frac{1}{\lambda} \left( F^{\mathrm{tree}} + F_\Sigma \right) \right\} \, .$$ That is, the total exponent $F$ of the amplitude consists of two contributions, $F = F^{\mathrm{tree}} + F_\Sigma$.
We now want to consider the double scaling limit $n \to \infty$ and $\lambda \to 0$ with $\epsilon = \lambda n$ fixed. In this regime the sign of $F$ is crucial. Any $\epsilon$ where $F$ is positive will ultimately lead to an inevitable growth of the amplitudes in the limit where $n \to \infty$. We are thus interested in the overall sign of the holy grail function for all $\epsilon$.
In order to establish the sign of $F$, let us discuss its tree-level part first. Using Stirling’s formula for the factorial as $n \to \infty$ in , the tree-level contribution can be written as $$F^{\mathrm{tree}} \left( \epsilon \right) \sim \frac{\epsilon}{2} \left( \ln \frac{\epsilon}{4 \sqrt{2}} - 1 \right) \, .
\label{eq:FTree}$$ This is illustrated in Fig. \[fig:F\_tree\].
![Tree-level contribution to the holy grail function $F$ in the double scaling limit $n \to \infty$, $\epsilon=\lambda n=const$. It exhibits a minimum at $\epsilon=4\sqrt{2}$ and a root at $\epsilon=4\sqrt{2}e$.[]{data-label="fig:F_tree"}](F_tree){width="50.00000%"}
$F^{\mathrm{tree}}$ indicates a serious issue. It exhibits a root at $\epsilon = 4 \sqrt{2} e$ where it changes to positive sign, leading to a growth of the amplitude as $n \to \infty$, as pointed out earlier. The single-well anharmonic oscillator shows a similar behavior of the tree-level contribution, but it turns out that a suitable resummation of $F_\Sigma$ resolves this issue – $F$ remains negative for any $\epsilon$ [@Jaeckel:2018ipq]. However, such a direct resummation is problematic for the double-well. In fact, if one computes $F_\Sigma$ in the $1/n$-expansion , the only contribution that will matter in the double scaling limit is the leading order correction $F_0$, which reads $$F_0 \left( \epsilon \right) = \frac{17}{32} \epsilon^2 + \frac{125}{64 \sqrt{2}} \epsilon^3 + \frac{17815}{3072} \epsilon^4 + \frac{87549}{2048 \sqrt{2}} \epsilon^5 + \mathcal{O} \left( \epsilon^6 \right) \, .$$
![Leading order holy grail function $F = F^{\mathrm{tree}} + F_0$ in the double scaling limit $n \to \infty$ with $\epsilon = \lambda n$ fixed. The label denotes the highest coefficient in $\epsilon$ that is included in the series representation of $F_0$.[]{data-label="fig:F_lead"}](F_lead){width="50.00000%"}
The functional form of $F_0$ is strongly governed by the series truncation, such that the root of $F$ is shifted towards smaller values of $\epsilon$ when more terms of the series are included, as can be seen in Fig. \[fig:F\_lead\]. In contrast to the single-well the series representation of $F_0$, however, has only positive coefficients instead of alternating ones. This can indicate problems with unambiguous[^6] Borel resummation. Therefore, the sign of $F$ in the case of vacuum transitions in the double-well potential remains unclear.
In summary, applying the same techniques we used for the single-well ($m^2>0$) anharmonic oscillator in [@Jaeckel:2018ipq] to a double-well ($m^2<0$) shows that the perturbative expansion of the transition amplitudes is still of exponential form and can in principle be recovered *exactly*. However, in contrast to the single-well the series expansion of the exponent $F$ for the double-well indicates potential problems with naive Borel resummation, leaving the overall sign unclear.
Exact Perturbation Theory and the Holy Grail Function {#sec:ept}
=====================================================
We have just seen that ordinary perturbation theory provides no clear resolution to the quickly growing high multiplicity amplitudes. Therefore we have to turn to more powerful methods.
The crucial difference to the single-well case is the presence of two degenerate vacua. This can lead, e.g., to instantons, which cannot be captured by perturbation theory but still have to be included in the quantum mechanical path integral as non-trivial saddles of the action. It is known that these quantum effects, not respected by our ansatz, can cause perturbative expansions (of e.g. the vacuum energy) to be non Borel resummable [@Brezin:1977gk; @Bogomolny:1977ty; @Bogomolny:1980ur; @Stone:1977au; @Achuthan:1988wh; @Liang:1995zq].
Non-perturbative effects (e.g. instantons) in quantum mechanics and quantum field theory have been extensively studied in the literature, e.g. [@ZinnJustin:1981dx; @ZinnJustin:1982td; @ZinnJustin:1983nr; @Balitsky:1985in; @Balitsky:1986qn; @Aoyama:1998nt; @Aoyama:1997qk; @ZinnJustin:2004ib; @ZinnJustin:2004cg; @Jentschura:2004jg; @Marino:2007te; @Marino:2008ya; @Unsal:2012zj; @Aniceto:2013fka]. However, instead of using instanton calculus we follow a novel approach put forward by Serone *et al.* in [@Serone:2016qog; @Serone:2017nmd]. The principal idea is to recover full non-perturbative results by smart deformations of the perturbative series. As we will see, this approach is well suited for our consideration, because it makes efficient use of our previous results on the single-well case [@Jaeckel:2018ipq]. In particular, suitable deformations exploit the non-trivial exponentiation of the amplitude, as they fall in the same class of theories where this powerful resummation is possible[^7].
Let us briefly outline the relevant steps of their approach. In general, we can consider a quantum mechanical potential $V(x;\lambda)$ with coupling $\lambda$ that admits bound states (more precisely $\lim_{\lvert x \rvert \to \infty} V(x;\lambda) = \infty$). If in addition the potential satisfies $V(x;\lambda) = V(x \sqrt{\lambda}; 1) / \lambda$ it is called *classical*, because the perturbative expansion in $\lambda$ is identical to the expansion in $\hbar$.
Now consider two such classical potentials, $V_0(x;\lambda)$ and $V_1(x;\lambda)$. The crucial insight is then, that if $V_0(x;\lambda)$ admits a perturbation theory that is Borel resummable to the exact result, the perturbative series of $V(x;\lambda) = V_0(x;\lambda) + \lambda V_1(x;\lambda)$ is also Borel resummable to the exact result, given that $\lim_{\lvert x \rvert \to \infty} V_1(x;1)/V_0(x;1)=0$. The key is that we treat the part of the potential that causes trouble with Borel summability as a “small” perturbation $\sim\lambda$, thereby rearranging the perturbative expansion. This was coined *exact perturbation theory* in [@Serone:2016qog; @Serone:2017nmd].
Each of these two classical potentials can depend on an additional parameter $\lambda_0$. We can now try to find potentials, depending suitably on $\lambda_{0}$, such that $$\hat{V}(x;\lambda,\lambda_0) = V_0(x;\lambda,\lambda_0) + \lambda V_1(x;\lambda,\lambda_0)$$ and $$\hat{V}(x;\lambda,\lambda) = V(x;\lambda),$$ i.e. the original potential is recovered for $\lambda=\lambda_0$. This will allow us to extract the full information of $V(x;\lambda)$ by a perturbative expansion in $\lambda$ of the *auxiliary* potential $\hat{V}(x;\lambda,\lambda_0)$ and setting $\lambda=\lambda_0$ after performing the Borel resummation. Serone *et al.* discuss a variety of quantum mechanical examples in [@Serone:2016qog; @Serone:2017nmd].
This method can be useful for potentials with a negative mass term, where standard perturbation theory is not applicable or does not admit an unambiguous Borel resummation.
Let us now apply these ideas to the double-well potential[^8], $$V(x;\lambda) = -x^2 + \lambda x^4 \, .
\label{eq:doublewell}$$ For this we want to find a potential $\hat{V}(x;\lambda,\lambda_0) = V_0(x;\lambda,\lambda_0) + \lambda V_1(x;\lambda,\lambda_0)$ that reproduces $V(x;\lambda)$ at finite coupling and where $V_0(x;\lambda)$ admits a perturbative expansion that is Borel resummable to the exact result.
Note that the potentials $V_0$ and $V_1$ (and thus $\hat{V}$) are by no means unique. Even though the final results will be the same after exact resummation, there is a plethora of choices of $V_0$ and $V_1$ which are more or less suited for the approximate computation of certain quantities. In fact, neglecting constant and linear terms of the potential, the condition on $V_0$ and $V_1$ being *classical* constrains the most general form of $\hat{V}$, $$\hat{V}(x;\lambda,\lambda_0) = \left( v_2 + \lambda w_2 \right) x^2 + \left( v_3 + \lambda w_3 \right) \sqrt{\lambda} x^3 + v_4 \lambda x^4 \, ,
\label{eq:vhatgeneral}$$ where the coefficients $v_i$ and $w_i$ that belong to $V_0$ and $V_1$, respectively, are functions of $\lambda_0$ only, $v_i = v_i \left( \lambda_0 \right)$ and $w_i = w_i \left( \lambda_0 \right)$. In order to reproduce the original double-well potential in at $\lambda_0 = \lambda$ they have to satisfy the conditions $$\begin{aligned}
v_2 (\lambda) + \lambda w_2 (\lambda) &= -1 \\
v_3 (\lambda) + \lambda w_3 (\lambda) &= 0 \, ,\end{aligned}$$ as well as $$v_{4}=1\, .$$ This implies that the only free parameters to set up perturbation theory are $v_i ( \lambda_0 )$ (up to additional terms that cancel at $\lambda_0 = \lambda$ in both $v_i$ and $w_i$). Note, that we have normalized everything to the mass, $m^2=1$. Furthermore, the $v_i$ have to be chosen such that $V_0$ admits a perturbative expansion that is Borel summable in the end.
While the above conditions yield the most general choice of the potentials $V_0$ and $V_1$, we will focus on a specific example choosing a simple but non-trivial $v_{2}$.
The simplest example presumably constitutes the most intuitive choice of $V_0$ and $V_1$, $$\begin{aligned}
V_0(x;\lambda,\lambda_0) &= x^2 + \lambda x^4 \\
V_1(x;\lambda,\lambda_0) &= -\frac{2}{\lambda_{0}} x^2 \, ,\end{aligned}$$ where the single-well anharmonic oscillator potential $V_0(x;\lambda)$ is known to be Borel resummable [@Loeffel:1970fe; @Graffi:1990pe]. We then define the potential $$\hat{V}(x;\lambda,\lambda_0) = V_0(x;\lambda,\lambda_0) + \lambda V_1(x;\lambda,\lambda_0) = \left(1 - 2\frac{\lambda}{\lambda_0}\right) x^2 + \lambda x^4 \, ,
\label{eq:ex1_hatpotential}$$ which reproduces the double-well potential $V(x;\lambda)$ when setting $\lambda_0 = \lambda$, $$\hat{V}(x;\lambda,\lambda) = -x^2 + \lambda x^4 \, .$$ According to the ideas introduced at the beginning of this section we can now compute any quantity of interest in the double-well $V(x;\lambda)$ by considering the potential $\hat{V}(x;\lambda,\lambda_0)$ instead. In this potential we can do a perturbative expansion in $\lambda$ (while keeping $\lambda_0$ fixed), perform its Borel resummation and in the end remove the deformation, $\lambda_0 = \lambda$.
In principle we could now plug the new potential into the Schroedinger operator and perform all steps of the original computation of Section \[sec:wavefunctions\] in order to obtain ${\langle n \lvert \hat{x} \rvert 0 \rangle}$. However, in view of the form of $\hat{V}(x; \lambda, \lambda_0)$ in the deformation of the original potential is effectively introducing a mass term that depends on the coupling, $m^2(\lambda) = 1 - 2 \lambda / \lambda_0$. This allows us to use previous results on ${\langle n \lvert \hat{x} \rvert 0 \rangle}$ obtained in the single-well potential [@Jaeckel:2018ipq]. In particular, for arbitrary $m^2>0$ we know $${\langle n \lvert \hat{x} \rvert 0 \rangle} = {\langle n \lvert \hat{x} \rvert 0 \rangle}_{\mathrm{tree}} \exp \left( \frac{1}{\lambda} F_\Sigma \right) \, ,$$ where the tree-level contribution reads $${\langle n \lvert \hat{x} \rvert 0 \rangle}_{\mathrm{tree}} = \sqrt{\frac{n!}{2m}} \left( \frac{\lambda}{8m^3} \right)^{\frac{n-1}{2}} \, .$$ Here $F_\Sigma$ can again be written as a series expansion in $1/n$ (cf. ). It is thus dominated[^9] at large $n$ by $F_0$ which is given by $$F_0 (\epsilon) = -\frac{17}{16} \frac{\epsilon^2}{m^3} + \frac{125}{64} \frac{\epsilon^3}{m^6} - \frac{17815}{3072} \frac{\epsilon^4}{m^9} + \frac{87549}{4096} \frac{\epsilon^5}{m^{12}} + \mathcal{O} \left( \epsilon^6 \right) \, .$$ We can now plug the mass term $m^2(\lambda) = 1 - 2 \lambda / \lambda_0 = 1 - 2 \epsilon / \epsilon_0$ into $F$ (including tree-level and higher order contributions), do a perturbative expansion in $\lambda$ and rearrange the result in the corresponding $1/n$-expansion. This yields $$\hat{F}_0 (\epsilon, \epsilon_0) = -\frac{17}{16} \epsilon^2 + \frac{125}{64} \epsilon^3 - \frac{17815}{3072} \epsilon^4 + \mathcal{O} \left( \epsilon^5 \right) + \frac{1}{\epsilon_0} \left( \frac{3}{2} \epsilon^2 - \frac{51}{16} \epsilon^3 + \mathcal{O}\left( \epsilon^4 \right) \right) + \mathcal{O} \left( \frac{\epsilon^3}{\epsilon_0^2} \right)\, .$$ In principle $\hat{F}_0$ can now be resummed in $\epsilon$ before the deformation of the potential is lifted[^10] by inserting $\epsilon_0 = \epsilon$. This yields the holy grail function $F$ associated to the double-well potential . However, by construction we only know a finite number of terms of the series in $\epsilon$. Thus, we need to make use of an appropriate technique to estimate the behavior from the perturbative coefficients. Similar to our earlier work on the symmetric case [@Jaeckel:2018ipq], we first tried to make use of *Padé approximation*. However, a crucial difference is that here we have to do a separate Padé approximation for every value of $\epsilon_0$ that we want to probe.
While the different Padé approximants appear to converge to negative values for large $\epsilon$, it turns out that the approximation is spoiled by several spurious poles in the small $\epsilon$ region.
Spurious poles in Padé approximants occur also for a number of well-behaved functions and sometimes question the validity of the approximation beyond the pole. In fact, looking more closely at our series expansion for small $\epsilon_{0}<1$, we find indications for bad behavior. Here, the series appears to forfeit its oscillating sign structure that typically indicates stability in a resummation with a finite number of known terms. We discuss the technical details of this feature in Appendix \[app:auxparam\]. The relevant sign structure can directly be seen from Fig. \[fig:ept\_ex1\_F0\_coeffsign\].
In order to circumvent these problems we have tried different approximation schemes. Good results are provided by a *Borel-Padé approximation*. The first few diagonal Borel-Padé approximants of $F$ are illustrated in Fig. \[fig:ept\_ex1\_F\_borelpade\].
![Diagonal Borel-Padé approximants of the holy grail function $F$ in the double scaling limit $n \to \infty, \epsilon = \lambda n=const$. Higher order corrections in the $1/n$-expansion are neglected. $F$ is obtained with the ansatz of exact perturbation theory using the auxiliary potential .[]{data-label="fig:ept_ex1_F_borelpade"}](ept_ex1_F_borelpade){width="50.00000%"}
We observe that $F$ is indeed negative for a range of $\epsilon$. In particular, the roots of the resummed $F$ are shifted towards larger $\epsilon$ compared to the naive tree-level contribution when going to higher order in the Borel-Padé approximation. This gives crucial hints that – similar to the single-well – suitably resummed perturbation theory of $F$ resolves the rapid growth ${\langle n \lvert \hat{x} \rvert 0 \rangle}$ for large $n$.
However, we also note that there are still potential problems with the approximation that are more pronounced at small $\epsilon$. At small values of $\epsilon$ the Padé approximant to the Borel sum inherits the problem of spurious poles due to the all positive signs of the power series expansion. In the Laplace transformation these poles do not contribute significantly if we take the principle value for the integral. This results in the smooth estimate for the function $F$ shown in Fig. \[fig:ept\_ex1\_F\_borelpade\]. The effect of the spurious poles is suppressed at large values of $\epsilon$ since the integrand in the Laplace transform is suppressed exponentially in the region containing the poles. We also remark that such liftable poles are a common feature of Borel-Padé approximations since Padé approximants often feature poles somewhere along the positive real axis.
To verify the result of the Borel-Padé approximation, we have tried a number of other resummation schemes. They are briefly discussed in Appendices \[app:approximation\] and \[app:largeorder\]. In general, all of them consistently share the same features of negative $F$ at large $\epsilon$ but also some instability. In Fig. \[fig:convergence\] we explicitly illustrate their behavior at the minimum and at the root of the tree-level holy grail function. The Borel-Padé resummation scheme is shown in yellow. The other colors correspond to different approximation schemes discussed in Appendices \[app:approximation\] and \[app:largeorder\]. While convergence is not completely monotonous for all approximation schemes, they generally agree well with each other. In particular, at the tree-level zero of the holy grail function the spread between the different results is far smaller than the distance to zero. This gives a good indication that the sign of the holy grail function is indeed negative at this point.
![Value of the different approximants of the holy grail function $F$ in the double scaling limit $n \to \infty, \, \epsilon = \lambda n=const$ at the minimum, $\epsilon=4\sqrt{2}$, (left panel) and at the root, $\epsilon=4\sqrt{2} e$, (right panel) of the tree-level holy grail function. The $k$ denotes the number of coefficients of the power series of $F$ taken into account in the corresponding approximation scheme. The different approximation schemes shown are Padé, Borel-Padé (BP), Meijer G (MG) and Shafer (Sh), all of which are discussed in Appendix \[app:approximation\]. The label $\alpha$-${\mathrm{Exp}}$ corresponds to the scheme proposed in Appendix \[app:largeorder\].[]{data-label="fig:convergence"}](ept_F_min "fig:"){width="46.00000%"} ![Value of the different approximants of the holy grail function $F$ in the double scaling limit $n \to \infty, \, \epsilon = \lambda n=const$ at the minimum, $\epsilon=4\sqrt{2}$, (left panel) and at the root, $\epsilon=4\sqrt{2} e$, (right panel) of the tree-level holy grail function. The $k$ denotes the number of coefficients of the power series of $F$ taken into account in the corresponding approximation scheme. The different approximation schemes shown are Padé, Borel-Padé (BP), Meijer G (MG) and Shafer (Sh), all of which are discussed in Appendix \[app:approximation\]. The label $\alpha$-${\mathrm{Exp}}$ corresponds to the scheme proposed in Appendix \[app:largeorder\].[]{data-label="fig:convergence"}](ept_F_root "fig:"){width="46.00000%"}
The different approximations of the holy grail function can also be compared to existing results from WKB estimates [@Cornwall:1993rh] and a rigorous bound derived by Bachas [@Bachas:1991fd]. They are shown in Fig. \[fig:ept\_ex1\_F\_bounds\]. It turns out that $F$ obtained by EPT is consistent with these results, providing evidence that the ansatz is valid and yields a good approximation to the holy grail function associated to the symmetric double-well potential.
![Different approximants (to highest available order) of the holy grail function $F$ in the double scaling limit $n \to \infty, \, \epsilon = \lambda n=const$ (as in Fig. \[fig:ept\_ex1\_F\_borelpade\]) compared to WKB estimates [@Cornwall:1993rh] and a rigorous bound derived by Bachas [@Bachas:1991fd], labelled WKB and B, respectively. The other labels are as in Fig. \[fig:convergence\].[]{data-label="fig:ept_ex1_F_bounds"}](ept_ex1_F_bounds){width="50.00000%"}
Before concluding let us note that this simple example is just a particular case of a parametrization of $v_2 (\lambda_0) = const$. In our example we use $v_2=1$. In Appendix \[app:auxparam\] we discuss the convergence as a function of $v_{2}$.
Conclusions {#sec:conclusions}
===========
The behavior of $1 \to n$ scattering amplitudes in $\phi^4$ scalar quantum field theory at high multiplicities remains not well understood. Calculations of these amplitudes return results that rapidly grow with increasing $n$. This raises questions about the applicability of the employed calculational techniques, but possibly also about the interpretation of the underlying quantum field theory. Applied to the Higgs it may even allow for an entirely different phenomenology such as “Higgsplosion” [@Khoze:2017tjt]. In order to address these questions we consider the quantum mechanical equivalent, vacuum transition amplitudes to highly excited states, ${\langle n \lvert \hat{x} \rvert 0 \rangle}$, in the anharmonic oscillator with quartic coupling $\lambda$.
We extend our previous work on the single-well potential [@Jaeckel:2018ipq] to the symmetric double-well, that resembles a theory with spontaneous symmetry breaking such as the Higgs sector of the Standard Model. We find many similarities between both cases.
In particular, using standard perturbation theory to high orders, we find that the amplitude again takes on an exponential form, $${\langle n \lvert \hat{x} \rvert 0 \rangle} = {\langle n \lvert \hat{x} \rvert 0 \rangle}_{\mathrm{tree}} \exp \left( \frac{1}{\lambda} F_\Sigma \right) \, ,$$ where the exponent can be constructed in a $1/n$-expansion beyond leading order, $$F_\Sigma (\lambda, n) = F_0 (\lambda n) + \frac{F_1(\lambda n)}{n} + \frac{F_2(\lambda n)}{n^2} + \mathcal{O} \left(\frac{1}{n^3}\right) \, .$$ As explained in [@Jaeckel:2018ipq] it is non-trivial that the above form reproduces the perturbative series of ${\langle n \lvert \hat{x} \rvert 0 \rangle}$ *exactly*.
However, in the standard perturbative approach we find that $F_\Sigma$ has a series representation with only positive, growing coefficients, such that Borel summability may be problematic. Consequently, we make use of *exact perturbation theory* [@Serone:2016qog; @Serone:2017nmd], a novel approach to (perturbatively) study QM and QFT systems governed by non-perturbative effects. Considering different resummation schemes, we are able to obtain the holy grail function $F$ associated to vacuum transitions in the double-well. This suggests that $F$ is indeed negative everywhere in the double scaling limit $n \to \infty$ with $\epsilon = \lambda n = const$, $$F(\epsilon) < 0 \quad \forall \epsilon \, .$$ That is, the vacuum transitions ${\langle n \lvert \hat{x} \rvert 0 \rangle}$ in the symmetric double-well potential are in line with unitarity bounds for $n \to \infty$.
In summary, our results in this quantum mechanical toy model indicate that – similar to the single-well potential – appropriate resummation of the perturbative expansion of vacuum transitions in the symmetric double-well prevents the growth of the amplitude at high multiplicities. Even though it is just its quantum mechanical analogue, it might still suggest a possible guideline for the resolution of the rapidly growing $1 \to n$ scattering amplitudes in (spontaneously broken) $\phi^4$-theory, including the case of the Standard Model Higgs.
However, we remark that $\phi^4$-theory is a higher dimensional quantum field theory that is subject to additional complications. Amongst others these include the non-trivial phase space and the existence of weakly coupled, asymptotic states that are not present in quantum mechanics [@Khoze:2018mey].
It would be very interesting to further investigate if and how our results on quantum mechanics can be extended to quantum field theory, paying special attention to the properties that are unique to higher dimensional theories.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Valya Khoze and Michael Spannowsky for interesting discussions. JJ would like to thank the IPPP for hospitality. SS gratefully acknowledges financial support by the Heidelberg Graduate School of Fundamental Physics.
Choice of Auxiliary Potentials {#app:auxparam}
==============================
In general the auxiliary potential $\hat{V}(x; \lambda, \lambda_0) = V_0 (x; \lambda, \lambda_0) + \lambda V_1(x; \lambda, \lambda_0)$ deformed by the parameter $\lambda_0$ has to satisfy a few requirements. For instance, apart from recovering the original potential $V(x; \lambda)$ when the deformation is removed, $\lambda_0 = \lambda$, $$\hat{V}(x; \lambda, \lambda) = V (x; \lambda) \, ,$$ the potential $V_0$ has to admit bound states and both $V_i$ have to be *classical*, such that their perturbative expansion in $\lambda$ coincides with the expansion in $\hbar$. Nevertheless, these conditions leave us with a plethora of possibilities to construct $\hat{V}$.
In this section we want to discuss the implications of different choices of the potential $V_0$ and $V_{1}$. It is intuitive that the physical result after resumming and removing the potential deformation must be independent of the choice of $V_0$ and $V_{1}$. But, the choice of the deformation can affect the convergence properties of the perturbative expansion and different choices might be useful when calculating different observables. When deciding which choice is *suitable* one has to deal with several subtleties that we want to discuss in the following.
In order to be in line with the example presented in Section \[sec:ept\] we focus on the auxiliary mass term $v_2$ (cf. ). In particular, we want to discuss the most simple case $v_2(\lambda_0)=v_2=const$. Such a parametrization corresponds to the potentials $$V_0 = v_2 x^2 + \lambda x^4$$ and $$V_1 = -\frac{(v_2+1)}{\lambda_{0}}x^2 \, .$$ Here we require $v_2 > 0$ such that $V_0$ admits a perturbation theory that is Borel resummable to the exact result. Combining $V_{0}$ and $V_{1}$ we have the auxiliary potential $$\hat{V} \left(x; \lambda, \lambda_0 \right) = \left[ v_2 - \frac{\lambda}{\lambda_0} \left( v_2 + 1 \right) \right] x^2 + \lambda x^4 \, .$$ By a suitable rescaling we find that the effective dimensionless coupling of the theory with potential $V_0$ is $\lambda / v_2^{3/2}$. Thus, different choices of $v_2$ might influence the convergence properties of its associated perturbative expansion. The case $v_2=1$ corresponds to the simple example considered in . Let us now consider the behavior as we move away from this point into the two regimes $v_2 \ll 1$ and $v_2 \gg 1$.
In the first case $v_2 \ll 1$ we can see from the effective coupling, $\lambda / v_2^{3/2}$, that already the theory given by $V_0$ is strongly coupled. Effectively, even at small $\lambda$ we are therefore trying to set up a perturbative expansion that is not well-defined to begin with.
In contrast, perturbation theory for $V_0$ with $v_2 \gg 1$ naively should work well since the effective coupling is very small. However, we also note that $V_1$ in this case is large, pointing towards potential trouble. As we will see, this is indeed the case.
In order for EPT to work the perturbative expansion of $F$ has to be Borel resummable at fixed values of both $v_2$ and $\lambda_0$. In general an alternating sign at high orders (i.e. up to a finite number of exceptions) indicates well-behaved Borel summability. A problem with this criterion arises, because by construction we only know a finite number of terms of the perturbative series of $F$.
![Sign of each coefficient of $\hat{F}_0 (\epsilon,\epsilon_{0}) = \sum_k \hat{F}_{0,k}(\epsilon_{0}) \epsilon^k$ for different values of $\epsilon_0$ shown on the vertical axes. Blue dots denote a positive while red dots denote a negative sign, respectively. The left panel corresponds to $v_{2}=1$ whereas the right panel is for $v_{2}=1/2$. All input parameters are normalized to the mass, $m^2=1$. Note the different scales for $\epsilon_{0}$.[]{data-label="fig:ept_ex1_F0_coeffsign"}](ept_ex1_v2signs1 "fig:"){width="46.00000%"} ![Sign of each coefficient of $\hat{F}_0 (\epsilon,\epsilon_{0}) = \sum_k \hat{F}_{0,k}(\epsilon_{0}) \epsilon^k$ for different values of $\epsilon_0$ shown on the vertical axes. Blue dots denote a positive while red dots denote a negative sign, respectively. The left panel corresponds to $v_{2}=1$ whereas the right panel is for $v_{2}=1/2$. All input parameters are normalized to the mass, $m^2=1$. Note the different scales for $\epsilon_{0}$.[]{data-label="fig:ept_ex1_F0_coeffsign"}](ept_ex1_v2signs2 "fig:"){width="46.00000%"}
Fig. \[fig:ept\_ex1\_F0\_coeffsign\] illustrates the signs of the $k$-th coefficient of the series $\hat{F}_0 (\epsilon,\epsilon_{0}) = \sum_k \hat{F}_{0,k}(\epsilon_{0}) \epsilon^k$ for various values of $\epsilon_0$ and two values of $v_2$. For $v_{2}=1$ (left panel) we can see that a fully alternating series occurs only for values of $\epsilon_{0}>1$. Thus, after setting $\epsilon_{0}=\epsilon$ we can expect good convergence with the first few approximants only for $\epsilon>1$.
When lowering $v_{2}$ the alternating sign pattern is preserved for smaller values of $\epsilon_{0}$. However, as argued above, in this case the effective coupling is larger and despite the alternating sign we need more terms of the series for good convergence.
However, increasing $v_{2}$ too much is problematic, too. In this case the alternating sign pattern only appears for large values of $\epsilon_{0}$. While the alternating sign will be restored at higher orders, it is clear that approximations based only on the first few terms cannot capture this and behave as if the theory were not Borel summable. In this sense such a choice of $v_{2}$ is expected to exhibit worse convergence properties.
In summary, we note that the choice $v_2 \ll 1$ is problematic because we start with a strongly coupled theory, while $v_2 \gg 1$ suffers from an apparent breakdown of Borel summability. For a reasonable range of $\epsilon$ the choice $v_2 \simeq \mathcal{O}(1)$ seems suitable. An optimal choice will likely depend on the desired range of $\epsilon$.
Standard Approximation Schemes {#app:approximation}
==============================
An essential point of exact perturbation theory is the resummation of a (divergent) power series expansion, before the deformation of the potential is lifted. In the following we give a brief overview of the different approximation schemes shown in Section \[sec:ept\].
All of them are designed to cope with situations where only a finite number of terms of a power series expansion is known. Let us consider the formal power series $Z(g) = \sum_{k=0}^\infty z_k g^k$ in the following.
[Padé approximation]{} Padé approximation [@Pade:1892] is probably the most widely used technique to resum divergent series expansions where only a finite number of terms is available. Its key idea is to approximate $Z(g)$ by a rational function constructed out of two polynomials $P_M(g)$ and $Q_N(g)$ of degree $M$ and $N$, respectively, such that their ratio coincides with the first few coefficients of $Z$, $$P_M + Q_N Z(g) = \mathcal{O} \left(g^{M+N+1} \right) \, .$$ The Padé approximant of order $[M,N]$, $Z_{[M,N]}$, is then defined by the condition $$P_M + Q_N Z_{[M,N]}(g) = 0 \, .$$
It is empirically known that in most examples where Padé approximation is applicable, the diagonal sequence of approximants, $Z_{[N,N]}$, exhibits the best convergence properties to reconstruct $Z$, $$Z_{[N,N]}(g) \to Z(g) \quad (N \to \infty) \, .$$ In particular, if the coefficients of the power series giving rise to the Padé approximants have an oscillating sign, the true value of $Z$ will typically[^11] lie in between the neighbouring approximants, $Z_{[N,N]}$ and $Z_{[N,N+1]}$.
[Borel-Padé approximation]{} Borel resummation [@Borel:1899] is typically used, if the large order asymptotics of the coefficients of $Z(g)$ are known. The method relies on the idea to cancel potential factorial growth of the coefficients by considering the Borel transform of $Z(g)$, $$\mathcal{B}Z(g) = \sum_{k=0} \frac{z_k}{k!} g^k \, ,$$ hoping for $\mathcal{B}Z$ to converge. In the end, the factorial factor can be reintroduced by a Laplace transform, such that $Z(g)$ is recovered, $$Z(g) = \int_0^\infty dt \, e^{-t} \mathcal{B}Z(gt) \, .$$
For all practical purposes, the Borel-Padé approximation now takes into account the fact that the large order asymptotics of $Z(g)$ might not be known. Instead of computing the Borel transform $\mathcal{B}Z$ exactly, one can try to reconstruct it by a Padé approximant (cf. previous paragraph), $\mathcal{B}Z_{[M,N]}$. One can then show that for instance the diagonal Padé sequence will converge to the exact result, $$\int_0^\infty dt \, e^{-t} \mathcal{B}Z_{[N,N]}(gt) \to Z(g) \quad (N \to \infty) \, .$$
In this sense Borel-Padé approximation literally combines the Padé approximation with a conventional Borel resummation technique.
[Shafer approximation]{} The Shafer approximation [@Shafer:1974] can be understood as the quadratic extension of Padé approximation. That is, one tries to construct polynomials $P_L(g)$, $Q_M(g)$ and $R_N(g)$ of degree $L, M, N$, respectively, such that $$P_L + Q_M Z(g) + R_N Z^2(g) = \mathcal{O} \left( x^{L+M+N+2} \right) \, .$$ The Shafer approximant of order $[L,M,N]$, $Z_{[L,M,N]}$, is then defined by the quadratic equation $$P_L + Q_M Z_{[L,M,N]}(g) + R_N Z_{[L,M,N]}^2(g) = 0 \, .$$ Similar to Padé approximation, the diagonal Shafer approximants $Z_{[N,N,N]}$ will typically have the best convergence properties to $Z$, $$Z_{[N,N,N]}(g) \to Z(g) \quad (N \to \infty) \, .$$
[Meijer G approximation]{} The recently proposed Meijer G approximation scheme [@Mera:2018qte] relies on a similar idea that the Borel-Padé approximation makes use of. However instead of reconstructing the Borel transform $\mathcal{B}Z(g)$ by means of Padé approximants, it tries to “guess" its large order asymptotics by representing it by generalized hypergeometric functions, $$\mathcal{B}Z (g) \sim {}_{N+1}F_{N} \left( \mathbf{x}, \mathbf{y}; g \right) \, ,$$ where the argument vectors $\mathbf{x}$ and $\mathbf{y}$ are defined by singular points of Padé approximants of the successive coefficient ratios of $\mathcal{B}Z(g)$. The Laplace transform of $\mathcal{B}Z(g)$ can be carried out analytically and yields a Meijer G function. For details we refer the reader to [@Mera:2018qte].
Approximation by Guessing the Large Order Behavior of the Borel Sum {#app:largeorder}
===================================================================
Including the large order behavior as done in the Meijer G approximation scheme seems very promising. However, explicitly calculating the ratios between the coefficients in the Borel series for our case we find that they are not well approximated by a constant. Instead they seem to behave approximately as, $$r_{n}=\frac{a_{n+1}}{a_{n}}\sim f(\epsilon_{0})n^{\alpha},$$ with $\alpha$ in the range $1/2-1$ and the function $f$ depending on $\alpha$.
Using this the large order behavior can be accounted for with a so-called $\alpha$-exponential, $$\exp_{\alpha}\left[f(\epsilon_0) \epsilon \right] = \sum_{k=0}^{\infty} \frac{f \left( \epsilon_0 \right)^k}{k!^{\alpha}} \epsilon^k \quad \mathrm{with} \quad 0 < \alpha \leq 1 \, .
\label{eq:BF_largeorder}$$ By fitting to the known coefficients the function $f$ can be accurately expressed in an expansion of $1 / \epsilon_0$.
This function by itself does not very well represent the low order behavior. To include this we can correct the Borel transform by explicitly including the known coefficients up to $n_{\rm max}=14$, $$\mathcal{B}F(\epsilon, \epsilon_0) = \sum_{k=0}^{\infty} \frac{f \left( \epsilon_0 \right)^k}{k!^{\alpha}} \epsilon^k + \sum_{k=0}^{n_{\rm max}} \left( \frac{F_{0,k} (\epsilon_0)}{k!} - \frac{f \left( \epsilon_0 \right)^k}{k!^{\alpha}} \right) \epsilon^{k} \, .
\label{eq:BF_fullinformation}$$
In principle this could now be directly Laplace transformed, with respect to $\epsilon$ (keeping $\epsilon_{0}$ fixed). However, the behavior can be significantly improved by applying a Padé approximation to the remainder function given by the second part on the right hand side of Eq. . For practical purposes we apply the same order of Padé approximation to the remainder as well as the $\alpha$-exponential.
We can now apply this to our problem at hand. Fitting the known coefficients for the remainder function we find, $$\label{largeapp}
f(\epsilon_0) \approx -1.476 + 0.66 / \epsilon_0 + 0.064 / \epsilon_0^2 \quad{\rm for}\quad\alpha=1/2\,.$$ The results are shown in Fig. \[fig:ept\_ex1\_F\_borelpade\_largeorder\]. Note, that this approximation is only good for reasonably large $\epsilon_{0}\gtrsim 2$. Hence, the spread in the different Padé orders possibly underestimates the true uncertainty.
![Diagonal Borel-Padé approximants of the holy grail function $F$ with an estimated large order behavior as discussed in this Appendix using Eq. . We can see that the higher approximants are already nicely converged. Note, however, that this does not represent the full error. There is an additional systematic uncertainty at small $\epsilon$ due to the use of the approximate formula , as well as a general uncertainty due to our guessing of the large order behavior.[]{data-label="fig:ept_ex1_F_borelpade_largeorder"}](ept_ex1_F_borelpade_largeorder){width="50.00000%"}
Finally let us finish this discussion with a few words of caution. By estimating the large order behavior of $\mathcal{B}F$ from the low order coefficients, we gain access to Borel-Padé approximants of higher order. However, there is a subtle issue when summing $\mathcal{B}F$ that has to be treated with care. First of all, even though the $\alpha$-exponential has an infinite radius of convergence, $\mathcal{B}F$ cannot be summed naively, i.e. term by term, as it still contains the low order coefficients $F_{0,k}$. That is, the remainder function is a finite polynomial. This will dominate over the quickly falling $\alpha$-exponential above a certain critical $\epsilon$, most likely yielding the wrong asymptotics of the Borel-transformed series. This also means, that the Laplace transform of $\mathcal{B}F$ then does not give a good approximation of the desired function $F$. This is why we think that the Padé approximation yields a better estimate to the large $\epsilon$ asymptotics of the remainder function. We have checked that it in fact does not exceed the $\alpha$-exponential part for a wide range of $\epsilon\gtrsim 2$, but still gives a significant contribution to the Laplace transform.
J. Jaeckel and S. Schenk, “Exploring High Multiplicity Amplitudes in Quantum Mechanics,” Phys. Rev. D [**98**]{} (2018) no.9, 096007 \[arXiv:1806.01857 \[hep-ph\]\].
M. Serone, G. Spada and G. Villadoro, “Instantons from Perturbation Theory,” Phys. Rev. D [**96**]{} (2017) no.2, 021701 \[arXiv:1612.04376 \[hep-th\]\].
M. Serone, G. Spada and G. Villadoro, “The Power of Perturbation Theory,” JHEP [**1705**]{} (2017) 056 \[arXiv:1702.04148 \[hep-th\]\].
C. Bachas, “A Proof of exponential suppression of high-energy transitions in the anharmonic oscillator,” Nucl. Phys. B [**377**]{} (1992) 622.
J. M. Cornwall, “On the High-energy Behavior of Weakly Coupled Gauge Theories,” Phys. Lett. B [**243**]{} (1990) 271.
H. Goldberg, “Breakdown of perturbation theory at tree level in theories with scalars,” Phys. Lett. B [**246**]{} (1990) 445.
L. S. Brown, “Summing tree graphs at threshold,” Phys. Rev. D [**46**]{} (1992) R4125 \[hep-ph/9209203\].
M. B. Voloshin, “Multiparticle amplitudes at zero energy and momentum in scalar theory,” Nucl. Phys. B [**383**]{} (1992) 233.
E. N. Argyres, R. H. P. Kleiss and C. G. Papadopoulos, “Amplitude estimates for multi - Higgs production at high-energies,” Nucl. Phys. B [**391**]{} (1993) 42.
B. H. Smith, “Properties of perturbative multiparticle amplitudes in phi\*\*k and O(N) theories,” Phys. Rev. D [**47**]{} (1993) 3521 \[hep-ph/9211238\].
B. H. Smith, “Summing one loop graphs in a theory with broken symmetry,” Phys. Rev. D [**47**]{} (1993) 3518 \[hep-ph/9209287\].
V. V. Khoze and J. Reiness, “Review of the semiclassical formalism for multiparticle production at high energies,” arXiv:1810.01722 \[hep-ph\].
S. V. Demidov and B. R. Farkhtdinov, “Numerical study of multiparticle scattering in $\lambda\phi^4$ theory,” JHEP [**1811**]{} (2018) 068 \[arXiv:1806.10996 \[hep-ph\]\].
D. T. Son, “Semiclassical approach for multiparticle production in scalar theories,” Nucl. Phys. B [**477**]{} (1996) 378 \[hep-ph/9505338\].
V. V. Khoze, “Multiparticle production in the large $\lambda n$ limit: realising Higgsplosion in a scalar QFT,” JHEP [**1706**]{} (2017) 148 \[arXiv:1705.04365 \[hep-ph\]\].
V. V. Khoze, “Semiclassical computation of quantum effects in multiparticle production at large lambda n,” arXiv:1806.05648 \[hep-ph\].
G. Aad [*et al.*]{} \[ATLAS Collaboration\], “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B [**716**]{} (2012) 1 \[arXiv:1207.7214 \[hep-ex\]\].
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B [**716**]{} (2012) 30 \[arXiv:1207.7235 \[hep-ex\]\].
M. B. Voloshin, “Estimate of the onset of nonperturbative particle production at high-energy in a scalar theory,” Phys. Lett. B [**293**]{} (1992) 389.
J. Jaeckel and V. V. Khoze, “Upper limit on the scale of new physics phenomena from rising cross sections in high multiplicity Higgs and vector boson events,” Phys. Rev. D [**91**]{} (2015) no.9, 093007 \[arXiv:1411.5633 \[hep-ph\]\].
V. V. Khoze and M. Spannowsky, “Higgsplosion: Solving the Hierarchy Problem via rapid decays of heavy states into multiple Higgs bosons,” Nucl. Phys. B [**926**]{} (2018) 95 \[arXiv:1704.03447 \[hep-ph\]\].
V. V. Khoze and M. Spannowsky, “Higgsploding universe,” Phys. Rev. D [**96**]{} (2017) no.7, 075042 \[arXiv:1707.01531 \[hep-ph\]\].
V. V. Khoze, J. Reiness, M. Spannowsky and P. Waite, “Precision measurements for the Higgsploding Standard Model,” arXiv:1709.08655 \[hep-ph\].
V. V. Khoze, J. Reiness, J. Scholtz and M. Spannowsky, “A Higgsploding Theory of Dark Matter,” arXiv:1803.05441 \[hep-ph\].
A. Belyaev, F. Bezrukov, C. Shepherd and D. Ross, “Problems with Higgsplosion,” Phys. Rev. D [**98**]{} (2018) no.11, 113001 \[arXiv:1808.05641 \[hep-ph\]\].
A. Monin, “Inconsistencies of higgsplosion,” arXiv:1808.05810 \[hep-th\].
V. V. Khoze and M. Spannowsky, “Consistency of Higgsplosion in Localizable QFT,” Phys. Lett. B [**790**]{} (2019) 466 \[arXiv:1809.11141 \[hep-ph\]\].
M. B. Voloshin, “Summing one loop graphs at multiparticle threshold,” Phys. Rev. D [**47**]{} (1993) R357 \[hep-ph/9209240\].
S. Y. Khlebnikov, “Semiclassical approach to multiparticle production,” Phys. Lett. B [**282**]{} (1992) 459.
M. V. Libanov, V. A. Rubakov, D. T. Son and S. V. Troitsky, “Exponentiation of multiparticle amplitudes in scalar theories,” Phys. Rev. D [**50**]{} (1994) 7553 \[hep-ph/9407381\].
M. V. Libanov, D. T. Son and S. V. Troitsky, “Exponentiation of multiparticle amplitudes in scalar theories. 2. Universality of the exponent,” Phys. Rev. D [**52**]{} (1995) 3679 \[hep-ph/9503412\].
F. L. Bezrukov, M. V. Libanov, D. T. Son and S. V. Troitsky, “Singular classical solutions and tree multiparticle cross-sections in scalar theories,” In \*Zvenigorod 1995, High energy physics and quantum field theory\* 228-238 \[hep-ph/9512342\].
M. V. Libanov, “Multiparticle threshold amplitudes exponentiate in arbitrary scalar theories,” Mod. Phys. Lett. A [**11**]{} (1996) 2539 \[hep-th/9610036\].
C. M. Bender and T. T. Wu, “Anharmonic oscillator,” Phys. Rev. [**184**]{} (1969) 1231.
C. M. Bender and T. T. Wu, “Anharmonic oscillator. 2: A Study of perturbation theory in large order,” Phys. Rev. D [**7**]{} (1973) 1620.
U. D. Jentschura and J. Zinn-Justin, “Instantons in quantum mechanics and resurgent expansions,” Phys. Lett. B [**596**]{} (2004) 138 \[hep-ph/0405279\].
G. V. Dunne and M. Unsal, “Uniform WKB, Multi-instantons, and Resurgent Trans-Series,” Phys. Rev. D [**89**]{} (2014) no.10, 105009 \[arXiv:1401.5202 \[hep-th\]\].
E. Brezin, G. Parisi and J. Zinn-Justin, “Perturbation Theory at Large Orders for Potential with Degenerate Minima,” Phys. Rev. D [**16**]{} (1977) 408.
E. B. Bogomolny and V. A. Fateev, “Large Orders Calculations in the Gauge Theories,” Phys. Lett. B [**71**]{} (1977) 93 \[Phys. Lett. [**71B**]{} (1977) 93\].
E. B. Bogomolny, “Calculation Of Instanton - Anti-instanton Contributions In Quantum Mechanics,” Phys. Lett. [**91B**]{} (1980) 431.
M. Stone and J. Reeve, “Late Terms in the Asymptotic Expansion for the Energy Levels of a Periodic Potential,” Phys. Rev. D [**18**]{} (1978) 4746.
P. Achuthan, H. J. W. Muller-Kirsten and A. Wiedemann, “Perturbation Theory And Boundary Conditions: Analogous Treatments Of Anharmonic Oscillators And Double Wells And Similarly Related Potentials And The Calculation Of Exponentially Small Contributions To Eigenvalues,” Fortsch. Phys. [**38**]{} (1990) 78.
J. Q. Liang and H. J. W. Muller-Kirsten, “Quantum tunneling for the sine-Gordon potential: Energy band structure and Bogomolny-Fateev relation,” Phys. Rev. D [**51**]{} (1995) 718.
J. Zinn-Justin, “Multi - Instanton Contributions in Quantum Mechanics,” Nucl. Phys. B [**192**]{} (1981) 125.
J. Zinn-Justin, “Multi - Instanton Contributions in Quantum Mechanics. 2.,” Nucl. Phys. B [**218**]{} (1983) 333.
J. Zinn-Justin, “Instantons in Quantum Mechanics: Numerical Evidence for a Conjecture,” J. Math. Phys. [**25**]{} (1984) 549.
I. I. Balitsky and A. V. Yung, “Instanton Molecular Vacuum in $N=1$ Supersymmetric Quantum Mechanics,” Nucl. Phys. B [**274**]{} (1986) 475.
I. I. Balitsky and A. V. Yung, “Collective - Coordinate Method for Quasizero Modes,” Phys. Lett. B [**168**]{} (1986) 113 \[Phys. Lett. [**168B**]{} (1986) 113\].
H. Aoyama, H. Kikuchi, I. Okouchi, M. Sato and S. Wada, “Valley views: Instantons, large order behaviors, and supersymmetry,” Nucl. Phys. B [**553**]{} (1999) 644 \[hep-th/9808034\].
H. Aoyama, H. Kikuchi, I. Okouchi, M. Sato and S. Wada, “Valleys in quantum mechanics,” Phys. Lett. B [**424**]{} (1998) 93 \[quant-ph/9710064\].
J. Zinn-Justin and U. D. Jentschura, “Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions,” Annals Phys. [**313**]{} (2004) 197 \[quant-ph/0501136\].
J. Zinn-Justin and U. D. Jentschura, “Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations,” Annals Phys. [**313**]{} (2004) 269 \[quant-ph/0501137\].
M. Marino, R. Schiappa and M. Weiss, “Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings,” Commun. Num. Theor. Phys. [**2**]{} (2008) 349 \[arXiv:0711.1954 \[hep-th\]\].
M. Marino, “Nonperturbative effects and nonperturbative definitions in matrix models and topological strings,” JHEP [**0812**]{} (2008) 114 \[arXiv:0805.3033 \[hep-th\]\].
M. Unsal, “Theta dependence, sign problems and topological interference,” Phys. Rev. D [**86**]{} (2012) 105012 \[arXiv:1201.6426 \[hep-th\]\].
I. Aniceto and R. Schiappa, “Nonperturbative Ambiguities and the Reality of Resurgent Transseries,” Commun. Math. Phys. [**335**]{} (2015) no.1, 183 \[arXiv:1308.1115 \[hep-th\]\].
J. J. Loeffel, A. Martin, B. Simon and A. S. Wightman, “Pade approximants and the anharmonic oscillator,” Phys. Lett. [**30B**]{} (1969) 656.
S. Graffi, V. Grecchi and B. Simon, “Borel summability: Application to the anharmonic oscillator,” Phys. Lett. [**32B**]{} (1970) 631.
J. M. Cornwall and G. Tiktopoulos, “Semiclassical matrix elements for the quartic oscillator,” Annals Phys. [**228**]{} (1993) 365.
H. Padé, “Sur la représentation approchée d’une fonction par des fractions rationnelles," Ann. Sci. Ec. Norm. Super. [**9**]{} (1892) 3.
C. M. Bender and S. A. Orszag, “Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory," Springer, 1999.
É. Borel, “Mémoire sur les séries divergentes," Ann. Sci. Ec. Norm. Super. [**16**]{} (1899) 9.
R. E. Shafer, “On quadratic approximation," SIAM J. Numer. Anal. [**11**]{} (1974) 447.
H. Mera, T. G. Pedersen and B. K. Nikolić, “Fast summation of divergent series and resurgent transseries from Meijer-G approximants,” Phys. Rev. D [**97**]{} (2018) no.10, 105027 \[arXiv:1802.06034 \[hep-th\]\].
[^1]: For a recent review of semiclassical techniques for multiparticle production see [@Khoze:2018mey].
[^2]: Recently the authors of [@Demidov:2018czx] have used classical simulations to study high multiplicity processes.
[^3]: For a discussion on the nature of the underlying quantum field theory, in particular, on aspects of localizability and unitarity in a theory featuring “Higgsplosion" see [@Belyaev:2018mtd; @Monin:2018cbi; @Khoze:2018qhz].
[^4]: Since the two minima are related by parity, both choices are equivalent.
[^5]: Note some technical detail here. Since the additional cubic term $\sqrt{\lambda} x^3$ appears with a fractional power of the coupling, we instead define $\Lambda \equiv \sqrt{\lambda}$ and solve the system by integer orders of $\Lambda$.
[^6]: These ambiguities can, e.g., be related to poles in the Borel plane, leading to imaginary contributions that are possibly lifted by including non-perturbative effects (see, e.g. [@Jentschura:2004jg; @Dunne:2014bca]).
[^7]: Nevertheless, it would be very interesting to apply instanton calculus and resurgent trans-series [@Dunne:2014bca] to the example at hand. This could provide a valuable cross-check and potentially also allow us to reach values of $\epsilon$ where the present approach is converging slowly. We leave this for future work.
[^8]: The double-well potential is the prime example where instanton solutions play an important role. For instance, they lift the vacuum degeneracy (see, e.g. [@ZinnJustin:2004ib; @ZinnJustin:2004cg]).
[^9]: Note that, in general, a mass term that depends on the coupling of the theory can introduce new factors in the $1/n$-expansion of $F_\Sigma$ which are not subdominant anymore. However, this does not happen in our case. This motivates our choice of deformation.
[^10]: Here one should be careful about the order of the resummation and the lifting of the deformation. If we perform a Borel resummation this requires that we calculate the following Laplace transformation, $F(\epsilon,\epsilon_{0})=\int^{\infty}_{0}dt\, \exp(-t) {\mathcal{B}}F(\epsilon t,\epsilon_{0})$. Here, ${\mathcal{B}}F(\epsilon t,\epsilon_{0})$ is the Borel sum of the power series in $\epsilon t$ while $\epsilon_{0}$ is treated as an external parameter. Note in particular that the argument in the integral is $\epsilon t$ while the external parameter $\epsilon_{0}$ is not multiplied by the integration variable $t$. Only then we can evaluate $F(\epsilon,\epsilon)$.
[^11]: Mathematically a sufficient condition is that the approximated function is a Stieltjes function (see, e.g. [@Bender:1999]). However, in our case this cannot be rigorously deduced from a finite number of coefficients.
|
---
abstract: 'We provide new sufficient conditions for Chebyshev estimates for Beurling generalized primes. It is shown that if the counting function $N$ of a generalized number system satisfies the $L^{1}$-condition $$\int_{1}^{\infty}\left|\frac{N(x)-ax}{x}\right|\frac{\mathrm{d}x}{x}<\infty$$ and $N(x)=ax+o\left(x/\log x\right),$ for some $a>0$, then $$0<\liminf_{x\to\infty}\frac{\psi(x)}{x}\ \ \ \mbox{and}\ \ \ \limsup_{x\to\infty}\frac{\psi(x)}{x}<\infty$$ hold. We give an analytic proof of this result. It is based on the Wiener division theorem. Our result extends those of Diamond (Proc. Amer. Math. Soc. 39 (1973), 503–508) and Zhang (Proc. Amer. Math. Soc. 101 (1987), 205–212).'
address: 'Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, B 9000 Gent, Belgium'
author:
- Jasson Vindas
title: 'Chebyshev Estimates for Beurling Generalized Prime Numbers. I'
---
[^1]
Introduction
============
This note reports on new conditions that ensure the validity of Chebyshev estimates for Beurling’s generalized primes. We will considerably improve earlier results by Diamond [@diamond3] and Zhang [@zhang]. In particular, we shall answer an open question posed by Diamond in [@diamond4 p. 10].
Let $P=\left\{p_k\right\}_{k=1}^{\infty}$ be a set of Beurling generalized primes, that is, a non-decreasing sequence of real numbers tending to infinity, where it is assumed $p_1>1$. The sequence $\left\{n_{k}\right\}_{k=1}^{\infty}$ denotes its associated set of generalized integers [@bateman-diamond; @beurling]. Set further, $$N(x)=N_{P}(x)=\sum_{n_{k}<x}1 \ \ \mbox{ and }\ \
\psi(x)=\psi_{P}(x)=\sum_{n_{k}<x}\Lambda(n_{k})\ ,$$ where $\Lambda=\Lambda_{P}$ is the von Mangoldt function of the generalized number system [@bateman-diamond]. Beurling [@beurling] investigated the truth of the prime number theorem (PNT) in this context, i.e., $$\psi(x)\sim x\ , \ \ \ x\to\infty\ .$$ He proved that $$\label{ibpneq4}
N(x)=ax+O\left(\frac{x}{\log^{\gamma}x}\right)\ , \ \ \ x\to\infty\ \ \ (a>0)\ ,$$ where $\gamma>3/2$, suffices for the PNT to hold. His condition is sharp: When $\gamma=3/2$, then the PNT need not hold, as exhibited by counterexamples in [@beurling; @diamond2]. See [@kahane1; @vindasGPNT] for the most recent extensions of Beurling’s PNT. Since the PNT breaks down for $\gamma\leq3/2$, a natural question arises: Under which conditions over $N$ do Chebyshev estimates hold true? A partial answer to this question was provided by Diamond [@diamond3], he showed that (\[ibpneq4\]) with $\gamma>1$ is enough to obtain Chebyshev estimates, namely, $$\label{pnbpneq5}
0<\liminf_{x\to\infty} \frac{\psi(x)}{x} \ \ \ \mbox{and} \ \ \ \limsup_{x\to\infty} \frac{\psi(x)}{x}<\infty\ .$$ On the other hand, (\[pnbpneq5\]) is not generally true when $\gamma<1$, as follows from an example of Hall [@hall].
Diamond conjectured [@diamond4] that $$\label{icepneq6}
\int_{1}^{\infty}\left|\frac{N(x)-ax}{x}\right|\frac{\mathrm{d}x}{x}<\infty\ , \ \ \ \mbox{with } a>0\ ,$$ would be enough for (\[pnbpneq5\]) to hold. His conjecture turned out to be false. Kahane established the negative answer to Diamond’s conjecture in [@kahane2].
Remarkably, as shown in this article, if one adds a side condition to (\[icepneq6\]), one can indeed achieve Chebyshev estimates. Our main goal is to prove the following theorem.
\[icepnth2\] Diamond’s $L^{1}$-condition and the asymptotic behavior $$\label{icepneq7}
N(x)=ax+o\left(\frac{x}{\log x}\right)\ , \ \ \ x\to\infty\ ,$$ suffice for Chebyshev estimates $(\ref{pnbpneq5})$.
Clearly, Theorem \[icepnth2\] extends the result of Diamond quoted above. It should be noticed that Zhang [@zhang] also gave an extension of Diamond’s theorem. Our result includes it as a particular instance:
\[icec1\] The Chebyshev estimates $(\ref{pnbpneq5})$ hold if $$\label{icepneq5} \int_{1}^{\infty}\left( \sup_{x\leq t}\left|\frac{N(t)-at}{t}\right|\right)\frac{\mathrm{d}x}{x}<\infty \ \ \ (a>0)\ .$$
Naturally, (\[icepneq5\]) implies (\[icepneq6\]). If $\omega$ is a non-increasing and non-negative function such that $\int_{1}^{\infty}\omega(x)x^{-1}\mathrm{d}x<\infty$, one must have $\omega(x)=o(1/\log x)$; thus, Zhang’s condition (\[icepneq5\]) always yields (\[icepneq7\]).
We shall give a proof of Theorem \[icepnth2\] in Section \[cebpnproof\]. We point out that the methods of Diamond and Zhang from [@diamond3; @zhang] are elementary. Furthermore, Diamond has asked in [@diamond4 p. 10] whether it is possible to find an analytic approach to Chebyshev inequalities. Our proof of Theorem \[icepnth2\] is non-elementary and it therefore gives an answer to Diamond’s question; it uses the zeta function of the generalized number system and the Wiener division theorem [@korevaar Chap. 2]. In addition, we make use of the operational calculus for the Laplace transform of distributions [@vladimirov]. It would also be interesting to find an elementary proof. Finally, it should be mentioned that Zhang has provided another related sufficient condition for Chebyshev estimates in [@zhang1993]; the author announces that is also possible to obtain substantial improvements of that result. However, we will not pursue those questions here and they will be treated elsewhere.
Notation
--------
The Schwartz spaces $\mathcal{D}(\mathbb{R})$, $\mathcal{S}(\mathbb{R})$, $\mathcal{D}'(\mathbb{R})$ and $\mathcal{S}'(\mathbb{R})$ are well known; we refer to [@vladimirov] for their properties. If $f\in\mathcal{S}'(\mathbb{R})$ has support in $[0,\infty)$, its Laplace transform is well defined as $$\mathcal{L}\left\{f;s\right\}=\left\langle f(u),e^{-su}\right\rangle\ , \ \ \ \Re e\:s>0\ ,$$ and the Fourier transform $\hat{f}$ is the distributional boundary value of $\mathcal{L}\left\{f;s\right\}$ on $\Re e\:s=0$.
We use the notation $H$ for the *Heaviside function*, it is simply the characteristic function of $(0,\infty)$.
Proof of Theorem \[icepnth2\] {#cebpnproof}
=============================
We assume (\[icepneq6\]) and (\[icepneq7\]). Our starting point is the identity $$\label{cepnpeq1}
\mathcal{L}\left\{\psi(e^{u});s\right\}=-\frac{\zeta'(s)}{s\zeta(s)}= \frac{1}{s}\cdot \frac{-(s-1)G'(s)}{(s-1)\zeta(s)}-\frac{G(s)}{s(s-1)\zeta(s)}-\frac{1}{s}+\frac{1}{s-1}\ ,$$ where $$G(s):=\zeta(s)-\frac{a}{s-1}\ .$$ We set $E_{1}(u):=e^{-u}N(e^{u})-aH(u)$. Our assumptions (\[icepneq6\]) and (\[icepneq7\]) translate into $E_{1}\in L^{1}(\mathbb{R})$ and $uE_{1}(u)=o(1)$, $u\to\infty$.
\[cepnl1\] $G(s)$ extends to a continuous function on $\Re e\:s=1$. Consequently, $(s-1)\zeta(s)$ is continuous on $\Re e\:s=1$ and there exists $c>0$ such that $it\zeta(1+it)\neq 0$ for all $t\in(-3c,3c)$.
Clearly, $$\label{cepnpeq0}
G(s)= s\mathcal{L}\left\{E_{1};s-1\right\}+a\ .$$ Taking distributional boundary values, we obtain $G(1+it)=(1+it)\hat{E}_{1}(t)+a$. Since $E_1\in L^{1}(\mathbb{R})$, $\hat{E}_{1}$ is continuous and the assertions follow at once.
Set $T(u)=e^{-u}\psi(e^{u})$, we must show that $$\label{cepnpeq2}
0<\liminf_{u\to\infty}T(u) \ \ \ \mbox{and} \ \ \ \limsup_{u\to\infty}T(u)<\infty\ .$$ The next step is to use the boundary behavior of (\[cepnpeq1\]) near $s=1$ to derive (\[cepnpeq2\]). We first study convolution averages of $T$.
\[cepnl2\] For any fixed $\phi\in \mathcal{D}(-c,c)$, $$\label{cepnpeq3}
\int_{-\infty}^{\infty}T(u)\hat{\phi}(u-h)\mathrm{d}u=\int_{-\infty}^{\infty}\hat{\phi}(u)\mathrm{d}u+o(1)\ , \ \ \ h\to\infty\ .$$
Fix $\phi\in\mathcal{D}(-c,c)$. We use (\[cepnpeq0\]) to decompose (\[cepnpeq1\]) further, $$\label{cepnpeq5}
-\frac{\zeta'(s)}{s\zeta(s)}= \frac{(s-1)\mathcal{L}\left\{uE_{1}(u);s-1\right\}}{(s-1)\zeta(s)}-\frac{\mathcal{L}\left\{E_{1};s-1\right\}}{s\zeta(s)}-\frac{G(s)}{s(s-1)\zeta(s)}-\frac{1}{s}+\frac{1}{s-1}\ .$$ Set now $$g_{1}(t):=\lim _{\sigma\to1^{+}}\frac{(\sigma-1+it)\mathcal{L}\left\{uE_{1}(u);\sigma-1+it\right\}}{(\sigma-1+it)\zeta(\sigma+it)} \ \ \ \mbox{in }\mathcal{S}'(\mathbb{R})\ ,$$ and $$g_{2}(t):=-\lim _{\sigma\to1^{+}}\left(\frac{(\sigma-1+it)\mathcal{L}\left\{E_{1};\sigma-1+it\right\}+G(\sigma+it)}{(\sigma+it)(\sigma-1+it)\zeta(\sigma+it)}+\frac{1}{\sigma+it}\right)$$ in $\mathcal{S}'(\mathbb{R})$. Taking boundary values in (\[cepnpeq5\]), we obtain $
\hat{T}(t)=g_{1}(t)+ g_{2}(t)+\hat{H}(t),
$ an equality that must be interpreted in the sense of distributions. Recall that $H$ is the Heaviside function. By Lemma \[cepnl1\], $g_{2}$ is continuous on $(-3c,3c)$. Next, applying the Riemann-Lebesgue lemma to the continuous function $\phi (t)g_{2}(t)$, we conclude that $$\begin{aligned}
\int_{-\infty}^{\infty}T(u)\hat{\phi}(u-h)\mathrm{d}u&=\left\langle \hat{T}(t),e^{iht}\phi(t)\right\rangle
\\
&
%=\left\langle g_{1}(t),e^{iht}\phi(t)\right\rangle+ o(1)+\int_{-h}^{\infty}\hat{\phi}(u)\mathrm{d}u
=\int_{-\infty}^{\infty}\hat{\phi}(u)\mathrm{d}u+ \left\langle g_{1}(t),e^{iht}\phi(t)\right\rangle+o(1)\ .\end{aligned}$$ Thus, it is enough to show that $$\lim_{h\to\infty} \left\langle g_{1}(t),e^{iht}\phi(t)\right\rangle=0\ .$$ Let $M\in\mathcal{S}'(\mathbb{R})$ be the distribution supported in the interval $[0,\infty)$ that satisfies $\mathcal{L}\left\{M;s-1\right\}=((s-1)\zeta(s))^{-1}$. Notice also that $(s-1)\mathcal{L}\left\{E_{2};s-1\right\}=\mathcal{L}\left\{E_{2}';s-1\right\}$, where $E_{2}(u)=uE_1(u)=o(1)$, so we have that $g_{1}=\widehat{( E_{2}'\ast M)}$, where $\ast$ denotes convolution [@vladimirov]. Consider an even function $\eta\in \mathcal{D}(-3c,3c)$ such that $\eta(t)=1$ for all $t\in(-2c,2c)$. Clearly $\eta(t)it\zeta(1+it)\neq0$ for all $t\in(-2c,2c)$; moreover, it is the Fourier transform of an $L^{1}$-function. Finally, we apply the Wiener division theorem [@korevaar p. 88] to $\eta(t)it\zeta(1+it)$ and $\phi(t)$ and conclude the existence of $f\in L^{1}(\mathbb{R})$ such that
$$\hat{f}(t)=\frac{\phi(t)}{\eta(t) it \zeta(1+it)}\ .$$ Therefore, as $h\to\infty$, $$\left\langle g_{1}(t),e^{iht}\phi(t)\right\rangle=\left\langle (E_{2}'\ast M)(u),\hat{\phi}(u-h)\right\rangle=(E_{2}\ast (\hat{\eta})'\ast f)(h)=o(1)\ ,$$ because $E_{2}(u)=o(1)$ and $(\hat{\eta})'\ast f\in L^{1}(\mathbb{R})$. Thus, (\[cepnpeq3\]) has been established.
The estimates (\[cepnpeq2\]) follow now easily from (\[cepnpeq3\]) in Lemma \[cepnl2\]. Choose $\phi\in\mathcal{D}(-c,c)$ in (\[cepnpeq3\]) such that $\hat{\phi}$ is non-negative. Using the fact that $\psi(e^{u})$ is non-decreasing, we have that $e^{-u}T(h)\leq T(u+h)$ whenever $u$ and $h$ are positive, setting $C_1=\int_{0}^{\infty}e^{-u}\hat{\phi}(u)\mathrm{d}u>0,$ $$T(h)=
C_1^{-1}\int_{0}^{\infty}e^{-u}T(h)\hat{\phi}(u)\mathrm{d}u\leq C_1^{-1}\int_{0}^{\infty}T(u+h)\hat{\phi}(u)\mathrm{d}u=O(1)\leq C_{2}\ ,$$ for some constant $C_{2}>0$. Fix now $A>0$; observe that if $u\leq A$, then $T(h)\geq e^{u-A}T(h-A+u)$, and hence $$\begin{aligned}
\liminf_{h\to\infty} T(h)&\geq \frac{e^{-A}}{\int_{-A}^{A}e^{-u}\hat{\phi}(u)\mathrm{d}u}\liminf_{h\to\infty}\int_{-A}^{A}T(h-A+u)\hat{\phi}(u)\mathrm{d}u\\
&
=\frac{e^{-A}}{\int_{-A}^{A}e^{-u}\hat{\phi}(u)\mathrm{d}u}\liminf_{h\to\infty}\left(\int_{-\infty}^{\infty}-\int_{\left|u\right|\geq A}\right)T(h-A+u)\hat{\phi}(u)\mathrm{d}u
\\
&
\geq
\frac{e^{-A}}{\int_{-A}^{A}e^{-u}\hat{\phi}(u)\mathrm{d}u}\left(\int_{-\infty}^{\infty}\hat{\phi}(u)\mathrm{d}u-C_2\int_{\left|u\right|\geq A}\hat{\phi}(u)\mathrm{d}u\right)\ .\end{aligned}$$ It remains to choose $A$ so large that $\int_{-\infty}^{\infty}\hat{\phi}(u)\mathrm{d}u-C_2\int_{\left|u\right|\geq A}\hat{\phi}(u)\mathrm{d}u>0$. The proof is complete.
[99]{}
P. T. Bateman, H. G. Diamond, *Asymptotic distribution of Beurling’s generalized prime numbers,* Studies in Number Theory, pp. 152–210, Math. Assoc. Amer., Prentice-Hall, Englewood Cliffs, N.J., 1969.
A. Beurling, *Analyse de la loi asymptotique de la distribution des nombres premiers généralisés,* Acta Math. **68** (1937), 255–291.
H. G. Diamond, *A set of generalized numbers showing Beurling’s theorem to be sharp,* Illinois J. Math. **14** (1970), 29–34.
H. G. Diamond, *Chebyshev estimates for Beurling generalized prime numbers,* Proc. Amer. Math. Soc. **39** (1973), 503–508.
H. G. Diamond, *Chebyshev type estimates in prime number theory,* in: Sémin. Théor. Nombres, 1973–1974, Univ. Bordeaux, Exposé 24, (1974).
R. S. Hall, *Beurling generalized prime number systems in which the Chebyshev inequalities fail,* Proc. Amer. Math. Soc. **40** (1973), 79–82.
J.-P. Kahane, *Sur les nombres premiers généralisés de Beurling. Preuve d’une conjecture de Bateman et Diamond,* J. Théor. Nombres Bordeaux **9** (1997), 251–266.
J.-P. Kahane, *Le rôle des algèbres $A$ de Wiener, $A\sp \infty$ de Beurling et $H\sp 1$ de Sobolev dans la théorie des nombres premiers généralisés de Beurling,* Ann. Inst. Fourier (Grenoble) **48** (1998), 611–648.
J. Korevaar, *Tauberian theory. A century of developments*, Grundlehren der Mathematischen Wissenschaften, 329, Springer-Verlag, Berlin, 2004.
J.-C. Schlage-Puchta, J. Vindas, *The prime number theorem for Beurling’s generalized numbers. New cases,* Acta Arith. **153** (2012), 299–324.
V. S. Vladimirov, *Methods of the theory of generalized functions,* Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002.
W.-B. Zhang, *Chebyshev type estimates for Beurling generalized prime numbers,* Proc. Amer. Math. Soc. **101** (1987), 205–212.
W.-B. Zhang, *Chebyshev type estimates for Beurling generalized prime numbers. II,* Trans. Amer. Math. Soc. **337** (1993), 651–675.
[^1]: The author gratefully acknowledges support by a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO, Belgium)
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---
abstract: 'The repulsive one-dimensional Hubbard model with bond-charge interaction (HBC) in the superconducting regime is mapped onto the spin-$1/2$ XY model with transverse field. We calculate correlations and phase boundaries, realizing an excellent agreement with numerical results. The critical line for the superconducting transition is shown to coincide with the analytical factorization line identifying the commensurate-incommensurate transition in the XY model.'
author:
- Marco Roncaglia
- Cristian Degli Esposti Boschi
- Arianna Montorsi
title: 'Hidden XY structure of the bond-charge Hubbard model'
---
The Hubbard Hamiltonian and its extensions are known to model several correlated quantum systems, ranging from high-$T_{c}$ superconductors to cold fermionic atoms trapped into optical lattices [@BDZ]. In particular, the HBC model describes the interaction between fermions located on bonds and on lattice sites [@HBC; @HBC-rev]. This extension is considered to be especially relevant to the field of high-$T_{c}$ superconductors [@HIRSCH]. In fact, it has recently been found [@ADMO; @AAA] that a superconducting phase takes place also for repulsive values of the on-site Coulomb interaction. The phase is characterized by incommensurate modulations in the charge structure factor. Its boundaries have been explored numerically, though their fundamental nature has not been understood yet.
We find that the explanation of the above features resides into the underlying effective model, which for the superconducting phase turns out to be the anisotropic XY chain in a transverse field. Such model is known to be equivalent to free spinless fermions and it is remarkable how it can faithfully describe quantities of a strongly correlated system like the HBC chain. Indeed, the mapping allows us to derive analytical expressions for both the critical line and correlations, reproducing with amazing accuracy the numerical data.
The model Hamiltonian for the HBC chain reads
$$\begin{aligned}
\mathcal{H}= & -\sum_{i\sigma}\left[1-X\left(n_{i\bar{\sigma}}+n_{i+1\bar{\sigma}}\right)\right](c_{i\sigma}^{\dagger}c_{i+1\sigma}+c_{i+1\sigma}^{\dagger}c_{i\sigma})\nonumber \\
& +U\sum_{i}n_{i\uparrow}n_{i\downarrow}-\frac{U}{2}\sum_{i\sigma}n_{i\sigma}\label{eq:Hirsch}\end{aligned}$$
where $\sigma=\uparrow,\downarrow$ ($\bar{\sigma}$ denoting the opposite of $\sigma$), and the operator $c_{i\sigma}^{\dagger}$ creates a fermion at site $i$ with spin $\sigma$. Moreover $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$. The parameters $U$ and $X,$ expressed in units of the hopping amplitude, are the on-site and bond-charge Coulomb repulsion respectively.
While the HBC model cannot be exactly solved for all $X$, there are two integrable point at $X=0$ and $X=1$, for all values of $U$. The former is the well-known Hubbard model which is solvable by Bethe Ansatz. The integrability of the case $X=1$ is due to the fact that the empty and the doubly occupied sites in this case are indistinguishable, and the same holds for the $\uparrow$ and $\downarrow$ spins in the singly occupied sites, so that the model can be rephrased in terms of tight-binding spinless fermions in 1D [@AA]. In addition, the number of double occupancies turns out to be a conserved quantity.
In the general case, Eq.(\[eq:Hirsch\]) can be fruitfully recasted passing to a slave boson representation. One can make the transformation $|0\rangle\to e_{i}|0\rangle$, $c_{i\sigma}^{\dagger}|0\rangle\to f_{i\sigma}^{\dagger}|0\rangle$ and $c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}|0\rangle\to d_{i}|0\rangle$, where empty and doubly occupies sites are bosons, while the single occupations are fermions. The hard-core constraint $e_{i}^{\dagger}e_{i}+d_{i}^{\dagger}d_{i}+\sum_{\sigma}f_{i\sigma}^{\dagger}f_{i\sigma}=1$ completes the identification. Then, the $c$-fermions are $c_{i\sigma}^{\dagger}=f_{i\sigma}^{\dagger}e_{i}+d_{i}^{\dagger}f_{i\bar{\sigma}}$ and $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}=f_{i\sigma}^{\dagger}f_{i\sigma}+d_{i}^{\dagger}d_{i}$. The total number of particles is $N=N_{f}+2N_{d}$. The filling factor is $\nu=N/L$, with $0\leq\nu\leq2$. Accordingly, we have $\nu_{e}+\nu_{f}+\nu_{d}=1$ and $\nu=\nu_{f}+2\nu_{d}$. After the substitution, the Hamiltonian becomes $\mathcal{H}=\sum_{i\sigma}\mathcal{H}_{i\sigma}$, where $$\begin{aligned}
\mathcal{H}_{i\sigma} & =-\frac{U}{2}f_{i\sigma}^{\dagger}f_{i\sigma}+\left[f_{i\sigma}^{\dagger}f_{i+1,\sigma}\left(t_{X}d_{i+1}^{\dagger}d_{i}-e_{i+1}^{\dagger}e_{i}\right)\right.\nonumber \\
& \left.-s_{X}f_{i\sigma}^{\dagger}f_{i+1,\bar{\sigma}}^{\dagger}\left(e_{i+1}d_{i}+d_{i+1}e_{i}\right)+\mathrm{H.c.}\right],\label{eq:H_slave}\end{aligned}$$ with $t_{X}=1-2X$, and $s_{X}=1-X$. It can be recognized that the first two terms describe the kinetic energy of a single electron (hole) with spin $\sigma$ in a background of empty (doubly occupied) sites, whereas the third term describes the transformation of two opposite spins into an empty and a doubly occupied site. Since the coefficient $s_{X}$ turns out to give the smallest contribution for $X>2/3$, it is not surprising that the exact solution obtained assuming $s_{X}=0\mbox{ }$ (and arbitrary $t_{X})$ [@MON] shares in this regime many features of the ground state of the true model, obtained by numerical investigation [@ADM2009]. To some extent, these features hold within the range $X>X_{c}=1/2$, where $X_{c}$ is the value at which $t_{X}$ changes sign. This is true in particular as for the presence of phase coexistence of domains formed by only empty or doubly occipied sites, in which the single particles move. On the other hand, fixing $s_{X}=0$ yields to a critical curve $U_{PS}=4X$ for the stabilility of the phase separated region, whereas the superconducting transition takes place (only for $s_{X}\neq0$) at a value $U_{SC}$ which is well below that line. Since the role of empty and doubly occupied sites, as well as the conservation of their number, appears to be the same as for $s_{X}=0$ [@ABM] also in the superconducting case, one can infer that it is just the motion of the single electrons and holes which determines the change $U_{PS}\rightarrow U_{SC}$ for $s_{X}\neq0$. In this paper, we assume this point of view: *treating the empty and doubly occupied states as the vacuum in which the single particles move*.
Let us go back to Eq.(\[eq:H\_slave\]) and consider what happens at $s_{X}\neq0$. The $SU(2)$ charge symmetry is broken down to $U(1)$, which merely describes the conservation of the number of fermions. The large spin degeneracy is removed and it is like as if the fermionic dynamics is influenced by the background imposed by the bosons and viceversa. This picture is correct as far as the spin and charge degrees of freedom are not separated. For $X\lesssim1$, the pair creation term in Eq.(\[eq:H\_slave\]) induces short-ranged antiferromagnetic (AFM) correlations in both spin and pseudospin degrees of freedom. Since at half-filling the probabilities of having an empty and a doubly occupied sites are identical and coincide with 1/2, we can approximate the term $\langle e_{i+1}^{\dagger}e_{i}-t_{X}d_{i+1}^{\dagger}d_{i}\rangle\approx X$. Thus, in this case the kinetic energy term in $\mathcal{H}_{i\sigma}$ becomes $-X\, f_{i\sigma}^{\dagger}f_{i+1\sigma}+(X-1)\, f_{i\sigma}^{\dagger}f_{i+1,\bar{\sigma}}^{\dagger}+\mathrm{H.c.}$ where the term $f_{i\sigma}^{\dagger}f_{i+1,\bar{\sigma}}^{\dagger}$ always takes place due to the bosonic AFM correlations. Both the bosonic species are considered as a unique vacuum for the fermions $f$.
Assuming the existence of AFM correlations also in the fermionic variables, we can drop the spin indices. The effect of $f_{i}^{\dagger}f_{i+1}^{\dagger}$ is to open a gap at the Fermi level, hence reducing considerably the ground state (GS) energy. This mechanism is analogous to what happens in the case of the Peierls instability (in that case the gap is opened by the dimerization) where the bosons here play the role of the phonons that distort the lattice. So, we obtain a free-spinless fermion model $\mathcal{H}^{(f)}=\sum_{i=1}^{L}\mathcal{H}_{i}^{(f)}$, where
$$\begin{aligned}
\mathcal{H}_{i}^{(f)}= & -X\left(f_{i}^{\dagger}f_{i+1}+\frac{1-X}{X}f_{i}^{\dagger}f_{i+1}^{\dagger}+\mathrm{H.c.}\right)-\frac{U}{2}f_{i}^{\dagger}f_{i}\:.\label{eq:Hf}\end{aligned}$$
It is instructive to notice that even in this form one can recover the exact solution of the case $X=1$. Indeed a straghtforward diagonalization in Fourier space gives $H=-2\sum_{k}[\cos k+U/4]f_{k}^{\dagger}f_{k}$. The fermions fill the negative energy states up to the Fermi point $k_{f}=\pi\nu_{f}$. The saturation occurs for $U_{c}=-4\cos(\pi\nu)$ for $0<\nu<2$.
In the general case, $\mathcal{H}^{(f)}$ can be easily shown to be equivalent to the following XY model in a transverse field$$\mathcal{H}_{XY}=E_{0}-\frac{1}{\zeta}\sum_{i=1}^{L}\left[\frac{1+\gamma}{2}\sigma_{i}^{x}\sigma_{i+1}^{x}+\frac{1-\gamma}{2}\sigma_{i}^{y}\sigma_{i+1}^{y}+h\sigma_{i}^{z}\right]\label{eq:Ising_map}$$ where $\gamma=\frac{1-X}{X}$, $h=\frac{U}{4X}$, $E_{0}=-\frac{UL}{4}$ and $\zeta=\frac{1}{X}$, at half filling. As usual we have applied the Jordan-Wigner transformation $\sigma_{i}^{z}=2f_{i}^{\dagger}f_{i}-\mathbb{I}$, $\sigma_{i}^{+}=f_{i}^{\dagger}K_{i-1}$, $\sigma_{i}^{-}=K_{i-1}^{\dagger}f_{i}$ with $K_{l}=\prod_{k=1}^{l}\left(-\sigma_{k}^{z}\right)=\exp[i\pi\sum_{k=1}^{l}n_{k}]$.
AFM correlations in both bosonic and fermionic particles are here assumed on the intuitive basic observation of the reduction of GS energy by means of the pair creation terms. A more rigorous approach would involve a self-consistent determination of the hopping coefficients in the quadratic model in which the spin labels are retained. Such approach allows to extend the analysis away from half filling and in magnetic field, and goes beyond the purpose of the present paper. We dedicate a forthcoming extended manuscript to a self-consistent approach.
In what follows, we examine some important consequences that can be derived from the exact solution of the XY model, written in Eq.(\[eq:Ising\_map\]).
As known, the Hamiltonian (\[eq:Ising\_map\]) can be diagonalized: $H=E_{0}+\frac{1}{\zeta}\sum_{k\in BZ}\Lambda_{k}\left(\beta_{k}^{\dagger}\beta_{k}-\frac{1}{2}\right),$ where the sum is performed in the Brillouin zone (BZ), and the dispersion relations are $\Lambda_{k}=2\sqrt{\left(\cos k+h\right)^{2}+\gamma^{2}\sin^{2}k}$. Given the positiveness of $\Lambda_{k}$, the GS energy $E_{GS}$ is determined by the vacuum of the Bogoliubov quasiparticles $\beta_{k}$, giving $E_{GS}=E_{0}-\frac{1}{2L}\sum_{k\in BZ}\Lambda_{k}$. By taking the thermodynamic limit $L\to\infty$, we get an energy density $e_{GS}=-\frac{U}{4}-\frac{X}{4\pi}\int_{-\pi}^{\pi}dk\Lambda_{k}$.
We have compared the outcomes of our mapping with numerical calculations using the density matrix renormalization group (DMRG) [@S2005]. In particular, we used extrapolations in $1/L$ of data collected by selecting 7 finite-system sweeps and 1024-1152 states. Numerical and analytical results of the energy density at $X=0.8$ are displayed in table \[tab:compx08\].
$U$ $e_{GS}^{num}$ $e_{GS}^{th}$ $\nu_{d}^{num}$ $\nu_{d}^{th}$ $q/\pi$ $\psi/\pi$
------- ---------------- --------------- ----------------- ---------------- --------- ------------
$0$ $-0.5390$ $-0.54612$ $0.2511$ $1/4$ $14/30$ $1/2$
$0.5$ $-0.670$ $-0.67708$ $0.216$ $0.22611$ $14/30$ $0.44841$
$1$ $-0.81544$ $-0.82011$ $0.19016$ $0.20164$ $12/30$ $0.39539$
$1.5$ $-0.9717$ $-0.97565$ $0.173$ $0.17588$ $10/30$ $0.33914$
$2.5$ $-1.3300$ $-1.3287$ $0.1063$ $0.11488$ $6/30$ $0.20116$
: Comparison between various quantities defined in the text computed either numerically (num) with DMRG or analytically by means of the equivalent XY model (th) for $X=0.8$, both for periodic boundary conditions. The latter is treated directly in the thermodynamic limit, while the former are extrapolated to $L\to\infty$ from finite-size data. The characteristic wavenumber $q$ is extracted from Fourier transforms at $L=30$.\[tab:compx08\]
An important feature of the XY chain is the presence of a factorization line $h^{2}+\gamma^{2}=1$, which corresponds to a commensurate-incommensurate (CIC) transition. In the HBC model this transition is mapped analytically into $$U_{SC}=4\sqrt{2X-1}.\label{Usc}$$
Such transition was discovered numerically in Ref.[@ADMO] and separates a incommensurate singlet superconducting (ICSS) phase from a bond ordered wave (BOW) phase [@AAA]. As seen in Fig.\[fig:pdhf\], the curve obtained with our mapping describes rather accurately the numerical data of the transition.
![Comparison between the phase diagram of the HBC chain, calculated numerically in Ref.[@AAA] (symbols with dashed lines) and the phase diagram obtained from the mapping onto the XY model in transverse field (continuous lines). The upper curves correspond to the spin gap transition where spin excitations become gapless, while the lower curves mark the transition into the ICSS phase where the charge compressibility diverges.\[fig:pdhf\]](UccUcs)
Along the factorization line the GS in the $S=1/2$ model is written as $\otimes_{i=1}^{L}|\phi\rangle$, where $|\phi\rangle=\cos\frac{\theta}{2}|\uparrow\rangle+\sin\frac{\theta}{2}|\downarrow\rangle$, with $\cos\theta=[(1-\gamma)/(1+\gamma)]^{1/2}=\alpha.$ Here the local magnetization is $2\nu_{f}-1=\alpha=\sqrt{2X-1}.$ Accordingly, the number of double occupations along the factorization line at half filling is $\nu_{d}=(1-\alpha)/4=(1-\sqrt{2X-1})/4.$ In the rest of the phase diagram, the transverse magnetization of the XY chain is given by $\langle\sigma_{i}^{z}\rangle=\frac{2}{L}\sum_{k\in BZ}(h+\cos k)\Lambda_{k}^{-1}$. The diverging charge compressibility of the ICSS phase is explainable simply by observing that adding two particles produce the conversion of an empty site onto a doubly occopied one, without changing the energy in the XY representation.
In addition, the XY model in 1D is known to undergo a quantum phase transition along line $h=1$, belonging to the universality class of the classical Ising model in 2D. This translates directly into the line $U=4X$ in the phase diagram of the HBC model (see Fig.\[fig:pdhf\]). The latter coincides with the critical line of stability of PS ($U_{PS}$) in the integrable case $s_{X}=0$, and is close to the numerical critical line between the spin density wave (SDW) and the BOW phase in Fig. 1, at least for $X$ close to 1. Moreover, the line $\gamma=1$, which is known to describe the Ising model in a transverse field, here corresponds to the case $X=1/2$. While it is questionable whether the assumptions that have originated our approximations for the ICSS phase are still valid in the above limiting cases, one can recognize that instead at the very crucial critical point $X=1/2$ and $U=0$, our system described in Eq.(\[eq:Hirsch\]) is mapped into nothing but the Ising model.
From the seminal paper of Barouch and McCoy on the statistical mechanics of the XY model [@BM1971], it is known that the oscillation wavenumber of the correlator $\rho_{xx}(R)=\langle\sigma_{i}^{x}\sigma_{i+R}^{x}\rangle$ in the incommensurate region $h^{2}+\gamma^{2}<1$ is $$\begin{aligned}
\tan\psi & =\frac{\sqrt{1-\gamma^{2}-h^{2}}}{h}=\sqrt{\frac{2X-1}{\left(U/4\right)^{2}}-1},\label{eq:peak}\end{aligned}$$ with a period $R_{0}=2\pi/\psi$.
A first striking observation is the fact that the correlations of the *total* density exhibit a peak very close to the characteristic wavenumber $\psi$ in Eq.(\[eq:peak\]): in the last two columns of table \[tab:compx08\] we report the wavevector $q$ at which the total density structure factor has a peak (see an example in fig. \[fig:FTnntds\]) and the corresponding value of $\psi$. For $X=0.9$ and $U=3$ with $L=32$ the peak is located at $q/\pi=0.1875$ while $\psi/\pi=0.18342$.
![Analysis of the various contributions [\[]{}see Eq.(\[eq:contnn\])[\]]{} to the static structure factor (Fourier transform) of the density correlation function for the HBC model ($L=30)$ with the parameters reported in the legend. The vertical line corresponds to $\psi/\pi$ (see table \[tab:compx08\]).\[fig:FTnntds\]](FTnntsd_x0\lyxdot 8U2\lyxdot 5)
The appearance of the peak at wavenumber $Q$ in the Fourier transform of a correlation function that decays as $\cos(QR)\exp(-R/\xi)/R^{a}$ is related also to the exponent $a$: the smaller is $a$ the sharper is the peak. In particular for $a=2$ which is the case for the correlation $\rho_{zz}(R)=\langle\sigma_{i}^{z}\sigma_{i+R}^{z}\rangle$ of the XY model, the peak it not visibile at all, despite the fact that the oscillations actually ** characteristic wavenumber $2\psi$. Hence, it is worth to inspect in more detail the origin of the peaks observed numerically. Since the local density operator $n_{i}$ in terms of single and double occupancies $n_{si}$ and $n_{di}$ is given by $n_{i}=n_{si}+2n_{di}$, the correlation function of the total density decomposes in the following parts$$\begin{aligned}
\langle n_{i}n_{i+R}\rangle= & \langle n_{si}n_{si+R}\rangle+4\langle n_{di}n_{di+R}\rangle\nonumber \\
& +2\langle n_{di}n_{si+R}\rangle+2\langle n_{si}n_{di+R}\rangle.\label{eq:contnn}\end{aligned}$$ It turns out that the peak in the static structure factor is not due to the first term, but it is instead provided by $\langle n_{di}n_{di+R}\rangle$, as shown in Fig.\[fig:FTnntds\] for the test case $X=0.8$ and $U=2.5$, although we obtained the same qualitative picture at $U=1$.
According to our mapping, we can compare directly the connected correlator $N_{s}(R)=\langle n_{si}n_{si+R}\rangle-\langle n_{si}\rangle^{2}$ in the HBC model with the density correlation function $\rho(R)=\langle f_{i}^{\dagger}f_{i}f_{i+R}^{\dagger}f_{i+R}\rangle-\langle f_{i}^{\dagger}f_{i}\rangle^{2}$ for the spinless fermions with Hamiltonian (\[eq:Hf\]). The calculation of the latter is omitted here since it is quite lengthy, though it simply involves a standard application the Wick theorem. The fully fermionic correlator $N_{s}(R)$ and the spinless fermions correlator $\rho(R)$ are compared in Fig.\[fig:corrs\_Hirsch\_vs\_XY\] for various choices of the parameters $U$ and $X$ in the ICSS phase of our starting system (i.e. the incommensurate one in the XY model); the agreement in real space is generally very good. Such behaviour of $N_{s}(R)$ is not obvious a priori in the HBC model and we interpret it as a remarkable nontrivial prediction of our mapping.
![Comparison of the connected real-space correlation functions $N_{s}(R)$ (at half-filling) and $\rho(R)$ for the HBC and XY model, respectively (see text for definitions based on singly occupied sites operators). The parameters of the two models are related by mapping as $\gamma=(1-X)/X$ and $h=U/4X$. All the DMRG calculations for the HBC model and the analytical experessions of the curves for the XY model refer to $L=50$. From top to bottom the data have been offset by +0.15, +0.10 and +0.05 for the sake of clarity.\[fig:corrs\_Hirsch\_vs\_XY\]](corrs_Hirsch_vs_XY)
In summary, we have studied the Hubbard model with bond-charge interaction in the superconducting regime, unveiling its underlying XY structure. We have shown that at half filling the numerical critical line for superconductivity coincides with remarcable accuracy to the analytical factorization curve that marks the CIC transition of the anisotropc XY model in a transverse field. Exploting the mapping for the calculation of correlations in the effective model has allowed us to predict rather accurately the peak in the charge structure factor of the original model. The results confirm a posteriori the crucial role of short range AF correlations and spin degrees of freedom as to the onset of superconductivity. The ultimate presence of the latter is however to be ascribed to the interplay of the spin with the charge degrees of freedom, the superconducting properties being absent from the incommensurate phase of the free fermions model.
Based on the success of the present mapping, a number of further result are now in order. First, since the one-dimensionality of the model is not crucial to the mapping, the latter should hold in higher dimension as well. In 2D, the numerical investigation of the XY model has been largely explored in the literature: this could provide useful hints on the type of phase diagram which characterizes the 2D HBC model. Moreover, it would be interesting to understand the implications on the HBC model of a non-vanishing string order parameter which is peculiar of the XY model in transverse field. Finally, we expect that a similar mapping should hold also in the strongly repulsive regime $U\rightarrow\infty,$ since in that case no doubly occupied sites occur, and it is still quite natural to assume short range AFM order of single particles.
We are grateful to Alberto Anfossi for useful discussions, and for providing us some data to compare. AM acknowledges the hospitality of Condensed Matter Theory Visitor’s Program at Boston University, where this work was completed. The Bologna Section of the INFN is also acknowledged for the computational resources. This work was partiall supported by national italian funds, PRIN2007JHLPEZ\_005.
[12]{} I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. **80**, 885 (2008).
J. T. Gammel and D. K. Campbell, Phys. Rev. B **60**, 71 (1988).
For reviews see G. I. Japaridze and A. P. Kampf, Phys. Rev. B $\textbf{59}$, 12822 (1999); M. Nakamura, T. Okano and K. Itoh, Phys. Rev. B **72** 115121 (2005); A. Aligia, and A. Dobry, preprint, arXiv:1009.4113, and references therein; as well as reference [@AAA] below.
J. E. Hirsch, Physica C **158** 326 (1989); J. E. Hirsch and F. Marsiglio, Phys. Rev. B $\textbf{39}$, 11515 (1989).
A. Anfossi, C. Degli Esposti Boschi, A. Montorsi, and F. Ortolani, Phys. Rev. B $\textbf{73 }$, 085113 (2006).
A.A. Aligia *et al.*, Phys. Rev. Lett. **99**, 206401 (2007).
L. Arrachea and A. A. Aligia, Phys. Rev. Lett. **73**, 2240 (1994).
A. Montorsi, J. Stat. Mech. L09001 (2008)
A. Anfossi, C. Degli Esposti Boschi and A. Montorsi, Phys. Rev. B **79**, 235117 (2009)
A. Anfossi, L. Barbiero, and A. Montorsi, Phys. Rev. A **80**, 043602 (2009)
E. Barouch and B. M. McCoy, Phys. Rev. A **3**, 786 (1971).
U. Schollwöck, Rev. Mod. Phys. **77**, 259 (2005).
|
---
abstract: 'It is shown that the tensor product $JH{\tilde{\otimes}_\epsilon}JH$ fails Pełczńyski’s property (u). The proof uses a result of Kwapień and Pełczńyski on the main triangle projection in matrix spaces.'
address: |
Department of Mathematics\
National University of Singapore\
Singapore 119260
author:
- 'Denny H. Leung'
title: 'Property $\text{(u)}$ in $JH{\tilde{\otimes}_\epsilon}JH$'
---
\[thm\][Lemma]{} \[thm\][Corollary]{}
The Banach space $JH$ constructed by Hagler [@H] has a number of interesting properties. For instance, it is known that $JH$ contains no isomorph of $\ell^1$, and has property (S): every normalized weakly null sequence has a subsequence equivalent to the $c_0$-basis. This easily implies that $JH$ is $c_0$-saturated, i.e., every infinite dimensional closed subspace contains an isomorph of $c_0$. In answer to a question raised originally in [@H], Knaust and Odell [@KO] showed that every Banach space which has property (S) also has Pełczyński’s property (u). In [@L], the author showed that the Banach space $JH{\tilde{\otimes}_\epsilon}JH$ is $c_0$-saturated. It is thus natural to ask whether $JH{\tilde{\otimes}_\epsilon}JH$ has also the related properties (S) and/or (u). In this note, we show that $JH{\tilde{\otimes}_\epsilon}JH$ fails property (u) (and hence property (S) as well). Our proof makes use of a result, due to Kwapień and Pełczyński, that the main triangle projection is unbounded in certain matrix spaces.
We use standard Banach space notation as may be found in [@LT]. Recall that a series $\sum x_n$ in a Banach space $E$ is called [*weakly unconditionally Cauchy*]{} (wuC) if there is a constant $K < \infty$ such that $\|\sum^k_{n=1}\epsilon_nx_n\| \leq K$ for all choices of signs $\epsilon_n = \pm 1$ and all $k \in {{\Bbb N}}$. A Banach space $E$ has [*property*]{} (u) if whenever $(x_n)$ is a weakly Cauchy sequence in $E$, there is a wuC series $\sum y_k$ in $E$ such that $x_n - \sum^n_{k=1}y_k \to 0$ weakly as $n \to \infty$. If $E$ and $F$ are Banach spaces, and $L(E',F)$ is the space of all bounded linear operators from $E'$ into $F$ endowed with the operator norm, then the tensor product $E{\tilde{\otimes}_\epsilon}F$ is the closed subspace of $L(E',F)$ generated by the weak\*-weakly continuous operators of finite rank. In particular, for any $x \in E$, and $y \in F$, one obtains an element $x\otimes y \in E{\tilde{\otimes}_\epsilon}F$ defined by $(x\otimes y)x' = {\langle}x,
x'{\rangle}y$ for all $x' \in E'$.
Let us also recall the definition of the space $JH$, as well as fix some terms and notation. Let $T = \cup^\infty_{n=0}\{0,1\}^n$ be the dyadic tree. The elements of $T$ are called [*nodes*]{}. If $\phi$ is a node of the form $({\epsilon}_i)^n_{i=1}$, we say that $\phi$ has [*length*]{} $n$ and write $|\phi| = n$. The length of the empty node is defined to be $0$. For $\phi, \psi \in T$ with $\phi = ({\epsilon}_i)^n_{i=1}$ and $\psi = (\delta_i)^m_{i=1}$, we say that $\phi \leq \psi$ if $n \leq
m$ and ${\epsilon}_i = \delta_i$ for $1 \leq i \leq n$. The empty node is $\leq \phi$ for all $\phi \in T$. Two nodes $\phi$ and $\psi$ are [*incomparable*]{} if neither $\phi \leq \psi$ nor $\psi \leq \phi$ hold. If $\phi \leq \psi$, we say that $\psi$ is a [*descendant*]{} of $\phi$, and we set $$S(\phi,\psi) = \{\xi: \phi \leq \xi \leq \psi\}.$$ A set of the form $S(\phi,\psi)$ is called a [*segment*]{}, or more specifically, an $m$-$n$ [*segment*]{} provided $|\phi| = m,$ and $|\psi| =
n$. A [*branch*]{} is a maximal totally ordered subset of $T$. The set of all branches is denoted by $\Gamma$. A branch $\gamma$ (respectively, a segment $S$) is said to [*pass through*]{} a node $\phi$ if $\phi \in \gamma$ (respectively, $\phi \in S$). If $x: T \to {{\Bbb R}}$ is a finitely supported function and $S$ is a segment, we define (with slight abuse of notation) $Sx = \sum_{\phi\in S}x(\phi)$. In case $S
= \{\phi\}$ is a singleton, we write simply $\phi x$ for $Sx$. Similarly, if $\gamma \in \Gamma$, we define $\gamma(x) = \sum_{\phi\in
\gamma}x(\phi)$. A set of segments $\{S_1,\dotsc,S_r\}$ is [*admissible*]{} if they are pairwise disjoint, and there are $m, n \in
{{\Bbb N}}\cup\{0\}$ such that each $S_i$ is an $m$-$n$ segment. The James Hagler space $JH$ is defined as the completion of the set of all finitely supported functions $x: T \to {{\Bbb R}}$ under the norm: $$\|x\| = \sup\left\{\sum^r_{i=1}|S_ix| : S_1,\dotsc ,S_r \mbox{ is an
admissible set of segments}\right\}.$$ Clearly, all $S$ and $\gamma$ extend to norm $1$ functionals on $JH$. It is known that the set $T$ of all node functionals, and the set $\Gamma$ of all branch functionals together span a dense subspace of $JH'$ (cf. p. 301 of [@H]). Finally, if $x: T \to {{\Bbb R}}$ is finitely supported, and $n \geq 0$, let $P_nx: T \to {{\Bbb R}}$ be defined by $$(P_nx)(\phi) =
\begin{cases}
x(\phi)& \text{if $|\phi| \geq n$} \\
0& \mbox{otherwise.}
\end{cases}$$ Obviously, $P_n$ extends uniquely to a norm $1$ projection on $JH$, which we denote again by $P_n$. The proof of the following lemma is left to the reader. We thank the referee for the succinct formulation.
\[proj\] For any $n \in {{\Bbb N}}$, construct a sequence $(\pi(1),\pi(2),\dotsc,\pi(n))$ by writing the odd integers in the set $\{1,\dotsc,n\}$ in increasing order, followed by the even integers in decreasing order. Then $$(-1)^{\min(\pi(i),\pi(j))+1} = 1 \quad {\Longleftrightarrow}\quad i + j \leq n+1.$$
For any $n \in {{\Bbb N}}$ and $n \times n$ real matrix $M =
[M(i,j)]^n_{i,j=1}$, let $E(M)$ be the matrix $[(-1)^{\min(i,j)+1}M(i,j)]$. Denote by $\sigma(M)$ the norm of $M$ considered as a linear map from $\ell^\infty(n)$ into $\ell^1(n)$, i.e., $$\sigma(M) = \sup\left\{ \sum^n_{i,j=1}a_ib_jM(i,j) : \sup_{1\leq
i,j\leq n}\{|a_i|, |b_j|\} \leq 1\right\}.$$
\[est\] There is a constant $C > 0$ such that for every $n \in {{\Bbb N}}$, there is an $n \times n$ real matrix $M_n$ such that $\sigma(M_n) = 1$ and $\sigma(E(M_n)) \geq C \log n$.
It follows easily from [@KP Proposition 1.2] that there are a constant $C > 0$ and real $n \times n$ matrices $N_n = [N_n(i,j)]$ for every $n$ such that $\sigma(N_n) = 1$, and $\sigma([\epsilon(i,j)N_n(i,j)]) \geq C
\log n$, where $$\epsilon(i,j) =
\begin{cases}
1& \text{if $i + j \leq n + 1$},\\
-1& \text{otherwise}.
\end{cases}$$ Let $\pi$ be the permutation in Lemma \[proj\]. Define $M_n(i,j) = N_n(\pi^{-1}(i), \pi^{-1}(j))$, $1 \leq i, j
\leq n$, and let $M_n = [M_n(i,j)]$. Clearly $\sigma(M_n) = \sigma(N_n)
= 1$ for all $n$. Also, $$\begin{gathered}
\sigma(E(M_n)) =
\sigma\left(\left[(-1)^{\min(\pi(i),\pi(j))+1}M_n(\pi(i),\pi(j))\right]\right)\\
= \sigma\left(\left[\epsilon(i,j)N_n(i,j)\right]\right)
\geq C \log n,\end{gathered}$$ as required.
Let $\psi_1$ denote the node $(0)$, and $\psi_n = ({\displaystyle \overbrace{1\dots
1}^{n-1}0})$ for $n \geq 2$. For convenience, define $s_0 = 0$ and $s_k = \sum^k_{i=1}i$ for $k \geq 1$. Now choose a strictly increasing sequence $(n_k)$ in ${{\Bbb N}}$, and a sequence of pairwise distinct nodes $(\phi_i)$ such that $\phi_i$ is a descendant of $\psi_k$ having length $n_k$ whenever $s_{k-1} < i \leq s_k$, $k \in {{\Bbb N}}$. For any $i
\in {{\Bbb N}}$, choose a branch $\gamma_i$ which passes through $\phi_i$. If $i \leq s_k$, denote by $\phi(i,k)$ the node of length $n_k$ which belongs to $\gamma_i$. Finally, let $R_k =
[R_k(i,j)]^{s_k}_{i,j=s_{k-1}+1}$ be $k \times k$ real matrices such that $\sum_k\sigma(R_k) < \infty$. Then define a sequence of elements in $JH{\tilde{\otimes}_\epsilon}JH$ as follows: $$U_l =
\sum^l_{k=1}\sum^{s_k}_{i,j=s_{k-1}+1}R_k(i,j)e_{\phi(i,l)}\otimes
e_{\phi(j,l)}$$ for $l \in {{\Bbb N}}$. Here, $e_\phi \in JH$ is the characteristic function of the singleton set $\{\phi\}$. Since the sequence $(e_{\phi(i,l)})^{s_k}_{i=s_{k-1}+1}$ is isometrically equivalent to the $\ell^1(k)$-basis whenever $k \leq l$, $$\left\|\sum^{s_k}_{i,j=s_{k-1}+1}R_k(i,j)e_{\phi(i,l)}\otimes
e_{\phi(j,l)}\right\| = \sigma(R_k),$$ and thus $\|U_l\| \leq
\sum_k\sigma(R_k) < \infty$ for any $l$.
The sequence $(U_l)$ is a weakly Cauchy sequence in $JH{\tilde{\otimes}_\epsilon}JH$.
It is well known that a bounded sequence $(W_n)$ in a tensor product $E{\tilde{\otimes}_\epsilon}F$ is weakly Cauchy if and only if $(W_nx')$ is weakly Cauchy in $F$ for all $x' \in E'$. Since $(U_l)$ is a bounded sequence, and $[T \cup \Gamma] = JH'$, it suffices to show that $(U_lx')$ is weakly Cauchy in $JH$ for every $x'$ in $T \cup
\Gamma$. Now for all $\phi \in T$, we clearly have $U_l\phi =
0$ for all large enough $l$. Next, consider any $\gamma \in
\Gamma$. If $\gamma$ does not pass through any $\psi_k$, then it cannot pass through any $\phi(i,k)$ either. So $U_l\gamma = 0$ for all $l$. Otherwise, due to the pairwise incomparability of $(\psi_k)$, there is a unique $k_0$ such that $\psi_{k_0} \in \gamma$. If $\gamma$ is distinct from $\gamma_i$ for all $s_{k_0-1} < i \leq
s_{k_0}$, then again $U_l\gamma = 0$ for all sufficiently large $l$. Now suppose $\gamma = \gamma_{i_0}$ , where $s_{k_0-1} < i_0 \leq
s_{k_0}$. Then, for $l \geq k_0$, $$U_l\gamma = \sum^{s_{k_0}}_{j=s_{k_0-1}+1}R_{k_0}(i_0,j)e_{\phi(j,l)} .$$ Since each sequence $(e_{\phi(j,l)})^\infty_{l=k_0}$ is weakly Cauchy in $JH$, so is $(U_l\gamma)$.
Now if $JH{\tilde{\otimes}_\epsilon}JH$ has property (u), then it is easy to observe that there must be a block sequence of convex combinations $(V_r)$ of $(U_l)$ such that $\sum (V_r - V_{r+1})$ is a wuC series. Write $V_r = \sum^{l_r}_{l=l_{r-1}+1}a_lU_l$ (convex combination), where $(l_r)$ is a strictly increasing sequence in ${{\Bbb N}}$. Fix $r \in {{\Bbb N}}$. For $s_{r-1}< i \leq s_r$, let $\xi_i$ be a branch such that $\phi(i,l_{r+i-s_{r-1}})$ is the node of maximal length which it shares with $\gamma_i$. Then if $s_{r-1} < i, j \leq s_r$, and $r
\leq l$, $${\langle}e_{\phi(i,l)}, \xi_j{\rangle}= 1 \qquad {\Longleftrightarrow}\qquad i = j \quad
\text{and} \quad l \leq l_{r+j-s_{r-1}}.$$ Hence, if $r \leq l$, $${\langle}U_l\xi_i, \xi_j{\rangle}=
\begin{cases}
R_r(i,j)& \text{if $l \leq
\min(l_{r+i-s_{r-1}},l_{r+j-s_{r-1}})$,}\\
0& \text{otherwise.}
\end{cases}$$ Thus, if $s_{r-1} < i, j ,k \leq s_r$, $${\langle}V_{r+k-s_{r-1}}\xi_i, \xi_j{\rangle}=
\begin{cases}
R_r(i,j)& \text{if $k \leq \min(i,j)$,} \\
0& \text{otherwise.}
\end{cases}$$ It follows that $$\begin{gathered}
\left{\langle}\left\{\sum^{s_r}_{k=s_{r-1}+1}(-1)^{k+1-s_{r-1}}\left(V_{r+k-s_{r-1}}
- V_{r+k+1-s_{r-1}}\right)\right\}\xi_i, \xi_j\right{\rangle}\\
= (-1)^{\min(i-s_{r-1},j-s_{r-1})+1}R_r(i,j). \end{gathered}$$ Notice that $k > s_{r-1}$ implies $l_{r+k-s_{r-1}} \geq l_{r+1} \geq
r$, hence $${\langle}V_{r+k-s_{r-1}}P'_{n_r}\xi_i, P'_{n_r}\xi_j{\rangle}= {\langle}V_{r+k-s_{r-1}}\xi_i, \xi_j{\rangle}.$$ Also, $(P'_{n_r}\xi_i)^{s_r}_{i=s_{r-1}+1}$ is isometrically equivalent to the $\ell^\infty(r)$-basis. Therefore, $$\begin{gathered}
\left\|\sum^{s_r}_{k=s_{r-1}+1}(-1)^{k+1-s_{r-1}}\left(V_{r+k-s_{r-1}}
- V_{r+k+1-s_{r-1}}\right)\right\| \\
\geq
\sigma\left(\left[(-1)^{\min(i-s_{r-1},j-s_{r-1})+1}R_r(i,j)\right]\right) =
\sigma(E(R_r)) . \end{gathered}$$ But since $\sum (V_r - V_{r+1})$ is a wuC series, there is a constant $K < \infty$ (which may depend on the sequence $(R_k)$) such that $$\left\|\sum^{s_r}_{k=s_{r-1}+1}(-1)^{k+1-s_{r-1}}\left(V_{r+k-s_{r-1}}
- V_{r+k+1-s_{r-1}}\right)\right\| \leq K$$ for any $r$. Consequently, $\sup_r\sigma(E(R_r)) \leq K$.
Now choose a strictly increasing sequence $(r_m)$ such that $\lim_m 2^{-m}\log r_m = \infty$. Then let $$R_k =
\begin{cases}
\frac{M_{r_m}}{2^m}& \text{if $k = r_m$ for some $m$,}\\
0& \text{otherwise,}
\end{cases}$$ where $M_{r_m}$ is the matrix given by Lemma \[est\]. Then $\sum_k\sigma(R_k) = \sum_m2^{-m}\sigma(M_{r_m}) = 1$. So the preceding argument yields a finite constant $K$ such that $$K \geq \sup_m\frac{\sigma(E(M_{r_m}))}{2^m} \geq C
\sup_m\frac{\log r_m}{2^m} ,$$ contrary to the choice of $(r_m)$. We have thus proved the following result.
The Banach space $JH{\tilde{\otimes}_\epsilon}JH$ fails property [(u)]{}.
[99]{}
, [*A counterexample to several questions about Banach spaces*]{}, Studia Math. [**60**]{}(1977), 289-308.
, [*On $c_0$ sequences in Banach spaces*]{}, Israel J. Math. [**67**]{}(1989), 153-169.
, [*The main triangle projection in matrix spaces and its applications*]{}, Studia Math. [**34**]{}(1970), 43-68.
, [*Some stability properties of $c_0$-saturated spaces*]{}, Math. Proc. Camb. Phil. Soc., [**118**]{}(1995), 287-301.
, Classical Banach Spaces I, Springer-Verlag, 1979.
|
---
abstract: 'Stability and convergence of full discretizations of various surface evolution equations are studied in this paper. The proposed discretization combines a higher-order evolving-surface finite element method (ESFEM) for space discretization with higher-order linearly implicit backward difference formulae (BDF) for time discretization. The stability of the full discretization is studied in the matrix–vector formulation of the numerical method. The geometry of the problem enters into the bounds of the consistency errors, but does not enter into the proof of stability. Numerical examples illustrate the convergence behaviour of the full discretization.'
author:
- Balázs Kovács
- Christian Lubich
date:
title: |
Linearly implicit full discretization of\
surface evolution
---
Introduction
============
In this paper we study full discretizations of geometric evolution equations using the evolving surface finite element method (ESFEM) for space discretization and linearly implicit backward differentiation formulae (BDF) for time discretization. We consider the situation where the velocity $v(x,t)$ of a point $x$ on an evolving two-dimensional closed surface $\Gamma(t)\subset{\mathbb{R}}^3$ at time $t$ is determined by one of the following velocity laws, for which finite element semi-discretization in space was studied in [@soldriven]:
\(i) [*Regularized mean curvature flow:*]{} for $x\in{\Gamma}(t)$, $$\label{v-eq}
v(x,t) - \alpha {\Delta}_{{\Gamma}(t)} v(x,t) = -\beta H_{{\Gamma}(t)}(x)\, \nu_{{\Gamma}(t)}(x) +g\bigl(x,t\bigr)\, \nu_{{\Gamma}(t)}(x),
$$ where ${\Delta}_{{\Gamma}(t)}$ is the Laplace–Beltrami operator on the surface $\Gamma(t)$, $H_{{\Gamma}(t)}$ is mean curvature, $\nu_{{\Gamma}(t)}$ is the outer normal, $g$ is a smooth real-valued function, and $\alpha>0$ and $\beta\ge 0$ are fixed parameters. This velocity law can be viewed as an elliptically regularized mean curvature flow with an additional driving term in the direction of the normal vector. In [@soldriven] this elliptic regularization allowed us to give a complete stability and convergence analysis of the ESFEM semi-discretization, for finite elements of polynomial degree at least two. In contrast, for pure mean curvature flow (that is, $\alpha=0$), no convergence results appear to be known for ESFEM on two-dimensional closed surfaces. (ii) [*A dynamic velocity law:*]{} for $x\in{\Gamma}(t)$, $$\label{v-dyn}
{\partial^{\bullet}}v(x,t) + v(x,t) \,{\nabla}_{{\Gamma}(t)} \cdot v (x,t) - \alpha {\Delta}_{{\Gamma}(t)} v(x,t) = g(x,t) \, \nu_{{\Gamma}(t)}(x),$$ where ${\partial^{\bullet}}v$ denotes the material time derivative of $v$ and ${\nabla}_{{\Gamma}} \cdot v$ denotes the surface divergence of $v$;
\(iii) The case where the velocity law (i) or (ii) is coupled to diffusion on the evolving surface, as in [@soldriven].
We note that in all these cases, the considered velocity $v$ is in general not normal to the surface, but contains tangential components.
The rigorous study of the stability and convergence properties of full discretizations obtained by combining the ESFEM with various time discretizations for problems on evolving surfaces was begun in the papers [@DziukElliott_fulldiscr] (implicit Euler method), [@DziukLubichMansour_rksurf] (implicit Runge–Kutta methods) and [@LubichMansourVenkataraman_bdsurf] (BDF methods). These papers studied a linear parabolic equation on a [*given*]{} moving closed surface $\Gamma(t)$. Convergence of full discretizations of that problem using higher-order evolving surface finite elements is studied in [@highorder]. Convergence properties of full discretizations for quasi- and semilinear parabolic equations on prescribed moving surfaces are studied in [@KovacsPower_quasilinear]. For curves instead of two-dimensional surfaces, convergence of full discretizations of curve-shortening flow coupled to diffusion is studied by Barrett, Deckelnick & Styles [@BDS].
The main difficulty in proving the convergence of the full discretization of the surface-evolution equation in (i)–(iii) is the proof of stability in the sense of bounding errors in terms of defects in the discrete equations. The proof requires some auxiliary results from [@soldriven], which relate different finite element surfaces. For , the stability proof just uses the zero-stability of the BDF methods up to order 6. For , it is based on energy estimates that become available for BDF methods up to order 5 by the multiplier technique of Nevanlinna and Odeh [@NevanlinnaOdeh], which in turn is based on the $G$-stability theory of Dahlquist [@Dahlquist]. These techniques were originally developed for stiff ordinary differential equations and have recently been used for linear parabolic equations on given moving surfaces in [@LubichMansourVenkataraman_bdsurf] and for various quasilinear parabolic problems in [@AkrivisLubich_quasilinBDF; @AkrivisLiLubich_quasilinBDF; @KovacsPower_quasilinear].
The paper is organized as follows.
In Section \[section: problem\] we describe the problem and the numerical methods. We recall the basics of the evolving surface finite element method and give its matrix–vector formulation, and we formulate the linearly implicit BDF time discretization.
In Section \[section: main result\] we present the main result for , which gives optimal-order convergence estimates for the full discretization by ESFEM of polynomial degree at least $2$ and linearly implicit BDF methods up to order 6. This result is proven in Sections 4 to 7.
Section \[section: aux\] contains auxiliary results for the stability analysis of the discretized velocity law . We collect results from [@soldriven] that relate different finite element surfaces to one another. We also include a new auxiliary result for the linearly implicit BDF time discretization.
Section \[section: proof of stability\] contains the stability analysis, which works with the matrix–vector formulation of the discrete equations. Like the proof of stability of the ESFEM spatial semi-discretization in [@soldriven], it does not use geometric arguments.
Section \[section: consistency\] gives estimates for the consistency errors, that is, for the defects on inserting the interpolated exact solution into the discrete equations.
Section \[section: proof completed\] proves the convergence result for the full discretization of by combining the results of the previous sections.
In Section \[section: dynamic\] we extend the convergence analysis to the full discretization of the dynamic velocity law . This is done for BDF methods up to order 5 using energy estimates based on the Nevanlinna–Odeh multiplier technique.
In Section \[section: coupled\] we extend the convergence result for the full discretization to the case where the velocity law or is coupled to diffusion on the evolving surface, as studied in [@soldriven] for the semi-discretization. The result is obtained by combining the techniques of [@soldriven] and [@LubichMansourVenkataraman_bdsurf] with those of Sections \[section: aux\] to \[section: proof completed\] of the present paper.
Section \[section: numerics\] presents numerical experiments using quadratic ESFEM that illustrate the numerical behaviour of the proposed full discretization.
We use the notational convention to denote vectors in ${\mathbb{R}}^3$ by italic letters, but to denote finite element nodal vectors in ${\mathbb{R}}^{3N}$ by boldface lowercase letters and finite element mass and stiffness matrices by boldface capitals. All boldface symbols in this paper will thus be related to the matrix–vector formulation of the ESFEM.
Problem formulation and ESFEM / BDF full discretization {#section: problem}
=======================================================
We use the same setting as in our previous work [@soldriven]. We recall basic notions, but refer to Section 2 of [@soldriven] for a more detailed description.
Basic notions and notation {#subsection: basic notions}
--------------------------
We consider the evolving two-dimensional closed surface $\Gamma(t)\subset{\mathbb{R}}^3$ as the image $${\Gamma}(t) = \{ X(q,t) \,:\, q \in {\Gamma}^0 \}$$ of a regular vector-valued function $X:{\Gamma}^0\times [0,T]\to {\mathbb{R}}^3$, where ${\Gamma}^0$ is the smooth closed initial surface, and $X(q,0)=q$. To indicate the dependence of the surface on $X$, we write $${\Gamma}(t) = {\Gamma}(X(\cdot,t)), \quad\hbox{ or briefly}\quad {\Gamma}(X)$$ when the time $t$ is clear from the context. The position $X(q,\cdot)$ is related to the [*velocity*]{} $v(x,t)\in{\mathbb{R}}^3$ at the point $x=X(q,t)\in\Gamma(t)$ via the ordinary differential equation $$\label{velocity}
\partial_t X(q,t)= v(X(q,t),t).$$ For $x\in\Gamma(t)$ and $0\le t \le T$, we denote by $\nu_{{\Gamma}(X)}(x)$ the outer normal, by $\nabla_{{\Gamma}(X)}u(x,t)$ the tangential gradient of a real-valued function $u$ on ${\Gamma}(t)$, and by ${\Delta}_{{\Gamma}(X)} u(x,t)$ the Laplace–Beltrami operator applied to $u$.
Weak formulation of the surface-evolution equation {#section: problem and weak form}
--------------------------------------------------
The space discretization is based on the weak formulation of the surface-evolution equation , which reads as follows: Find $v(\cdot,t) \in W^{1,\infty}({\Gamma}(X(\cdot,t)))^3 $ such that for all test functions $\psi(\cdot,t) \in H^1({\Gamma}(X(\cdot,t) ))^3$, $$\label{weak form}
\begin{aligned}
\int_{{\Gamma}(X)} \!\!\!\! v \cdot \psi
&\ + \alpha \int_{{\Gamma}(X)} \!\!\!\! \nabla_{\Gamma(X)} v \cdot \nabla_{\Gamma(X)} \psi \\
&\ + \beta \int_{{\Gamma}(X)} \!\!\!\! \nabla_{\Gamma(X)} X \cdot \nabla_{\Gamma(X)} \psi
= \int_{{\Gamma}(X)} \!\! g \,\nu_{{\Gamma}(X)} \cdot \psi ,
\end{aligned}$$ alongside with the ordinary differential equation for the positions $X$ determining the surface $\Gamma(X)$. (More precisely, the term $\nabla_{\Gamma(X)} X$ should read $\nabla_{\Gamma(X)} {\rm id}_{\Gamma(X)}$.)
We assume throughout this paper that the problem or admits a unique solution with sufficiently high Sobolev regularity on the time interval $[0,T]$ for the given initial data $X(\cdot,0)$. We assume further that the flow map $X(\cdot,t):\Gamma_0\to \Gamma(t)\subset{\mathbb{R}}^3$ is non-degenerate for $0\le t \le T$, so that $\Gamma(t)$ is a regular surface.
Evolving surface finite elements {#section:ESFEM}
--------------------------------
From Section 2.3 of [@soldriven] we recall the description of the surface finite element discretization of our problem, which is based on [@Dziuk88] and [@Demlow2009]. We use simplicial elements and continuous piecewise polynomial basis functions of degree $k$, as defined in [@Demlow2009 Section 2.5].
We triangulate the given smooth surface $\Gamma^0$ by an admissible family of triangulations $\mathcal{T}_h$ of decreasing maximal element diameter $h$; see [@DziukElliott_ESFEM] for the notion of an admissible triangulation, which includes quasi-uniformity and shape regularity. For a momentarily fixed $h$, we denote by ${{\mathbf x}}^0=(x_1^0,\dots,x_N^0)$ the vector in ${\mathbb{R}}^{3N}$ that collects all $N$ nodes of the triangulation. By piecewise polynomial interpolation of degree $k$, the nodal vector defines an approximate surface $\Gamma_h^0$ that interpolates $\Gamma^0$ in the nodes $x_j^0$. We will evolve the $j$th node in time, denoted $x_j(t)$ with $x_j(0)=x_j^0$, and collect the nodes at time $t$ in a column vector in ${\mathbb{R}}^{3N}$, $${{\mathbf x}}(t) = (x_1(t),\dots,x_N(t)) \in {\mathbb{R}}^{3N}.$$ We just write ${{\mathbf x}}$ for ${{\mathbf x}}(t)$ when the dependence on $t$ is not important.
By piecewise polynomial interpolation on the plane reference triangle that corresponds to every curved triangle of the triangulation, the nodal vector ${{\mathbf x}}$ defines a closed surface denoted by $\Gamma_h[{{\mathbf x}}]$. We can then define finite element [*basis functions*]{} $$\phi_j[{{\mathbf x}}]:\Gamma_h[{{\mathbf x}}]\to{\mathbb{R}}, \qquad j=1,\dotsc,N,$$ which have the property that on every triangle their pullback to the reference triangle is polynomial of degree $k$, and which satisfy $$\phi_j[{{\mathbf x}}](x_k) = \delta_{jk} \quad \text{ for all } j,k = 1, \dotsc, N .$$ These functions span the finite element space on $\Gamma_h[{{\mathbf x}}]$, $$S_h[{{\mathbf x}}] = S_h(\Gamma_h[{{\mathbf x}}])={\textnormal{span}}\big\{ \phi_1[{{\mathbf x}}], \phi_2[{{\mathbf x}}], \dotsc, \phi_N[{{\mathbf x}}] \big\} .$$ For a finite element function $u_h\in S_h[{{\mathbf x}}]$ the tangential gradient $\nabla_{\Gamma_h[{{\mathbf x}}]}u_h$ is defined piecewise on each element. We set $$X_h(q_h,t) = \sum_{j=1}^N x_j(t) \, \phi_j[{{\mathbf x}}(0)](q_h), \qquad q_h \in \Gamma_h^0,$$ which has the properties that $X_h(q_j,t)=x_j(t)$ for $j=1,\dots,N$, that $X_h(q_h,0)=q_h$ for all $q_h\in\Gamma_h^0$, and $$\Gamma_h[{{\mathbf x}}(t)]=\Gamma(X_h(\cdot,t)).$$ The [*discrete velocity*]{} $v_h(x,t)\in{\mathbb{R}}^3$ at a point $x=X_h(q_h,t)\in \Gamma(X_h(\cdot,t))$ is given by $$\partial_t X_h(q_h,t) = v_h(X_h(q_h,t),t).$$ In view of the transport property of the basis functions [@DziukElliott_ESFEM], $$\frac{\hbox{\rm d}}{{\hbox{\rm d}}t} \Bigl( \phi_j[{{\mathbf x}}(t)](X_h(q_h,t)) \Bigr) =0 ,$$ the discrete velocity equals, for $x \in \Gamma_h[{{\mathbf x}}(t)]$, $$v_h(x,t) = \sum_{j=1}^N v_j(t) \, \phi_j[{{\mathbf x}}(t)](x)
\qquad \hbox{with } \ v_j(t)=\dot x_j(t),$$ where the dot denotes the time derivative ${\hbox{\rm d}}/{\hbox{\rm d}}t$. Hence, the nodal vector of the discrete velocity is ${{\mathbf v}}=\dot{{\mathbf x}}$.
ESFEM spatial semi-discretization of the evolving-surface problem {#subsection:semi-discretization}
-----------------------------------------------------------------
The finite element spatial semi-discretization of the problem reads as follows: Find the unknown nodal vector ${{\mathbf x}}(t)\in {\mathbb{R}}^{3N}$ and the unknown finite element function $v_h(\cdot,t)\in S_h[{{\mathbf x}}(t)]^3$ such that, for all $\psi_h(\cdot,t)\in S_h[{{\mathbf x}}(t)]^3$, $$\label{uh-vh-equation}
\begin{aligned}
\int_{{\Gamma}_h[{{\mathbf x}}]}\!\!\! v_h \cdot \psi_h
&\ + \alpha \int_{{\Gamma}_h[{{\mathbf x}}]} \!\!\!\!\! \nabla_{{\Gamma}_h[{{\mathbf x}}]} v_h \cdot \nabla_{{\Gamma}_h[{{\mathbf x}}]} \psi_h \\
&\ + \beta \int_{{\Gamma}_h[{{\mathbf x}}]} \!\!\!\!\! \nabla_{{\Gamma}_h[{{\mathbf x}}]} X_h \cdot \nabla_{{\Gamma}_h[{{\mathbf x}}]} \psi_h
= \int_{{\Gamma}_h[{{\mathbf x}}]} \!\! g \,\nu_{{\Gamma}_h[{{\mathbf x}}]} \cdot \psi_h,
\end{aligned}
$$ and $$\label{xh}
\partial_t X_h(q_h,t) = v_h(X_h(q_h,t),t), \qquad q_h\in{\Gamma}_h^0.$$ The initial values for the nodal vector ${{\mathbf x}}$ of the initial positions are taken as the exact initial values at the nodes $x_j^0$ of the triangulation of the given initial surface $\Gamma^0$: $$x_j(0) = x_j^0, \qquad j=1,\dotsc,N.$$
Matrix–vector formulation {#subsection:DAE}
-------------------------
We define the surface-dependent mass matrix ${{\mathbf M}}({{\mathbf x}})$ and stiffness matrix ${{\mathbf A}}({{\mathbf x}})$ on the surface determined by the nodal vector ${{\mathbf x}}$ (cf. [@soldriven Section 2.5]): $$\begin{aligned}
{{\mathbf M}}({{\mathbf x}})\vert_{jk} =&\ \int_{{\Gamma}_h[{{\mathbf x}}]} \! \phi_j[{{\mathbf x}}] \phi_k[{{\mathbf x}}] , \\
{{\mathbf A}}({{\mathbf x}})\vert_{jk} =&\ \int_{{\Gamma}_h[{{\mathbf x}}]} \! {\nabla}_{{\Gamma}_h} \phi_j[{{\mathbf x}}] \cdot {\nabla}_{{\Gamma}_h} \phi_k[{{\mathbf x}}] ,
\end{aligned}
\qquad (j,k = 1, \dotsc, N) .$$ We further let (with the identity matrix $I_3 \in {\mathbb{R}}^{3 \times 3}$) $${{\mathbf M}}^{[3]}({{\mathbf x}})= I_3 \otimes {{\mathbf M}}({{\mathbf x}}) \quad \textnormal{and} \quad {{\mathbf A}}^{[3]}({{\mathbf x}}) = I_3 \otimes {{\mathbf A}}({{\mathbf x}}) ,$$ and then define $$\label{eq: K matrix def}
{{\mathbf K}}({{\mathbf x}})= {{\mathbf M}}^{[3]}({{\mathbf x}}) + \alpha {{\mathbf A}}^{[3]}({{\mathbf x}}) .$$ When no confusion can arise, we write in the following ${{\mathbf M}}({{\mathbf x}})$ for ${{\mathbf M}}^{[3]}({{\mathbf x}})$, ${{\mathbf A}}({{\mathbf x}})$ for ${{\mathbf A}}^{[3]}({{\mathbf x}})$ and $\| \cdot \|_{H^1(\Gamma)}$ for $\| \cdot \|_{H^1(\Gamma)^3}$, etc.
The right-hand side vector ${{\mathbf g}}({{\mathbf x}},t)\in{\mathbb{R}}^{3N}$ is given by $$\begin{aligned}
{{\mathbf g}}({{\mathbf x}},t)\vert_{j+N(\ell-1)} &= \int_{{\Gamma}_h[{{\mathbf x}}]} g(\cdot,t) \,\bigl(\nu_{{\Gamma}_h[{{\mathbf x}}]}\bigr)_\ell \, \phi_j[{{\mathbf x}}],
\end{aligned}$$ for $j = 1, \dotsc, N$ and $\ell=1,2,3$.
We then obtain from – the following system of ordinary differential equations (ODEs) for the nodal vectors ${{\mathbf x}}(t)\in{\mathbb{R}}^{3N}$: $$\label{eq: ODE form}
{{\mathbf K}}({{\mathbf x}}) \dot {{\mathbf x}}+ {\beta {{\mathbf A}}}({{\mathbf x}}) {{\mathbf x}}= {{\mathbf g}}({{\mathbf x}},t) .$$
Linearly implicit BDF time discretization
-----------------------------------------
We apply a $p$-step linearly implicit backward difference formula (BDF) for $p \leq 6$ as a time discretization to the ODE system . For a step size $\tau>0$, and with $t_n = n \tau \leq T$, we determine the approximation ${{\mathbf x}}^n$ to ${{\mathbf x}}(t_n)$ by the fully discrete system of linear equations $$\label{BDF}
\begin{aligned}
{{\mathbf K}}(\widetilde {{\mathbf x}}^n) {{\mathbf v}}^n + {\beta {{\mathbf A}}}(\widetilde {{\mathbf x}}^n) {{\mathbf x}}^n = &\ {{\mathbf g}}(\widetilde {{\mathbf x}}^n,t_n) , \\
{{\mathbf v}}^n =&\ \frac{1}{\tau} \sum_{j=0}^p \delta_j {{\mathbf x}}^{n-j} ,
\end{aligned}
\qquad\ n \geq p ,$$ where the extrapolated position vector $\widetilde {{\mathbf x}}^n$ is defined by $$\label{eq: extrapolation of u def}
\widetilde {{\mathbf x}}^n = \sum_{j=0}^{p-1} \gamma_j {{\mathbf x}}^{n-1-j} , \qquad n \geq p .$$ The starting values ${{\mathbf x}}^0, {{\mathbf x}}^1, \dotsc, {{\mathbf x}}^{p-1}$ are assumed to be given. They can be precomputed in a way as is usual with multistep methods: using lower-order methods with smaller step sizes or using an implicit Runge–Kutta method.
The coefficients are given by $\delta(\zeta)=\sum_{j=0}^p \delta_j \zeta^j=\sum_{\ell=1}^p \frac{1}{\ell}(1-\zeta)^\ell$ and $\gamma(\zeta) = \sum_{j=0}^{p-1} \gamma_j \zeta^j = (1 - (1-\zeta)^p)/\zeta$. The classical BDF method is known to be zero-stable for $p\leq6$ and to have order $p$; see [@HairerWannerII Chapter V]. This order is retained by the linearly implicit variant using the above coefficients $\gamma_j$; cf. [@AkrivisLubich_quasilinBDF; @AkrivisLiLubich_quasilinBDF].
We note that the method requires solving a linear system with the symmetric positive definite matrix $\frac{\delta_0}\tau {{\mathbf K}}(\widetilde {{\mathbf x}}^n) + {\beta {{\mathbf A}}}(\widetilde {{\mathbf x}}^n)$ in the $n$th time step.
From the vectors ${{\mathbf x}}^n =(x_j^n)$ and ${{\mathbf v}}^n = (v_j^n)$ we obtain position and velocity approximations to $X(\cdot,t_n)$ and $v(\cdot,t_n)$ as $$\label{x-v-approx}
\begin{split}
X_h^n(q_h) &= \sum_{j=1}^N x_j^n \, \phi_j[{{\mathbf x}}(0)](q_h) \quad\hbox{ for } q_h \in \Gamma_h^0,\\
v_h^n(x) &= \sum_{j=1}^N v_j^n \, \phi_j[{{\mathbf x}}^n](x) \qquad\hbox{ for } x \in \Gamma_h[{{\mathbf x}}^n].
\end{split}$$
Lifts {#section:lifts}
-----
Here we recapitulate [@soldriven Section 2.6]. In the error analysis we need to compare functions on three different surfaces: the [*exact surface*]{} $\Gamma(t)=\Gamma(X(\cdot,t))$, the [*discrete surface*]{} $\Gamma_h(t)=\Gamma_h[{{\mathbf x}}(t)]$, and the [*interpolated surface*]{} $\Gamma_h^*(t)=\Gamma_h[{{{\mathbf x}}_\ast}(t)]$, where ${{{\mathbf x}}_\ast}(t)$ is the nodal vector collecting the grid points $x_{*,j}(t)=X(q_j,t)$ on the exact surface. In the following definitions we omit the argument $t$ in the notation.
For a finite element function $w_h:\Gamma_h\to{\mathbb{R}}^m$ ($m=1$ or 3) on the discrete surface, with nodal values $w_j$, we denote by $\widehat w_h:\Gamma_h^*\to{\mathbb{R}}^m$ the finite element function on the interpolated surface that has the same nodal values: $$\widehat w_h = \sum_{j=1}^N w_j \phi_j[{{{\mathbf x}}_\ast}].$$ The transition between the interpolated surface and the exact surface is done by the *lift operator*, which was introduced for linear surface approximations in [@Dziuk88]; see also [@DziukElliott_ESFEM; @DziukElliott_L2]. Higher-order generalizations have been studied in [@Demlow2009]. The lift operator $l$ maps a function on the interpolated surface $\Gamma_h^*$ to a function on the exact surface $\Gamma$, provided that $\Gamma_h^*$ is sufficiently close to $\Gamma$.
The exact regular surface ${\Gamma}(X(\cdot,t))$ can be represented by a (sufficiently smooth) signed distance function $d : {\mathbb{R}}^3 \times [0,T] \to {\mathbb{R}}$, cf. [@DziukElliott_ESFEM Section 2.1], such that ${\Gamma}(X(\cdot,t)) = \big\{ x\in {\mathbb{R}}^3 \mid d(x,t) = 0 \big\} \subset {\mathbb{R}}^3$ . Using this distance function, the lift of a continuous function $\eta_h \colon {\Gamma}_h^* \to {\mathbb{R}}^m$ is defined as $$\eta_{h}^{l}(y) := \eta_h(x), \qquad x\in{\Gamma}_h^*,$$ where for every $x\in {\Gamma}_h^*$ the point $y=y(x)\in{\Gamma}$ is uniquely defined via $y = x - \nu(y) d(x)$. We denote the composed lift $L$ from finite element functions on $\Gamma_h$ to functions on $\Gamma$ via $\Gamma_h^*$ by $$w_h^L = (\widehat w_h)^l.$$
Statement of the main result: fully discrete error bound {#section: main result}
========================================================
We formulate the main result of this paper, which yields optimal-order error bounds for the ESFEM / BDF full discretization of the surface-evolution equation , for finite elements of polynomial degree $k\ge 2$ and BDF methods of order $p\le 6$. We denote by $\Gamma(t_n)=\Gamma(X(\cdot,t_n))$ the exact surface and by $\Gamma_h^n=\Gamma(X_h^n)=\Gamma_h[{{\mathbf x}}^n]$ the discrete surface at time $t_n$. For the lifted position function we introduce the notation $$(x_h^n)^L (x) = (X_h^n)^L(q) \in \Gamma_h^n \qquad\hbox{for}\quad x=X(q,t_n)\in\Gamma(t_n).$$
\[theorem: main\] Consider the ESFEM / BDF linearly implicit full discretization of the surface-evolution equation , using finite elements of polynomial degree $k\ge 2$ and BDF methods of order $p\le 6$. We assume quasi-uniform admissible triangulations of the initial surface and initial values chosen by finite element interpolation of the initial data for $X$. Suppose that the problem admits an exact solution $(X,v)$ that is sufficiently smooth (say, of class $C([0,T],H^{k+1})\cap C^{p+1}([0,T],W^{1,\infty})$) on the time interval $0\le t \le T$, and that the flow map $X(\cdot,t):\Gamma_0\to \Gamma(t)\subset{\mathbb{R}}^3$ is non-degenerate for $0\le t \le T$, so that $\Gamma(t)$ is a regular surface. Suppose further that the starting values are sufficiently accurate: $$\| (X_h^i)^L - X(\cdot,i\tau) \|_{H^1({\Gamma}^0)^3} \le C_0 (h^k+\tau^{p}), \qquad i=0,1,\dots,p-1.$$ Then, there exist $h_0 >0$, $\tau_0>0$ and $c_0>0$ such that for all mesh widths $h \leq h_0$ and step sizes $\tau\le\tau_0$ satisfying the mild stepsize restriction $$\tau^p \le c_0 h,$$ the following error bounds hold over the exact surface ${\Gamma}(t_n)={\Gamma}(X(\cdot,t_n))$ uniformly for $0\le t_n=n\tau \le T$: $$\begin{aligned}
\|(x_h^n)^{L} - \mathrm{id}_{\Gamma(t_n)}\|_{H^1({\Gamma}(t_n))^3} &\leq C(h^k+\tau^p),\\
\|(v_h^n)^{L} - v(\cdot,t_n)\|_{H^1({\Gamma}(t_n))^3} &\leq C(h^k+\tau^p).
\end{aligned}$$ The constant $C$ is independent of $h$ and $\tau$ and $n$ with $n\tau\le T$, but depends on bounds of higher derivatives of the solution $(X,v)$, and on the length $T$ of the time interval.
We note that the first error bound is equivalent to $$\| (X_h^n)^L - X(\cdot,t_n) \|_{H^1({\Gamma}^0)^3} \le C' (h^k + \tau^{p}) ,$$ and we mention that the remarks after Theorem 3.1 in [@soldriven] (the convergence theorem of the ESFEM semi-discretization) apply also to the fully discretized situation considered here.
The proof of Theorem \[theorem: main\] is given in the course of the next four sections.
Preparation: Estimates relating different surfaces {#section: aux}
==================================================
In our previous work [@soldriven Section 4] we have shown some auxiliary results relating different finite element surfaces, which we recapitulate here.
The finite element matrices of Section \[subsection:DAE\] induce discrete versions of Sobolev norms. For any ${{\mathbf w}}=(w_j) \in {\mathbb{R}}^N$ with corresponding finite element function $w_h= \sum_{j=1}^N w_j \phi_j[{{\mathbf x}}] \in S_h[{{\mathbf x}}]$ we note $$\begin{aligned}
\label{M-L2}
& \|{{\mathbf w}}\|_{{{\mathbf M}}({{\mathbf x}})}^{2} = {{\mathbf w}}^T {{\mathbf M}}({{\mathbf x}}) {{\mathbf w}}= \|w_h\|_{L^2({\Gamma}_h[{{\mathbf x}}])}^2, \\
\label{A-H1}
& \|{{\mathbf w}}\|_{{{\mathbf A}}({{\mathbf x}})}^{2} = {{\mathbf w}}^T {{\mathbf A}}({{\mathbf x}}) {{\mathbf w}}= \|{\nabla}_{{\Gamma}_h[{{\mathbf x}}]} w_h\|_{L^2({\Gamma}_h[{{\mathbf x}}])}^2.
$$
We use the following setting. Let ${{\mathbf x}},{{\mathbf y}}\in {\mathbb{R}}^{3 N}$ be two nodal vectors defining discrete surfaces $\Gamma_h[{{\mathbf x}}]$ and $\Gamma_h[{{\mathbf y}}]$, respectively. We let ${{\mathbf e}}= (e_j)={{\mathbf x}}-{{\mathbf y}}\in {\mathbb{R}}^{ 3 N}$. For $\theta\in[0,1]$, we consider the intermediate surface $\Gamma_h^\theta=\Gamma_h[{{\mathbf y}}+\theta{{\mathbf e}}]$ and the corresponding finite element functions given as $$e_h^\theta=\sum_{j=1}^N e_j \phi_j[{{\mathbf y}}+\theta{{\mathbf e}}]$$ and in the same way, for any vectors ${{\mathbf w}},{{\mathbf z}}\in {\mathbb{R}}^N$, $$w_h^\theta=\sum_{j=1}^N w_j \phi_j[{{\mathbf y}}+\theta{{\mathbf e}}] {\quad\hbox{ and }\quad}z_h^\theta=\sum_{j=1}^N z_j \phi_j[{{\mathbf y}}+\theta{{\mathbf e}}] .$$ The following lemma collects results from [@soldriven Section 4].
\[lemma: technicals\] (i) \[matrix differences\] In the above setting the following identities hold: $$\begin{aligned}
{{\mathbf w}}^T ({{\mathbf M}}({{\mathbf x}})-{{\mathbf M}}({{\mathbf y}})) {{\mathbf z}}=&\ \int_0^1 \int_{{\Gamma}_h^\theta} w_h^\theta (\nabla_{{\Gamma}_h^\theta} \cdot e_h^\theta) z_h^\theta \; {\hbox{\rm d}}\theta, \\
{{\mathbf w}}^T ({{\mathbf A}}({{\mathbf x}})-{{\mathbf A}}({{\mathbf y}})) {{\mathbf z}}=&\ \int_0^1 \int_{{\Gamma}_h^\theta} {\nabla}_{{\Gamma}_h^\theta} w_h^\theta \cdot (D_{{\Gamma}_h^\theta} e_h^\theta){\nabla}_{{\Gamma}_h^\theta} z_h^\theta \; {\hbox{\rm d}}\theta ,
\end{aligned}$$ with $D_{{\Gamma}_h^\theta} e_h^\theta = \textnormal{trace}(E) I_3 - (E+E^T)$ for $E=\nabla_{{\Gamma}_h^\theta} e_h^\theta \in {\mathbb{R}}^{3\times 3}$.
\(ii) \[lemma:cond-equiv\] If $\| \nabla_{\Gamma_h^\theta} \cdot e_h^\theta \|_{L^\infty(\Gamma_h^\theta)} \leq \mu $ and $\| D_{\Gamma_h^\theta} e_h^\theta \|_{L^\infty(\Gamma_h^\theta)} \leq \rho {\color{black}}$ for $0 \leq \theta \leq 1$, then $
\| {{\mathbf w}}\|_{{{\mathbf M}}({{\mathbf y}}+\theta{{\mathbf e}})} \le e^{\mu/2} \, \| {{\mathbf w}}\|_{{{\mathbf M}}({{\mathbf y}})}
$ and $
\| {{\mathbf w}}\|_{{{\mathbf A}}({{\mathbf y}}+\theta{{\mathbf e}})} \le e^{\rho {\color{black}}/2} \, \| {{\mathbf w}}\|_{{{\mathbf A}}({{\mathbf y}})}
$.
\(iii) \[lemma:theta-independence\] If $ \| \nabla_{\Gamma_h[{{\mathbf y}}]} e_h^0 \|_{L^\infty(\Gamma_h[{{\mathbf y}}])} \le \frac12,
$ then, for $0\le\theta\le 1$, the function $w_h^\theta=\sum_{j=1}^N w_j \phi_j[{{\mathbf y}}+\theta{{\mathbf e}}]$ on $\Gamma_h^\theta=\Gamma_h[{{\mathbf y}}+\theta{{\mathbf e}}]$ is bounded by $$\| \nabla_{\Gamma_h^\theta} w_h^\theta \|_{L^p(\Gamma_h^\theta)} \le c_p \, \| \nabla_{\Gamma_h^0} w_h^0 \|_{L^p(\Gamma_h^0)}
{\quad\hbox{ for }\quad}1\le p \le \infty,$$ where $c_p$ depends only on $p$ (we have $c_\infty=2$).
\(iv) \[lemma: normal vector perturbation\] Let $y_h^\theta\in\Gamma_h^\theta$ be defined as $y_h^\theta=\sum_{j=1}^N (y_j +\theta e_j)\phi_j[{{\mathbf y}}](q_h)$ for $q_h\in \Gamma_h[{{\mathbf y}}]$. If $ \| \nabla_{\Gamma_h[{{\mathbf y}}]} e_h^0 \|_{L^\infty(\Gamma_h[{{\mathbf y}}])} \le \frac12,
$ then the corresponding unit normal vectors differ by no more than $$|\nu_{\Gamma_h^\theta}(y_h^\theta) - \nu_{\Gamma_h^0}(y_h^0)| \le C\theta | \nabla_{\Gamma_h^0} e_h^0(y_h^0) | ,
$$ where $C$ is independent of $h$ and of $q_h\in \Gamma_h[{{\mathbf y}}]$.
The following result is shown in Lemma 4.1 of [@DziukLubichMansour_rksurf].
\[lemma: matrix derivatives\] Let $\Gamma(t)=\Gamma(X(\cdot,t))$, $t \in [0,T]$, be a smoothly evolving family of smooth closed surfaces, and let the vector ${{{\mathbf x}}_\ast}(t) \in {\mathbb{R}}^{3N}$ collect the nodes $x_j^*(t)=X(q_j,t)$. Then, for $0\le s, t \le T$ and for all ${{\mathbf w}},{{\mathbf z}}\in {\mathbb{R}}^N$, $$\begin{aligned}
{{\mathbf w}}^T \bigl({{\mathbf M}}({{{\mathbf x}}_\ast}(t)) - {{\mathbf M}}({{{\mathbf x}}_\ast}(s))\bigr){{\mathbf z}}\leq&\ C (t-s) \|{{\mathbf w}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}(t))}\|{{\mathbf z}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}(t))} , \\
{{\mathbf w}}^T \bigl({{\mathbf A}}({{{\mathbf x}}_\ast}(t)) - {{\mathbf A}}({{{\mathbf x}}_\ast}(s))\bigr){{\mathbf z}}\leq&\ C (t-s) \|{{\mathbf w}}\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}(t))}\|{{\mathbf z}}\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}(t))}
\end{aligned}$$ and the norms for different times are uniformly equivalent for $0\le s, t \le T$: $$\| {{\mathbf w}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}(t))} \le C \| {{\mathbf w}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}(s))}, \quad\
\| {{\mathbf w}}\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}(t))} \le C \| {{\mathbf w}}\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}(s))}.$$ The constant $C$ depends only on a bound of the $W^{1,\infty}\!$ norm of the surface velocity.
We also need a result which compares the finite element surfaces with exact and extrapolated nodes.
\[lemma: matrix identity for extrapolation\] Let $\Gamma(t)=\Gamma(X(\cdot,t))$, $t \in [0,T]$, be a smoothly evolving family of smooth closed surfaces. We denote the nodal vectors of exact solution values by ${{\mathbf x}}_*^n ={{\mathbf x}}_*(t_n)$ and of the extrapolated values by $\widetilde {{\mathbf x}}_*^n = \sum_{j=0}^{p-1} \gamma_j {{\mathbf x}}_*^{n-1-j}$. Then, the following estimates hold for all ${{\mathbf w}},{{\mathbf z}}\in{\mathbb{R}}^N$: $$\begin{aligned}
{{\mathbf w}}^T ({{\mathbf M}}(\widetilde{{\mathbf x}}_\ast^n)-{{\mathbf M}}({{{\mathbf x}}_\ast}^n)) {{\mathbf z}}&\leq C\tau^p \,\|{{\mathbf w}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)} \|{{\mathbf z}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)} , \\
{{\mathbf w}}^T ({{\mathbf A}}(\widetilde{{\mathbf x}}_\ast^n)-{{\mathbf A}}({{{\mathbf x}}_\ast}^n)) {{\mathbf z}}&\leq C\tau^p \, \|{{\mathbf w}}\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)} \|{{\mathbf z}}\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)} ,
\end{aligned}$$ where $C$ is independent of $h$, $\tau$ and $n$ with $0\le n\tau\le T$.
For the extrapolated value $\widetilde X(q,t) = \sum_{j=0}^{p-1} \gamma_j X(q, t-(j+1)\tau)$, we use the error formula with Peano kernel representation, see e.g. [@Gautschi Section 3.2.6], $$\label{peano}
\widetilde X(q,t) - X(q,t) = \tau^p \int_0^p \kappa_p(\lambda) \, \partial_t^{p+1} X(q,t-\lambda\tau)\, {\hbox{\rm d}}\lambda$$ with a bounded Peano kernel $\kappa_p$. We note that we have $$\widetilde x_{*,j}^n - x_{*,j}^n = \widetilde X(q_j,t_n) - X(q_j,t_n).$$ Since $X$ is assumed smooth, we obtain from the above error formula that for $0 \leq \theta \leq 1$, the finite element function ${\widetilde e}_h^{n,\theta}$ in $S_h({\Gamma}_h^{\theta})$ with the nodal vector $\widetilde{{\mathbf x}}_\ast^n - {{{\mathbf x}}_\ast}^n$, for ${\Gamma}_h^{\theta} = {\Gamma}_h[{{{\mathbf x}}_\ast}^n+\theta (\widetilde{{\mathbf x}}_\ast^n - {{{\mathbf x}}_\ast}^n)]$, has a gradient bounded in the maximum norm by $c\tau^p$, where $c$ is independent of $\tau$ and $h$. So we have the bound $$ \|\nabla_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \cdot {\widetilde e}_h^{n,0} \|_{L^\infty({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \le c\tau^p.$$Together with Lemma \[lemma: technicals\] and an $L^2 - L^\infty - L^2$ estimate, we thus obtain $$\begin{aligned}
{{\mathbf w}}^T ({{\mathbf M}}(\widetilde{{\mathbf x}}_\ast^n)-{{\mathbf M}}({{{\mathbf x}}_\ast}^n)) {{\mathbf z}}=&\ \int_0^1 \int_{\Gamma_h^{n,\theta}} w_h^\theta (\nabla_{\Gamma_h^{n,\theta}} \cdot {\widetilde e}_h^{n,\theta} ) z_h^\theta {\hbox{\rm d}}\theta \\
\leq &\ \int_0^1 \|w_h^\theta\|_{L^2(\Gamma_h^{n,\theta})}
\|\nabla_{\Gamma_h^{n,\theta}} \cdot {\widetilde e}_h^{n,\theta} \|_{L^\infty(\Gamma_h^{n,\theta})}
\|z_h^\theta\|_{L^2(\Gamma_h^{n,\theta})} {\hbox{\rm d}}\theta \\
\leq &\ c\tau^p
\|w_h^0\|_{L^2({\Gamma}_h^{0,n})} \|z_h^0\|_{L^2({\Gamma}_h^{0,n})} \\ \leq &\ c\tau^p \|{{\mathbf w}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)} \|{{\mathbf z}}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)} .\end{aligned}$$ The second estimate is proved in the same way.
The above lemma immediately implies the following norm equivalence, for sufficiently small step size $\tau$, $$\label{eq: extrap norm equivalence}
\half \|{{\mathbf w}}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2
\leq \|{{\mathbf w}}\|_{{{\mathbf K}}(\widetilde{{\mathbf x}}_\ast^n)}^2
\leq \tfrac{3}{2} \|{{\mathbf w}}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 .$$
Stability {#section: proof of stability}
=========
We denote by $$ {{{\mathbf x}}_\ast}(t)=\bigl( x_{*,j}(t)\bigr)\in {\mathbb{R}}^{3N}\quad\hbox{with}\quad x_{*,j}(t)=X(q_j,t) , \qquad (j=1,\dots,N)$$ the nodal vector of the *exact* positions on the surface $\Gamma(X(\cdot,t))$. This defines a discrete surface $\Gamma_h[{{{\mathbf x}}_\ast}(t)]$ that interpolates the exact surface $\Gamma(X(\cdot,t))$.
We consider the interpolated exact velocity $$v_{*,h}(\cdot, t)=\sum_{j=1}^N v_{*,j}(t) \phi_j[{{{\mathbf x}}_\ast}(t)] \quad\ \hbox{ with }\quad\ v_{*,j}(t)=\dot x_{*,j}(t),$$ with the corresponding nodal vector $${{{\mathbf v}}_\ast}(t)=\bigl( v_{*,j}(t)\bigr)= {\dot{{\mathbf x}}_\ast}(t) \in {\mathbb{R}}^{3N}.$$ We write $${{\mathbf x}}_*^n={{\mathbf x}}_*(t_n), \ \ {{\mathbf v}}_*^n={{\mathbf v}}_*(t_n).$$ The errors of the numerical solution values ${{\mathbf x}}^n$ and ${{\mathbf v}}^n$ are marked with their respective subscript, hence are denoted by $$\begin{aligned}
{{{\mathbf e}}_{{\mathbf v}}}^n = {{\mathbf v}}^n - {{{\mathbf v}}_\ast}^n, \qquad {{{\mathbf e}}_{{\mathbf x}}}^n = {{\mathbf x}}^n - {{{\mathbf x}}_\ast}^n .\end{aligned}$$
Error equations
---------------
The nodal vectors of the exact solution satisfy the equations of the linearly implicit BDF method only up to defects ${{{\mathbf d}}_{{\mathbf v}}}^n$ and ${{{\mathbf d}}_{{\mathbf x}}}^n$ that, for $n \geq p$, are defined by the equations
$$\label{eq: BDF for exact sol}
\begin{aligned}
{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) {{{\mathbf v}}_\ast}^n + {\beta {{\mathbf A}}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) {{{\mathbf x}}_\ast}^n =&\ {{\mathbf g}}({\widetilde{{{\mathbf x}}}_{\ast}}^n,t_n) + {{\mathbf M}}({{{\mathbf x}}_\ast}^n){{{\mathbf d}}_{{\mathbf v}}}^n , \\
\frac{1}{\tau} \sum_{j=0}^p \delta_j {{{\mathbf x}}_\ast}^{n-j} =&\ {{{\mathbf v}}_\ast}^n + {{{\mathbf d}}_{{\mathbf x}}}^n .
\end{aligned}$$
We subtract from to obtain the error equations $$\label{eq: error equations for BDF}
\begin{aligned}
{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) {{{\mathbf e}}_{{\mathbf v}}}^n + &\ {\beta {{\mathbf A}}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) {{{\mathbf e}}_{{\mathbf x}}}^n \\
=&\ - \big({{\mathbf K}}(\widetilde {{\mathbf x}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf e}}_{{\mathbf v}}}^n
- \big({{\mathbf K}}(\widetilde {{\mathbf x}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf v}}_\ast}^n \\
&\ - \beta \big({{\mathbf A}}(\widetilde {{\mathbf x}}^n) - {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf e}}_{{\mathbf x}}}^n
- \beta \big({{\mathbf A}}(\widetilde {{\mathbf x}}^n) - {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf x}}_\ast}^n \\
&\ + {{\mathbf g}}(\widetilde {{\mathbf x}}^n,t_n) - {{\mathbf g}}({\widetilde{{{\mathbf x}}}_{\ast}}^n,t_n) - {{\mathbf M}}({\widetilde{{{\mathbf x}}}_{\ast}}^n){{{\mathbf d}}_{{\mathbf v}}}^n , \\
\frac{1}{\tau} \sum_{j=0}^p \delta_j {{{\mathbf e}}_{{\mathbf x}}}^{n-j} &=\ {{{\mathbf e}}_{{\mathbf v}}}^n - {{{\mathbf d}}_{{\mathbf x}}}^n .
\end{aligned}$$
Stability bound
---------------
We recall that the matrix ${{\mathbf K}}({{{\mathbf x}}_\ast})$ defines a norm which is equivalent to the $H^1$ norm on ${\Gamma}_h[{{{\mathbf x}}_\ast}]$. The defect ${{\mathbf d}}_{{\mathbf v}}\in {\mathbb{R}}^{3N}$ will be measured in the dual norm defined by $$\|{{\mathbf d}}\|_{\star,{{{\mathbf x}}_\ast}}^2 := {{\mathbf d}}^T {{\mathbf M}}({{{\mathbf x}}_\ast}){{\mathbf K}}({{{\mathbf x}}_\ast}){^{-1}}{{\mathbf M}}({{{\mathbf x}}_\ast}){{\mathbf d}},$$ which is such that for the finite element function $d_h\in S_h[{{\mathbf x}}^*]^3$ with nodal vector ${{\mathbf d}}$ we have, from [@LubichMansourVenkataraman_bdsurf Proof of Theorem 5.1] or [@soldriven Formula (5.5)], $$\label{dual-norm-h}
\|{{\mathbf d}}\|_{\star,{{{\mathbf x}}_\ast}} = \|d_h\|_{H_h{^{-1}}(\Gamma_h[{{\mathbf x}}^*])} := \sup_{0\neq \psi_h \in S_h[{{\mathbf x}}^*]^3} \frac{\int_{\Gamma_h[{{\mathbf x}}^*]} d_h \cdot \psi_h}{\|\psi_h\|_{H^1(\Gamma_h[{{\mathbf x}}^*])^3}}.$$ In these norms we have the following stability result.
\[propostion: stability - regularised velocity law\] Suppose that the defects of the $p$-step linearly implicit BDF method are bounded as follows, with a sufficiently small $\vartheta > 0$ (that is independent of $h$ and $\tau$ and $n$): for $n\ge p$ with $n\tau \le T$, $$\label{eq: assume small defects}
\|{{{\mathbf d}}_{{\mathbf x}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^k)} \leq \vartheta h
{\quad\hbox{ and }\quad}\|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^k} \leq \vartheta h \quad \textnormal{ for } \, k \tau \leq T .$$ Further, assume that the initial values are chosen such that $$\label{eq: small initial errors}
\|{{{\mathbf e}}_{{\mathbf x}}}^k\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^k)} \leq \vartheta h
{\quad\hbox{ and }\quad}\|{{{\mathbf e}}_{{\mathbf v}}}^k\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^k)} \leq \vartheta h \quad \textnormal{for } \, k=0,\dotsc,p-1.$$ Then, the following error bounds hold, for $n\ge p$ such that $ n \tau \leq T$, $$\label{eq: error estimate}
\begin{aligned}
\|{{{\mathbf e}}_{{\mathbf x}}}^{n}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^{n})}^2
\leq &\ C \tau \sum_{j=p}^{n} \Big( \|{{{\mathbf d}}_{{\mathbf x}}}^j\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^j)}^2 + \|{{{\mathbf d}}_{{\mathbf v}}}^j\|_{\star,{{{\mathbf x}}_\ast}^j}^2 \Big) + C\sum_{i=0}^{p-1} \|{{{\mathbf e}}_{{\mathbf x}}}^i\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^i)}^2 , \\
\|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2
\leq &\ C \tau \sum_{j=p}^n \Big( \|{{{\mathbf d}}_{{\mathbf x}}}^j\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^j)}^2 + \|{{{\mathbf d}}_{{\mathbf v}}}^j\|_{\star,{{{\mathbf x}}_\ast}^j}^2 \Big) + C \|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n}^2 + C\sum_{i=0}^{p-1} \|{{{\mathbf e}}_{{\mathbf x}}}^i\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^i)}^2,
\end{aligned}$$ where $C$ is independent of $h$, $\tau$ and $n$ with $n\tau\le T$, but depends on $T$.
In Section \[section: consistency\] we will show that the defects obtained on inserting the exact solution values into the BDF scheme satisfy the bounds $$\|{{{\mathbf d}}_{{\mathbf x}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)} \le C(h^k+\tau^p),\quad\
\|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n} \le C(h^k+\tau^p).$$ Hence, condition is satisfied under the mild stepsize restriction $$\label{stepsize-restriction}
\tau^p \le c_0 h$$ for a sufficiently small $c_0$ that is independent of $h$ and $\tau$. We note that the error functions $e_x^n,e_v^n \in S_h[{{{\mathbf x}}_\ast}^n]^3$ with nodal vectors ${{{\mathbf e}}_{{\mathbf x}}}^{n}$ and ${{{\mathbf e}}_{{\mathbf v}}}^{n}$, respectively, are then bounded by $$\begin{aligned}
&\| e_x^n \|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \leq C(h^k+\tau^p), \\
&\| e_v^n \|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \leq C(h^k+\tau^p) ,
\end{aligned}
\qquad \textnormal{ for } n \tau \leq T.$$
The proof is based on energy estimates for the matrix–vector formulation of the error equations and relies on the results of Section \[section: aux\]. In the proof, $c$ will be a generic constant independent of $h$ and $\tau$ and $n$ with $n\tau\le T$, which assumes different values on different occurrences. For many estimates we use similar techniques of proof as for the corresponding time-continuous results in [@soldriven]. However, to keep the paper fairly self-contained we include some detailed arguments.
In view of the condition in (iii) of Lemma \[lemma: technicals\] for ${{\mathbf y}}={\widetilde{{{\mathbf x}}}_{\ast}}^n$ and ${{\mathbf x}}={\widetilde{{\mathbf x}}}^n$, we need to control the $W^{1,\infty}$ norm of the position error $\widetilde{e}_x^n$. Let us assume that the error estimate holds for $ p, \dotsc, n-1$. Then, using an inverse inequality and the norm equivalence and the definition of ${\widetilde{{{\mathbf e}}}_{{{\mathbf x}}}}^n$ (cf. ), we obtain $$\label{eq: assumed bounds}
\begin{aligned}
\|\nabla_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \widetilde{e}_x^n\|_{L^\infty({\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}])}
\leq &\ c h^{-1} \|\nabla_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \widetilde{e}_x^n\|_{L^2({\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}])} \\
\leq &\ c h^{-1} \| {\widetilde{{{\mathbf e}}}_{{{\mathbf x}}}}^n \|_{{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^{n})} \leq c h^{-1} \| {\widetilde{{{\mathbf e}}}_{{{\mathbf x}}}}^n \|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^{n})} \\
\leq &\ c h^{-1} \sum_{j=1}^p \| {{{\mathbf e}}_{{\mathbf x}}}^{n-j} \|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^{n})} \\
\leq &\ c h^{-1} \cdot c\vartheta h \le c\vartheta,
\end{aligned}$$ where the last but one estimate follows from for the past, and the assumption on small defects . For sufficiently small $\vartheta$, we are thus in the position to use the bounds given in Lemma \[lemma: technicals\]. We estimate the two error equations separately, and then combine them to yield the final estimate.
\(a) *Estimates for the velocity law.* By testing the first line of the error equations with ${{{\mathbf e}}_{{\mathbf v}}}^n$ we obtain $$\begin{aligned}
&\ \half \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 \leq \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n)}^2 \\
=&\ - ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf K}}(\widetilde {{\mathbf x}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf v}}_\ast}^n
- ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf K}}(\widetilde {{\mathbf x}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf e}}_{{\mathbf v}}}^n \\
&\ - \beta ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf A}}(\widetilde {{\mathbf x}}^n) - {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf x}}_\ast}^n
- \beta ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf A}}(\widetilde {{\mathbf x}}^n) - {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf e}}_{{\mathbf x}}}^n \\
&\ + ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf g}}(\widetilde {{\mathbf x}}^n,t_n) - {{\mathbf g}}({\widetilde{{{\mathbf x}}}_{\ast}}^n,t_n)\big) - \beta ({{{\mathbf e}}_{{\mathbf v}}}^n)^T {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) {{{\mathbf e}}_{{\mathbf x}}}^n - ({{{\mathbf e}}_{{\mathbf v}}}^n)^T {{\mathbf M}}({\widetilde{{{\mathbf x}}}_{\ast}}^n){{{\mathbf d}}_{{\mathbf v}}}^n ,\end{aligned}$$ where the inequality follows from . To bound the right-hand side, we use arguments of the proof of Proposition 10.1 (and that of Proposition 5.1) of [@soldriven], using the results of Lemma \[lemma: technicals\].
\(i) For $0 \leq \theta \leq 1$, we denote $\Gamma_h^{n,\theta} = \Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n + \theta{\widetilde{{{\mathbf e}}}_{{{\mathbf x}}}}^n]$, where ${\widetilde{{{\mathbf e}}}_{{{\mathbf x}}}}^n = {\widetilde{{\mathbf x}}}^n - {\widetilde{{{\mathbf x}}}_{\ast}}^n = \sum_{j=0}^{p-1} \gamma_j {{{\mathbf e}}_{{\mathbf x}}}^{n-p+j}$. We denote the finite element functions in $S_h(\Gamma_h^{n,\theta})^3$ with nodal vectors ${\widetilde{{{\mathbf e}}}_{{{\mathbf x}}}}^n$, ${{{\mathbf e}}_{{\mathbf v}}}^n$ and ${{{\mathbf v}}_\ast}^n$ by ${\widetilde{e}_{x}}^{n,\theta}$,$e_v^{n,\theta}$ and $v_{*}^{n,\theta}$, respectively. The definition and Lemma \[lemma: technicals\] then give us $$\begin{aligned}
&\ ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf K}}({\widetilde{{\mathbf x}}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf v}}_\ast}^n
= \ \int_0^1 \!\! \int_{\Gamma_h^{n,\theta}} \!\!\! e_v^{n,\theta} \cdot \bigl( \nabla_{\Gamma_h^{n,\theta}}\cdot {\widetilde{e}_{x}}^{n,\theta} \bigr) v_{*}^{n,\theta} \, {\hbox{\rm d}}\theta
\\ &\qquad\qquad \qquad
+ \alpha \int_0^1 \!\! \int_{\Gamma_h^{n,\theta}} \!\!\! \nabla_{\Gamma_h^{n,\theta}} e_v^{n,\theta} \cdot \bigl( D_{\Gamma_h^{n,\theta}} {\widetilde{e}_{x}}^{n,\theta} \bigr) \nabla_{\Gamma_h^{n,\theta}} v_{*}^{n,\theta} \, {\hbox{\rm d}}\theta .
$$ Using the Cauchy–Schwarz inequality, we estimate the integral with the product of the $L^2-L^2-L^\infty$ norms of the three factors. We thus have $$\begin{aligned}
&({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf K}}({\widetilde{{\mathbf x}}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf v}}_\ast}^n \\
&\leq \ \int_0^1 \| e_v^{n,\theta} \|_{L^2(\Gamma_h^{n,\theta})} \,
\| \nabla_{\Gamma_h^{n,\theta}}\cdot {\widetilde{e}_{x}}^{n,\theta} \|_{L^2(\Gamma_h^{n,\theta})} \,
\| v_{*}^{n,\theta} \|_{L^{\infty}(\Gamma_h^{n,\theta})}\, {\hbox{\rm d}}\theta \\
&\ \ \ + \alpha \int_0^1 \| \nabla_{\Gamma_h^{n,\theta}} e_v^{n,\theta} \|_{L^2(\Gamma_h^{n,\theta}) }\, \| D_{\Gamma_h^{n,\theta}} {\widetilde{e}_{x}}^{n,\theta} \|_{L^2(\Gamma_h^{n,\theta})} \,
\| \nabla_{\Gamma_h^{n,\theta}} v_{*}^{n,\theta} \|_{L^{\infty}(\Gamma_h^{n,\theta})}\, {\hbox{\rm d}}\theta
\\
&\leq \ c \int_0^1 \| e_v^{n,\theta} \|_{H^1(\Gamma_h^{n,\theta})} \,
\| {\widetilde{e}_{x}}^{n,\theta} \|_{H^1(\Gamma_h^{n,\theta})} \,
\| v_{*}^{n,\theta} \|_{W^{1,\infty}(\Gamma_h^{n,\theta})}\, {\hbox{\rm d}}\theta.\end{aligned}$$ By and Lemma \[lemma:theta-independence\], this is bounded by $$\begin{aligned}
&({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf K}}({\widetilde{{\mathbf x}}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf v}}_\ast}^n \\
&\leq \ c \| e_v^n \|_{H^1(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])} \,
\| {\widetilde{e}_{x}}^{n} \|_{H^1(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])} \,
\| v_*^n \|_{W^{1,\infty}(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])} ,\end{aligned}$$ where the last factor is bounded independently of $h$ and $\tau$. By Young’s inequality, we thus obtain $$\begin{aligned}
({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf K}}({\widetilde{{\mathbf x}}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf v}}_\ast}^n
\leq & \ \tfrac{1}{48}\| e_v^n \|_{H^1(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])}^2
+ c \sum_{j=1}^{p} \| e_x^{n-j} \|_{H^1(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])}^2 \\
= &\ \tfrac{1}{48} \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n)}^2
+ c \sum_{j=1}^{p} \|{{{\mathbf e}}_{{\mathbf x}}}^{n-j}\|_{{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n)}^2 \\
\leq &\ \tfrac{1}{24} \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2
+ c \sum_{j=1}^{p} \|{{{\mathbf e}}_{{\mathbf x}}}^{n-j}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 ,\end{aligned}$$ where the last inequality follows from the norm equivalence .
\(ii) Similarly, estimating the three factors in the integrals by $L^2-L^\infty-L^2$, we obtain $$\begin{aligned}
({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf K}}(\widetilde {{\mathbf x}}^n) - {{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf e}}_{{\mathbf v}}}^n
\leq &\ c \| e_v^n \|_{L^2(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])}^2 \,
\| {\nabla_{\Gamma_h}}\cdot {\widetilde{e}_{x}}^{n} \|_{L^{\infty}(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])} \\
&\ + c \| {\nabla_{\Gamma_h}}e_v^n \|_{L^2(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])}^2 \,
\| D_{{\Gamma}_h}{\widetilde{e}_{x}}^{n} \|_{L^{\infty}(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])} \\
\leq &\ c\vartheta \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 \leq \tfrac{1}{24} \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2,\end{aligned}$$ where we used the estimate in the last but one inequality.
(iii)–(iv) The estimates involving the mean curvature term $\beta {{\mathbf A}}$ (in view of ) can be shown analogously as (i) and (ii): $$\begin{aligned}
& ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf A}}(\widetilde {{\mathbf x}}^n) - {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf x}}_\ast}^n
+ ({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf A}}(\widetilde {{\mathbf x}}^n) - {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) \big) {{{\mathbf e}}_{{\mathbf x}}}^n
\\
&\leq \tfrac{1}{24} \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2
+ c \|{{{\mathbf e}}_{{\mathbf x}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2
+ c \sum_{j=1}^{p} \|{{{\mathbf e}}_{{\mathbf x}}}^{n-j}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 , \\
& ({{{\mathbf e}}_{{\mathbf v}}}^n)^T {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) {{{\mathbf e}}_{{\mathbf x}}}^n \leq \tfrac{1}{24} \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 + c \|{{{\mathbf e}}_{{\mathbf x}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 .\end{aligned}$$
\(v) Similarly as in (i) we rewrite $$\begin{aligned}
({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf g}}(\widetilde {{\mathbf x}}^n,t_n) - {{\mathbf g}}({\widetilde{{{\mathbf x}}}_{\ast}}^n,t_n)\big)
=&\ \int_{{\Gamma}_h^{1,n}} g^n \nu_{\Gamma_h^{1,n}}\cdot e_v^{1,n}
- \int_{{\Gamma}_h^{0,n}} g^n \nu_{\Gamma_h^{0,n}}\cdot e_v^{0,n} \\
=&\ \int_0^1 \frac{{\hbox{\rm d}}}{{\hbox{\rm d}}\theta} \int_{\Gamma_h^{n,\theta}} g^n \nu_{\Gamma_h^{n,\theta}}\cdot e_v^{n,\theta} {\hbox{\rm d}}\theta .\end{aligned}$$ We use the Leibniz formula and ${\partial^{\bullet}}_\theta e_v^{0,n} = 0$ just as in (iii) of the proof of [@soldriven Proposition 5.1], to finally obtain $$\begin{aligned}
({{{\mathbf e}}_{{\mathbf v}}}^n)^T \big({{\mathbf g}}(\widetilde {{\mathbf x}}^n,t_n) - {{\mathbf g}}({\widetilde{{{\mathbf x}}}_{\ast}}^n,t_n)\big)
\leq &\ c \| e_v^n \|_{L^2(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])} \,
\| {\widetilde{e}_{x}}^{n} \|_{H^1(\Gamma_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n])} \\
\leq &\ c \| {{{\mathbf e}}_{{\mathbf v}}}^n \|_{{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n)} \|{\widetilde{{{\mathbf e}}}_{{{\mathbf x}}}}^n\|_{{{\mathbf K}}({\widetilde{{{\mathbf x}}}_{\ast}}^n)}^2 \\
\leq &\ \tfrac{1}{24} \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2
+ c \sum_{j=1}^{p} \|{{{\mathbf e}}_{{\mathbf x}}}^{n-j}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 .\end{aligned}$$
\(vi) The term with the defect is estimated as $$\begin{aligned}
({{{\mathbf e}}_{{\mathbf v}}}^n)^T {{\mathbf M}}({{{\mathbf x}}_\ast}^n) {{{\mathbf d}}_{{\mathbf v}}}^n
= &\ ({{{\mathbf e}}_{{\mathbf v}}}^n)^T {{\mathbf K}}({{{\mathbf x}}_\ast}^n)^{1/2} {{\mathbf K}}({{{\mathbf x}}_\ast}^n)^{-1/2} {{\mathbf M}}({{{\mathbf x}}_\ast}^n) {{{\mathbf d}}_{{\mathbf v}}}^n \\
\leq &\ \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)} \|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n}
\leq \tfrac{1}{24} \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 + c \|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n}^2 .\end{aligned}$$
Finally, by combining all these estimates, using multiple absorptions, with sufficiently small $\vartheta$ we finally obtain $$\label{eq: final estimate for error in v}
\|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 \leq c \|{{{\mathbf e}}_{{\mathbf x}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 + c \sum_{j=1}^{p} \|{{{\mathbf e}}_{{\mathbf x}}}^{n-j}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}^2 + c \|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n}^2 .$$
\(b) *Estimates for ODE.* We rewrite the second equation of as $$\frac{1}{\tau} \sum_{j=p}^n \delta_{n-j} {{{\mathbf e}}_{{\mathbf x}}}^{j} =\ {{{\mathbf e}}_{{\mathbf v}}}^n - \widehat{{{\mathbf d}}_{{\mathbf x}}}^n,$$ with $\delta_j=0$ for $j>p$ and $$\widehat{{{\mathbf d}}_{{\mathbf x}}}^n = {{{\mathbf d}}_{{\mathbf x}}}^n + \frac{1}{\tau} \sum_{j=0}^{p-1} \delta_{n-j} {{{\mathbf e}}_{{\mathbf x}}}^{j},$$ where we note that $\widehat{{{\mathbf d}}_{{\mathbf x}}}^n = {{{\mathbf d}}_{{\mathbf x}}}^n$ for $n\ge2p$. With the coefficients of the power series $$\mu(\zeta) = \sum_{n=0}^\infty \mu_n \zeta^n = \frac 1{\delta(\zeta)}$$ we then have, for $n\ge p$, $${{{\mathbf e}}_{{\mathbf x}}}^n = \tau \sum_{j=p}^n \mu_{n-j} ({{{\mathbf e}}_{{\mathbf v}}}^j-\widehat{{{\mathbf d}}_{{\mathbf x}}}^j).$$ By the zero-stability of the BDF method of order $p\le 6$ (which states that all zeros of $\delta(\zeta)$ are outside the unit circle with the exception of the simple zero at $\zeta=1$), the coefficients $\mu_n$ are bounded: $|\mu_n| \le c$ for all $n$.
Taking the $K({{{\mathbf x}}_\ast}^n)$ norm on both sides and recalling that by Lemma \[lemma: matrix derivatives\] all these norms are uniformly equivalent for $0\le n\tau\le T$, we obtain with the Cauchy–Schwarz inequality $$\begin{aligned}
\| {{{\mathbf e}}_{{\mathbf x}}}^n \|_{K({{{\mathbf x}}_\ast}^n)}^2 &\le c \tau \sum_{j=p}^n \| {{{\mathbf e}}_{{\mathbf v}}}^j-\widehat{{{\mathbf d}}_{{\mathbf x}}}^j \|_{K({{{\mathbf x}}_\ast}^j)}^2 \\
&\le c \tau \sum_{j=p}^n \| {{{\mathbf e}}_{{\mathbf v}}}^j \|_{K({{{\mathbf x}}_\ast}^j)}^2 +c \tau \sum_{j=p}^n \| {{{\mathbf d}}_{{\mathbf x}}}^j \|_{K({{{\mathbf x}}_\ast}^j)}^2 + c \sum_{i=0}^{p-1} \| {{{\mathbf e}}_{{\mathbf x}}}^i\|_{K({{{\mathbf x}}_\ast}^i)}^2.\end{aligned}$$ Combining this inequality with and using a discrete Gronwall inequality then yields the result.
Consistency error {#section: consistency}
=================
In this section we show that the consistency errors, that is, the defects defined by and obtained by inserting the interpolated exact solution into the numerical method, are bounded in the required norms by $ C(h^k+\tau^p)$ for the finite element method of polynomial degree $k$ and the $p$-step BDF method.
Let us first recall the formula for the defect of the spatial semi-discretization $d_{h,v}(\cdot,t)$ from Section 8 of [@soldriven], for $\psi_h \in S_h[{{{\mathbf x}}_\ast}(t)]^3$: $$\begin{aligned}
&\int_{{\Gamma}_h[{{{\mathbf x}}_\ast}\t]} \!\!\! d_{h,v}(\cdot,t) \cdot \psi_h =\ \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}\t]}\!\! {\widetilde{I}_h}v(\cdot,t) \cdot \psi_h
+ \alpha \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}\t]} \!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}v(\cdot,t) \cdot {\nabla_{\Gamma_h}}\psi_h \\
&\hskip 2cm + \beta \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}\t]} \!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}X(\cdot,t) \cdot {\nabla_{\Gamma_h}}\psi_h - \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}\t]} \!\!\! g(\cdot,t) \,\nu_{{\Gamma}_h[{{{\mathbf x}}_\ast}\t]} \cdot \psi_h ,\end{aligned}$$ which satisfies the following bounds.
[[@soldriven Lemma 8.1]]{} Let the surface $X$ and its velocity $v$ be sufficiently smooth. Then there exists a constant $c>0$ (independent of $t$) such that for all $h\leq h_0$, with a sufficiently small $h_0>0$, and for all $t\in[0,T]$, the defects $d_{h,v}$ of the $k$th-degree finite element interpolation are bounded as $$\begin{aligned}
\|d_{h,v}(\cdot,t)\|_{H_h{^{-1}}({\Gamma}({X_h^\ast}))} \leq c h^k .
\end{aligned}$$
We will now bound the defect of the full discretization.
\[lemma: defect estimates for BDF\] Let the surface $X$ and its velocity $v$ be sufficiently smooth. Then there exist $h_0>0$ and $\tau_0>0$ such that for all $h\leq h_0$ and for all $\tau\le\tau_0$, the consistency errors are bounded as $$\begin{aligned}
&\|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n}=\|d_v^n\|_{H_h{^{-1}}({\Gamma}({X_h^\ast}(t_n)))} \, \leq c \big( \tau^p + h^k \big) , \\
&\|{{{\mathbf d}}_{{\mathbf x}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)}= \|d_x^n\|_{H^1({\Gamma}({X_h^\ast}(t_n)))} \leq c \tau^p ,
\end{aligned}$$ where $c$ is independent of $h$, $\tau$ and $n$ with $n\tau \leq T$.
For the defect in $v$, the corresponding finite element function $d_v^n \in S_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n]$ with nodal values ${{{\mathbf d}}_{{\mathbf v}}}^n$ satisfies the following: for all finite element functions $\bar\psi_h \in S_h[{{{\mathbf x}}_\ast}^n]$ and the corresponding $\psi_h\in S_h[{\widetilde{{{\mathbf x}}}_{\ast}}^n]$ with the same nodal values, $$\label{eq: defect def for v}
\begin{aligned}
\int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! d_v^n \cdot \bar\psi_h
=&\ \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h
+ \alpha \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}v(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h \\
&\ + \beta \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}X(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h
- \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! g(\cdot,t_n) \nu_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \cdot \psi_h ,
\end{aligned}$$ where ${\widetilde{I}_h}v(\cdot,t_n), {\widetilde{I}_h}X(\cdot,t_n)\in S_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]^3$ denote the finite element interpolation of $v(\cdot,t_n)$ and $X(\cdot,t_n)$, respectively, on ${\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]$. Let us first rewrite , by subtracting the weak form of the problem . For the first term on the right-hand side, by adding and subtracting, this yields $$\begin{aligned}
&\int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h - \int_{{\Gamma}(X(t_n))} \!\!\!\!\!\!\!\! v(\cdot,t_n) \cdot \psi_h^l \\
&=\ \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h - \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h \\
&\ \ \ + \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h - \int_{{\Gamma}(X(t_n))} \!\!\!\!\!\!\!\! v(\cdot,t_n) \cdot \psi_h^l .\end{aligned}$$ Note that the last pair is simply a spatial defect, therefore repeating the same process for all four terms, and using the spatial defect $d_{h,v}$ from Section 8 of [@soldriven], we obtain $$\begin{aligned}
&\int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! d_v^n \cdot \psi_h = \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h - \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h \\
&\ + \alpha \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}v(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h
- \alpha \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}v(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h \\
&\ + \beta \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}X(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h - \beta \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}X(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h\\
&\ - \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! g(\cdot,t_n) \nu_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \cdot \psi_h + \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! g(\cdot,t_n) \nu_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \cdot \psi_h \\
&\ + \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! d_{h,v}(\cdot,t_n) \cdot \psi_h.\end{aligned}$$ We estimate the defect $d_v^n$ pairwise, using similar tools as in part (a) of the proof of Proposition \[propostion: stability - regularised velocity law\] and recalling .
For the first pair, we use the setting of Lemma \[lemma: matrix identity for extrapolation\], and then a Cauchy–Schwarz inequality and an $L^2 - L^2 - L^\infty$ estimate yield $$\begin{aligned}
&\hspace{-3mm} \Big| \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h - \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! {\widetilde{I}_h}v(\cdot,t_n) \cdot \psi_h \Big| \\
=&\ \Big| \int_0^1 \int_{\Gamma_h^{n,\theta}} \psi_h^\theta (\nabla_{\Gamma_h^{n,\theta}} \cdot {\widetilde e}_h^{n,\theta} ) v_{\ast,h}^{n,\theta} {\hbox{\rm d}}\theta \Big| \\
\leq &\ \int_0^1 \|\psi_h^\theta\|_{L^2(\Gamma_h^{n,\theta})}
\|\nabla_{\Gamma_h^{n,\theta}} \cdot {\widetilde e}_h^{n,\theta} \|_{L^2(\Gamma_h^{n,\theta})}
\|v_{\ast,h}^{n,\theta}\|_{L^\infty(\Gamma_h^{n,\theta})} {\hbox{\rm d}}\theta \\
\leq &\ c \|\psi_h^0\|_{L^2({\Gamma}_h^{0,n})}
\| {\widetilde e}_h^{n,0} \|_{H^1({\Gamma}_h^{0,n})}
\| v_{\ast,h}^{n,0}\|_{L^{\infty}({\Gamma}_h^{0,n})} \\
\leq &\ c \|\psi_h\|_{L^2({\Gamma}_h[{{{\mathbf x}}_\ast}^n])}
\| {\widetilde e}_h^{n} \|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \\
&\ \cdot
\Big(\| v_{\ast}(\cdot,t_n)\|_{L^{\infty}({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} + \| v_{\ast,h}(\cdot,t_n)-v_{\ast}(\cdot,t_n)\|_{L^{\infty}({\Gamma}_h[{{{\mathbf x}}_\ast}^n])}\Big) \\
\leq &\ c \|\psi_h\|_{L^2({\Gamma}_h[{{{\mathbf x}}_\ast}^n])}
\| {\widetilde e}_h^{n} \|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} (1+ch^2)
\| v_{\ast}(\cdot,t_n)\|_{W^{1,\infty}({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \\
\leq &\ c \|{\widetilde{{{\mathbf x}}}_{\ast}}^n-{{{\mathbf x}}_\ast}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)} \|\psi_h\|_{L^2({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \\
\leq &\ c \tau^p \|\psi_h\|_{L^2({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} ,\end{aligned}$$ where we used a $W^{1,\infty}$ interpolation estimate from [@Demlow2009 Proposition 2.7], and the last inequality follows from .
The other three pairs are again estimated similarly as above, and we finally obtain the bounds $$\begin{aligned}
\Big| \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}v(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h
- &\ \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}v(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h \Big| \\
\leq &\ c \tau^p \|\psi_h\|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])}\\[2mm]
\Big| \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}X(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h - &\ \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! {\nabla_{\Gamma_h}}{\widetilde{I}_h}X(\cdot,t_n) \cdot {\nabla_{\Gamma_h}}\psi_h \Big| \\
\leq &\ c \tau^p \|\psi_h\|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])}\\[2mm]
\Big| \int_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \!\!\!\! g(\cdot,t_n) \nu_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \cdot \psi_h - &\ \int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! g(\cdot,t_n) \nu_{{\Gamma}_h[{\widetilde{{{\mathbf x}}}_{\ast}}^{n}]} \cdot \psi_h \Big| \\
\leq &\ c \tau^p \|\psi_h\|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} .\end{aligned}$$ Furthermore, as shown in Lemma 8.1 of [@soldriven], the spatial defect $d_{h,v}(\cdot,t_n)$ is bounded by $$\int_{{\Gamma}_h[{{{\mathbf x}}_\ast}^n]} \!\!\!\! d_{h,v}(\cdot,t_n) \cdot \psi_h \leq c h^k \|\psi_h\|_{H^1({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} .$$ Combining the above estimates, we obtain the bound $\|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n} \leq c \big( \tau^p + h^k \big)$. The defect in $X$ is given by $${{{\mathbf d}}_{{\mathbf x}}}^n = \frac{1}{\tau} \sum_{j=0}^p \delta_j {{{\mathbf x}}_\ast}(t_{n-j}) - {\dot{{\mathbf x}}_\ast}(t_n)$$ and is solely due to temporal discretization. The bound $\|{{{\mathbf d}}_{{\mathbf x}}}^n\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^n)} \leq c \tau^p $ then follows by Taylor expansion.
Proof of Theorem \[theorem: main\] {#section: proof completed}
==================================
The errors are decomposed using interpolations and the definition of lifts from Section \[section:lifts\]. We denote by $\widehat I_h v\in S_h[{{{\mathbf x}}_\ast}]$ the finite element interpolation of $v$ on the interpolated surface $\Gamma_h[{{{\mathbf x}}_\ast}]$ and by $I_hv =(\widehat I_h v)^l$ its lift to the exact surface $\Gamma(X)$. We write $$\begin{aligned}
(v_h^n)^{L} - v(\cdot,t_n) =&\ \big(\widehat v_h^n - \widehat I_h v(\cdot,t_n) \big)^{l} + \big(I_h v(\cdot,t_n) - v(\cdot,t_n) \big) , \\
(X_h^n)^L - X(\cdot,t_n) =&\ \big( \widehat X_h^n - \widehat I_h X(\cdot,t_n) \big)^{l} + \big(I_h X(\cdot,t_n) - X(\cdot,t_n) \big).\end{aligned}$$ The last terms in these formulas can be bounded in the $H^1(\Gamma)$ norm by $Ch^k$, using the interpolation bounds of [@highorder].
To bound the first terms on the right-hand sides, we first use the defect bounds of Lemma \[lemma: defect estimates for BDF\], which then, under the mild stepsize restriction, together with the stability estimate of Proposition \[propostion: stability - regularised velocity law\] proves the result, since by the norm equivalences from Lemma \[lemma: technicals\] and equations – we have $$\begin{aligned}
\| \big(\widehat v_h^n - \widehat I_h v(\cdot,t_n)\big)^{l} \|_{L^2({\Gamma}(\cdot,t_n))} \leq &\
c \| \widehat v_h^n - \widehat I_h v(\cdot,t_n) \|_{L^2({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \\
= &\ c \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)} ,
\\
\| \nabla_{\Gamma} \big(\widehat v_h^n - \widehat I_h v(\cdot,t_n)\big)^{l} \|_{L^2({\Gamma}_h[{{{\mathbf x}}_\ast}^n]))} \leq &\
c \| \nabla_{\Gamma_h^*} \big(\widehat v_h^n - \widehat I_h v(\cdot,t_n) \big) \|_{L^2({\Gamma}_h[{{{\mathbf x}}_\ast}^n])} \\
= &\ c \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)},\end{aligned}$$ and similarly for $\widehat X_h^n - \widehat I_h X(\cdot,t_n)$.
A dynamic velocity law {#section: dynamic}
======================
Weak formulation and ESFEM / BDF full discretization
----------------------------------------------------
We now consider the dynamic velocity law , viz., $${\partial^{\bullet}}v + v {\nabla}_{{\Gamma}(X)} \cdot v - \alpha {\Delta}_{{\Gamma}(X)} v = g(\cdot,t) \, \nu_{{\Gamma}(X)},$$ where again $g:{\mathbb{R}}^3\times{\mathbb{R}}\to{\mathbb{R}}$ is a given smooth function of $(x,t)$, and $\alpha>0$ is a fixed parameter. This problem is considered together with the ordinary differential equation for the positions $X$ determining the surface $\Gamma(X)$. Initial values are specified for $X$ and $v$.
The weak formulation of the dynamic velocity law reads as follows: Find $v(\cdot,t) \in W^{1,\infty}({\Gamma}(X(\cdot,t)))^3 $ such that for all test functions $\psi(\cdot,t) \in H^1({\Gamma}(X(\cdot,t) ))^3$ with vanishing material derivative, $$\label{weak form - dynamic}
\begin{aligned}
\frac{{\hbox{\rm d}}}{{\hbox{\rm d}}t} \int_{{\Gamma}(X)} \!\!v \cdot \psi
&\ + \alpha \int_{{\Gamma}(X)} \!\! \nabla_{\Gamma(X)} v \cdot \nabla_{\Gamma(X)} \psi
= \int_{{\Gamma}(X)} \!\! g \,\nu_{{\Gamma}(X)} \cdot \psi ,
\end{aligned}$$ together with the ordinary differential equation for the positions $X$ determining the surface $\Gamma(X)$. The finite element space discretization is done in the usual way. We forego the straightforward formulation and immediately present the matrix–vector formulation of the semi-discretization. As in Section \[subsection:DAE\], the nodal vectors ${{\mathbf v}}(t)\in{\mathbb{R}}^{3N}$ of the finite element function $v_h(\cdot,t)$, together with the surface nodal vector ${{\mathbf x}}(t)\in{\mathbb{R}}^{3N}$ satisfy a system of ordinary differential equations with matrices and driving term as in Section \[subsection:DAE\]: $$\label{eq: DAE form - dynamic}
\begin{aligned}
\operatorname{\frac{{\hbox{\rm d}}}{{\hbox{\rm d}}t}}\Big({{\mathbf M}}({{\mathbf x}}) {{\mathbf v}}\Big) + {{\mathbf A}}({{\mathbf x}}){{\mathbf v}}=&\ {{\mathbf g}}({{\mathbf x}},t) , \\
\dot{{\mathbf x}}=&\ {{\mathbf v}}.
\end{aligned}$$ We apply a $p$-step linearly implicit BDF method to the above ODE system with a step size $\tau >0 $: with $t_n= n \tau \leq T $ and with the extrapolated nodal vector ${\widetilde{{{\mathbf x}}}_{\ast}}^n$ defined by , the new nodal vectors of velocity and position, ${{\mathbf v}}^n$ and ${{\mathbf x}}^n$, respectively, are determined from the following system of linear equations: $$\label{eq: BDF for dynamic}
\begin{aligned}
\frac{1}{\tau} \sum_{j=0}^p &\,\delta_j {{\mathbf M}}({\widetilde{{\mathbf x}}}^{n-j}){{\mathbf v}}^{n-j} + \ {{\mathbf A}}({\widetilde{{\mathbf x}}}^n){{\mathbf v}}^n ={{\mathbf g}}({\widetilde{{\mathbf x}}}^n,t)\\
\frac{1}{\tau} \sum_{j=0}^p &\,\delta_j {{\mathbf x}}^{n-j} =\ {{\mathbf v}}^n .
\end{aligned}$$ As in Section \[section: problem\], the nodal vector ${{\mathbf x}}^n$ defines the discrete surface $\Gamma_h[{{\mathbf x}}^n]=\Gamma(X_h^n)$, which is to approximate the exact surface $\Gamma(X)$, and we obtain the position and velocity approximations .
Statement of the error bound
----------------------------
The following result is the analogue of Theorem \[theorem: main\] for the dynamic velocity law. We use the same notation for the lifted approximations.
\[theorem: main-dyn\] Consider the ESFEM / BDF linearly implicit full discretization of the dynamic velocity equation , using finite elements of polynomial degree $k\ge 2$ and BDF methods of order $p\le 5$. We assume quasi-uniform admissible triangulations of the initial surface and initial values chosen by finite element interpolation of the initial data for $X$. Suppose that the problem admits an exact solution $X,v$ that is sufficiently smooth (say, of class $C([0,T],H^{k+1})\cap C^{p+1}([0,T],W^{1,\infty})$) on the time interval $0\le t \le T$, and that the flow map $X(\cdot,t):\Gamma_0\to \Gamma(t)\subset{\mathbb{R}}^3$ is non-degenerate for $0\le t \le T$, so that $\Gamma(t)$ is a regular surface. Suppose further that the starting values are sufficiently accurate: for $i=0,\dots, p-1$, $$\| (X_h^i)^L - X(\cdot,i\tau) \|_{H^1({\Gamma}^0)^3} + \| (v_h^i)^L - v(\cdot,i\tau) \|_{H^1({\Gamma}^0)^3}\le C_0 (h^k+\tau^{p}).$$ Then, there exist $h_0 >0$, $\tau_0>0$ and $c_0>0$ such that for all mesh widths $h \leq h_0$ and step sizes $\tau\le\tau_0$ satisfying the mild stepsize restriction $
\tau^p \le c_0 h,
$ the following error bounds hold over the exact surface ${\Gamma}(t_n)={\Gamma}(X(\cdot,t_n))$ uniformly for $0\le t_n=n\tau \le T$: $$\begin{aligned}
\|(x_h^n)^{L} - \mathrm{id}_{\Gamma(t_n)}\|_{H^1({\Gamma}(t_n))^3} &\leq C(h^k+\tau^p),\\
\|(v_h^n)^{L} - v(\cdot,t_n)\|_{L^2({\Gamma}(t_n))^3} + \biggl( \sum_{j=p}^n \| (v_h^j)^{L} - v(\cdot,t_j)&\|_{H^1({\Gamma}(t_j))^3}^2 \biggr)^{1/2}\\
&\leq C(h^k+\tau^p).
\end{aligned}$$ The constant $C$ is independent of $h$ and $\tau$ and $n$ with $n\tau\le T$, but depends on bounds of higher derivatives of the solution $(X,v)$, and on the length $T$ of the time interval.
Auxiliary results by Dahlquist and Nevanlinna & Odeh
----------------------------------------------------
While the formulations of Theorems \[theorem: main\] and \[theorem: main-dyn\] are very similar, the proofs differ substantially in the stability analysis. In this subsection we recall two important results that combined permit us to use energy estimates for BDF methods up to order 5: the first result is from Dahlquist’s $G$-stability theory, and the second one from the multiplier technique of Nevanlinna and Odeh. These results have previously been used in the error analysis of BDF methods for various parabolic problems in [@AkrivisLubich_quasilinBDF; @AkrivisLiLubich_quasilinBDF; @KovacsPower_quasilinear; @LubichMansourVenkataraman_bdsurf].
\[lemma: Dahlquist\] Let $\delta(\zeta)=\sum_{j=1}^p\delta_j\zeta^j$ and $\mu(\zeta)=\sum_{j=1}^p\mu_j\zeta^j$ be polynomials of degree at most $p$ (at least one of them of degree $p$) that have no common divisor. Let ${\langle}\ \cdot , \cdot \ {\rangle}$ denote an inner product on ${\mathbb{R}}^N$. If $$\textnormal{Re} \frac{\delta(\zeta)}{\mu(\zeta)} > 0, \qquad \textrm{for} \quad |\zeta|<1,$$ then there exists a symmetric positive definite matrix $G = (g_{ij}) \in {\mathbb{R}}^{p\times p}$ such that for all ${{\mathbf w}}_0,\dotsc,{{\mathbf w}}_p\in{\mathbb{R}}^N$ $$\Big{\langle}\sum_{i=0}^p \delta_i {{\mathbf w}}_{p-i} , \sum_{i=0}^p \mu_i {{\mathbf w}}_{p-i} \Big{\rangle}\ge \sum_{i,j=1}^p g_{ij} {\langle}{{\mathbf w}}_i , {{\mathbf w}}_j {\rangle}- \sum_{i,j=1}^p g_{ij} {\langle}{{\mathbf w}}_{i-1} , {{\mathbf w}}_{j-1} {\rangle}.$$
In view of the following result, the choice $\mu(\zeta)=1-\eta\zeta$ together with the polynomial $\delta(\zeta)$ of the BDF methods will play an important role later on.
\[lemma: NevanlinnaOdeh multiplier\] If $p\leq5$, then there exists $0\leq\eta<1$ [such that]{} for $\delta(\zeta)=\sum_{\ell=1}^p \frac{1}{\ell}(1-\zeta)^\ell$, $$\textnormal{Re} \,\frac{\delta(\zeta)}{1-\eta\zeta} > 0, \qquad \textrm{for} \quad |\zeta|<1.$$ The smallest possible values of $\eta$ are found to be $\eta= 0, 0, 0.0836, 0.2878, 0.8160$ for $p=1,\dotsc,5$, respectively.
Error equations
---------------
By using the same notations as in the previous sections for the nodal vectors of the exact positions ${{{\mathbf x}}_\ast}^n\in{\mathbb{R}}^{3N}$ and of the exact velocity ${{{\mathbf v}}_\ast}^n\in{\mathbb{R}}^{3N}$, and for their defects ${{{\mathbf d}}_{{\mathbf v}}}^n$ and ${{{\mathbf d}}_{{\mathbf x}}}^n$, we obtain that they fulfil the following equations: $$\begin{aligned}
\frac{1}{\tau} \sum_{j=0}^p \delta_j {{\mathbf M}}({\widetilde{{{\mathbf x}}}_{\ast}}^{n-j}){{{\mathbf v}}_\ast}^{n-j} + \ {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n){{{\mathbf v}}_\ast}^n &={{\mathbf g}}({\widetilde{{\mathbf x}}}^n,t)+ {{\mathbf M}}({{{\mathbf x}}_\ast}^n) {{{\mathbf d}}_{{\mathbf v}}}^n,\\
\frac{1}{\tau} \sum_{j=0}^p \delta_j {{{\mathbf x}}_\ast}^{n-j} &={{{\mathbf v}}_\ast}^n + {{{\mathbf d}}_{{\mathbf x}}}^n .
\end{aligned}
$$ By subtracting the above equations from , we obtain the error equations for the surface nodes and velocity: $$\label{eq:exact solution for dyn velocity law}
\begin{aligned}
&\ \hspace{-.5cm} {{\mathbf M}}({{{\mathbf x}}_\ast}^n) \frac{1}{\tau} \sum_{j=0}^p \delta_j {{{\mathbf e}}_{{\mathbf v}}}^{n-j}
+ {{\mathbf A}}({{{\mathbf x}}_\ast}^n){{{\mathbf e}}_{{\mathbf v}}}^n \\
=&\ - \frac{1}{\tau}\sum_{j=1}^p \delta_j \big( {{\mathbf M}}({{{\mathbf x}}_\ast}^{n-j}) - {{\mathbf M}}({{{\mathbf x}}_\ast}^{n})\bigr) {{{\mathbf e}}_{{\mathbf v}}}^{n-j}
- \frac{1}{\tau} \sum_{j=0}^p \delta_j \big( {{\mathbf M}}({\widetilde{{{\mathbf x}}}_{\ast}}^{n-j}) - {{\mathbf M}}({{{\mathbf x}}_\ast}^{n-j})\bigr) {{{\mathbf e}}_{{\mathbf v}}}^{n-j}\\
&\ - \frac{1}{\tau} \sum_{j=0}^p \delta_j \big( {{\mathbf M}}({\widetilde{{\mathbf x}}}^{n-j}) - {{\mathbf M}}({\widetilde{{{\mathbf x}}}_{\ast}}^{n-j})\bigr)({{{\mathbf v}}_\ast}^{n-j} + {{{\mathbf e}}_{{\mathbf v}}}^{n-j})\\
&\ - \big({{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n) - {{\mathbf A}}({{{\mathbf x}}_\ast}^n)\big) {{{\mathbf e}}_{{\mathbf v}}}^n - \big({{\mathbf A}}(\widetilde{{\mathbf x}}^n) - {{\mathbf A}}({\widetilde{{{\mathbf x}}}_{\ast}}^n)\big) ({{{\mathbf v}}_\ast}^n+{{{\mathbf e}}_{{\mathbf v}}}^n) \\[1mm]
&\ + {{\mathbf g}}({\widetilde{{\mathbf x}}}^n,t_n) - {{\mathbf g}}({\widetilde{{{\mathbf x}}}_{\ast}}^n,t_n) - {{\mathbf M}}({{{\mathbf x}}_\ast}^n) {{{\mathbf d}}_{{\mathbf v}}}^n \\[2mm]
&\ \hspace{-.5cm} \frac{1}{\tau} \sum_{j=0}^p \delta_j {{{\mathbf e}}_{{\mathbf x}}}^{n-j} = {{{\mathbf e}}_{{\mathbf v}}}^n - {{{\mathbf d}}_{{\mathbf x}}}^n .
\end{aligned}$$
Stability {#stability}
---------
We then have the following stability result.
\[proposition: stability - dynamic velocity law\] Under the smallness assumptions of Proposition \[propostion: stability - regularised velocity law\] for the defects and the errors in the initial values, the following error bound holds for BDF methods of order $p\le 5$ for $n \tau \leq T$: $$\label{eq: error estimate - dynamic}
\begin{aligned}
& \|{{{\mathbf e}}_{{\mathbf x}}}^{n}\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^{n})}^2
+ \|{{{\mathbf e}}_{{\mathbf v}}}^n\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)}^2
+ \tau \sum_{j=p}^n \|{{{\mathbf e}}_{{\mathbf v}}}^j\|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^j)}^2
\\
&\leq C \tau \sum_{j=p}^n \Big( \|{{{\mathbf d}}_{{\mathbf x}}}^j\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^j)}^2 + \|{{{\mathbf d}}_{{\mathbf v}}}^j\|_{\star,{{{\mathbf x}}_\ast}^j}^2 \Big) + c \|{{{\mathbf d}}_{{\mathbf v}}}^n\|_{\star,{{{\mathbf x}}_\ast}^n}^2
\\
&\quad + C\sum_{i=0}^{p-1}\Big( \|{{{\mathbf e}}_{{\mathbf x}}}^i\|_{{{\mathbf K}}({{{\mathbf x}}_\ast}^i)}^2 + \|{{{\mathbf e}}_{{\mathbf v}}}^i\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^i)}^2 \Big).
\end{aligned}$$ The constant $C$ is independent of $h$, $\tau$ and $n$, but depends on $T$.
We test the first error equation in with ${{{\mathbf e}}_{{\mathbf v}}}^n-\eta {{{\mathbf e}}_{{\mathbf v}}}^{n-1}$ to obtain $$({{{\mathbf e}}_{{\mathbf v}}}^n-\eta {{{\mathbf e}}_{{\mathbf v}}}^{n-1})^T {{\mathbf M}}({{{\mathbf x}}_\ast}^n) \frac{1}{\tau} \sum_{j=0}^p \delta_j {{{\mathbf e}}_{{\mathbf v}}}^{n-j} + ({{{\mathbf e}}_{{\mathbf v}}}^n-\eta {{{\mathbf e}}_{{\mathbf v}}}^{n-1})^T{{\mathbf A}}({{{\mathbf x}}_\ast}^n){{{\mathbf e}}_{{\mathbf v}}}^n =\rho^n,$$ where the right-hand term $\rho^n$ can be estimated by the same arguments as in part (a) of the proof of Proposition \[propostion: stability - regularised velocity law\]. On the left-hand side we have a term containing the stiffness matrix ${{\mathbf A}}({{{\mathbf x}}_\ast}^n)$, which is estimated from below as follows using Lemmas \[lemma: matrix derivatives\] and \[lemma: matrix identity for extrapolation\]: $$\begin{aligned}
({{{\mathbf e}}_{{\mathbf v}}}^n-&\eta {{{\mathbf e}}_{{\mathbf v}}}^{n-1})^T{{\mathbf A}}({{{\mathbf x}}_\ast}^n){{{\mathbf e}}_{{\mathbf v}}}^n \ge \| {{{\mathbf e}}_{{\mathbf v}}}^n \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)}^2 - \eta \| {{{\mathbf e}}_{{\mathbf v}}}^{n-1} \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)} \| {{{\mathbf e}}_{{\mathbf v}}}^n \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)}
\\
&\ge \| {{{\mathbf e}}_{{\mathbf v}}}^n \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)}^2 - \eta (1+c\tau)\| {{{\mathbf e}}_{{\mathbf v}}}^{n-1} \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^{n-1})} \| {{{\mathbf e}}_{{\mathbf v}}}^n \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)}
\\
&\ge (1-\tfrac12\eta-c\tau) \| {{{\mathbf e}}_{{\mathbf v}}}^n \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)}^2 - (\tfrac12\eta+c\tau) \| {{{\mathbf e}}_{{\mathbf v}}}^{n-1} \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^{n-1})}^2.\end{aligned}$$ The other term on the left-hand side, which contains the mass matrix ${{\mathbf M}}({{{\mathbf x}}_\ast}^n)$, is estimated from below using Lemmas \[lemma: Dahlquist\] and \[lemma: NevanlinnaOdeh multiplier\]. Let us introduce $${{\mathbf E}}_{{{\mathbf v}}}^n = \big({{{\mathbf e}}_{{\mathbf v}}}^{n-p+1}, \dotsc, {{{\mathbf e}}_{{\mathbf v}}}^{n-1}, {{{\mathbf e}}_{{\mathbf v}}}^n \big)$$ and the norm $$|{{\mathbf E}}_{{{\mathbf v}}}^n|_{G,{{{\mathbf x}}_\ast}^n}^2 = \sum_{i,j=1}^p g_{ij} ({{{\mathbf e}}_{{\mathbf v}}}^{n-p+i})^T {{\mathbf M}}({{{\mathbf x}}_\ast}^n) {{{\mathbf e}}_{{\mathbf v}}}^{n-p+j} ,$$ which satisfies the norm equivalence relation $$\label{eq: G norm equivalence}
\lambda_{\min} \sum_{i=1}^p \|{{{\mathbf e}}_{{\mathbf v}}}^{n-p+i}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)}^2
\leq |{{\mathbf E}}_{{{\mathbf x}}}^n|_{G,{{{\mathbf x}}_\ast}^n}^2
\leq \lambda_{\max} \sum_{i=1}^p \|{{{\mathbf e}}_{{\mathbf v}}}^{n-p+i}\|_{{{\mathbf M}}({{{\mathbf x}}_\ast}^n)}^2 ,$$ where $\lambda_{\min}$ and $\lambda_{\max}$ are the smallest and largest eigenvalue of the symmetric positive definite matrix $G=(g_{ij})$ of Lemma \[lemma: Dahlquist\]. Hence we obtain from Lemmas \[lemma: Dahlquist\] and \[lemma: NevanlinnaOdeh multiplier\] $$({{{\mathbf e}}_{{\mathbf v}}}^n-\eta {{{\mathbf e}}_{{\mathbf v}}}^{n-1})^T {{\mathbf M}}({{{\mathbf x}}_\ast}^n) \sum_{j=0}^p \delta_j {{{\mathbf e}}_{{\mathbf v}}}^{n-j} \ge
|{{\mathbf E}}_{{{\mathbf v}}}^n|_{G,{{{\mathbf x}}_\ast}^n}^2 - |{{\mathbf E}}_{{{\mathbf v}}}^{n-1}|_{G,{{{\mathbf x}}_\ast}^{n}}^2\,,$$ where we note that by Lemma \[lemma: matrix derivatives\], $$|{{\mathbf E}}_{{{\mathbf v}}}^{n-1}|_{G,{{{\mathbf x}}_\ast}^{n}}^2 \le (1+c\tau)|{{\mathbf E}}_{{{\mathbf v}}}^{n-1}|_{G,{{{\mathbf x}}_\ast}^{n-1}}^2,$$ so that altogether we have $$\begin{aligned}
&|{{\mathbf E}}_{{{\mathbf v}}}^n|_{G,{{{\mathbf x}}_\ast}^n}^2 - (1+c\tau)|{{\mathbf E}}_{{{\mathbf v}}}^{n-1}|_{G,{{{\mathbf x}}_\ast}^{n-1}}^2 \\
&+ \tau(1-\tfrac12\eta-c\tau) \| {{{\mathbf e}}_{{\mathbf v}}}^n \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^n)}^2 - \tau (\tfrac12\eta+c\tau) \| {{{\mathbf e}}_{{\mathbf v}}}^{n-1} \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^{n-1})}^2 \le \tau\rho^n.\end{aligned}$$ Using these inequalities from 1 to $n$ yields for sufficiently small $\tau$, with a positive constant $\gamma$, $$|{{\mathbf E}}_{{{\mathbf v}}}^n|_{G,{{{\mathbf x}}_\ast}^n}^2 + \gamma\tau \sum_{j=0}^n e^{c(n-j)\tau} \| {{{\mathbf e}}_{{\mathbf v}}}^j \|_{{{\mathbf A}}({{{\mathbf x}}_\ast}^j)}^2
\le e^{cn\tau} |{{\mathbf E}}_{{{\mathbf v}}}^{0}|_{G,{{{\mathbf x}}_\ast}^{0}}^2 + \tau \sum_{j=0}^n e^{c(n-j)\tau} \rho^j.$$ Using this bound together with estimates for $\rho^j$ and ${{{\mathbf e}}_{{\mathbf x}}}^j$ obtained in the same way as in the proof of Proposition \[propostion: stability - regularised velocity law\] then yields the stated result.
Together with bounds for the consistency errors ${{{\mathbf d}}_{{\mathbf v}}}^n$ and ${{{\mathbf d}}_{{\mathbf x}}}^n$, which are proven in the same way as in Section \[section: consistency\], the stability bounds of Proposition \[proposition: stability - dynamic velocity law\] then yield the $O(h^k+\tau^p)$ error bounds of Theorem \[theorem: main-dyn\].
Coupling with diffusion on the surface {#section: coupled}
======================================
Let us now turn to the parabolic surface PDE coupled with the regularised velocity law. We consider the following coupled problem of an evolving surface driven by diffusion on the surface, for which the ESFEM semi-discretization was studied in [@soldriven]: $$\label{eq: coupled problem}
\begin{aligned}
{\partial^{\bullet}}u + u {\nabla}_{{\Gamma}(X)} \cdot v - {\Delta}_{{\Gamma}(X)} u =&\ f(u, \nabla_{{\Gamma}(X)} u), \\
v - \alpha {\Delta}_{{\Gamma}(X)} v + \beta H_{{\Gamma}(X)}\nu_{{\Gamma}(X)}=&\ g(u, {\nabla}_{{\Gamma}(X)} u ) \nu_{{\Gamma}(X)} \\
{\partial}_t X(q,t) =&\ v(X(q,t),t),
\end{aligned}$$ with $\alpha>0$ and $\beta\ge 0$. The weak formulation and the ESFEM spatial semi-discretization, also in its matrix–vector formulation, are given in Section 2 of [@soldriven]. The finally obtained coupled system of differential-algebraic equations for the vectors of nodal values ${{\mathbf u}}(t) \in {\mathbb{R}}^N$, ${{\mathbf v}}(t) \in {\mathbb{R}}^{3N}$, and ${{\mathbf x}}(t) \in {\mathbb{R}}^{3N}$ reads, with the matrices of Section \[subsection:DAE\]: $$\label{eq: DAE form - coupled}
\begin{aligned}
\operatorname{\frac{{\hbox{\rm d}}}{{\hbox{\rm d}}t}}\Big({{\mathbf M}}({{\mathbf x}}){{\mathbf u}}\Big) + {{\mathbf A}}({{\mathbf x}}){{\mathbf u}}=&\ {{\mathbf f}}({{\mathbf x}},{{\mathbf u}}), \\
{{\mathbf K}}({{\mathbf x}}) {{\mathbf v}}+\beta {{\mathbf A}}({{\mathbf x}}){{\mathbf x}}=&\ {{\mathbf g}}({{\mathbf x}},{{\mathbf u}}),\\
\dot {{\mathbf x}}=&\ {{\mathbf v}}.
\end{aligned}$$ The right-hand side vectors are defined slightly differently from Section \[subsection:DAE\]. They are given by $$\begin{aligned}
{{\mathbf f}}({{\mathbf x}},{{\mathbf u}})\vert_j &= \int_{{\Gamma}_h[{{\mathbf x}}]} f(u_h,{\nabla_{\Gamma_h}}u_h) \, \phi_j[{{\mathbf x}}],
\\
{{\mathbf g}}({{\mathbf x}},{{\mathbf u}})\vert_{3(j-1)+\ell} &= \int_{{\Gamma}_h[{{\mathbf x}}]} g(u_h,{\nabla_{\Gamma_h}}u_h) \,\bigl(\nu_{{\Gamma}_h[{{\mathbf x}}]}\bigr)_\ell \, \phi_j[{{\mathbf x}}],
\end{aligned}$$ for $j = 1, \dotsc, N,$ and $\ell=1,2,3$.
The linearly implicit BDF discretization then reads as follows: with the extrapolated position vectors $\widetilde {{\mathbf x}}^n$ defined by ,
$$\label{eq: BDF for coupled}
\begin{aligned}
\frac{1}{\tau} \sum_{j=0}^p \delta_j {{\mathbf M}}({\widetilde{{\mathbf x}}}^{n-j}){{\mathbf u}}^{n-j} + {{\mathbf A}}({\widetilde{{\mathbf x}}}^n){{\mathbf u}}^n &= {{\mathbf f}}({\widetilde{{\mathbf x}}}^n,{\widetilde{{\mathbf u}}}^n) , \\
{{\mathbf K}}({\widetilde{{\mathbf x}}}^n) {{\mathbf v}}^n +\beta {{\mathbf A}}({\widetilde{{\mathbf x}}}^n){{\mathbf x}}^n &= {{\mathbf g}}({\widetilde{{\mathbf x}}}^n,{\widetilde{{\mathbf u}}}^n), \\
\frac{1}{\tau} \sum_{j=0}^p \delta_j {{\mathbf x}}^{n-j} &= {{\mathbf v}}^n .
\end{aligned}$$
Full discretizations using BDF methods of parabolic PDEs on an evolving surface with a [*given*]{} velocity have been studied in [@LubichMansourVenkataraman_bdsurf]. The combination of the proofs of Lemma 4.1 and Theorem 5.1 of [@LubichMansourVenkataraman_bdsurf] with the error analysis of the ESFEM semi-discretization in [@soldriven] and with the proof of Theorem \[theorem: main\] in the present paper yields the following convergence theorem. We omit the details of the proof.
\[theorem: main-coupled\] Consider the ESFEM / BDF linearly implicit full discretization of the coupled surface-evolution equation , using finite elements of polynomial degree $k\ge 2$ and BDF methods of order $p\le 5$. We assume quasi-uniform admissible triangulations of the initial surface and initial values chosen by finite element interpolation of the initial data for $X$. Suppose that the problem admits an exact solution $u,X,v$ that is sufficiently smooth (say, of class $C([0,T],H^{k+1})\cap C^{p+1}([0,T],W^{1,\infty})$) on the time interval $0\le t \le T$, and that the flow map $X(\cdot,t):\Gamma_0\to \Gamma(t)\subset{\mathbb{R}}^3$ is non-degenerate for $0\le t \le T$, so that $\Gamma(t)$ is a regular surface. Suppose further that the starting values are sufficiently accurate. Then, there exist $h_0 >0$, $\tau_0>0$ and $c_0>0$ such that for all mesh widths $h \leq h_0$ and step sizes $\tau\le\tau_0$ satisfying the mild stepsize restriction $
\tau^p \le c_0 h,
$ the following error bounds hold over the exact surface ${\Gamma}(t_n)={\Gamma}(X(\cdot,t_n))$ uniformly for $0\le t_n=n\tau \le T$: $$\begin{aligned}
\|(u_h^n)^{L} - u(\cdot,t_n)\|_{L^2({\Gamma}(t_n))^3} + \biggl( \sum_{j=p}^n \| (u_h^j)^{L} - u(\cdot,t_j)&\|_{H^1({\Gamma}(t_j))^3}^2 \biggr)^{1/2}\\
&\leq C(h^k+\tau^p),\\
\|(v_h^n)^{L} - v(\cdot,t_n)\|_{L^2({\Gamma}(t_n))^3} + \biggl( \sum_{j=p}^n \| (v_h^j)^{L} - v(\cdot,t_j)&\|_{H^1({\Gamma}(t_j))^3}^2 \biggr)^{1/2}\\
&\leq C(h^k+\tau^p),\\[1mm]
\|(x_h^n)^{L} - \mathrm{id}_{\Gamma(t_n)}\|_{H^1({\Gamma}(t_n))^3} &\leq C(h^k+\tau^p).\\
\end{aligned}$$ The constant $C$ is independent of $h$ and $\tau$ and $n$ with $n\tau\le T$, but depends on bounds of higher derivatives of the solution $(u,v,X)$, and on the length $T$ of the time interval.
Numerical experiments {#section: numerics}
=====================
Forced mean curvature flow
--------------------------
We performed numerical experiments for the velocity law : for $x=X(q,t) \in \Gamma(t)$ with $q \in {\Gamma}_0$, $$\label{eq: numerics problem}
\begin{aligned}
v(x,t) - \alpha {\Delta}_{{\Gamma}(t)} v(x,t) =&\ -\beta H_{{\Gamma}(t)}(x)\, \nu_{{\Gamma}(t)}(x) +g\bigl(x,t\bigr)\, \nu_{{\Gamma}(t)}(x), \\
\partial_t X(q,t) =&\ v(X(q,t),t) ,
\end{aligned}$$ where the inhomogeneity $g:R^3 \times [0,T] \to {\mathbb{R}}$ is chosen such that the exact solution is $X(q,t)=r(t) q$, with $q$ on the unit sphere $\Gamma_0$. The function $r$ satisfies the logistic differential equation: $$\begin{aligned}
\dot r\t =&\ \Big(1 - \tfrac{r_1}{r\t}\Big) r\t, \qquad t \in [0,T], \\
r(0) =&\ r_0 ,\end{aligned}$$ with $r_1 \geq r_0 =1$, i.e. $r\t=r_0r_1 \big( r_0(1-e^{-t}) + r_1 e^{-t}\big){^{-1}}$.
Therefore, the velocity is simply given by, for $x\t = X(q,t)$, $$\begin{aligned}
v(x\t,t) = &\ \dot x\t = \dot r \t p = \Big(1 - \tfrac{r_1}{r\t}\Big) r\t p = \Big(1 - \tfrac{r_1}{r\t}\Big) x\t .\end{aligned}$$
The numerical experiments were performed in Matlab, using a quadratic approximation of the initial surface ${\Gamma}_0$ and using the quadratic ESFEM implementation from [@highorder], and linearly implicit BDF methods of various orders.
Let $(\mathcal{T}_k)_{k=1,2,\dotsc,m}$ and $(\tau_k)_{k=1,2,\dotsc,n}$ be a series of quadratic initial meshes and time steps, respectively, such that $2 \tau_k = \tau_{k-1}$, with $\tau_1=0.1$, where the meshes are generated independently. We computed the fully discrete numerical solution of the above problem, with parameters $\alpha=1$ and $\beta=1$, for each mesh and stepsize using the second order BDF method and second order ESFEM. In Figures \[fig: timeconv\] and \[fig: spaceconv\] we report on the following errors of the quadratic ESFEM / BDF2 full discretization $$\|(x_h^n)^{L} - \mathrm{id}_{\Gamma(t_n)}\|_{L^2({\Gamma}(t_n))^3} {\quad\hbox{ and }\quad}\|{\nabla_{\Gamma}}\big( (x_h^n)^{L} - \mathrm{id}_{\Gamma(t_n)} \big) \|_{L^2({\Gamma}(t_n))^3}$$ at time $T=N\tau=5$. The logarithmic plots show the errors against time step size $\tau$ (in Figure \[fig: timeconv\]), and against the mesh width $h$ (in Figure \[fig: spaceconv\]).
The different lines correspond to different mesh refinements and to different time step sizes in Figure \[fig: timeconv\] and Figure \[fig: spaceconv\], respectively. In both figures we can observe two regions: In Figure \[fig: timeconv\], a region where the temporal discretization error dominates, matching to the $O(\tau^2)$ order of convergence of our theoretical result, and a region, with small stepsizes, where the space discretization error dominates (the error curves are flattening out). In Figure \[fig: spaceconv\], the same description applies, but with reversed roles. First the space discretization error dominates, while for finer meshes the temporal error dominates. The convergence in time, see Figure \[fig: timeconv\], can be nicely observed in agreement with the theoretical results (note the reference line), whereas we observe better $L^2$ norm convergence rates ($O(h^3)$) for the space discretization, see Figure \[fig: spaceconv\], than shown in Theorem \[theorem: main\] for the $H^1$ norm (only $O(h^2)$). This phenomenon is due to the fact that in the defect estimates we use the interpolation instead of a Ritz projection (which is hard to define in this setting), therefore have a defect estimate of order two. However, the classical optimal $L^2$ norm convergence rates of $O(h^3)$ are nevertheless observed.
![Spatial convergence of the BDF2 / quadratic ESFEM discretization for the surface-evolution equation []{data-label="fig: spaceconv"}](BDF2_soldriven_timeconv_revised){width="\textwidth" height=".42\textheight"}
![Spatial convergence of the BDF2 / quadratic ESFEM discretization for the surface-evolution equation []{data-label="fig: spaceconv"}](BDF2_soldriven_spaceconv_revised){width="\textwidth" height=".42\textheight"}
Figure \[fig: BDF4\_timeconv\] shows the same errors for the BDF method of order 4. It is clearly seen that in this problem the BDF4 method gives much better accuracy than BDF2, at nearly the same computational cost.
![Temporal convergence of the BDF4 / quadratic ESFEM discretization for the surface-evolution equation []{data-label="fig: BDF4_timeconv"}](BDF4_soldriven_timeconv_revised){width="\textwidth" height=".42\textheight"}
Numerical experiments for a semi-linear parabolic PDE system coupled to a velocity law on a surface with less symmetry, illustrating the coupled problem of Theorem \[theorem: main-coupled\], are discussed in detail in our previous work [@soldriven], where linearly implicit BDF methods have also been used.
Mean curvature flow
-------------------
We also performed some numerical experiments, using mean curvature flow (MCF), to illustrate the effect of the elliptic regularisation. We again consider the problem , however without a forcing term, i.e. the following form of mean curvature flow: $$\begin{aligned}
v(x,t) - \alpha {\Delta}_{{\Gamma}(t)} v(x,t) =&\ - \beta H_{{\Gamma}(t)}(x)\, \nu_{{\Gamma}(t)}(x) , \\
\partial_t X(q,t) =&\ v(X(q,t),t) .
\end{aligned}$$ The initial surface is a rounded cube, the parameter $\beta$ is fixed to one. Figure \[fig:MCF compare\] shows the results of different numerical experiments (using quadratic finite elements and BDF method of order $4$) at times $t=0, 0.2, 0.4, 0.5$ from top to bottom, while the parameter $\alpha$ is set to $0.1, 0.01, 0.001$ and $0$, from left to right, respectively. We note that our convergence results apply only to the case of a fixed positive $\alpha$, but the numerical experiments show good behaviour also for $\alpha\to 0$.
![MCF with different values of $\alpha$ at different times[]{data-label="fig:MCF compare"}](MCF){width="\textwidth" height="\textwidth"}
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is supported by Deutsche Forschungsgemeinschaft, SFB 1173.
[KLLP17]{}
G. Akrivis and C. Lubich. Fully implicit, linearly implicit and implicit–explicit backward difference formulae for quasi-linear parabolic equations. , 131(4):713–735, 2015.
G. Akrivis, B. Li, and C. Lubich. Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. , 86(306):1527–1552, 2017.
J.W. Barrett, K. Deckelnick, and V. Styles. Numerical analysis for a system coupling curve evolution to reaction diffusion on the curve. , 55(2):1080–1100, 2017.
G. Dahlquist. –stability is equivalent to [A]{}–stability. , 18:384–401, 1978.
A. Demlow. Higher–order finite element methods and pointwise error estimates for elliptic problems on surfaces. , 47(2):805–807, 2009.
G. Dziuk. Finite elements for the [B]{}eltrami operator on arbitrary surfaces. , pages 142–155, 1988.
G. Dziuk and [C.M.]{} Elliott. Finite elements on evolving surfaces. , 27(2):262–292, 2007.
G. Dziuk and [C.M.]{} Elliott. Fully discrete evolving surface finite element method. , 50(5):2677–2694, 2012.
G. Dziuk and [C.M.]{} Elliott. –estimates for the evolving surface finite element method. , 82(281):1–24, 2013.
G. Dziuk, C. Lubich, and D.E. Mansour. unge–[K]{}utta time discretization of parabolic differential equations on evolving surfaces. , 32(2):394–416, 2012.
W. Gautschi. . Birkauser Boston, [F]{}irst edition, 1997.
E. Hairer and G. Wanner. . Springer, Berlin, [S]{}econd edition, 1996.
B. Kovács. High-order evolving surface finite element method for parabolic problems on evolving surfaces. , 38(1):430–459, 2018. doi:10.1093/imanum/drx013.
B. Kovács, B. Li, C. Lubich, and [C.A.]{} [Power Guerra]{}. Convergence of finite elements on an evolving surface driven by diffusion on the surface. , 137(3):643–689, 2017. doi:10.1007/s00211-017-0888-4.
B. Kovács and [C.A.]{} [Power Guerra]{}. Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces. , 32(4):1200–1231, 2016.
C. Lubich, D.E. Mansour, and C. Venkataraman. Backward difference time discretization of parabolic differential equations on evolving surfaces. , 33(4):1365–1385, 2013.
O. Nevanlinna and F. Odeh. Multiplier techniques for linear multistep methods. , 3(4):377–423, 1981.
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---
abstract: 'Consider a compact Riemannian manifold with boundary. If all maximally extended geodesics intersect the boundary at both ends, then to each geodesic $\gamma(t)$ we can form the triple $(\dot\gamma(0),\dot\gamma(T),T)$, consisting of the initial and final vectors of the segment as well as the length between them. The collection of all such triples comprises the lens data. In this paper, it is shown that in the category of analytic Riemannian manifolds, the lens data uniquely determine the metric up to isometry. There are no convexity assumptions on the boundary, and conjugate points are allowed, but with some restriction.'
address: 'Department of Mathematics, University of Washington, Seattle, WA 98195'
author:
- James Vargo
title: A proof of Lens Rigidity in the category of Analytic Metrics
---
An introduction including the result proved
===========================================
Let $(M,g)$ be a compact, Riemannian manifold with boundary ${\partial M}$, and let it be non-trapping. That means all geodesics, when maximally extended, terminate at the boundary at both their ends. Let $SM$ denote its sphere bundle. Then for any vector $v\in {\partial}SM$, the geodesic $\gamma_v$ originating at $v$ eventually leaves the manifold after some distance $T$. Let $\ell(v)$ denote the length of the geodesic, and let $\Sigma(v)=\dot \gamma_v(T)$ denote its terminal vector. $$\Sigma:{\partial}SM\rightarrow {\partial}SM$$ is called the scattering map. Together, $\Sigma$ and $\ell$ comprise the lens data of $(M,g)$.
The lens rigidity conjecture states that one may recover a Riemannian manifold up to isometry from its lens data $(\Sigma,\ell)$. To be more precise, suppose we have two non-trapping Riemannian manifolds $(M_i,g_i), i=1,2$ which share the same boundary. That is, ${\partial M}_1={\partial M}_2$ (henceforth both will be denoted ${\partial M}$). Then for any point $x\in{\partial M}$, there is a natural correspondence $$\Lambda_x:{\partial}S_xM_1{\rightarrow}{\partial}S_xM_2.$$ Indeed, a unit vector at the boundary of a Riemannian manifold is uniquely characterized by its inward normal component and the direction of its tangential projection. So we shall say that $v_2=\Lambda(v_1)$ if these two quantities agree for $v_1$ and $v_2$, respectively.
Let $(\Sigma_i,\ell_i)$ denote the lens data for the manifold $(M_i,g_i): i=1,2$. We shall say that the two manifolds have the same lens data if $\Lambda\circ\Sigma_1=\Sigma_2\circ\Lambda$ and $\ell_1=\ell_2\circ
\Lambda$.
\[cnj lrc\] If $(M_1,g_1)$ and $(M_2,g_2)$ are non-trapping with the same lens data, then the two manifolds are related by an isometry that fixes the points of the boundary ${\partial M}$. That is, there exists a diffeomorphism ${\varphi}:M_1\rightarrow M_2$ satisfying ${\varphi}|_{{\partial M}}=id$ and ${\varphi}^*g_2=g_1$.
The lens rigidity problem is a generalization of the boundary rigidity problem. In that problem, the initial data is taken to be the boundary distance function $$\rho_g:{\partial M}\times{\partial M}\rightarrow \mathbb{R}.$$ $\rho_g(x,y)$ is equal to the length of the shortest curve joining $x$ to $y$. Of course, metrics related by an isometry fixing the boundary will also yield the same boundary distance function. The boundary rigidity problem is whether this is the only obstruction to unique recovery of $g$ from $\rho_g$.
If a metric $g$ has the property that the only other metrics with the same boundary distance function are isometric to $g$, then $g$ is called boundary rigid. There are many examples of metrics that are not boundary rigid. Indeed, $\rho_g$ only records the lengths of the shortest paths, and it is not hard to construct metrics for which the shortest paths do not enter certain open subsets of the manifold. To circumvent this problem, the assumption of simplicity is usually made on the metric.
The Riemannian manifold $(M,g)$ is simple, if $\partial M$ is strictly convex with respect to $g$, and for any $x\in M$, the exponential map $\exp_x :\exp^{-1}_x(M) \to
M$ is a diffeomorphism.
A simple manifold has the property of being geodesically convex. That is, every pair of points is connected by a unique geodesic and that geodesic is length minimizing. Topologically, a simple manifold is a ball. Michel [@M] was the first to conjecture that simple Riemannian manifolds are boundary rigid. This has been proved recently in two dimensions [@PU]. It has also been proved for subdomains of Euclidean space [@Gr], for metrics close to the Euclidean [@BI], and symmetric spaces of negative curvature [@BCG]. In [@SU2], Stefanov and Uhlmann proved a local boundary rigidity result. If $g$ belongs to a certain generic set which includes real-analytic metrics, and $g'$ is sufficiently close to $g$, then $\rho_g=\rho_{g'}$ implies that $g$ and $g'$ be isometric. For other local results see [@CDS], [@E], [@LSU], [@SU1]. It is shown in [@SU3] that the lens rigidity problem is equivalent to the boundary rigidity problem if the manifold is simple.
If the manifold is not simple, the lens data carries more information than the boundary distance function. Indeed, it includes the lengths of all geodesics, so in the case that $g$ be non-trapping, these geodesics pass through every point of the manifold in every direction. However, if the manifold is trapping, there are examples in which the lens data is not sufficient to determine the metric, (see [@CK]). There are not many results on the lens rigidity problem, but the following are notable. If a manifold is lens rigid a finite quotient of it is also lens rigid [@C2]. In [@SU3], Stefanov and Uhlmann generalized their local result for simple metrics to obtain a local lens rigidity result. There are some assumptions on conjugate points and a topological assumption. Assuming these, if $g$ belongs to a certain generic set which includes real-analytic metrics, and $g'$ is another metric with the same lens data that, a-priori, is known to be sufficiently close to $g$, then $g'$ is isometric to $g$.
In this paper, the following statement is proved.
\[thm 1\] Let $(M_i,g_i), i=1,2$ be non-trapping analytic Riemannian manifolds with a common, analytic boundary ${\partial}M$. Further, assume that in each connected component of $S({\partial M}_1)$, there exists $(x_0,\xi_0)$ such that $x_0$ is not conjugate to any points of ${\partial M}$ that lie along the geodesic $\gamma_{x_0,\xi_0}$. Then if the two manifolds have the same lens data, there must exist an analytic diffeomorphism ${\varphi}:M_1\rightarrow M_2$ with ${\varphi}|_{{\partial M}}=id$ and ${\varphi}^*g_2=g_1$.
Note the slightly asymmetric nature of the second hypothesis. This property is used in the proof of Theorem 2 to guarantee the possibility of a certain construction on the lens data. Since $g_2$ has the same lens data, the same construction will work automatically, even though, a priori, there is no reason why the condition of the theorem should also be true for $g_2$.
Constructing an isometry on a band about the boundary
=====================================================
Let $(M,g)$ be a general compact Riemannian manifold with boundary. Let $\nu$ be the field of inward unit normal vectors at the boundary ${\partial M}$, and let $x_0 \in {\partial M}$ be a boundary point. Then there is a small neighborhood $N\subset {\partial M}$ of $x_0$ and a number $\epsilon >
0$ such that the mapping $$\exp_{\nu}:N\times [0,\epsilon){\rightarrow}M$$ given by $(x',x^n)\mapsto \exp_{x'}(x^n\nu)$ gives a local coordinate system. These are the boundary normal coordinates. Through them, the metric has the form $$ds^2=g_{\alpha\beta}dx'^{\alpha}dx'^{\beta}+(dx^n)^2,$$ where $\alpha,\beta$ are indices running over the tangential directions. Now let $\bar{M}$ be an open manifold slightly extending $M$ and extend $g$ smoothly to $\bar{M}$ (extend by analytic continuation in the case that $(M,g)$ is analytic). By choosing a smaller $\epsilon$ if necessary, we may now use our boundary normal coordinates as a coordinate system in $\bar{M}$ by allowing the coordinate $x^n$ to vary over the set $(-\epsilon,\epsilon)$.
By compactness, we may choose $\epsilon$ uniformly over the whole boundary. We may also select $\epsilon$ sufficiently small so that our boundary normal coordinates give a global diffeomorphism $$\exp_{\nu}:{\partial}M \times (-\epsilon,\epsilon)\rightarrow V,$$ where $V$ is a neighborhood of ${\partial}M$ in $\bar{M}$.
To show this, it is only necessary to prove the above mapping injective. Around each point of ${\partial}M$, let $N$ be a connected open neighborhood such that $\exp_{\nu}$ defined on $N\times
(-\epsilon,\epsilon)$ is injective. By compactness, ${\partial}M$ is covered by a finite number of such neighborhoods $N_1,\dots, N_m$. There exists a number $\delta>0$ such that for any two points $x,y\in {\partial M}$, $x,y$ must belong to a common neighborhood $N_i$ if $d(x,y)<\delta$. Take $\epsilon$ to be less than half of $\delta$.
Suppose there are points $x,y\in {\partial M}$ and numbers $s,t\in
(-\epsilon,\epsilon)$ such that $$\exp_{\nu}(x,s)=\exp_{\nu}(y,t).$$ Then by the triangle, inequality, $d(x,y)<s+t<\delta$ which shows that $x,y$ belong to a common neighborhood $N_i$. But on $N_i\times (-\epsilon,\epsilon)$, $\exp_{\nu}$ is injective. Hence $x=y,s=t$.
We define the manifold ${\tilde}M$ to be $M\cup U$, where $U$ is a collar defined by:
$$U=\{x:-\epsilon \leq x^n \leq 0\}.$$
Next, note that the set $V$ is a subset of ${\tilde}M$, and can be described as the set of points in ${\tilde}M$ whose distance from ${\partial M}$ is less than $\epsilon$:
$$V=\{x\in{\tilde}{M}:d(x,{\partial}M)<\epsilon\}.$$ See Figure \[UV\].
![${\tilde}{M}$[]{data-label="UV"}](AL5)
Theorem 1 relies principally on the following theorem proved by Stefanov and Uhlmann in [@SU3].
\[thm\_jet\] Let $(M,g)$ be a compact Riemannian manifold with boundary. Let $(x_0,\xi_0)\in S(\partial M)$ be such that the maximal geodesic $\gamma_{x_0,\xi_0}$ through it is of finite length, and assume that $x_0$ is not conjugate to any point in $\gamma_{x_0,\xi_0}\cap
{\partial M}$. Then the jet of $g$ at $x_0$ in boundary normal coordinates is uniquely determined by the lens data $(\Sigma,\ell)$.
\[cor bnc\] Assume $(M,g)$ is analytic with analytic boundary and that, in each connected component of $S({\partial M})$, there is at least one vector $(x_0,\xi_0)$ satisfying the conditions of the theorem. Then the lens data uniquely determine the metric $g$ in boundary normal coordinates.
As above we let $V$ denote the set of points $\{x\in{\tilde}{M}:d(x,{\partial M})\leq\epsilon\}$. Then by hypothesis, in each component of $V$ there is at least one point at which the jet of the metric is determined. Since the metric is analytic, it must be uniquely determined on all of $V$.
Now we apply this to our two Riemannian manifolds $(M_i,g_i)$, taking an $\epsilon$ sufficiently small to work for both. We obtain $\exp_{\nu_i}:{\partial M}\times[-\epsilon,\epsilon]{\rightarrow}V_i\subset {\tilde}{M_i}$. Using these coordinates, both metrics have the form $g_{\alpha
\beta}dx'^{\alpha}dx'^{\beta}+(dx^n)^2$. By the corollary, the functions $g_{\alpha\beta}$ coincide for the two metrics throughout the domains ${\partial M}\times[-\epsilon,\epsilon]$, which means that the mapping ${\varphi}_0:V_1{\rightarrow}V_2$ defined by ${\varphi}_0=\exp_{\nu_2}\circ\exp_{\nu_1}^{-1}$ is an isometry.
Note that ${\varphi}_0|_{{\partial M}}=id$, and ${\varphi}_{0*}(\nu_1)=\nu_2$. In particular, ${\varphi}_{0*}$ must preserve directions in $T{\partial M}$ and must preserve components in the normal direction. Thus ${\varphi}_{0*}|_{SM_1}=\Lambda$, the mapping that relates the lens data of our two manifolds.
Extension of the isometry to the entire manifold
================================================
The rest of this paper shall be concerned with extending ${\varphi}_0$ to an isometry ${\varphi}:{\tilde}{M}_1{\rightarrow}{\tilde}{M_2}$. If the extension exists, then it must be uniquely defined. Indeed, given a point $x_0\in U_1$ and a unit vector $\xi$ at $x_0$, we must require
$${\varphi}(\exp^{g_1}_{x_0}(t\xi))=\exp^{g_2}_{{\varphi}_0(x_0)}(t{\varphi}_{0*}\xi).$$ See Figure \[phi\].
![[]{data-label="phi"}](AL1 "fig:") ![[]{data-label="phi"}](AL2 "fig:")
By the non-trapping assumption, all points $x\in M_1$ lie on a geodesic originating in $U_1$. Therefore this equation uniquely determines the extended mapping ${\varphi}$. However, it is not at all clear that the equation yields a well-defined mapping. To get around this problem, we shall first define a mapping ${\tilde}{{\varphi}}:SM_1{\rightarrow}M_2$ and then show that the values of ${\tilde}{{\varphi}}$ only depend on the basepoint $x\in M_1$.
Choose $(x,v)\in SM_1$, and consider the geodesic $\gamma_{x,-v}$ (note the reversal of $v$). Let $T_0=T_0(v)\geq 0$ be the time at which this curve first leaves $M_1$ and enters $U_1$. That is, $$T_0=T_0(v)=\inf \{t\geq 0:\gamma_{x,-v}(t)\notin M_1\}.$$ This value exists because of the nontrapping assumption. Similarly, we define $T_1=T_1(v)$ to be the first time after $T_0(v)$ at which the curve leaves the interior of $U_1$: $$T_1=T_1(v)=\inf \{t> T_0(v):\gamma_{x,-v}(t)\in {\partial}U_1\}.$$ If, somehow, the curve never leaves the interior of $U_1$, then we set $T_1=\infty$.
We claim that $T_1>T_0$. This follows from the assumption that ${\partial}M$ and the metric are both analytic. Consequently, a geodesic cannot have contact of infinite order with the boundary without being trapped in the boundary. Therefore, at $t=T_0$, we conclude that there exists $m\geq 1$ for which $$\begin{array}{cc} {\partial}_t^k (x^n\circ \gamma_{x,-v})(T_0)=0, & 0\leq k <
m;
\\ {\partial}_t^m (x^n\circ \gamma_{x,-v})(T_0)<0. & \end{array}$$ This shows that for some small amount of time after $T_0$, the geodesic must remain entirely outside $M_1$. Hence $T_1>T_0$.
Now we let $T=T(v)$ be an arbitrarily chosen number strictly between $T_0$ and $T_1$, and we let $\xi_v=-\dot\gamma_{x,-v}(T)$. By construction, $x=\exp^{g_1}(T\xi_v)$. We define ${\tilde}{{\varphi}}(x,v)$ by:
$${\tilde}{{\varphi}}(x,v)=\exp^{g_2}(T{\varphi}_{0*}\xi_v)$$ See Figure \[prim\].
![The Construction of ${\tilde}{{\varphi}}$[]{data-label="prim"}](AL3)
${\tilde}{{\varphi}}(x,v)$ is a well defined function on $SM_1$ with values in $M_2$.
We must show two things: first, that $\exp^{g_2}(T{\varphi}_{0*}\xi_v)$ is a point in $M_2$; second, that the value of ${\tilde}{{\varphi}}$ is independent of the choice of $T$.
The curve $\gamma_{\xi_v}(t)=\exp^{g_1}(t\xi_v):0\leq t\leq T$ is composed of two segments; the first of which lies within $U_1$, the second of which lies within $M_1$. The break between the two occurs at $t=T-T_0$. We conclude that $\ell(\dot\gamma_{\xi_v}(T-T_0))\geq
T_0$.
The curve $\exp^{g_2}(t{\varphi}_{0*}\xi_v):t\in[0,T]$ is also composed of two segments of length $T-T_0$ and $T_0$ lying in $U_2$ and $M_2$ respectively. Indeed, for $t\in [0,T-T_0]$, we have $$\label{corndog}
\exp^{g_2}(t{\varphi}_{0*}\xi_v)={\varphi}_0(\exp^{g_1}(t\xi_v))$$ from the fact that ${\varphi}_0$ is an isometry on $U_1$. Hence the left side belongs to $U_2$.
To show that the remainder of the curve $\exp^{g_2}(t{\varphi}_{0*}\xi_v)$ lies in $M_2$, we look at the lens data. If we differentiate at $t=T-T_0$, we obtain from equation , ${\varphi}_{0*}(\dot\gamma_{\xi_v}(T-T_0))$ which equals $\Lambda(\dot\gamma_{\xi_v}(T-T_0))$.
By the fact that $M_1$ and $M_2$ have the same lens data, we conclude that $$\ell({\varphi}_{0*}(\dot\gamma_{\xi_v}(T-T_0)))\geq
T_0,$$ so the point $\exp(T{\varphi}_{0*}\xi_v)$ does indeed exist and lie in $M_2$.
Now let $T'$ be another time in between $T_0$ and $T_1$, and let $\xi_v'$ be the corresponding vector. Without loss of generality we may assume that $\Delta T= T-T'>0$. Then we have the following identity
$$\exp^{g_1}(t\xi_v)=\exp^{g_1}((t-\Delta T)\xi_v').$$
By the definitions of $T_0$ and $T_1$, the curve $\exp^{g_1}(t\xi_v):0\leq t\leq \Delta T$ is a geodesic segment lying entirely within the interior of $U_1$. Since ${\varphi}_0$ is an isometry on $U_1$, the vectors ${\varphi}_{0*}\xi_v$ and ${\varphi}_{0*}\xi_v'$ must also be tangent to a common geodesic at a distance of $\Delta T$. Hence
$$\exp^{g_2}(t{\varphi}_{0*}\xi_v)=\exp^{g_2}((t-\Delta T){\varphi}_{0*}\xi_v').$$ Setting $t=T$, we obtain the needed result.
For fixed $x_0$, ${\tilde}{{\varphi}}(x_0,v)$ is constant.
The strategy here is to prove that ${\tilde}{{\varphi}}(x_0,v)$ is locally constant. Then the statement follows from the connectedness of the sphere. First, we need a lemma.
For a pair of points in ${\tilde}{M}$, let $d(x,y)$ denote the distance between them. This function is not necessarily smooth, even off the diagonal. However, the next lemma shows that the squared distance function $d(x,y)^2$ is as smooth as the metric for $(x,y)$ sufficiently close to each other.
Let ${\tilde}{M}$ be as above (with subscript omitted). For every $x_0$ in the interior of ${\tilde}{M}$, there exists a positive number $r$ such that the squared distance function is analytic on the set $$\Delta_r(x_0)=\{(x,y):d(x,x_0)<r, d(x,y)<r\}.$$ If $K$ is a compact set contained within the interior of ${\tilde}{M}$, then there is an open $\sc{O}\subset {\tilde}{M}$ containing $K$ and a positive number $r$ such that the squared distance function is analytic on the set $$\Delta_{\sc{O},r}(K)=\{(x,y):x\in\sc{O}, d(x,y)<r\}.$$
We choose $r>0$ so that the ball $B_{2r}(x_0)$ is contained within ${\tilde}{M}$ and is geodesically convex (see [@St], Theorem 6.2, noting that the restriction on the radius is only that it be sufficiently small). By definition every pair of points in $B_{2r}(x_0)$ is joined by a unique geodesic segment contained entirely within $B_{2r}(x_0)$. Moreover, that segment is length-minimizing.
Now assume that $(x,y)\in \Delta_r(x_0)$. Then there is exactly one geodesic segment connecting them whose length is less than $r$. Indeed, there is at least one, since the two points lie within $B_{2r}(x_0)$. If there were another geodesic segment connecting them, it would have to leave $B_{2r}(x_0)$ at some point. Since $d(x_0,x)<r$, such a segment would necessarily have length greater than $r$.
This shows that the mapping $$\{(x,v):d(x,x_0)<r,|v|_g<r\}\rightarrow \Delta_r(x_0)$$ given by $(x,v)\mapsto (x,\exp_x(v))$ is bijective. Naturally, the exponential map is analytic as long as the metric is analytic. By the inverse function theorem, $\exp^g$ gives a diffeomorphism between these two sets. Through this diffeomorphism, the squared distance function is expressed $d(x,y)^2=g_{ij}v^iv^j$, which is analytic as long as $g$ is analytic.
The second statement of the lemma follows from the first by a compactness argument. Indeed, for every $x_0\in K$ we take the number $r$ from the first statement and form the ball $B_r(x_0)$. All such balls form an open cover of $K$. We take a finite subcover, let $\sc{O}$ be the union of its members, and let $r$ be the smallest radius in that subcover.
Now fix a vector $(x_0,v)$ and choose $T=T(v)$ and $\xi_v$. Let $y_0\in U_1$ be the basepoint of the vector $\xi_v$ so that $x_0=\exp^{g_1}_{y_0}(T\xi_v)$. Also, let $\gamma_1=\gamma_{x_0y_0}$ denote the geodesic segment connecting the two points. Let $\sc{O_1}$, $r_1$ be the open set and radius corresponding to the compact set $\gamma_1$ as in the lemma.
In ${\tilde}{M_2}$, we have a corresponding segment $\gamma_2$ between the points ${\varphi}_0(y_0)$ and ${\tilde}{{\varphi}}(x_0,v)$. It is given by the curve $$\exp^{g_2}(t{\varphi}_{0*}\xi_v):,\,\,0\leq t \leq T.$$ Let $r_2$ be the radius corresponding to $\gamma_2$ as in the lemma.
Let $r$ denote the positive number $$r=\inf\{d(\gamma_1,{\partial}{\tilde}{M_1}), d(y_0,{\partial}U_1), r_1,r_2\}.$$
By continuity, there exists a neighborhood $N$ of $v$ in $S_{x_0}M_1$ sufficiently small such that for all $w\in N$, $$d(\gamma_{x_0,-v}(t),\gamma_{x_0,-w}(t))<r$$ for all $t$ in the interval $[0,T]$. The restrictions on $r$ guarantee that the curve $\gamma_{x_0,-w}(t)$ remain within ${\tilde}{M_1}$ and that its endpoint, $\gamma_{x_0,-w}(T)$, be in the interior of $U_1$. For each $w$, let $\eta_w=-\dot\gamma_{x_0,-w}(T)$. We prove ${\tilde}{{\varphi}}(x_0,v)={\tilde}{{\varphi}}(x_0,w)$ by breaking this into the two equations: $$\label{puppydog}
\exp^{g_2}(T{\varphi}_{0*}\xi_v)=\exp^{g_2}(T{\varphi}_{0*}\eta_w);$$ $$\label{hounddog}
\exp^{g_2}(T{\varphi}_{0*}\eta_w)=\exp^{g_2}(T(w){\varphi}_{0*}\xi_w).$$ See Figure \[locally constant\].
![[]{data-label="locally constant"}](AL4)
Consider the function $$\rho_1(t)=d^2_{g_1}(\exp(t\xi_v),\exp(t\eta_w)).$$ By our choices of $r$ and $N$, and by the lemma, this is an analytic function for $t\in [0,T]$.
Now we consider ${\tilde}{M_2}$, and define $$\rho_2(t)=d_{g_2}^2(\exp^{g_2}(t{\varphi}_{0*}\xi_v),\exp^{g_2}(t{\varphi}_{0*}\eta_w)).$$ Since ${\varphi}_0|_{U_1}:U_1{\rightarrow}U_2$ is an isometry, the functions $\rho_1$ and $\rho_2$ must coincide for small values of $t$. Also, we note that the function $\rho_2(t)$ is analytic as long as $d_{g_2}(\exp^{g_2}(t{\varphi}_{0*}\xi_v),\exp^{g_2}(t{\varphi}_{0*}\eta_w))<r$, since $r$ was chosen to be smaller than $r_2$. Therefore, by analytic continuation, the functions $\rho_1$ and $\rho_2$ are equal up to the first point $t_0$ where $\rho_2=r^2$. But by continuity, we would then have $\rho_1(t_0)=r^2$, which does not occur. Therefore, we see that $\rho_1(t)=\rho_2(t)$ throughout the interval $0\leq t \leq T$. In particular, we find that $\rho_2(T)=0$, which verifies equation .
If $T$ lies between $T_0(w)$ and $T_1(w)$, then equation is nothing but a restatement that ${\tilde}{{\varphi}}(x_0,w)$ is well defined. Clearly, $T>T_0(w)$, so assume that it is also greater than $T_1(w)$ (The possiblity that $T=T_1(w)$ is ruled out by the fact that $\gamma_{x,-w}(T_1(w))\in
{\partial}M$). Fix a number $T'=T(w)$, and a corresponding $\xi_w$. If we let $\Delta T=T-T'$, then we have the equation $$\exp^{g_1}(t\eta_w)=\exp^{g_1}((t-\Delta T)\xi_w).$$
Let $\gamma(t)=\exp^{g_1}(t\eta_w)$, for $0\leq t\leq \Delta T$. If it happens that it lies entirely within $U_1$, then the same proof that we used to show that ${\tilde}{{\varphi}}$ is well-defined will verify equation . So assume that $\gamma(t)$ does not lie entirely within $U_1$. Then we can uniquely partition it into subsegments which alternately lie in $U_1$ and $M_1$. Indeed, we define $$\begin{array}{cc} t_0= & 0, \\
t_1= & \inf \{t>0:\gamma(t)\notin U_1\}, \\
t_2= & \inf \{t> t_1:\gamma(t)\notin M_1\},\\
\vdots & \vdots \\
t_{m-1}= & \inf \{t> t_{m-2}:\gamma(t)\notin M_1\}, \\
t_m= & \Delta T.\end{array}$$ The fact that $0=t_0<t_1<...<t_m=\Delta T$ is true follows from the same reasoning that was used above to prove that $T_1(v)>T_0(v)$. The fact that this partition is finite also follows from the analytic assumption. Indeed an analytic curve segment cannot intersect the analytic ${\partial M}$ more than a finite number of times without being entirely contained within it. Note that the segment $\gamma|_{[t_k,t_{k+1}]}$ lies in $U_1$ or $M_1$ according to whether $k$ is even or odd, respectively. In particular, $m$ is odd.
To prove equation , we will use induction to show that for all $k=1,2,\dots,m$, and all $t\in [0,T]$, $$\label{wienerdog}
\exp^{g_2}(t{\varphi}_{0*}\dot\gamma(0))=\exp^{g_2}((t-t_k){\varphi}_{0*}\dot\gamma(t_k)).$$ Then setting $k=m$ and $t=T$ yields the result.
Step 1: $\eta_w=\dot\gamma(0)$ and $\dot\gamma(t_1)$ lie on the geodesic $\gamma$ at a distance of $t_1$ from each other. Since this segment lies within $U_1$ and since ${\varphi}_0$ is an isometry of $U_1$ to $U_2$, we see that ${\varphi}_{0*}\dot\gamma(0)$ and ${\varphi}_{0*}\dot\gamma(t_1)$ also lie on a common geodesic at the same distance apart. Hence equation is established for $k=1$.
Step 2: The next segment of $\gamma$ lies within $M_1$. Indeed we have the following: $$\dot\gamma(t_2)=\Sigma_{g_1}(\dot\gamma(t_1)),\,\,\ell_{g_1}(\dot\gamma(t_1))=t_2-t_1.$$ Since $M_2$ has the same lens data as $M_1$, we see that ${\varphi}_{0*}\dot\gamma(t_1)$ and ${\varphi}_{0*}\dot\gamma(t_2)$ are connected by a geodesic across $M_2$ with the same length $t_2-t_1$. Together with step 1, this shows that ${\varphi}_{0*}\dot\gamma(t_2)$ lies tangent to the same geodesic as ${\varphi}_{0*}\dot\gamma(0)$ at a distance of $t_2$. Hence Equation 8 is established for $k=2$.
Step 3: By induction, we may repeat these steps, establishing equation for all $k$ up to $k=m$.
For $x\in M_1$, set ${\varphi}(x)={\tilde}{{\varphi}}(x,v)$. If $x\in V_1$, then we are in the domain of the boundary normal coordinates. We choose $v=\frac{{\partial}}{{\partial}x^n}$. Then $\gamma_{x,-v}$ is the geodesic segment from $x$ to $U_1$ normal to ${\partial M}$. We choose $T(v),\xi_v$ so that $x=\exp^{g_1}(T(v)\xi_v)$ and note that the segment $\exp^{g_1}(t\xi_v): 0\leq t\leq T_v$ lies entirely in $V$. We have $${\varphi}_0(x)=\exp(T(v){\varphi}_{0*}\xi_v)={\tilde}{{\varphi}}(x,v)={\varphi}(x)$$ The first equation is true by the fact that ${\varphi}_0$ is an isometry on $V_1$. Hence ${\varphi}$ and ${\varphi}_0$ agree on their common domains. Gluing them together, we form $${\varphi}:{\tilde}{M_1}{\rightarrow}{\tilde}{M_2}.$$
That ${\varphi}$ is a diffeomorphism and an isometry
====================================================
Let $(x,v)$ lie in the interior of $S_xM_1$. We choose $T=T(v)$ and $\xi_v\in SU_1$ so that $\gamma(t)=\exp^{g_1}(t\xi_v)$ is a geodesic in ${\tilde}{M_1}$ that reaches $x$ at time $T$. In the segment $[0,T-T_0]$, $\gamma$ lies entirely within $U_1$; whereas on the segment $[T-T_0,T+\delta]$, it lies entirely within $M_1$ for $\delta$ sufficiently small. For the first segment, we see that $${\varphi}(\gamma(t))={\varphi}_0(\gamma(t))=\exp^{g_2}(t{\varphi}_{0*}\xi_v);$$ where the second equality holds by the fact that ${\varphi}_0$ is an isometry on $U_1$.
On the second segment, we see that for each pair $(\gamma(t),\dot\gamma(t))$, we can choose the same $\xi_v$ for $\xi_{\dot\gamma(t)}$ with the corresponding $T(\dot\gamma(t))=t$. So, for all $t\in [0,T+\delta]$,
$$\label{bulldog}
{\varphi}(\gamma(t))=\exp^{g_2}(t{\varphi}_{0*}\xi_v).$$
This can be rewritten in the form: $$\label{maddog}
{\varphi}(\gamma(t))=\exp^{g_2}(t{\varphi}_{0*}\dot\gamma(0)).$$
In fact, the latter equation is true for any geodesic segment $\gamma|{[0,T]}$ that can be partitioned into $\gamma|{[0,a]}$ and $\gamma|{[a,T]}$ with the two subsegments lying entirely in $U_1$ and $M_1$ respectively.
${\varphi}:{\tilde}{M_1}{\rightarrow}{\tilde}{M_2}$ is bijective.
Reversing the roles of ${\tilde}{M_1}$ and ${\tilde}{M_2}$, we can define a mapping $\psi:{\tilde}{M_2}{\rightarrow}{\tilde}{M_1}$ by the same process by which we defined ${\varphi}$. In particular it would extend ${\varphi}_0^{-1}$.
The analog to equation is: $$\label{hangdog}
\psi(\beta(t))=\exp^{g_1}(t{\varphi}^{-1}_{0*}\dot\beta(0)),$$ where $\beta$ is any geodesic segment composed of two subsegments contained in $U_2$ and $M_2$, respectively.
Using the notation from above, we would like to prove that $\psi\circ{\varphi}(\gamma(t))=\gamma(t)$ for $t\in[0,T]$. To that end, we will first show that the geodesic segment $\beta(t)={\varphi}(\gamma(t))$ is of the type valid for equation .
We note that for $t\in[0,T-T_0]$, $\gamma(t)\in U_1$ so ${\varphi}(\gamma(t))$ must lie in $U_2$. For $t\in[T-T_0,T]$, $\gamma(t)\in M_1$, so by Proposition 1, ${\varphi}(\gamma(t))\in M_2$.
Therefore, we may apply equation , which yields: $$\begin{array}{ccc}
\psi\circ{\varphi}(\gamma(t)) & = &
\exp^{g_1}(t{\varphi}_{0*}^{-1}\dot\beta(0))
\\
& = & \exp^{g_1}(t{\varphi}_{0*}^{-1}{\varphi}_{0*}\dot\gamma(0)) \\
& = & \exp^{g_1}(t\dot\gamma(0)) \\ & = & \gamma(t).\end{array}$$
Since every point in $M_1$ lies on some such curve $\gamma(t)$, we conclude that $\psi\circ{\varphi}=id$ on $M_1$. But we know that the same identity is true on $V_1$, so it is true on all of ${\tilde}{M_1}$. By the symmetry of the construction, we conclude ${\varphi}\circ\psi$ is also the identity.
${\varphi}:{\tilde}{M_1}{\rightarrow}{\tilde}{M_2}$ is an analytic isometry.
Since ${\varphi}$ is bijective, it is sufficient to prove the statement locally. These properties are already known on $V_1$ where ${\varphi}={\varphi}_0$, so we assume $x$ lies in the interior $M_1$. First we show that all directional derivatives of ${\varphi}$ exist. Indeed, by differentiating equation at $t=T(v)$, we obtain: $$D_v{\varphi}(x)={\partial}_t({\varphi}\circ\gamma)(T(v))={\partial}_t
\exp^{g_2}(t{\varphi}_{0*}\xi_v)|_{t=T(v)}.$$ Clearly, the quantity on the right side exists. What’s more, it is a vector of length $1$. We conclude that ${\varphi}_*$ exists and preserves lengths of vectors. In particular it is nonsingular.
From the fact that it preserves lengths, we derive smoothness. Indeed, $g_1={\varphi}^*g_2$, which has the coordinate form: $$g_1(x)_{ij}={\varphi}^k_{,\,i}(x)\,g_2({\varphi}(x))_{kj}.$$ This yields: $$g_1(x)_{ij}g_2({\varphi}(x))^{jl}={\varphi}^k_{,\,i}(x).$$ The left side is once differentiable; hence ${\varphi}$ is twice differentiable. But then that implies the left side to be twice differentiable which shows ${\varphi}$ to be three-times differentiable. By an obvious application of induction, ${\varphi}$ must be smooth.
It only remains to prove that ${\varphi}$ is analytic. Of course this is already known in $V_1$. Since ${\varphi}$ is a smooth isometry, we can state $${\varphi}(\exp^{g_1} \xi)=\exp^{g_2}({\varphi}_*\xi)$$for any vector $\xi\in
T{\tilde}{M_1}$. Given $x_0$ in the interior of $M_1$, consider a ball of radius $r$, centered at $x_0$, which is strictly geodesically convex, and choose any point $y_0$ within this ball. Then there is a unique $\xi_0\in T_yM_1$ satisfying $|\xi_0|_{g_1}<r$ and $\exp_{y_0}\xi_0=x$. Moreover, $x_0$ and $y_0$ are not conjugate along the corresponding geodesic, so $$\exp_{y_0}^{g_1}:\xi\mapsto x$$ is a local diffeomorphism about $\xi_0$. It is analytic by the fact that $g_1$ is analytic. Consequently, it’s inverse function is analytic. Let $\xi(x)$ denote the inverse, which is defined for $x$ in some neighborhood of $x_0$. Then we see that $${\varphi}(x)=\exp^{g_2}_{{\varphi}(y_0)}({\varphi}_*\xi(x)).$$ $y_0$ is constant, so ${\varphi}_*$ is a constant linear mapping. From the fact that $g_2$ is analytic, we see that this mapping is also analytic.
, Boundary rigidity and filling volume minimality of metrics close to a flat one, manuscript, 2005.
, Entropies et rigidités des espaces localement symétriques de courbure strictment négative, [*Geom. Funct. Anal.*]{}, [**5**]{}(1995), 731-799.
, Rigidity and the distance between boundary points, [*J. Differential Geom.*]{}, [**33**]{}(1991), no. 2, 445–464.
, Boundary and lens rigidity of finite quotients, [*Proc. AMS*]{}, [**133**]{}(2005), no. 12, 3663–3668.
, Local boundary rigidity of a compact Riemannian manifold with curvature bounded above, [*Trans. Amer. Math. Soc.*]{} [**352**]{}(2000), no. 9, 3937–3956.
, Conjugacy and rigidity for manifolds with a parallel vector field, [*J. Diff. Geom.*]{} 39(1994), 659–680.
, Inverse scattering problem in anisotropic media, [*Comm. Math. Phys.*]{} [**199**]{}(1998), no. 2, 471–491.
, Filling Riemannian manifolds, [*J. Diff. Geometry*]{} [**18**]{}(1983), no. 1, 1–148.
, Semiglobal boundary rigidity for Riemannian metrics, [*Math. Ann.*]{}, [**325**]{}(2003), 767–793.
, Sur la rigidité imposée par la longueur des géodésiques, [*Invent. Math.*]{} [**65**]{}(1981), 71–83.
, Boundary rigidity and holography, [*J. High Energy Phys.*]{} [**0401**]{}, 034-057.
, Geometry VI: Riemannian Geometry (2001).
, Two dimensional simple compact manifolds with boundary are boundary rigid, [*Ann. Math.*]{} [**161**]{}(2)(2005), 1089–1106.
, Rigidity for metrics with the same lengths of geodesics, [*Math. Res. Lett.*]{} [**5**]{}(1998), 83–96.
, Boundary rigidity and stability for generic simple metrics, [*J. Amer. Math. Soc.*]{} [**18**]{}(2005), 975–1003.
, Local Lens Rigidity with Incomplete Data for a class of non-Simple Riemannian Manifolds.
, Lectures on Differential Geometry (1964).
Cauchy data, [*IMA publications*]{} [**Vol 137**]{}, “Geometric methods in inverse problems and PDE control" (2003), 263-288.
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---
abstract: 'The production of particles with double, triple and hidden charm in heavy ion collisions is studied in the framework of the statistical coalescence model. According to the postulates of the model, the charm quark-antiquark pairs are created at the initial stage of a heavy ion reaction in hard parton collisions. The amount of charm is assumed to be unchanged at later stages. The charm (anti)quarks are distributed among different hadron species at hadronization according to the laws of statistical physics. Several approaches to the statistical treatment of charm hadronization are considered. The grand canonical approach is appropriate for systems containing large number of charm (anti)quarks. The exact charm conservation and Poissonian fluctuations of the number of charm quark-antiquark pairs should be taken into account, if the average number of these pairs is of oder of unity or smaller. The charm hadronization in a subsystem of a larger system is discussed. It is explained why the canonical approach is not appropriate for the description of charm hadronization. The obtained formulas can be used to calculate the production of charm in heavy ion collisions in a wide energy range.'
author:
- 'A.P. Kostyuk'
title: 'Double, Triple and Hidden Charm Production in the Statistical Coalescence Model'
---
Introduction
============
The thermal hadron gas (HG) model has demonstrated an obvious success in describing the chemical composition of light-flavored hadrons produced in heavy ion [@HG] and even in elementary hadron-hadron [@HGh] as well as in electron-positron [@HGe] collisions. The experimental data can be well fitted with only three free parameters: the temperature $T$, the volume $V$ and the baryonic chemical potential $\mu_b$ of the hadron gas at the point of the chemical freeze-out. (Sometimes the fit is improved by introducing one more parameter — the strangeness suppression factor $\gamma_s$.) This success motivated attempts to extend the applicability domain of the thermal model also to heavy-flavored hadrons, for instance, to the description of $J/\psi$ meson production [@gago1].
A straightforward application of the equilibrium HG model to hadrons with open and hidden charm is not, however, justified. Partons with rather large momenta are needed to produce a heavy quark-antiquark pair. Consequently, the time of charm equilibration in a thermal hadronic or even quark-gluon medium is large and definitely exceeds the lifetime of the fireball. Production of charm can take place only at the initial stage of the heavy ion reaction, when hard partons are available. The charm production/annihilation rate is too low to keep the number of heavy quark-antiquark pairs at its chemical equilibrium value at later stages. Therefore, the [*total amount*]{} of heavy flavor should be out of equilibrium at the point of hadronization and chemical freeze-out. It is reasonable to expect, however, the [*distribution*]{} of heavy quarks and antiquarks among different hadrons with open and hidden charm to be thermal and controlled by the same values of thermodynamic parameters that fit the chemical composition of light-flavored hadrons. These ideas are implemented in the statistical coalescence model (SCM) [@BMS; @We; @WeBMS].
In the present paper, I consider SCM, which is based on the following postulates:
- The charm quarks, $c$, and antiquarks, $\overline{c}$, are created at the initial stage of A+A reaction in hard parton collisions.
- Creation and annihilation of $c\overline{c}$ pairs at later stages can be neglected.
- The formation of observed hadrons with open and hidden charm takes place near the point of chemical freeze-out in accordance with the laws of statistical physics.
This approach appeared to be quite successful describing the $J/\psi$ and $\psi'$ production in (semi)central Pb+Pb collisions at Super Proton Synchrotron (SPS) [@Kost; @psi-prime]. Although the role of the statistical coalescence at SPS energies is still under discussion [@Comb], its dominance at the energies of the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) is more certain. The SCM-based predictions for the quarkonium production at RHIC [@RHIC; @RHICa] are consistent with transport calculations [@brat].
Quarkonia are not the only type of hadrons whose production can be described by the statistical coalescence model. If two, three or more charm quark-antiquark pairs are created in a nucleus-nucleus collision, hadrons with double and triple charm can be formed. These hadronic states were long time ago predicted by the quark model (see [@Multi] and references therein), but most of them have not been observed experimentally yet.[^1]
The hadrons with double and triple charm are of special interest from both theoretical end experimental point of view. Due to a rather large mass of charm (anti)quarks, their interactions within a hadron are close to the perturbative regime of Quantum Cromodynamics (QCD). A study of the properties of doubly and especially triply charmed baryons would allow to test QCD-based models of quark forces. An observation of more exotic hadronic states, like multi-charm tetra- and pentaquarks will open a new window into the structure of the hadronic matter. The intention to detect the double and triple charm poses a new challenge to the experimentalists and will demand a further development of the experimental technique.
In the present paper, I derive the formulas that allows to calculate the yield of double and triple charm particles. The formulas for the hidden charm are also included as their detailed derivation has not been published yet.
The article is organized as follows. In the Section \[Grand\] I consider the grand canonical approach. A system with exactly fixed numbers of charm quarks and antiquarks is studied in Section \[Fixed\]. A more realistic situation, the system with Poissonian fluctuations of the number of charm quark-antiquark pairs, is considered in Section \[Poisson\]. Section \[Subs\] is devoted to the case, when only a part of the total system is available for the observation: the double, triple and hidden charm yield in a subsystem is studied. Summary is given in Section \[Summ\].
The grand canonical approach to the statistical coalescence {#Grand}
===========================================================
The grand canonical version of the statistical coalescence model was proposed in [@BMS]. It can be applied to the systems containing large ($N_{c\overline{c}} \gg 1$) number of $c\overline{c}$ pairs.
Let us start from the grand canonical partition function for the ideal hadron gas in the Boltzmann approximation: $$\label{gcpf}
{\cal Z}(V,T,\{ \lambda \} ) = \sum_{i_1 = 0}^{\infty} \sum_{i_2 = 0}^{\infty}
\dots \sum_{i_L = 1}^{\infty} \prod_{l=0}^L
\frac{\left[ \lambda_l \phi(T,m_l,g_l) V \right]^{i_l}}{i_l !},$$ where $L$ is the total number of hadron species (including resonances), $V$ and $T$ are the volume and temperature, $\phi(T;m,g)$ is the one-particle partition function $$\begin{aligned}
\label{phi}
\phi(T;m,g) &=& \frac{g}{2 \pi^2} \int_0^{\infty}p^2 dp~
\exp \left( - \frac{\sqrt{p^{2}+m^{2}}}{T} \right)~ \\
&=& g\frac{m^{2} T}{2 \pi^{2}}~ K_{2}\left( \frac{m}{T} \right)~.
\nonumber\end{aligned}$$ Here $m$ is the particle mass, $g$ is the degeneration factor (the number of spin states), $K_{2}$ is the modified Bessel function. In the nonrelativistic limit $m >> T$ the expression (\[phi\]) takes the form $$\label{phinr}
\phi(T;m,g)~ \simeq~ g \left(\frac{mT}{2 \pi} \right)^{3/2} \exp(-m/T).$$
The fugacity $\lambda_l$ is expressed via the chemical potentials (electric — $\mu_e$, baryonic — $\mu_b$, strangeness — $\mu_s$ and charm — $\mu_c$) and suppression (enhancement) factors (for strangeness — $\gamma_s$, and charm — $\gamma_c$): $$\begin{aligned}
\label{lambda}
\lambda_l &=& \gamma_s^{|s|_l} \gamma_c^{|c|_l}
\exp\left( \frac{\mu_l}{T} \right), \\\label{mu}
\mu_l &=& q_l \mu_e + b_l \mu_b + s_l \mu_s + c_l \mu_c \ , \label{mul}\end{aligned}$$ where $q_l$, $b_l$, $s_l$, $c_l$, are, respectively, the electric charge, baryon number, strangeness and charm of the hadron species $l$, $|s|_l$ and $|c|_l$ are the numbers of valence strange and charmed (anti-)quarks. The chemical potentials in the right-hand-side of (\[mul\]) are responsible for keeping the correct [*average*]{} values of the corresponding charges in the system. The suppression (enhancement) factors $\gamma_s$ and $\gamma_c$ are introduced to take into account a deviation of the total number of strangeness and charm from their equilibrium values[^2].
The average number of particles is given for each species as $$\label{Nl}
N_l = \lambda_l
\frac{\partial \log {\cal Z}(V,T,\{ \lambda \} )}
{\partial \lambda_l} = \lambda_l \phi(T,m_l,g_l) V.$$
The total number of hadrons $$\label{Nt}
N_{tot} = \sum_l N_l, \mbox{ ($l$ runs over all hadron species)},$$ can be broken up into several pieces: $N_0$, the number of zero charm hadrons (excluding hidden charm), $N_H$, the number of hidden charm mesons, $N_1$, $N_{\bar{1}}$, $N_2$, $N_{\bar{2}}$, $N_3$, $N_{\bar{3}}$, the numbers of hadrons with, respectively, single, double and triple charm and anticharm: $$\label{Ntc}
N_{tot} = N_0+N_H+N_1+N_{\bar{1}}+N_2+N_{\bar{2}}+N_3+N_{\bar{3}} .$$
Let us consider a system containing [*in average*]{} $N_{c}$ charm quark and $N_{\bar{c}}$ charm antiquarks. Then, as far as charm creation and annihilation are neglected, the following equalities should be satisfied: $$\begin{aligned}
\label{gceq1} \langle N_{c} \rangle &=& N_1 + N_H + 2 N_2 + 3 N_3,\\
\label{gceq2} \langle N_{\bar{c}} \rangle &=& N_{\bar{1}} + N_H +
2 N_{\bar{2}} + 3 N_{\bar{3}}.\end{aligned}$$ It is easy to see from (\[lambda\]),(\[mu\]) and (\[Nl\]) that the above equations can be rewritten as $$\begin{aligned}
\label{gceqt1} \langle N_{c} \rangle &=& \gamma_c \lambda_c \tilde{N}_1 + \gamma_c^2
\tilde{N}_H + 2 \gamma_c^2 \lambda_c^2 \tilde{N}_2 + 3 \gamma_c^3
\lambda_c^3 \tilde{N}_3, \label{eq1} \\
\label{gceqt2} \langle N_{\bar{c}} \rangle &=& \gamma_c
\lambda_c^{-1} \tilde{N}_{\bar{1}} + \gamma_c^2 \tilde{N}_H + 2
\gamma_c^2 \lambda_c^{-2} \tilde{N}_{\bar{2}} + 3 \gamma_c^3
\lambda_c^{-3} \tilde{N}_{\bar{3}}, \nonumber \\
\label{eq2}\end{aligned}$$ where $$\begin{aligned}
\label{Ntilde}
\tilde{N}_k &=& \left. N_k \right|_{\mu_c=0, \gamma_c=1},\ \
k= H, 1, \bar{1}, 2, \bar{2}, 3, \bar{3},\end{aligned}$$ and $$\begin{aligned}
\lambda_c = \exp\left( \frac{\mu_c}{T} \right).\end{aligned}$$
The constituent charm quark mass is about an order of magnitude larger than the typical temperature of chemical freeze-out. From this reason, $$\label{inequ}
\tilde{N}_1,\tilde{N}_{\bar{1}} \gg \tilde{N}_H,
\tilde{N}_2,\tilde{N}_{\bar{2}} \gg
\tilde{N}_3,\tilde{N}_{\bar{3}}$$ due to the exponential factor in (\[phinr\]). Therefore, if the factor $\gamma_c$ is not extraordinary large, the terms in (\[gceqt1\]) and (\[gceqt2\]) corresponding to hidden, double and triple (anti)charm can be neglected in a zero approximation and the coupled equations (\[gceqt1\]) and (\[gceqt2\]) can be simplified: $$\begin{aligned}
\label{gceqt10} \langle N_{c} \rangle &=& \gamma_c^{(0)} \lambda_c^{(0)} \tilde{N}_1 ,
\label{eq1o} \\
\label{gceqt20} \langle N_{\bar{c}} \rangle &=&
\gamma_c^{(0)} (\lambda_c^{(0)})^{-1} \tilde{N}_{\bar{1}} \label{eq2o}.\end{aligned}$$ The solution can be easily found: $$\begin{aligned}
\gamma_c^{(0)} &=& \sqrt{\frac{ \langle N_{c} \rangle
\langle N_{\bar{c}} \rangle }{\tilde{N}_1
\tilde{N}_{\bar{1}}}},\label{gamcG}\\
\lambda_c^{(0)} &=& \sqrt{ \frac{ \langle N_{c} \rangle
\tilde{N}_{\bar{1}} }{\langle N_{\bar{c}} \rangle \tilde{N}_1}}.
\label{lamcG}\end{aligned}$$ Now one can calculate the number of single, hidden, double and triple charm particles in the zero approximation: $$\begin{aligned}
\langle N_1 \rangle_{GCE}^{(0)} &=& \gamma_c^{(0)} \lambda_c^{(0)}
\tilde{N}_1 = \langle N_{c} \rangle , \label{N1G}\\
\langle N_H \rangle_{GCE}^{(0)} &=& (\gamma_c^{(0)})^2
\tilde{N}_H = \langle N_{c} \rangle
\langle N_{\bar{c}} \rangle
\frac{\tilde{N}_H}{\tilde{N}_1 \tilde{N}_{\bar{1}}}, \label{NHG} \\
\langle N_2 \rangle_{GCE}^{(0)} &=& (\gamma_c^{(0)})^2 (\lambda_c^{(0)})^2
\tilde{N}_2 = \langle N_{c} \rangle^2
\frac{\tilde{N}_2}{\tilde{N}_1^2}, \label{N2G} \\
\langle N_3 \rangle_{GCE}^{(0)} &=& (\gamma_c^{(0)})^2 (\lambda_c^{(0)})^2
\tilde{N}_3 =
\langle N_{c} \rangle^3 \frac{\tilde{N}_3}{\tilde{N}_1^3}. \label{N3G} \end{aligned}$$ The formulas for anticharm can be obtained by the obvious replacement $c \rightarrow \bar{c}$, $1 \rightarrow \bar{1}$ etc.
The formulas (\[N1G\]–\[N3G\]) are obtained under the assumption that the charm enhancement factor $\gamma_s$ is not large, so that only a tiny fraction of the charm quarks and antiquarks hadronizes into hidden, double and triple charm particles. This may be not true at very high energies. Indeed, let us consider for example the leading order expression (\[N2G\]) for the number of double charm particles. The fraction of charm quarks that hadronize into double charm particles is proportional to the number of charm quarks in the system and inversely proportional to the system volume at freeze-out: $$\label{ratio2}
\frac{\langle N_2 \rangle_{GCE}^{(0)}}{\langle N_{c} \rangle}
\propto \frac{\langle N_{c} \rangle}{V}.$$ The volume is proportional to the multiplicity of light hadrons. The charm production cross section grows with the collision energy faster than the multiplicity of light hadrons. Therefore, the ratio (\[ratio2\]) becomes comparable to unity at some point. In this case the above approximation does not work.
The accuracy of the leading-order approximation (\[N1G\])–(\[N3G\]) can be estimated by calculating the next-to-leading order corrections. Let us substitute $\gamma_c \rightarrow \gamma_c^{(0)} + \gamma_c^{(1)}$ and $\lambda_c \rightarrow \lambda_c^{(0)} + \lambda_c^{(1)}$ into (\[eq1\]) and (\[eq2\]). Neglecting higher order terms like $\gamma_c^{(1)} \lambda_c^{(1)}$, $\gamma_c^{(1)} \tilde{N}_k$ ($k= H, 2, \bar{2}, 3, \bar{3}$), etc. and taking into account (\[eq1o\]),(\[eq2o\]) and (\[NHG\])–(\[N3G\]) one gets
$$\begin{aligned}
\lambda_c^{(0)} \gamma_c^{(1)} + \gamma_c^{(0)} \lambda_c^{(1)} &=&
- \frac{\langle N_H \rangle_{GCE}^{(0)} +
2 \langle N_2 \rangle_{GCE}^{(0)} +
3 \langle N_3 \rangle_{GCE}^{(0)}}{\tilde{N}_1} , \label{eq1o1}\\
\left( \lambda_c^{(0)} \right)^{-2} \left(
\lambda_c^{(0)} \gamma_c^{(1)} - \gamma_c^{(0)} \lambda_c^{(1)}
\right)
&=&
- \frac{\langle N_H \rangle_{GCE}^{(0)} +
2 \langle N_{\bar{2}} \rangle_{GCE}^{(0)} +
3 \langle N_{\bar{3}} \rangle_{GCE}^{(0)}}{\tilde{N}_{\bar{1}}}.
\label{eq2o1}\end{aligned}$$
These coupled linear equations can be easily solved: $$\begin{aligned}
\gamma_c^{(1)} &=& - \frac{1}{2} \gamma_c^{(0)}
\left(
\frac{\langle N_H \rangle_{GCE}^{(0)} +
2 \langle N_2 \rangle_{GCE}^{(0)} +
3 \langle N_3 \rangle_{GCE}^{(0)}}{\langle N_{c} \rangle}
+
\frac{\langle N_H \rangle_{GCE}^{(0)} +
2 \langle N_{\bar{2}} \rangle_{GCE}^{(0)} +
3 \langle N_{\bar{3}} \rangle_{GCE}^{(0)}}{\langle N_{\bar{c}} \rangle}
\right), \\
\lambda_c^{(1)} &=& - \frac{1}{2} \lambda_c^{(0)}
\left(
\frac{\langle N_H \rangle_{GCE}^{(0)} +
2 \langle N_2 \rangle_{GCE}^{(0)} +
3 \langle N_3 \rangle_{GCE}^{(0)}}{\langle N_{c} \rangle}
-
\frac{\langle N_H \rangle_{GCE}^{(0)} +
2 \langle N_{\bar{2}} \rangle_{GCE}^{(0)} +
3 \langle N_{\bar{3}} \rangle_{GCE}^{(0)}}{\langle N_{\bar{c}} \rangle}
\right).\end{aligned}$$
Now the next-to-leading order corrections to the number of particles with different charm content can be calculated: $$\begin{aligned}
\langle N_1 \rangle_{GCE}^{(1)} &=& - \left(
\langle N_H \rangle_{GCE}^{(0)} +
2 \langle N_2 \rangle_{GCE}^{(0)} +
3 \langle N_3 \rangle_{GCE}^{(0)} \right), \nonumber \\
\label{N1Go1} \\
\langle N_H \rangle_{GCE}^{(1)} &=& \langle N_H \rangle_{GCE}^{(0)}
\left(
\frac{\langle N_1 \rangle_{GCE}^{(1)}}{\langle N_{c} \rangle} +
\frac{\langle N_{\bar{1}} \rangle_{GCE}^{(1)}}
{\langle N_{\bar{c}} \rangle}
\right), \label{NHGo1} \\
\langle N_2 \rangle_{GCE}^{(1)} &=& \langle N_2 \rangle_{GCE}^{(0)}
\frac{2 \langle N_1 \rangle_{GCE}^{(1)}}{\langle N_{c} \rangle} ,
\label{N2Go1} \\
\langle N_3 \rangle_{GCE}^{(1)} &=& \langle N_3 \rangle_{GCE}^{(0)}
\frac{3 \langle N_1 \rangle_{GCE}^{(1)}}{\langle N_{c} \rangle} .
\label{N3Go1}\end{aligned}$$ The higher order corrections can be obtained in a similar way. Still, the expressions become rather unwieldy. It is more reasonable to solve the coupled equations (\[eq1\]) and (\[eq2\]) numerically, if the next-to-leading order corrections (\[N1Go1\])–(\[N3Go1\]) become too large.
The above formulas allow to calculate the total number of particles with given (anti)charm content (hidden, single, double or triple). If the particle number of a single species is needed, it can be found in the following way. One can deduce from equations (\[lambda\]), (\[mu\]) and (\[Nl\]) that the number of particles of a single species $l$ can be found from $$\label{Nlc}
N_{l} = \frac{\tilde{N}_l}{\tilde{N}_k} N_{k},$$ where $k= H, 1, \bar{1}, 2, \bar{2}, 3$ or $\bar{3}$, depending on the charm content of the species $l$.
The grand canonical approach is the simplest version of the statistical coalescence model. However, it does not reflect the real experimental situation sufficiently well. Indeed, we operated only with average values $\langle N_{c} \rangle$ and $\langle N_{\bar{c}} \rangle$ in the above consideration. The correlations between $N_{c}$ and $N_{\bar{c}}$ were ignored. In reality, however, the quarks and antiquarks are produced in pairs. So that not only average numbers coincide $\langle N_{c} \rangle = \langle N_{\bar{c}} \rangle$, but also the numbers of quarks and antiquarks [*in every single event*]{} are equal: $N_{c} = N_{\bar{c}}$. The influence of this fact on the results of the SCM will be studied in the subsequent sections.
System with fixed numbers of charm quarks and antiquarks {#Fixed}
========================================================
In this section, a system with fixed numbers of charm quarks and antiquarks is considered. This does not correspond exactly to what is observed in the experiment. Still a study of such a system would allow us to obtain an important intermediate result. Later, it will be used in more realistic calculations.
Using Eq.(\[Nl\]), one can rewrite the partition function (\[gcpf\]) in a more compact form: $$\begin{aligned}
\label{gcpfc}
{\cal Z}(V,T,\{ \lambda \} ) &=& \sum_{i_1 = 0}^{\infty} \sum_{i_2 = 0}^{\infty}
\dots \sum_{i_L = 1}^{\infty} \prod_{l=0}^L
\frac{N_l^{i_l}}{i_l !} \nonumber \\
&=& \exp \left( \sum_{l=0}^L N_l \right) = \exp \left( N_{tot} \right) .\end{aligned}$$
The total number of particles in the last expression can be represented according to (\[Ntc\]), then $$\begin{aligned}
\label{gcpfc1}
{\cal Z}(V,T,\{ \lambda \} ) &=& \exp \left( \sum_{k} N_k \right)
= \prod_{k} \exp \left( N_k \right) , \\
& & k= 0, H, 1, \bar{1}, 2, \bar{2}, 3, \bar{3}. \nonumber\end{aligned}$$ Expanding all the exponents in the product, except the first one, into the Taylor series one gets $$\begin{aligned}
\label{gcpf1}
{\cal Z}(V,T,\{ \lambda \} ) &=&
e^{N_0} \sum_{i_H = 0}^{\infty} \sum_{i_1 = 0}^{\infty}
\dots \sum_{i_{\bar{3}} = 1}^{\infty} \prod_{k}
\frac{N_k^{i_k}}{i_k !}, \\
& & k= H, 1, \bar{1}, 2, \bar{2}, 3, \bar{3}. \nonumber\end{aligned}$$
Due to the property of The Kronecker delta $\sum_m \delta(m,n) = 1$, nothing changes if we multiply an expression by a Kronecker delta and sum over one of its indices. Doing this twice on the expression (\[gcpf1\]), one gets
$$\begin{aligned}
\label{gcpf1a}
{\cal Z}(V,T,\{ \lambda \} ) &=&
e^{N_0} \sum_{i_H = 0}^{\infty} \sum_{i_1 = 0}^{\infty}
\dots \sum_{i_{\bar{3}} = 1}^{\infty}
\sum_{N_{c}} \sum_{N_{\bar{c}}}
\delta(N_{c},i_H+i_1+i_2+i_3)
\delta(N_{\bar{c}},i_H+i_{\bar{1}}+i_{\bar{2}}+i_{\bar{3}})
\prod_{k}
\frac{N_k^{i_k}}{i_k !}, \\
& & k= H, 1, \bar{1}, 2, \bar{2}, 3, \bar{3}. \nonumber\end{aligned}$$
Then, after changing the summation order and using (\[lambda\]), (\[mul\]), (\[Nl\]) and (\[Ntilde\]), the above expression can be rewritten as $$\begin{aligned}
\label{gcpf2}
{\cal Z}(V,T,\{ \lambda \} ) &=& e^{N_0} \sum_{N_{c}} \sum_{N_{\bar{c}}}
\gamma_c^{N_{c} + N_{\bar{c}}}
\exp\left[ (N_{c} - N_{\bar{c}}) \frac{\mu_c}{T} \right]
Z_{N_{c} N_{\bar{c}}}(V,T,\{ \tilde{\lambda} \}) .\end{aligned}$$ Here $Z_{N_{c} N_{\bar{c}}}$ is the partition functions for the systems containing exactly $N_{c}$ charm quarks and $N_{\bar{c}}$ antiquarks: $$\begin{aligned}
\label{cpf}
Z_{N_{c} N_{\bar{c}}}(V,T,\{ \tilde{\lambda} \})
&=& \sum_{i_H = 0}^{\infty} \sum_{i_1 = 0}^{\infty}
\dots \sum_{i_{\bar{3}} = 1}^{\infty}
\delta(N_{c},i_H+i_1+i_2+i_3) \,
\delta(N_{\bar{c}},i_H+i_{\bar{1}}+i_{\bar{2}}+i_{\bar{3}})
\prod_{k}
\frac{\tilde{N}_k^{i_k}}{i_k !},\\
& & \hspace{5cm} k= H, 1, \bar{1}, 2, \bar{2}, 3, \bar{3}. \nonumber\end{aligned}$$ This function is [*canonical*]{} with respect to the [*exact*]{} conservation of the number of charm quarks and antiquarks and [*grand canonical*]{} with respect to the conservation [*in average*]{} of all other charges, whose values are controlled by the activities $$\tilde{\lambda}_k = \left. \lambda_k \right|_{\mu_c=0, \gamma_c=1} .$$
The number of summations in (\[cpf\]) can be reduced due to the Kronecker deltas: $$\label{gcpf2a}
Z_{N_{c} N_{\bar{c}}}(V,T,\{ \tilde{\lambda} \})
=
\sum_{i_H = 0}^{i_H^{max}} \frac{\tilde{N}_H^{i_H}}{i_H !}
\sum_{i_2 = 0}^{i_2^{max}} \frac{\tilde{N}_2^{i_2}}{i_2 !}
\sum_{i_{\bar{2}} = 0}^{i_{\bar{2}}^{max}}
\frac{\tilde{N}_{\bar{2}}^{i_{\bar{2}}}}{i_{\bar{2}} !}
\sum_{i_3 = 0}^{i_3^{max}} \frac{\tilde{N}_3^{i_3}}{i_3 !}
\sum_{i_{\bar{3}} = 0}^{i_{\bar{3}}^{max}}
\frac{\tilde{N}_{\bar{3}}^{i_{\bar{3}}}}{i_{\bar{3}} !}
\frac{\tilde{N}_1^{i_1}}{i_1 !}
\frac{\tilde{N}_{\bar{1}}^{i_{\bar{1}}}}{i_{\bar{1}} !}.$$
Here $$\begin{aligned}
i_H^{max} &=& min(N_{c},N_{\bar{c}}),\\
i_2^{max}(i_H) &=& [(N_{c}-i_H)/2],\\
i_{\bar{2}}^{max}(i_H) &=& [(N_{\bar{c}}-i_H)/2],\\
i_3^{max}(i_H,i_2) &=& [(N_{c}-i_H-2 i_2)/3],\\
i_{\bar{3}}^{max}(i_H,i_{\bar{2}}) &=& [(N_{\bar{c}}-i_H - 2 i_{\bar{2}})/3],\\
i_1(i_H,i_2,i_3) &=& [N_{c} - i_H - 2 i_2 - 3 i_3],\\
i_{\bar{1}}(i_H,i_{\bar{2}},i_{\bar{3}}) &=&
[N_{\bar{c}}-i_H - 2 i_{\bar{2}}- 3 i_{\bar{3}}].\end{aligned}$$ (The square brackets mean here the integer part, i.e. $[x]$ is the largest integer number that does not exceed $x$.)
The average number of hidden (single, double, triple) (anti)charm particles in the system with [*fixed*]{} numbers of charm quarks and antiquarks is found as $$\begin{aligned}
\label{Nk}
& &
\langle N_{k} \rangle_{fix} = \sum_l \tilde{\lambda}_l
\frac{\partial \log Z_{N_{c}N_{\bar{c}}} }{\partial \tilde{\lambda}_l} =
\sum_l \tilde{\lambda}_l \frac{\partial \tilde{N}_k}{\partial
\tilde{\lambda}_l} \frac{\partial \log Z_{N_{c}N_{\bar{c}}}}{\partial
\tilde{N}_k}, \nonumber \\
& & k= H, 1, \bar{1}, 2, \bar{2}, 3, \bar{3}; \\
& & \mbox{$l$ runs over all hadron species of the type $k$}.
\nonumber\end{aligned}$$ From (\[Nl\]) one sees that $$\tilde{\lambda}_l \frac{\partial \tilde{N}_l}{\partial
\tilde{\lambda}_l} = \tilde{N}_l,$$ therefore $$\sum_l \tilde{\lambda}_l
\frac{\partial \tilde{N}_k}{\partial \tilde{\lambda}_l}
= \tilde{N}_k$$ and finally $$\label{Nkf}
\langle N_{k} \rangle_{fix} = \tilde{N}_k
\frac{1}{Z_{N_{c}N_{\bar{c}}}}
\frac{\partial Z_{N_{c}N_{\bar{c}}}}{\partial \tilde{N}_k}.$$
I restrict my further consideration to the case, when the partition function (\[gcpf2a\]) is dominated[^3] by the term with $i_H = i_2= i_{\bar{2}} = i_3 = i_{\bar{3}} = 0$, $i_1 = N_{c}$ and $i_{\bar{1}} = N_{\bar{c}}$: $$\label{Z0}
Z_{N_{c}N_{\bar{c}}}^{(0)} \approx
\frac{\tilde{N}_1^{N_{c}}}{N_{c} !}
\frac{\tilde{N}_{\bar{1}}^{N_{\bar{c}}}}{N_{\bar{c}}!} .$$ The same term dominates also the derivatives of $Z_{N_{c}N_{\bar{c}}}$ with respect to $N_{c}$ and $N_{\bar{c}}$. It is easy to see that in this case, most of the charm hadronizes into hadrons containing only one $c$-quark or antiquark: $$\begin{aligned}
\langle N_{1} \rangle_{fix}^{(0)} &\approx& N_{c}, \label{N1f} \\
\langle N_{\bar{1}} \rangle_{fix}^{(0)}
&\approx& N_{\bar{c}}. \label{N1bf}\end{aligned}$$ Only a tiny fraction of the total charm is accommodated into hidden, double and triple charm hadrons.
The leading term (\[Z0\]) does not depend on $i_H$. Therefore, the derivative ${\partial Z_{N_{c}N_{\bar{c}}}}/{\partial \tilde{N}_H}$ is dominated by the term of (\[gcpf2a\]) with $i_H = 1$, $i_2= i_{\bar{2}} = i_3 = i_{\bar{3}} = 0$, $i_1 = N_{c}-1$ and $i_{\bar{1}} = N_{\bar{c}}-1$: $$\label{dZH}
\frac{\partial Z_{N_{c}N_{\bar{c}}}^{(0)}}{\partial \tilde{N}_H}
\approx
\frac{\tilde{N}_1^{N_{c}-1}}{(N_{c}-1) !}
\frac{\tilde{N}_{\bar{1}}^{N_{\bar{c}}-1}}{(N_{\bar{c}}-1)!} .$$ From (\[Nkf\]) one finds the average number of hidden charm particles in the system with fixed numbers of charm quarks and antiquarks: $$\label{NHf}
\langle N_{H} \rangle_{fix}^{(0)} =
N_{c} N_{\bar{c}} \frac{\tilde{N}_H}{\tilde{N}_1 \tilde{N}_{\bar{1}}} .$$
Similarly, the average numbers of double and triple charm particles are given by: $$\label{N2f}
\langle N_{2} \rangle_{fix}^{(0)} =
N_{c} (N_{c} - 1) \frac{\tilde{N}_2}{\tilde{N}_1^2}$$ and $$\label{N3f}
\langle N_{3} \rangle_{fix}^{(0)} =
N_{c} (N_{c} - 1) (N_{c} - 2) \frac{\tilde{N}_3}{\tilde{N}_1^3},$$ respectively. The corresponding formulas for anticharm can be obtained from (\[N2f\]) and (\[N3f\]) by the obvious replacement $c \rightarrow \bar{c}$, $1 \rightarrow \bar{1}$ etc.
We have considered the charm hadron production by quark coalescence in the thermal hadron system with [*fixed*]{} numbers of charm quarks and antiquarks. In reality, this number is not fixed. It fluctuates from event to event. These fluctuations have to be taken into account in more realistic calculations.
The system with Poissonian fluctuations of the number of charm quark-antiquark pairs {#Poisson}
====================================================================================
In a relativistic collision of two nuclei, the charm quarks and antiquarks are created in pairs in independent nucleon-nucleon collisions. Therefore, the number of quarks in the system is always equal to the number of antiquarks: $$\label{Nccbar}
N_{c} = N_{\bar{c}} \equiv N_{c\bar{c}}$$ and the fluctuations of the number of pairs approximately[^4] conforms the Poissonian law: $$\label{prob}
w_{P}(N_{c\bar{c}}) = e^{- \langle N_{c\bar{c}} \rangle }
\frac{\langle N_{c\bar{c}} \rangle^{N_{c\bar{c}}}}{N_{c\bar{c}}!}.$$ Here $w(N_{c\bar{c}})$ is the probability to observe $N_{c\bar{c}}$ charm quark-antiquark pairs in an event, provided that the average number of $c\bar{c}$ pairs in this type of events is $\langle N_{c\bar{c}} \rangle$.
The average number of particles with hidden, single, double, and triple charm is found by the convolution of the results of the last section (\[N1f\]),(\[NHf\])–(\[N3f\]) with the probability (\[prob\]): $$\begin{aligned}
\label{NkP}
\langle N_{k} \rangle_{P} &=& \sum_{N_{c\bar{c}}=1}^{\infty}
w_{P}(N_{c\bar{c}})
\langle N_{k} \rangle_{fix} \\
& & k= H, 1, \bar{1}, 2, \bar{2}, 3, \bar{3} \nonumber\end{aligned}$$ (The subscript “$P$” stands for “Poisson”). Here we again see that the most of the charm is accommodated into the single charm hadrons: $$\label{N1P}
\langle N_{1} \rangle_{P}^{(0)} \approx
\langle N_{\bar{1}} \rangle_{P}^{(0)} \approx
\langle N_{c\bar{c}} \rangle.$$ The rest (a tiny fraction) is distributed over hidden, double and triple charm particles whose number can be easily found: $$\begin{aligned}
\langle N_H \rangle_{P}^{(0)} &=& \langle N_{c\bar{c}} \rangle
(\langle N_{c\bar{c}} \rangle +1)
\frac{\tilde{N}_H}{\tilde{N}_1 \tilde{N}_{\bar{1}}}, \label{NHP} \\
\langle N_2 \rangle_{P}^{(0)} &=& \langle N_{c\bar{c}} \rangle^2
\frac{\tilde{N}_2}{\tilde{N}_1^2}, \label{N2P}\\
\langle N_3 \rangle_{P}^{(0)} &=&
\langle N_{c\bar{c}} \rangle^3 \frac{\tilde{N}_3}{\tilde{N}_1^3}. \label{N3P}\end{aligned}$$ Surprisingly, the formulas for the number of double and triple charm particles (\[N2P\]),(\[N3P\]) appeared to be exactly the same as in the grand canonical approach (\[N2G\]),(\[N3G\]).
However, this is not the case for the hidden charm. If the average number of $c\bar{c}$ pairs is small, $\langle N_{c\bar{c}} \rangle \alt 1$, the average number of produced hidden charm hadrons (\[NHP\]) is essentially larger than it would be naïvely expected from the grand canonical formula (\[NHG\]). The two formulas (\[NHG\]) and (\[NHP\]) give similar results only when the number $c\bar{c}$ pairs is large, $\langle N_{c\bar{c}} \rangle \gg 1$. The reason for this similarity is that the Poissonian distribution becomes narrow at $\langle N_{c\bar{c}} \rangle \gg 1$: $\langle (N_{c\bar{c}} - \langle N_{c\bar{c}} \rangle)^2 \rangle =
\langle N_{c\bar{c}} \rangle
\ll \langle N_{c\bar{c}} \rangle^2$.
It is easy to see that any narrow probability distribution of the number of charm quarks and antiquarks would give the same result as the grand canonical approach, provided that $\langle N_{c\bar{c}} \rangle \gg 1$.
Charm coalescence in a subsystem {#Subs}
================================
The formulas of the previous section allow to calculate the number of charm particles in the entire system. It often happens, however, that only a part of the phase space is observed in the experiment: a limited rapidity interval, for instance. Formulas for charm production in a subsystem are necessary in this case. The charm and non-charm hadrons are distributed inhomogeneously in the (phase) space. Therefore, the number of observed charm particles in a subsystem cannot be calculated merely as a fraction of their total number in the system, proportional to the ratio of the volume of the subsystem to the total volume of the system. Moreover, the total volume of the system may be even unknown. The formulas derived in this section are based on the thermodynamic parameters of the subsystem without any reference to those of the entire system.
Let $\xi \le 1$ is the probability to find a charm quark in the subsystem, provided that exactly one $c\bar{c}$ pair is present in the entire system. Then, if the number of $c\bar{c}$ pairs in the entire system is $N_{c\bar{c}}$, the probability to find $N_{c}$ charm quarks in the subsystem is given by the binomial law: $$\label{binom}
w(N_{c} | N_{c\bar{c}}) =
\frac{N_{c\bar{c}}!}{N_{c}! (N_{c\bar{c}} - N_{c})!}
\xi^{N_{c}} (1-\xi)^{N_{c\bar{c}}-N_{c}}.$$ It is assumed that the distributions of quarks and antiquarks are uncorrelated, and the probability distribution of the number of antiquarks $N_{\bar{c}}$ is given by the same binomial law.
Then, if the total number of $c\bar{c}$ pairs in the system is fixed, the average number of the hidden charm particles produced in the system is given by a convolution of two probability distributions (\[binom\]) with the right-hand side of (\[NHf\]): $$\begin{aligned}
\label{NHfs}
\langle N_{H} \rangle_{fix}^{(0)} &=&
\sum_{N_{c}=0}^{\infty} w(N_{c} | N_{c\bar{c}})
\sum_{N_{c}=0}^{\infty} w(N_{\bar{c}} | N_{c\bar{c}})
N_{c} N_{\bar{c}}
\frac{\tilde{N}_H}{\tilde{N}_1 \tilde{N}_{\bar{1}}}
\nonumber \\
&=&
\xi^2
(N_{c\bar{c}})^2 \frac{\tilde{N}_H}{\tilde{N}_1 \tilde{N}_{\bar{1}}} .\end{aligned}$$
In a more realistic situation, if the total number of $c\bar{c}$ pairs in the entire system fluctuates according to the Poissonian law (\[prob\]), the average number of hidden charm in the subsystem is given by $$\langle N_H \rangle_{sub}^{(0)} = \xi^2 \langle N_{c\bar{c}} \rangle
(\langle N_{c\bar{c}} \rangle +1)
\frac{\tilde{N}_H}{\tilde{N}_1 \tilde{N}_{\bar{1}}}. \label{NHPs}$$ Similarly for the double and triple charm: $$\begin{aligned}
\langle N_2 \rangle_{sub}^{(0)} &=& \xi^2 \langle N_{c\bar{c}} \rangle^2
\frac{\tilde{N}_2}{\tilde{N}_1^2} , \label{N2Ps}\\
\langle N_3 \rangle_{sub}^{(0)} &=& \xi^3
\langle N_{c\bar{c}} \rangle^3 \frac{\tilde{N}_3}{\tilde{N}_1^3}. \label{N3Ps}\end{aligned}$$ Note, that only the average number of the charm quark-antiquark pairs refers to the entire system in the above equations. The thermal quantities ${N}_H$, $\tilde{N}_1$ and $\tilde{N}_{\bar{1}}$ are related to the subsystem under consideration. Therefore, one can calculate the number of charm particles produced in the subsystem, even if the thermodynamic parameters of the entire system are not known. Moreover, to calculate the number of double and triple charm, one does not actually need the total number of $c\bar{c}$ pairs in the entire system. It suffices to know this number for the subsystem $\xi \langle N_{c\bar{c}} \rangle$. Only the calculation of the hidden charm requires the total number of charm pairs in the entire system, if this number is comparable to $1$ or smaller.
The canonical approach and why it is not appropriate for the charm coalescence {#Canon}
==============================================================================
The canonical ensemble (CE) approach was initially proposed for a thermal treatment of the strangeness productions, when the average number of the strange particles in the system is small ($\alt 1$) and the exact conservation of the net strangeness becomes important [@CanonStr]. It was also applied to the baryonic charge [@CanonBar].
Although, as it will be explained later, this approach is not appropriate for the charm coalescence, I consider it in details because of two reasons. First, it has been widely used for the description of the charmonium production [@We; @WeBMS; @RHICa; @Rep]. Second, it would be instructive to see, what changes, if the event-by-event fluctuations of the number of charm pairs is different from the Poissonian one.
From the formal point of view, there is no problem to apply the canonical approach to the charm coalescence. Let us choose the probability distribution of the event-by-event fluctuations of the number of $c\bar{c}$ pairs as $$\label{probCE}
w_{CE}(N_{c\bar{c}}) = \frac{\gamma_c^2
Z_{N_{c\bar{c}}}(V,T,\{ \tilde{\lambda} \}) }
{\sum_{N_{c\bar{c}}=0}^{\infty} \gamma_c^{2 N_{c\bar{c}}}
Z_{N_{c\bar{c}}}(V,T,\{ \tilde{\lambda} \})},$$ where $$\label{ZNcc}
Z_{N_{c\bar{c}}} (V,T,\{ \tilde{\lambda} \})
\equiv \left. Z_{N_{c} N_{\bar{c}}} (V,T,\{ \tilde{\lambda} \})
\right|_{N_{c}=N_{\bar{c}} \equiv N_{c\bar{c}}}.$$ In the zero approximation (\[Z0\]), the sum in the denominator can be expressed via the modified Bessel function $I_0$: $$\begin{aligned}
\label{ZNccr}
\sum_{N_{c\bar{c}}=0}^{\infty} \gamma_c^{2 N_{c\bar{c}}}
Z_{N_{c\bar{c}}} &\approx&
\sum_{N_{c\bar{c}}=0}^{\infty}
\frac{\left( \gamma_c^{2} \tilde{N}_1 \tilde{N}_{\bar{1}}
\right)^{N_{c\bar{c}}}}{\left( N_{c\bar{c}} ! \right)^2} \\
&=&
I_{0} \left( 2 \gamma_c \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \right).
\nonumber\end{aligned}$$
Again, to find the number of charm particles with different charm content we have to convolute the probability (\[probCE\]) with the expressions (\[N1f\]) and (\[NHf\])–(\[N3f\]) similarly to (\[NkP\]). The zero approximation results are expressed via the modified Bessel functions $I_k$, $k=0,\dots,3$: $$\begin{aligned}
\langle N_{1} \rangle_{CE}^{(0)} &\approx&
\langle N_{\bar{1}} \rangle_{CE}^{(0)} \approx
\langle N_{c\bar{c}} \rangle \label{N1CE} \\
&\approx&
\gamma_c^{(0)} \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \,
\frac{I_{1} \left( 2 \gamma_c^{(0)} \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \right)}
{I_{0} \left( 2 \gamma_c^{(0)} \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \right)},
\nonumber \\
\langle N_H \rangle_{CE}^{(0)} &=& \left( \gamma_c^{(0)}
\right)^2 \tilde{N}_H, \label{NHCE} \\
\langle N_2 \rangle_{CE}^{(0)} &=&
\left( \gamma_c^{(0)} \right)^2 \tilde{N}_2 \frac{\tilde{N}_{\bar{1}}}{\tilde{N}_1}
\frac{I_{2} \left( 2 \gamma_c^{(0)} \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \right)}
{I_{0} \left( 2 \gamma_c^{(0)} \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \right)},
\label{N2CE}\\
\langle N_3 \rangle_{CE}^{(0)} &=&
\left( \gamma_c^{(0)} \right)^{ 3} \tilde{N}_3 \left(
\sqrt{ \frac{\tilde{N}_{\bar{1}}}{\tilde{N}_1} }
\right)^{ 3}
\frac{I_{3} \left( 2 \gamma_c^{(0)} \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \right)}
{I_{0} \left( 2 \gamma_c^{(0)} \sqrt{\tilde{N}_1 \tilde{N}_{\bar{1}}} \right)}
. \nonumber \\
\label{N3CE}\end{aligned}$$ The formula (\[NHCE\]) for the hidden charm in CE has the same form as the corresponding formula in GCE approach (see the middle part of (\[NHG\])), but the result is different, because the value of $\gamma_c^{(0)}$ is not the same. Indeed, this value has to be found at given $N_{c\bar{c}}$ from the transcendental equation (\[N1CE\]). Its solution may be quite different from (\[gamcG\]).
The equation (\[N1CE\]) can be solved analytically in two limiting cases: $\langle N_{c\bar{c}} \rangle \gg 1$ and $\langle N_{c\bar{c}} \rangle \ll 1$. In the first case, the ratios of the Bessel functions tends to $1$: $$\frac{I_{k} \left( x \right)}
{I_{0} \left( x \right)}
\simeq 1 \mbox{\ \ at \ \ }
x \gg 1,
\mbox{ $k=1,2,\dots$}$$ and the formulas are reduced to those of the grand canonical approach.
In the second case, $\langle N_{c\bar{c}} \rangle \ll 1$, the Bessel functions can be replaced by the leading terms of their Taylor expansions: $$I_{k} \left( x \right)
\simeq \frac{x^k}{2^k k!} \mbox{\ \ at \ \ }
x \ll 1,
\mbox{ $k=0,2,\dots$}.$$ The equation (\[N1CE\]) then becomes $$\langle N_{c\bar{c}} \rangle \simeq \left( \gamma_c^{(0)} \right)^2
\tilde{N}_1 \tilde{N}_{\bar{1}}.$$ The factor $\gamma_c^{(0)}$ can be easily found: $$\gamma_c^{(0)} \simeq \sqrt{
\frac{\langle N_{c\bar{c}} \rangle}{\tilde{N}_1 \tilde{N}_{\bar{1}}}
}.$$ Then the number of hidden, double and triple charm particles can be calculated: $$\begin{aligned}
\langle N_H \rangle_{CE}^{(0)} & \simeq & \langle N_{c\bar{c}} \rangle
\frac{\tilde{N}_H}{\tilde{N}_1 \tilde{N}_{\bar{1}}}, \label{NHCEa} \\
\langle N_2 \rangle_{CE}^{(0)} & \simeq & \frac{1}{2} \langle N_{c\bar{c}} \rangle^2
\frac{\tilde{N}_2}{\tilde{N}_1^2}, \label{N2CEa}\\
\langle N_3 \rangle_{CE}^{(0)} & \simeq & \frac{1}{6}
\langle N_{c\bar{c}} \rangle^3 \frac{\tilde{N}_3}{\tilde{N}_1^3}. \label{N3CEa}\end{aligned}$$ The result (\[NHCEa\]) for the hidden charm coincides with that for the Poissonian fluctuation case (\[NHP\]) in the limit $\langle N_{c\bar{c}} \rangle \ll 1$. The reason for this coincidence is the following property of the both distributions: $$w(n) \ll w(1) \mbox{\ \ at \ \ } \langle N_{c\bar{c}} \rangle \ll 1,
\mbox{\ \ } n=2,3,\dots \ .$$ Therefore, the hidden charm hadrons are mostly produced in the systems containing a single $c\bar{c}$ pair. The probabilities to observe exactly one $c\bar{c}$ pair becomes approximately equal to each other for such distributions, if the average number of the pairs is the same and is small: $$w_{P}(1) \simeq w_{CE}(1) \simeq \langle N_{c\bar{c}} \rangle
\mbox{\ \ at \ \ } \langle N_{c\bar{c}} \rangle \ll 1.$$
In contrast, the double and triple charm particles cannot be produced in the system containing only one $c\bar{c}$ pair. Two and three pairs at least are needed to produce, respectively, a double and a triple charm particle. The probability to have two or three pairs in the system are different in the Poissonian and the canonical cases. From this reason, the results are essentially different.
Although the Poissonian and canonical probability laws give the same result for the hidden charm in two limiting cases: $\langle N_{c\bar{c}} \rangle \ll 1$ and $\langle N_{c\bar{c}} \rangle \gg 1$, they are up to about 10% different in the intermediate region $\langle N_{c\bar{c}} \rangle \sim 1$ [@We].
Which of two approaches, the Poissonian or the Canonical one, is appropriate for the description of the charm coalescence in a system containing a small number of $c\bar{c}$ pairs? In fact, the key assumption (\[probCE\]) of the canonical approach relates the fluctuations of the number of charm pairs to the thermodynamic parameters of the system at chemical freeze-out. This is in an obvious contradiction with the basic postulates of the statistical coalescence model. Indeed, we have postulated that charm quarks are produced exclusively at the initial stage of the reaction in hard parton collisions. Creation and annihilation at later stages are neglected. Therefore, the fluctuations of the number of the $c\bar{c}$ pairs cannot have any relation to the properties of the system at the thermal stage. From these reasons, the canonical approach is not appropriate for the treatment of charm coalescence.
The case of strangeness hadronization is essentially different: strange quarks and antiquarks can be produced at the thermal stage, as far as the temperature is larger or comparable to the strange quark mass. Therefore, in the case of full strangeness thermalization (if not only the momenta, but the number of strange quark pairs is thermal), the canonical approach is the most appropriate. Still, it should be used with care if the full strangeness thermalization is not reached: $\gamma_s \not= 1$.
Summary and Outlook {#Summ}
===================
The production of particles with single, double, triple and hidden charm in the framework of the statistical coalescence model has been considered. The grand canonical approach (Section \[Grand\]) is appropriate for systems containing a large number of charm quark-antiquark pairs: $\langle N_{c\bar{c}}\rangle \gg 1$. The solution can be found analytically (\[N1G\])–(\[N3G\]), if the number of hidden, double and triple charm particles is small comparing to the total charm. This is the case at presently available collision energies. At higher energies, the result can be found numerically from the coupled equations (\[eq1\]) and (\[eq2\]).
The grand canonical approach cannot be applied to the system containing a small number of charm quark-antiquark pairs: $\langle N_{c\bar{c}}\rangle \alt 1$. It has been shown that the canonical approach is also inappropriate in this case. It is in variance with the basic postulates of the statistical coalescence model. As far as $c\bar{c}$ pairs are created in mutually independent nucleon-nucleon collisions, the fluctuations of $N_{c\bar{c}}$ follows the Poissonian law. In this case, the result coincides with the grand canonical one for the double and triple charm (\[N2P\]),(\[N3P\]), but differs essentially for the number of the hidden charm (\[NHP\]).
The charm coalescence in a part of a large system has been also studied. It is sufficient to know the thermal parameters of the subsystem under consideration to find the number of particles with different charm content. The thermal properties of the entire system are not needed. Still one has to know the total number of the charm pairs in the entire system to calculate the number of hidden charm particles in the subsystem. For double and triple charm, the information on the number of $c\bar{c}$ pairs in the subsystem suffices.
The obtained derived allow to obtain predictions for double, triple and hidden charm production in heavy ion collisions at all collision energies. At very high energies, like those of the Large Hadron Collider (LHC) in CERN (Switzerland), the expected number of produced charm quark-antiquarks is large (of order of hundreds or even more). Under these conditions, a sizable number of double and triple charm is expected. On the other hand, it would be interesting to see, whether the statistical coalescence model works at low energies. The possibility to study this at the accelerator facility with a very high luminosity which is planned to be build in GSI (Germany) is worth to be checked. A thermal or nonthermal behavior of heavy quarks can tell us much about the properties of the medium created during the heavy ion collision [@thermaliz].
I acknowledge the financial support of the Deutsche Forshunggemanschaft (DFG), Germany.
[99]{}
P. Braun-Munzinger, I. Heppe and J. Stachel, Phys. Lett. B [**465**]{}, 15 (1999) \[arXiv:nucl-th/9903010\];\
G. D. Yen and M. I. Gorenstein, Phys. Rev. C [**59**]{}, 2788 (1999) \[arXiv:nucl-th/9808012\];\
F. Becattini [*et al.*]{}, Phys. Rev. C [**64**]{}, 024901 (2001) \[arXiv:hep-ph/0002267\];\
W. Florkowski, W. Broniowski and M. Michalec, Acta Phys. Polon. B [**33**]{}, 761 (2002) \[arXiv:nucl-th/0106009\]. F. Becattini and U. W. Heinz, Z. Phys. C [**76**]{}, 269 (1997) \[Erratum-ibid. C [**76**]{}, 578 (1997)\] \[arXiv:hep-ph/9702274\]. F. Becattini, Z. Phys. C [**69**]{}, 485 (1996). M. Gaździcki and M. I. Gorenstein, Phys. Rev. Lett. [**83**]{}, 4009 (1999) \[hep-ph/9905515\]. P. Braun-Munzinger and J. Stachel, Phys. Lett. B [**490**]{}, 196 (2000) \[arXiv:nucl-th/0007059\]. M. I. Gorenstein, A. P. Kostyuk, H. Stöcker and W. Greiner, Phys. Lett. B [**509**]{}, 277 (2001) \[arXiv:hep-ph/0010148\]. M. I. Gorenstein, A. P. Kostyuk, H. Stöcker and W. Greiner, J. Phys. G [**27**]{}, L47 (2001) \[arXiv:hep-ph/0012015\];\
P. Braun-Munzinger and J. Stachel, Nucl. Phys. A [**690**]{}, 119c (2001) \[arXiv:nucl-th/0012064\]. A. P. Kostyuk, M. I. Gorenstein, H. Stöcker and W. Greiner, Phys. Lett. B [**531**]{}, 195 (2002) \[arXiv:hep-ph/0110269\]. J. Phys. G [**28**]{}, 2297 (2002) \[arXiv:hep-ph/0204180\];\
A. P. Kostyuk, J. Phys. G [**28**]{}, 2047 (2002) \[arXiv:hep-ph/0111096\]; Nucl. Phys. A [**715**]{}, 569 (2003) \[arXiv:hep-ph/0209139\]; arXiv:hep-ph/0306123. A. Kostyuk and H. Stoecker, GSI Scientific Report 2003, p. 86 \[arXiv:nucl-th/0501077\]. L. Grandchamp and R. Rapp, Phys. Lett. B [**523**]{}, 60 (2001) \[arXiv:hep-ph/0103124\]; Nucl. Phys. A [**709**]{}, 415 (2002) \[arXiv:hep-ph/0205305\];\
L. Grandchamp, R. Rapp and G. E. Brown, Phys. Rev. Lett. [**92**]{}, 212301 (2004) \[arXiv:hep-ph/0306077\]. M. I. Gorenstein, A. P. Kostyuk, H. Stöcker and W. Greiner, Phys. Lett. B [**524**]{}, 265 (2002) \[arXiv:hep-ph/0104071\];\
M. I. Gorenstein, A. P. Kostyuk, L. McLerran, H. Stöcker and W. Greiner, J. Phys. G [**28**]{}, 2151 (2002) \[arXiv:hep-ph/0012292\];\
A. P. Kostyuk, M. I. Gorenstein, H. Stöcker and W. Greiner, Phys. Rev. C [**68**]{}, 041902 (2003) \[arXiv:hep-ph/0305277\]. A. Andronic, P. Braun-Munzinger, K. Redlich and J. Stachel, Phys. Lett. B [**571**]{}, 36 (2003) \[arXiv:nucl-th/0303036\]; Nucl. Phys. A [**715**]{}, 529 (2003) \[arXiv:nucl-th/0209035\]. E. L. Bratkovskaya, A. P. Kostyuk, W. Cassing and H. Stöcker, Phys. Rev. C [**69**]{}, 054903 (2004) \[arXiv:nucl-th/0402042\]. M. A. Moinester, Z. Phys. A [**355**]{}, 349 (1996) \[arXiv:hep-ph/9506405\];\
V. V. Kiselev and A. K. Likhoded, Phys. Usp. [**45**]{}, 455 (2002) \[arXiv:hep-ph/0103169\]. M. Mattson [*et al.*]{} \[SELEX Collaboration\], Phys. Rev. Lett. [**89**]{}, 112001 (2002) \[arXiv:hep-ex/0208014\];\
A. Ocherashvili [*et al.*]{} \[SELEX Collaboration\], arXiv:hep-ex/0406033. V. V. Kiselev and A. K. Likhoded, arXiv:hep-ph/0208231. K. Redlich and L. Turko, Z. Phys. C [**5**]{}, 201 (1980);\
J. Rafelski and M. Danos, Phys. Lett. B [**97**]{}, 279 (1980);\
J. Cleymans, K. Redlich and E. Suhonen, Z. Phys. C [**51**]{}, 137 (1991). M. I. Gorenstein, M. Gazdzicki and W. Greiner, Phys. Lett. B [**483**]{}, 60 (2000) \[arXiv:hep-ph/0001112\]. N. Brambilla [*et al.*]{}, arXiv:hep-ph/0412158. Y. L. Dokshitzer and D. E. Kharzeev, Phys. Lett. B [**519**]{}, 199 (2001) \[arXiv:hep-ph/0106202\];\
H. van Hees and R. Rapp, arXiv:nucl-th/0412015.
[^1]: The recent observation of the doubly charmed baryon $\Xi_{cc}^{+}$ [@Xicc] is still under discussion [@XiccComm].
[^2]: The suppression (enhancement) factor $\gamma_s$ (the same for $\gamma_c$) is equivalent to an additional chemical potential $\mu_{|s|}$ which, in contrast to $\mu_{s}$, has the same influence on the strangeness as on the antistrangeness: $\gamma_s = \exp\left( \frac{\mu_{|s|}}{T} \right)$
[^3]: Due to the inequality (\[inequ\]), the contribution of other terms becomes sizable only at very large $N_{c}$ and/or $N_{\bar{c}}$. This case can be treated within the grand canonical approach and there is no reason to study it here.
[^4]: This approximation obviously breaks down at very large $N_{c\bar{c}}$ when the energy of the produced $c\bar{c}$ pairs becomes comparable with the total energy of the system. But the probability of such events is clearly negligible. Only a tiny fraction of the total energy of the system is accumulated into the charm particles.
|
---
title: Notes on Orc
---
|
---
address:
- 'Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Av.Libertador Bernardo O’Higgins 3363, Santiago, Chile'
- |
CNRS (French National Center for Scientific Research), LIG, F-38000 Grenoble, France\
and Univ. Grenoble Alpes, LIG, F-38000 Grenoble, France
author:
- Mario Bravo
- Panayotis Mertikopoulos
bibliography:
- 'IEEEabrv.bib'
- 'Bibliography.bib'
title: On the robustness of learning in games with stochastically perturbed payoff observations
---
[^1]
[^2]
Introduction {#sec:introduction}
============
The model {#sec:model}
=========
Regret minimization {#sec:regret}
===================
Extinction of dominated strategies {#sec:dominated}
==================================
Long-term stability and convergence analysis {#sec:folk}
============================================
Time averages in $2$-player games {#sec:averages}
=================================
Discussion {#sec:discussion}
==========
Appendix {#appendix .unnumbered}
========
Technical proofs {#app:main}
================
Auxiliary results {#app:auxiliary}
=================
[^1]: The authors are greatly indebted to Roberto Cominetti for arranging the visit of the second author to the University of Chile and for his many constructive comments. The authors would also like to express their gratitude to Mathias Staudigl for his many insightful comments and suggestions, to Bill Sandholm, Josef Hofbauer, and Yannick Viossat for helpful discussions, and to two anonymous referees for their detailed remarks and recommendations.
[^2]: Part of this work was carried out during the authors’ visit to the Hausdorff Research Institute for Mathematics at the University of Bonn in the framework of the Trimester Program “Stochastic Dynamics in Economics and Finance” and during the second author’s visit to the University of Chile. MB was partially supported by Fondecyt grant No. 11151003, and the Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003. PM was partially supported by the French National Research Agency (grant nos. NETLEARN–13–INFR–004 and GAGA–13–JS01–0004–01) and the French National Center for Scientific Research (grant no. REAL.NET–PEPS–JCJC–INS2I–2014.)
|
---
author:
- |
Paolo d’Alessandro\
Dept. Of Mathematics\
Third University of Rome\
e-mail:dalex@math.uniroma3.it
title: A new conical internal evolutive LP algorithm
---
Introduction
============
In this paper we develop extensions of the conical techniques exposed in the book [@dacoap97], and we develop new algorithms as well. We make an effort to keep this paper enough self-contained, however, the book [@dacoap97] (hereafter often referred to as ”the book” for the sake of brevity) can be useful for further details. Such book is an outgrow of two papers published on the journal Optimization, namely [@dadualcon] and [@daprimcon92].
To begin with the conical approach has his own nomenclature. Terms like duality have a different meaning than those of standard approaches (see e.g. [@bertsimas1997]). Other terms like strict tangency are peculiar to the conical approach. The term bounded refers to a LP problem (maximization of a linear functional on a polyhedron) that is feasible and has maximum. The specification internal for a primal algorithm has also a meaning related to the conical context, and, as explained below, refers to the particular approach followed here to reach optimality, as opposed to the primal algorithm presented in the book.
In the book we stressed the particular interest of conical conditions that are at one time purely conical ( that is, involve only cones) and purely pointed (that is, the involved cones are pointed). Dual conical feasibility and optimality conditions are of this kind. Although they have lead in the book to an enumerative algorithm, the dual conical methodology is not only useful for its peculiar properties (e.g. it gives a closed form expression of the maximum,solves a whole class of problems and is particularly efficient in handling parameter variations), but it has provided an useful tool for applications in fields like control and game theory. The related literature is by now very extended and relevant, and has provided many interesting new results (see e.g [@dade01] and the bibliography therein).
As to the primal conical conditions given in the book, the first is conical but not pointed. The second involves a pointed cone and an affine space, and has been the starting point for the development an evolutive primal external conical algorithm, enjoying exact finite convergence.
We shall take here this whole business to a more advanced stage. First because we shall give a further primal condition, that is purely conical and purely pointed. Secondly because, intertwining this new primal approach with the generators machinery used in the book to develop the dual conical algorithm, we provide a further algorithm, which although based on generators is evolutive. Thirdly we give a closed form expression of the maximum in the primal setting too. More than that we give explicit closed forms of the solution both on the range space of the coefficient matrix and in terms of unknowns of the LP problem. Finally we stress that the present result complete the theory presented in the book from yet another point of view. In fact the second primal optimality condition is a tangency condition (of an affine space to the non-negative orthant) and the previous primal algorithm reaches tangency landing the affine onto the cone. Thus we knew that one could in principle try to reach tangency from the other side: that is, starting with the two sets that meet each other and then taking the affine to touch the only boundary of the cone (thereby achieving tangency). However, developing such an algorithm (emerging submarine instead of landing hydroplane metaphor) has defeated us for a while. It is finally presented here.
As to notations we stick to those adopted in the book with only one variation in the interest of simplicity. The variation is that when we shall get to the parameterized feasibility formulation of optimality we shall denote the augmented coefficient matrix and bound vector in the same way as the non-augmented case, leaving to the context the specification of which is which. Recall also that we use always the same symbol $P$ for the non-negative orthant of any euclidean space. Again the space to which such symbol is referred is specified by the context.
Mathematical base
=================
We will assume thorough the hypothesis of strict tangency of the feasibility and linear programming problems we study. Whereas this is no restriction of generality as proved in the above book, it has a great geometrical importance and it yields a much more neat and elegant path to the development of algorithms. Note that actually such an assumption is made in the primal conical LP algorithm introduced in [@daprimcon92], and reported in the book, within the theorem that states exact finite convergence. Here however, we simplify matters avoiding to postpone the use of such hypothesis as much as possible in an effort to achieve maximum generality.
Let us start considering a linear feasibility problem. That is, the problem of determining whether or not a polyhedron is non-void and in the positive case finding at least a point in the polyhedron. Better yet would be finding more points, i.e., having the possibility of exploring the polyhedron, even, ideally, finding all of its points. A polyhedron is the intersection of a finite set of semispaces. Thus a set $\mathcal{G}$ of the form:
$$\mathcal{G}=\{x:Gx\leq \mathbf{v}\}$$
where $G$ is an $n\times m$ real matrix, $x$ (the unknown vector) is in $R^{m}$ and $\mathbf{v}$ (the bound vector) is in $R^{n}$. The polyhedron is a cone if and only if $\mathbf{v}=0$.We shall often denote $\mathcal{R}(G)$ (the range of $G$) by $F$, for brevity. Recall also that the vector $\mathbf{v}-Gx$ is called the slack vector (a feasible slack vector if $\mathcal{G}\neq \phi $ and $x\in \mathcal{G}$).
We now make a simple but important remark, independent of the hypothesis of strict tangency. If we decompose $\mathbf{v}$ as $\mathbf{v}=\mathbf{v}_{F}+\upsilon $, where $\mathbf{v}_{F}$ is the orthogonal projection of $\mathbf{v}$ on $F$ and $\upsilon $ is the orthogonal projection of $\mathbf{v}$ on $F^{^{\perp }}$, we can write for the inequality defining the polyhedron:
$$Gx\leq \mathbf{v}_{F}+\upsilon$$
Because vector inequalities are translation invariant, assuming $Gz=\mathbf{v}_{F}$, the latter is true if and only if:
$$Gx-\mathbf{v}_{F}=G(x-z)\leq \upsilon$$
Thus in passing from one to the other condition, feasibility is invariant, the slack vector too, and the solution is varied by a constant vector $z$. Note that the polyhedron is a cone (when $\mathbf{v}_{F}=0$) or a translated cone (when $\mathbf{v}_{F}\neq 0$). if and only if $\upsilon =0$. In what follows we assume that this is not the case i.e.: $\upsilon \neq 0$. Otherwise we always have the trivial solution $x=0$ or, respectively, $x=z$, and our investigation would become pointless. Also notice that under this assumption the slack vector cannot be zero because $\upsilon \neq
0\Longrightarrow \forall w\in R^{m},\upsilon -Gw\neq 0$.
Notice that if $\mathbf{v\in P}$, then $\mathcal{G}\neq \phi $, because $x=0$ is obviously a solution. Similarly if $\upsilon \mathbf{\in }P$, then $\mathcal{G}\neq \phi $, because $x=z$ is obviously a solution.
To say that the associated problem is strictly tangent [@dacoap97] means that the subspace $\mathcal{R}(G)$ is strictly tangent to the non-negative orthant $P$, that is, $\mathcal{R}(G)$ meets $P$ in the only origin:
$$\mathcal{R}(G)\cap P=\{0\}$$
We now recall the second primal conical feasibility (necessary and sufficient) condition [@dacoap97]:
$$\mathcal{G}\neq \phi \Longleftrightarrow (\mathbf{v}+\mathcal{R}(G)\cap
P\neq \phi$$
This is a conical condition, but there is an affine set appearing in it, namely $\mathbf{v}+\mathcal{R}(G)$.
Recall also that, if feasibility prevails, to derive a feasible slack vector in the range space of $G$ we can find any vector $y$ in the intersection:
$$y\in (\mathbf{v}+\mathcal{R}(G)\cap P$$
Consequently, if we want a solution in the domain space, we can solve in $x$ the system:
$$Gx=\mathbf{v}-y$$
Obviously if we solve this equation for all $y$ in $(\mathbf{v}+\mathcal{R}(G)\cap P$ we get all and nothing but the points of the polyhedron $\mathcal{G}$.
New primal conical feasibility conditions
=========================================
We shall now start elaborating further on this second primal feasibility condition. One of the dividends will be the introduction of a new purely conical and pointed primal condition. As a first remark note that (as it is immediate to prove):
$$\mathbf{v}+\mathcal{R}(G)=\upsilon +\mathcal{R}(G)$$
and therefore we can write:
$$\mathcal{G}\neq \phi \Longleftrightarrow (\upsilon +\mathcal{R}(G)\cap P\neq
\phi$$
Next we consider a subspace that is extended with respect to $\mathcal{R}(G)$, namely (with $\mathcal{L}(.)$ we denote linear extensions and with $Co(.)$ conical extensions):
$$F_{e}=\mathcal{L}(\mathbf{v})+F=\mathcal{L}(\upsilon )+F=\mathcal{L(}\upsilon +\mathcal{R}(G))$$
Finally we need the cone:
$$C_{e}=Co(\upsilon )+F=Co(\upsilon +F)$$
Note that clearly:
$$F_{e}=C_{e}+(-C_{e})$$
We are now in a position that allows to state the following new primal conical feasibility condition, which requires that a pointed cone do not degenerate into the trivial (singleton of the origin) cone:
The following primal conical feasibility conditions hold:
$$\mathcal{G}\neq \phi \Longleftrightarrow C_{e}\text{ }\cap \text{ }P\neq
\{0\}$$
or equivalently:
$$\mathcal{G}=\phi \Longleftrightarrow C_{e}\text{ }\cap \text{ }P=\{0\}$$
If $\mathcal{G}\neq \phi $, by the above version of the second primal feasibility condition, $(\upsilon +\mathcal{R}(G)\cap P\neq \phi $ and this intersection is made up of non-zero vectors. Let $y$ be one of those, so that $y=\upsilon +z$ with $z$ in $\mathcal{R}(G)$. Therefore there are non-zero points in $C_{e}$ $\cap $ $P$ and the condition is necessary.
Conversely take $y\neq 0$ in $C_{e}$ $\cap $ $P.$ It will have the expression $y=\beta \upsilon +z$ with $z$ in $\mathcal{R}(G)$ for some real $\beta >0$ (if $\beta $ were zero $y$ would be zero too by virtue of strict tangency). Thus the non-zero vector $w=1/\beta (\beta \upsilon +z)$ satisfies: $$w=1/\beta (\beta \upsilon +z)=\upsilon +\frac{1}{\beta }z\in C_{e}$$
But evidently it is also true that $w\in (\upsilon +\mathcal{R}(G))\cap P$ and this completes the proof.
The procedure explained in the proof to pass from the vector $y$ in the cone $C_{e}$ $\cap $ $P$ to the vector $w$ moving along the ray generated by $y$, will be called calibration. Notice that such procedure is viable numerically in a very simple way. In fact, if we denote by $P_{F^{\bot }}$ the orthogonal projection onto $F^{\bot },$ it is obvious that:
$$\beta =(P_{F^{\bot }}y)_{i}/\upsilon _{i};\forall i\text{ s.t. }\upsilon
_{i}\neq 0$$
Another important fact that follows immediately from the above proof is that if we let, in the feasibility case $P_{c}=(v+\mathcal{R}(G)\cap P$ (more on this set later) we can state the following:
The following relation holds whenever feasibility prevails
$$Co(P_{c})=C_{e}\text{ }\cap \text{ }P$$
Next notice that if $\mathcal{G}=\phi $ so that $C_{e}$ $\cap $ $P=\{0\}$ it may either be the case that:
$$-C_{e}\text{ }\cap \text{ }P=\{0\}$$
in which case it is clear that $F_{e}$ is strictly tangent to $P$; or that
$$-C_{e}\text{ }\cap \text{ }P\neq \{0\}$$
In this respect, the next natural question is to find out whether it can be the case that $-C_{e}$ $\cap $ $P\neq \{0\}$ and $C_{e}$ $\cap $ $P\neq
\{0\} $ at the same time. We shall show in the proof of the next result that this cannot be the case, because the two cones $C_{e}$ and $-$ $C_{e}$ live in opposite semispaces and the non-negative orthant, with the exception of the origin, is entirely contained in the interior of one of them. And this means that we can establish the following further feasibility condition, which is more handy in that it substitutes the subspace $F_{e}$ to the cone $C_{e}$.
If $F_{e}$ $\cap $ $P\neq \{0\}$ then the sign of $\beta $ is constant $\forall y\neq 0$ in $F_{e}\cap P$. Moreover:
$$\mathcal{G}\neq \phi \Longleftrightarrow \ F_{e}\text{ }\cap \text{ }P\neq
\{0\}\text{ and }\beta >0$$
Suppose $F_{e}$ $\cap $ $P\neq \{0\}$ and consider an $y\neq 0$ in $F_{e}$ $\cap $ $P$. Because $F$ is strictly tangent to $P$, we know from theorem 6.2.1 in [@dacoap97] that $F$ is contained in an hyperplane $H$, which is strictly tangent to $P$ and there is a vector $n$, normal to this hyperplane, which is internal to $P$. Then for some $z\in $:$F,\gamma \neq 0$ :
$$0<(n,y)=(n,(\gamma \upsilon +z))=\gamma (n,\upsilon )$$
Assume to fix the ideas that $\gamma >0$ so that $(n,\upsilon )>0$, $y\in
C_{e}$ and therefore $y\in $.$C_{e}$ $\cap $ $P$. In this case by the previous condition feasibility prevails. Clearly by the same argument and because $n$ is internal to $P$:
$$\begin{aligned}
(n,C_{e}) &\subset &[0,+\infty ) \\
(n,P) &\subset &[0,+\infty )\end{aligned}$$
Thus both the cones $C_{e}$ and $P$ are contained in the semispace $\{x:(n,x)\geq 0\},$ delimited by the hyperplane $H$, and, in addition, $P$ is in the interior of the semispace, with the only exception of the origin, because $n$ is in the interior of $P$.
Moreover::
$$(n,-C_{e})\subset (-\infty ,0]$$
and this latter implies that $-C_{e}$ is in the opposite semispace (i.e. the semispace $\{x:(n,x)\leq 0\}$) with respect to $P,$ so that $-C_{e}$ $\cap $ $P=\{0\}$. Therefore the sign of $\beta $ in $F_{e}$ $\cap $ $P=(C_{e}+(-C_{e}))\cap $ $P$ is constant.
Assuming instead that $\gamma <0$ and reasoning along the same lines $(n,\upsilon )<0$, $y\in -C_{e}$ and therefore $y\in $.$-C_{e}$ $\cap $ $P$. Moreover in this case:
$$(n,C_{e})\subset (-\infty ,0]$$
and therefore $C_{e}$ $\cap $ $P=\{0\}$ and feasibility does not prevail. In this case $-C_{e}$ $\cap $ $P\neq \{0\}$ and the sign of $\beta $ is constant in $-C_{e}$ $\cap $ $P$. Thus the proof is finished.
There are three mutually exclusive and exhaustive cases possible for the feasibility problem:
a\) $F_{e}$ is strictly tangent to $P$. That is $F_{e}$ $\cap $ $P=\{0\}$: In this case the problem is unfeasible.
b\) $F_{e}$ $\cap $ $P\neq \{0\}$ and $\beta <0$. In this case too the problem is unfeasible.
c\) $F_{e}$ $\cap $ $P\neq \{0\}$ and $\beta >0$. In this case the problem is feasible
As in all algorithms exposed in [@dacoap97], once it is determined that the problem is feasible and an $y\in C_{e}$ $\cap $ $P$ is found, a solution in the domain space can be found solving the equation $Gx=v-w$, where $w$ is obtained from $y$ by the calibration procedure.
Feasibility algorithm
=====================
We next introduce a feasibility algorithm, based on the results obtained so far. To this purpose we have to devise a method to find (in the feasible case) one or more feasible solutions in the range space $y\in C_{e}$ $\cap $ $P$ . Or, which is more easily viable, find an $y\in F_{e}$ $\cap P$ with $\beta >0$ and then calibrate it. The method should fail if and only if unfeasibility prevails so to completely solve the problem. One possible and useful way to go is to exploit the generators machinery for pointed polyhedral cones (and we got purely conical and pointed conditions primarily to that effect) in this primal conical setting. The advantages of this approach will be more and more evident in the sequel. Because $y$ must be in a pointed polyhedral cone given by the intersection of a subspace with the non-negative orthant, we can find one or more of the extreme rays of this cone or, equivalently, of its generators. A good part of the book [@dacoap97] is devoted to the development of a sophisticated machinery to solve this problem, with basic algorithms and two levels of more advanced algorithms as well as an implementation and numerical results. The fundamental results on which the generators techniques are based are given in Chapter 9.
To carry out this program, we only need to find an operator whose null space is $F_{e}$. This is not difficult. In fact:
$$F_{e}=\mathcal{L}(\upsilon )+F$$
Where the sum is a direct orthogonal sum. We introduce the notations $V=$ $\mathcal{L}(\upsilon )$ and $P_{V}$ and $P_{F}$ to denote, respectively, the orthogonal projections onto $\ V$ and $F$. Then the orthogonal projection onto $\mathcal{L}(\upsilon )+F$ is given by $P_{V}$ +$P_{F}$ so that $I-P_{V} $ $-P_{F}$ is the orthogonal projection onto $(V+F)^{\perp }$ Therefore:
$$F_{e}=\mathcal{L}(\upsilon )+F=\mathcal{N}(I-P_{V}-P_{F})$$
Thus we can state the following feasibility algorithm. In the feasible case one can compute either a single or more solutions in the range space. This is not a detail it is a major point on which we shall expand in the next section.
: New primal conical feasibility algorithm
1- Preliminary step: verify whether** **$\mathbf{v}$ or $\upsilon $ are in $P$. If either of those is the case trivial solution are immediately available as explained in Section 2. In such a case STOP.
2- Compute $P_{V}$ and $P_{F}$ and hence $I-P_{V}-P_{F}$.
3- Use the techniques of [@dacoap97] to compute a first generator of $\mathcal{N}(I-P_{V}-P_{F})\cap P$. If none is found the problem is unfeasible. Hence STOP. Otherwise go to step 4.
4- Calibrate the generator. (Recall that $\beta =0$ cannot happen) If $\beta
<0$ the problem is unfeasible. In such case STOP. If $\beta >0$ the problem is feasible. Proceed to the next steps.
5- Facultative step. Can be executed or skipped. Go on to calculate more or all the generators and calibrate each of them.
6- For each calibrated generator $g_{c}$ a solution can be obtained solving $Gx=v-g_{c}$.
In certain applications, it may be convenient (of course in the feasible case) to find a point in relative interior of the pointed polyhedral cone $F_{e}\cap P=\mathcal{N}(I-P_{V}-P_{F})\cap P$. To this effect one can compute all the calibrated generators and then the sum of them (Theorem 6.1.2 in [@dacoap97]) is in the relative interior of the cone.
The significance of the set of all the calibrated generator goes way beyond this simple remark and is of crucial importance in the present context. This issue is fully explored in the next section.
The contact polytope
====================
Suppose that the problem is feasible. Then, as recalled above, the polyhedron $P_{c}=(\mathbf{v}+\mathcal{R}(G)\cap P=(\upsilon +\mathcal{R}(G)\cap P$ is non-void. In the book, leaning on a result by Ben Israel it is shown that this set is actually a polytope, which we called the contact polytope. The properties of the contact polytope play an important role in the book, but it was not determined explicitly. Here we complete the theory with a detailed description. In fact the set $\{g_{ci}:i=1,..,k\}$ of calibrated generators of the pointed polyhedral cone $C_{e}$ $\cap $ $P$ is exactly the set of extreme points of $P_{c}$. This is stated in the next theorem:
The set of calibrated generators of the cone $C_{e}$ $\cap $ $P$ is equal to the set of extreme points of the contact polytope $P_{c}=(\mathbf{v}+\mathcal{R}(G)\cap P$. Thus:
$$P_{c}=\mathcal{C(\{}g_{ci}\})$$
First we prove that any ray of the cone $C_{e}$ $\cap $ $P$ can intersect $P_{c}$ only in a single point. For suppose that two points of a ray, say $z_{1}$ and $\ z_{2}$, be in the intersection, so that we may assume, without restriction of generality, that $z_{2}=\alpha z_{1}$ with $\alpha >1.$ Then $z_{2}-z_{1}\neq 0$ would be both in $\mathcal{R}(G)$ and in $P$. But this contradicts the hypothesis that $\mathcal{R}(G)$ is strictly tangent to $P$. Therefore $z_{2}=z_{1}$ as we wanted to prove.
We know from the previous section that all the points of $\{g_{ci}:i=1,..,k\} $ are in $P_{c}$ and $P_{c}$ is contained in $C_{e}$ $\cap $ $P$. Let now $\{z_{j}:j=1,..,p\}$ be the set of extreme points of $P_{c}$.
We start showing that each calibrated generator is an extreme point of $P_{c}$. In fact suppose that is not so for a certain $\ g_{ci}$. Then we can write (in what follow we drop indexes in sums to keep the notation simple):
$$\ g_{ci}=\sum \alpha _{r}z_{r}$$
with at least two non-zero coefficients. Isolating one term, we can write without restriction of generality (possibly the sum reduces to a single term):
$$\ g_{ci}=\alpha _{1}z_{1}+\sum \alpha _{\eta }z_{\eta }=\alpha
_{1}z_{1}+(\sum \alpha _{\eta })\sum (\alpha _{\eta }/\sum \alpha _{\eta
})z_{\eta }=\alpha _{1}z_{1}+\beta w$$
where $\alpha _{1}+\beta =1$ and $w\in P_{c}$. Because $z_{1}$ is an extreme point, $z_{1}\neq w$, and by what we proved at the beginning, these two vectors cannot be proportional. It follows that this equality contradicts that $g_{ci}$ lies on an extreme ray and therefore we have reached the conclusion that all the points $\ g_{ci}$ are actually extreme points of $P_{c}$.
Finally assume that there is some extreme points that are not in the set of calibrated generators. Let one of these be $z_{j}$. Then because $z_{j}$.$\in C_{e}$ $\cap $ $P$:
$$z_{j}=\sum \alpha _{i}g_{ci}$$
where $\alpha _{i}\geq 0$ $\forall $ $i$ and $\sum \alpha _{i}>0$ If just one $\alpha _{i}$ is non-zero a contradiction is immediate, because if $\alpha _{i}=1$, then $z_{j}$ is a calibrated generator and if $\alpha
_{i}\neq 1$ there are two proportional vectors in $P_{c}$. Excluding this case, either $\sum \alpha _{i}=1$, and then we get the contradiction that an extreme point is a non-zero convex combination of a set of other extreme points or $\sum \alpha _{i}\neq 1$. If that is so, dividing both sides of the above expression of $z_{j}$ by $\sum \alpha _{i}$ we get again the contradiction that there are two points in a ray that belong to the polytope $P_{c}$. The proof is thereby completed.
The determination of the contact polytope and, in particular, the identification of the extreme rays of $C_{e}$ $\cap $ $P$ with the rays generated by the extreme points of $P_{c}$ is a very important addition to our theory. We can score on this result immediately.
We start with a remark on solutions. In fact $P_{c}$ can be also viewed as the set of all feasible slack vectors, in the sense that it contains all and nothing but the slack vectors $y$ for which all solution of $Gx+y=\mathbf{v}
$ is a feasible solution. Consequently the set of vectors $x$ obtained in this way is the set of all feasible solutions. We can summarize this in the following::
if we let the above feasibility algorithm run to compute all the calibrated generators, then all and nothing but the solution of the problem are given by $\{x:Gx=\mathbf{v}-y,$ $y\in \mathcal{C(\{}g_{ci}\})\}$.
It should be stressed at this point that the present theory yields in a primal conical setting an explicit expression for both the polytope of feasible solution in the range space (slack vectors) and the polyhedron of the feasible solutions of the problem
Another important consequence of this result is that we are now in the position of introducing an internal primal conical LP algorithms.
Before getting into this (in the next section), let us recall briefly the well-known parameterized feasibility formulation on which we based all LP methods (see the book for more details).
Consider the problem of maximizing a linear functional on a polyhedron (Problem LP)
$$\begin{aligned}
&&\max \text{ }fx\text{ } \\
subject\text{ }to &:&\text{ }Gx\leq \mathbf{v}\end{aligned}$$
We can rewrite this as:
$$\begin{aligned}
&&\max \text{ }h \\
subject\text{ }to &:&\widehat{G}x\leq \widehat{\mathbf{v}}(h)\end{aligned}$$
where the augmented coefficient matrix $\widehat{G}$ is obtained adding to $G $ a row with the entries of $-f$ and the augmented bound vector $\widehat{\mathbf{v}}(h)$ is obtained adding to $\mathbf{v}$ a last entry equal to $-h$.
In what follows we shall soon use for this problem the same notations of the feasibility problem in order to simplify our presentation. Thus the reader is advised that it is the context to determine whether we refer to plain or augmented coefficient matrix and to plain or augmented bound vector. The same rule we apply to all the other mathematical entities related to the problem. Thus, for example we still denote by $P_{c}$ the contact polytope corresponding to the polyhedron defined by $\widehat{G}x\leq \widehat{\mathbf{v}}(h)$. Also the non-negative orthant is still denoted by $P$ in the augmented euclidean space. However, whenever beneficial to clarity, we explicitly denote dependence of the appropriate items on $h$ (writing, e.g., $P_{c}(h))$.
In this parameterized feasibility setting, to solve the problem, we have to find an $h_{o}$ such that if $h>h_{o}$ then the polyhedron $\{x:\widehat{G}x\leq \widehat{\mathbf{v}}(h)\}$ is void and, if $h\leq h_{o}$, the same polyhedron is non-void. Such an $h_{o}$, if it exists at all, is the optimum value of the functional. In terms of the second primal conical feasibility condition we have to find $h_{o}$ that verifies the following tangency condition:
$$(\widehat{\mathbf{v}}(h)+\mathcal{R}(\widehat{G}))\cap P=\phi \text{ if }h>h_{o}$$
$$(\widehat{\mathbf{v}}(h)+\mathcal{R}(\widehat{G}))\cap P=P_{c}\neq \phi
\text{ if }h\leq h_{o}$$
The primal conical LP algorithm in the book (under conditions that insure feasibility and boundedness - see next section) started from a large value of $h$ , so to insure that $P_{c}$ be void and diminished such an $h$ until the tangency of the affine space to the non-negative orthant is reached, so that such value of $h$ is just the optimum value $h_{o}$of the functional . This is the external approach.
Here we take the dual (how many meaning of this word!) view: we start from low values of $h,$ so that the contact polytope is non-void, and increment $\ h$, until the contact polytope is squeezed into the maximal face of the non-negative orthant, that corresponds to a zero last component. At that point tangency prevails and, therefore, the corresponding value of $h$ is equal to the optimum $h_{o}$. Actually we give an algorithm of this sort in two versions. The second one will realize a further particularly important advance within our methodology. To each of them we devote the next two sections.
Primal conical internal algorithm: first version.
=================================================
As in the book, and without restriction of generality, we assume strict tangency Feasibility can be ascertained as illustrated in the previous section. We recall from the book that, in view of Theorem 6.3.1, under strict tangency, feasibility implies boundedness. Thus we can now work on optimality with all three hypothesis (strict tangency feasibility and boundedness) in force. Keep in mind that we refer to the augmented problem now, although we do not change notations.
In view of the results of the foregoing section, in the feasible case, to know the calibrated generators of $C_{e}$ $\cap $ $P$ is the same as knowing the extreme points of the contact polytope $P_{c}$. As we shall see momentarily the knowledge of certain extreme points of $P_{c}.$ is equivalent to the knowledge of the maximum value of the functional . It cannot be overemphasized the importance of the fact that, by the following theorem, we obtain a closed form expression for the maximum in a primal setting, just as a closed expression for the maximum was already given in the dual setting (Theorem 5.3.1 in the book).
\[maxformula\]Suppose to choose an $h$ such that the augmented problem is feasible($h\leq h_{o}$) and let $\{g_{ci}:i=1,..,k\}$ be the set of extreme points of $P_{c}(h)$ (or, what is the same, calibrated generators of .$C_{e}(h)$ $\cap $ $P$) then, letting $h_{m}=\max \{g_{ci_{n+1}}:i=1,..,k\}$:
$$h_{o}=h+h_{m}$$
Passing from an $h$ to $h+\Delta h$ corresponds to adding $-\Delta h$ to the last component of all points of $\mathbf{v}+\mathcal{R}(G)$. Thus all points in the contact polytope with the last component equal to $h_{m}$ are still in the new contact polytope $P_{c}(h+h_{m})$ and fall in the maximal face of $P,$ $M_{n+1}=\{y:y\in P$ and $y_{n+1}=0\}$. Suppose now that in $P_{c}(h+h_{m})$ there is a point $z$ with $z_{n+1}=\rho >0.$ Then passing from $h+h_{m}$ to $h$ (that is, incrementing $h$ by $-h_{m}$) this point $z$ translates to a point with all the same components but the last, which is equal to $\rho +h_{m}>h_{m}$ and, in addition, it clearly is in $P_{c}(h)$. However the last component $\gamma $ of all the points in $P_{c}(h)$ satisfies $h_{m}=\max \{g_{ci_{n+1}}:i=1,..,k\}$ $\geq \gamma \geq $ $\min
\{g_{ci_{n+1}}:i=1,..,k\}$ by fact that a polytope is the convex extension of the set of its extreme points. Therefore we have found a contradiction and it follows that $P_{c}(h+h_{m})\subset M_{n+1}$, or, in other words, $\mathbf{v}+\mathcal{R}(G)$ is tangent to $P$. By the second primal conical optimality condition, $h_{o}=h+h_{m}$ is the optimum value of the functional and solves the problem in the range space of $G$.
In the proof of the theorem the procedure to find solutions is already built in. We record such procedure in the following Corollary (by $e_{i}$ we denote the vector that has all zero components but the $ith$, which is equal to $1$):
\(i) Let $g_{c}$ be an extreme point of $P_{c}(h)$ such that $g_{c_{n+1}}=h_{m}$. Then an optimum slack vector is given by $y_{o}=g_{c}-h_{m}e_{n+1}$, $y_{o}$ is an extreme point of $P_{c}(h_{o})$ and all solutions of the equation $Gx=\mathbf{v}-y_{o}$ (that necessarily exist) are optimum solutions.
Let $\{g_{ci}\}$ be the set of extreme points of $P_{c}(h_{o})$ . Then the set of all optimum solutions is obtained solving the equation $Gx=\mathbf{v}-y$ where $y\in \mathcal{C}(\{g_{ci}\})$.
With this Corollary we have completed the picture of the conical approach giving also the explicit closed form for the sets of optimal solutions both in the range space and in the domain space.
The proof of the Corollary requires but trivial new verifications ans can be safely omitted.
We can now structure a PL algorithm.. First set an $h$ small enough (it can be arbitrarily small) to ensure that $h<h_{o}$ (a remark on this is given right after the statement).. Then apply the following
(Primal conical internal algorithm)
Step1 Find all the calibrated generators with positive last component.
Step2. Set $h_{o}=h+h_{m},$ where $h_{m}$ is the maximum of last components of the found calibrated generators
Step3 Consider any calibrated generator $g_{ci}$ such that $g_{ci_{n+1}}=h_{m}$. Then an optimum slack vector is given by $y_{o}=g_{ci}-h_{m}e_{n+1}$ and an optimum solution $x_{o}$ is given by any solution of the equation $Gx=\mathbf{v}-y_{o}$.
The proof of the algorithm is given in the theory so far developed. The above Corollary also illustrate how to find the set of all the solutions, if needed.
Note that the internal and external algorithm (Described in Ch 11 of the book) complete each other. In fact if no calibrated generator with positive last component is found we are either at the optimum i.e., $h=h_{o}$, or $h>h_{o}$. In any case we can revert to the external algorithm [@dacoap97] and find the solution.
An interesting aspect of this algorithm is that it is not purely enumerative because we do not look for all the generators. We may easily reformulate it requiring that in STEP 1 all the generators be found. In this case the remark is changed accordingly. If we find generators, but none has a positive last component then $h=h_{o}.$ If no generator is found $h>h_{o}$ and we can revert to the external algorithm.
We can pursue that feature further and introduce an evolutive version of the algorithm. Although the evolutive character could be exhibited in abstract terms, it becomes more evident if we take to the fore .the techniques illustrated in the book to find the generators of a polyhedral pointed cone, which is the intersection of a subspace and the non-negative orthant. We conjugate in this way the generators technique used to deploy the dual conical methods (which were essentially enumerative) with the present new primal approach and fulfill the quest for evolutiveness mentioned in the book within the dual conical framework.
Computation of generators and the evolutive version of the algorithm
====================================================================
We assume the same hypotheses and, in particular, feasibility and $h<h_{o}$ Our first purpose is to show that it is possible to apply to the present problem the machinery developed in the book for the computation of generators. The peculiarities of our method will then allow us to derive an evolutive algorithm.
Recall that the orthogonal projection $P_{F_{e}}$ of the space onto $F_{e}=\mathcal{L}(\upsilon )+F=V+F,$ is given by:
$$P_{F_{e}}=P_{V}+P_{F}$$
and the orthogonal projection of the space onto $F_{e}^{\bot }$ is given by:
$$P_{F_{e}^{\bot }}=I-(P_{V}+P_{F})$$
It follows that we can express $F_{e}$ as:
$$F_{e}=\mathcal{N}(P_{F_{e}^{\bot }})=\mathcal{N}(I-(P_{V}+P_{F}))$$
At this point we can apply all the machinery developed in the book to find the generators of
$$F_{e}\cap P=\mathcal{N}(I-(P_{V}+P_{F}))\cap P$$
Notice that in this formula only $P_{V}$ depends on $h$.
However, with respect to the case of the dual conical method, there are numerous simplifications. First because we want those generators that have a non-zero last component. In this respect we can state the following:
Under the present hypotheses, deleting the last column of the matrix $I-(P_{\mathcal{L}(\upsilon )}+P_{F})$, we obtain a matrix with the same rank as the original matrix.
In view of theorem 15.1.1 of the book, if it were not so, we would not get any generator with non-zero last component, and hence in view of Theorem \[maxformula\] above a contradiction would arise.
In the procedure for search of generators of $\mathcal{N}(I-(P_{V}+P_{F}))\cap P$ with non-zero last component given in the book we can constantly use the last column as test column.
We are now ready to introduce an evolutive conical algorithm
(Primal conical internal evolutive algorithm)
Put $h(0)=h$. Put $T(0)=I-(P_{V(h)}+P_{F})$. Repeat the following step for i=1,2,..:
STEP i: If i=1 perform the procedure of the book modified fixing the last column as test column to find generators of .$\mathcal{N}(T(0))\cap P.$ If i1 resume the search from the sequence of basic column subsequent to the last of step i-1. If a generator is found, do not verify it was already found, and proceed to calibrate it. Let $g(i)$ be the calibrated generator. Set $h(i)=h(i-1)+$ $g(i)|_{n+1}.$ Compute $T(i)$ setting $h=h(i).$
Until the procedure introduced in the book terminates.
If the loop is exited at $i=j,$ set $h_{o}=h(j).$ Optimal solutions can be obtained solving for $x$ the equation $G(x)=v(h_{o})-g(j)$.
The evolutive algorithm enjoys exact finite convergence. That is it converges in a finite number of step and if the loop is exited at $i=j,$ $h_{o}=h(j).$
The proof is essentially contained in the proof of the enumerative version. What we do is to pass from a calibrated generator to the next with increasing last component. Because the number of calibrated generators is finite it is granted the algorithm converges in a finite number of steps. The only thing that remains to be proved is that the technique to find the sequence of calibrated generators is correct. But this too is rather obvious. In fact each time we increase $h$ the current calibrated generator (as well as any calibrated generator with lower last component) is eliminated. Thus any new calibrated generator has a larger last component with respect to the former ones. and, consequently, we never have to verify that we find already known calibrated generators. Moreover we can resume the search from where it left at each step, because if we started from scratch and found a calibrated generator, the same calibrated generator would have appeared before, by the argument used in the proof of the first version of the algorithm, and would have appeared with a lower last component. And we know from the theory developed hitherto that this is a contradiction. The rest of the algorithm (computation of solutions) should by now obvious.
Notice that, as made clear by the above proof, the algorithm will usually get rid of some of the generators that are computed in the non-evolutive version. In other words evolutiveness is not just adjourning the value of the maximum, but, in general, avoiding the necessity of visiting the whole set of extreme points of the contact polytope.
Conclusion
==========
As for the previous conical algorithms we deferred submission until we had evidence that the algorithm performs correctly numerically. During the development of the implementation we used the same example of [@dadualcon]. Of course, as in the previous cases, timeliness was priviledged, so to arrive to a first straightforward implementation, that in the present case was written in Pascal, within the Delphi environment..
The previous experience showed that optimization of the code is a lengthy and painful endevour, that required the derivation of further results that are accounted for in the book. However, we got the divident of entire orders of magnitude improvings in computing time. The same process is in its inception for the present algorithm. An uprise of its numerical efficiency will be given as soon as we will feel that the level optimization of the code is satisfactory, and further improvements will have marginal effects only.
[9]{} P. d’Alessandro, M. Dalla Mora and E. De Santis, ”Techniques of linear programming based on the theory of convex cones”, Optimization, 1989, vol.20 no.6, pp 761-777.
D. Bertsimas and G.N. Tsitsiklis, ”Introduction to linear optimization”, Athena Scientific, Belmont 1997
P. d’Alessandro, ” A conical approach to linear programming - scalar and vector optimizations problems”, Gordon and Breach, Amsterdam, 1997.
P. d’Alessandro, ”A primal conical linear programming algorithm”, Optimization, 1992, Vol. 25, pp. 197-207.
P. d’Alessandro and E. De Santis ”Controlled invariance and feedback laws” IEEE Trans. on AC, vol.46, no 7, July 2001, pp 1141-1146
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abstract: 'Using the Spitzer Space Telescope, we have obtained 3.6–24$\mu$m photometry of 38 radio galaxies and 24 quasars from the 3CR catalog at redshift $1<z<2.5$. This 178 MHz-selected sample is unbiased with respect to orientation and therefore suited to study orientation-dependent effects in the most powerful active galactic nuclei (AGN). Quasar and radio galaxy subsamples matched in isotropic radio luminosity are compared. The quasars all have similar spectral energy distributions (SEDs), nearly constant in $\nu$F$_{\nu}$ through the rest 1.6–10 range, consistent with a centrally heated dust distribution which outshines the host galaxy contribution. The radio galaxy SEDs show larger dispersion, but the mean radio galaxy SED declines from rest 1.6 to 3$\mu$m and then rises from 3 to 8$\mu$m. The radio galaxies are on average a factor 3–10 less luminous in this spectral range than the quasars. These characteristics are consistent with composite emission from a heavily reddened AGN plus starlight from the host galaxy. The mid-infrared colors and radio to mid-infrared spectral slopes of individual galaxies are also consistent with this picture. Individual galaxies show different amounts of extinction and host galaxy starlight, consistent with the orientation-dependent unified scheme.'
author:
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Martin Haas, S. P. Willner, Frank Heymann, M. L. N. Ashby,\
G. G. Fazio, Belinda J. Wilkes, Rolf Chini, and Ralf Siebenmorgen
title: 'Near- and mid-infrared photometry of high-redshift 3CR sources '
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Introduction
============
When exploring the general evolution of galaxy populations across cosmic times, a particular challenge is to distinguish between black hole and star-forming activity. Star formation and obscuring dust go hand in hand, and black-hole-driven active galactic nuclei (AGN) are also surrounded by dust mainly distributed in a disk/torus-like geometry (Antonucci 1993). There is evidence that AGN mainly power the near- and mid-IR emission (NIR, $\sim$2$\mu$m; MIR, $\sim$10$\mu$m) from hot nuclear dust, while starbursts contribute mainly to the far-infrared (FIR, $\sim$60$\mu$m) luminosity (e.g., Rowan-Robinson 1995, Vernet et al. 2001, Schweitzer et al. 2006). Using the MIR/FIR luminosity ratio as an indicator for the relative AGN and starburst contributions, numerous studies have found an increase of AGN/starburst activity with total luminosity and redshift, but the validity of this trend is still under discussion because of selection effects on the various samples. More seriously, an unfavorable AGN orientation could cause MIR obscuration (e.g. Pier & Krolik 1993), leading to a fundamental observational degeneracy: a low MIR/FIR luminosity ratio can be due to either a high star-forming contribution or to an AGN in which the hot dust is obscured. The spectral energy distribution (SED) of an obscured AGN may thus mimic that of a starburst-powered source. While this degeneracy has now been widely examined at low/intermediate luminosity and redshift ($z<1$), it has still to be explored for the most luminous sources at high redshift ($z>1$). In order to assess galaxy and AGN evolution in the universe, we therefore need to understand this AGN/starburst degeneracy for a population of luminous high-redshift sources. A crucial step towards this is to study the orientation dependence of the NIR and MIR emission of high-redshift AGN.
Orientation-dependent effects can only be tested and quantified with AGN samples having type 1 (unobscured) and type 2 (obscured) subsamples matched in isotropic emission. The clean AGN tracers — optical, \[\] $\lambda$5007Å, NIR, and X-ray ($\la$10 keV) — all fail to fulfill this requirement. The \[\] $\lambda$3727Å emission, while isotropic (Hes et al. 1993), is probably dominated by extended starbursts and shocks (Best et al. 2000) rather than by the AGN. Therefore, the only feasible way is low-frequency (meter-wavelength) radio selection because the integrated emission from the radio lobes is optically thin and essentially isotropic. This makes radio-loud AGN particularly attractive for studying orientation-dependent properties at other wavelengths and, after sorting out the influence of radio jets/lobes on the emission, for generalizing conclusions about orientation-dependent effects to the much larger population of radio-quiet AGN.
The brightest low-frequency-selected AGN sample is the 3CR compilation (Spinrad et al. 1985). The powerful double-lobed radio galaxies (henceforth simply called radio galaxies) are supposed to be misaligned quasars (Barthel 1989). Based on [*IRAS*]{} coadded scans and a few individual detections, Heckman et al. (1992, 1994) already noted an average MIR/FIR difference between 3CR quasars and radio galaxies. More comprehensive MIR and FIR spectrophotometry from [*ISO*]{} is in hand (as compiled by Siebenmorgen et al. 2004 and by Haas et al. 2004) as well as from [*Spitzer*]{} (e.g., Shi et al. 2005, Haas et al. 2005, Ogle et al. 2006, Cleary et al. 2007), providing a basis to study the $z < 1$ 3CR objects. These sources are, however, a factor of five less radio-luminous on average than the most powerful radio sources seen at higher redshift, and the lower indicated accretion power may reflect different source physics. The higher-luminosity population can be sampled by the 3CR sources at $1<z<2.5$, which have radio luminosities similar to those of the most powerful radio sources at even higher redshift ($2.5<z<6$). However, with the exception of a few targets (Siebenmorgen et al. 2004, Seymour et al. 2007), the high-$z$ 3CR sample has not been well observed in the rest frame NIR and MIR.
This paper is based on [*Spitzer*]{} observations of 62 of the 64 high-$z$ 3CR sources. It focuses on the hot nuclear dust emission and its obscuration in the most luminous type-2 AGN. We use a $\Lambda$CDM cosmology with $H_0 =71$ km/s/Mpc, $\Omega_{{\rm m}} = 0.27$ and $\Omega_{\Lambda} = 0.73$.
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Observations and Data
=====================
With the [*Spitzer Space Telescope*]{} (Werner et al. 2004), we are observing the entire sample of 64 high-$z$ 3CR sources using the instruments IRAC (3.6–8.0$\mu$m, Fazio et al. 2004), the IRS blue peak-up array (16$\mu$m, Houck et al. 2004) and MIPS (24$\mu$m, Rieke et al. 2004). Most observations are performed in our guaranteed time program (PID 40072; PI G. Fazio) with on-source exposure times 4$\times$30s (each IRAC band), 4$\times$14s (IRS), and 10$\times$10s (MIPS). A few sources have been observed in other programs, and we use the published photometry if available (e.g., PID 3329; PI D. Stern, Seymour et al. 2007).
For IRAC, we used the basic calibrated data products (BCD, version S16) and coadded them to 0$\farcs$869 pixels using the latest version of [IRACProc]{} (Schuster et al. 2006). This optimally handles the slightly undersampled IRAC PSF in order to assure accurate point-source photometry. For IRS, we used the post-BCD pipeline product, version S16. For MIPS, we used custom routines to modify the version S17 BCD files to remove instrumental artifacts (e.g., residual images) before shifting and coadding to create the final mosaics. All sources are well seen on the images in all filters. The sources were extracted and matched using the SExtractor tool (Bertin & Arnouts 1996). We used sufficiently large apertures so that aperture corrections are small ($<$5%). The photometric errors are typically smaller than 10% but increase for faint sources; exceptions are 3C225A and 3C294, where nearby bright stars make the photometry uncertain in the shorter IRAC bands.
As of 2008 April, 24 quasars and 38 radio galaxies have been observed, covering the complete high-$z$ 3CR sample with the exception of the quasar 3C245 and the radio galaxy 3C325. All 62 sources have IRAC measurements and are observed in at least one of the 16 and 24$\mu$m bands (54 sources at 16$\mu$m and 60 sources at 24$\mu$m).
For the analysis it is desirable to compare rest frame SEDs with the same wavelength sampling. Depending on the redshift of our sources ($1<z<2.5$) the observations sample different rest wavelengths between 1.6 and 10$\mu$m. Before resampling and interpolating the SEDs, we checked that spectral features do not affect the interpolation. In principle, prominent spectral features in this wavelength range could be PAH emission bands around 7.7$\mu$m and the 9.7$\mu$m silicate absorption feature. IRS spectra of several low-$z$ and even a few high-$z$ FRII radio sources are available. PAH features are weak and usually undetected, and the continua are generally smooth. Strong silicate absorption is, however, present in some objects (Haas et al. 2005, Ogle et al. 2006, Cleary et al. 2007, Seymour et al. 2008). Our broadband SEDs therefore represent the smooth continua for $\lambda\la8$ but are uncertain at rest wavelengths near 10 . In practice, the de-redshifted SEDs were interpolated in log-log space at 12 wavelengths between rest 1.6 and 10$\mu$m to produce the figures shown.
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The quasar and radio galaxy samples match reasonably well in redshift and rest-frame 178MHz flux density. Rest-frame 178MHz radio flux densities were derived from data listed in the NASA Extragalactic Database, NED. The quasars have mean redshift $\langle z\rangle = 1.44\pm0.31$ and mean flux density $\langle S_{178}\rangle = 27.8\pm15.1$ Jy, while the values for the radio galaxies are $\langle z\rangle = 1.42\pm0.31$, $\langle S_{178}\rangle = 22.2\pm6.2$ Jy. Thus over the whole sample, quasars are about 30% more luminous than radio galaxies as shown in Figure \[fig\_ir\_radio\_lum\]. In order to improve the luminosity match, we have excluded the sources at the low and high ends of the distribution, $L_{178}<1.8\times10^{44}$erg/s and $L_{178}>1.5\times10^{45}$erg/s, respectively. We have also excluded the quasar 3C418 because of its flat radio spectrum (low-frequency spectral index $\alpha_{178}\approx0$), while all other sources have steep radio spectra ($-1.1\la \alpha_{178} \la -0.6$). The resulting mean radio luminosities of the sample galaxies are $\langle L_{178}\rangle = (5.35\pm2.53)\times 10^{44}$ erg/s for quasars and $(5.55\pm2.34)\times 10^{44}$ for radio galaxies.
While the quasar and radio galaxy distributions match very well in $L_{178}$, a proper analysis of orientation-dependent effects requires also that the individual SEDs are normalized by the radio luminosity, which serves as a tracer for the intrinsic AGN strength. Therefore, we have normalized each SED to the sample mean 178MHz luminosity; after normalization each object has $L_{178} = 5.4\times10^{44}$ erg/s. Because of the good $L_{178}$ match of the samples, it turned out that the net effect of this normalization on the results is small.
Results and Discussion
======================
Radio galaxies as obscured quasars {#sec_rg}
----------------------------------
The NIR–MIR SEDs of quasars are all very similar in shape, as shown in Figure \[fig\_qso\_gal\_seds\]. The SEDs can be described by a single power law $L_\nu \propto \nu^{- 1}$, consistent with previous results for lower-redshift objects (e.g., Elvis et al. 1994). The dispersion of the SEDs is essentially caused by differing ratios of MIR to radio luminosity. Some quasars exhibit small (10-20%) bumps around 5$\mu$m explainable by distinct hot dust components.[^1] The power law shape of the quasar SED can naturally be explained by the superposition of centrally-heated dust components with a radial temperature gradient (1500K $>T>300$K) as has been found also in lower luminosity type-1 AGN (e.g. Ward et al. 1987, Barvainis 1987; see also Rowan-Robinson 1980). Any contribution of the quasar host galaxies to the NIR–MIR SEDs appears to be outshone (factor $\ga$5–10) by the AGN dust emission.
In contrast to quasars, radio galaxies display a diversity of SED shapes leading to a 50% larger dispersion around their mean SED (Fig. \[fig\_qso\_gal\_seds\]). Despite the dispersion, nearly all radio galaxy SEDs show a decline from rest 1.6$\mu$m to 3$\mu$m and a rise from 3$\mu$m to 8$\mu$m ($L_\nu \propto \nu^{- 1.9}$). In addition, the average radio galaxy SED is fainter by a factor of three at 8$\mu$m and a factor of eight at 2$\mu$m relative to the quasar SED. Unlike the quasars, hot ($T>750$K) dust emission is not seen in the radio galaxy SEDs. Its absence can be explained by absorption (screen $A_{V}\approx50$)[^2] of the central dust emission. The short wavelength ($\lambda< 3$ ) component can then be explained by emission from stars in the host galaxy. Extrapolation of the mean 3–8$\mu$m SED slopes towards longer wavelengths suggests that the radio galaxy and quasar SEDs meet each other at about 25–40$\mu$m, and beyond these wavelengths extinction may be no longer relevant.
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As noted above, the quasar NIR–MIR SED shapes are extremely homogeneous. This is reflected in the narrow range of the quasars’ NIR and MIR colors. The color-color diagram shown in Figure \[fig\_nir\_mir\_cc\] illustrates the differing SED types. In this diagram, quasars populate a distinct locus (“E”), while radio galaxies show wider dispersion as mentioned above. According to their location in the color-color diagram, we have grouped the radio galaxies into five classes described below. Their SED shapes are illustrated in Figure\[fig\_gal\_seds\].
- Four sources at the high end of the 3$\mu$m/1.6$\mu$m ratio (above the dotted and dashed boxes in Fig.\[fig\_nir\_mir\_cc\]): basically, they have quasar-like SEDs, but the hottest dust emission at about 1–2$\mu$m appears to be absorbed (screen A$_{\rm V}$$\approx$5) leading to a redder 3$\mu$m/1.6$\mu$m color compared to quasars.
- The bulk of radio galaxies (20 sources) have declining 1–3$\mu$m SEDs with a steep 3–8$\mu$m rise. Their colors can be explained by a heavily reddened AGN plus an added component of starlight contributing at 1.6$\mu$m. If this explanation is correct, the direction of the extinction vector A$_{\rm V}$ becomes meaningless because host galaxy starlight will not be affected by extinction near the nucleus. Instead, vertical position in the plot is determined by the relative contributions of starlight and AGN light, while horizontal position measures the amount of extinction (to the extent the underlying AGN SEDs are the same). As noted above, $A_V \sim 50$ mag is required to explain the colors.
- Three sources at the low end of the 3$\mu$m/1.6$\mu$m ratio (below the long dashed line in Fig.\[fig\_nir\_mir\_cc\]): Their SEDs show a very strong host galaxy contribution at 1.6$\mu$m, and starlight exceeds the dust luminosity even at wavelengths as long as 3.5$\mu$m. In principle, class C is similar to class B but with stronger host galaxy contribution.
- Three sources immediately below the dotted box in Fig.\[fig\_nir\_mir\_cc\]: Their SEDs can be explained by a slightly reddened AGN (similar to class A) plus an added component of starlight contributing significantly at 1.6–3$\mu$m.
- Three sources with quasar-like SED colors (inside the dotted box in Fig.\[fig\_nir\_mir\_cc\]): Their SEDs overlap with the low luminosity end of the quasar SEDs. In the orientation-based unified scheme, these sources could be borderline so that most of the dust torus is visible but the broad line region and the UV-optical continuum are obscured.
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While the rest 8–10 range is poorly sampled, eight galaxies show declines in this range that could be caused by silicate absorption. (The MIPS-24 filter, 50% transmission at 20.8–29.3$\mu$m, requires $z\la1.8$ for the silicate feature to fall into its range.) One of these sources (3C469.1, $z=1.336$) has an IRS spectrum available. It shows a broad silicate absorption with optical depth $\tau_{ 9.7} \approx 0.55$ corresponding to $A_{V}\approx10$, consistent with its position in Fig.\[fig\_nir\_mir\_cc\]. This supports the view that the SED declines in the other radio galaxies are due to silicate absorption, too. The photometric silicate absorption sources show a wide range of colors (Fig.\[fig\_nir\_mir\_cc\]), but only one galaxy (3C469.1) is on the blue (right) side, where low-extinction sources reside. The important conclusion is that the silicate feature requires considerable extinction to be present in at least some of the radio galaxies, and this is largely independent of the SED class.[^3]
If radio galaxies are misaligned quasars, as proposed in the unified scheme, reddening of individual galaxies should be correlated with their extinction. Figure \[fig\_ir\_radio\_cc\] shows that this is indeed the case. Quasars populate a distinct region of this diagram characterized by high MIR/radio and blue NIR–MIR colors. Most radio galaxies spread towards fainter MIR/radio and redder NIR/MIR. Under the reasonable assumption that the emission at 5–8$\mu$m is not affected by the host galaxy, de-reddening along the direction of the extinction vector can place each radio galaxy inside the region populated by quasars. Thus individual radio galaxies can be explained as reddened quasars, consistent with the orientation-dependent unified scheme.
The typical amount of radio galaxy reddening, $A_V \approx 50$ for an obscuring screen (Fig. \[fig\_ir\_radio\_cc\]), corresponds to a hydrogen column density $N_{H} \approx 9\times10^{22}$ cm$^{-2}$ (for $A_V = 5.6\times10^{-22}$ mag/cm$^{-2}$ — Seward et al. 1999). This is close to but below the Compton-thick limit ($N_{H} = 10^{24}$ cm$^{-2}$). Screen extinction is a simplification, and one may expect a more complex geometry. If emitting dust particles are spatially mixed with the absorbing ones, the amount of dust has to be higher for the same observed reddening, typically by a factor 3–5.[^4] Thus there could very well be enough gas present to render the AGN Compton-thick. There is, unfortunately, no independent measurement of reddening for individual galaxies, nor is it certain that a Galactic reddening curve applies to AGN. Thus it is still an open question whether after de-reddening there will remain a difference in the 8 $\mu$m/178 MHz ratio between radio galaxies and quasars. If such a difference remains, with quasars having a higher 8 $\mu$m/178 MHz ratio than radio galaxies, then either our screen extinction premise is too simple or the MIR luminosity of quasars is enhanced by an additional — potentially non-thermal — contribution. Our [*Chandra*]{} X-ray observations of a subset of the sample will provide independent estimates of the extinction towards the nuclei (Wilkes et al., in prep.).
To summarize, while quasars exhibit a uniform SED shape which can be explained by a centrally heated dust distribution, radio galaxies show a diversity of SED shapes. In all cases, however, the radio galaxy SEDs are consistent with being intrinsically a quasar modified by absorption of the dust emission and addition of some amount of host galaxy starlight.
Evolutionary effects and non-thermal contributions
--------------------------------------------------
Studying powerful 3CR sources at $z<1$, Ogle et al. (2006) found evidence for a population of accretion-inefficient radio galaxies, in which the jet/lobe may be powered by extraction of rotational black hole energy. These sources, mainly optically-classified low-excitation radio galaxies (LERGs), have a 15$\mu$m luminosity below 8$\times$10$^{\rm 43}$erg/s and a luminosity ratio $L_{\rm 15\mu m}$/L$_{\rm 178MHz}$$<$10. In contrast, with the reaonable assumption that $L_{\rm 8\mu m}\la L_{\rm 15\mu m}$, all our $z>1$ sources have observed MIR luminosity $L_{\rm 8\mu m}>5\times10^{\rm 44}$erg/s, which is expected to be even higher after de-reddening. Also, the two LERGs (3C68.2, 3C469.1) in our sample show a high luminosity ratio L$_{\rm 8\mu m}$/L$_{\rm 178MHz}$$>$10 comparable to that of quasars (Fig.\[fig\_ir\_radio\_lum\]). From this, our data do not support the existence of an accretion-inefficient population among the powerful 3CR sources at $z>1$. A possible explanation for the deficit of optical high-excitation line luminosity (for instance \[\] $\lambda$5007Å) in our two LERGs may be extinction of the narrow-line region. On the other hand, some of our radio galaxies with very strong host contribution (plotted as squares in Fig.\[fig\_ir\_radio\_cc\]) are expected after de-reddening to lie at the low end of the $L_{\rm 8\mu m}/L_{\rm 178MHz}$ distribution. Hence compared with the strength of both the host and the radio lobes, these galaxies are relatively weak in the MIR and may represent a population at the beginning of a different evolutionary state.
Some authors have attributed the excess emission of quasars compared to radio galaxies to nonthermal emission from synchrotron jets. For example, Cleary et al. (2007) fitted the SEDs and spectra of 3CR sources at $0.5<z<1$ with a combination of a spherically symmetric dust model and a jet+lobe synchrotron component. They attributed half of the factor of four excess in the 15$\mu$m luminosities of steep-spectrum quasars relative to radio galaxies to a non-thermal component. If such a non-thermal component were present in our 3CR sources at $z>1$, it would show up in Fig.\[fig\_ir\_radio\_cc\] as an offset by about a factor of two between dereddened radio galaxies and quasars. This conclusion is, however, dependent on both the reddening law and on the radiative transfer and thus the geometry of the emitting region. In order to draw definite conclusions about any MIR luminosity excess, detailed radiative transfer modelling is required (Heymann et al., in prep.). Spherically symmetric models are wholly inadequate for this purpose. In an inclined disk-like system, some fraction of the MIR emission is likely to have very little obscuration while the bulk of the MIR emission is heavily obscured, and no spherical model can properly account for this geometry. All we can say at the moment is that our data appear consistent with a simple thermal interpretation and show no evidence for a non-thermal component.
Conclusions
===========
The 3CR sample at $1<z<2.5$ represents the most luminous steep-spectrum quasars (type 1 AGN) and powerful double-lobed radio galaxies (type 2 AGN). This sample is nearly unbiased by orientation. We have defined subsamples of 19 quasars and 33 radio galaxies matched in isotropic rest 178MHz luminosity and have obtained [*Spitzer*]{} 3.6–24$\mu$m photometry. The main results are:
- Quasars all have similar energy distributions in the rest frame 1.6–10$\mu$m range, and their ratio of MIR to radio luminosity is also nearly constant. This is consistent with results seen previously in lower-redshift samples.
- The rest frame 1.6–10$\mu$m SEDs of radio galaxies can be explained as reddened quasars, consistent with orientation-dependent unification. Various amounts of extinction of the AGN emission combined with addition of host galaxy starlight can explain the diversity of radio galaxy SEDs.
- If the extinction is sufficiently large, there is no need to invoke a beamed synchrotron contribution to explain the MIR luminosity difference between quasars and radio galaxies. The actual amount of extinction has to be derived from additional observations.
- The above results hold also for splitting our sample in redshift and luminosity; within our sample we do not find any trends with redshift or luminosity.
- At rest frame 8$\mu$m, quasars are 3 times more luminous than radio galaxies. If this difference applies also to high-redshift, radio-quiet AGN, then MIR (24$\mu$m) surveys are strongly biased in favour of type-1 and against type-2 AGN. This will make it very difficult to resolve the AGN/starburst degeneracy with only broadband SEDs, and spectral line diagnosis will be required.
While our near-mid-IR SEDs provide a fundamental set of high luminosity AGN templates, we expect to derive more stringent conclusions from proper two-dimensional radiative transfer modelling in combination with [*Spitzer*]{} MIR spectra, [*Chandra*]{} X-ray observations, and [*Herschel*]{} far-IR/sub-mm data.
This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. M.H. is supported by the Nordrhein–Westfälische Akademie der Wissenschaften.
[*Facilities:*]{} .
Antonucci, R.R.J. 1993, ARA&A, 31, 473
Barthel P. 1989, ApJ 336, 606
Barvainis, R. 1987, , 320, 537
Bertin E., Arnouts S. 1996, A&AS 117, 393
Best, P.N.; Röttgering, H.J.A.; Longair, M.S. 2000, MNRAS 311, 23
Cleary, K.; Lawrence, C.R.; Marshall, J.A.; et al. 2007, ApJ 660, 117
Disney, M.; Davies, J.; Phillipps, S. 1989, MNRAS 239, 939
Elvis, M.; Wilkes, B.J.; McDowell, J.C.; et al. 1994, , 95, 1
Fazio G.G.; Hora, J. L.; Allen, L. E.; et al. 2004, ApJS 154, 10
Haas, M.; Müller, S.A.H.; Bertoldi, F.; et al. 2004, A&A 424, 531
Haas, M.; Siebenmorgen, R.; Schulz, B.; et al. 2005, A&A 442, L39
Heckman, T.M.; Chambers, K.C.; Postman, M. 1992, ApJ 391, 39
Heckman, T.M.; O’Dea, Ch.P.; Baum, S.A.; Laurikainen, E. 1994, ApJ 428, 65
Hes, R.; Barthel, P.D.; Fosbury, R.A.E. 1993, Nature 362, 326
Houck, J.R.; Roellig, T.L.; van Cleve, J.; et al. 2004, ApJS 154, 18
Indebetouw, R.; Mathis, J. S.; Babler, B. L.; et al. 2005, ApJ 619, 931
Ogle, P.; Whysong D.; Antonucci, R.R.J. 2006, ApJ 647, 161
Pier, E. A.; Krolik, J. H. 1993, ApJ 418, 673
Rieke, G.H.; Young, E. T.; Engelbracht, C. W.; et al. 2004, ApJS 154, 25
Rowan-Robinson, M., 1980, ApJS 44, 403
Rowan-Robinson, M., 1995, MNRAS 272, 737
Seward, F.D., 1999, in “Astrophysical Quantities”, AIP Press, ed. A.N. Cox, p. 183
Schuster, M. T., Marengo, M., & Patten, B. M. 2006, , 6270,
Schweitzer, M.; Lutz, D.; Sturm, E.; et al. 2006, ApJ 649, 79
Seymour, N.; Stern, D.; De Breuck, C.; et al. 2007, ApJS 171, 353
Seymour, N.; Ogle, P.; De Breuck, C., et al. 2008, ApJL in press, arXiv:0805.2143
Shi, Y.; Rieke, G.H.; Hines, D.C.; et al. 2005, ApJ 629, 88
Siebenmorgen, R.; Freudling W.; Krügel E.; Haas, M. 2004, A&A 421, 129
Spinrad, H.; Marr, J.; Aguilar, L.; Djorgovski, S. 1985, PASP 97, 932 Vernet, J., Fosbury, R. A. E., Villar-Mart[í]{}n, M., Cohen, M. H., Cimatti, A., di Serego Alighieri, S., & Goodrich, R. W. 2001, , 366, 7
Ward, M.; Elvis, M.; Fabbiano, G.; et al. 1987, ApJ 315, 74
Werner, M.W.; Roellig, T.L.; Low, F.J.; et al. 2003, ApJS 154, 1
[^1]: Despite the similarity of the infrared SEDs, the quasar population is not homogeneous at optical wavelengths: there are quasars like 3C186 with blue optical SED and 3C68.1 with red optical SED, as listed in NED. In the orientation-based unified scheme, 3C68.1 could be borderline so that the broad lines are detected, but most of the UV-optical continuum is absorbed.
[^2]: The reddening curve used is a compromise between the latest results for Milky Way reddening and earlier data (summarized by Indebetouw et al. 2005): A$_{V}$/A$_{H}$/A$_{\rm 3\mu m}$/A$_{\rm 5\mu
m}$/A$_{\rm 8\mu m}$/A$_{\rm 10\mu m}$ = 1/0.184/0.070/0.037/0.028/0.040.
[^3]: The photometric silicate absorption sources are 3C68.2, 3C222, 3C249, 3C250, 3C266, 3C305.1, 3C324, and 3C469.1. These galaxies lie in the redshift range $1.08<z<1.83$, suggesting that in this range the broad band 16$\mu$m/24$\mu$m filter combination is able to register silicate absorption, if strong enough. For comparison, this redshift range contains 20 more radio galaxies with 16 and 24 photometry available but without silicate absorption signatures in their broadband SEDs. Four of these sources (3C13, 3C266, 3C267, 3C356) have IRS spectra available, but significant silicate absorption is detected only in one of them (3C267, $z=1.14$, $\tau_{9.7}\approx0.2$). At low redshift, the detection of silicate absorption appears not to be directly correlated with other absorption signatures, perhaps because of complex geometry and/or varying silicate dust abundance (e.g., Haas et al. 2005, Ogle et al. 2006, Cleary et al. 2007). A detailed analysis of the high-$z$ 3CR spectra and the photometric detectability of silicate features will be presented elsewhere (Leipski et al. in prep.).
[^4]: The transmission factors are $\exp(-\tau_\lambda)$ and $(1-\exp(-\tau_\lambda))/\tau_\lambda$ for the screen and the mixed case, respectively, with $\tau_\lambda = 0.916\times A_{\lambda}$ (Disney et al. 1989).
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abstract: 'We investigate the effect of charge self-consistency (CSC) in density functional theory plus dynamical mean-field theory (DFT+DMFT) calculations compared to simpler “one-shot” calculations for materials where interaction effects lead to a strong redistribution of electronic charges between different orbitals or between different sites. We focus on two systems close to a metal-insulator transition, for which the importance of CSC is currently not well understood. Specifically, we analyze the strain-related orbital polarization in the correlated metal CaVO$_3$ and the spontaneous electronic charge disproportionation in the rare-earth nickelate LuNiO$_3$. In both cases, we find that the CSC treatment reduces the charge redistribution compared to cheaper one-shot calculations. However, while the MIT in CaVO$_3$ is only slightly shifted due to the reduced orbital polarization, the effect of the site polarization on the MIT in LuNiO$_3$ is more subtle. Furthermore, we highlight the role of the double-counting correction in CSC calculations containing different inequivalent sites.'
author:
- Alexander Hampel
- Sophie Beck
- Claude Ederer
bibliography:
- 'bibfile.bib'
title: 'On the effect of charge self-consistency in DFT+DMFT calculations for complex transition metal oxides'
---
Introduction
============
During recent years, the combination of density functional theory (DFT) and dynamical mean-field theory (DMFT) has become a widespread tool to calculate properties of so-called “correlated materials”, i.e., materials where the strong Coulomb repulsion between electrons in partially filled $d$ or $f$ shells leads to effects that cannot easily be treated within effective non-interacting electron theories [@held2007]. The basic idea in combining DFT and DMFT is the assumption that for the relevant materials the electronic degrees of freedom can be separated into a “weakly interacting” part, for which a standard DFT treatment is adequate, and a “correlated subspace”, which requires a more elaborate treatment of the electron-electron interaction. The latter leads, in general, to a redistribution of electrons within the correlated subspace compared to the DFT result. This change should then enter, in a self-consistent way, the effective potential felt by the weakly interacting electrons, which is achieved by iterating between DFT and DMFT steps. However, such a charge self-consistent (CSC) DFT+DMFT calculation leads to a higher computational cost compared to simpler “one-shot” (OS) calculations, where the charge rearrangement within the correlated subspace is neglected in the DFT calculation.
Thus, as DFT+DMFT develops further towards a standard [*ab initio*]{}-based computational method for materials science [@Grieger_et_al:2012; @Adler_et_al:2019], it becomes essential to know in which cases it is possible to reduce the required computational effort by using more approximate variants of the method, e.g., by neglecting charge self-consistency. While CSC DFT+DMFT calculations have become more common recently, the DFT+DMFT method has also been applied to larger and more complex systems, such as, e.g., oxide heterostructures, [@Ishida:2009; @Zhong:2015; @Lechermann:2018; @Beck:2019] defective systems, [@Backes_et_al:2016; @Sing:2017; @Souto-Casares:2019] or large molecules [@Jacob:2010; @Turkowski:2012]. Therefore, a detailed understanding of the effect of charge self-consistency is desirable in order to better assess in which cases a CSC calculation is crucial or, more importantly, in which circumstances a one-shot calculation is sufficient. Unfortunately, there are currently very few studies available that provide a systematic quantitative comparison between CSC and one-shot calculations. It is the purpose of the present work to start filling this gap.
It can be assumed that charge self-consistency is particularly relevant for systems where correlation effects lead to a redistribution of electrons, e.g., for systems with charge-, and/or orbital-ordering. For example, existing studies of epitaxially strained SrVO$_3$ monolayers demonstrate a reduced orbital polarization in CSC calculations compared to OS [@Bhandary:2016; @Schuler_Aichhorn:2018]. Here, we therefore analyze the effect of charge self-consistency for two specific but representative cases. First, strained CaVO$_3$, where orbital polarization is considered relevant for a reported strain-induced metal-insulator transition [@Gu_et_al:2013; @Beck:2018]. And second, LuNiO$_3$, which is representative for a whole series of rare earth nickelates that exhibit a transition to a charge-ordered (or charge-disproportionated) insulating state, which is also strongly coupled to a structural distortion [@catalano.review].
Most previous work addressing the influence of charge self-consistency in DFT+DMFT calculations in transition metal (TM) oxides employed a so-called “[$p\textrm{-}d$]{}”-model to define the correlated subspace, [@Wang:2012; @Dang:2014; @Park:2014; @Aichhorn:2009; @Aichhorn:2011; @Karolak:2010] i.e., using a basis of rather localized, atomic-like orbitals constructed from a broad energy window that includes the TM $d$ as well as all oxygen $p$ bands. This appears conceptually appealing, since a wider energy window corresponds to a larger, and thus more complete, basis set, and since the use of rather localized orbitals provides better justification for the DMFT assumption of a purely local self-energy and Coulomb interaction [@Aichhorn:2011]. On the other hand, this also increases the computational load compared to using a minimal correlated subspace of “frontier” orbitals, corresponding to only a narrow energy window around the Fermi level. In TM oxides, the latter typically includes either $t_{2g}$ or $e_g$ bands.
In the present work, we focus on DFT+DMFT calculations that employ such a minimal correlated subspace, corresponding to only a small number of near-Fermi-surface bands. These are expressed in a localized basis through a suitable transformation in terms of Wannier functions [@PhysRevB.74.125120]. By including only the minimal number of orbitals needed to describe the dominant low-energy physics within DMFT, this scheme requires a comparatively small computational cost. Furthermore, it often allows for an intuitive interpretation of Wannier occupations in terms of formal charge states, since the corresponding Wannier functions include the hybridization with the oxygen $p$ states as “tails” located on the oxygen sites.
Another critical point arising in DFT+DMFT calculations using a [$p\textrm{-}d$]{}-type orbital subspace, is that the physically very important charge transfer energy, $\Delta_{{\ensuremath{p\textrm{-}d}}{}}$, which describes the energy difference between oxygen $p$ and transition metal $d$ states, effectively becomes controlled by the so-called double-counting correction [@Wang:2012; @Karolak:2010; @Solovyev/Terakura:1998]. The latter is required to account for the electron-electron interaction within the correlated subspace that is already included on the DFT level, and is notoriously ill-defined [@Kotliar:2006]. Different expressions to account for the double counting (DC) have been suggested [@Karolak:2010; @Haule:2015_exactDC], but in some cases the double-counting needs to be adjusted manually, in order to obtain satisfactory results [@Dang:2014; @Wang:2012]. It was shown that CSC calculations for such [$p\textrm{-}d$]{}-type calculations produce essentially the same spectral properties as OS calculations, if one tunes the DC correction to yield the same $d$-state occupancy [@Wang:2012]. It is, however, not clear a priori, that more complex observables, e.g., the total energy, need to agree within both approaches. We note that the use of a minimal correlated subspace avoids the problem that the DC correction critically affects the important charge transfer energy, because charge-neutrality between DFT and DMFT is ensured, and thus the DC potential shift can be absorbed in the chemical potential in DMFT [@Schuler_Aichhorn:2018]. However, as we show in the following, the DC correction can still have a strong effect for systems with multiple inequivalent correlated sites.
In the next section (Sec. \[sec:theory\]), we provide a detailed description of the DFT+DMFT method as used in this work, specifying also all important computational parameters. We then discuss the two specific cases of CaVO$_3$ and LuNiO$_3$ in Sec. \[sec:results\], where we also provide a brief introduction in the relevant physical background for each of these materials. Finally, our main conclusions are summarized in Sec. \[sec:summary\].
Theoretical framework {#sec:theory}
=====================
DFT calculations
----------------
Structural relaxations for CaVO$_3$ within the 20 atom unit cell in $Pbnm$ space group symmetry are performed using the <span style="font-variant:small-caps;">Quantum ESPRESSO</span> package [@Giannozzi_et_al:2009]. We employ scalar-relativistic ultrasoft pseudopotentials, with the $3s$ and $3p$ semicore states included in the valence for both V and Ca, together with the exchange-correlation functional according to Perdew, Burke, and Ernzerhof [@Perdew:1996iq]. Cell parameters and internal coordinates are relaxed until all force components are smaller than 0.1 mRy/$a_0$ ($a_0$: Bohr radius) and all components of the stress tensor are smaller than 0.1 kbar. The plane-wave energy cutoff is set to 70 Ry for the wavefunctions and 840 Ry for the charge density. A $6 \times 6 \times 4$ Monkhorst-Pack $k$-point grid is used to sample the Brillouin zone, and the Methfessel-Paxton scheme with a smearing parameter of 0.02 Ry is used to broaden electron occupations. For the calculation of epitaxially strained CaVO$_3$, the in-plane lattice parameters are increased by 4% and kept fixed, while the $c$-component of the cell and all atomic positions are relaxed.
All DFT calculations for LuNiO$_3$ as well as the DFT parts of all our CSC DFT+DMFT calculations are performed using the projector augmented wave (PAW) method [@Blochl:1994dx], implemented in the “Vienna Ab initio Simulation Package”(VASP) [@Kresse:1993bz; @Kresse:1996kl; @Kresse:1999dk], and also using the exchange correlation functional according to Perdew, Burke, and Ernzerhof [@Perdew:1996iq]. For Ni, we use the PAW potential where the 3$p$ semi-core states are included as valence electrons, while for Lu, we use the PAW potential corresponding to a $3+$ valence state with $f$-electrons frozen into the core. For the CaVO$_3$ calculations with VASP, we use the PAW potentials where the $s$ and $p$ semi-core states are included as valence electrons for both Ca and V. Furthermore, a $k$-point mesh with $9 \times 9 \times 7$ grid points along the three reciprocal lattice directions is used and a plane wave energy cut-off of 550 eV is chosen for LuNiO$_3$ and 600 eV for CaVO$_3$. The structure of LuNiO$_3$ is fully relaxed within $Pbnm$ symmetry, both internal and lattice parameters, until the forces acting on all atoms are smaller than $10^{-4}$ eV/.
DFT+DMFT calculations
---------------------
### Construction of the correlated subspace
In the DFT+DMFT method, the Kohn-Sham (KS) Hamiltonian within the chosen energy window is mapped onto a basis of localized states, spanning the correlated subspace $\mathcal{C}$, then a local Coulomb interaction is added, and the resulting Hubbard-like lattice Hamiltonian is solved via the DMFT approximation [@Georges:1996; @held2007]. Without feedback to the DFT part, this corresponds to a OS calculation. To perform CSC calculations, one computes a correction to the charge density, $\Delta \rho = \rho^\text{DMFT} - \rho^\text{DFT}$, which is then passed back to the DFT code (here VASP) to calculate new KS wave-functions and hence, update the correlated subspace [@Amadon:2008; @Lechermann:2018]. In a fully CSC calculation, this is repeated until $\Delta \rho$ does not change compared to the previous iteration.
For the DMFT calculation, the electronic degrees of freedom within the chosen energy window are described via the interacting lattice Green’s function: $$\begin{aligned}
\hat{G}(\mathbf{k}, i \omega_n) = \left[ (i \omega_n + \mu) \mathds{1} - \hat{H}_{\text{KS}}(\mathbf{k}) - \hat{\Sigma}(\mathbf{k}, i \omega_n) \right]^{-1}
\label{eq:Glat}\end{aligned}$$ where $\mu$ is the chemical potential and $\hat{H}_{\text{KS}}(\mathbf{k})$ is the Kohn-Sham (KS) Hamiltonian. The lattice self-energy $\hat{\Sigma}(\mathbf{k}, i \omega_n)$ is obtained by solving the effective DMFT impurity problem (see next sub-section).
The lattice Green’s function in Eq. is expressed in the KS (Bloch) basis. To achieve the up/down-folding between the quantities defined within the correlated subspace and the Green’s function in the KS basis, $$\begin{aligned}
{G}^{\mathcal{C}}_{L L'} (i \omega_n) = \sum_{k, \nu
\nu'} {P}_{L \nu}(\mathbf{k}) \ {G}_{\nu \nu'}(\mathbf{k}, i \omega_n) \ {P}^{*}_{\nu' L'}(\mathbf{k}) \ ,\end{aligned}$$ projector functions ${P}_{L \nu}(\mathbf{k})$ are introduced. The projector functions are defined as projections of the KS eigenstates $\ket{\Psi_{\nu \mathbf{k}}}$ onto localized orbitals
$\ket{\chi_L}$, ${P}_{L \nu}(\mathbf{k}) \equiv \braket{\chi_L | \Psi_{\nu \mathbf{k}}}$. Here, $L$ serves as compound index for all local quantum numbers (site, orbital, and spin-character).
In our VASP-based OS and CSC calculations, the local basis functions $\ket{\chi_L}$ are constructed from projection to localized orbitals (PLO) [@Amadon:2008; @Schuler_Aichhorn:2018]. To construct optimal projector functions, we apply the scheme introduced in Ref. , choosing a linear combination of the PAW partial wave augmentation channels that maximizes the overlap with the KS states inside a chosen energy window, which matches that of the correlated subspace $\mathcal{C}$. We use initial projections on [$t_{2g}$]{}- or $e_g$-like orbitals within the energy window of the correlated subspace $\mathcal{C}$. The resulting projectors $\tilde P_{L, \nu} (\mathbf{k})$ are in general not orthogonal to each other, and need to be orthonormalized: $$O_{L L'}(\mathbf{k}) = \sum_{\nu} {\tilde P}_{L \nu}(\mathbf{k}) {\tilde P}^*_{\nu L'}(\mathbf{k}) \quad ,$$ $$P_{L \nu}(\mathbf{k}) = \sum_{L'} [O^{-1/2}(\mathbf{k})]_{L L'} \ {\tilde P}_{L' \nu}(\mathbf{k}) \quad ,$$ to define an orthonormal set of PLO-based Wannier functions describing the correlated subspace $\mathcal{C}$. The orthonormalization of these PLO-based Wannier functions, as well as the whole DFT+DMFT self-consistency cycle has been implemented using the <span style="font-variant:small-caps;">TRIQS/DFTTools</span> software package [@aichhorn_dfttools_2016; @parcollet_triqs_2015].
The projectors ${P}_{L \nu}(\mathbf{k})$ are updated after each DMFT cycle from the new KS states. Thereby, the energy window defining the correlated subspace is kept fixed, while monitoring that the change in the KS energies due to the charge density correction does not move the relevant bands outside of this energy window. The observed change of the KS eigenvalues is relatively small for all cases considered in this work, e.g., the maximum bandwidth reduction in LuNiO$_3$ is smaller than $\sim5$%.
We note that the strong octahedral rotations present within $Pbnm$ symmetry lead to large off-diagonal crystal-field terms in the KS Hamiltonian, and the non-interacting Green’s function for the effective impurity problem is no longer diagonal. Since this can induce a severe sign problem in the quantum Monte Carlo (QMC) method [@Gull:2011] used to solve the effective impurity problem (see next sub-section), we perform a local unitary transformation of each impurity Green’s function after the down- respectively before the up-folding, which diagonalizes the initial non-interacting local Hamiltonian on each site transforming the system into the crystal field basis. We note that the transformation is optimized in the first CSC cycle, and is kept fixed in consecutive iterations to maintain a consistent orbital basis.
For [CaVO$_3$]{} we also perform OS DFT+DMFT calculations based on the electronic structure obtained with <span style="font-variant:small-caps;">Quantum ESPRESSO</span>. In this case, the low-energy tight-binding Hamiltonian, used as input for the OS DMFT calculation, is formulated in the basis of maximally localized Wannier functions (MLWFs) [@Marzari_et_al:2012] using the <span style="font-variant:small-caps;">Wannier90</span> code [@Mostofi_et_al:2014]. Note that the PLO basis functions used in our VASP-based DFT+DMFT calculations essentially correspond to the initial Wannier functions constructed by <span style="font-variant:small-caps;">Wannier90</span> before the spread minimization, which are based on orthogonalized projections of (pseudo-) atomic orbitals on the Bloch bands [@Mostofi_et_al:2014].
The code used for all DFT+DMFT calculations in this paper is publicly available on github [@soliDMFT].
### Solving the impurity problem
For both CaVO$_3$ ([$t_{2g}$]{} subspace) and LuNiO$_3$ ($e_g$ subspace) the effective impurity problem within the DMFT cycle is solved with a continuous-time QMC hybridization-expansion solver [@Gull:2011] implemented in <span style="font-variant:small-caps;">TRIQS/cthyb</span> [@Seth2016274], taking into account all off-diagonal elements of the local Green’s function in the crystal-field basis. For each impurity we add a local Coulomb interaction in the form of the Hubbard-Kanamori Hamiltonian [@vaugier2012], $$\begin{aligned}
\begin{split}
\hat{H}_{\mathrm{int}}\,&=\,U\sum_m \hnmu\hnmd\,+\,(U-2J)\sum_{m\neq\mp} \hnmu\hnpmd \\
&+(U-3J) \sum_{m<\mp,\sigma} \hn_{m\sigma}\hn_{\mp\sigma} \\ &+ J\, \sum_{m\neq\mp} \hat{c}^{\dagger}_{m\spinup}\hat{c}^{\dagger}_{m\spindown}\,\hat{c}_{\mp\spindown}\hat{c}_{\mp\spinup}
-J\,\sum_{m\neq\mp} \hat{c}^{\dagger}_{m\spinup}\hat{c}_{m\spindown}\,\hat{c}^{\dagger}_{\mp\spindown}\hat{c}_{\mp\spinup} \ ,
\end{split}
\label{eq:ham_kanamori}\end{aligned}$$ including all spin-flip and pair-hopping terms. Here, the operator $\hat{c}^{\dagger}_{m\sigma}$ creates an electron in the atom-centered Wannier orbitals of type $m$ and spin $\sigma$. The interaction parameters are given by the local intra-orbital Coulomb repulsion $U$, and the Hund’s coupling $J$. To reduce the QMC noise in the high-frequency regime of the impurity self-energy $\Sigma_\text{imp}$ and $G_\text{imp}$, we represent both quantities in the Legendre basis [@boehnke:2011] and sample the Legendre coefficients $G_l$ directly within the <span style="font-variant:small-caps;">TRIQS/cthyb</span> solver.
### Double counting correction {#par:DC}
To correct the electron-electron interaction within the correlated subspace already accounted for within VASP, we use the fully-localized limit DC correction scheme [@Solovyev:1994; @anisimov1997]. Specifically, we use the parameterization given in Ref. for the DC potential, $$\begin{aligned}
\label{eq:dcimp}
\Sigma_{dc,\alpha}^\text{imp} = \bar{U} (n_{\alpha}-\frac{1}{2}) \quad ,\end{aligned}$$ where $n_{\alpha}$ is the occupation of impurity site $\alpha$, and the average Coulomb interaction between $M$ orbitals, $\bar{U}$, is defined as [@held2007] $$\begin{aligned}
\label{eq:barU}
\bar{U} = \frac{U+(M-1)(U-2J)+(M-1)(U-3J)}{2M-1} \ .\end{aligned}$$ The potential shift of Eq. is added to the impurity self-energy; its form is directly tailored to the Hubbard-Kanamori interaction Hamiltonian in Eq. for a [$t_{2g}$]{}- or $e_g$-model resulting from an octahedral crystal-field environment of $M$ interacting orbitals ($M=3$ for [CaVO$_3$]{} and $M=2$ for LuNiO$_3$).
In this work, we draw particular attention on how the occupations $n_\alpha$ used for the DC correction are evaluated, i.e., whether they correspond to: a) the occupations of the Wannier functions as obtained from DFT, or b) the occupations corresponding to the impurity Green’s function $G_\text{imp}$ calculated by the QMC solver within the DMFT step. It can be misleading to assume that these quantities are the same, even within a CSC calculation. Indeed, when the system is in a charge-ordered phase, such as, e.g., in heterostructures or nickelates, or in any other case with several inequivalent impurity problems, different impurities can exchange charge within the DMFT loop, potentially leading to drastic changes of the local occupations compared to the ones calculated within the DFT step. In principle, only the occupations evaluated for the impurity problem within DMFT that are used to define the charge density correction, have physical meaning within a CSC DFT+DMFT calculation. By contrast, the occupations obtained in the DFT part, by filling up the lowest energy KS states, do not correspond to the charge density that is used to evaluate the Kohn-Sham potential in a CSC calculation. However, in the case of a OS DFT+DMFT calculation, the question of whether to use DFT or DMFT occupations for the DC correction is ambiguous. An informal (and perhaps unrepresentative) community survey conducted by us, has shown that both variants are currently used in different studies. Here, we show that in certain systems the question of how to extract $n_{\alpha}$ can have a strong influence on the results, and that one should be aware of this issue when evaluating the DC correction.
### Calculation of observables
From the imaginary-time Green’s function, we calculate the spectral weight around the Fermi level, , which indicates whether the system is metallic or insulating [@Fuchs:2011]. For $T=0$ ($\beta \rightarrow \infty$), $\bar{A}$ is identical to the spectral function at $\omega=0$. For finite temperatures, it represents a weighted average around $\omega=0$ with a width of $\sim k_\text{B}T$ [@Fuchs:2011].The full real-frequency spectral function $A(\omega)$ is obtained via analytic continuation using the maximum entropy method [@Jarrel:2010]. The on-site density matrix can be obtained directly from the local Matsubara Green’s function as .
To extract the total energy of the system we use the following formula [@PhysRevB.74.125120]: $$\begin{aligned}
\begin{split}
E_{\text{DFT+DMFT}} &= E_{\text{DFT}}[\rho] \\ &- \frac{1}{N_k} \sum_{\nu \in \mathcal{C} ,\mathbf{k}} \epsilon_{\nu,\mathbf{k}}^{\text{KS}} \ f_{\nu \mathbf{k}} + \langle \hat{H}_{\text{KS}} \rangle_{\text{DMFT}} \\ & + \langle \hat{H}_{\text{int}} \rangle_{\text{DMFT}} - E_{\text{DC}}^\text{imp} \quad ,
\end{split}
\label{eq:dmft-dft-tot-en-2}\end{aligned}$$ where $\epsilon_{\nu,\mathbf{k}}^{\text{KS}}$ are the KS eigenvalues with corresponding weights $f_{\nu \mathbf{k}}$ within the correlated subspace $\mathcal{C}$, and $\langle \cdot \rangle_{\text{DMFT}}$ denotes quantities evaluated from the DMFT solution. The interaction energy $\langle \hat{H}_{\text{int}} \rangle_{\text{DMFT}}$ is calculated using the Galitskii-Migdal formula [@abrikosov2012methods; @galitskii1958], and the last term in Eq. subtracts the DC energy. To ensure high accuracy, we sample the total energy over a minimum of additional 60 converged DMFT iterations after the CSC DFT+DMFT loop is already converged. Convergence is reached when the standard error of the site occupation during the last 10 DFT+DMFT loops is smaller than $1.5 \times 10^{-3}$. This way, we achieve an accuracy in the total energy of $<5$meV. All DMFT calculations are performed for $\beta=40$ eV$^{-1}$, which corresponds to a temperature of 290 K.
Materials & Results {#sec:results}
===================
{width="0.9\linewidth"}
To analyze the effect of CSC within DFT+DMFT, we study two representative examples of TM oxides with different levels of complexity. First, we consider the case of unstrained and strained [CaVO$_3$]{}. While in the former case this material is a correlated metal [@Nekrasov:2005; @Pavarini_et_al:2004], it has recently been demonstrated that tensile epitaxial strain leads to a transition towards the Mott insulating state within OS DFT+DMFT calculations [@Beck:2018]. An important aspect in this transition is the strain-induced crystal-field splitting between the partially filled $t_{2g}$ orbitals, leading to a strong orbital polarization, and thus a local charge redistribution, which can potentially affect the result of a CSC compared to a OS DFT+DMFT calculation. However, in [CaVO$_3$]{}, all correlated sites are symmetry-equivalent and thus the DC correction is irrelevant when using a minimal “$t_{2g}$-only” correlated subspace.
Second, we consider the rare earth nickelate LuNiO$_3$, which exhibits a complex interplay between a specific structural distortion and an associated charge ordering, resulting in a transition from a paramagnetic metallic towards an also paramagnetic but insulating phase [@catalano.review]. In previous works, it was shown that DFT+DMFT is well suited to describe this correlated insulating state [@Park:2012hg; @Park:2014; @Subedi:2015en]. Since two symmetry-inequivalent types of Ni sites appear in the insulating state, this case allows to analyze the effect of a site-dependent DC correction within CSC DFT+DMFT calculations.
Both materials, [CaVO$_3$]{} and LuNiO$_3$, exhibit a distorted perovskite structure with $Pbnm$ space group (in the case of LuNiO$_3$ this corresponds to the high symmetry metallic phase). The corresponding unit cell contains four TM atoms surrounded by edge-connected oxygen octahedra, that are tilted and rotated around the Cartesian axes, corresponding to the so-called GdFeO$_3$-type distortion ($a^-a^-c^+$ tilt system in Glazer notation), as depicted in Fig. \[fig:structure\]. The $d$-levels of the TM ions are split into $e_g$ and $t_{2g}$ manifolds by the octahedral crystal field, and the remaining degeneracies can be further lifted by additional distortions of the oxygen octahedra (also shown schematically in Fig. \[fig:structure\]).
[CaVO$_3$]{} - orbital polarization {#sec:cvo}
-----------------------------------
As stated above, bulk [CaVO$_3$]{} is a moderately correlated metal with weak [orbital polarization]{} that can undergo a transition to the Mott-insulating state under tensile epitaxial strain or in ultra-thin films [@Beck:2018; @Gu_et_al:2013; @Mcnally:2019]. As has been pointed out in Ref. , the [orbital polarization]{} resulting from the orthorhombic distortion of the perovskite structure (related to the tilts and rotations of the octahedral network) is an important factor in the metal-insulator transition (MIT). Several examples suggest that by an appropriate tuning of the bandwidth and the [crystal-field splitting]{} via, for example, strain or dimensional confinement, the resulting charge redistribution enhances the [orbital polarization]{}, eventually leading to a MIT [@Gu_et_al:2013; @Sclauzero/Dymkowski/Ederer:2016; @Beck:2018]. For example, as depicted in Fig. \[fig:structure\], tensile epitaxial strain will lift the degeneracy of the $t_{2g}$-states, lowering the energy of one orbital compared to the other two. Since the [orbital polarization]{} in [CaVO$_3$]{} can be seen as a measure for the likelihood of the Mott-insulating state, it is clear that describing this quantity accurately is essential for the success of the chosen method.
As described in Sec. \[sec:theory\], we perform DFT+DMFT calculations for the bulk structure of [CaVO$_3$]{} using three different schemes, i.e., OS calculations using either MLWFs (magenta line in Fig. \[fig:dmft\_cvo\]) or PLOs (blue lines in Fig. \[fig:dmft\_cvo\]) to represent the correlated subspace, as well as CSC calculations using PLOs (green lines in Fig. \[fig:dmft\_cvo\]). From this we obtain the orbital occupations and spectral weight at the Fermi level, shown in Fig. \[fig:dmft\_cvo\], as a function of the Coulomb interaction parameter $U$. In all cases, the spectral weight is finite for small values of $U$, where the system is metallic, and then becomes zero in the insulating phase for large $U$, with a rather sharp transition at $U_\text{MIT}$. For the unstrained bulk system, all three approaches give identical results for the spectral weight as function of $U$, with a critical value of $U_\text{MIT}$=5.5eV. Thus, at $U \approx 5$ eV, which is typically considered as realistic value for $3d^1$ transition metal oxides [@Pavarini_et_al:2004], we find a finite weight corresponding to metallic behaviour, in agreement with experimental observations. This shows that the obtained results do not depend on details of the implementation, such as small differences in the basis used to represent the correlated subspace.
From the occupations shown in Fig. \[fig:dmft\_cvo\] (top left), it can be seen that the orbital polarization is weak in the metallic regime, but is significantly enhanced above [$U_{\textrm{\sc mit}}$]{}, where the occcupation of one orbital is decreased compared to the other two orbitals. This is in line with the crystal-field splitting of the bulk on-site Wannier energies, where one orbital is energetically higher than the other two, with only a small difference between the latter [@Beck:2018]. Here, the two different OS calculations agree extremely well, while the orbital polarization is slightly reduced in the CSC calculation, however with no apparent effect on the predicted [$U_{\textrm{\sc mit}}$]{}.
Under 4% tensile strain (right panels in Fig. \[fig:dmft\_cvo\]), the MIT is shifted to lower $U$ values, below the realistic value of $U \approx 5$eV. Here, both the MLWF- and PLO-type OS calculations agree within the accuracy of the method, and give exactly the same value for the critical interaction parameter of $U_\text{MIT}=4.7$eV. The CSC calculation, however, places the MIT at a slightly higher value of $U_\text{MIT}=4.9$ eV.
An even stronger difference between OS and CSC calculations can be seen in the orbital polarization, which is generally strongly enhanced compared to the unstrained case, due to a large strain-induced crystal-field splitting [@Sclauzero/Dymkowski/Ederer:2016; @Beck:2018] (see Fig. \[fig:structure\]).
![DFT+DMFT results obtained from OS and CSC calculations with VASP (PLO basis), compared to OS calculations using <span style="font-variant:small-caps;">Quantum ESPRESSO</span> (QE, MLWF basis), for bulk (left) and strained (right) CaVO$_3$. Top panels: Orbitally-resolved occupations as a function of the interaction parameter $U$. Bottom panels: averaged spectral weight at the Fermi level, $\bar A(0)$. []{data-label="fig:dmft_cvo"}](dmftset_gb-occ_b40_update){width="\linewidth"}
Within the OS calculations, both PLO- and MLWF-based, we find that in the insulating regime two orbitals become completely empty, while the third one is essentially fully occupied by a single electron, i.e., the system exhibits full orbital polarization. In the CSC calculation this orbital polarization is significantly reduced, with a maximal occupation of $\sim0.7$ in the preferential orbital. The crystal-field-induced orbital polarization, enhanced by electronic interaction effects, has previously been suggested to be an important factor supporting the insulating phase [@Pavarini_et_al:2004], since the resulting effective half-filling of only one orbital promotes the MIT as opposed to fractional occupation of three degenerate levels. This is consistent with our results, since the lower orbital polarization in the CSC calculation correlates with a higher [$U_{\textrm{\sc mit}}$]{} compared to the OS case.
![Orbitally-resolved spectral functions for CaVO$_3$ under 4% tensile epitaxial strain, obtained from (PLO-based) OS (left) and CSC (right) DFT+DMFT calculations.[]{data-label="fig:spec_cvo"}](dmftset_spec_b40){width="\linewidth"}
To illustrate the difference between OS and CSC calculations in the strained case, we plot the spectral function $A(\omega)$ at $U=5.0$eV for both cases in Fig. \[fig:spec\_cvo\]. Here, the three different line-styles correspond to the three different [$t_{2g}$]{}-like orbitals. As discussed previously, in the OS calculation one of the orbitals is essentially completely occupied, while the remaining two are empty. In contrast to this, the CSC calculation shows a correlation-induced charge redistribution from the occupied orbital to the previously empty orbitals. Furthermore, comparing the gap sizes of both cases, it is clearly visible that in the CSC case the gap is reduced compared to OS, similar to what has been reported in earlier studies on SrVO$_3$ [@Bhandary:2016].
Overall, we conclude that charge self-consistency plays only a minor role for systems with weak or vanishing orbital polarization, where the corresponding charge redistribution obtained within DMFT compared to the DFT calculation is small. In contrast, for systems with strong differences in orbital occupations, the OS calculation can lead to an overestimation of the orbital polarization, which in turn can affect the tendency of the system to undergo a MIT. While the effect on [$U_{\textrm{\sc mit}}$]{} is not too strong in the present case, the corresponding differences in spectral properties can be more pronounced. Nevertheless, it appears that for the present case, OS calculations can at least give reliable qualitative information about the overall system behavior, such as, e.g., the effect of tensile epitaxial strain on [$U_{\textrm{\sc mit}}$]{}, favoring the insulating state.
Furthermore, we note that in our calculations using frontier orbitals, we find very good agreement between the PLO- and MLWF-based method, both in the spectral properties and for the orbital occupations. This is in contrast to previous studies, reporting that projector-based methods require a larger U in [$p\textrm{-}d$]{} models due to larger hybridization effects [@Dang:2014].
LuNiO$_3$ — charge-ordering and structural energetics
-----------------------------------------------------
The second case that we analyze is LuNiO$_3$. This material belongs to the family of rare-earth nickelates, $R$NiO$_3$, where $R$ can be any rare-earth ion ranging from Lu to Pr, including Y. All members of the series exhibit a MIT, which is accompanied by a structural transition, lowering the space group symmetry from $Pbnm$ to $P2_1/n$. The corresponding structural distortion results in a three dimensional checkerboard-like arrangement of long bond (LB) and short bond (SB) NiO$_6$ octahedra, referred to as breathing mode distortion [@Medarde2008], and schematically shown on the right side of Fig. \[fig:structure\]. Recent theoretical work indicates that this transition is related to an electronic instability towards spontaneous charge disproportionation on the Ni sites, which couples to the breathing mode, leading to a first-order coupled structural-electronic transition into a paramagnetic charge-disproportionated insulator (CDI) [@peil:2019; @hampel:2019]. Furthermore, the choice of the $R$ site cation determines the degree of octahedral rotations in the corresponding high symmetry $Pbnm$ structure, and thus the bandwidth. The latter then controls how close the system is to the electronic instability, driving trends across the series [@peil:2019; @hampel:2019; @Mercy2017; @Varignon:2017is; @hampel2017].
Here, we use the case of LuNiO$_3$ to analyze if, and how, the charge disproportionation, as a specific example for charge-ordering phenomena in general, is affected by the inclusion of charge self-consistency in DFT+DMFT. Earlier studies by @Park2014short also investigated the effect of CSC and DC for LuNiO$_3$ using a [$p\textrm{-}d$]{}-type subspace. They found only a small effect due to CSC on total energy calculations, but had to adjust the DC correction to obtain a stable finite equilibrium breathing mode distortion. Here, we use only the two $e_g$-like frontier orbitals per Ni site for our DFT+DMFT calculations. As shown in Ref. , the electronic instability towards charge disproportionation and the resulting *site-selective Mott transition* [@Park:2012hg] occurring in the paramagnetic state is well described within DFT+DMFT using such a minimal subspace.
@Subedi:2015en found that the CDI state emerges in the frontier $e_g$ model for nickelates for rather large values of the Hund’s coupling $J$, and is very sensitive to its variation. The fact that the Hund’s coupling $J$ is the critical ingredient in the occurence of the CDI state was first proposed by @Mazin:2007. They showed in an atomic picture that when $U-3J$ becomes small and is overcome by the potential energy difference between the Ni sites, $\Delta_s$, which results from the breathing mode distortion and the charge disproportionation, the CDI state is favored. This regime is accessible in systems with small or negative charge-transfer gap, which results in a strong screening of the Coulomb interaction in the effective $d$ bands, whereas the Hund’s coupling is less sensitive to screening [@Mazin:2007]. A strong screening of $U$ in nickelates has been confirmed by recent studies using the constrained random phase approximation (cRPA) [@Seth:2017; @hampel:2019]. Moreover, in Ref. it is shown, that such a CDI regime for small or negative $U-3J$ is also accessible in a general three orbital Hubbard model, and is thus not limited to nickelate systems.
To isolate the effect of the structural breathing mode distortion on the electronic charge disproportionation and the total energy of the system, we distinguish the various structural distortions present in LuNiO$_3$ by employing a symmetry-based mode decomposition [@PerezMato:2010ix], as outlined in Refs. . This allows to add the breathing mode distortion, with symmetry label $R_1^+$, on top of the relaxed $Pbnm$ structure, and systematically vary its amplitude without changing any other parameter of the unit cell. We use the software ISODISTORT [@Campbell:2006] to perform the mode decomposition.
### Results for fixed structure
![Results of different DFT+DMFT calculations for LuNiO$_3$ using the experimental $R_1^+$ amplitude, $U=1.85$eV, and varying $J$. CSC and OS calculations are labeled accordingly. For calculations labeled $n_\alpha^\text{DFT}$ ($n_{\alpha}^\text{DMFT}$) the DFT (DMFT) occupations have been used to evaluate the DC corrcetion. The dashed vertical line marks the cRPA value of $J$ [@hampel:2019]. Top: charge disproportionation $\nu$; bottom: corresponding spectral weight at the Fermi level.[]{data-label="fig:lno_dc_comparison"}](lno-jscan-dc-comparison.pdf){width="0.8\linewidth"}
First, we calculate the properties of LuNiO$_3$ for a fixed structure, using the experimentally observed breathing mode amplitude, $R_1^+=0.075$ [@Alonso:2001bs], thereby varying the strength of the Hund’s coupling $J$. As discussed above and shown in Ref. , the charge disproportionation and the resulting MIT depend sensitively on $J$, which thus allows us to critically examine the influence of CSC on the most crucial system properties. We use a fixed $U$ value of 1.85 eV, which corresponds to the value calculated for LuNiO$_3$ using the cRPA [@hampel:2019; @Seth_Georges:2017]. The results are depicted in Fig. \[fig:lno\_dc\_comparison\], where in the top panel the charge disproportionation, $\nu \equiv \langle n_{\text{LB}} \rangle - \langle n_{\text{SB}} \rangle$, i.e. the difference of the $e_g$ occupation between the LB and SB Ni sites, is shown as function of $J$. The bottom panel shows the corresponding value for $\bar{A}(0)$, indicating whether the system is metallic or insulating. The dashed vertical line corresponds to the $J$ value obtained within cRPA [@hampel:2019; @Seth_Georges:2017]. Different data-sets in Fig. \[fig:lno\_dc\_comparison\] correspond to DFT+DMFT calculations with different treatments of the DC correction, both OS and CSC, which we discuss in the following.
We first focus on the data-set labeled “CSC $n_\alpha^\text{DMFT}$” (shown in red), which corresponds to the CSC calculation where the occupations entering the DC correction are calculated from the impurity occupations, and are updated in each DMFT iteration. As discussed in Sec. \[par:DC\], this is the correct way to perform such CSC DFT+DMFT calculations, since the converged $n_\alpha^\text{DMFT}$ give rise to the corrected charge density from which the KS potential is constructed within the DFT step. It can be seen, that the transition to the CDI occurs at $J=0.2$ eV, indicated by clear jumps in $\nu$ and $\bar{A}(0)$. The jump in $\nu$ is related to a drastic change in the DC potential difference between the Ni sites, since, for not too large $J$ (see also below), the DC correction tends to increase the charge disproportionation by further lowering the $e_g$ states on the more occupied LB site compared to the less occupied SB site. This is discussed and analyzed in more detail in Appendix \[appendix\].
For further increasing $J$, $\nu$ stays almost constant until $J \approx 0.8$ eV, where $\nu$ decreases again. Finally, at around $J=1.2$ eV, the system becomes metallic again. This can be explained by the fact that for increasing $J$, the DC potential, proportional to $\bar{U}=U-\tfrac{5}{3}J$ (see Eq. for $M=2$), decreases, and eventually changes sign for $J=1.11$ eV where $\bar{U}=0$. Thus, above $J=1.11$eV the DC correction opposes the charge disproportionation by lowering the $e_g$ levels of the SB sites relative to the LB sites.
Next we compare the CSC calculations with the simpler OS calculations. As discussed in Sec. \[par:DC\], it is ambiguous whether to use the DMFT impurity occupations or the occupations of the Wannier functions obtained within DFT, $n_\alpha^\text{DFT}$, to evaluate the DC correction. We first compare to the OS calculations where $n_\alpha^\text{DMFT}$ has been used for the DC correction (shown in green). It can be observed that in these OS calculations the system is already in the CDI state even for $J=0.2$ eV. In addition, a small shift to larger $\nu$ can be observed compared to the CSC case. Thus, the tendency towards the CDI state is slightly stronger than in the CSC calculations.
In contrast, the OS calculations using $n_\alpha^\text{DFT}$ (shown in orange) leads to a significantly reduced $\nu$, which increases slowly with increasing $J$. Moreover, for small $J<0.5$ eV, clear metallic behavior is observed, while from $J=0.5$ to 1.0 eV, the system undergoes the MIT, where eventually at $J=1.0$ eV the system is completely in the CDI state with $\nu > 1.0$. The occupations obtained in the initial DFT step are $n^\text{DFT}_\text{LB}\approx 1.15$ and $n^\text{DFT}_\text{SB}\approx 0.85$.
For comparison, we also perform CSC calculations where the DFT occupations are used for the DC correction (shown in purple). However, one should note, that these calculations are somewhat artificial, since the DFT Wannier orbital occupations loose their physical meaning in a CSC calculation, and are used here just to allow for a more systematic comparison between OS and CSC calculations. One can see that overall the results of these calculations show similar behavior than the corresponding OS calculation using $n_\alpha^\text{DFT}$, albeit with a small further reduction of $\nu$.
The fixed structure calculations for LuNiO$_3$, show that performing CSC calculations leads to a small reduction of the charge disproportionation compared to OS calculations, if in both calculations the DMFT impurity occupations are used to determine the DC potential. Moreover, we find that the DC has a very strong effect, so that a OS calculation with DFT occupations significantly underestimates the tendency towards charge disproportionation compared to the “correct” CSC calculation.
Overall, we conclude that CSC has a small, but certainly not negligible influence on the DFT+DMFT calculations for LuNiO$_3$, reducing $\nu$ by approx. 10%. However, this only holds if DMFT occupations are used in the OS calculation to evaluate the DC correction. If DFT occupations are used in the OS calculation, then the tendency towards the CDI state is significantly weakened, indicated by the much smaller $\nu$, which is related to the smaller difference in the DC potential shift. However, compared to a hypothetical CSC calculation also using $n_\alpha^\text{DFT}$ for the DC correction, $\nu$ is again slightly enhanced in the OS calculation. Thus, one can clearly distinguish between the effect of the DC correction, and the effect of the charge density correction in the CSC calculation. The latter tends to reduce the charge disproportionation, independently of the chosen DC scheme, and analogous to reducing the orbital polarization in the case of [CaVO$_3$]{} discussed in Sec. \[sec:cvo\]. Finally, our results also indicate that the OS calculations using DMFT occupations for the DC correction already provide a good approximation for the CSC calculation, even though they slightly overestimate the SB/LB splitting and thus the tendency towards the CDI state.
### Influence on energetics
![Comparison of energetics from DFT+DMFT for LuNiO$_3$ as function of the $R_1^+$ amplitude. Calculations without CSC are labeled OS, both in combination with DC occupations obtained from DFT ($n_\alpha^{\text{DFT}}$) or with occupations obtained from each DMFT step ($n_\alpha^{\text{DMFT}}$). The left panels show results for small $J=0.42$ eV and the right panels for large $J=1.1$ eV, where the upper panel shows the energy as function of the $R_1^+$ amplitude and the panels at the bottom the corresponding spectral weight at the Fermi level.[]{data-label="fig:lno_energetics"}](dmft_energetics_comp.pdf){width="1.0\linewidth"}
Another important aspect is the influence of charge self-consistency in total energy calculations for different $R_1^+$ amplitudes, i.e., for different amplitudes of the structural breathing mode distortion. As the $R_1^+$ amplitude, and thus $\nu$, changes, the DC potential and energy correction change accordingly. In addition, within the CSC calculation, the Hartree energy and other DFT energy contributions are evaluated from the corrected, self-consistent charge density. Strictly speaking, only a full CSC calculation gives physical meaningful total energies [@Kotliar:2006], but nevertheless we discuss the difference here to better understand the influence of performing full CSC calculation [@Park2014short; @PhysRevB.76.235101]. To analyze the resulting effects, we again use $U=1.85$ eV and two different values for $J$, 0.42 eV (the cRPA value) and 1.1 eV (where $\bar{U}\approx 0$ and thus the DC correction vanishes). For both cases, we compare OS with CSC calculations with different treatments of the DC correction, as introduced above. The results are shown in Fig. \[fig:lno\_energetics\], where the top panels show the total energy as function of the $R_1^+$ amplitude, and the bottom panels show the corresponding $\bar{A}(0)$.
For the smaller value, $J=0.42$eV, both the OS (green) and CSC (red) result in an energy minimum at a finite $R_1^+$ amplitude close to the experimental value (indicated by the vertical line). However, the OS calculation exhibits a much stronger response on the $R_1^+$ amplitude, and hence shows a significantly deeper energy minimum. In contrast, the “artificial” CSC calculation using $n_\alpha^{\text{DFT}}$ for the DC correction (purple), exhibits no energy minimum for $R_1^+>0$. Furthermore, the “correct” CSC calculation using $n_\alpha^\text{DMFT}$ undergoes a MIT to the CDI between $R_1^+=0$ and $R_1^+=0.03$Å, while the corresponding OS calculation is already insulating without structural distortion and the CSC calculation with $n_\alpha^\text{DFT}$ remains metallic for any calculated $R_1^+$ amplitude.
For $J=1.1$ eV, both CSC calculations, done either with DFT (purple) or DMFT occupations (red), agree very well (due to $\bar{U} \approx 0$ in the DC) and do not result in a stable finite breathing mode amplitude, even though both undergo a MIT at around $R_1^+=0.03$ and exhibit a large charge disproportionation $\nu$ in the insulating state. In contrast, the OS calculation (orange), shows a stronger response, and predicts a breathing mode amplitude of $R_1^+=0.06$ . Note that here we used $n_\alpha^\text{DFT}$ for the DC correction, but the same result would be obtained using $n_\alpha^\text{DMFT}$, due to $\bar{U}\approx 0$.
These results show that, even though the effect of charge self-consistency on $\nu$ for fixed crystal structure seems to be relatively minor, the effect on the energetics can be quite drastic, such that one can obtain a finite breathing mode distortion within a OS calculation, while the CSC calculation does not exhibit an energy minimum for $R_1^+>0$.
Summary {#sec:summary}
=======
We have studied the effect of charge self-consistency and the role of the DC correction within CSC DFT+DMFT calculations in two representative examples of transition metal oxides, using only a minimal correlated subspace corresponding to few frontier bands around the Fermi level. Our goal is to better understand in which cases charge self-consistency is really required in order to obtain accurate results, and in which cases a computationally much cheaper OS calculation might be sufficient.
For [CaVO$_3$]{}, we find that the strong orbital polarization in the insulating phase under tensile strain is significantly overestimated by about 30% in OS compared to CSC calculations, in agreement with similar calculations for SrVO$_3$ in Refs. . This has a small but noticeable effect on $U_{\text{MIT}}$, the critical $U$ for the MIT, which is slightly underestimated in the OS calculations. In contrast, for the unstrained system, where the orbital polarization is much smaller, the difference between CSC and OS calculations is nearly negligible, even though also in this case the orbital polarization is slightly overestimated in OS calculations. Furthermore, we also compared OS calculations using PLO-based and MLWF-based schemes for constructing the correlated subspace, and found very good agreement between the two methods.
While for [CaVO$_3$]{} all TM sites are symmetry-equivalent, and thus the site-dependent but orbitally-independent DC correction does not affect the results, for the second example investigated in this work, LuNiO$_3$, the DC correction becomes rather important. Here, we find that if DMFT occupations are used to evaluate the DC correction in the OS calculation, one can obtain results that are in rather good agreement with the CSC calculation, even though the charge disproportionation $\nu$ is overestimated by $~\sim 10$%. Thus, similar to reducing the orbital polarization for strained [CaVO$_3$]{}, including charge self-consistency leads to a somewhat more homogeneous charge distribution compared to a OS calculation. Nevertheless, it appears that in order to obtain qualitative insights or general trends, OS calculations can be a reasonable approximation, even in charge ordered systems, if the DMFT occupations are used for the DC. However, our analysis of the energetics of the breathing mode distortion shows that for certain observables, such as the total energy and resulting structural distortions, charge self-consistency can be crucial. For example in the case of LuNiO$_3$, OS calculations overestimate the response on the $R_1^+$ mode, in the most extreme case leading to a stable finite breathing mode amplitude, which is absent in the CSC calculation. In this case it is is inevitable to perform a full CSC calculation to obtain reliable results.
In summary, the effect of charge self-consistency is mainly to reduce a potential site or orbital polarization by favoring a more “homogeneous” distribution of electrons over all sites and/or orbitals. For the cases studied in this work, this results in a weak to moderate charge redistribution, which can be quantitatively relevant, depending on the specific application. In particular for total energy calculations, which depend on a subtle balance between different contributions, charge self-consistency can be crucial to obtain quantitatively and even qualitatively correct results. Nevertheless, it appears that cheaper OS calculations are often sufficient to gain insight into the system properties on a qualitative level, even though the, in principle ambiguous, choice of DFT or DMFT occupations to evaluate the DC correction in the OS calculations can become crucial. In the present examples, the use of DMFT occupations provided better agreement with the full CSC calculation, but in other cases this approach might also severely overestimate the electron transfer between inequivalent sites.
We hope that our detailed analysis of two specifically selected cases, provides useful insights for future DFT+DMFT studies of related material systems, thus allowing the treatment of larger and more complex materials systems by avoiding the higher computational cost of a CSC calculation when possible.
This work was supported by ETH Zurich and the Swiss National Science Foundation through NCCR-MARVEL. Calculations have been performed on the cluster “Piz Daint” hosted by the Swiss National Supercomputing Centre.
Influence of the DC on the effective inter-site splitting {#appendix}
=========================================================
In this appendix we explicitly show, how the DC corrections affects the $e_g$ level splitting between the two inequivalent Ni sites in the charge disproportionated state, which in turn controls the tendency to form a CDI state in the rare earth nickelates.
As outlined in the main text, @Subedi:2015en found that the CDI state emerges in the frontier $e_g$ model for nickelates when the following inequality is satisfied (derived from the the atomic limit): $$\begin{aligned}
U - 3J \lesssim \Delta_s \quad .
\label{eq:neg-charge-transfer}\end{aligned}$$ Here, $\Delta_s$ is the “bare” site splitting, i.e., the difference in the average $e_g$ orbital energy between the SB and LB Ni sites, and is given as: $$\begin{aligned}
\Delta_s = \Delta_s^{\text{DFT}} - \Delta_s^{\text{DC}} \quad ,
\label{eq:deltas}\end{aligned}$$ where the first term, $\Delta_s^{\text{DFT}}$, denotes the corresponding splitting obtained within DFT from the on-site energies of the Wannier functions, and is found to be $\approx 0.25$ eV for $R_1^+=0.075$ in LuNiO$_3$. The second term, $\Delta_s^{\text{DC}}$, arises from the difference in the DC potential between the SB and LB sites: $$\begin{aligned}
\Delta_s^{\text{DC}} = \Sigma_{\text{dc,SB}} - \Sigma_{\text{dc,LB}} \quad .
\label{eq:deltas_dc}\end{aligned}$$
![$U-3J-\Delta_s$ as function of $J$ for LuNiO$_3$ with the experimental $R_1^+$ amplitude [@Alonso:2001bs], corresponding to the calculations shown in Fig. \[fig:lno\_dc\_comparison\]. $U-3J-\Delta_s$ is shown for the different flavors of DC ($n_\alpha^{\text{DFT}}$ vs. $n_\alpha^{\text{DMFT}}$) and for OS and CSC calculations. If $U-3J-\Delta_s<0$ the CDI state is favored (magenta shaded area).[]{data-label="fig:lno_u3j_ds"}](U-3J-deltas.pdf){width="0.8\linewidth"}
To further elucidate the interplay of the site-dependent DC potential in our DFT+DMFT calculations for LuNiO$_3$, we analyzed the behavior of the critical quantity $U - 3J - \Delta_s$ for obtaining a CDI state as function of $J$ for the different DC schemes applied in our study. The behavior of $U-3J-\Delta_s$ is depicted in Fig. \[fig:lno\_u3j\_ds\] for the different flavors of DC, and for OS and CSC calculations at fixed $U=1.85$ eV and $\Delta_s^{\text{DFT}}=0.25$ eV. The parameter regime which corresponds to the CDI in the atomic limit is highlighted in magenta. To directly compare our calculations with Ref. we also performed OS calculations without applying any DC correction.
For the OS calculation without DC, $\Delta_s^{\text{DC}}=0$ (blue circles), $U-3J-\Delta_s$ becomes negative for $J \leq 0.53$ eV. Then, for the calculations with DC evaluated with DFT occupations (OS: orange squares, CSC: purple crosses) this happens at a slightly smaller $J$ value, leading to an increased response in $\nu(J)$ compared to the calculations without DC correction. This shows, that the DC correction enhances the CDI state for positive values of $\bar{U}$ ($J<1.11$eV), since then $\Delta_s^\text{DC}$ is negative thus $\Delta_s > \Delta_s^\text{DFT}$.
The strong tendency to form the CDI state in the calculations with $n_\alpha^{\text{DMFT}}$ can be explained along the same lines. It can be seen that in the CSC calculations (red crosses), the CDI regime is entered already for $J=0.2$ eV, and in the OS calculations (green stars) for even smaller $J$. Importantly, it can be seen that in these cases the DC potential jumps at the MIT, strongly favoring the CDI state. We note that of course the underlying atomic limit consideration neglects the important effect of the bandwidth, but it nevertheless can give a transparent pciture of the underlying physics.
|
---
address:
- '$^1$Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan'
- '$^2$Theoretische Physik, Eidgenössische Technische Hochschule, CH-8093 Zürich, Switzerland'
author:
- 'Munehisa Matsumoto,$^1$ Chitoshi Yasuda,$^1$ Synge Todo,$^{1,2}$ and Hajime Takayama$^1$'
title: |
Ground-State Phase Diagram of Quantum Heisenberg Antiferromagnets\
on the Anisotropic Dimerized Square Lattice
---
Introduction
============
Recently many low-dimensional antiferromagnets with excitation modes separated from a ground state by a finite energy gap have been synthesized, and effects of impurities or magnetic fields on those materials have been investigated experimentally in relation to the impurity-induced long-range order (LRO) and magnetic-field-induced LRO. For example, there are the $S=1/2$ quasi-one-dimensional (Q1D) Heisenberg antiferromagnet (HAF) with bond dimerization, CuGeO$_3$, [@hase] $S=1$ Q1D HAF’s, NENP, [@meyer] NDMAZ, [@honda1] NDMAP, [@honda2] and PbNi$_2$V$_2$O$_8$, [@uchiyama] and $S=1$ Q1D HAF’s with bond alternation, NTEAP [@hagiwara] and NTENP. [@narumi1] Those materials have attracted our interest since they reveal various aspects of the quantum phase transition between the quantum-disordered spin-gapped phase and the classical (Néel) long-range-ordered phase.
Intrachain spin interaction with or without bond alternation and interchain interaction are considered to be the most basic ingredients to understand the quantum phase transitions mentioned above. More explicitly, they are expected to be modeled effectively by the quantum HAF on the anisotropic dimerized square lattice, which is described by the Hamiltonian: $$\begin{aligned}
\label{ham}
{\cal H}&=&\sum_{i,j}{\bf S}_{2i,j}\cdot{\bf S}_{2i+1,j}
+\alpha\sum_{i,j}{\bf S}_{2i+1,j}\cdot{\bf S}_{2i+2,j} \\
&+&J'\sum_{i,j}{\bf S}_{i,j}\cdot{\bf S}_{i,j+1} \ . \nonumber\end{aligned}$$ Here ${\bf S}_{i,j}$ is the quantum spin operator at site $(i,j)$ on the square lattice. The first two terms in r.h.s. represent the one-dimensional antiferromagnetic (AF) Heisenberg chains with alternating coupling constants, 1 and $\alpha$ $(0\leq \alpha \leq 1)$, and the last term does the AF interchain exchange interaction ($J' \geq
0$). We choose the $x$-axis as being along the chain direction and the $y$-axis as in the perpendicular one. The bond arrangement of this model is shown in Fig. \[notation\].
The ground state of decoupled chains, i.e., $J'=0$, has been well established. In particular, that of the $S=1/2$ chain [@bulaevskii] is the dimer state with a finite spin gap except for the uniform case ($\alpha=1$), which has a critical ground state. On the other hand, for the $S=1$ chain there exist two spin-gapped phases: the Haldane phase [@haldane] at $\alpha > \alpha_{\rm c}$ and the dimer phase at $\alpha < \alpha_{\rm c}$.[@singh] At the critical point $\alpha=\alpha_{\rm c}$ between these two phases the gap vanishes. The value of $\alpha_{\rm c}$ has been estimated to be 0.5879(6).[@singh; @kohno] In both cases, the critical point is considered to belong to the Gaussian universality class.[@singh]
For the AF LRO to appear the higher-dimensionality effect, i.e., the interchain interaction $J'$, is indispensable. In most of the numerical works reported so far, the effect of interchain coupling has been examined in certain approximated or perturbed ways. For example, Sakai and Takahashi [@sakai] estimated the critical strength, $J'_{\rm
c}$, for the uniform case ($\alpha=1$) by the exact diagonalization method for the intrachain interactions combined with the mean-field approximation for the interchain interaction, and obtained $J'^{(S=1/2)}_{\rm c}=0$ and $J'^{(S=1)}_{\rm c}\geq 0.025$. More recently, Koga and Kawakami [@koga] investigated the $S=1$ model by the cluster-expansion method, and obtained $J'_{\rm c} = 0.056(1)$ for $\alpha = 1$. However, there have been only a very limited number of numerical works, in which both of the interchain and intrachain interactions are treated on an equal footing. [@katoh; @kim] Such numerical analyses are certainly required, since the mean-field-like approximation is not necessarily appropriate even in the Q1D regime. [@affleck]
In the present paper, we report the results of quantum Monte Carlo (QMC) simulations by using the continuous-imaginary-time loop algorithm [@evertz1; @evertz2; @harada; @todo2] on the $S=1/2$ and $S=1$ HAF model described by Eq. (\[ham\]). The present paper is organized as follows. In Sec. \[method-section\], the method of our numerical analyses is explained. In Secs. \[result-section-spin-half\] and \[result-section-spin-one\], the ground-state phase diagram parameterized by the strength of the bond alternation, $\alpha$, and that of the interchain coupling, $J'$, is determined precisely for $S=1/2$ and $S=1$, respectively. Especially, for the $S=1$ system with $\alpha=1$, we obtain $J'_{\rm c} = 0.043648(8)$, which is consistent with $0.040(5)$ suggested by the recent QMC work,[@kim] but is much more accurate. Furthermore, both in the $S=1/2$ and $S=1$ systems, the quantum phase transitions between the spin-gapped phases and the AF-LRO phase are confirmed to belong to the same universality class with that of the 3D classical Heisenberg model: the exponent of the correlation lengths is $\nu=0.71(3)$ for $S=1/2$ and $\nu=0.70(1)$ for $S=1$, which coincides fairly well with that of the latter model, $\nu=0.7048(30)$. [@chen] We also show the results on the correlation length and the gap in the spin-gapped phase. In Sec. \[discussions-section\], the topology of the phase diagram is discussed in detail based on the result of the present QMC calculation. We show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected with each other in the extended phase parameter space. This is in a sharp constrast to the strict 1D case, in which the spin-gapped phases are classified into different classes in terms of the so-called string-order parameter.[@denNijs] It is of interest that the 1D spin-gap phases, which have different hidden symmetry, are connected without encountering any singularity in the 2D phase diagram. The final section is devoted to the concluding remarks.
=0.22
Method {#method-section}
======
We consider the system descrived by the Hamiltonian (\[ham\]) with $S=1/2$ and $S=1$. The real-space size is $L_x \times L_y$ and the inverse temperature, i.e., the imaginary-time size, is $\beta=1/T$. Periodic boundary conditions are imposed in the $x$- and $y$-directions. We use the continuous-imaginary-time loop algorithm with multi-cluster update.[@evertz1; @evertz2] Especially, for the $S=1$ system we adopt the subspin-representation technique,[@harada; @todo2] in which the $S=1$ system is represented by an $S=1/2$ system with special boundary conditions in the imaginary-time direction. By using these techniques, we can perform the simulatation up to $L_x \times L_y = 336 \times 48$ with $\beta = 100$ for the $S=1$ case without encountering any difficulty.
The imaginary-time dynamical structure factor is defined by $$\begin{aligned}
&& S_{\rm d}(q_{x},q_{y},\omega) \\
&& \mbox{} =
\frac{1}{L_{x}L_{y}\beta}
\sum_{i,j}\int_{0}^{\beta} \!\! dt \, dt'\,
{\rm e}^{- i {\bf q}\cdot({\bf r}_{i}-{\bf r}_{j})- i \omega(t-t')}
\langle S_{i}^{z}(t)S_{j}^{z}(t') \rangle, \nonumber\end{aligned}$$ where ${\bf q}=(q_{x},q_{y})$ is the wave-number vector, $S_{i}^{z}(t)$ is the $z$-component of the spin on site $i$ at imaginary time $\tau$, and $\langle\cdots\rangle$ is the thermal average. By using $S_{\rm d}(q_x,q_y,\omega)$, the staggered correlation length along the $x$-direction, $\xi_x$, is then evaluated by the second-moment method, [@todo2; @cooper] $$\xi_x = {L_x \over 2\pi}\sqrt{ {S_{\rm d}(\pi,\pi,0) \over
S_{\rm d}(\pi+2\pi/L_{x},\pi,0) } -1}.$$ The correlation length in the $y$-direction, $\xi_y$, and that in the imaginary-time ($\tau$) direction, $\xi_\tau$, which is related to the gap $\Delta$ by $\Delta = 1/\xi_\tau$, are calculated similarly. Finally the staggered susceptibility, $\chi_{\rm s}$, is evaluated by $$\begin{aligned}
\chi_{\rm s} &=&S_{\rm d}(\pi,\pi,0) \\
\mbox{} &=&
\frac{1}{L_{x}L_{y}\beta}
\sum_{i,j}\int_{0}^{\beta} \!\! dt \, dt'\,
e^{-i {\bf \pi} \cdot ({\bf r}_i - {\bf r}_j)}
\langle S_{i}^{z}(t)S_{j}^{z}(t') \rangle. \nonumber\end{aligned}$$ All the structure factors are calculated by using the improved estimators.[@baker] The period of $10^2$–$10^3$ Monte Carlo steps (MCS) is used for thermalization and that of $10^3$–$10^5$ MCS for the evaluation of physical quantities.
Near the critical point $(\alpha_{\rm c},J'_{\rm c})$ of the ground-state transition, the correlation lengths diverge as $$\begin{aligned}
\xi_x, \xi_y\sim t^{-\nu}\\
\xi_{\tau}\sim t^{-z\nu}= t^{-\nu},\end{aligned}$$ where $t$ is the distance from the critical point and $\nu$ is the critical exponent for the correlation length. Here we have put $z=1$ assuming the Lorenz invariance.[@chakravarty] Furthermore, the following finite-size-scaling (FSS) formula[@barber] holds near $(\alpha_{\rm c},J'_{\rm c})$ and $T=0$ for systems with the fixed ratio $L_x:L_y:\beta$, $$\xi_x/L_x \simeq f(t L_x^{1/\nu}, L_x^{z}T) =f(t L_x^{1/\nu}),
\label{fss-formula-for-xi}$$ and similar ones for $\xi_y$ and $\xi_{\tau}$, and $$\chi_{\rm s}\simeq L_x^{\gamma/\nu}g(t L_x^{1/\nu},L_x^{z}T)=
L_x^{\gamma/\nu}g(t L_x^{1/\nu}).
\label{fss-formula-for-chi-s}$$ Here $f$ and $g$ are scaling functions and $\gamma$ the exponent for $\chi_{\rm s}\ (\sim t^{-\gamma})$. Note that $L_xT$ is put constant in the above equations. We assume a polynomial up to the second order for the scaling functions. By using least-squares fitting, we obtain the critical point $(\alpha_{\rm c}, J'_{\rm c})$ and the associated critical exponents $\nu$ and $\gamma$.
In addition, at some points in the spin-gapped phase, we explicitly evaluate the correlation lengths, $\xi_x$ and $\xi_y$, and the gap, $\Delta$, at $T=0$ in the thermodynamic limit $L_x, L_y \rightarrow
\infty$. For this purpose we extrapolate the simulated data first to the ground state $T\rightarrow 0$ and then to the thermodynamic limit $L_x, L_y \rightarrow \infty$.
Results for $\bf S=1/2$ {#result-section-spin-half}
=======================
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Ground-state phase diagram
--------------------------
In Fig. \[gs\_phase\_diagram\_spin\_half\] we show the ground-state phase diagram of the $S=1/2$ system obtained by the FSS analysis explained in the previous section. As an example of the FSS analysis, we show in Fig. \[fss\_spin\_half\] that of $\xi_\tau$ against $\alpha$ for $J'=1$ (dotted line in Fig. \[gs\_phase\_diagram\_spin\_half\]). The aspect ratio of the $(2+1)$-dimensional system is taken as $L_x:L_y:\beta =
1:1:1$. By the least-squares fitting, the exponent $\nu$ and the critical coupling $\alpha_{\rm c}$ are estimated as 0.71(1) and 0.31407(5), respectively. Here, the figure in parentheses denotes the statistical error ($1\sigma$) in the last digit. We also perform the same analyses on other lines in the $\alpha$-$J'$ plane, whose results are presented by the solid circles in Fig. \[gs\_phase\_diagram\_spin\_half\]. For example, on the line $\alpha
= J'$ (dashed line in Fig. \[gs\_phase\_diagram\_spin\_half\]), we obtain $\alpha_{\rm c} = J'_{\rm c} = 0.52337(3)$ and $\nu = 0.71(3)$. In the phase diagram we can see that the ground state of the chain ($J'=0$) is the dimer state with a spin gap except for $\alpha=1$, [@sakai; @aoki; @affleck2; @sandvik] and that the region of the AF-LRO phase enlarges monotonically as $J'$ increases.
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Our phase diagram is qualitatively the same as that of Katoh and Imada (KI), [@katoh] but not quantitatively. In particular, we obtain the critical point, $\alpha_{\rm c}=0.31407(5)$, on the line $J'=1$ (dotted line in Fig. \[gs\_phase\_diagram\_spin\_half\]), which is significantly smaller than their estimate $\alpha_{\rm c} = 0.398$. More importantly, in the present simulation, the critical exponent $\nu$ on the transition points is evaluated as $\nu = 0.71(1)$, which is consistent with $\nu = 0.7048(30)$ for the 3D classical Heisenberg model. [@chen] The similar results have been obtained for the 2D $1/5$-depleted HAF model.[@troyer] On the other hand, KI concluded $\nu=1$. The reason of the these discrepancies might be due to the smallness of the system sizes and the inverse temperature used in the study by KI.
Correlation lengths and the gap
-------------------------------
We also evaluate explicitly the ground-state correlation lengths and the gap on some points in the dimer phase by using the dynamic structure factors. Unless the points are very close to the critical line, these quantities in each systems with $L$ ($=L_x=L_y$) saturate to the ground-state values at temperatures we have simulated. For example, the $T$-dependences of these quantities are not to be discernible at $T=0.05$ and $0.01$ for $\alpha=J'=0.4$ and 0.5, respectively. On the other hand, the $L$-dependence still remains in sizes we have calculated. The $L$-dependences of the ground-state spatial correlation lengths and the gap are shown in Fig. \[fig\_corr\_one\_half\] for $\alpha=J'=0.5$, which is close to the critical point $\alpha_{\rm
c}=J'_{\rm c}=0.52337(3)$. Their values in the thermodynamic limit are estimated by fitting $\xi_k(L)$ to $\xi_k(L)= \xi-b
\exp (-c L)$, where $k=x$, $y$, or $\tau$, $\xi$ is the value in the thermodynamic limit, and $b$ and $c$ are fitting parameters. As a result, we obtain $\xi_{x}=3.0089(9)$, $\xi_{y}=2.2097(6)$ and $\Delta=0.32261(4)$ for $\alpha=J'=0.4$ and $\xi_{x}=11.998(9)$, $\xi_{y}=9.312(10)$, and $\Delta=0.0913(2)$ for $\alpha=J'=0.5$. As $\alpha$ ($=J'$) becomes smaller, i.e., the system becomes more distant from the critical point, $\Delta$ becomes larger and $\xi$ smaller.
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Results for $\bf S=1$ {#result-section-spin-one}
=====================
Overview on the phase diagram
-----------------------------
Before going into the QMC analysis, let us here summarize the ground-state phase diagram of the $S=1$ system argued so far, which is shown in Fig. \[schematic-phase-diagram\]. For some points in the phase diagram the ground state is well understood by the previous theoretical and numerical studies: (1) $(\alpha,J')=(0,0)$: The system consists of a set of the isolated antiferromagnetically-coupled spin pairs. The ground state is a trivial tensor product of dimer singlets sitting on each bond. (2) $(\alpha,J')=(1,0)$: The system consists of isolated $x$-parallel Haldane chains. (3) $(\alpha,J')=(1,1)$: The system is a uniform and isotropic 2D HAF. There exists an AF LRO in the ground state.[@kubo] (4) $J'=\infty$: The system consists of $y$-parallel Haldane chains. Note that in this limit the value of $\alpha$ becomes irrelevant (see also discussions in Sec. \[discussions-section\]).
In their analysis by the cluster expansion method, Koga and Kawakami[@koga] derived three phases in the Q1D region, and they called the regions which includes point (1), (2) and (3) the dimer phase, the Haldane phase, and the AF-LRO phase, respectively. The region near point (4) is another Haldane phase. Therefore we call here the region which includes point (2) the Haldane I (H-I) phase and the one which includes the line (4) the Haldane II (H-II) phase. For the uniform systems with $\alpha=1$ the H-I and H-II phases are equivalent when we exchange the roles of the $x$-axis and the $y$-axis, and of $J'$ and $1/J'$.
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Haldane-AF phase transition in the non-dimerized system
-------------------------------------------------------
To demonstrate our FSS analyses, let us begin with critical behavior near $(1,J'_{\rm c})$, which separates the H-I and AF phases ($J'_{\rm c}
\simeq 0.04$ due to Ref. ). We sweep $J'$ near supposed $J'_{\rm c}$ with $\alpha$ fixed to unity. In Fig. \[raw\_data\], $\xi_x/L_x$, $\xi_y/L_y$, and $\xi_{\tau}/\beta$ of the systems with $L_x=L_y=\beta\equiv L$ ($=16$, 24,$\cdots$, 64) are plotted. As one sees immediately, the data suffer from quite large corrections to scaling, i.e., the crossing point of the scaled correlation lengths with two different $L$’s clearly shifts to larger $J'$ as the system size increases. We attribute these large corrections to the strong spatial anisotropy in the coupling constants ($J' \ll 1$). Indeed, the value of $\xi_y$ at $J'=0.0435$ is $9.29(7)$ (see Fig. \[raw\_data\] (b)) even for the largest system size $L=64$, which is quite smaller than those in the other directions ($\xi_x=60.4(3)$ and $\xi_{\tau}=24.7(2)$). Among the three correlation lengths, $\xi_x$ is the largest: the growth of the correlation is dominated only by the system size in the $x$-direction. This indicates that we need larger lattices, especially in the $x$-direction, in order to perform a precise FSS analysis.
To simulate larger lattices with minimal costs, we therefore optimize the aspect ratio, $L_x:L_y:\beta$, as explained below. Expecting that the scaled correlation lengths, $\xi_x/L_x$, $\xi_y/L_y$, and $\xi_\tau/\beta$, become nearly equal with each other at the critical point, we set the aspect ratio $L_x:L_y:\beta$ as $7:1:2$ based on the data presented in Fig. \[raw\_data\]. With this ratio we simulate systems with $L_x=168$, 224, 280, and 336 and perform the FSS analyses. The raw data of $\xi_x/L_x$, $\xi_y/L_y$, and $\xi_{\tau}/\beta$ with this aspect ratio are shown in Fig. \[raw\_data\_large\_system\]. Now $\xi_y=12.02(5)$ even for the smallest system size ($L_x=168$ and $L_y=24$) at $J'=0.0435$. The ratios $\xi_x/L_x$, $\xi_y/L_y$ and $\xi_\tau/\beta$ in a common range of $J'$ become nearly equal, and corrections to scaling become much smaller than in the previous ones as we expected. The FSS plot for $\xi_{x}$ is shown in Fig. \[fss\_plot\_corrx\_large\_system\]. The resultant $J'_{\rm c}$ and $\nu$ are as follows: $(J'_{\rm c},\nu)=(0.043648(9),0.69(1))$ from $\xi_x$, $(0.043649(8),0.71(1))$ from $\xi_y$, and $(0.043648(7),0.69(1))$ from $\xi_{\tau}$.
Averaging these three values we conclude with $$\begin{aligned}
J'_{\rm c}& = & 0.043648(8)\label{precise_Jc}\end{aligned}$$ and $$\begin{aligned}
\nu &=& 0.70(1).\label{nu_S1}\end{aligned}$$
Fixing the value of $J'_{\rm c}$ and $\nu$ thus determined, we next perform the FSS analysis on the staggered susceptibility $\chi_{\rm
s}$. The raw data of $\chi_{\rm s}$ vs. $J'$ are shown in Fig. \[raw\_data\_ssus\] and the fitting result is shown in Fig. \[fss\_plot\_ssus\]. The latter yields $$\gamma=1.373(3).
\label{gamma_S1}$$ The exponents $\nu$ and $\gamma$ we have obtained for the $S=1$ system again agree with those of the 3D classical Heisenberg model. [@chen]
The critical point obtained just above is consistent with the previous result by the method involving the mean-field approximation, $J'_{\rm
c}\geq 0.025$,[@sakai] and also with that by the recent QMC method, $J'_{\rm c}=0.040(5)$.[@kim] On the other hand, the present result significantly differs from that obtained by the cluster-expansion method, $J'_{\rm c} = 0.56(1)$.[@koga] The present FSS analyses on our extensive QMC results make us possible to obtain the critical value with much higher accuracy than the other methods.
Ground-state phase diagram
--------------------------
In a similar way as described in the previous subsection, we obtain other critical points on the $\alpha$-$J'$ phase diagram as shown in Fig. \[gs\_phase\_diagram\_spin\_one\]. First, the scaled correlation lengths, $\xi_x/L_x$, $\xi_y/L_y$, and $\xi_{\tau}/\beta$, are calculated up to $L_x = 64$ with either $\alpha$ or $J'$ fixed. Sweeping $J'$ or $\alpha$ with sufficiently high resolution, we regard a crossing point of these $\xi$’s as the critical point. Note that the optimal aspect ratio depends strongly on the value of $\alpha$ and $J'$. However, we adopt $L_x:L_y:\beta=1:1:1$ for simplicity. Although the results thus obtained suffer from relatively larger systematic corrections than those presented in the last subsection, the absolute magnitude of the systematic error in the estimates should be still smaller enough than the symbol size in Fig. \[gs\_phase\_diagram\_spin\_one\].
For some critical points the FSS analysis as in the previous subsection is also carried out. We obtain the exponents $\nu$ and $\gamma$ which are consistent with Eqs. (\[nu\_S1\]) and (\[gamma\_S1\]), respectively. This supports that the quantum critical phenomena in the $S=1$ system also belong to the same universality class as that of the 3D classical Heisenberg model. An exception is the 1D critical point located at $(\alpha_{\rm c},J_{\rm c}')=(0.5879(6),0),$[@kohno] which separates the dimer phase from the Haldane phase. The apparent value of $\nu$ starts to deviate from (\[nu\_S1\]) when $\alpha$ becomes closer to $\alpha_{\rm c}$. This is attributed to the crossover to the critical phenomena belonging to the Gaussian universality class.[@singh] We confirm that the AF-LRO phase exists between the two spin-gap phases at least down to $J'=0.01$ at $\alpha=\alpha_{\rm
c}$. Although in the present simulation it is quite difficult to prove the existence of the AF-LRO phase at smaller $J'$, we believe that the the point $(\alpha_{\rm c},0)$ is tricritical: the 1D critical point is unstable against an infinitesimal interchain coupling and the AF LRO immediately appears as the same as in the $S=1/2$ uniform chain. [@sakai; @aoki; @affleck2; @sandvik]
Interestingly, the H-II and D phases are adiabatically connected with each other. The gapless AF-LRO phase does not touch the line of $\alpha=0$ as seen in Fig. \[phase\_diagram\_near\_the\_connected\_part\], where the part of the whole phase diagram (Fig. \[gs\_phase\_diagram\_spin\_one\]) near $\alpha=0$ is magnified. Indeed, on the $\alpha=0$ line, which corresponds to the $S=1$ two-leg ladder, it is shown that there exists no critical point by the recent QMC study.[@ladder-paper] Thus, the D phase can be identified with the H-II phase, and there are only two distinct spin-gap phases, H-I and H-II in Fig. \[gs\_phase\_diagram\_spin\_one\]. The closeness of the critical line to the $\alpha=0$ line is due to the strong AF fluctuations, which already exist in the two-leg ladder system.[@ladder-paper]
Correlation lengths and the gap
-------------------------------
We obtain the explicit values of $\xi_{x}$, $\xi_{y}$, and $\Delta$ in the ground state at $(\alpha,J')=(1,0.04)$. They are calculated for systems with sizes $L_{x}=168$, 224, 280, and 336 and with the aspect ratio $L_{x}:L_{y}=7:1$ at $T$ regarded as zero temperature. Their $T$-dependences are negligible at $T=0.01$. We extrapolate the finite-size data to the thermodynamic limit in the same way as explained for the $S=1/2$ system. We obtain $\xi_{x}=39.2(1)$, $\xi_{y}=5.67(1)$, and $\Delta=0.0632(2)$. As $J'$ becomes smaller, $\xi_{x}$ becomes smaller and $\Delta$ larger to reach at $J'=0$ the single chain values $\xi_{x}=6.0153(3)$ and $\Delta=0.41048(6)$.[@todo2]
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Discussions {#discussions-section}
===========
The analyses presented in the preceding section revealed that the ground-state phase diagram of the $S=1$ system has a rather complicated topology, i.e., the H-II and D phases are adiabatically connected with each other, though the channel between them is quite narrow (Fig. \[gs\_phase\_diagram\_spin\_one\]). On the other hand, as for the H-I and D phases, in the 1D system ($J'=0$) these two spin-gapped phases are distinctively separated by the critical point at $\alpha_{\rm c}=0.5879(6)$,[@singh; @kohno] and they are distinguished by the string-order parameter,[@denNijs] which is zero in the former phase and finite in the latter one. The transition can be viewed as a rearrangement of dimer-singlet pattern between the (1,1)- and (2,0)-valence-bond-solid (VBS) states.[@AKLT; @arovas] We emphasize that once $J'$ is introduced, however, the string-order parameter should vanish even in the H-I phase, being similar to the $S=1$ ladder.[@ladder-paper] Still one may consider that the two phases essentially differ with each other since they are separated by the AF-LRO phase. If, however, we introduce the bond alternation also in the $y$-direction, the two phases can be connected without passing the gapless state as explained below.
\(a) (b)
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Let us consider the 2D HAF model defined in the extended parameter space: $$\begin{aligned}
\label{ham2}
{\cal H}&=&J_x\left\{\sum_{i,j}{\bf S}_{2i,j}\cdot{\bf S}_{2i+1,j}
+\alpha_x\sum_{i,j}{\bf S}_{2i+1,j}\cdot{\bf S}_{2i+2,j}\right\} \nonumber \\
&+&J_y\left\{\sum_{i,j}{\bf S}_{i,2j}\cdot{\bf S}_{i,2j+1}
+\alpha_y\sum_{i,j}{\bf S}_{i,2j+1}\cdot{\bf S}_{i,2j+2}\right\} \ .
\nonumber \\\end{aligned}$$ The original Hamiltonian (\[ham\]) corresponds to the case with $J_x=1$, $\alpha_x = \alpha$, $J_y = J'$, and $\alpha_y=1$. The Hamiltonian (\[ham2\]) is invariant under the exchange between $(J_x,\alpha_x)$ and $(J_y,\alpha_y)$.
To draw the phase diagram in this extended parameter space, it is convenient to introduce a parameter, $R \equiv J_y/(J_x+J_y)$. The limits $J_y \rightarrow 0$ and $J_y \rightarrow \infty$ correspond to $R=0$ and 1, respectively. Since $\alpha_x$ ($\alpha_y$) becomes irrelevant in the limit $J_x \rightarrow 0$ ($J_x \rightarrow \infty$), the whole phase diagram in the three-dimensional parameter space is shaped as a tetrahedron. In Fig. \[tetrapack\_phase\_diagram\], we present the ground-state phase diagram parametrized by $R$, $\alpha_x (1-R)$, and $\alpha_y R$ for $S=1/2$ (a) and $S=1$ (b). It should be noted that the phase diagram should be invariant under the transformation, $(R,\alpha_x
(1-R),\alpha_y R) \leftrightarrow (1-R,\alpha_y R,\alpha_x (1-R))$, reflecting the symmetry in the Hamiltonian explained above. In the phase diagram, the edge AB (CD) corresponds to isolated $x$-parallel ($y$-parallel) decoupled chains, the edge AD isolated four-spin plaquettes, and the edge AC (BD) the two-leg ladders in $y$-direction ($x$-direction).
The face ABC (and also CDB) in Fig. \[tetrapack\_phase\_diagram\] corresponds to the original phase diagram shown in Figs. \[gs\_phase\_diagram\_spin\_half\] and \[gs\_phase\_diagram\_spin\_one\], though the $y$-parallel-chain limit $J'\rightarrow\infty$ in the original diagram is represented by one vertex C (B) in the new ones. In the extended phase diagram the shaded (unshaded) area represents the AF-LRO (spin-gapped) phase on the ABC- and CDB-faces.
It should be emphasized that on the ACD- and DBA-faces there is no AF-LRO phase, since the system is one-dimensional dimerized two-leg ladder. There exist only the 1D critical points discussed already. Especially, there is no critical point on the edge AD. Therefore, in the $S=1$ case, the three spin-gapped phases, H-I, D, and H-II, are connected by the path C$\rightarrow$A$\rightarrow$D$\rightarrow$B. Similarly, in the $S=1/2$ case, the two dimer phases, which correspond to the vertex A and D, respectively, are connected directly by the path A$\rightarrow$D. Thus, in both cases, there are only two phases, namely, the spin-gapped phase and the AF-LRO one.
Concluding Remarks {#conclusion-section}
==================
In this paper, we have investigated the ground-state phase diagram of $S=1/2$ and $S=1$ HAF on the anisotropic dimerized square lattice by means of the extensive QMC simulation with the continuous-imaginary-time loop algorithm and the FSS analyses. It is confirmed that, for both $S=$1/2 and 1, the quantum critical phenomena in the model belong to the same universality class as that of the 3D classical Heisenberg model, except for the 1D critical points, which belong to the same universality class as that of the Gaussian model. We have also demonstrated that the spin-gapped phases of the 1D chain are connected when we introduce the interchain couplings with bond alternation. In the 2D system, only one spin-gapped phase exists in both of the $S=1/2$ and $S=1$ systems.
The results obtained in the present work are considered to be the proper basis for investigation of peculiar phenomena observed in the Q1D HAF’s mentioned at the beginning of this paper. We have already reported the QMC analysis on the site-dilution-induced AF LRO in these materials based on the Hamiltonian (\[ham\]). [@yasuda] In order to discuss various experimental results quantitatively, it is certainly necessary to take into account other ingredients than in Eq. (\[ham\]), such as the next-nearest-neighbor intrachain interaction and the single-ion anisotropy. They are beyond the scope of the present work. The present results, however, demonstrate the role of the higher dimensionality which has been overlooked so far.
Most of the numerical calculations in the present work have been performed on the DEC Alpha, SGI ORIGIN 2000, SGI 2800, and RANDOM at the Materials Design and Characterization Laboratory, Institute for Solid State Physics, University of Tokyo and on the Hitachi SR-2201 at the Supercomputer Center, University of Tokyo. The program used in the present simulation was based on the library ‘Looper version 2’ developed by S.T. and K. Kato and also on the ‘PARAPACK version 2’ by S.T. The present work was supported by the “Research for the Future Program” (JSPS-RFTF97P01103) of the Japan Society for the Promotion of Science. S.T’s work was partly supported by the Swiss National Science Foundation.
[99]{}
M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. [**70**]{}, 3651 (1993). A. Meyer, A. Gleizes, J. Girerd, M. Verdaguer, and O. Kahn, Inorg. Chem. [**21**]{}, 1729 (1982); J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C. Erkelens, J. Rossat-Mignod, and W. G. Stirling, Europhys. Lett. [**3**]{}, 945 (1987). M. Yamashita, K. Inoue, T. Ohishi, T. Takeuchi, T. Yosida, and W. Mori, Mol. Cryst. Liq. Cryst. [**274**]{}, 25 (1995); Z. Honda, K. Katsumata, H. Aruga Katori, K. Yamada, T. Ohishi, T. Manabe, and M. Yamashita, J. Phys.: Condens. Matter [**9**]{}, L83 (1997); 3487 (1997). M. Monfort, J. Ribas, X. Solans, and M. F. Bardia, Inorg. Chem. [**35**]{}, 7633 (1996); Z. Honda, H. Asakawa, and K. Katsumata, Phys. Rev. Lett. [**81**]{}, 2566 (1998). Y. Uchiyama, Y. Sasago, I. Tsukada, K. Uchinokura, A. Zheludev, T. Hayashi, N. Miura, and P. Böni, Phys. Rev. Lett. [**83**]{}, 632 (1999). M. Hagiwara, Y. Narumi, K. Kindo, M. Kohno, H. Nakano, R. Sato, and M. Takahashi, Phys. Rev. Lett. [ **80**]{}, 1312 (1998). A. Escuer, R. Vincente, and X. Solans, J. Chem. Soc. Dalton Trans. 531 (1997); Y. Narumi, M. Hagiwara, M. Kohno, and K. Kindo, Phys. Rev. Lett. [**86**]{}, 324 (2001). L. N. Bulaevskii, Sov. Phys. JETP [**17**]{}, 684 (1963); J. C. Bonner, H. W. J. Blöte, J. W. Bray, and I. S. Jacobs, J. Appl. Phys. [**50**]{}, 1810 (1979). F. D. M. Haldane, Phys. Lett. [**A93**]{}, 464 (1983); Phys. Rev. Lett. [**50**]{}, 1153 (1983). I. Affleck and F. D. M. Haldane, Phys. Rev. B [**36**]{}, 5291 (1987); R. R. P. Singh and M. P. Gelfand, Phys. Rev. Lett. [**61**]{}, 2133 (1988); Y. Kato and A. Tanaka, J. Phys. Soc. Jpn. [**63**]{}, 1277 (1994); S. Yamamoto, Phys. Rev. B [**51**]{}, 16128 (1995); [**52**]{}, 10170 (1995); A. Kitazawa, K. Nomura, and K. Okamoto, Phys. Rev. Lett. [**76**]{}, 4038 (1996); A. Kitazawa and K. Nomura, J. Phys. Soc. Jpn. [**66**]{}, 3379 (1997); [**66**]{}, 3944 (1997); M. Kohno, M. Takahashi, and M. Hagiwara, Phys. Rev. B [**57**]{}, 1046 (1998). T. Sakai and M. Takahashi, J. Phys. Soc. Jpn. [**58**]{}, 3131 (1989). A. Koga and N. Kawakami, Phys. Rev. B [**61**]{}, 6133 (2000). N. Katoh and M. Imada, J. Phys. Soc. Jpn. [**62**]{}, 3728 (1993); [**63**]{}, 4529 (1994); [**64**]{}, 1437 (1995). Y. J. Kim and R. J. Birgeneau, Phys. Rev. B [**62**]{}, 6378 (2000). I. Affleck, M. P. Gelfand, and R. R. P. Singh, J. Phys. A: Math. Gen. [**27**]{}, 7313 (1994). H. G. Evertz, cond-mat/9707221 (unpublished). H. G. Evertz, G. Lana, and M. Marcu, Phys. Rev. Lett. [**70**]{}, 875 (1993); U.-J. Wiese and H.-P. Ying, Z. Phys. B [**93**]{}, 147 (1994); B. B. Beard and U.-J. Wiese, Phys. Rev. Lett. [**77**]{}, 5130 (1996); N. Kawashima and J. E. Gubernatis, Phys. Rev. Lett. [**73**]{}, 1295 (1994). K. Harada, M. Troyer, and N. Kawashima, J. Phys. Soc. Jpn. [**67**]{}, 1130 (1998). S. Todo and K. Kato, Phys. Rev. Lett. [**87**]{}, 047203 (2001). K. Chen, A. M. Ferrenberg, and D. P. Landau, Phys. Rev. B [**48**]{}, 3249 (1993). M. Troyer, H. Kontani, and K. Ueda, Phys. Rev. Lett. [**76**]{}, 3822 (1996). M. den Nijs and K. Rommelse, Phys. Rev. B [**40**]{}, 4709 (1989). F. Cooper, B. Freedman, and D. Preston, Nucl. Phys. B [**210**]{} \[FS6\], 210 (1989). G. A. Baker, Jr., and N. Kawashima, Phys. Rev. Lett. [**75**]{}, 994 (1995). S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. Lett. [**60**]{}, 1057 (1988); Phys. Rev. B [**39**]{}, 2344 (1989); M. Troyer and M. Imada, in [*Computer Simulations in Condensed Matter Physics X*]{}, edited by D. P. Landau, K. K. Mon, and H.-B. Schütttler (Springer Verlag, Heidelberg, 1998), p.146. M. N. Barber, in [*Phase Transitions and Critical Phenomena*]{} edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1983) Vol.8, p.145. T. Aoki, J. Phys. Soc. Jpn. [**64**]{}, 605 (1995). I. Affleck and B. I. Halperin, J. Phys. A: Math. Gen. [**29**]{}, 2627 (1996). A. W. Sandvik, Phys. Rev. Lett. [**83**]{}, 3069 (1999). K. Kubo and T. Kishi, Phys. Rev. Lett. [**61**]{}, 2585 (1988). I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. [**59**]{}, 799 (1987). D. P. Arovas, A. Auerbach, and F. D. M. Haldane, Phys. Rev. Lett. [**60**]{}, 531 (1988); M. Oshikawa, J. Phys.: Condens. Matter [**4**]{}, 7469 (1992). S. Todo, M. Matsumoto, C. Yasuda, and H. Takayama, cond-mat/0106073 (unpublished). C. Yasuda, S. Todo, M. Matsumoto, and H. Takayama, to appear in Phys. Rev. B (cond-mat/0101337).
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abstract: 'The category of quasi frames (or qframes) is introduced and studied. In the context of qframes we can jointly study problems related to the L-Surjunctivity and Stable Finiteness Conjectures. As a consequence of our main results, we can generalize some of the known results on these conjectures. In particular, let $R$ be a ring, let $G$ be a sofic group, fix a crossed product $\RG$ and let $N$ be a right $R$-module. It is proved that: (1) the endomorphism ring $\End_{\RG}(N\otimes_R \RG)$ is stably finite, provided $N$ is finitely generated and has Krull dimension; (2) any linear cellular automaton $\phi:N^G\to N^G$ is surjunctive, provided $N$ is Artinian.'
author:
- 'Simone Virili[^1]'
bibliography:
- '/Users/simonevirili/Dropbox/refs.bib'
title: 'A point-free approach to L-Surjunctivity and Stable Finiteness'
---
——————————–
[**2010 Mathematics Subject Classification.**]{} 16S35, 37B15, 06C99.
[**Key words and phrases.**]{} Stably finite, surjunctive, sofic group, qframe, crossed product.
Introduction
============
In this paper we describe a [*point-free strategy*]{} to partially solve two classical problems about the representations of a given group $G$. Let us introduce first the problems we are interested in, and then give an idea of our approach to their solution.
A map is [*surjunctive*]{} if it is non-injective or surjective. Let $A$ be a finite set and equip $A^G=\prod_{g\in G} A$ with the product of the discrete topologies on each copy of $A$. There is a canonical left action of $G$ on $A^G$ defined by $$gx(h) = x(g^{-1}h) \text{ for all $g, h \in G$ and $x \in A^G$}\,.$$ A long standing open problem by Gottschalk [@Gott] is that of determining whether or not any continuous and $G$-equivariant map $\phi:A^G\to A^G$ is surjunctive, we refer to this problem as the [*Surjunctivity Conjecture*]{}. When $G$ is amenable this problem has been known for a long time to have a positive solution but it was just in 1999 when Gromov [@Gro] came out with a general theorem solving the problem in the positive for the large class of sofic groups (see also [@Weiss]). The general case remains open.
An analogous problem is as follows. Let $\K$ be a field, let $V$ be a finite dimensional $\K$-vector space, endow $V^G$ with the product of the discrete topologies and consider the canonical left $G$-action on $V^G$. The [*L-Surjunctivity Conjecture*]{} states that any $G$-equivariant continuous and [*$\K$-linear*]{} map $V^G\to V^G$ is surjunctive. This conjecture is known for the case of sofic groups, that follows again by Gromov’s general surjunctivity theorem in [@Gro] (see also [@Ceccherini]). Again, the general case is unknown.
A ring $R$ is [*directly finite*]{} if $xy=1$ implies $yx=1$ for all $x,y\in R$. Furthermore, $R$ is [*stably finite*]{} if the ring of square $k\times k$ matrices $\Mat_k(R)$ is directly finite for all $k\in {\mathbb N}_+$. A long-standing open problem due to Kaplansky [@Kap] is to determine whether the group ring $\K[G]$ is stably finite for any field $\K$, we refer to this problem as the [*Stable Finiteness Conjecture*]{}.\
Notice that $\Mat_k(\K[G])\cong \End_{\K[G]}(\K[G]^k)$, so an equivalent way to state the Stable Finiteness Conjecture is to say that any surjective endomorphism of a free right (or left) $\K[G]$-module of finite rank is injective. In case the field $\K$ is commutative and has characteristic $0$, then the problem was solved in the positive by Kaplansky. There was no progress in the positive characteristic case until 2002, when Ara, O’Meara and Perera [@Ara] proved that a group algebra $D[G]$ is stably finite whenever $G$ is residually amenable and $D$ is any division ring. This last result was generalized by Elek and Szabó [@Elek], that proved the same result for $G$ a sofic group (see also [@Ceccherini] and [@Goulnara_L] for alternative proofs). We remark that in [@Ara] one can also find a proof of the fact that any crossed product $D\asterisk G$ (see Section \[crossed\]) of a division ring $D$ and an amenable group $G$ is stably finite.
In the introduction of [@Elek], it is observed that the Surjunctivity Conjecture implies the Stable Finiteness Conjecture, in case $\K$ is a finite field. Roughly speaking, the idea is to consider $\K$ as a discrete compact Abelian group, view $(\K[G])^k$ as a dense subgroup of the compact group $(\K^k)^{G}$ and extend maps by continuity. Let us give a different argument. In brief, consider the finite field $\K$ as a finite discrete Abelian group; then, applying Pontryagin-Van Kampen’s duality to a $G$-equivariant endomorphism $\phi$ of the discrete group $(\K^k)^{(G)}$ (with the right $G$-action) we get a continuous $G$-equivariant endomorphism $\widehat \phi$ of the compact group $(\K^k)^{G}$ (with the left $G$-action) endowed with the product of the discrete topologies, and viceversa. In fact, Pontryagin-Van Kampen’s duality induces an anti-isomorphism between the ring of $G$-equivariant $\K$-endomorphisms of $(\K^k)^{(G)}$ and the ring of $G$-equivariant continuous $\K$-endomorphisms of $(\K^k)^G$. Ceccherini-Silberstein and Coornaert [@Ceccherini] give a different argument that shows that the same ring anti-isomorphism holds for arbitrary fields (they compose their map with the usual anti-involution on $\Mat_k(\K[G])$ to make it an actual ring isomorphism). This proves that the $L$-Surjunctivity Conjecture is equivalent to the Stable Finiteness Conjecture. The Appendix of this paper is devoted to recall the classical theory of duality between categories of discrete and linearly compact modules. As a consequence we can generalize the ring anti-isomorphism described above (see Corollary \[coro\_dual\_anti\]), clarifying the relation between Stable Finiteness and L-Surjunctivity Conjectures.
Let $G$ be a group, let $R$ be a ring and fix a crossed product $\RG$. Let $N$ be a right $R$-module, let $M_{\RG}=N\otimes_R\RG$ and consider an endomorphism $\phi:M\to M$. It is well known that the poset $\L(M)$ of $R$-submodules of $M$ (ordered by inclusion) is a lattice with very good properties, furthermore, $\phi$ induces a semi-lattice homomorphism $\Phi:\L(M)\to \L(M)$, that associates to a submodule $K\leq M$ the submodule $\phi(K)$. There is a natural right action of $G$ on $\L(M)$, induced by the $\RG$-module structure of $M$ (even if there is no natural $G$-action on $M$ in general), and $\Phi$ is $G$-equivariant with respect to this action. It turns out that $\phi$ is injective (resp., surjective) if and only if $\Phi$ has the same property. Using this construction we can translate (a general form of) the Stable Finiteness Conjecture in terms of some “well-behaved" lattices with a $G$-action and semi-lattice $G$-equivariant endomorphisms on them.
Similarly, consider a left $R$-module $N$, take the product $N^G$ endowed with the product of the discrete topologies and the usual left $G$-action, and consider a $G$-equivariant continuous endomorphism $\phi:N^G\to N^G$. If $N$ is linearly compact in the discrete topology (e.g., it is Artinian), one can show that the poset $\mathcal N(N^G)$ of closed submodules of $N^G$, ordered by [*reverse*]{} inclusion, is a lattice with many common features with a lattice of submodules of a discrete module. Furthermore, $\phi$ induces a semi-lattice homomorphism $\Phi:\mathcal N(N^G)\to \mathcal N(N^G)$, that associates to a closed submodule $K\leq N^G$ its preimage $\phi^{-1}(K)$. There is a natural right action of $G$ on $\mathcal N(N^G)$, induced by the left $G$-action on $N^G$, and $\Phi$ is $G$-equivariant with respect to this action. It turns out that $\phi$ is injective (resp., surjective) if and only if $\Phi$ is surjective (resp., injective). Thus, with this construction we can translate (a general form of) the L-Surjectivity Conjecture in terms of lattices with a $G$-action and $G$-equivariant semi-lattice endomorphisms on them, exactly as we did for the Stable Finiteness Conjecture.
In Section \[QFrame\_Sec\], we introduce the category $\Qframe$ of [*quasi frames*]{} (or [*qframes*]{}). Roughly speaking, a qframe is a lattice with properties analogous to a lattice of sub-modules. We study many constructions in that category, mimicking similar constructions on modules.
In Section \[Krull\_Gabriel\] we introduce and study two cardinal invariants attached to a qframe: its Krull and its Gabriel dimension. In the final part of the section we introduce the concepts of torsion and localization of a qframe.
In Section \[Main\_Sec\], we prove a general theorem for a $G$-equivariant endormorphism of qframes, where $G$ is a sofic group. We work first in semi-Artinian qframes (that are exactly the qframes with Gabriel dimension $1$), see Theroem \[main\], and then, using torsion and localization, we lift our result to higher Gabriel dimension, see Theorem \[main\_higher\].
In Section \[Appl\_Sec\], we apply the general theorem to the above conjectures. In particular, we can prove a general version of the L-Surjunctivity Conjecture for sofic groups:
[**Theorem \[main\_automata\]**]{} [*Let $R$ be a ring, let $G$ be a sofic group and let $_RN$ be a left $R$-module. If $_RN$ is Artinian, then any $G$-equivariant continuous morphism $\phi:N^G\to N^G$ is surjunctive.*]{}
Notice that the above theorem generalizes in different directions the main result of [@Cecc_Art] and [@Cecc_finite_length]. Furthermore, we prove a general version of the Stable Finiteness Conjecture in the sofic case, generalizing results of [@Elek] and [@Ara]:
[**Theorem \[Sofic\_Kap\_conj\]**]{}
*Let $R$ be a ring, let $G$ be a sofic group, fix a crossed product $R\asterisk G$, let $N_R$ be a finitely generated right $R$-module and let $M_{\RG}=N\otimes_RR\asterisk G$.*
1. If $N_R$ is Noetherian, then any surjective homomorphism $\phi:M\to M$ is injective;
2. If $N_R$ is finitely generated and has Krull dimension, then $\End_{R\asterisk G}(N\otimes_RR\asterisk G)$ is stably finite.
As a consequence of the above results we obtain that both the L-Srjunctivity and the Stable Finiteness Conjectures hold for free-by-sofic and for (finite-by-polycyclic)-by-sofic groups.
[**Acknowledgements.**]{} It is a pleasure for me to thank Dolors Herbera and Pere Ara for some interesting conversations that inspired in part this work, and for some concrete comments correcting or pointing out some inaccuracies in previous versions of this work. I am grateful to Federico Berlai for many useful discussions and for evidencing a number of misprints. I would like to acknowledge Toma Albu for some useful remarks on Sections 2 and 3, and for pointing me to relevant references.
Quasi-frames {#QFrame_Sec}
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In this section we introduce the category of quasi-frames and we introduce the technical machinery that will be used in the proof of our main results.
The category of quasi-frames
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Recall that a poset $(L,\leq)$ is a [*lattice*]{} if any finite subset $F=\{f_1,\dots,f_k\}\subseteq L$ has a [*least upper bound*]{} (also called the [*join of $F$*]{}), denoted by $\bigvee F$ or $f_1\vee \dots\vee f_k$, and a [*greatest lower bound*]{} (also called the [*meet*]{} of $F$), denoted by $\bigwedge F$ or $f_1\wedge\dots\wedge f_k$.
Given a lattice $(L,\leq)$ and two elements $x$, $y\in L$, the [*segment*]{} between $x$ and $y$ is $$[x,y]=\{s\in L:x\leq s\leq y\}\,.$$ We also let $(x,y]=[x,y]\setminus\{x\}$, $[x,y)=[x,y]\setminus\{y\}$ and $(x,y)=[x,y]\setminus\{x,y\}$. Notice that $[x,y]$ is itself a lattice with the partial order induced by $L$.
Let us recall the following properties that a lattice $(L,\leq)$ may (or may not) have:
1. $(L,\leq)$ is [*bounded*]{} if it has a maximum (denoted by $1$) and a minimum (denoted by $0$);
2. $(L,\leq)$ is [*modular*]{} if, for all $a,$ $b$ and $c\in L$ with $a\leq c$, $$a\vee(b\wedge c)=(a\vee b)\wedge c\,;$$
3. $(L,\leq)$ is [*distributive*]{} if, for all $a,$ $b$ and $c\in L$, $$a\vee(b\wedge c)=(a\vee b)\wedge (a\vee c) \ \text{ and }\ a\wedge(b\vee c)=(a\wedge b)\vee (a\wedge c)\,;$$
4. $(L,\leq)$ is [*complete*]{} if it has joins and meets for any of its subsets (finite or infinite). By convention we put $\bigvee \emptyset=0$;
5. $(L,\leq)$ is [*upper-continuous*]{} if it is complete and, for any directed subset $\{x_i:i\in I\}$ of $L$ (or, equivalently, for any chain in $L$) and any $x\in L$, $$x\wedge\bigvee_{i\in I}x_i=\bigvee_{i\in I}(x\wedge x_i)\,;$$
6. given $a\in L$, and element $c$ is a [*pseudo-complement*]{} for $a$ if it is maximal with respect to the property that $a\wedge c=0$. $(L,\leq)$ is [*pseudo-complemented*]{} if, for any choice of $a\leq b\leq c$ in $L$, there is a pseudo-complement of $b$ in the lattice $[a,c]$;
7. $(L,\leq)$ is a [*frame*]{} if it is complete and, for any subset $\{x_i:i\in I\}$ of $L$ and any $x\in L$, $$x\wedge\bigvee_{i\in I}x_i=\bigvee_{i\in I}(x\wedge x_i)\,;$$
8. $(L,\leq)$ is [*compact*]{} if it has a maximum $1\in L$ and, for any subset $S\subseteq L$ such that $\bigvee S=1$, there exists a finite subset $F\subseteq S$ such that $\bigvee F=1$.
If a lattice is distributive, then it is also modular, an upper-continuous modular lattice is always pseudo-complemented, furthermore, any complete lattice is bounded. Notice also that a frame has all the properties listed in (1)–(6).
1. Let $X$ be a set. The power set $\P(X)$, ordered by inclusion, is a frame;
2. the family of open sets $\Open(X)$ of a topological spaces $(X,\tau)$, ordered by inclusion, is a frame;
3. given a ring $R$ and a right $R$-module $M$, the family $\L(M)$ of submodules of $M$, ordered by inclusion, is an upper-continuous modular lattice. Furthermore, $\L(M)$ is compact if and only if $M$ is finitely generated.
Let $(L_1,\leq)$ and $(L_2,\leq)$ be two lattices and consider a map $\phi:L_1\to L_2$ between them. Then,
1. $\phi$ is a [*semi-lattice homomorphism*]{} provided $\phi(x\vee y)=\phi(x)\vee \phi(y)$, for all $x$ and $y\in L_1$;
2. $\phi$ is a [*lattice homomorphism*]{} if it is a semi-lattice homomorphism and $\phi(x\wedge y)=\phi(x)\wedge \phi(y)$, for all $x$ and $y\in L_1$;
3. $\phi$ [*commutes with arbitrary joins*]{} if, for any subset $S$ of $L_1$, $\phi(\bigvee S)=\bigvee_{s\in S} \phi(s)$, meaning that, if $\bigvee S$ exists, then also $\bigvee_{s\in S} \phi(s)$ exists and it coincides with $\phi(\bigvee S)$;
4. $\phi$ [*preserves segments*]{} if $\phi([a,b])=[\phi(a),\phi(b)]$ for all $x\leq y\in L_1$.
Notice that, if $L_1$ has a minimum element $0=\bigvee \emptyset$ and $\phi$ commutes with arbitrary joins, then $\phi(0)=\bigvee \emptyset=0\in L_2$ is a minimum element in $L_2$. Furthermore, a map which preserves segments is surjective if and only if its image contains $0$ and $1$.
\[modules\] Let $R$ be a ring and consider a homomorphism of right $R$ modules $\phi:M\to N$. Then, $\phi$ induces a map $$\Phi:\L(M)\to \L(N)\,,$$ such that $\Phi(K)=\phi(K)\leq N$, for all $K\in\L(M)$. One can show that $\Phi$ is a semi-lattice homomorphism that commutes with arbitrary joins and that preserves segments, while in general $\Phi$ is not a lattice homomorphism.
A [*quasi-frame*]{} (or [*qframe*]{}) is an upper-continuous modular lattice. A map between two quasi-frames is a [*homomorphism of quasi-frames*]{} if it is a homomorphism of semi-lattices that preserves segments and commutes with arbitrary joins. We denote by $\Qframe$ the category of quasi-frames and homomorphisms of quasi-frames with the obvious composition.
Given a qframe $(L,\leq)$ and $x\in L$, a segment of the form $[0,x]$ is said to be a [*sub-qframe*]{} of $L$.
Constructions in $\Qframe$
--------------------------
Let $\phi:L_1\to L_2$ be homomorphism of qframes. The element $${\mathrm{Ker}}(\phi)=\bigvee_{\phi(x)=0}x\in L_1$$ is said to be the [*kernel of $\phi$*]{}. We say that $\phi$ is [*algebraic*]{} provided the restriction $\phi:[{\mathrm{Ker}}(\phi),1]\to L_2$ of $\phi$ to $[{\mathrm{Ker}}(\phi),1]$ is injective.
Notice that an algebraic homomorphism of qframes is injective if and only if its kernel is $0$. It is a useful exercise to prove that the morphism $\Phi$ in Example \[modules\] is algebraic. Let us remark that, as noticed in [@latt_prerad Proposition 0.8], there is quite a strong relation between algebraic homomorphisms of qframes and linear morphisms as introduced in [@linear_lattice]. In [@latt_prerad Example 0.9] one can also find an easy example of a non-algebraic homomorphism of qframes.
Let $(L,\leq)$ be a qframe, let $I$ be a set and consider a subset $\F=\{x_i:i\in I\}\subseteq L$ such that $x_i\neq 0$ for all $i\in I$. We say that $\F$ is a [*join-independent family*]{} if, for any $i\in I$, $$\left(\bigvee_{j\in I\setminus\{i\}} x_j\right) \wedge x_i=0\,.$$ Furthermore $\F$ is a [*basis*]{} for $L$ if it is join-independent and $\bigvee_{i\in I}x_i=1$.
As an example one can consider a family $\{M_i:i\in I\}$ of right $R$-modules and the direct sum $M=\bigoplus_IM_i$. Then, identifying each $M_i$ with a submodule of $M$ in the obvious way, the family $\{M_i:i\in I\}$ is a basis of the qframe $\L(M)$.
The following lemma will be useful later on.
\[finite\_base\]\[fg\_base\] Let $(L,\leq)$ be a qframe, let $x\in L$ and let $\{y_i:i \in I\}$ be a basis of $L$. Then,
1. if $x\neq 0$, there exists a finite subset of $I$ such that $x\wedge\bigvee_{i\in F}y_i \neq 0$;
2. if $[0,x]$ is compact, there exists a finite subset $F$ of $I$ such that $x\leq \bigvee_{i\in F}y_i$.
\(1) Notice that $0\neq x=x\wedge\bigvee\{\bigvee_{i\in F}y_i:{F\subseteq I \text{ finite}}\}=\bigvee\{x\wedge\bigvee_{i\in F}y_i:{F\subseteq I \text{ finite}}\}$, so for at least one finite subset $F$ of $I$, $x\wedge\bigvee_{i\in F}y_i\neq 0$.
\(2) Notice that $x=x\wedge\bigvee_{i\in I}\{\bigvee_{i\in G}y_i:{G\subseteq I \text{ finite}}\}=\bigvee\{x\wedge\bigvee_{i\in G}y_i:{G\subseteq I \text{ finite}}\}$. By the definition of compact lattice, there exists a finite subset $K$ of the set of finite subsets of $I$ such that $x=\bigvee\{x\wedge\bigvee_{i\in G}y_i:G\in K\}$. Taking $F=\bigcup_{G\in K}G$ we get $$x=\bigvee\left\{x\wedge\bigvee_{i\in G}y_i:G\in K\right\}\leq x\wedge\bigvee_{i\in F}y_i\leq x\,.$$ Thus, $x=x\wedge\bigvee_{i\in F}y_i$, which means exactly that $x\leq \bigvee_{i\in F}y_i$.
Let $I$ be a set and, for all $i\in I$, let $(L_i,\leq)$ be a qframe. The [*product*]{} of this family is $$\prod_{i\in I} L_i=\{\underline x=(x_i)_{I}: x_i\in L_i\, , \text{ for all } i\in I\}$$ with the partial order relation defined by $$(\ \underline x\leq \underline y\ ) \ \Longleftrightarrow\ (\ x_i\leq y_i\,, \text{ for all $i\in I$}\ )\,.$$
One can verify that $(\prod_{i\in I}L_i,\leq)$ is a qframe. Furthermore, for any subset $J\subseteq I$ the usual projection $\pi_J:\prod_{i\in I} L_i\to \prod_{j\in J}L_j$ is a surjective homomorphism of qframes, and the usual inclusion $\epsilon_J:\prod_{j\in J}L_j\to \prod_{i\in I} L_i$ is an injective homomorphism of qframes.
We conclude this subsection describing the quotient objects in the category of qframes.
A [*congruence*]{} on a qframe $(L,\leq)$ is a subset $\mathcal R\subseteq L\times L$ that satisfies the following properties:
1. $\mathcal R$ is an equivalence relation;
2. for all $a,$ $b$ and $c\in L$, $(a ,b)\in \mathcal R$ implies $(a \vee c, b \vee c)\in \mathcal R$;
3. for all $a,$ $b$ and $c\in L$, $(a, b)\in \mathcal R$ implies $(a \wedge c , b \wedge c)\in \mathcal R$.
Furthermore, if $\mathcal R$ satisfies the following condition (Cong.4), then $\mathcal R$ is a [*strong congruence*]{}:
1. for all $a\in L$, the equivalence class $[a]=\{b\in L:(a,b)\in\mathcal R\}$ has a maximum.
Usually, given a congruence $\mathcal R$ on a qframe $(L,\leq)$, we write $a\sim b$ to denote that $(a,b)\in\mathcal R$.
The following lemma is analogous to [@AS2 Proposition 2.2], anyway we prefer to give a complete proof to make the paper reasonably self-contained.
\[proj\_comm\_join\] Let $(L,\leq)$ be a qframe and let $\mathcal R$ be a strong congruence on $L$. Let $L/\mathcal R$ be the set of equivalence classes in $L$ and endow it with the following binary relation: $$(\ [a]\preceq [b]\ )\ \Longleftrightarrow \ (\ \exists a'\in [a] \text{ and } b'\in [b]\text{ such that } a'\leq b'\ )\,.$$ Then, $\preceq$ is a partial order and $(L/\mathcal R,\preceq)$ is a qframe. Furthermore, the canonical map $\pi:L\to L/\mathcal R$ such that $x\mapsto [x]$ is a surjective homomorphism of qframes.
[*$\preceq$ is a partial order.*]{} The unique non-obvious think is to verify that $[a]\preceq [b]\preceq [c]\in L/\mathcal R$ implies that $[a]\preceq [c]$. To do so, take $a'\in [a]$, $b'$ and $b''\in [b]$, and $c'\in [c]$ such that $a'\leq b'$ and $b''\leq c'$. It is then clear that $a'\leq b'\vee b''\leq b'\vee c'$, and also that $c'=c'\vee b''\sim c'\vee b'$, thus $[a]\preceq [c]$.\
[*Lattice structure.*]{} Let $a$ and $b\in L$ and let us show that $[a\wedge b]$ is a greatest lower bound for $[a]$ and $[b]$ in $L/\mathcal R$. Indeed, it is clear that $[a\wedge b]$ is $\preceq$ of both $[a]$ and $[b]$. Furthermore, given $c\in L$ such that $[c]\preceq [a]$ and $[c]\preceq [b]$, there exit $a'\in[a]$, $b'\in [b]$ and $c',$ $c''\in [c]$, such that $c'\leq a'$ and $c''\leq b'$. Thus, $[c]=[c'\wedge c'']\preceq [a'\wedge b']=Ê[a\wedge b]$. One can show analogously that $[a\vee b]$ is a least upper bound for $[a]$ and $[b]$.\
[*Modularity.*]{} Let $a$, $b$ and $c\in L$ be elements and suppose $[a]\preceq [c]$. Choose $a'\in [a]$ and $c'\in [c]$ such that $a'\leq c'$, then, by the modularity of $L$, we have that $a'\vee(b\wedge c')=(a'\vee b)\wedge c'$ which implies $$[a]\vee([b]\wedge [c])=[a']\vee([b]\wedge [c'])=([a']\vee [b])\wedge [c']=([a]\vee [b])\wedge [c]\,.$$ [*Completeness.*]{} Consider a family $\F=\{[x_i]:i\in I\}$ in $L/\mathcal R$, we claim that $\left[\bigvee_{i\in I} x_i\right]$ is a least upper bound for $\F$. In fact, it is clear that $\left[\bigvee_{i\in I} x_i\right]\succeq [x_j]$ for all $j\in I$. Furthermore, given $c\in L$ such that $[c]\succeq [x_i]$ for all $i\in I$, we can choose $x_i'\in [x_i]$ such that $x_i'\leq \bar c$ for all $i\in I$, where $\bar c$ is the maximum of $[c]$, for all $i\in I$. Letting $\bar x_i$ be the maximum of $[x_i]$, for all $i\in I$, $\bar c=\bar c\vee x_i'\sim \bar c\vee\bar x_i$ and so $\bar x_i\leq \bar c$, for all $i\in I$. Thus, $[c]=[\bar c]\succeq \left[\bigvee_{i\in I} \bar x_i\right]\succeq\left[\bigvee_{i\in I} x_i\right]$.\
[*$(L/\mathcal R,\preceq)$ is a qframe.*]{} We have just to verify upper continuity. Let $\{[x_i]:i\in I\}$ be a directed family in $L/\mathcal R$ and let $\bar x_i=\bigvee [x_i]$, for all $i\in I$. The set $\{\bar x_i:i\in I\}$ is directed and so, for all $x\in L$, $ x\wedge\bigvee_{i\in I} \bar x_i=\bigvee_{i\in I}( x\wedge \bar x_i)$. Thus, by our description of lattice operations, $$[x]\wedge\bigvee_{i\in I} [x_i]= [x]\wedge\bigvee_{i\in I} [\bar x_i]=\bigvee_{i\in I}( [x]\wedge [\bar x_i])=\bigvee_{i\in I}( [x]\wedge [x_i])\,.$$\
[*$\pi$ is a surjective homomorphism of qframes.*]{} It is all clear from the description of the lattice operation in $L/\mathcal R$ a part the fact that $\pi$ preserves segments. So take $x\leq y\in L$ and consider $[z]\in [[x],[y]]$. Let $x'\in [x]$ and $z'\in [z]$ be such that $x'\leq z'$. Clearly, $x\leq z'\vee x\in [z]$, in fact, $x\sim x'$ implies $z'\vee x\sim z'\vee x'=z'$. Furthermore, $y\geq (z'\vee x) \wedge y\in [z]$, in fact, given $z''\in [z]$ and $y'\in [y]$ such that $z''\leq y'$, we obtain $(z'\vee x)\wedge y\sim z''\wedge y\sim z''\wedge y'=z''$. Thus, $(z'\vee x)\wedge y\in [x,y]$ and $\pi((z'\vee x)\wedge y)=[z]$.
Composition length
------------------
Let $(L,\leq)$ be a qframe. Given a finite chain $$\sigma:\, x_0\leq x_1\leq \dots\leq x_n$$ of elements of $L$, we say that the length $\ell (\sigma)$ of $\sigma$ is the number of strict inequalities in the chain.
Let $(L,\leq)$ be a qframe. The [*length*]{} of $L$ is $$\ell(L)=\sup\{\ell(\sigma):\text{ $\sigma$ a finite chain of elements of $L$}\}\in{\mathbb N}\cup\{\infty\}\,.$$ For any element $x\in L$ we use the notation $\ell(x)$ to denote the length of the segment $[0,x]$.
A qframe $(L,\leq)$ is said to be [*trivial*]{} if it has just one element. In what follows, by [*non-trivial*]{} qframe we mean a qframe which contains at least two elements. Furthermore, $(L,\leq)$ is said to be an [*atom*]{} (or to be [*simple*]{}) if it has two elements.
\[rem\_trivial\] A qframe $(L,\leq)$ is trivial if and only if $\ell(L)=0$, while it is an atom if and only if $\ell(L)=1$.
Let $(L,\leq)$ be a qframe and consider a finite chain $$\sigma:\, 0=x_0\leq x_1\leq \dots\leq x_n=1\,.$$ If all the segments $[x_i,x_{i+1}]$, with $i=0,\dots,n-1$, are simple, then we say that $\sigma$ is a [*composition series*]{}.
The following lemma is known as Artin-Schreier’s Refinement Theorem.
[[@Ste Proposition 3.1, Ch. III]]{}\[A-S\] Let $(L,\leq)$ be a qframe, let $a\leq b\in L$ and let $$\sigma_1:\, a=x_0\leq x_1\leq \dots\leq x_n=b\ \ \text{ and }\ \ \sigma_2:\, a=y_0\leq y_1\leq \dots\leq y_m=b\,.$$ Then, there exists a series $\sigma:\, a=z_0\leq z_1\leq \dots\leq z_t=b$ which is equivalent to a refinement of both $\sigma_1$ and $\sigma_2$.
Using Lemma \[A-S\], one can deduce the following lemma, which is usually known as Jordan-Hölder Theorem.
\[composition\_length\] Let $(L,\leq)$ be a qframe of finite length. Then,
1. any finite chain in $L$ can be refined to a composition series;
2. any two composition series in $L$ have the same length;
3. $\ell(L)=n$ if and only if there exists a composition series of length $n$ in $L$.
A qframe $(L,\leq)$ is
1. [*Noetherian*]{} if any ascending chain $x_1\leq x_2\leq \dots\leq x_n\leq \dots$ stabilizes at some point;
2. [*Artinian*]{} if any descending chain $x_1\geq x_2\geq \dots\geq x_n\geq \dots$ stabilizes at some point.
Using Lemma \[composition\_length\], one can prove that $\ell(L)<\infty$ if and only if $L$ is both Noetherian and Artinian (see also Lemma \[semi-art\]). The following lemma is well-known.
\[Noetherian\_implies\_fg\] Let $(L,\leq)$ be a qframe. Then, $L$ is Noetherian if and only if $[0,x]$ is compact for all $x\in L$.
\[length\_sum\][[@latt_conc Lemma 3.2]]{} Let $(L,\leq)$ be a qframe of finite length and let $x$, $y\in L$. Then, $$\ell(x\vee y)+\ell(x\wedge y)= \ell(x)+\ell(y)\,.$$
\[length\_inj\_surj\] Let $\phi:L_1\to L_2$ be a homomorphism of qframes:
1. if $\phi$ is injective, then $\ell (L_1)\leq \ell(L_2)$;
2. if $\phi$ is surjective, then $\ell (L_2)\leq \ell(L_1)$.
\(1) Let $\sigma:\ x_{1}\leq x_2\leq \dots\leq x_n$ be a chain in $L_1$, then $\phi(\sigma):\ \phi(x_{1})\leq \phi(x_2)\leq \dots\leq \phi(x_n)$ is a chain in $L_2$. Furthermore, if $x_i\neq x_j$, then $\phi(x_i)\neq \phi(x_j)$ by injectivity. Thus, $\ell (\phi(\sigma))=\ell(\sigma)$ and so $\ell (L_1)\leq \ell(L_2)$.
\(2) Let $\sigma:\ x_{1}\leq x_2\leq \dots\leq x_n$ be a chain in $L_2$. Since $\phi$ is surjective, there exist $y_1,\dots,y_n\in L_1$ such that $\phi(y_i)=x_i$ for all $i=1,\dots, n$. Clearly, $\sigma':\ y_1\leq (y_1\vee y_2) \leq \dots \leq (y_1\vee y_2\vee \ldots\vee y_n)$ and, for all $i=1,\dots, n$, $\phi(y_1\vee\ldots\vee y_i)=\phi(y_1)\vee\ldots\vee \phi(y_i)=x_1\vee \ldots\vee x_i=x_i$. If $x_i\neq x_{i+1}$, then $y_1\vee\ldots \vee y_i\neq y_1\vee\ldots \vee y_i\vee y_{i+1}$ and so $\ell (\sigma)\leq \ell (\sigma')$. Thus, $\ell(L_2)\leq \ell(L_1)$.
\[length\_prod\] Let $I$ be a set. For all $i\in I$, let $(L_i,\leq)$ be a non-trivial qframe and let $L=\prod_IL_i$. Then, $$\ell (L)=\begin{cases}\sum_{i\in I}\ell(L_i)&\text{if $I$ is finite;}\\
\infty&\text{otherwise.}\end{cases}$$
If $\ell(L_i)=\infty$ for some $i\in I$ there is nothing to prove, so we suppose that $\ell(L_i)$ is finite for all $i\in I$. Let $\epsilon_i:L_i\to L$ be the canonical inclusion and let $1_i=\bigvee \epsilon (L_i)$, for all $i\in I$. Notice that $\epsilon_i(L_i)=[0,1_i]$, so $\ell (L_i)=\ell(1_i)$, and $L=[0,\bigvee_{i\in I}1_i]$, so $\ell(L)=\ell(\bigvee_{i\in I}1_i)$.\
When $I$ is finite, the proof follows by Lemma \[length\_sum\] and the fact that, $1_i\wedge\bigvee_{j\neq i}1_j=0$.\
If $I$ is not finite, then for any finite subset $J\subseteq I$, we have $\ell (\prod_J L_j)=\sum_{J}\ell(L_j)\geq |J|$ by the first part of the proof. Furthermore, $\ell(\prod_IL_i)\geq\ell (\prod_J L_j)$, by Lemma \[length\_inj\_surj\] applied to the maps $\pi_J:\prod_IL_i\to \prod_J L_j$. Thus, $\ell\left(\prod_IL_i\right)\geq \sup\{|J|:J\subseteq I\text{ finite}\}=\infty$.
In the following lemma we verify that the qframes with finite length are Hopfian and coHopfian objects in $\Qframe$.
\[hopf\_lattice\] Let $(L,\leq)$ be a qframe of finite length, let $(L',\leq)$ be a qframe, and let $\phi:L\to L'$ be a homomorphism of qframes. Then,
1. $\phi$ is injective if and only if $\ell(L)=\ell(\phi(L))$;
2. $\phi$ is surjective if and only if $\ell(\phi(L))=\ell(L')$.
In particular, if $\ell(L)=\ell(L')$, then $\phi$ is injective if and only if it is surjective.
\(1) Suppose that $\ell(L)=\ell(\phi(L))$ and let $x,$ $y\in L$ be such that $\phi(x)=\phi(y)$. If, looking for a contradiction $x\neq y$, then either $x< x\vee y$ or $y< x\vee y$. Without loss of generality, we suppose that $x< x\vee y$. Take the chain $0\leq x< x\vee y\leq 1$ between $0$ and $1$ and refine it to a composition chain $$\sigma:\ 0\leq \dots\leq x< \dots < x\vee y< \dots\leq 1\,,$$ thus $\ell(\sigma)=\ell(L)$ (see Lemma \[composition\_length\]). The image via a homomorphism of qframes of a composition chain is a (eventually shorter) composition chain in the image. Thus, $\ell(\phi(\sigma))=\ell(\phi(L))=\ell(L)=\ell(\sigma)$, in particular, $\phi(x)\neq \phi(x\vee y)=\phi(x)\vee \phi(y)$, which contradicts the fact that $\phi(x)=\phi(y)$. The converse is trivial since, if $\phi$ is injective, then $L\cong \phi(L)$ and then clearly $\ell(L)=\ell(\phi(L))$ (use, for example, Lemma \[length\_inj\_surj\]).
\(2) Suppose that $\phi$ is not surjective and consider a composition chain $\sigma:\, 0=x_0\leq x_1\leq \ldots\leq x_n=\phi(1)$ in $\phi(L)$. We can define a longer chain $\sigma':\, 0=x_0\leq x_1\leq \ldots\leq x_n<1$ in $L'$. Hence, $\ell(\phi(L))=\ell(\sigma)<\ell(\sigma')\leq \ell(L')$. The converse is trivial since $\phi(L)=L'$ clearly implies that $\ell(\phi(L))=\ell(L')$.
Socle series
------------
Let $(L,\leq)$ be a qframe. The [*socle*]{} $s(L)$ of $L$ is the join of all the atoms in $L$. For all $x\in L$, we let $s(x)=s([0,x])$.
\[basic\_soc\] Let $(L,\leq)$ be a qframe and let $I$ be a set. Then,
1. $s(x)\leq x$ and $s(x_1)\leq s(x_2)$, for all $x\in L$ and $x_1\leq x_2\in L$;
2. $s(\bigvee_{i\in I} x_i)\geq \bigvee_{i\in I} s(x_i)$, where $x_i\in L$ for all $i\in I$. Furthermore equality holds if $\{x_i:i\in I\}$ is join-independent;
3. $s(\bigwedge_{i\in I} x_i)\leq \bigwedge_{i\in I} s(x_i)$, where $x_i\in L$ for all $i\in I$;
4. if $\phi:L\to L'$ is a homomorphism of qframes, then $\phi(s(L))\leq s(L')$.
Parts (1) and (3) follow by the properties described in [@latt_conc page 47]. Part (2) follows by [@latt_prerad Proposition 1.4]. For part (4), notice that $\phi(s(L))=\phi(\bigvee\{x\in L:[0,x] \text{ is an atom}\}=\bigvee\{\phi(x):[0,x] \text{ is an atom}\}\leq \bigvee\{y\in L':[0,y] \text{ is an atom}\}$ (use the fact that $\phi$ takes intervals to intervals).
Thanks to part (4) of Lemma \[basic\_soc\], we can give the following
Let $(L,\leq)$ be a qframe and let $\Soc(L)=[0,s(L)]$. Furthermore, given a homomorphism $\phi: L\to L'$ of qframes, we denote by $\Soc(\phi):\Soc(L)\to \Soc(L')$ the restriction of $\phi$. This defines a covariant functor $\Soc:\Qframe\to \Qframe$.
It is not difficult to show that $\Soc$ is compatible with the composition of morphisms, so that the above definition is correct.
We can iterate the procedure that defines the socle as follows:
Let $(L,\leq)$ be qframe. Then,
1. $s_0(L)=s(L)$;
2. for any ordinal $\alpha$, $s_{\alpha+1}(L)=s([s_{\alpha}(L),1])$;
3. for any limit ordinal $\alpha$, $s_\lambda(L)=\bigvee_{\alpha<\lambda}s_\alpha(L)$.
$L$ is [*semi-Artinian*]{} if $s_\tau(L)=1$ for some ordinal $\tau$.
One can show that the uniform dimension (see [@latt_conc Chapter 8]) of a semi-Artinian qframe is the length of its socle.
\[semi-art\][[@latt_conc Theorem 5.2 and Proposition 5.3]]{} Let $(L,\leq)$ be a qframe. Then,
1. $L$ is semi-Artinian if and only if $[0,x]$ and $[x,1]$ are semi-Artinian for all $x\in L$;
2. $L$ is semi-Artinian and Noetherian if and only if $\ell(L)<\infty$.
Krull and Gabriel dimension {#Krull_Gabriel}
===========================
In this section we introduce the notions of Krull and Gabriel dimension of a qframe and we compare these two concepts. The notion of Gabriel dimension is then used to define torsion and localization endofunctors of the category of qframes.
Krull and Gabriel dimension {#krull-and-gabriel-dimension}
---------------------------
Let $(L,\leq)$ be a qframe. The [*Krull dimension*]{} $\Kdim(L)$ of $L$ is defined as follows:
1. $\Kdim(L)=-1$ if and only if $L$ is trivial;
2. if $\alpha$ is an ordinal and we already defined what it means to have Krull dimension $\beta$ for any ordinal $\beta<\alpha$, $\Kdim(L)=\alpha$ if and only if $\Kdim(L)\neq \beta$ for all $\beta<\alpha$ and, for any descending chain $$x_1\geq x_2\geq x_3 \geq \ldots \geq x_n\geq \dots$$ in $L$, there exists $\bar n\in {\mathbb N}_+$ such that $\Kdim([x_n,x_{n+1}])=\beta_n$ for all $n\geq \bar n$ and $\beta_n<\alpha$.
If $\Kdim(L)\neq \alpha$ for any ordinal $\alpha$ we set $\Kdim(L)=\infty$.
Notice that the qframes with $0$ Krull dimension are precisely the Artinian qframes.
A subclass $\X\subseteq \Ob(\Qframe)$ is a [*Serre class*]{} if it is closed under isomorphisms and, given $L\in \Ob(\Qframe)$ and $x\leq y\leq z\in L$, $[x,y]$, $[y,z]\in\X$ if and only if $[x,z]\in\X$.
The class of all qframes with Krull dimension $\leq \alpha$ for some ordinal $\alpha$ is a Serre class (see [@latt_conc Proposition 13.5]).
Let $(L_1,\leq)$ and $(L_2,\leq)$ be qframes. If $\Kdim(L_1)$ exists and if there exists a surjective homomorphism of qframes $\phi:L_1\to L_2$, then $\Kdim(L_1)\geq \Kdim(L_2)$.
Let us proceed by induction on $\Kdim(L_1)=\alpha$. If $\alpha=-1$, then clearly also $\Kdim(L_2)=-1$. Suppose now that $\alpha>-1$ and that we already proved our result for all $\beta<\alpha$. If $\Kdim(L_2)<\Kdim(L_1)$ there is nothing to prove, so suppose that $\Kdim(L_2)\not<\Kdim(L_1)$ and let us show that $\Kdim(L_2)=\Kdim(L_1)$. Indeed, consider a descending chain in $L_2$ $$x_0\geq x_1\geq \dots \geq x_n\geq \dots$$ By the surjectivity of $\phi$, we can choose $y_i\in L_1$ so that $\phi(y_i)=x_i$, for all $i\in {\mathbb N}$, let also $y_i'=\bigvee_{j\geq i}y_j$. It is not difficult to see that $$y_0'\geq y_1'\geq\dots\geq y_n'\geq\dots$$ and that $\phi(y_i')=\bigvee_{j\geq i}\phi(y_j)=x_i$. By definition of Krull dimension, there exists $\bar n\in {\mathbb N}_+$ such that $\Kdim([y'_n,y'_{n+1}])=\beta_n$ for all $n\geq \bar n$ and $\beta_n<\alpha$. By inductive hypothesis, $\Kdim([x_n,x_{n+1}])\leq\Kdim([y'_n,y'_{n+1}])=\beta_n$, showing that $\Kdim(L_2)\leq \alpha$, and so, $\Kdim(L_2)=\alpha$.
Let $(L,\leq)$ be a qframe. We define the [*Gabriel dimension*]{} $\Gdim(L)$ of $L$ by transfinite induction:
1. $\Gdim(L)=0$ if and only if $L$ is trivial. A qframe $S$ is $0$-simple (or just simple) if it is an atom;
2. let $\alpha$ be an ordinal for which we already know what it means to have Gabriel dimension $\beta$, for all $\beta\leq\alpha$. A qframe $S$ is $\alpha$-simple if, for all $0\neq a\in S$, $\Gdim([0,a])\nleq \alpha$ and $\Gdim([a,1])\leq\alpha$;
3. let $\sigma$ be an ordinal for which we already know what it means to have Gabriel dimension $\beta$, for all $\beta<\sigma$. Then, $\Gdim(L)=\sigma$ if $\Gdim(L)\not\less \sigma$ and, for all $1\neq a\in L$, there exists $b>a$ such that $[a,b]$ is $\beta$-simple for some ordinal $\beta<\sigma$.
If $\Gdim(L)\neq \alpha$ for any ordinal $\alpha$ we set $\Gdim(L)=\infty$.
Notice that the qframes with Gabriel dimension $1$ are precisely the semi-Artinian qframes. Also the class of all qframes with Gabriel dimension $\leq \alpha$ for some ordinal $\alpha$ is a Serre class (see part (1) of Lemma \[basic\_gabriel\]). For any ordinal $\alpha$, $\Gdim(S)=\alpha+1$, for any $\alpha$-simple qframe $S$.
\[sub\_simp\] Let $\alpha$ be an ordinal and let $(L,\leq)$ be an $\alpha$-simple qframe. Any non-trivial sub-qframe of $L$ is $\alpha$-simple.
We proceed by transfinite induction on $\alpha$. If $\alpha=0$, then $L$ is an atom and there is no non-trivial sub-qframe but $L$ itself. Let $\alpha>0$ and choose $0\neq b\leq a\in L$. By definition, $\Gdim([0,b])\not \leq\alpha$ so, to prove that $[0,a]$ is $\alpha$-simple, it is enough to show that $\Gdim([b,a])\leq \alpha$. If $\Gdim([b,a])< \alpha$ there is nothing to prove, so let us consider the case when $\Gdim([b,a])\not< \alpha$. Let $a'\in (b,a]$, choose a pseudo-complement $c$ of $a$ in $[a',1]$ and let $d\in [c,1]$ be such that $[c,d]$ is $\beta$-simple for some $\beta<\alpha$. Let $b'=d\wedge a$, then $[a',b']$ is non-trivial by the maximality included in the definition of pseudo-complement, furthermore, by modularity, $[a',b']$ is isomorphic to $[c,b'\vee c]$, which is a sub-qframe of $[c,d]$. By inductive hypothesis, $[a',b']$ is $\beta$-simple. This proves that $\Gdim([b,a])= \alpha$, as desired.
\[GvK\] Let $(L,\leq)$ be a qframe. The following statements hold true:
1. $L$ has Krull dimension if and only if any segment of $L$ has finite uniform dimension and $L$ has Gabriel dimension;
2. if $L$ has Krull dimension, then $\Kdim(L)\leq \Gdim(L)\leq \Kdim(L)+1$;
3. if $L$ is Noetherian, then there exists a finite chain $0=x_0\leq x_1\leq \dots\leq x_n=1$ such that $[x_{i-1},x_{i}]$ is $\alpha_i$-simple for some ordinal $\alpha_i$, for all $i=1,\dots,n$. Furthermore, $L$ has Krull dimension and $\Gdim(L)=\Kdim(L)+1$.
For (1), see Exercise (116) in [@Dim_ring_th] (an argument to solve that exercise can be found in [@GR]). For parts (2) and (3) see respectively [@latt_conc Theorem 13.9] and (statement and proof of) [@latt_conc Theorem 13.10].
In the following lemmas we collect some properties of Gabriel dimension. Their proof is inspired by the treatment in [@Dim_ring_th] but we prefer to give complete proofs also here.
\[basic\_gabriel\] Let $L$ be a qframe with Gabriel dimension. The following statements hold true:
1. if $a\leq b\in L$, then $\Gdim([a,b])\leq \Gdim(L)$;
2. if $a\in L$, then $\Gdim(L)=\max\{\Gdim([0,a]),\Gdim([a,1])\}$;
3. given a subset $\F\subseteq L$ such that $\bigvee \F=1$, $\Gdim(L)=\sup\{\Gdim([0,x]):x\in \F\}.$
4. if $L$ is not trivial, then $\Gdim(L)=\sup\{\Gdim([a,b]):[a,b] \text{ $\beta$-simple for some $\beta$}\}$;
5. $\Gdim(L)\leq \beta+1$, where $\beta=\sup\{\Gdim([x,1]):x\neq 0\}$.
Let $\Gdim(L)= \alpha$.
\(1) We proceed by transfinite induction on $\alpha$. If $\alpha=0$, there is nothing to prove, as well as when $\alpha>0$ and $\Gdim([a,b])<\alpha$. Consider the case when $\alpha>0$ and $\Gdim([a,b])\not<\alpha$. Let $a'\in [a,b)$ and let us find $b'\in (a',b]$ such that $[a',b']$ is $\beta$-simple for some $\beta<\alpha$. Indeed, we consider a pseudo-complement $c$ of $b$ in $[a',1]$ and we let $d\in [c,1]$ be such that $[c,d]$ is $\beta$-simple for some $\beta<\alpha$. Let $b'=d\wedge b$. By modularity, $[a',b']\cong [c,(d\wedge b)\vee c]$, which is a sub-qframe of $[c,d]$. By Lemma \[sub\_simp\], $[a',b']$ is $\beta$-simple.
\(2) Let $\beta_1=\Gdim([0,a])$ and $\beta_2=\Gdim([a,1])$. By part (1), $\alpha\geq\max\{\beta_1,\beta_2\}$. Let us show that $\alpha\leq \max\{\beta_1,\beta_2\}$, that is, given $1\neq b\in L$ we need to find $c\in (b,1]$ such that $[b,c]$ is $\gamma$-simple for some $\gamma < \max\{\beta_1,\beta_2\}$. Indeed, given $1\neq b\in L$, we distinguish two cases. If $a\leq b$, then $b\in [a,1]$ and so there is $c\in (b,1]$ such that $[b,c]$ is $\gamma$-simple for some $\gamma<\beta_2$. If $a\not\leq b$, then there is $c\in [a\wedge b,a]$ such that $[b\wedge a,c]$ is $\gamma$-simple for some $\gamma<\beta_1$ and, by modularity, $[b,b\vee c]\cong [a\wedge b,c]$.
\(3) Let $\sup\{\Gdim([0,x]):x\in \F\}=\beta$. Given $1\neq a\in L$, we have to show that there exists $b\in[a,1]$ such that $[a,b]$ is $\gamma$-simple for some $\gamma<\beta$. By hypothesis, there exists $x\in \F$ such that $x\not\leq a$. Thus, $x\neq x\wedge a\in [0,x]$ and so there exists $b'\in [x\wedge a,x]$ such that $[x\wedge a,b']$ is $\gamma$-simple for some $\gamma<\Gdim([0,x])\leq \beta$. Let $b=b'\vee a$; by modularity $[a,b]\cong [x\wedge a, b']$ is $\gamma$-simple as desired.
\(4) Consider a continuous chain in $L$ defined as follows:
1. $x_0=0$;
2. if $\sigma=\tau+1$ is a successor ordinal, then $x_{\sigma}=1$ if $x_{\tau}=1$, while $x_{\sigma}$ is an element $\geq x_\tau$ such that $[x_\sigma,x_\tau]$ is $\beta$-simple for some $\beta$;
3. $x_\sigma=\bigvee_{\tau<\sigma}x_\tau$ if $\sigma$ is a limit ordinal.
Since we supposed that $L$ has Gabriel dimension, then the above definition is correct and there exists an ordinal $\bar \sigma$ such that $x_{\bar \sigma}=1$. Let us prove our statement by induction on $\bar \sigma$. If $\bar \sigma=1$, there is nothing to prove. Furthermore, if $\bar\sigma=\tau+1$, then by part (2), $\Gdim(L)=\max\{\Gdim([0,x_\tau]),\Gdim([x_\tau,x_{\bar\sigma}])$ and we can conclude by inductive hypothesis. If $\bar\sigma$ is a limit ordinal, one concludes similarly using part (3).
\(5) It is enough to prove the statement for $\gamma$-simple qframes for all ordinals $\gamma$ and then apply part (4). So, let $\gamma$ be an ordinal and let $L$ be a $\gamma$-simple qframe. Then, $\Gdim(L)=\gamma+1$ and we should prove that $\sup\{\Gdim([x,1]):x\neq 0\}\geq \gamma$. If, looking for a contradiction, $\sup\{\Gdim([x,1]):x\neq 0\}=\beta<\gamma$ then, just by definition, $L$ is $\beta$-simple, that is a contradiction.
\[coro\_gab\] Let $(L,\leq)$ be a qframe and let $\alpha$ be an ordinal. Then, $\Gdim(L)\leq \alpha$ if and only if, for any element $x\neq 1$, there exists $y> x$ such that $\Gdim([x,y])\leq \alpha$.
Let $x_0=0$, for any ordinal $\gamma$ let $x_{\gamma+1}=1$ if $x_\gamma=1$, otherwise we let $x_{\gamma+1}$ be an element $>x_\gamma$ such that $\Gdim([x_\gamma,x_{\gamma+1}])\leq \alpha$. Furthermore, for any limit ordinal $\lambda$ we let $x_\lambda=\bigvee_{\gamma<\lambda} x_{\gamma}$. Let us prove by transfinite induction that $\Gdim([0,x_\gamma])\leq \alpha$ for all $\gamma$, this will conclude the proof since there exists $\gamma$ such that $x_\gamma=1$. Our claim is clear when $\gamma=0$. Furthermore, if $\gamma=\beta+1$ and $\Gdim([0,x_\beta])\leq \alpha$, then by Lemma \[basic\_gabriel\] (2), $\Gdim([0,x_\gamma])\leq \alpha$. If $\gamma$ is a limit ordinal and $\Gdim([0,x_\beta])\leq \alpha$ for all $\beta<\gamma$, one concludes by Lemma \[basic\_gabriel\] (3).
\[mono\_gabriel\] Let $(L,\leq)$ be a qframe with Gabriel dimension, let $(L',\leq)$ be a qframe and let $\phi:L\to L'$ be a surjective homomorphism of qframes. Then, $\Gdim(L')\leq \Gdim(L)$.
Let us proceed by transfinite induction on $\Gdim(L)$.\
If $\Gdim(L)=0$, then $L$ is a trivial as well as $L'$, so there is nothing to prove.\
Suppose now that $\Gdim(L)=\alpha>0$ and that we have already verified our claim for all $\beta<\alpha$. Let first $\alpha=\gamma+1$ be a successor ordinal and let $L$ be $\gamma$-simple. Then, for all $0\neq a\in L$, $\Gdim([a,1])\leq\gamma$ and so, by inductive hypothesis, $\Gdim(\phi([a,1]))\leq \gamma$. By Lemma \[basic\_gabriel\] (5), $\Gdim(\phi(L))\leq \gamma+1=\alpha$.\
Let now $x'\in L'$, consider the set $\mathcal S=\{x\in L:\phi(x)=x'\}$ and let $\bar x=\bigvee \mathcal S$, so that $\phi(\bar x)=\bigvee_{x\in \mathcal S}\phi( x)=x'$. Let also $\bar y\geq \bar x$ be such that $[\bar x,\bar y]$ is $\beta$-simple for some $\beta< \alpha$ and let $y'=\phi(\bar y)\in L'$. Then, $y'\geq x'$, furthermore $y'\neq x'$ (since $y'=x'$ would imply that $\bar y\in \mathcal S$, that is, $\bar y=\bar x$, which is a contradiction). By the first part of the proof, $\Gdim([x',y'])\leq \beta+1\leq \alpha$. To conclude apply Corollary \[coro\_gab\].
Torsion and localization
------------------------
Let $(L,\leq)$ be a qframe, and let $\alpha$ be an ordinal. We define the [*$\alpha$-torsion part*]{} of $L$ as $$t_\alpha (L)=\bigvee\{x\in L:\Gdim([0,x])\leq \alpha\}\,.$$ For any given $a\in L$, we let $t_\alpha (a)=t_\alpha ([0,a])$.
\[prop\_tors\] Let $(L,\leq)$ be a qframe, let $a$, $b\in L$ and let $\alpha$ be an ordinal. Then,
1. $t_\alpha (a)=a\wedge t_\alpha (1)$;
2. $t_\alpha (a\vee b)\leq t_\alpha (a)\vee b$, provided $a\wedge b=0$;
3. $t_\alpha (a\vee b)= t_\alpha (a)\vee t_\alpha (b)$, provided $a\wedge b=0$;
In particular, $t_\alpha (\bigvee_{i\in I}x_i)=\bigvee_{i\in I}t_\alpha (x_i)$ for any join-independent set $\{x_i:i\in I\}$ in $L$.
\(1) By definition, $t_\alpha (a)\leq a\wedge t_\alpha (1)$. On the other hand, by upper continuity, $$\begin{aligned}
a\wedge\bigvee\{x\in L:\Gdim([0,x])\leq \alpha\}&=\bigvee\{a\wedge x\in L:\Gdim([0,x])\leq \alpha\}\\
&=\bigvee\{x\in [0,a]:\Gdim([0,x])\leq \alpha\}=t_\alpha (a)\,.\end{aligned}$$ This works since the family $\{x\in L:\Gdim([0,x])\leq \alpha\}$ is directed by part (2) of Lemma \[basic\_gabriel\].
\(2) Let $x\in [0,a\vee b]$ be such that $\Gdim([0,x])\leq \alpha$, then $x\vee b\in [0,a\vee b]$ and $\Gdim([b,x\vee b])=\Gdim([b\wedge x,x])\leq \Gdim([0,x])\leq \alpha$. This shows ($*$) below: $$\begin{aligned}
t_\alpha (a\vee b)&=\bigvee\{x\in [0,a\vee b]:\Gdim([0,x])\leq \alpha\}\\
&\overset{(*)}{\leq} \bigvee\{x\vee b\in [0,a\vee b]:\Gdim([b,x\vee b])\leq \alpha\}\\
&=\bigvee\{x\in [b,a\vee b]:\Gdim([b,x])\leq \alpha\}\\
&\overset{(**)}{=}\bigvee\{x\vee b:x\in [0,a]\text{ and }\Gdim([0,x])\leq \alpha\}\\
&\overset{\binom{*}{**}}{=}b\vee\bigvee\{x:x\in [0,a]\text{ and }\Gdim([0,x])\leq \alpha\}=b\vee t_\alpha (a)\,,\end{aligned}$$ where ($**$) holds since te map $x\mapsto x\vee b$ is an isomorphism between $[0,a]$ and $[b,b\vee a]$ (use the fact that $a\wedge b=0$), and in $\binom{*}{**}$ we used upper-continuity.
\(3) It is clear that $t_\alpha (b)\vee t_\alpha (a)\leq t_\alpha(a\vee b)$. Using twice part (2) and the modularity of $L$, $$\begin{aligned}
t_\alpha (b)\vee t_\alpha (a)\leq t_\alpha (a\vee b)&\leq (t_\alpha (a)\vee b)\wedge (t_\alpha (b)\vee a)= t_\alpha (a)\vee (b\wedge(t_\alpha (b)\vee a))\\
&=t_\alpha (a)\vee ((b\wedge a)\vee t_\alpha (b))=t_\alpha (a)\vee t_\alpha (b)\,.\end{aligned}$$ where the last equality holds since $a\wedge b=0$.
For the last part of the statement, notice that $$\bigvee_{i\in I}x_i=\bigvee_{F\subseteq I\text{ finite}}\left(\bigvee_{i\in F}x_i\right)\,.$$ Thus, using upper-continuity and part (3) of the lemma, $$\begin{aligned}
t_\alpha \left(\bigvee_{i\in I}x_i\right)&=t_{\alpha}(1)\wedge \bigvee_{F\subseteq I\text{ finite}}\left(\bigvee_{i\in F}x_i\right)\\
&=\bigvee_{F\subseteq I\text{ finite}}\left(t_{\alpha}(1)\wedge\bigvee_{i\in F}x_i\right)=\bigvee_{F\subseteq I\text{ finite}}\left(\bigvee_{i\in F}t_{\alpha}(x_i)\right)=\bigvee_{i\in I}t_{\alpha}(x_i)\,.\end{aligned}$$
\[succ\_ord\] Let $L$ be a qframe, let $x\in L$ and let $\{y_s:s\in S\}\subseteq L$. Suppose that
1. $[0,y_s]\cong [0,y_t]$ for all $s$, $t\in S$;
2. $[0,y_s]$ is Noetherian for some (hence all) $s\in S$;
3. $\{y_s:s\in S\}$ is a basis for $L$.
Then, $\Gdim ([0,x])$ is a successor ordinal.
A consequence of Theorem \[GvK\] (3) is that, for all $s\in S$, $t_{\alpha+1}(y_s)\neq t_\alpha (y_s)$ for just finitely many ordinals $\alpha$ (the same $\alpha$’s for all $s\in S$). Furthermore, $\bigvee_{s\in S} t_{\alpha} (y_s)=t_{\alpha}(1)$ for all $\alpha$, by the above lemma. Thus, $t_{\alpha+1}( 1)\neq t_\alpha ( 1)$ for finitely many ordinals $\alpha$. Notice also that $t_\alpha (x)=t_\alpha( 1) \wedge x$ for all $\alpha$, thus $t_{\alpha+1}(x)\neq t_{\alpha}(x)$ implies $t_{\alpha+1}(1)\neq t_{\alpha}(1)$ and so, $t_{\alpha+1}(x)\neq t_{\alpha}(x)$ for finitely many ordinals $\alpha$. Hence, $\Gdim([0,x])=\sup\{\alpha+1:t_{\alpha+1}(x)\neq t_\alpha (x)\}=\max\{\alpha+1:t_{\alpha+1}(x)\neq t_\alpha (x)\}$ is a successor ordinal.
Let $(L,\leq)$ be a qframe and let $\alpha$ be an ordinal. Then,
1. $x\in [0,t_\alpha (1)]$ if and only if $\Gdim([0,x])\leq \alpha$;
2. given a qframe $(L',\leq)$ and a homomorphism of qframes $\phi:L\to L'$, $\phi(t_\alpha (L))\leq t_\alpha (L')$.
\(1) By part (3) of Lemma \[basic\_gabriel\], $\Gdim([0,t_{\alpha}(1)])\leq \alpha$ and so, by part (1) of the same lemma, $\Gdim([0,x])\leq\Gdim([0,t_{\alpha}(1)])\leq \alpha$ for all $x\in [0,t_\alpha (1)]$. On the other hand, if $\Gdim([0,x])\leq \alpha$, then $x\leq t_\alpha (1)$ by construction.
\(2) is an application of part (1) and Lemma \[mono\_gabriel\].
By part (2) of the above proposition we can give the following:
Let $\alpha$ be an ordinal. Given a qframe $(L,\leq)$, we let $T_\alpha (L)=[0,t_\alpha (1)]$, while, given a homomorphism of qframes $\phi:L\to L'$, we let $T_\alpha (\phi):T_{\alpha}(L)\to T_\alpha (L')$ be the restriction of $\phi$. This defines a covariant functor $T_\alpha :\Qframe\to \Qframe$ that we call [*$\alpha$-torsion functor*]{}.
It is not difficult to show that $T_\alpha$ is compatible with the composition of morphisms, so that the above definition is correct. Notice that the $\alpha$-torsion functor is an idempotent and hereditary preradical in the terminology of [@latt_prerad].
In the rest of this section we study the following equivalence relation induced by Gabriel dimension:
Let $(L,\leq)$ be a qframe, let $\alpha$ be an ordinal and define the following relation between two elements $x$ and $y$ in $L$: $$(x, y)\in\mathcal R_\alpha\ \ \text{ if and only if }\ \ (\Gdim([x\wedge y,x\vee y])\leq \alpha)\,.$$ We also use the notation $x\sim_\alpha y$ to say that $(x, y)\in\mathcal R_\alpha$.
Let $(L,\leq)$ be a qframe and let $\alpha$ be an ordinal, then $\mathcal R_\alpha$ is a strong congruence on $L$.
The fact that $\mathcal R_\alpha$ is a congruence follows by Lemma \[basic\_gabriel\] (2) and [@AS1 Proposition 2.4]. Furthermore, given $x\in L$, let us show that $\bigvee[x]\in [x]$. In fact, $$\Gdim\left(\left[x,\ \bigvee_{y\in [x]} y\right]\right)=\Gdim\left(\left[x,\bigvee_{y\in [x]} x\vee y\right]\right)=\sup\{\Gdim([x,x\vee y]):y\in [x]\}\leq \alpha\,,$$ by Lemma \[basic\_gabriel\] (3). Thus, $\mathcal R_\alpha$ is a strong congruence.
We denote by $Q_\alpha(L)$ the quotient of $L$ over $\mathcal R_\alpha$ and by $\pi_\alpha:L\to Q_\alpha(L)$ the canonical surjective homomorphism.
\[loc\_alpha\] Let $(L,\leq)$ and $(L',\leq)$ be qframes, let $\phi:L\to L'$ be a homomorphism of qframes, and let $\alpha$ be an ordinal.
1. If $x\sim_\alpha y$ in $L$, then $\phi(x)\sim_\alpha \phi(y)$ in $L'$;
2. $\Gdim(Q_{\alpha}(T_{\alpha+1}(L)))\leq 1$, that is, $Q_{\alpha}(T_{\alpha+1}(L))$ is semi-Artinian for any ordinal $\alpha$.
\(1) By Lemma \[mono\_gabriel\], $\Gdim([x\wedge y,x\vee y])\geq \Gdim(\phi([x\wedge y,x\vee y]))=\Gdim([\phi(x\wedge y),\phi(x)\vee \phi(y)])$. Furthermore, $\phi(x)\wedge \phi(y)\geq \phi(x\wedge y)$ and so $\Gdim ([\phi(x)\wedge \phi(y),\phi(x)\vee \phi(y)])\leq \Gdim[\phi(x\wedge y),\phi(x)\vee \phi(y)]\leq \alpha$, by Lemma \[basic\_gabriel\] (1).
\(2) Let $S=[a,b]$ be an $\alpha$-simple segment of $T_{\alpha+1}(L)$. Then, $\pi_\alpha(S)$ is an atom since $a$ is not $\alpha$-equivalent to $b$ (as $\Gdim(S)=\alpha+1$) and $b$ is $\alpha$-equivalent to any $c\in(a,b]$ (as $\Gdim([c,b])\leq \alpha$).\
If $Q_{\alpha}(T_{\alpha+1}(L))=0$ there is nothing to prove, otherwise choose an element $x\in T_{\alpha+1}(L)$ such that $\pi_\alpha(t_{\alpha+1}(1))\neq \pi_\alpha(x)\in Q_\alpha(T_{\alpha+1}(L))$ and let $\bar x=\bigvee [x]\in T_{\alpha+1}(L)$. Notice that $\Gdim([\bar x, t_{\alpha+1}(1)])=\alpha+1$ (otherwise $[x]=[t_{\alpha+1}(1)]$). By definition of Gabriel dimension, there exists $\bar y\in T_{\alpha+1}(L)$ such that $[\bar x,\bar y]$ is $\beta$-simple for some $\beta< \alpha+1$ and, since $\bar y\notin [\bar x]$, we have $\beta=\alpha$. By the previous discussion, $[\pi_\alpha(x),\pi_\alpha(\bar y)]$ is $0$-simple. One can conclude by Corollary \[coro\_gab\].
By part (1) of the above proposition, we can give the following:
Let $\alpha$ be an ordinal. Given a qframe $(L,\leq)$, we let $Q_\alpha(L)=L_{/\mathcal R_{\alpha}}$, while, given a homomorphism of qframes $\phi:L\to L'$, we let $Q_\alpha(\phi):Q_{\alpha}(L)\to Q_\alpha(L')$ be the induced homomorphism. This defines a functor $Q_\alpha:\Qframe\to \Qframe$ that we call [*$\alpha$-localization functor*]{}.
It is not difficult to show that $Q_\alpha$ is compatible with the composition of morphisms, so that the above definition is correct.
Main Theorems {#Main_Sec}
=============
Let $V$ be a nonempty finite set and denote by $S_V$ the symmetric group on $V$. Given two permutations $\sigma_1$ and $\sigma_2\in S_V$ we let $$d_V(\sigma_1,\sigma_2)=\frac{|\{v\in V:\sigma_1(v)\neq \sigma_2(v)\}|}{|V|}\,,$$ be the [*normalized Hamming distance*]{} between $\sigma_1$ and $\sigma_2$.
Let $G$ be a group, let $K\subseteq G$ be a subset and let $\varepsilon>0$. Given a finite set $V$, a [*$(K,\varepsilon)$-quasi-action*]{} of $G$ on $V$ is a map $\varphi:G\to S_V$ such that:
1. $\varphi(e)=\id_V$;
2. for all $k_1$ and $k_2\in K$, $d_V(\varphi(k_1k_2),\varphi(k_1)\varphi(k_2))\leq \varepsilon$;
3. for all $k_1\neq k_2\in K$, $d_V(\varphi(k_1),\varphi(k_2))\geq 1- \varepsilon$.
Whenever we have a quasi-action we adopt the following notation. Given two subsets $V'\subseteq V$ and $G'\subseteq G$, we let $G'V'=\{\varphi_g(v):g\in G',\ v\in V'\}$. In case $V'=\{v\}$ is a singleton set we let $G'v=G'\{v\}$. Similarly, if $G'=\{g\}$ is a singleton, $gV'=\{g\}V'$. Furthermore, $gv=\varphi_g(v)$ for all $v\in V$ and $g\in G'$.
For finitely generated groups, the following definition of sofic group is equivalent to the definition given in [@Weiss] and [@Gro] (see [@Capraro]).
A group $G$ is sofic if, for any subset $K\subseteq G$ and for any positive constant $\varepsilon$, there exists a finite set $V$ and a $(K,\varepsilon)$-quasi-action of $G$ on $V$.
The $1$-dimensional case
------------------------
I’ve learnt the arguments used in the proof of the following lemma while reading [@Elek proof of Proposition 4.4] and [@Weiss proof of Lemma 3.1]. Also Lemma \[key\] is inspired to the argument used by Weiss to show surjunctivity of sofic groups.
\[setting\] Let $G$ be a group, let $K$ be a finite symmetric subset of $G$ and let $H=KK$. Choose $n\in {\mathbb N}_{\geq 2}$, let $\varepsilon$ be a positive constant such that $\varepsilon<\frac{1}{2n|H|^2}$, let $V$ be a finite set, let $\varphi:G\to S_V$ be an $(H,\varepsilon)$-quasi-action of $G$ on $V$ and define the following set: $${\bar V}=\{v\in V:hv\neq h'v \text{ and } (h_1h_2)v=h_1(h_2v),\text{ for all } h\neq h'\in H,\ h_1,h_2\in H\}\,.$$ Then, the following statements hold true:
1. $| {\bar V}|\geq (1-1/n)|V|$;
2. there is a subset $W\subseteq {\bar V}$ such that $Kv\cap Kw=\emptyset$ for all $v\neq w\in W$ and $|W|\geq |V|/2|H|$.
\(1) A given $v\in V$ belongs to ${\bar V}$ if and only if it satisfies the following two conditions:
1. $\varphi_{h_1}(v)\neq \varphi_{h_2}(v)$ for all $h_1\neq h_2\in H$;
2. $\varphi_{h_1h_2}(v)=\varphi_{h_1}(\varphi_{h_2}(v))$ for all $h_1,h_2\in H$.
There are less than $|H|^2$ equations in (a) and each of these equations can fail for at most $\varepsilon|V|$ elements $v$ in $V$. Similarly, there are $|H|^2$ equations in (b) and each of these equations can fail for at most $\varepsilon |V|$ elements $v\in V$. Thus, the cardinality of ${\bar V}$ is at least $$|V|- (|H|^2\varepsilon |V|+|H|^2\varepsilon|V|)\geq |V|(1- 2|H|^2\varepsilon)\geq |V|(1-1/n)\,.$$ (2) Let $W$ be a maximal subset of ${\bar V}$ with the property that $Kv\cap Kw=\emptyset$ for all $v\neq w\in W$. We claim that $HW$ contains ${\bar V}$. In fact, if there is $v\in {\bar V}$ such that $v\notin HW$, this means that, for all $w\in W$, $Kv\cap Kw=\emptyset$, contradicting the maximality of $W$. Thus, $|{\bar V}|\leq |WH|\leq |W||H|$. To conclude, use that $2|{\bar V}|\geq |V|$ by part (1) and the choice of $n$.
\[key\] In the same setting of Lemma \[setting\], let $(L_1,\leq)$ and $(L_2,\leq)$ be two qframes of finite length and consider a homomorphism of qframes $\Phi:L_1\to L_2$. Let $l\in {\mathbb N}_{\geq 1}$ and suppose that
1. there is distinguished family of elements $\{\bar x_v:v\in K{\bar V}\}$ such that
1. $\bigvee_{K{\bar V}} \bar x_v=1$ ;
2. $\ell(\bar x_v)=l$, for all $v\in K{\bar V}$;
2. $\ell\left(\bigvee_{v\in Kw}\Phi(\bar x_v)\right)\leq |K|l-1$, for all $w\in \bar V$.
Then, $\ell(\Im(\Phi))\leq \left(1-\frac{1}{2|H|l}\right)|V|l$.
Choose a $W\subseteq \bar V$ as in part (2) of Lemma \[setting\]. By Lemma \[length\_sum\], $$\ell(\Phi(L_1))=\ell\left( \bigvee_{v\in K\bar{ V}} \Phi(\bar x_v)\right)\leq \ell\left( \bigvee_{v\in K\bar{ V}\setminus KW} \Phi(\bar x_v)\right)+\ell\left( \bigvee_{v\in KW} \Phi(\bar x_v)\right)\,.$$ Furthermore, $$\ell\left( \bigvee_{v\in KW} \Phi(\bar x_v)\right)\leq\sum_{w\in W}\ell\left(\bigvee_{v\in Kw}\Phi(\bar x_v)\right)\leq |W|(|K|l-1)
\,.$$ By the choice of $W$, $|K{\bar V}\setminus KW|=|K{\bar V}|- \sum_{w\in W}|Kw|=|K{\bar V}|-|W| |K|$ and, by Lemma \[length\_inj\_surj\], $\ell\left( \bigvee_{v\in K{\bar V}\setminus KW} \Phi(\bar x_v)\right)\leq \ell\left( \bigvee_{v\in K{\bar V}\setminus KW} \bar x_v\right)$, thus $$\ell\left( \bigvee_{v\in K{\bar V}\setminus KW} \bar x_v\right)\leq \sum_{v\in K{\bar V}\setminus KW}\ell(\bar x_v)=|K{\bar V}\setminus KW|l=(|K{\bar V}|-|W| |K|)l\leq (|V|-|W| |K|)l\,,$$ Putting together all these data, we get $$\begin{aligned}
\ell(\Phi(L_1))&\leq |W|(|K|l-1)+(|V|-|W||K|)l=-|W|+|V|l\leq \left(1-\frac{1}{2|H|l}\right)|V|l\,.\end{aligned}$$
\[main\] Let $M$ be a qframe, let $G$ be a sofic group, let $\rho:G\to \Aut(M)$ be a right action of $G$ on $M$ (we let $\rho(g)=\rho_g$ for all $g\in G$) and let $\phi:M\to M$ be a $G$-equivariant homomorphism of qframes, that is, $\rho_g\phi=\phi\rho_g$, for all $g\in G$. Choose an element $\bar y\in M$ such that
1. $\ell(\bar y)=l<\infty$;
2. the family $\{\bar y_g:g\in G\}$ is join-independent, where $\bar y_g=\rho_g(\bar y)$ for all $g\in G$;
3. there exists a finite symmetric subset $F\subseteq G$ such that $\phi(\bar y)\leq \bigvee_{g\in F}\bar y_g$ and $e\in F$.
Fix an $F$ as in (c) and let $K$ be a finite symmetric subset of $G$ containing $F$. Then, the following conditions are mutually exclusive:
1. $\bar y\leq \bigvee_{g\in K}\phi(\bar y_g)$;
2. $\ell\left(\bigvee_{g\in K}\phi(\bar y_g)\right)\leq |K|l-1$.
Assume, looking for a contradiction, that both (1) and (2) are verified. We start by constructing some objects to which we want to apply Lemmas \[setting\] and \[key\].
First we construct the objects mentioned in Lemma \[setting\]. Choose a positive integer $n\geq 2|H|l$, let $H=KK$, let $\varepsilon$ be a positive constant such that $\varepsilon<\frac{1}{2n|H|^2}$, let $V$ be a finite set, let $\varphi:G\to S_V$ be an $(H,\varepsilon)$-quasi-action of $G$ on $V$ and define $${\bar V}=\{v\in V:hv\neq h'v \text{ and } (h_1h_2)v=h_1(h_2v),\text{ for all } h\neq h'\in H,\,h_1,h_2\in H\}\,.$$
Let us construct now the objects mentioned in Lemma \[key\]. For a subset $G'\subseteq G$, we use the notation $\bar y_{G'}=\bigvee_{g\in G'}\bar y_g$ and, for all $v\in V$, we let $Q^{G'}_{v}$ be a qframe isomorphic to $[0,\bar y_{G'}]$. For all $v\in V$, we identify both $Q^{e}_v=Q^{\{e\}}_v$ and $Q^{K}_v$ with sub-qframes of $Q^{H}_v$ in such a way that there is an isomorphism of qframes $$q_{v}:Q^{H}_v\longrightarrow [0,\bar y_H]\,,$$ such that $q_v(Q_v^e)=[0,\bar y]$ and $q_v(Q_v^K)=[0,\bar y_K]$. For all $v\in {\bar V}$, the map $\sigma_v:Hv\to H$ such that $\sigma_v(hv)=h$ is well defined and bijective. So, given $v\in {\bar V}$ and $w\in Hv$ we let $$q_{v}^{w}:Q_{w}^{H}\tilde\longrightarrow [0,\bar y_{H\sigma_{v}(w)}]$$ be the composition $q_{v}^{w}=\rho_{\sigma_{v}(w)}q_{w}$. Let us introduce the following notation for all $G'\subseteq G$: $$Q^{G'}=\prod_{v\in {\bar V}}Q^{G'}_v\,,\ \ \forall G'\subseteq G\,.$$ For all $v\in {\bar V}$, we denote by $\iota^{G'}_v:Q^{G'}_v\to Q^{G'}$ the canonical inclusion in the product. Consider, for all $v\in {\bar V}$, the following homomorphism of qframes: $$\Psi_{v}:Q^{H}\longrightarrow [0,\bar y_{HH}]\ \text{ such that } \ (a_{w})_{w\in {\bar V}}\mapsto \bigvee_{w\in Hv\cap {\bar V}}q_{v}^{w}(a_{w})\,.$$ We define a relation $\mathcal R\subseteq Q^{H}\times Q^H$ as follows: $$(a,b)\in\mathcal R \ \ \ \Longleftrightarrow \ \ \ \Psi_{v}( a)=\Psi_{v}( b)\ \forall v\in {\bar V}\,.$$ This defines a strong congruence on $Q^{H}$ and, by restriction, on $Q^{K}$. Let $L_1=Q^{K}/\mathcal R$ and $L_2=Q^{H}/\mathcal R$ and let $\pi_1:Q^{K}\to L_1$ and $\pi_2:Q^{H}\to L_2$ be the canonical projections. For all $v\in\bar{\bar V}$, let $\Phi_{v}:Q_v^{K}\to Q_v^{H}$ be the unique map such that $q_v\Phi_v(x)=\phi(q_v(x))$, for all $x\in Q^{K}_v$. We let $\Phi:Q^{K}\to Q^{H}$ be the product of these maps, that is, $ \Phi(x_{v})_{v\in {\bar V}}=(\Phi_{v}(x_{v}))_{v\in {\bar V}}$. Given two elements $a\sim b\in Q^{K}$, $\Phi( a)\sim \Phi( b)$. In fact, for all $v\in { \bar{V}}$, $$\begin{aligned}
\Psi_{v}\Phi( a)&= \bigvee_{w\in Hv\cap {\bar V}} \rho_{\sigma_{v}(w)}q_{w}\Phi_{w}(a_{w})= \bigvee_{w\in Hv\cap {\bar V}} \rho_{\sigma_{v}(w)}\phi q_{w}(a_{w})= \phi \left(\bigvee_{w\in Hv\cap {\bar V}} q^{w}_{v}(a_{w})\right)\\
&=\phi \left(\bigvee_{w\in Hv\cap {\bar V}} q^{w}_{v}(b_{w})\right)=\ldots=\Psi_{v}\Phi( b)\,.\end{aligned}$$ Let $\bar\Phi:L_1\to L_2$ be the unique map such that $\bar \Phi\pi_1=\pi_2\Phi$. One verifies that $\bar \Phi$ is a morphism of qframes.
Now that the setting is constructed we need to verify that the hypotheses (1) and (2) of Lemma \[key\] are satisfied. For all $v\in {\bar V}$ and $k\in K$ we let $x^{v}_k=\iota^{K}_v (q_{v}^{-1}(\bar y_{k}))$. Let us show that $x_k^v\sim x_{k'}^{v'}$ if and only if $kv=kv'$. Indeed, given $v,\, v'\in \bar V$ and $k,\, k'\in K$ such that $x_k^v\sim x_{k'}^{v'}$, notice that $$\Psi_{v}(x_k^v)=\bar y_k \ \ \text{ and }\ \ \Psi_{v}(x_{k'}^{v'})=\rho_{\sigma_v(v')}\bar y_{k'}$$ if $v'\in Hv$, otherwise it is $0$. Thus, $\sigma_v(v')k'=k$, that is, $v'=\sigma_v(v')v=(k')^{-1}k v$, so $k'v'=kv$ (here we are using that $v,\, v'\in\bar V$). Hence, given $w=kv\in K{\bar V}$, we can define $\bar x_w=\pi_1(x_k^v)$ without any ambiguity. Clearly $\bigvee_{v\in K {\bar V}} \bar x_v=1$, let us show that the family $\{\bar x_v:v\in K {\bar V}\}\subseteq L_1$ is join-independent. Indeed, given $k'v'\in K {\bar V}$, $$\bar x_{k'v'}\wedge \bigvee_{k'v'\neq kv\in K\bar V} \bar x_{kv}=\pi_1\left( x_{k'}^{v'}\wedge \bigvee_{k'v'\neq kw\in K\bar V} x_k^v\right)=\pi_1(0)=0\,,$$ where the first equality comes from the definition of the $\bar x_w$ and the properties of $\pi_1$ (see Lemma \[proj\_comm\_join\]), while the second equality holds since the family $\{ x_k^v:kv\in K {\bar V}\}\subseteq Q^K$ is join-independent.
Furthermore, for all $w\in {\bar V}$: $$\begin{aligned}
\ell \left(\bigvee_{v\in Kw}\bar\Phi(\bar x_v)\right)&=\ell \left(\bigvee_{v\in Kw}\bar\Phi\pi_1(Q_v^{K})\right)=\ell \left(\bigvee_{v\in Kw}\pi_2\Phi_v(Q_v^{K})\right)\leq \ell \left(\bigvee_{v\in Kw}\Phi(Q_v^{K})\right)\\
&\leq \ell(\phi([0,\bar y_K]))\leq |K|l-1\,.\end{aligned}$$
In the last part of the proof we obtain the contradiction we were looking for. Indeed, we claim that the restriction of $\pi_2$ to $Q^{e}$ is injective and that $\pi_2(Q^{e})\subseteq \bar\Phi(L_1)$. In fact, let $a=(a_{v})_{v\in {\bar V}}$ and $ b=(b_{v})_{v\in {\bar{V}}}\in Q^{e}$ and suppose that $\pi_2(a)=\pi_2(b)$, that is, $ a\sim b$. For all $v\in { {\bar V}}$ and $w\in Hv\cap {\bar{V}}$, by construction, $q_{v}^{w}(a_{w}),\ q_{v}^{w}(b_{w})\leq \bar y_{\sigma_{v}(w)}$. So, using modularity and the independence of the family $\{\bar y_g: g\in G\}$, $$\begin{aligned}
q_{v}(a_{v})=q_{v}(a_{v})\vee 0&=h_{v}(a_{v})\vee\left(\bigvee_{v\neq w\in Hv\cap {\bar{ V}}}q_{v}^{w}(a_{w})\wedge \bar y\right)=\bar y\wedge\left(q_{v}(a_{v})\vee\bigvee_{v\neq w\in Hv\cap {{\bar V}}}q_{v}^{w}(a_{w})\right)\\
&=\bar y\wedge \Psi_{v}( a)=\bar y\wedge \Psi_{v}( b)=\ldots= q_{v}(b_{v})\,,\end{aligned}$$ that is, $a_{v}=b_{v}$, for all $v\in {\bar V}$. Our second claim follows by construction and the hypothesis (1). Also recalling the estimate for $|{\bar V}|$ given in Lemma \[setting\], the two claims we just verified imply that $$\ell(\Im(\bar \Phi))\geq \ell(\pi_2(Q^{(e)}))=\ell(Q^{(e)})=|{\bar V}|l \geq \left(1-\frac{1}{n}\right)|V|l\,.$$ Furthermore, by Lemma \[key\], $\ell(\Im(\bar\Phi))< \left(1-\frac{1}{2|H|l}\right)|V|l$. Thus, $n<2|H|l$, which is a contradiction.
Higher dimensions
-----------------
\[prel\_high\] Let $(M,\leq)$ be a qframe, let $G$ be a group, let $\rho:G\to \Aut(M)$ be a right action of $G$ on $M$ and consider an algebraic $G$-equivariant homomorphism of qframes $\phi:M\to M$. Suppose that there exists an element $y\in M$ such that $[0,y]$ is finitely generated and such that, letting $y_g=\rho_g( y)$ for all $g\in G$, the family $\{y_g:g\in G\}$ is a basis for $M$. Then,
1. $\phi$ is surjective if and only if there exists a finite subset $K\subseteq G$ such that $y\leq \bigvee_{g\in K}\phi(y_g)$;
2. $\phi$ is not injective if and only if there exist a finite subset $K\subseteq G$ and $0\neq x\leq \bigvee_{g\in K}y_g$ such that $\phi(x)=0$.
\(1) Suppose that $\phi$ is surjective, then $\bigvee_{g\in G} \phi(y_g)=\phi(1)=1$. By Lemma \[fg\_base\], one can find a finite subset $K\subseteq G$ such that $y\leq\bigvee_{g\in G} \phi(y_g)$ . On the other hand, if there exists $K\subseteq G$ such that $y\leq \bigvee_{g\in K}\phi(y_g)$, then $y_h\leq \bigvee_{g\in Kh^{-1}}\phi(y_g)\leq \phi(1)$ for all $h\in G$. Thus, $1=\bigvee_{h\in G}y_h\leq \phi(1)$ and so $\phi$ is surjective.
\(2) By the algebraicity of $\phi$, if $\phi$ is not injective, there is a non-trivial element $x'\in {\mathrm{Ker}}(\phi)$. By Lemma \[finite\_base\], there exists a finite subset $K\subseteq G$ such that $x'\wedge \bigvee_{g\in K}y_g\neq 0$, so that $x=x'\wedge \bigvee_{g\in K}y_g$ is the element we were looking for. The converse is trivial.
\[main\_higher\] Let $(M,\leq)$ be a qframe, let $G$ be a sofic group, let $\rho:G\to \Aut(M)$ be a right action of $G$ on $M$ and consider a surjective algebraic $G$-equivariant homomorphism of qframes $\phi:M\to M$. For a given element $ y\in M$ such that $[0,y]$ is compact, consider the following conditions:
1. $[0, y]$ is Noetherian;
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1. $\Kdim([0, y])$ exists and there is a homomorphism of qframes $\psi:M\to M$ such that $\phi\psi=\id$;
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1. letting $y_g=\rho_g( y)$ for all $g\in G$, the family $\{y_g:g\in G\}$ is a basis for $M$.
If (b$_*$) and either (a$_*$) or (a$_*'$) hold, then $\phi$ is injective.
Suppose, looking for a contradiction, that $\phi$ is not injective. Suppose that (b$_*$) is verified, so by Lemma \[prel\_high\], there exists a finite subset $K$ of $G$ such that
1. $ y\leq \bigvee_{g\in K}\phi(y_g)$;
2. there exists $0\neq x\leq \bigvee_{g\in K}y_g$ such that $\phi(x)=0$.
Furthermore, since $[0,y]$ is compact, also $[0,\phi(y)]$ is compact and so there exists a finite subset $F\subseteq G$ such that
1. $\phi(y)\leq \bigvee_{g\in F} y_g$.
In case (a$_*$) is verified, by Lemma \[succ\_ord\] there exists an ordinal $\alpha$ such that $\Gdim([0,{\mathrm{Ker}}(\phi)])=\alpha+1$. On the other hand, if (a’$_*$) is verified, we let $\alpha$ be any ordinal such that $t_\alpha (x)\neq t_{\alpha+1}(x)$. In both cases, let $\bar M=Q_\alpha(T_{\alpha+1}(M))$ and denote by $\pi:T_{\alpha+1}(M)\to \bar M$ the canonical projection. We let $\bar x=\pi(t_{\alpha+1}(x))$ and $\bar y=\pi(t_{\alpha+1}(y))$. There is an induced right action of $G$ on $\bar M$, $\bar \rho:G\to \Aut(\bar M)$, where $\bar \rho_g=Q_\alpha(T_{\alpha+1}(\rho_g))$ for all $g\in G$. Of course, the map $\bar \phi=Q_\alpha(T_{\alpha+1}(\phi)):\bar M\to \bar M$ is $G$-equivariant. One can prove that $\bar \rho_g(\bar y)=\pi(t_{\alpha+1}(y_g))$, for all $g\in G$, and so, whenever (b$_*$) is verified, the family $\{\bar y_g:g\in G\}$, where $\bar y_g=\bar \rho_g(\bar y)$, is a basis of $\bar M$ (it is clear that $\bigvee \bar y_g=1$, to see that this family is join-independent use that the canonical projection commutes with joins and finite meets by Lemma \[proj\_comm\_join\]).
Suppose that (a$_*$) is verified. By Proposition \[loc\_alpha\] (2), $[0,\bar y]$ is semi-Artinian and, by (a$_*$), it is also Noetherian. Thus, $\ell(\bar y)=l<\infty$. Notice that, by (3$_*$), $\bar \phi(\bar y)\leq \bigvee_{g\in F}\bar y_g$ and, by (1$_{*}$), $t_{\alpha+1}(y)\in [0,\bigvee_{g\in K}\phi(y_g)]$, thus there exists $z\leq \bigvee_{g\in K}y_g$ such that $\phi(z)=t_{\alpha+1}(y)$. By the algebraicity of $\phi$ and Lemma \[basic\_gabriel\] (2), $\Gdim ([0,z])=\max\{\Gdim([0,{\mathrm{Ker}}(\phi)\wedge z]),\Gdim([0,t_{\alpha+1}(y)])\}=\alpha+1$, thus $z\in [0,\bigvee_{g\in K}t_{\alpha+1}(y_g)]$. Applying $\pi$, we obtain an element $\pi(z)\in [0,\bigvee_{g\in K}\bar y_g]$ such that $\bar \phi(\pi(z))=\pi(\phi(z))=\bar y$. Thus, $\bar y\leq \bigvee_{g\in K}\bar\phi(\bar y_g)$. By the choice of $\alpha$, ${\mathrm{Ker}}(\bar \phi)\neq 0$ and so, by Lemma \[finite\_base\], there exists a finite subset $F'\subseteq G$ such that ${\mathrm{Ker}}(\bar \phi)\wedge \bigvee_{g\in F'}\bar y_g\neq 0$. Let $K'$ be a finite symmetric subset of $G$ which contains both $F'$ and $K$, then $$\bar y\leq \bigvee_{g\in K'}\bar\phi(\bar y_g)\ \ \text{ and }\ \ \ell\left(\bigvee_{g\in K'}\bar\phi(\bar y_g)\right)\leq |K'|l-1\,,$$ by the above discussion and Lemma \[hopf\_lattice\]. These two conditions cannot happen for the same $K'$ by Theorem \[main\], so we get a contradiction.
Suppose now that (a$_*'$) is verified. We define $\bar \psi=Q_\alpha(T_{\alpha+1}(\psi)):\bar M\to \bar M$, so that $\bar \phi\bar \psi=\id$. Consider the socle $\Soc(\bar M)=[0,s(\bar M)]$ and notice that, since $\{\bar y_g:g\in G\}$ is join-independent, $s(\bar M)=\bigvee_{g\in G} s([0,\bar y_g])$. Since $[0,\bar y]$ is semi-Artinian and it has Krull dimension, then it is Artinian, thus, it has a socle of finite length: let $l=\ell(s(\bar y))$. By the choice of $\alpha$, $\bar x\neq 0$ and $\bar x\wedge s(\bar M)\neq 0$, since, being $\bar M$ semi-Artinian, $s(\bar M)$ is essential in $\bar M$. Since $\Soc(\bar M)$ is fully invariant (see Lemma \[basic\_soc\] (4)), $\bar\phi\restriction_{\Soc(\bar M)}\bar\psi\restriction_{\Soc(\bar M)}=id_{\Soc(\bar M)}$. The family $\{s(\bar y_g):g\in G\}$ is clearly join-independent. Furthermore, using the fact that $[0,s(\bar y)]$ is compact (since it has finite length), also $[0,\bar \phi(s(\bar y))]$ and $[0,\bar \psi(s(\bar y))]$ are compact, so there exists a finite subset $F'\subseteq G$ such that $\bar \phi(s(\bar y)),\,\bar \psi(s(\bar y))\leq \bigvee_{g\in F'}s(\bar y_g)$. Let $K'\subseteq G$ be a finite symmetric subset that contains both $F'$ and $K$, then $$s(\bar y)=\bar \phi(\bar \psi(s(\bar y)))\leq\bar \phi\left(
\bigvee_{g\in K'}s(\bar y_g)\right)\leq \bigvee_{g\in K'}\bar\phi(s(\bar y_g))\ \ \text{ and }\ \ \ell\left(\bigvee_{g\in K'}\bar\phi(s(\bar y_g))\right)\leq |K'|l-1\,,$$ by Lemma \[hopf\_lattice\] and the fact that $\bar \phi(\bar x\wedge s(\bar M))=0$. This is a contradiction by Theorem \[main\].
Applications {#Appl_Sec}
============
Stable Finiteness of crossed products {#crossed}
-------------------------------------
Given a group $G$ and a ring $R$, a [*crossed product*]{} $\RG $ of $R$ with $G$ is a ring constructed as follows: as a set, $\RG $ is the collection of all the formal sums $$\sum_{g\in G}r_g\underline g\, ,$$ with $r_g\in R$ and $r_g=0$ for all but finite $g\in G$, and where any $\underline g$ is a symbol uniquely assigned to $g\in G$. The sum in $\RG $ is defined component-wise exploiting the addition in $R$: $$\left(\sum_{g\in G}r_g\underline g\right)+\left(\sum_{g\in G}s_g\underline g\right)=\sum_{g\in G}(r_g+s_g)\underline g\, .$$ In order to define a product in $\RG $, we need to specify two maps $$\sigma:G\to \Aut_{ring}(R)\ \ \text{ and }\ \ \tau:G\times G\to U(R) \, ,$$ where $\Aut_{ring}(R)$ is the group of ring automorphisms of $R$ and $U(R)$ is the group of units of $R$. Given $g\in G$ and $r\in R$ we denote the image of $r$ via the automorphism $\sigma(g)$ by $r^{\sigma(g)}$. We suppose also that $\sigma$ and $\tau$ satisfy the following conditions for all $r\in R$ and $g$, $g_1,$ $g_2$ and $g_3\in G$:
1. $\sigma(e)=1$ and $\tau(g,e)=\tau(e,g)=1$;
2. $\tau(g_1,g_2)\tau(g_1g_2,g_3)=\tau(g_2,g_3)^{\sigma(g_1)}\tau(g_1,g_2g_3)$;
3. $r^{\sigma(g_2)\sigma(g_1)}=\tau(g_1,g_2)r^{\sigma(g_1g_2)}\tau(g_1,g_2)^{-1}$.
The product in $\RG $ is defined by the rule $(r\underline g)(s\underline h)=rs^{\sigma(g)}\tau(g,h)\underline{gh}$, together with bilinearity, that is $$\left(\sum_{g\in G}r_g\underline g\right) \left(\sum_{g\in G}s_g\underline g\right)=\sum_{g\in G}\left(\sum_{h_1h_2=g}r_{h_1}s_{h_2}^{\sigma(h_1)}\tau(h_1,h_2)\right)\underline g\, .$$ $\RG $ is an associative and unitary ring. Of course the definition of $\RG $ does not depend only on $R$ and $G$, but also on $\sigma$ and $\tau$. Anyway we prefer the compact (though imprecise) notation $\RG$ to something like $R[G,\rho,\sigma]$. The easiest example of crossed group ring is the group ring $R[G]$, which corresponds to trivial maps $\sigma$ and $\tau$. For more details on this kind of construction we refer to [@Passman].
\[crossed\_action\_1\] Let $R$ be a ring, let $M_{R}$ and $N_{R}$ be right $R$-modules and let $\phi:M\to N$ be a homomorphism of right $R$-modules. Then, $(\L(M),\leq)$ and $(\L(N),\leq)$ are qframes and the map $$\Phi:\L(M)\to \L(N)\ \text{ such that }\ \Phi(K)=\phi(K)$$ is a homomorphism of qframes. Furthermore, if $\phi$ is surjective, then $\Phi$ is surjective and algebraic, and, in this case, $\Phi$ is injective if and only if $\phi$ is injective.
It is well-known that the family of submodules with the usual ordering is a qframe (the maximum of $\L(M)$ is $M$, while its minimum is $0$, furthermore, join and meet are given by sum and intersection respectively). Furthermore, it is easy to verify that $\Phi$ is a semi-lattice homomorphism which commutes with arbitrary joins. To show that $\Phi$ sends segments to segments, let $K_1\leq K_2\in \L(M)$ and consider $K\in [\Phi(K_1),\Phi(K_2)]$. Then, $$\begin{aligned}
K&=\Phi(\phi^{-1}K)=\Phi(\phi^{-1}K\cap \phi^{-1}\phi (K_2))\\
&=\Phi(\phi^{-1}K\cap (K_2+{\mathrm{Ker}}(\phi)))=\Phi((\phi^{-1}K\cap K_2)+{\mathrm{Ker}}(\phi))\\
&=\Phi(\phi^{-1}K\cap K_2)+\Phi({\mathrm{Ker}}(\phi))=\Phi(\phi^{-1}K\cap K_2)\,,\end{aligned}$$ where in the first line we used that $K$ is contained in the image of $\phi$, while in the second line we used the modularity of $\L(M)$. Since $\phi^{-1}(K)\cap K_2\in [K_1,K_2]$ we proved that $\Phi$ sends segments to segments, thus it is a morphism of qframes.
Suppose now that $\phi$ is surjective. Then, $\Phi$ is surjective as $\Phi(1)=\phi(M)=N$, which is the maximum of $\L(N)$. To show that $\Phi$ is algebraic, notice that ${\mathrm{Ker}}(\Phi)={\mathrm{Ker}}(\phi)$ and that, given $K_1,\, K_2\in [{\mathrm{Ker}}(\phi),1]$ such that $\Phi(K_1)=\Phi(K_2)$, we get $$K_1=K_1+{\mathrm{Ker}}(\phi)=\phi^{-1}(\phi(K_1))=\phi^{-1}(\phi(K_2))=K_2+{\mathrm{Ker}}(\phi)=K_2\,.$$ Finally, notice that $\phi$ is injective if and only if ${\mathrm{Ker}}(\phi)={\mathrm{Ker}}(\Phi)=0$, which happen, by the algebraicity of $\Phi$, if and only if $\Phi$ is injective.
Given a crossed product $\RG$, there is a canonical injective ring homomorphism $R\to \RG$ such that $r\mapsto r\underline e$, which allows to identify $R$ with a subring of $\RG$ and to consider $\RG$-modules also as $R$-modules in a natural way. On the other hand, there is no natural map $G\to \RG $ which is compatible with the operations, anyway the obvious assignment $g\mapsto \underline g$ respects the operations “modulo some units of $R$". This is enough to obtain a canonical right action of $G$ on $\L_R(M)$, for any right $R$-module $M$:
\[crossed\_action\_2\] Let $R$ be a ring, let $G$ be a group, fix a crossed product $R\asterisk G$, let $M_{\RG}$ be a right $\RG$-module and let $\phi:M_{\RG}\to M_{\RG}$ be an endomorphism of right $\RG$-modules. Letting $\L_R(M)$ denote the qframe of $R$-submodules of $M$, the following map $$\rho:G\to \Aut(\L_R(M))\ \ \ \rho\mapsto \rho_g:\L_R(M)\to \L_R(M)\,,$$ where $\rho_g(K)=K\underline g$, for all $g\in G$ and $K\in \L_R(M)$ is a group anti-homomorphism. Furthermore, the endomorphism of qframes $$\Phi:\L_R(M)\to \L_R(M)\ \text{ such that }\ \Phi(K)=\phi(K)$$ is $G$-equivariant.
Let $N\in \L_R(M)$, $r\in R$ and $g\in G$. Then, $\rho_g(N)r=N\underline{g}r=Nr^{\sigma(g)}\underline g\subseteq N\underline g$ and so $\rho_g(N)\in \L_R(M)$. Let now $\{N_i:i\in I\}$ a family of elements in $\L_R(M)$, then $$\rho_g\left(\sum_{i\in I}N_i\right)=\left(\sum_{i\in I}N_i\right)\overline g=\sum_{i\in I}(N_i\overline g)=\sum_{i\in I}\rho_g(N_i)$$ so $\rho_g$ is a semi-lattice homomorphism which commutes with arbitrary joins. Furthermore, given $g$, $h\in G$ and $N\in \L_R(M)$, $$\rho_g(\rho_h(N))=\rho_g(N\underline h)=N\underline h\underline g=N\tau(h,g)\underline{ h g}=N\underline{hg}=\rho_{hg}(N)\,,$$ where the fourth equality holds since $\tau(h,g)\in U(R)$. In particular, $\rho_g\rho_{g^{-1}}=\rho_{g^-1}\rho_g=\id_{\L_R(M)}$ so, given a segment $[N_1,N_2]$ in $\L_R(M)$ and $N\in [\rho_g(N_1),\rho_g(N_2)]$, then $N=\rho_{g}(\rho_{g^{-1}}N)$ and $\rho_{g^{-1}}N\in [N_1,N_2]$. Thus we proved that each $\rho_g$ is a homomorphism of qframes and that $\rho$ is a group anti-homomorphism.
Finally, let us show that $\rho_g\Phi=\Phi\rho_g$. Indeed, given $N\in\L_R(M)$, $$\rho_g\Phi(N)=\phi(N)\underline g=\phi(N\underline g)=\Phi(\rho_g(N))\,,$$ where the third equality holds since $\phi$ is a homomorphism of left $\RG$-modules.
\[Sofic\_Kap\_conj\] Let $R$ be a ring, let $G$ be a sofic group, fix a crossed product $R\asterisk G$, let $N_R$ be a finitely generated right $R$-module and let $M=N\otimes \RG$ . Then,
1. if $N_R$ is Noetherian, then any surjective $\RG$-linear endomorphism of $M$ is injective;
2. if $N_R$ has Krull dimension, then $\End_{R\asterisk G}(M)$ is stably finite.
The proof is an application of Theorem \[main\_higher\] and consists in translating the statement in a problem about qframes using the above lemmas.
Suppose first (1) and let $\phi:M\to M$ be a surjective endomorphism of right $\RG$-modules. Consider the qframe $\L_R(M)$ of all the right $R$-submodules of $M$ (which is described in Lemma \[crossed\_action\_1\]), with the right $G$-action described in Lemma \[crossed\_action\_2\]. By the same lemma, $\phi$ induces a $G$-equivariant surjective algebraic homomorphism of qframes $\Phi:\L_R(M)\to \L_R(M)$.\
Let $y=N\otimes \underline e\in \L_R(M)$, and notice that conditions (a$_*$) and (b$_*$) in Theorem \[main\_higher\] are verified for this choice of $y$. Thus, by the theorem, $\Phi$ is injective and this is equivalent to say that $\phi$ is injective by Lemma \[crossed\_action\_1\].
The proof of part (2) is analogous.
The above theorem can be used to verify the conjecture for classes of groups that are not known to be sofic. Remember that, given two classes ${\mathcal C}_1$ and ${\mathcal C}_2$ of groups, a group is said to be ${\mathcal C}_1$-by-${\mathcal C}_2$ provided there exists a short exact sequence $$1\to C_1\to G\to C_2\to 1$$ with $C_1\in {\mathcal C}_1$ and $C_2\in {\mathcal C}_2$.
\[coro\_poly\] Let $R$ be a right Noetherian ring an let $G$ be a (finite-by-polycyclic)-by-sofic group. Then any crossed product $\RG$ is stably finite.
Consider a short exact sequence $1\to C_1\to G\to C_2\to 1$ with $C_1$ a finite-by-polycyclic group and $C_2$ sofic. Then, $\RG\cong (R\asterisk C_1)\asterisk C_2$ for suitable choices of the crossed products on the right. It is well-known (see, for example, [@Bell Proposition 2.5]) that $R\asterisk C_1$ is a right Noetherian ring, thus $(R\asterisk C_1)\asterisk C_2$ is stably finite by Theorem \[Sofic\_Kap\_conj\].
\[coro\_free\] Let $R$ be a division ring and let $G$ be a free-by-sofic group. Then $R[G]$ is stably finite.
Consider a short exact sequence $1\to C_1\to G\to C_2\to 1$ with $C_1$ a free group and $C_2$ sofic. It is known (see [@Cohn Theorem 5.3.9]) that $R[C_1]$ embeds in a division ring $D$ and so, $R[G]\cong R[C_1]\asterisk C_2$ embeds in $D\asterisk C_2$. One concludes applying Theorem \[Sofic\_Kap\_conj\].
In the same line of the above corollaries, Federico Berlai [@federico] will use Theorem \[Sofic\_Kap\_conj\] to provide examples of groups that are not known to be sofic but that satisfy the Stable Finiteness Conjecture.
Let us conclude this subsection with an open question:
\[quest\] Let $D$ be a division ring and let $G$ be a group. Is it possible to find a sofic group $H$ and a suitable crossed product such that $D\asterisk H\cong D[G]$? If this is not possible, can we choose $D\asterisk H$ (with $H$ sofic) to be Morita equivalent to $D[G]$?
Given a stably finite ring $R$, any subring of $\Mat_n(R)$, for any $n\in{\mathbb N}_+$, is stably finite, thus a positive answer to the above question would solve the Kaplansky Stable Finiteness Conjecture for all groups.
L-Surjunctivity {#surj_sec}
---------------
Let $G$ be a group and let $A$ be a set. The set of [*configurations*]{} over $G$ in the alphabet $A$ is the cartesian product $A^G = \{x: G \to A\}$. The left action of $G$ on $A^G$ defined by $$gx(h) = x(g^{-1}h) \text{ for all $g, h \in G$ and $x \in A^G$}\,,$$ is called the (left) $G$-shift on $A^G$. Given a configuration $x \in A^G$ and a subset $F\subseteq G$, the element $x\restriction_{F} \in A^{F}$ defined by $x\restriction_{F}(g) = x(g)$ for all $g\in F$ is called the [*restriction*]{} of $x$ to $F$. For any subset $G'\subseteq G$, we let $\pi_{G'}:A^G\to A^{G'}$ be the map such that $\pi(x)=x\restriction_{G'}$, for all $x\in A^G$.
A [*cellular automaton*]{} over the group $G$ and the alphabet $A$ is a map $\phi : A^G \to A^G$ satisfying the following condition: there exist a finite subset $F \subseteq G$ and a map $\alpha: A^F \to A$ such that $$\phi (x)(g) = \alpha((g^{-1}x)\restriction_F)$$ for all $x \in A^G$ and $g \in G$. In this case, $F$ is a [*memory set*]{} of $\phi$ and $\alpha$ is the [*local defining map*]{} for $\phi$ associated with $F$.
Let $R$ be a ring, let $N$ be a left $R$-module, let $G$ be a group and consider a cellular automaton $\phi:N^G\to N^G$. We say that $\phi$ is a [*linear cellular automaton*]{} if there is a memory set $F$ and a local defining map $\alpha:N^F\to N$ that is a homomorphism of left $R$-modules.
The following lemma is a particular case of [@Ceccherini_libro Theorem 1.9.1].
Let $R$ be a ring, let $N$ be a left $R$-module, let $G$ be a group and consider a map $\phi:N^G\to N^G$. Endow $N^G$ with the product of the discrete topologies on each copy of $N$. The following are equivalent:
1. $\phi$ is a linear cellular automaton;
2. $\phi$ is a continuous $G$-equivariant homomorphism.
In this subsection we use the general results we proved for qframes to deduce a surjunctivity theorem for a suitable family of linear cellular automaton. Let us start defining a natural qframe associated with strictly linearly compact modules (see Definition \[def\_SLC\]).
Let $R$ be a discrete ring and let $M$ be a linearly topologized left $R$-module. We let $(\mathcal N(M),\leq )$ be the poset of submodules of $M$, ordered by [*reverse*]{} inclusion.
\[lin\_top\_qframe\] Let $R$ be a discrete ring, let $M$ and $N$ be strictly linearly compact left $R$-modules and let $\phi:M\to N$ be a continuous homomorphism of left $R$-modules. Then, $\mathcal N(M)$ and $\mathcal N(N)$ are qframes and the map $$\Phi:\mathcal N(N)\to \mathcal N(M) \ \text{ such that }\ \Phi(C)=\phi^{-1}(C)$$ is a homomorphism of qframes. Furthermore, if $\phi$ is injective then $\Phi$ is surjective and algebraic, and, under these hypotheses, $\Phi$ is injective if and only if $\phi$ is surjective.
It is easy to check that $\mathcal N(M)$ and $\mathcal N(N)$ are complete lattices (in fact, the maximum of $\mathcal N(M)$ is $0$, while its minimum is $M$; furthermore the meet of two closed submodules is the closure of their sum, while the join of a family (finite or infinite) of closed submodules is their intersection). To show that $\mathcal N(M)$ is modular take $A,\, B,\, C\in \mathcal N(M)$ such that $A\leq C$ (that is, $C\subseteq A$). Using, the modularity of the lattice of all submodules $\L(M)$ of $M$ with the usual order, one gets $C+(B\cap A)=(C+B)\cap A$, thus $$\overline{C+(B\cap A)}=\overline{(C+B)\cap A}=\overline{(C+B)}\cap A\,,$$ which is the modular law in $\mathcal N(M)$. The fact that $\mathcal N(M)$ and $\mathcal N(N)$ are upper continuous is proved for example in [@Warner Theorem 28.20].
The map $\Phi$ is well-defined by the continuity of $\phi$, that ensures that $\phi^{-1}(C)\in\mathcal N(M)$, for all $C\in \mathcal N(N)$. Since $\phi^{-1}$ commutes with arbitrary intersections, $\Phi$ commutes with arbitrary joins. Let now $C_1\leq C_2\in \mathcal N(N)$ and let us show that $\Phi([C_1,C_2])=[\Phi(C_1),\Phi(C_2)]$. Indeed, given $C\in [\Phi(C_1),\Phi(C_2)]$, $\phi^{-1}(C_2)\subseteq C\subseteq \phi^{-1}(C_1)$, so that $C_2\cap \phi(M)\subseteq \phi(C)\subseteq C_1\cap \phi(M)$. Thus, $$\begin{aligned}
C&=\Phi(\phi(C))=\Phi(\overline{\phi(C)+(C_2\cap \phi(M))})\\
&=\Phi(\overline{(\phi(C)+C_2)}\cap \phi(M))=\Phi\overline{(\phi(C)+C_2)}\,,\end{aligned}$$ where in the first line we used that $C$ contains the kernel of $\phi$, while in the second line we applied the modular law. Since $\overline{\phi(C)+C_2}\in [C_1,C_2]$, $\Phi$ sends segments to segments and so it is a morphism of qframes.
Suppose now that $\phi$ is injective. To show that $\Phi$ is surjective notice that, by the injectivity of $\phi$, $\Phi([0,1])=[0,\Phi(1)]=[0,{\mathrm{Ker}}(\phi)]=\mathcal N(M)$. It remains to show that $\Phi$ is algebraic: it is enough to notice that ${\mathrm{Ker}}(\Phi)=\phi(M)$ and that, given $C_1,\, C_2\in [\phi(M),1]$ such that $\Phi(C_1)=\Phi(C_2)$, then $$C_1=C_1\cap \phi(M)=\phi(\phi^{-1}(C_1))=\phi(\phi^{-1}(C_2))=C_2\cap \phi(M)=C_2\,.$$
Finally, since $\Phi$ is algebraic, $\Phi$ is injective if and only if ${\mathrm{Ker}}(\Phi)=0$, that is, $\phi(M)=M$, which is equivalent to say that $\phi$ is surjective.
\[main\_automata\] Let $R$ be a ring, let $G$ be a sofic group and let $_RN$ be an Artinian left $R$-module. Then, any linear cellular automaton $\phi:N^G\to N^G$ is surjunctive.
Suppose that $\phi:N^G\to N^G$ is an injective linear cellular automaton and let us prove that it is surjective.
By Lemmas \[ex\_art\] and \[fact\_lc\] (2), $N^G$ is strictly linearly compact so, by Lemma \[lin\_top\_qframe\], $\mathcal N(N^G)$ is a qframe. Furthermore, the map $$\rho:G\to \Aut(\mathcal N(N^G))\ \ \ \rho(g)=\rho_g\,,$$ such that $\rho_g(K)=\lambda_g^{-1}(K)$, for all $K\in\mathcal N(N^G)$ and $g\in G$, is a right action and the map $$\Phi:\mathcal N(N^G)\to \mathcal N(N^G)\ \ \ \Phi(K)=\phi^{-1}(K)\,,$$ for all $K\in \mathcal N(N^G)$, is a $G$-equivariant surjective algebraic homomorphism of qframes.
Let $y=\pi_e^{-1}(\{0\})$, where $\pi_e:N^G\to N^{e}$ is the usual projection, notice that $[0,y]\cong \mathcal N(N)$ is a Noetherian lattice and let $y_g=\rho_g(y)$, for all $g\in G$. It is clear that $\{y_g:g\in G\}$ is a basis for $\mathcal N(N^G)$.
By the above discussion, hypotheses (a$_*$) and (b$_*$) of Theorem \[main\_higher\] are satisfied and so $\Phi$ is injective. By Lemma \[lin\_top\_qframe\], $\phi$ is surjective.
Let $D$ be a division ring, the $V$ be a finite dimensional left vector space over $D$, let $G$ be a group and let $\phi:V^G\to V^G$ be a linear cellular automaton. If $G$ is either (finite-by-polycyclic)-by-sofic or free-by-sofic, then $\phi$ is surjunctive.
The result follows by the duality between strictly linearly compact and discrete vector spaces (see the Duality Theorem in the Appendix), Corollary \[coro\_free\] and Corollary \[coro\_poly\].
Let us remark that, by the same argument of the above corollary, a positive solution to Question \[quest\] would solve also the L-Surjunctivity Conjecture.
Topological modules and duality {#Sec_dual}
===============================
Let $R$ be a topopological ring (addition and multiplication are continuous functions $R\times R\to R$), let $M$ be a left $R$-module and let $\tau$ be a topology on $M$. The pair $(M,\tau)$ is said to be a [*topological module*]{} if it is a topological group and the scalar multiplication $R\times M\to M$ is a continuous map.
Let $G$ be a group, let $R$ be a topological ring, let $N$ be a topological left $R$-module and consider a cellular automaton $\phi:N^G\to N^G$. We say that $\phi$ is a [*linear cellular automaton*]{} if there is a memory set $F$ and a local defining map $\alpha:N^F\to N$ that is a continuous homomorphism of left $R$-modules (where $N^F$ is endowed with the product topology).
\[top\_def\_aut\] Let $G$ be a group, let $R$ be a topological ring, let $N$ be a topological left $R$-module and consider a map $\phi:N^G\to N^G$. Endow $N^G$ with the product topology and consider the following statements:
1. $\phi$ is a linear cellular automaton;
2. $\phi$ is a continuous and $G$-equivariant homomorphism.
Then, (1)$\Rightarrow$(2). If $N$ is discrete, then also (2)$\Rightarrow$(1).
(1)$\Rightarrow$(2). Let $F\subseteq G$ be a memory set and let $\alpha:N^F\to N$ be the associated local defining map. For any subset $G'\subseteq G$ we let $\pi:N^G\to N^{G'}$ be the canonical projection $\pi(x)=x\restriction_{G'}$. Recall that a typical basic neighborhood of $0$ for the product topology on $N^{G}$ is of the form $\pi_{G'}^{-1}(A)$ for some finite subset $G'\subseteq G$ and some open neighborhood $A$ of $0$ in $N^{G'}$.
For any open neighborhood $A$ of $0$ in $N$, $\phi^{-1}(\pi_{\{g\}}^{-1}(A))=\pi_{gF}^{-1}(\alpha^{-1}(A))$ is an open neighborhood in $N^G$. This is enough to show that $\phi$ is continuous since $\{\pi_{\{g\}}^{-1}(A):g\in G\}$ is a prebase of the topology. It is not difficult to show that $\phi$ is $G$-equivariant.
(2)$\Rightarrow$(1). When $N$ is discrete this follows as in [@Ceccherini_libro Theorem 1.9.1]
(Strictly) Linearly compact modules
-----------------------------------
From now on, we fix a discrete ring $R$ (i.e., a topological ring endowed with the discrete topology). A topological left $R$-module $(M,\tau)$ is [*linearly topologized*]{} if the filter of neighborhoods of $0$ has a filter base consisting of open submodules, that is, there exists a family of open submodules $\mathcal B=\{B_i\}_{i\in I}$ such that a subset $S\subseteq M$ is a neighborhood of $0$ if and only if it contains $B_i$, for some $i\in I$. We call $\mathcal B$ a [*linear base*]{} of $\tau$. In this setting, $(M,\tau)$ is a Hausdorff space if and only if $0$ is a closed point, that is, $\bigcap_{i\in I} B_i=\{0\}$.
Let $(M_1,\tau_1)$ and $(M_2,\tau_2)$ be linearly topologized left $R$-modules and let $\phi:M_1\to M_2$ be a homomorphism of left $R$-modules. Then, $\phi$ is continuous if and only if $\phi^{-1}(B)$ is an open neighborhood of $M_1$, for any open neighborhood $B$, belonging to a fixed base for the filter of neighborhoods of $0$ in $M_2$. We denote by $\Chom_R(M_1,M_2)$ the group of continuous homomorphisms from $M_1$ to $M_2$.
\[def\_SLC\] Let $R$ be a ring. We denote by $\LT R$ the category of linearly topologized Hausdorff left $R$-modules and continuous homomorphisms.
Let $(M,\tau)\in \LT R$. A [*(open, closed) linear variety*]{} is a subset of $M$ of the form $x+N$ where $N$ is a (open, closed) submodule. Given a set $I$ and a family $\F=\{x_i+N_i:i\in I\}$ of linear varieties, $\F$ has the [*finite intersection property*]{} (f.i.p.) if $\bigcap_{i\in J}x_i+N_i\neq \emptyset$, for any finite subset $J\subseteq I$.
Let $(M,\tau)\in \LT R$. Then,
1. $(M,\tau)$ is [*linearly compact*]{} if any family $\F$ of open linear varieties with the f.i.p. has non-empty intersection;
2. $(M,\tau)$ is [*strictly linearly compact*]{} if it is linearly compact and, given any $(M',\tau')\in \LT R$ and any surjective continuous homomorphism $\phi:M\to M'$, $\phi$ is open.
In the following example we work out the above definition in the discrete case.
\[ex\_art\] Let $(M,\tau)$ be a discrete module. It is not difficult to show that if $M$ is Artinian, then it is discrete if and only if it is Hausdorff. Furthermore, if $\tau$ is the discrete topology, then $M$ is Artinian if and only if $M$ is strictly linearly compact.
The proof of the following properties can be found in [@Warner Chapter VII].
\[fact\_lc\] Let $R$ be a ring and let $(M,\tau)\in \LT R$.
1. $M$ is (strictly) linearly compact if and only if both $N$ and $M/N$ are (strictly) linearly compact (with respect to the induced topologies), for any closed $N\leq M$.
2. If $M$ is the product of a family $\{(N_i,\tau_i):i\in I\}$, then $M$ is (strictly) linearly compact if and only if $N_i$ if (strictly) linearly compact for all $i\in I$;
3. $M$ is (strictly) linearly compact if and only if $M$ is complete and $M/B_i$ is (strictly) linearly compact discrete, where $\mathcal B=\{B_i:i\in I\}$ is a linear base for $M$.
If $R$ is a field, by part (3) of the above proposition, a linearly topologized Hausdorff $R$-vector space is linearly compact if and only if it is strictly linearly compact, if and only if it is complete and it has a base of neighborhoods made of vector subspaces of finite codimension.
We will need also the following fact, which can be found again in [@Warner Chapter VII]:
\[closed\] Let $(M_1,\tau_1)$ and $(M_2,\tau_2)\in \LT R$. If $M_1$ is (strictly) linearly compact and is a continuous morphism, then $\phi(M_1)$ is (strictly) linearly compact.
Let $R$ be a ring. We denote by $\SLC R$ the full subcategory of $\LT R$ whose objects are the strictly linearly compact modules.
Duality
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We start fixing the setting that we will maintain all along this section.
1. $R$ is a ring that is linearly compact as a left $R$-module endowed with the discrete topology;
2. ${}_RK$ is a minimal injective cogenerator, that is, ${}_RK$ is the injective envelope of the direct sum of a family of representatives of the simple left $R$-modules. We assume ${}_RK$ is Artinian;
3. we denote by $A$ the endomorphism ring of ${}_RK$.
The above setting for duality happens, for example, when $R$ is a (skew) field or a commutative local complete Noetherian ring (see [@Macdonald]).
We define two contravariant functors: $$\label{funct}
\Chom_R(-,K)=(-)^*:\SLC R\to \mod A\,,\ \ \ {\mathrm{Hom}}_A(-,K)=(-)^*:\mod A\to \SLC R$$ where, given a left $A$ module $N$, the right $R$-module $N^*={\mathrm{Hom}}_A(A,K)$ is endowed with the [*finite topology*]{}, that is, we take the following submodules as basic neighborhoods of $0$: $$\mathcal V(F)=\{f\in N^*: f(x)=0\,,\ \forall x\in F\}\ \ \ \text{ for a finite subset $F\subseteq N$}\,.$$ The following result can be deduced by the main results of [@Muller2] and [@Muller].
[**Duality Theorem.**]{} [*Let $R$ be a ring, let ${}_RK$ be a minimal injective cogenerator and let $A=\End_R(K)$. Suppose that $R$ is linearly compact discrete and that ${}_RK$ is Artinian. Then, the above functors define a duality between $\mod A$ and $\SLC R$.*]{} The above theorem is a particular case of the results discussed in [@Muller], that was also generalized by many authors (see for example the bibliography of [@Menini_Orsatti]). The particular statement above is enough for our needs and it allows us not to define “canonical choices" of topologies.
The above Duality Theorem can be used to recover Sections 4 and 5 in [@Elek_dual]. In particular, the weak exactness of the duality functors described in [@Elek_dual Section 5] can be improved to real exactness.
Applications {#applications}
------------
We state the following definition for a general category ${\mathcal C}$ but, in what follows, ${\mathcal C}$ will always be $\mod A$ or $\SLC R$ for some rings $R$ and $A$.
Let $\mathcal C$ be a category and let $G$ be a group. A [*left*]{} (resp., [*right*]{}) [*representation*]{} of $G$ on $\mathcal C$ is a (anti)homomorphism $G\to \Aut_{\mathcal C}(C)$ for some object $C\in {\mathcal C}$. A [*homomorphism*]{} $\phi:\mu_1\to \mu_2$ between two left (resp., right) representations $\mu_1:G\to \Aut_{{\mathcal C}}(C_1)$ and $\mu_2:G\to \Aut_{{\mathcal C}}(C_2)$ is a $G$-equivariant morphism $\phi:C_1\to C_2$ in ${\mathcal C}$. We denote by $\LRep(G,{\mathcal C})$ and $\RRep(G,{\mathcal C})$ respectively the categories of left and right representations of $G$ on ${\mathcal C}$.
It is a classical observation that $\RRep(G,\mod A)$ is canonically isomorphic to $\mod {A[G]}$. Notice also that, by Lemma \[top\_def\_aut\] and Proposition \[fact\_lc\], a linear cellular automaton whose alphabet is a discrete Artinian left $R$-module is a morphism in $\LRep(G,\SLC R)$.
Let $N\in \LT R$ and endow $N^G$ with the product topology. A [*subshift*]{} of $N^G$ is a closed $G$-invariant submodule.
Let $G$ be a group and let $R$ be a ring. Let $\lambda_1:G\to \Aut_{\SLC R}(N_1)$ and $\lambda_2:G\to \Aut_{\SLC R}(N_2)$ be two left representations of $G$ on strictly linearly compact left $R$-modules. Given a morphism of representations $\phi:N_1\to N_2$, the image $\phi(N_1)$ is closed and invariant under the action of $G$ on $N_2$.\
In particular, given $N\in \SLC R$ and a linear cellular automaton $\phi:N^G\to N^G$, the image of $\phi$ is a subshift.
Apply Lemma \[closed\].
The following corollary of the Duality Theorem provides a “bridge" between automata and homomorphisms of right $A[G]$-modules.
Let $G$ be a group and consider the setting described in (Dual.1, 2, 3). The duality described in the Duality Theorem induces a duality between $\mod {A[G]}$ and $\LRep(G,\SLC R)$.
It is enough to notice that a right action $\rho:G\to \Aut_{A}(M)$ of $G$ on a right $A$-module $M$ corresponds to a left action $\rho^*:G\to \Aut_{\SLC R}(M^*)$ of $G$ on the dual module $M^*\in \SLC R$ (just letting $\rho^*(g)=(\rho(g))^*$ for all $g\in G$) and that a left action $\lambda:G\to \Aut_{\SLC R}(N)$ on a strictly linearly compact left $R$-module $N$ (notice that $G$ acts via topological automorphisms) corresponds to a right action $\lambda^*:G\to \Aut_A(N^*)$ of $G$ on $N^*\in \mod A$. To conclude one applies the Duality Theorem.
Let $N\in \LT R$ and consider a subshift $X\subseteq N^G$. A $G$-equivariant continuous morphism $\phi:X\to X$ is [*reversible*]{} if there is a continuous $G$-equivariant morphism $\psi:X\to X$ such that $\psi\phi=\id_X$. The following corollary generalizes [@Ceccherini_libro Theorem 8.12.1]
\[reverse\] Let $G$ be a group, let $R$ be a ring and let $N\in \SLC R$. Let $X\subseteq N^G$ be a subshift. Then, any bijective $G$-equivariant continuous morphism $\phi:X\to X$ is reversible.\
Furthermore, in the setting described in (Dual.1, 2, 3) and letting $H=K^n$ for some positive integer $n$, any injective linear cellular automaton $\phi:H^G\to H^G$ is reversible.
By Proposition \[fact\_lc\], $X$ is strictly linearly compact, so $\phi$ is a topological automorphism and thus its inverse $\psi:X\to X$ is automatically a topological automorphism. The fact that $\psi$ is $G$-equivariant can be deduced from the fact that it is the inverse of a $G$-equivariant map.
For the second part, just notice that the dual of $H^G$ is the projective right $A[G]$-module $A[G]^n$ and so $H^G$ is an injective object in $\LRep(G,\SLC R)$.
The following corollary improves [@Ceccherini Theorem 1.3].
\[coro\_dual\_anti\] Let $G$ be a group, consider the setting described in (Dual.1, 2, 3), let $\lambda:G\to \Aut_{\SLC R}(N)\in \LRep(G,\SLC R)$ and let $M\in\mod {A[G]}$. There is an anti-isomorphism of rings $$\xymatrix@R=1pt{\End_{\LRep(G,\SLC R)}(N)\ar[r]& \End_{A[G]}(M)\\
\phi\ar@{|->}[r]&\phi^*\,.}$$ In particular, $\End_{\LRep(G,\SLC R)}((K^n)^G)$ is anti-isomorphic to $\Mat_n(A[G])$ for any positive integer $n$. Hence, $A[G]$ is stably finite if and only if any linear cellular automaton $\phi:(K^n)^G\to (K^n)^G$ is surjunctive, for any positive integer $n$.
The first statement is an easy consequence of duality, while the fact that $\End_{\LRep(G,\SLC R)}(K^n)$ is anti-isomorphic to $\Mat_k(A[G])$ follows noticing that the dual of $(K^n)^G$ is exactly $A[G]^n$ and that $\End_{A[G]}(A[G]^n)\cong \Mat_n(A[G])$. The last statement follows by the previous one recalling that linear cellular automata $(K^n)^G\to (K^n)^G$ are exactly the continuous $G$-equivariant endomorphisms of $(K^n)^G$ and using the second part of Corollary \[reverse\].
Simone Virili - [simone@mat.uab.cat]{}
Departament de Matemàtiques, Universitat Autònoma de Barcelona
Edifici C - 08193 Bellaterra (Barcelona), Spain.
[^1]: The author was partially supported by DGI MINECO MTM2011-28992-C02-01 and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. Part of this work was written during a research stay in the University of Vienna, where the author was partially supported by the ERC grant of Prof. Goulnara Arzhantseva “ANALYTIC" no. 259527.
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abstract: 'In [@lur1] Lurie published an expository article outlining a proof for a higher version of the cobordism hypothesis conjectured by Baez and Dolan in [@bd]. In this note we give a proof for the 1-dimensional case of this conjecture. The proof follows most of the outline given in [@lur1], but differs in a few crucial details. In particular, the proof makes use of the theory of quasi-unital $\infty$-categories as developed by the author in [@har].'
author:
- Yonatan Harpaz
title: The Cobordism Hypothesis in Dimension $1$
---
Introduction
==============
Let ${\mathcal{B}}^{\operatorname{or}}_1$ denote the $1$-dimensional oriented cobordism $\infty$-category, i.e. the symmetric monoidal $\infty$-category whose objects are oriented $0$-dimensional closed manifolds and whose morphisms are oriented $1$-dimensional cobordisms between them.
Let ${\mathcal{D}}$ be a symmetric monoidal $\infty$-category with duals. The $1$-dimensional cobordism hypothesis concerns the $\infty$-category $$\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$$ of symmetric monoidal functors $\vphi: {\mathcal{B}}^{\operatorname{or}}_1 \lrar {\mathcal{D}}$. If $X_+ \in {\mathcal{B}}^{\operatorname{or}}_1$ is the object corresponding to a point with positive orientation then the evaluation map $Z \mapsto Z(X_+)$ induces a functor $$\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}}) \lrar {\mathcal{D}}$$
It is not hard to show that since ${\mathcal{B}}^{\operatorname{or}}_1$ has duals the $\infty$-category $\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$ is in fact an $\infty$-groupoid, i.e. every natural transformation between two functors $F,G: {\mathcal{B}}^{\operatorname{or}}_1 \lrar {\mathcal{D}}$ is a natural equivalence. This means that the evaluation map $Z \mapsto Z(X_+)$ actually factors through a map $$\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}}) \lrar \wtl{{\mathcal{D}}}$$ where $\wtl{{\mathcal{D}}}$ is the maximal $\infty$-groupoid of ${\mathcal{D}}$. The cobordism hypothesis then states
\[cobordism-hypothesis\] The evaluation map $$\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}}) \lrar \wtl{{\mathcal{D}}}$$ is an equivalence of $\infty$-categories.
From the consideration above we see that we could have written the cobordism hypothesis as an equivalence $$\wtl{\operatorname{Fun}}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}}) \x{\simeq}{\lrar} \wtl{{\mathcal{D}}}$$ where $\wtl{\operatorname{Fun}}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$ is the maximal $\infty$-groupoid of $\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$ (which in this case happens to coincide with $\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$). This $\infty$-groupoid is the fundamental groupoid of the space of maps from ${\mathcal{B}}^{\operatorname{or}}_1$ to ${\mathcal{D}}$ in the $\infty$-category $\operatorname{Cat}^{\otimes}$ of symmetric monoidal $\infty$-categories.
In his paper [@lur1] Lurie gives an elaborate sketch of proof for a higher dimensional generalization of the $1$-dimensional cobordism hypothesis. For this one needs to generalize the notion of $\infty$-categories to $(\infty,n)$-categories. The strategy of proof described in [@lur1] is inductive in nature. In particular in order to understand the $n=1$ case, one should start by considering the $n=0$ case.
Let ${\mathcal{B}}^{\operatorname{un}}_0$ be the $0$-dimensional unoriented cobordism category, i.e. the objects of ${\mathcal{B}}^{\operatorname{un}}_0$ are $0$-dimensional closed manifolds (or equivalently, finite sets) and the morphisms are diffeomorphisms (or equivalently, isomorphisms of finite sets). Note that ${\mathcal{B}}^{\operatorname{un}}_0$ is a (discrete) $\infty$-groupoid.
Let $X \in {\mathcal{B}}^{\operatorname{un}}_0$ be the object corresponding to one point. Then the $0$-dimensional cobordism hypothesis states that ${\mathcal{B}}^{\operatorname{un}}_0$ is in fact the free $\infty$-groupoid (or $(\infty,0)$-category) on one object, i.e. if ${\mathcal{G}}$ is any other $\infty$-groupoid then the evaluation map $Z \mapsto Z(X)$ induces an equivalence of $\infty$-groupoids
$$\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{un}}_0,{\mathcal{G}}) \x{\simeq}{\lrar} {\mathcal{G}}$$
At this point one can wonder what is the justification for considering non-oriented manifolds in the $n=0$ case oriented ones in the $n=1$ case. As is explained in [@lur1] the desired notion when working in the $n$-dimensional cobordism $(\infty,n)$-category is that of **$n$-framed** manifolds. One then observes that $0$-framed $0$-manifolds are unoriented manifolds, while taking $1$-framed $1$-manifolds (and $1$-framed $0$-manifolds) is equivalent to taking the respective manifolds with orientation.
Now the $0$-dimensional cobordism hypothesis is not hard to verify. In fact, it holds in a slightly more general context - we do not have to assume that ${\mathcal{G}}$ is an $\infty$-groupoid. In fact, if ${\mathcal{G}}$ is **any symmetric monoidal $\infty$-category** then the evaluation map induces an equivalence of $\infty$-categories $$\operatorname{Fun}^{\otimes}({\mathcal{B}}^{\operatorname{un}}_0,{\mathcal{G}}) \x{\simeq}{\lrar} {\mathcal{G}}$$ and hence also an equivalence of $\infty$-groupoids $$\wtl{\operatorname{Fun}}^{\otimes}({\mathcal{B}}^{\operatorname{un}}_0,{\mathcal{G}}) \x{\simeq}{\lrar} \wtl{{\mathcal{G}}}$$
Now consider the under-category $\operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0/}$ of symmetric monoidal $\infty$-categories ${\mathcal{D}}$ equipped with a functor ${\mathcal{B}}^{\operatorname{un}}_0 \lrar {\mathcal{D}}$. Since ${\mathcal{B}}^{\operatorname{un}}_0$ is free on one generator this category can be identified with the $\infty$-category of **pointed** symmetric monoidal $\infty$-categories, i.e. symmetric monoidal $\infty$-categories with a chosen object. We will often not distinguish between these two notions.
Now the point of positive orientation $X_+ \in {\mathcal{B}}^{\operatorname{or}}_1$ determines a functor ${\mathcal{B}}^{\operatorname{un}}_0 \lrar {\mathcal{B}}^{\operatorname{or}}_1$, i.e. an object in $\operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0/}$, which we shall denote by ${\mathcal{B}}^+_1$. The $1$-dimensional coborodism hypothesis is then equivalent to the following statement:
\[0-to-1\]\[Cobordism Hypothesis $0$-to-$1$\] Let ${\mathcal{D}}\in \operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}$ be a pointed symmetric monoidal $\infty$-category with duals. Then the $\infty$-groupoid $$\wtl{\operatorname{Fun}}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}({\mathcal{B}}^+_1,{\mathcal{D}})$$ is **contractible**.
Theorem \[0-to-1\] can be considered as the inductive step from the $0$-dimensional cobordism hypothesis to the $1$-dimensional one. Now the strategy outlines in [@lur1] proceeds to bridge the gap between ${\mathcal{B}}^{\operatorname{un}}_0$ to ${\mathcal{B}}^{\operatorname{or}}_1$ by considering an intermediate $\infty$-category $${\mathcal{B}}^{\operatorname{un}}_0 \hrar {\mathcal{B}}^{\operatorname{ev}}_1 \hrar {\mathcal{B}}^{\operatorname{or}}_1$$ This intermediate $\infty$-category is defined in [@lur1] in terms of framed functions and index restriction. However in the $1$-dimensional case one can describe it without going into the theory of framed functors. In particular we will use the following definition:
Let $\iota: {\mathcal{B}}^{\operatorname{ev}}_1 \hrar {\mathcal{B}}^{\operatorname{or}}_1$ be the subcategory containing all objects and only the cobordisms $M$ in which every connected component $M_0 \subseteq M$ is either an identity segment or an evaluation segment.
Let us now describe how to bridge the gap between ${\mathcal{B}}^{\operatorname{un}}_0$ and ${\mathcal{B}}^{\operatorname{ev}}_1$. Let ${\mathcal{D}}$ be an $\infty$-category with duals and let $$\vphi:{\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$$ be a symmetric monoidal functor. We will say that $\vphi$ is **non-degenerate** if for each $X \in {\mathcal{B}}^{\operatorname{ev}}_1$ the map $$\vphi(\operatorname{ev}_X): \vphi(X) \otimes \vphi(\check{X}) \simeq \vphi(X \otimes \check{X}) \lrar \vphi(1) \simeq 1$$ is **non-degenerate**, i.e. identifies $\vphi(\check{X})$ with a dual of $\vphi(X)$. We will denote by $$\operatorname{Cat}^{\operatorname{nd}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /} \subseteq \operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}$$ the full subcategory spanned by objects $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ such that ${\mathcal{D}}$ has duals and $\vphi$ is non-degenerate.
Let $X_+ \in {\mathcal{B}}^{\operatorname{ev}}_1$ be the point with positive orientation. Then $X_+$ determines a functor $${\mathcal{B}}^{\operatorname{un}}_0 \lrar {\mathcal{B}}^{\operatorname{ev}}_1$$ The restriction map $\vphi \mapsto \vphi|_{{\mathcal{B}}^{\operatorname{un}}_0}$ then induces a functor $$\operatorname{Cat}^{\operatorname{nd}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /} \lrar \operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}$$
Now the gap between ${\mathcal{B}}^{\operatorname{ev}}_1$ and ${\mathcal{B}}^{\operatorname{un}}_0$ can be climbed using the following lemma (see [@lur1]):
\[0-to-1-ev\] The functor $$\operatorname{Cat}^{\operatorname{nd}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /} \lrar \operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}$$ is fully faithful.
First note that if $F:{\mathcal{D}}\lrar {\mathcal{D}}'$ is a symmetric monoidal functor where ${\mathcal{D}},{\mathcal{D}}'$ have duals and $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ is non-degenerate then $f \circ \vphi$ will be non-degenerate as well. Hence it will be enough to show that if ${\mathcal{D}}$ has duals then the restriction map induces an equivalence between the $\infty$-groupoid of non-degenerate symmetric monoidal functors $${\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$$ and the $\infty$-groupoid of symmetric monoidal functors $${\mathcal{B}}^{\operatorname{un}}_0 \lrar {\mathcal{D}}$$
Now specifying a non-degenerate functor $${\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$$ is equivalent to specifying a pair of objects $D_+,D_- \in {\mathcal{D}}$ (the images of $X_+,X_-$ respectively) and a non-degenerate morphism $$e: D_+ \otimes D_- \lrar 1$$ which is the image of $\operatorname{ev}_{X_+}$. Since ${\mathcal{D}}$ has duals the $\infty$-groupoid of triples $(D_+,D_-,e)$ in which $e$ is non-degenerate is equivalent to the $\infty$-groupoid of triples $(D_+,\check{D}_-,f)$ where $f: D_+ \lrar \check{D}_-$ is an equivalence. Hence the forgetful map $(D_+,D_-,e) \mapsto D_+$ is an equivalence.
Now consider the natural inclusion $\iota: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{B}}^{\operatorname{or}}_1$ as an object in $\operatorname{Cat}^{\operatorname{nd}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}$. Then by Lemma \[0-to-1-ev\] we see that the $1$-dimensional cobordism hypothesis will be established once we make the following last step:
\[cobordism-last-step\] Let ${\mathcal{D}}$ be a symmetric monoidal $\infty$-category with duals and let $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ be a **non-degenerate** functor. Then the $\infty$-groupoid $$\wtl{\operatorname{Fun}}^{\otimes}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$$ is contractible.
Note that since ${\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{B}}^{\operatorname{or}}_1$ is essentially surjective all the functors in $$\wtl{\operatorname{Fun}}^{\otimes}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$$ will have the same essential image of $\vphi$. Hence it will be enough to prove for the claim for the case where $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ is **essentially surjective**. We will denote by $$\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /} \subseteq \operatorname{Cat}^{\operatorname{nd}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}$$ the full subcategory spanned by essentially surjective functors $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$. Hence we can phrase Theorem \[cobordism-last-step\] as follows:
\[cobordism-last-step-2\] Let ${\mathcal{D}}$ be a symmetric monoidal $\infty$-category with duals and let $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ be an **essentially surjective non-degenerate** functor. Then the space of maps $$\operatorname{Map}_{\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}}(\iota,\vphi)$$ is contractible.
The purpose of this paper is to provide a formal proof for this last step. This paper is constructed as follows. In § \[s-qu-cobordism\] we prove a variant of Theorem \[cobordism-last-step-2\] which we call the quasi-unital cobordism hypothesis (Theorem \[qu-cobordism\]). Then in § \[s-from-qu-to-regular\] we explain how to deduce Theorem \[cobordism-last-step-2\] from Theorem \[qu-cobordism\]. Section § \[s-from-qu-to-regular\] relies on the notion of **quasi-unital $\infty$-categories** which is developed rigourously in [@har] (however § \[s-qu-cobordism\] is completely independent of [@har]).
The Quasi-Unital Cobordism Hypothesis {#s-qu-cobordism}
=======================================
Let $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ be a non-degenerate functor and let $\operatorname{Grp}_\infty$ denote the $\infty$-category of $\infty$-groupoids. We can define a lax symmetric functor $M_\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar \operatorname{Grp}_{\infty}$ by setting $$M_\vphi(X) = \operatorname{Map}_{{\mathcal{D}}}(1,\vphi(X))$$ We will refer to $M_\vphi$ as the **fiber functor** of $\vphi$. Now if ${\mathcal{D}}$ has duals and $\vphi$ is non-degenerate, then one can expect this to be reflected in $M_\vphi$ somehow. More precisely, we have the following notion:
Let $M: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar \operatorname{Grp}_{\infty}$ be a lax symmetric monoidal functor. An object $Z \in M(X \otimes \check{X})$ is called **non-degenerate** if for each object $Y \in {\mathcal{B}}^{\operatorname{ev}}_1$ the natural map $$M(Y \otimes \check{X}) \x{Id \times Z}{\lrar} M(Y \otimes \check{X}) \times M(X \otimes \check{X}) \lrar M(Y \otimes \check{X} \otimes X \otimes \check{X}) \x{M(Id \otimes \operatorname{ev}\otimes Id)}{\lrar} M(Y \otimes \check{X})$$ is an equivalence of $\infty$-groupoids.
\[uniqueness\] If a non-degenerate element $Z \in M(X \otimes \check{X})$ exists then it is unique up to a (non-canonical) equivalence.
\[unit\] Let $M: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar \operatorname{Grp}_{\infty}$ be a lax symmetric monoidal functor. The lax symmetric structure of $M$ includes a structure map $1_{\operatorname{Grp}_{\infty}} \lrar M(1)$ which can be described by choosing an object $Z_1 \in M(1)$. The axioms of lax monoidality then ensure that $Z_1$ is non-degenerate.
A lax symmetric monoidal functor $M: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar \operatorname{Grp}_{\infty}$ will be called **non-degenerate** if for each object $X \in {\mathcal{B}}^{\operatorname{ev}}_1$ there exists a non-degenerate object $Z \in M(X \otimes \check{X})$.
Let $M_1,M_2: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar \operatorname{Grp}_{\infty}$ be two non-degenerate lax symmetric monoidal functors. A lax symmetric natural transformation $T: M_1 \lrar M_2$ will be called **non-degenerate** if for each object $X \in \operatorname{Bord}^{\operatorname{ev}}$ and each non-degenerate object $Z \in M(X \otimes \check{X})$ the objects $T(Z) \in M_2(X \otimes \check{X})$ is non-degerate.
From remark \[uniqueness\] we see that if $T(Z) \in M_2(X \otimes \check{X})$ is non-degenerate for **at least one** non-degenerate $Z \in M_1(X \otimes \check{X})$ then it will be true for all non-degenerate $Z \in M_1(X \otimes \check{X})$.
Now we claim that if ${\mathcal{D}}$ has duals and $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ is non-degenerate then the fiber functor $M_\vphi$ will be non-degenerate: for each object $X \in {\mathcal{B}}^{\operatorname{ev}}_1$ there exists a coevaluation morphism $$\operatorname{coev}_{\vphi(X)}: 1 \lrar \vphi(X) \otimes \vphi(\check{X}) \simeq \vphi(X \otimes \check{X})$$ which determines an element in $Z_X \in M_\vphi(X \otimes \check{X})$. It is not hard to see that this element is non-degenerate.
Let $\operatorname{Fun}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty})$ denote the $\infty$-category of lax symmetric monoidal functors ${\mathcal{B}}^{\operatorname{ev}}_1 \lrar \operatorname{Grp}_{\infty}$ and by $$\operatorname{Fun}_{\operatorname{nd}}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty}) \subseteq \operatorname{Fun}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty})$$ the subcategory spanned by non-degenerate functors and non-degenerate natural transformations. Now the construction $\vphi \mapsto M_\vphi$ determines a functor
$$\operatorname{Cat}^{\operatorname{nd}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /} \lrar \operatorname{Fun}_{\operatorname{nd}}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty})$$ In particular if $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{C}}$ and $\psi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ are non-degenerate then any functor $T:{\mathcal{C}}\lrar {\mathcal{D}}$ under ${\mathcal{B}}^{\operatorname{ev}}_1$ will induce a non-degenerate natural transformation $$T_*: M_{\vphi} \lrar M_{\psi}$$
The rest of this section is devoted to proving the following result, which we call the “quasi-unital cobordism hypothesis”:
\[qu-cobordism\] Let ${\mathcal{D}}$ be a symmetric monoidal $\infty$-category with duals, let $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ be a non-degenerate functor and let $\iota: {\mathcal{B}}^{\operatorname{ev}}_1 \hrar {\mathcal{B}}^{\operatorname{or}}_1$ be the natural inclusion. Let $M_\iota,M_\vphi \in \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nd}}$ be the corresponding fiber functors. Them the space of maps $$\operatorname{Map}_{\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nd}}}(M_\iota, M_\vphi)$$ is contractible.
We start by transforming the lax symmetric monoidal functors $M_\iota,M_\vphi$ to **left fibrations** over ${\mathcal{B}}^{\operatorname{ev}}_1$ using the symmetric monoidal analogue of Grothendieck’s construction, as described in [@lur1], page $67-68$.
Let $M: {\mathcal{B}}\lrar \operatorname{Grp}_\infty$ be a lax symmetric monoidal functor. We can construct a symmetric monoidal $\infty$-category $\operatorname{Groth}({\mathcal{B}},M)$ as follows:
1. The objects of $\operatorname{Groth}({\mathcal{B}},M)$ are pairs $(X, \eta)$ where $X \in {\mathcal{B}}$ is an object and $\eta$ is an object of $M(X)$.
2. The space of maps from $(X,\eta)$ to $(X',\eta')$ in $\operatorname{Groth}({\mathcal{B}},M)$ is defined to be the classifying space of the $\infty$-groupoid of pairs $(f,\alp)$ where $f: X \lrar X'$ is a morphism in $B$ and $\alp: f_*\eta \lrar \eta$ is a morphism in $M(X')$. Composition is defined in a straightforward way.
3. The symmetric monoidal structure on $\operatorname{Groth}({\mathcal{B}},M)$ is obtained by defining $$(X,\eta) \otimes (X',\eta') = (X \otimes X',\beta_{X,Y}(\eta \otimes \eta'))$$ where $\beta_{X,Y}: M(X) \times M(Y) \lrar M(X \otimes Y)$ is given by the lax symmetric structure of $M$.
The forgetful functor $(X,\eta) \mapsto X$ induces a **left fibration** $$\operatorname{Groth}({\mathcal{B}},M) \lrar {\mathcal{B}}$$
\[unstraightening\] The association $M \mapsto \operatorname{Groth}({\mathcal{B}},M)$ induces an equivalence between the $\infty$-category of lax-symmetric monoidal functors ${\mathcal{B}}\lrar \operatorname{Grp}_\infty$ and the full subcategory of the over $\infty$-category $ \operatorname{Cat}^{\otimes}_{/{\mathcal{B}}} $ spanned by left fibrations.
This follows from the more general statement given in [@lur1] Proposition $3.3.26$. Note that any map of left fibrations over ${\mathcal{B}}$ is in particular a map of coCartesian fibrations because if $p: {\mathcal{C}}\lrar {\mathcal{B}}$ is a left fibration then any edge in ${\mathcal{C}}$ is $p$-coCartesian.
Note that if ${\mathcal{C}}\lrar {\mathcal{B}}$ is a left fibration of symmetric monoidal $\infty$-categories and ${\mathcal{A}}\lrar {\mathcal{B}}$ is a symmetric monoidal functor then the $\infty$-category $$\operatorname{Fun}^{\otimes}_{/ {\mathcal{B}}}({\mathcal{A}},{\mathcal{C}})$$ is actually an **$\infty$-groupoid**, and by Theorem \[unstraightening\] is equivalent to the $\infty$-groupoid of lax-monoidal natural transformations between the corresponding lax monoidal functors from ${\mathcal{B}}$ to $\operatorname{Grp}_\infty$.
Now set $${\mathcal{F}}_\iota \x{\operatorname{def}}{=} \operatorname{Groth}({\mathcal{B}}^{\operatorname{ev}}_1,M_{\iota})$$ $${\mathcal{F}}_\vphi \x{\operatorname{def}}{=} \operatorname{Groth}({\mathcal{B}}^{\operatorname{ev}}_1,M_{\vphi})$$
Let $$\operatorname{Fun}^{\operatorname{nd}}_{/{\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi}) \subseteq \operatorname{Fun}^{\otimes}_{/{\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi})$$ denote the full sub $\infty$-groupoid of functors which correspond to **non-degenerate** natural transformations $$M_\iota \lrar M_\vphi$$ under the Grothendieck construction. Note that $\operatorname{Fun}^{\operatorname{nd}}_{/{\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi})$ is a union of connected components of the $\infty$-groupoid $\operatorname{Fun}^{\otimes}_{/{\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi})$.
We now need to show that the $\infty$-groupoid $$\operatorname{Fun}^{\operatorname{nd}}_{/{\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi})$$ is contractible.
Unwinding the definitions we see that the objects of ${\mathcal{F}}_{\iota}$ are pairs $(X,M)$ where $X \in {\mathcal{B}}^{\operatorname{ev}}_1$ is a $0$-manifold and $M \in \operatorname{Map}_{{\mathcal{B}}^{\operatorname{or}}_1}(\emptyset,X)$ is a cobordism from $\emptyset$ to $X$. A morphism in $\vphi$ from $(X,M)$ to $(X',M')$ consists of a morphism in ${\mathcal{B}}^{\operatorname{ev}}_1$ $$N:X \lrar X'$$ and a diffeomorphism $$T:M \coprod_{X} N \cong M'$$ respecting $X'$. Note that for each $(X,M) \in {\mathcal{F}}_{\iota}$ we have an identification $X \simeq \partial M$. Further more the space of morphisms from $(\partial M,M)$ to $(\partial M',M')$ is **homotopy equivalent to the space of orientation-preserving $\pi_0$-surjective embeddings of $M$ in $M'$** (which are not required to respect the boundaries in any way).
Now in order to analyze the symmetric monoidal $\infty$-category ${\mathcal{F}}_\iota$ we are going to use the theory of **$\infty$-operads**, as developed in [@lur2]. Recall that the category $\operatorname{Cat}^{\otimes}$ of symmetric monoidal $\infty$-categories admits a forgetful functor $$\operatorname{Cat}^{\otimes} \lrar \operatorname{Op}^{\infty}$$ to the $\infty$-category of **$\infty$-operads**. This functor has a left adjoint $$\operatorname{Env}: \operatorname{Op}^{\infty} \lrar \operatorname{Cat}^{\otimes}$$ called the **monoidal envelope** functor (see [@lur2] §$2.2.4$). In particular, if $\mcal{C}^{\otimes}$ is an $\infty$-operad and ${\mathcal{D}}$ is a symmetric monoidal $\infty$-category with corresponding $\infty$-operad ${\mathcal{D}}^{\otimes} \lrar \operatorname{N}(\Gam_*)$ then there is an **equivalence of $\infty$-categories** $$\operatorname{Fun}^{\otimes}(\operatorname{Env}({\mathcal{C}}^{\otimes}),{\mathcal{D}}) \simeq \operatorname{Alg}_{\mcal{C}}({\mathcal{D}}^{\otimes})$$ Where $\operatorname{Alg}_{\mcal{C}}\left({\mathcal{D}}^{\otimes}\right) \subseteq \operatorname{Fun}_{/\operatorname{N}(\Gam_*)}(\mcal{C}^{\otimes},{\mathcal{D}}^{\otimes})$ denotes the full subcategory spanned by $\infty$-operad maps (see Proposition $2.2.4.9$ of [@lur2]).
Now observing the definition of monoidal envelop (see Remark $2.2.4.3$ in [@lur2]) we see that ${\mathcal{F}}_{\iota}$ is equivalent to the monoidal envelope of a certain simple $\infty$-operad $$F_\iota \simeq \operatorname{Env}\left({\mathcal{OF}}^{\otimes}\right)$$ which can be described as follows: the underlying $\infty$-category ${\mathcal{OF}}$ of ${\mathcal{OF}}^{\otimes}$ is the $\infty$-category of **connected** $1$-manifolds (i.e. either the segment or the circle) and the morphisms are **orientation-preserving embeddings** between them. The (active) $n$-to-$1$ operations of ${\mathcal{OF}}$ (for $n\geq 1$) from $(M_1,...,M_n)$ to $M$ are the orientation-preserving embeddings $$M_1 \coprod ... \coprod M_n \lrar M$$ and there are no $0$-to-$1$ operations.
Now observe that the induced map ${\mathcal{OF}}^{\otimes} \lrar ({\mathcal{B}}^{\operatorname{ev}}_1)^{\infty}$ is a fibration of $\infty$-operads. We claim that ${\mathcal{F}}_{\iota}$ is not only the enveloping symmetric monoidal $\infty$-category of ${\mathcal{OF}}^{\otimes}$, but that ${\mathcal{F}}_{\iota} \lrar {\mathcal{B}}^{\operatorname{ev}}_1$ is the enveloping **left fibration** of ${\mathcal{OF}}\lrar {\mathcal{B}}^{\operatorname{ev}}_1$. More precisely we claim that for any left fibration ${\mathcal{D}}\lrar {\mathcal{B}}^{\operatorname{ev}}_1$ of symmetric monoidal $\infty$-categories the natural map $$\operatorname{Fun}^{\otimes}_{/{\mathcal{B}}^{\operatorname{ev}}_1}\left(F_{\iota},{\mathcal{D}}\right) \lrar \operatorname{Alg}_{{\mathcal{OF}}/ {\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{D}}^{\otimes})$$ is an equivalence if $\infty$-groupoids (where both terms denote mapping objects in the respective **over-categories**). This is in fact not a special property of $F_{\iota}$:
\[left-envelope\] Let ${\mathcal{O}}$ be a symmetric monoidal $\infty$-category with corresponding $\infty$-operad ${\mathcal{O}}^{\otimes} \lrar \operatorname{N}(\Gam_*)$ and let $p:{\mathcal{C}}^{\otimes} \lrar {\mathcal{O}}^{\otimes}$ be a fibration of $\infty$-operads such that the induced map $$\ovl{p}:\operatorname{Env}\left({\mathcal{C}}^{\otimes}\right) \lrar {\mathcal{O}}$$ is a left fibration. Let ${\mathcal{D}}\lrar {\mathcal{O}}$ be some other left fibration of symmetric monoidal categories. Then the natural map $$\operatorname{Fun}^{\otimes}_{/{\mathcal{O}}}\left(\operatorname{Env}\left({\mathcal{C}}^{\otimes}\right),{\mathcal{D}}\right) \lrar \operatorname{Alg}_{{\mathcal{C}}/ {\mathcal{O}}}({\mathcal{D}}^{\otimes})$$ is an equivalence of $\infty$-categories. Further more both sides are in fact $\infty$-groupoids.
Consider the diagram $$\xymatrix{
\operatorname{Fun}^{\otimes}(\operatorname{Env}\left({\mathcal{C}}^{\otimes}\right),{\mathcal{D}}) \ar^{\simeq}[r]\ar[d] & \operatorname{Alg}_{{\mathcal{C}}}\left({\mathcal{D}}^{\otimes}\right) \ar[d] \\
\operatorname{Fun}^{\otimes}(\operatorname{Env}\left({\mathcal{C}}^{\otimes}\right),{\mathcal{O}}) \ar^{\simeq}[r] & \operatorname{Alg}_{{\mathcal{C}}}\left({\mathcal{O}}^{\otimes}\right) \\
}$$ Now the vertical maps are left fibrations and by adjunction the horizontal maps are equivalences. By [@lur3] Proposition $3.3.1.5$ we get that the induced map on the fibers of $p$ and $\ovl{p}$ respectively $$\operatorname{Fun}^{\otimes}_{/{\mathcal{O}}}\left(\operatorname{Env}\left({\mathcal{C}}^{\otimes}\right),{\mathcal{D}}\right) \lrar \operatorname{Alg}_{{\mathcal{C}}/ {\mathcal{O}}}({\mathcal{D}}^{\otimes})$$ is a weak equivalence of $\infty$-groupoids.
In [@lur2] a relative variant $\operatorname{Env}_{{\mathcal{B}}^{\operatorname{ev}}_1}$ of $\operatorname{Env}$ is introduced which sends a fibration of $\infty$-operads ${\mathcal{C}}^{\otimes} \lrar ({\mathcal{B}}^{\operatorname{ev}}_1)^{\otimes}$ to its enveloping coCartesin fibration $\operatorname{Env}_{{\mathcal{O}}}\left({\mathcal{C}}^{\otimes}\right) \lrar {\mathcal{B}}^{\operatorname{ev}}_1$. Note that in our case the map $${\mathcal{F}}_{\iota} \lrar {\mathcal{B}}^{\operatorname{ev}}_1$$ is **not** the enveloping coCartesian fibration of ${\mathcal{OF}}^{\otimes} \lrar ({\mathcal{B}}^{\operatorname{ev}}_1)^{\otimes}$. However from Lemma \[left-envelope\] it follows that the map $$\xymatrix{
{\mathcal{F}}_{\iota} \ar[rr]\ar[dr] && \operatorname{Env}_{{\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{OF}}^{\otimes}\right) \ar[dl] \\
& {\mathcal{B}}^{\operatorname{ev}}_1 & \\
}$$ is a **covariant equivalence** over ${\mathcal{B}}^{\operatorname{ev}}_1$, i.e. induces a weak equivalence of simplicial sets on the fibers (where the fibers on the left are $\infty$-groupoids and the fibers on the right are $\infty$-categories). This claim can also be verified directly by unwinding the definition of $\operatorname{Env}_{{\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{OF}}^{\otimes}\right)$.
Summing up the discussion so far we observe that we have a weak equivalence of $\infty$-groupoids $$\operatorname{Fun}^{\otimes}_{/{\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi}\right) \x{\simeq}{\lrar} \operatorname{Alg}_{{\mathcal{OF}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right)$$
Let $$\operatorname{Alg}^{\operatorname{nd}}_{{\mathcal{OF}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right) \subseteq \operatorname{Alg}_{{\mathcal{OF}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right)$$ denote the full sub $\infty$-groupoid corresponding to $$\operatorname{Fun}^{\operatorname{nd}}_{/{\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi}) \subseteq \operatorname{Fun}^{\otimes}_{/{\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\iota},{\mathcal{F}}_{\vphi})$$ under the adjunction. We are now reduced to prove that the $\infty$-groupoid $$\operatorname{Alg}^{\operatorname{nd}}_{{\mathcal{OF}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right)$$ is contractible.
Let ${\mathcal{OI}}^{\otimes} \subseteq {\mathcal{OF}}^{\otimes}$ be the full sub $\infty$-operad of ${\mathcal{OF}}^{\otimes}$ spanned by connected $1$-manifolds which are diffeomorphic to the segment (and all $n$-to-$1$ operations between them). In particular we see that ${\mathcal{OI}}^{\otimes}$ is equivalent to the **non-unital associative $\infty$-operad**.
We begin with the following theorem which reduces the handling of ${\mathcal{OF}}^{\otimes}$ to ${\mathcal{OI}}^{\otimes}$.
\[removing-circles\] Let $q:{\mathcal{C}}^{\otimes} \lrar {\mathcal{O}}^{\otimes}$ be a left fibration of $\infty$-operads. Then the restriction map $$\operatorname{Alg}_{{\mathcal{OF}}/ {\mathcal{O}}}({\mathcal{C}}^{\otimes}) \lrar \operatorname{Alg}_{{\mathcal{OI}}/ {\mathcal{O}}}({\mathcal{C}}^{\otimes})$$ is a weak equivalence.
We will base our claim on the following general lemma:
\[free-algebra\] Let ${\mathcal{A}}^{\otimes} \lrar {\mathcal{B}}^{\otimes}$ be a map of $\infty$-groupoids and let $q:{\mathcal{C}}^{\otimes} \lrar {\mathcal{O}}^{\otimes}$ be **left fibration** of $\infty$-operads. Suppose that for every object $B \in {\mathcal{B}}$, the category $${\mathcal{F}}_B = {\mathcal{A}}^{\otimes}_{\operatorname{act}} \times_{{\mathcal{B}}^{\otimes}_{\operatorname{act}}} {\mathcal{B}}^{\otimes}_{/B}$$ is weakly contractible (see [@lur2] for the terminology). Then the natural restriction map $$\operatorname{Alg}_{{\mathcal{A}}/ {\mathcal{O}}}({\mathcal{C}}^{\otimes}) \lrar \operatorname{Alg}_{{\mathcal{B}}/ {\mathcal{O}}}({\mathcal{C}}^{\otimes})$$ is a weak equivalence.
In [@lur2] §$3.1.3$ it is explained how under certain conditions the forgetful functor (i.e. restriction map) $$\operatorname{Alg}_{{\mathcal{A}}/ {\mathcal{O}}}({\mathcal{C}}^{\otimes}) \lrar \operatorname{Alg}_{{\mathcal{B}}/ {\mathcal{O}}}({\mathcal{C}}^{\otimes})$$ admits a left adjoint, called the **free algebra functor**. Since ${\mathcal{C}}^{\otimes} \lrar {\mathcal{O}}^{\otimes}$ is a left fibration both these $\infty$-categories are $\infty$-groupoids, and so any adjunction between them will be an equivalence. Hence it will suffice to show that the conditions for existence of left adjoint are satisfies in this case.
Since $q: {\mathcal{C}}^{\otimes} \lrar {\mathcal{O}}^{\otimes}$ is a left fibration $q$ is **compatible with colimits indexed by weakly contractible diagrams** in the sense of [@lur2] Definition $3.1.1.18$ (because weakly contractible colimits exists in every $\infty$-groupoid and are preserved by any functor between $\infty$-groupoids). Combining Corollary $3.1.3.4$ and Proposition $3.1.1.20$ of [@lur2] we see that the desired free algebra functor exists.
In view of Lemma \[free-algebra\] it will be enough to check that for every object $M \in {\mathcal{OF}}$ (i.e. every connected $1$-manifolds) the $\infty$-category $${\mathcal{F}}_M \x{\operatorname{def}}{=} {\mathcal{OI}}^{\otimes}_{\operatorname{act}} \times_{{\mathcal{OF}}^{\otimes}_{\operatorname{act}}} \left({\mathcal{OF}}^{\otimes}_{\operatorname{act}}\right)_{/M}$$ is weakly contractible.
Unwinding the definitions we see that the objects of ${\mathcal{F}}_M$ are tuples of $1$-manifolds $(M_1,...,M_n)$ ($n \geq 1$), such that each $M_i$ is diffeomorphic to a segment, together with an orientation preserving embedding $$f: M_1 \coprod ... \coprod M_n \hrar M$$ A morphisms in ${\mathcal{F}}_M$ from $$f: M_1 \coprod ... \coprod M_n \hrar M$$ to $$g: M_1' \coprod ... \coprod M_m' \hrar M$$ is a $\pi_0$-surjective orientation-preserving embedding $$T:M_1 \coprod ... \coprod M_n \lrar M_1' \coprod ... \coprod M_m'$$ together with an **isotopy** $g \circ T \sim f$.
Now when $M$ is the segment then ${\mathcal{F}}_M$ contains a terminal object and so is weakly contractible. Hence we only need to take care of the case of the circle $M=S^1$.
It is not hard to verify that the category $F_{S^1}$ is in fact discrete - the space of self isotopies of any embedding $f:M_1 \coprod ... \coprod M_n \hrar M $ is equivalent to the loop space of $S^1$ and hence discrete. In fact one can even describe $F_{S^1}$ in completely combinatorial terms. In order to do that we will need some terminology.
Let $\Lam_{\infty}$ be the category whose objects correspond to the natural numbers $1,2,3,...$ and the morphisms from $n$ to $m$ are (weak) order preserving maps $f: {\mathbb{Z}}\lrar {\mathbb{Z}}$ such that $f(x+n) = f(x)+m$.
The category $\Lam_{\infty}$ is a model for the the universal fibration over the cyclic category, i.e., there is a left fibration $\Lam_\infty \lrar \Lam$ (where $\Lam$ is connes’ cyclic category) such that the fibers are connected groupoids with a single object having automorphism group ${\mathbb{Z}}$ (or in other words circles). In particular the category $\Lam_{\infty}$ is known to be weakly contractible. See [@kal] for a detailed introduction and proof (Lemma $4.8$).
Let $\Lam^{\operatorname{sur}}_{\infty}$ be the sub category of $\Lam_\infty$ which contains all the objects and only **surjective** maps between. It is not hard to verify explicitly that the map $\Lam^{\operatorname{sur}}_\infty \lrar \Lam_\infty$ is cofinal and so $\Lam^{\operatorname{sur}}_{\infty}$ is contractible as well. Now we claim that $F_{S^1}$ is in fact equivalent to $\Lam^{\operatorname{sur}}_{\infty}$.
Let $\Lam^{\operatorname{sur}}_{\operatorname{big}}$ be the category whose objects are linearly ordered sets $S$ with an order preserving automorphisms $\sig: S \lrar S$ and whose morphisms are surjective order preserving maps which commute with the respective automorphisms. Then $\Lam^{\operatorname{sur}}_{\infty}$ can be considered as a full subcategory of $\Lam^{\operatorname{sur}}_{\operatorname{big}}$ such that $n$ corresponds to the object $({\mathbb{Z}},\sig_n)$ where $\sig_n: {\mathbb{Z}}\lrar {\mathbb{Z}}$ is the automorphism $x \mapsto x+n$.
Now let $p:{\mathbb{R}}\lrar S^1$ be the universal covering. We construct a functor $F_{S^1} \lrar \Lam^{\operatorname{sur}}_{\operatorname{big}}$ as follows: given an object $$f: M_1 \coprod ... \coprod M_n \hrar S^1$$ of $F_{S^1}$ consider the fiber product $$P = \left[M_1 \coprod ... \coprod M_n\right] \times_{S^1} {\mathbb{R}}$$ note that $P$ is homeomorphic to an infinite union of segments and the projection $$P \lrar {\mathbb{R}}$$ is injective (because $f$ is injective) giving us a well defined linear order on $P$. The automorphism $\sig: {\mathbb{R}}\lrar {\mathbb{R}}$ of ${\mathbb{R}}$ over $S^1$ given by $x \mapsto x + 1$ gives an order preserving automorphism $\wtl{\sig}: P \lrar P$.
Now suppose that $((M_1,...,M_n),f)$ and $((M_1',...,M_m'),g)$ are two objects and we have a morphism between them, i.e. an embedding
$$T:M_1 \coprod ... \coprod M_n \lrar M_1' \coprod ... \coprod M_m'$$ and an isotopy $\psi: g \circ T \sim f$. Then we see that the pair $(T,\psi)$ determine a well defined order preserving map $$\left[M_1 \coprod ... \coprod M_n\right] \times_{S^1} {\mathbb{R}}\lrar \left[M_1' \coprod ... \coprod M_m'\right] \times_{S^1} {\mathbb{R}}$$ which commutes with the respective automorphisms. Clearly we obtain in this way a functor $u:F_{S^1} \lrar \Lam^{\operatorname{sur}}_{\operatorname{big}}$ whose essential image is the same as the essential image of $\Lam^{\operatorname{sur}}_\infty$. It is also not hard to see that $u$ is fully faithful. Hence $F_{S^1}$ is equivalent to $\Lam^{\operatorname{sur}}_\infty$ which is weakly contractible. This finishes the proof of the theorem.
Let $$\operatorname{Alg}^{\operatorname{nd}}_{{\mathcal{OI}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right) \subseteq \operatorname{Alg}_{{\mathcal{OI}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right)$$ denote the full sub $\infty$-groupoid corresponding to the full sub $\infty$-groupoid $$\operatorname{Alg}^{\operatorname{nd}}_{{\mathcal{OF}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right) \subseteq \operatorname{Alg}_{{\mathcal{OF}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right)$$ under the equivalence of Theorem \[removing-circles\].
Now the last step of the cobordism hypothesis will be complete once we show the following:
\[final-lemma\] The $\infty$-groupoid $$\operatorname{Alg}^{\operatorname{nd}}_{{\mathcal{OI}}/ {\mathcal{B}}^{\operatorname{ev}}_1}\left({\mathcal{F}}_{\vphi}^{\otimes}\right)$$ is contractible.
Let $$q: p^*{\mathcal{F}}_{\vphi} \lrar {\mathcal{OI}}^{\otimes}$$ be the pullback of left fibration ${\mathcal{F}}_\vphi \lrar {\mathcal{B}}^{\operatorname{ev}}_1$ via the map $p: {\mathcal{OI}}^{\otimes} \lrar B^{\operatorname{ev}}_1$, so that $q$ is a left fibration as well. In particular, since ${\mathcal{OI}}^{\otimes}$ is the non-unital associative $\infty$-operad, we see that $q$ classifies an $\infty$-groupoid $q^{-1}({\mathcal{OI}})$ with a non-unital monoidal structure. Unwinding the definitions one sees that this $\infty$-groupoid is the fundamental groupoid of the space $$\operatorname{Map}_{{\mathcal{C}}}(1,\vphi(X_+) \otimes \vphi(X_-))$$ where $X_+,X_- \in {\mathcal{B}}^{\operatorname{ev}_1}$ are the points with positive and negative orientations respectively. The monoidal structure sends a pair of maps $$f,f': 1 \lrar \vphi(X_+) \otimes \vphi(X_-)$$ to the composition $$1 \x{f \otimes f'}{\lrar} \left[\vphi(X_+) \otimes \vphi(X_-)\right] \otimes \left[\vphi(X_+) \otimes \vphi(X_-)\right] \x{\simeq}{\lrar}$$ $$\vphi(X_+) \otimes \left[\vphi(X_-) \otimes \vphi(X_+)\right] \otimes \vphi(X_-) \x{Id \otimes \vphi(\operatorname{ev}) \otimes Id}{\lrar} \vphi(X_+) \otimes \vphi(X_-)$$ Since ${\mathcal{C}}$ has duals we see that this monoidal $\infty$-groupoid is equivalent to the fundamental $\infty$-groupoid of the space $$\operatorname{Map}_{{\mathcal{C}}}(\vphi(X_+),\vphi(X_+))$$ with the monoidal product coming from **composition**.
Now $$\operatorname{Alg}_{{\mathcal{OI}}/ {\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\vphi}) \simeq \operatorname{Alg}_{{\mathcal{OI}}/ {\mathcal{OI}}}(p^*{\mathcal{F}}_{\vphi})$$ classifies ${\mathcal{OI}}^{\otimes}$-algebra objects in $p^*{\mathcal{F}}_{\vphi}$, i.e. non-unital algebra objects in $$\operatorname{Map}_{{\mathcal{C}}}(\vphi(X_+),\vphi(X_+))$$ with respect to composition. The full sub $\infty$-groupoid $$\operatorname{Alg}^{\operatorname{nd}}_{{\mathcal{OI}}/ {\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\vphi}) \subseteq \operatorname{Alg}_{{\mathcal{OI}}/ {\mathcal{B}}^{\operatorname{ev}}_1}({\mathcal{F}}_{\vphi})$$ will then classify non-unital algebra objects $A$ which correspond to **self equivalences** $$\vphi(X_+) \lrar \vphi(X_+)$$ It is left to prove the following lemma:
Let ${\mathcal{C}}$ be an $\infty$-category. Let $X \in {\mathcal{C}}$ be an object and let ${\mathcal{E}}_X$ denote the $\infty$-groupoid of self equivalences $u: X \lrar X$ with the monoidal product induced from composition. Then the $\infty$-groupoid of non-unital algebra objects in ${\mathcal{E}}_X$ is contractible.
Let ${\mathcal{A}ss}_{\operatorname{nu}}$ denote the non-unital associative $\infty$-operad. The identity map ${\mathcal{A}ss}_{\operatorname{nu}} \lrar {\mathcal{A}ss}_{\operatorname{nu}}$ which is in particular a left fibration of $\infty$-operads classifies the terminal non-unital monoidal $\infty$-groupoid ${\mathcal{A}}$ which consists of single automorphismless idempotent object $a \in {\mathcal{A}}$. The non-unital algebra objects in ${\mathcal{E}}_X$ are then classified by non-unital lax monoidal functors $${\mathcal{A}}\lrar {\mathcal{E}}_X$$ Since ${\mathcal{E}}_X$ is an $\infty$-groupoid this is same as non-unital monoidal functors (without the lax) $${\mathcal{A}}\lrar {\mathcal{E}}_X$$ Now the forgetful functor from unital to non-unital monoidal $\infty$-groupoids has a left adjoint. Applying this left adjoint to ${\mathcal{A}}$ we obtain the $\infty$-groupoid ${\mathcal{UA}}$ with two automorphismless objects $${\mathcal{UA}}= \{1,a\}$$ such that $1$ is the unit of the monoidal structure and $a$ is an idempotent object.
Hence we need to show that the $\infty$-groupoids of monoidal functors $${\mathcal{UA}}\lrar {\mathcal{E}}_X$$ is contractible. Now given a monoidal $\infty$-groupoid ${\mathcal{G}}$ we can form the $\infty$-category ${\mathcal{B}}({\mathcal{G}})$ having a single object with endomorphism space ${\mathcal{G}}$ (the monoidal structure on ${\mathcal{G}}$ will then give the composition structure). This construction determines a fully faithful functor from the $\infty$-category of monoidal $\infty$-groupoids and the $\infty$-category of pointed $\infty$-categories (see [@lur1] Remark $4.4.6$ for a much more general statement). In particular it will be enough to show that the $\infty$-groupoid of **pointed functors** $${\mathcal{B}}({\mathcal{UA}}) \lrar {\mathcal{B}}({\mathcal{E}}_X)$$ is contractible. Since ${\mathcal{B}}({\mathcal{E}}_X)$ is an $\infty$-groupoid it will be enough to show that ${\mathcal{B}}({\mathcal{UA}})$ is weakly contractible.
Now the nerve $\operatorname{N}{\mathcal{B}}({\mathcal{UA}})$ of ${\mathcal{B}}({\mathcal{UA}})$ is the simplicial set in which for each $n$ there exists a single **non-degenerate** $n$-simplex $\sig_n \in \operatorname{N}{\mathcal{B}}({\mathcal{UA}})_n$ such that $d_i(\sig_n) = \sig_{n-1}$ for all $i=0,...,n$. By Van-Kampen it follows that $\operatorname{N}{\mathcal{B}}({\mathcal{UA}})$ is simply connected and by direct computation all the homology groups vanish.
This finishes the proof of Lemma \[final-lemma\].
This finishes the proof of Theorem \[qu-cobordism\].
From Quasi-Unital to Unital Cobordism Hypothesis {#s-from-qu-to-regular}
==================================================
In this section we will show how the quasi-unital cobordism hypothesis (Theorem \[qu-cobordism\]) implies the last step in the proof of the $1$-dimensional cobordism hypothesis (Theorem \[cobordism-last-step-2\]).
Let $M: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar \operatorname{Grp}_{\infty}$ be a non-degenerate lax symmetric monoidal functor. We can construct a pointed **non-unital** symmetric monoidal $\infty$-category ${\mathcal{C}}_M$ as follows:
1. The objects of ${\mathcal{C}}_M$ are the objects of ${\mathcal{B}}^{\operatorname{ev}}_1$. The marked point is the object $X_+$.
2. Given a pair of objects $X, Y \in {\mathcal{C}}_M$ we define $$\operatorname{Map}_{{\mathcal{C}}_M}(X, Y) = M(\check{X} \otimes Y)$$ Given a triple of objects $X, Y, Z \in {\mathcal{C}}_M$ the composition law $$\operatorname{Map}_{{\mathcal{C}}_M}(\check{X}, Y) \times \operatorname{Map}_{{\mathcal{C}}_M}(\check{Y},Z) \lrar \operatorname{Map}_{{\mathcal{C}}_M}(\check{X},Z)$$ is given by the composition $$M(\check{X} \otimes Y) \times M(\check{Y} \otimes Z) \lrar M(\check{X} \otimes Y \otimes \check{Y} \otimes Z) \lrar M(\check{X} \otimes Z)$$ where the first map is given by the lax symmetric monoidal structure on the functor $M$ and the second is induced by the evaluation map $$\operatorname{ev}_Y : \check{Y} \otimes Y \lrar 1$$ in ${\mathcal{B}}^{\operatorname{ev}}_1 $.
3. The symmetric monoidal structure is defined in a straight forward way using the lax monoidal structure of $M$.
It is not hard to see that if $M$ is non-degenerate then ${\mathcal{C}}_M$ is **quasi-unital**, i.e. each object contains a morphism which **behaves** like an identity map (see [@har]). This construction determines a functor $$G: \operatorname{Fun}_{\operatorname{nd}}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty}) \lrar \operatorname{Cat}^{\operatorname{qu},\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}$$ where $\operatorname{Cat}^{\operatorname{qu},\otimes}$ is the $\infty$-category of symmetric monoidal quasi-unital categories (i.e. commutative algebra objects in the $\infty$-category $\operatorname{Cat}^{\operatorname{qu}}$ of quasi-unital $\infty$-categories). In [@har] it is proved that the forgetful functor $$S:\operatorname{Cat}\lrar\operatorname{Cat}^{\operatorname{qu}}$$ From $\infty$-categories to quasi-unital $\infty$-categories is an **equivalence** and so the forgetful functor $$S^{\otimes}:\operatorname{Cat}^{\otimes} \lrar \operatorname{Cat}^{\operatorname{qu},\otimes}$$ is an equivalence as well.
Now recall that $$\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /} \subseteq \operatorname{Cat}^{\operatorname{nd}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}$$ is the full subcategory spanned by essentially surjective functors $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{C}}$. The fiber functor construction $\vphi \mapsto M_\vphi$ induces a functor $$F: \operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /} \lrar \operatorname{Fun}_{\operatorname{nd}}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty})$$
The composition $G \circ F$ gives a functor $$\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 / } \lrar \operatorname{Cat}^{\operatorname{qu},\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}$$
We claim that $G \circ F$ is in fact **equivalent** to the composition $$\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 / } \x{T}{\lrar} \operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 / } \x{S}{\lrar} \operatorname{Cat}^{\operatorname{qu},\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 / }$$ where $T$ is given by the restriction along $X_+:{\mathcal{B}}^{\operatorname{un}}_0 \hrar {\mathcal{B}}^{\operatorname{ev}}_1$ and $S$ is the forgetful functor.
Explicitly, we will construct a natural transformation $$N:G \circ F \x{\simeq}{\lrar} S \circ T$$ In order to construct $N$ we need to construct for each non-degenerate functor $\vphi: {\mathcal{B}}^{\operatorname{ev}}_1 \lrar {\mathcal{D}}$ a natural pointed functor $$N_\vphi: {\mathcal{C}}_{M_\vphi} \lrar {\mathcal{D}}$$ The functor $N_\vphi$ will map the objects of ${\mathcal{C}}_{M_\vphi}$ (which are the objects of ${\mathcal{B}}^{\operatorname{ev}}_1$) to ${\mathcal{D}}$ via $\vphi$. Then for each $X,Y \in {\mathcal{B}}^{\operatorname{ev}}_1$ we can map the morphisms $$\operatorname{Map}_{{\mathcal{C}}_{M_{\vphi}}}(X,Y) = \operatorname{Map}_{{\mathcal{D}}}(1,\check{X} \otimes Y) \lrar \operatorname{Map}_{{\mathcal{D}}}(X,Y)$$ via the duality structure - to a morphism $f: 1 \lrar \check{X} \otimes Y$ one associates the morphism $\what{f}: X \lrar Y$ given as the composition $$X \x{Id \otimes f}{\lrar} X \otimes \check{X} \otimes Y \x{\vphi(\operatorname{ev}_X) \otimes Y}{\lrar} Y$$ Since ${\mathcal{D}}$ has duals we get that $N_\vphi$ is fully faithful and since we have restricted to essentially surjective $\vphi$ we get that $N_\vphi$ is essentially surjective. Hence $N_\vphi$ is an equivalence of quasi-unital symmetric monoidal $\infty$-categories and $N$ is a natural equivalence of functors.
In particular we have a homotopy commutative diagram: $$\xymatrix{
& \operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 / } \ar_{F}[dl] \ar^{T}[dr] & \\
\operatorname{Fun}_{\operatorname{nd}}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty}) \ar_{G}[dr] & & \operatorname{Cat}^{\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /} \ar^{S}[dl] \\
& \operatorname{Cat}^{\operatorname{qu},\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /} & \\
}$$
Now from Lemma \[0-to-1-ev\] we see that $T$ is fully faithful. Since $S$ is an equivalence of $\infty$-categories we get
\[retract\] The functor $G \circ F$ is fully faithful.
We are now ready to complete the proof of \[cobordism-last-step-2\]. Let ${\mathcal{D}}$ be a symmetric monoidal $\infty$-category with duals and let $\vphi: {\mathcal{B}}\lrar {\mathcal{D}}$ be a non-degenerate functor. We wish to show that the space of maps $$\operatorname{Map}_{\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}}(\iota,\vphi)$$ is contractible. Consider the sequence
$$\operatorname{Map}_{\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}}(\iota,\vphi) \lrar \operatorname{Map}_{\operatorname{Fun}_{\operatorname{nd}}^{\operatorname{lax}}({\mathcal{B}}^{\operatorname{ev}}_1,\operatorname{Grp}_{\infty})}(M_\iota,M_\vphi) \lrar
\operatorname{Map}_{\operatorname{Cat}^{\operatorname{qu},\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$$ By Theorem \[qu-cobordism\] the middle space is contractible and by lemma \[retract\] the composition $$\operatorname{Map}_{\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}}(\iota,\vphi) \lrar \operatorname{Map}_{\operatorname{Cat}^{\operatorname{qu},\otimes}_{{\mathcal{B}}^{\operatorname{un}}_0 /}}({\mathcal{B}}^{\operatorname{or}}_1,{\mathcal{D}})$$ is a weak equivalence. Hence we get that $$\operatorname{Map}_{\operatorname{Cat}^{\operatorname{sur}}_{{\mathcal{B}}^{\operatorname{ev}}_1 /}}(\iota,\vphi)$$ is contractible. This completes the proof of Theorem \[cobordism-last-step-2\].
[cobordism]{}
Baez, J., Dolan, J., Higher-dimensional algebra and topological qauntum field theory, Journal of Mathematical Physics, 36 (11), 1995, 6073–6105.
Harpaz, Y. Quasi-unital $\infty$-categories, PhD Thesis.
Lurie, J., On the classification of topological field theories, Current Developments in Mathematics, 2009, p. 129-–280, <http://www.math.harvard.edu/~lurie/papers/cobordism.pdf>.
Lurie, J. *Higher Algebra*, <http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf>.
Lurie, J., *Higher Topos Theory*, Annals of Mathematics Studies, 170, Princeton University Press, 2009, <http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf>.
Kaledin, D., Homological methods in non-commutative geometry, preprint, <http://imperium.lenin.ru/~kaledin/math/tokyo/final.pdf>
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abstract: 'Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function $C(\omega)$ of the spin operator $\sigma_z$ for the biased sub-Ohmic spin-boson model. The small-$\omega$ behavior $C(\omega) \propto \omega^s$ is found to be universal and independent of the bias $\epsilon$ and the coupling strength $\alpha$ (except at the quantum critical point $\alpha =\alpha_c$ and $\epsilon=0$). Our NRG data also show $C(\omega) \propto \chi^{2}\omega^{s}$ for a wide range of parameters, including the biased strong coupling regime ($\epsilon \neq 0$ and $\alpha > \alpha_c$), supporting the general validity of the Shiba relation. Close to the quantum critical point $\alpha_c$, the dependence of $C(\omega)$ on $\alpha$ and $\epsilon$ is understood in terms of the competition between $\epsilon$ and the crossover energy scale $\omega_{0}^{\ast}$ of the unbiased case. $C(\omega)$ is stable with respect to $\epsilon$ for $\epsilon \ll \epsilon^{\ast}$. For $\epsilon \gg \epsilon^{\ast}$, it is suppressed by $\epsilon$ in the low frequency regime. We establish that $\epsilon^{\ast} \propto (\omega_0^{\ast})^{1/\theta}$ holds for all sub-Ohmic regime $0 \leqslant s < 1$, with $\theta=2/(3s)$ for $0 < s \leqslant 1/2$ and $\theta = 2/(1+s)$ for $1/2 < s < 1$. The variation of $C(\omega)$ with $\alpha$ and $\epsilon$ is summarized into a crossover phase diagram on the $\alpha-\epsilon$ plane.'
author:
- 'Da-Chuan Zheng'
- 'Ning-Hua Tong'
title: ' Equilibrium Dynamics of the Sub-Ohmic Spin-boson Model Under Bias '
---
[Introduction ]{}
The spin-boson model (SBM) is one of the simplest models to describe a quantum two-level system coupled to the environmental noise [@Leggett; @Weiss1]. It has realizations in various fields of physics, including the superconducting qubit [@Shnirman], mesoscopic metal ring penetrated by an Aharonov-Bohm flux [@Tong1], ultraslow glass dynamics [@Rosenberg], heavy fermion metals [@Si], and nanomechanical oscillators [@Seoanez], [*etc.*]{}. In this model, the environmental noise is represented by a bath of harmonic oscillators with the low frequency spectral function characterized by $J(\omega) \propto \alpha\omega^s$. Here $\alpha$ is the coupling constant and $s$ is a dimensionless parameter that discriminates the Ohmic ($s=1$), super-Ohmic ($s>1$), and sub-Ohmic ($0\leqslant s <1$) bath.
The SBM has been studied in detail for the Ohmic and weak-coupling regime to understand the quantum decoherence and dissipation, which are the key issues for making a long-lived quantum bit [@Shnirman], chemical reaction [@Mulhbacher], as well as in the optical absorption in quantum dot [@Borri]. The SBM without a bias field also contains non-trivial quantum phase transitions (QPTs) between a weak-coupling delocalized phase and a strong-coupling localized phase for $0 \leqslant s \leqslant 1$. It is an established result that a Kosterlitz-Thouless type QPT exists for $s=1$ and the ground state is always delocalized for $s>1$ [@Leggett; @Weiss1]. In the sub-Ohmic regime $0 \leqslant s <1$ [@Leggett; @Weiss1; @Kehrein], numerical methods such as numerical renormalization group (NRG) [@Bulla1; @Guo], quantum Monte Carlo (QMC) [@Winter], and exact diagonalization [@Alvermann] show that a continuous QPT occurs at certain critical coupling $\alpha_c$, at which both the static and the dynamic quantities show critical behavior [@Chin; @Anders1; @Lu; @Hur1; @Kast; @Liu; @Yao; @Nalbach; @Florens1].
For the study of both decoherence and QPT, the equilibrium dynamical correlation function is an important quantity. Most of the previous studies concentrate on the case of zero bias and weak coupling, [*i.e.*]{}, $\epsilon=0$ and $\alpha < \alpha_c$. In this regime, the $\sigma_z - \sigma_z$ correlation function $C(\omega)$ (to be defined below) shows a power law behaviour $C(\omega) \sim \omega^{s}$ in small frequency $\omega \ll \omega_{0}^{\ast}$ and critical behavior $C(\omega) \sim \omega^{-s}$ in the intermediate frequency $\omega_{0}^{\ast} \ll \omega \ll \omega_0$. Here $\omega_0$ is a non-universal high energy scale and $\omega_{0}^{\ast}$ is the crossover energy scale between the delocalized fixed point and the quantum critical fixed point.
The situation with a finite bias $\epsilon \neq 0$, however, receives less attention. For $\epsilon \neq 0$, the parity symmetry of the unbiased SBM is broken and hence the localize-delocalize QPT of the sub-Ohmic symmetric SBM no longer exists. Instead, as the coupling strength increases, the ground state changes smoothly from the biased weak-coupling delocalized-like state to the strong-coupling localized-like state with broken symmetry. The strong coupling regime is difficult to described for the perturbation-based theory [@Lu]. For the numerical approaches such as NRG [@Bulla1] and exact diagonalaization [@Alvermann], a finite bias increases the number of bosons in the ground state and makes an accurate calculation more difficult. As a result, systematic study of the dynamical correlation function for the sub-Ohmic bath under a finite bias, especially in the strong coupling regime $\alpha > \alpha_c$, is still lacking.
In this paper, we use NRG to study the equilibrium state dynamical correlation function of the sub-Ohmic SBM with a finite bias field $\epsilon \neq 0$, with emphasis on the strong-coupling regime $\alpha > \alpha_c$. By carefully extrapolating the boson-state truncation to infinity, we find that except for the exact QPT point, $C(\omega)$ always obeys $\chi^{2} \omega^s$ behavior in the small frequency limit, irrespective of the values of $\epsilon$ and $\alpha$. In the vicinity of the quantum critical point (QCP), $C(\omega)$ is characterized by two different power law regimes: $C(\omega) \propto \omega^{s}$ in the low frequency regime $ \omega \ll \omega^{\ast}$ and $C(\omega) \propto \omega^{-s}$ in the intermediate frequency regime $ \omega^{\ast} \ll \omega \ll \omega_0$, with $\omega_0$ being a non-universal high energy scale. The crossover frequency $\omega^{\ast}$ is tuned by $\epsilon$. For weak bias $\epsilon \ll \epsilon^{\ast}$, $\omega^{\ast} = \omega_{0}^{\ast} \propto |\alpha - \alpha_c|^{z\nu}$ and $C(\omega)$ is not changed significantly from the zero bias case. For strong bias $\epsilon \gg \epsilon^{\ast}$, $\omega^{\ast} \propto \epsilon^{\theta}$ and $C(\omega)$ is suppressed in $\omega \ll \omega^{\ast}$. The behavior of $\omega^{\ast}$ is understood in terms of the competition between the unbiased crossover scale $\omega_0^{\ast}$ and $\epsilon$, and we have $\omega^{\ast} \sim {\text max}\left[ \epsilon^{\theta}, \omega_{0}^{\ast} \right]$. We obtained the critical exponent $\theta$ as a function of $s$. The crossover bias $\epsilon^{\ast}$ is thus determined as $\epsilon^{\ast} = \left( \omega_0^{\ast} \right)^{1/\theta}$. We finally summarize the behavior of $C(\omega)$ in a crossover $\epsilon-\alpha$ phase diagram.
This paper is organized as the following. In section II we introduce SBM and the formalism we used to calculate the equilibrium correlation function with bosonic NRG method. Section III presents results from our NRG study. A conclusion is given in section IV.
[Model and Method ]{}
The Hamiltonian of SBM reads $$\begin{aligned}
&& H = -\frac{\Delta}{2} \sigma_x + \frac{\epsilon}{2} \sigma_z + \sum_{i} \omega_{i} a_{i}^{\dagger}a_{i} + \frac{\sigma_z}{2} \ \sum_{i} \lambda_{i}(a_{i}+a_{i}^{\dagger}). \nonumber \\
&&\end{aligned}$$ The first two terms describe a two level system with bias $\epsilon$ and tunnelling strength $\Delta$. The bosonic bath is described by the third term, where $\omega_{i}$ is the frequency of the $i$-th boson mode. In the last term, the two level system is coupled to the bosonic bath through $\sigma_z$ and the boson displacement operator. Various experimental realizations of this Hamiltonian have been proposed with tunable parameters $\epsilon$ and $\Delta$ [@Tong1; @Hur2; @Goldstein; @Egger]. The influence of bath on the two level system is characterized by the spectral function $$J(\omega) = \pi \sum_{i} \lambda_{i}^{2} \delta(\omega - \omega_{i}).$$ In this paper, we use a power law form of $J(\omega)$ with a hard cut-off at $\omega_c$, $$J(\omega) = 2\pi\alpha \omega^{s} \omega_{c}^{1-s}, \,\,\,\,\, \,\,\,\, (0 \leqslant \omega \leqslant \omega_c).$$ We set $\omega_c = 1.0$ as the unit of energy and fix $\Delta=0.1$ to study the dependence of equilibrium dynamics on $\epsilon$ and $\alpha$ for a sub-Ohmic bath.
At $\epsilon=0$, Eq.(1) is invariant under the combined boson and spin parity transformation $U a_{i} U^{-1} = -a_i$ and $U \sigma_{z} U^{-1} = - \sigma_z$. For the sub-Ohmic ($0 \leqslant s<1$) and the Ohmic ($s=1$) baths, a spontaneous breaking of this symmetry may occur in the regime $\alpha > \alpha_c$ and the system enters the localized phase, in which the quantum system is trapped to one of the two states and the local bosons has a finite displacement [@Kehrein; @Bulla1; @Guo; @Winter; @Alvermann]. This is the delocalize-localize quantum phase transition of SBM. With a finite bias $\epsilon \neq 0$, the above parity symmetry is broken from the outset and the phase transition no longer exists. As $\alpha$ increases, the ground state crosses over smoothly from the weak-coupling to the strong-coupling biased states.
In this paper, we focus on the $\sigma_z-\sigma_z$ dynamical correlation function defined as $$C(\omega)=\frac{1}{2\pi} \int_{-\infty}^{+\infty} C(t) \,dt,$$ where $C(t)=(1/2)\langle [ \sigma_z(t),\sigma_z(0) ]_{+}\rangle$ and $[\hat{A}, \hat{B} ]_{+}$ is the anti-commutator of $\hat{A}$ and $\hat{B}$. For a non-degenerate ground state, the Lehman representation of the correlation function at $T=0$ is written as $$\begin{aligned}
C(\omega)&=& \frac{1}{2} \sum_{n} | \langle 0|\sigma_z|n \rangle |^2 \delta(\omega+ E_{0}-E_{n}) \nonumber \\
&& + \frac{1}{2} \sum_{n} | \langle n|\sigma_z|0 \rangle |^2 \delta(\omega+ E_{n}- E_{0}) .\end{aligned}$$ Here $|n \rangle$ and $E_n$ are the $n$-th eigen state and energy of the Hamiltonian, respectively. $C(\omega)$ has the general form $C(\omega)= A \delta(\omega) + C^{\prime}(\omega)$, where $A= |\langle 0| \sigma_z |0 \rangle|^{2}$ and $|0 \rangle$ is the ground state. It is an even function of $\omega$ and fulfils the sum rule $$\int_{-\infty}^{+\infty} C(\omega) \, d \omega = 1.$$ The bosonic NRG method is regarded as one of the most accurate numerical techniques for studying SBM due to its non-perturbative nature and the applicability in the whole range of parameters [@Wilson; @Bulla1; @Bulla2]. The success of NRG relies on the energy scale separation due to the logarithmic discretization, and on the RG transformation which is carried out by iterative diagonalization in each energy shell. It is technically composed of three steps: logarithmic discretization, transforming the Hamiltonian into a semi-infinite chain, and the iterative diagonalization. In general, the errors in the NRG calculation come from two sources. One is the approximation of using one bath mode to represent each energy shell, whic is controlled by the logarithmic discretization parameter $\Lambda \geqslant 1$. The other is the truncation of the energy spectrum after each diagonalization to overcome the exponential increase of the Hilbert space, which is controlled by the number of kept states $M_s$. For the bosonic NRG, an additional source of error is the truncation of infinite dimensional Hilbert space of each boson mode into $N_b$ states on the occupation basis. Exact results are obtained in the limit $\Lambda=1$, $M_s= \infty$, and $N_b = \infty$.
In practice, however, one cannot do calculation directly at the above limit and extrapolating NRG data to the above limit could be difficult. In the strong coupling regime with a bias, the ground state has a large boson number and hence requires a large $N_b$ for an accurate calculation. A large $N_b$, however, will increase the truncation error because in NRG calculation only the lowest $1/N_b$ fraction of the eigen states of previous energy shell are kept for constructing the Hamiltonian of the lower energy shell. To keep the accuracy, one has to use a larger $\Lambda$ to increase the energy separation of successive energy shells. This in turn will lead to a larger discretization error of the bath. Fortunately, we found that a larger $\Lambda$ only induces quantitative changes to the NRG results and the qualitative conclusion can still be obtained reliably. Therefore, using a large $\Lambda = 2 \sim 10$, a moderate $M_s \sim 100 \sim 200$, and $N_b = 12 \sim 60$, we can obtain qualitatively correct results for the biased SBM in the strong coupling regime. Combining the knowledge gained from the finite size scaling method of $N_b$ (Ref. ), we checked that our final conclusions are stable when extrapolated to the exact limit. To calculate $C(\omega)$, we use the patching method of Bulla [@Bulla3] with which the sum rule Eq.(6) is fulfilled reasonably well. The discrete $\delta$ peaks obtained are broadened with a Gaussian function on the logarithmic scale [@Bulla3], $$\delta(\omega - \omega_n) \rightarrow \frac{e^{-b^2/4}}{b\omega_n\sqrt{\pi}} \exp { \left[ -\frac{( \ln{\omega}-\ln{\omega_n} )^2}{b^2} \right] }.$$ We choose the broadening parameter $b=0.7$ for $\Lambda < 4$ and $b=2.0$ for $\Lambda \geqslant 4$.
[Results ]{}
In this work, we study the dynamical correlation function $C(\omega)$ for all range of $\alpha$ and $\epsilon \neq 0$. As $\epsilon$ tends to zero, the strong coupling phase of $\alpha > \alpha_c$ is continuously connected to the localized state at $\epsilon =0$ with spontaneously broken symmetry. At $\epsilon=0$ and $\alpha > \alpha_c$, the localized state without symmetry breaking is special in that the ground state has a two-fold degeneracy and it is unstable in the thermodynamical limit [@Cao]. In this paper we hence confine our study to the symmetry-broken phase (either spontaneously or by a finite bias) and do not study that special case.
[$\omega^s$-behavior in the small frequency regime]{}
![(color online) NRG flow diagrams and $C(\omega)$ at $s=0.8$, $\Delta=0.1$, $\alpha=0.5 > \alpha_c$, and $\epsilon=10^{-3}$. (a) Flow diagrams obtained using $N_b=8$, $18$, and $40$ (symbols with eye-guiding lines, from left to right). The dashed lines are the flow of $\alpha=0$ and $\epsilon=0$. (b) Dynamical correlation function $C(\omega)$ for $N_{b}=8$, $18$, and $40$ (solid lines, from bottom to top), and $N_{b}=\infty $ from extrapolation (dashed lines). Fitting the small $\omega$ regime, we obtain the exponents $y_{l}=2.07$ for $N_b=8$, and $y_{h} = 0.799$ for $N_b = \infty$. The $\delta(\omega)$ peak is not shown. NRG parameters are $\Lambda=4.0$ and $M_s=120$. []{data-label="fig1"}](fig1_cw_nb.eps){width="280pt" height="200pt"}
![(color online) NRG flow diagrams and $C(\omega)$ at $s=0.3$, $\Delta=0.1$, $\alpha=0.045 > \alpha_c$. $\epsilon=10^{-2}$ (solid lines) and $\epsilon=10^{-3}$ (dashed lines). (a) Flow diagrams obtained using $N_b=12$, $27$, and $60$ (from left to right). The horizontal dashed line are the flow of $\alpha=0$ and $\epsilon=0$. (b) $C(\omega)$ for $N_b=12$, $27$, and $60$ (from bottom to top). The functions $\omega^{0.3}$ and $\omega^{-0.3}$ are marked by the dashed straight lines for guiding the eye. The $\delta(\omega)$ peak is not shown. NRG parameters are $\Lambda=4.0$ and $M_s=120$. []{data-label="fig2"}](fig2_cw_nb2.eps){width="320pt" height="240pt"}
For the Ohmic ($s=1$) and the super-Ohmic ($s>1$) baths, the small frequency behavior of $C(\omega)$ in the delocalized phase obeys the Shiba relation [@Weiss2], which in our notation reads[@Bulla1] $$C(w)= \frac{\alpha}{4} {\chi}^2 \omega^{s}, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\omega \to 0).$$ Here $\chi \equiv 2 \partial \langle \sigma_z \rangle / \partial \epsilon|_{\epsilon=0}$ is the local spin susceptibility. The proof in Ref. applies also to the case with a finite bias, but not to the localized phase at $\alpha > \alpha_c$ for the Ohmic bath. This exact relation was used to test the quality of various approximate results for SBM [@Florens1; @Egger2; @Costi; @Volker; @Keil; @Bulla1].
There have been attempts to generalize the Shiba relation to (i) sub-Ohmic bath, and (ii) with finite bias, and (iii) strong coupling regime $\alpha > \alpha_c$. Such a generalized relation, if exists, would imply an universal long time behavior $C(t) \propto t^{-(1+s)}$ for any $s$ values, coupling strength $\alpha$ (except at the critical point $\alpha = \alpha_c$ and $\epsilon=0$), and spin parameters $\Delta$ and $\epsilon$. Up to now these activities received partial success only. Florens and others [@Florens2] proved Eq.(8) for the sub-Ohmic bath, based on an exact relation from the perturbation theory in Majorana representation. However, their proof applies only to delocalized phase at the symmetry point $\epsilon=0$. The numerical results from the approximate perturbation theory built on an unitary transformation fulfils Eq.(8) exactly for the sub-Ohmic bath with [@Gan] or without [@Lu] bias in the weak to intermediate coupling. In the strong coupling regime $\alpha > \alpha_c$ the fulfilment is good but not exact. This leaves the question open whether Shiba relation holds in the biased strong coupling regime of the sub-Ohmic SBM.
As stated above, the inherent truncation errors of NRG hinder it from the quantitative confirmation/falsification of the generalized Shiba relation. However, qualitatively, it is possible to check the factors on the right-hand side of Eq.(8) each for a time. In this section, we will show that $C(\omega) \propto \omega^{s}$ holds for the sub-Ohmic SBM with general $\alpha$ and $\epsilon$. In Fig.6, data are presented to further support that $C(\omega)/\omega^{s} \propto \chi^{2}$ for general $\alpha$ and $\epsilon$ values.
In Fig.1, we first present the $N_b$-dependence of $C(\omega)$ obtained from NRG under bias, in order to gauge the choice of $N_b$ in our study. The energy flow (Fig1.(a)) and $C(\omega)$ (Fig1.(b)) are shown for a sub-Ohmic bath $s=0.8$ in the strong coupling regime $\alpha=0.5 > \alpha_c$, with a finite bias $\epsilon=10^{-3}$. We use a series of $N_b$ values from $N_b=8$ to $N_b=40$ with $\Lambda = 4$ and $M_s=120$. In Fig.1(a), for a given $N_b$, the excitation energies $\Lambda^{N}\left[ E_{i}(N) - E_{0}(N) \right]$ ($i=1, 2, 3$) first flow to an unstable fixed point (with $\Lambda^{N}\left[ E_{1}(N) - E_{0}(N) \right] \approx 0.61$) in the small $N$ regime. After a crossover $N^{\ast}$, they flow to the stable fixed point ( with $\Lambda^{N}\left[ E_{1}(N) - E_{0}(N) \right] \approx 0.77$) in the large $N$ limit. We found that the unstable fixed point in the small $N$ regime is same as the weak-coupling fixed point of the SBM in the non-bias case (dashed lines obtained using $\alpha=0$ and $\epsilon=0$). With $N_b$ increasing geometrically, the crossover $N^{\ast}$ increases linearly, showing that the associated energy scale $T^{\ast} \sim \Lambda^{-N^{\ast}}$ decreases to zero as a negative power of $N_b$. This supports that the unstable fixed point (dashed lines in Fig.1(a)) will extend to infinitely large $N$ in the limit $N_b = \infty$ and it is the true fixed point of the biased SBM. In this biased fixed point, $\epsilon$ flows to infinity and the spin is effectively decoupled from the bath, leading to the same excitation levels as the free boson chain.
Fig.1(b) shows the corresponding $C(\omega)$, which has two different power-law regimes: $C(\omega) \sim \omega^{y_l}$ in the low frequency regime $\omega \ll \omega^{\ast}$, and $C(\omega) \sim \omega^{y_h}$ in the high frequency regime $\omega \gg \omega^{\ast}$. NRG data for $s=0.8$ gives $y_l=2.07$ and $y_{h}=0.799$. With increasing $N_b$, the crossover frequency $\omega^{\ast}$ decreases to zero as a negative power of $N_b$, proving that $C(\omega) \sim \omega^{y_h}$ is the correct result of the biased SBM (dashed line in Fig.1(b)). Our results for other $s$ values in $1/2 \leqslant s \leq 1$ agree with $y_h = s$ within an error of $2 \%$ (not shown).
For $s<1/2$, the NRG results converge much more difficult with increasing $N_b$. In Fig.2, we show the strong coupling data ($\alpha=0.045 > \alpha_c$) for $s=0.3$. Both the energy flow and $C(\omega)$ have $N_b$-dependent crossover scales. With $N_b$ increasing to $60$, a clear trend can be seen that the $N_b$-converged levels flow towards the free boson energy levels (Fig.2(a)). For both $\epsilon=10^{-2}$ and $\epsilon=10^{-3}$, the section of $C(\omega)$ with $\omega^{s}$ behavior extends to smaller $\omega$ with increasing $N_b$ (Fig.2(b)). The shift of $C(\omega)$ with $N_b$ has an apparent scaling form and an analysis for $N_b$ could be carried out to extract the correct exponent, as done in Ref. . Here in Fig.2(b), we only mark out the expected asymptotic $\omega^{s}$ line for guiding the eye. With this understanding of the $N_b$-dependence of $C(\omega)$, in the rest part of this paper, we only show the physically correct results $C(\omega > \omega^{\ast})$ obtained using sufficiently large $N_b$.
![(color online) $C(\omega)$ for various $\alpha$ and $\epsilon$ at $s=0.8$, $\Delta=0.1$. (a) $\alpha < \alpha_c \approx 0.482$: $\alpha=0.48$ (solid lines), $0.4$ (dashed lines), and $0.1$ (dash-dotted lines). (b) $\alpha > \alpha_c$: $\alpha=0.49$ (solid lines), $0.57$ (dashed lines), and $0.7$ (dash-dotted lines). For each $\alpha$ value, from top to bottom are curves for $\epsilon=0$, $10^{-5}$, and $10^{-3}$. All curves show $C(\omega) \propto \omega^s$ behavior in the small $\omega$ limit. NRG parameters are $\Lambda=4$, $M_s=120$, and $N_b=60$. []{data-label="fig3"}](fig3_v_epi.eps){width="300pt" height="210pt"}
In Fig.3, we investigate the influence of $\epsilon$ on $C(\omega)$ for $\alpha$ values ranging from weak coupling $\alpha < \alpha_c$ (Fig.3(a)) to strong coupling $\alpha > \alpha_c$ (Fig.3(b)) for $s=0.8$ and $\Delta=0.1$. It is seen that $C(\omega) \sim \omega^{s}$ in the small frequency limit for all parameters. For the small coupling $\alpha=0.1$, $C(\omega)$ does not change for $\epsilon$ as large as $10^{-3}$. While for a moderate coupling $\alpha=0.4$, a slight downward shift is observed between $\epsilon=10^{-3}$ and $\epsilon=10^{-5}$. $C(\omega)$ is most sensitive to $\epsilon$ when $\alpha$ is close to $\alpha_c \approx 0.482$.
The universal $C(\omega) \propto \omega^{s}$ behavior observed so far in Figs.1, 2, and 3 has a natural understanding. When the parity symmetry already broken by a finite bias, the ground state of SBM can be tuned continuously on the $\alpha-\epsilon$ plane, going from a delocalized state at $\alpha < \alpha_c$, $\epsilon=0^{+}$, through a half circle in finite $\epsilon$ region, to the symmetry-spontaneously-broken state at $\alpha > \alpha_c$, $\epsilon=0^{+}$, without passing the critical point. Therefore, the ground state has same nature and no qualitative change is expected in the small $\omega$ limit of $C(\omega)$. Indeed, our NRG results for different $s$, $\alpha$, and $\epsilon$ confirm that $C(\omega) \propto \omega^{s}$ is a universal feature of SBM.
![(color online) (a)-(c): the flow diagrams for three different $\alpha$ values. For each $\alpha$, from right to left $\epsilon=0$, $10^{-7}$, $10^{-6}$, $10^{-5}$, $10^{-4}$, and $10^{-3}$. The empty squares make out the crossover $N^{\ast}$ for the case $\epsilon >0$. The solid circle marks out the crossover scale in the unbiased case $N_{0}^{\ast}$. In (c), $N_{0}^{\ast} = \infty$. NRG parameters are $\Lambda=2.0$, $M_s=200$, and $N_b=16$. []{data-label="Fig4"}](fig4_flows.eps){width="320pt" height="230pt"}
![(color online) (a)-(c): $C(\omega)$ for the same $\alpha$ values. For each $\alpha$, from top to bottom, $\epsilon=0$, $10^{-7}$, $10^{-6}$, $10^{-5}$, $10^{-4}$, and $10^{-3}$. NRG parameters are same as Fig.4. []{data-label="Fig5"}](fig5_cws.eps){width="320pt" height="230pt"}
In the strong coupling case shown in Fig.3(b), the same tendency is observed, [i.e.]{}, with increasing $\epsilon$, $C(\omega)$ shifts downwards and the most prominant change occurs near $\alpha_c$. The suppression of $C(\omega)$ can be understood as the weight transfer: with increasing $\epsilon$, $|\langle \sigma_z \rangle|$ increases and so does the weight of the zero-frequency $\delta$ peak. Due to the sum rule, $C(\omega > 0)$ decreases uniformly. Comparing Fig.3(a) and (b), one sees that a broad peak forms for $\alpha > \alpha_c$ and $\epsilon \sim 10^{-3}$.
[Equilibrium Dynamics near QCP]{}
In the above section, we established the universal $\omega^{s}$-behaviour of $C(\omega)$ in the full parameter space of SBM. In this section, we focus on the parameter regime near the QCP $\alpha \sim \alpha_c$ and study the critical properties of $C(\omega)$ under a bias. For the unbiased SBM, $T_{0}^{\ast} \propto |\alpha - \alpha_c|^{z\nu}$ is the only energy scale that controls the crossover between the stable fixed points (localized and delocalized fixed points) and the critical fixed point. Here $\nu$ is the critical exponent of correlation length and $z=1$ is the dynamical critical exponent. $T_{0}^{\ast}$ plays an important role in the temperature dependence of physical quantities close to QCP. At zero temperature, it also appears in the dynamical correlation function in the delocalized phase: $C(\omega) \sim \omega^{s}$ for $\omega \ll \omega_{0}^{\ast}$ and $C(\omega) \sim \omega^{-s}$ for $\omega \gg \omega_{0}^{\ast}$, and $\omega_0^{\ast} = T_{0}^{\ast}$.
For the biased case, $\epsilon >0$ is another energy scale that influences the crossover between different behaviors of $C(\omega)$. Our NRG results for $C(\omega)$ at different $\alpha$ and $\epsilon$ can be understood in terms of the competition between $\epsilon$ and $T_{0}^{\ast}$. Due to the difficulty of $N_b$ convergence for $0 \leqslant s < 1/2$, here we show NRG data for $s=0.8$, representing a typical case for $1/2 < s < 1$. Our conclusion also applies to $0 \leqslant s < 1/2$, as will be discussed below.
In Fig.4 and Fig.5, we show the flow diagrams (Fig.4) and $C(\omega)$ (Fig.5) for $\alpha=0.3868 < \alpha_c$ (a), $\alpha=0.4008 > \alpha_c$ (b), and $\alpha=0.393647 \approx \alpha_c$ (c), respectively. For each $\alpha$, $\epsilon$ varies from zero to $10^{-3}$. The purpose is to observe the influence of $\epsilon$ in the delocalized, localzied, and critical phases. Fig.4(a) and Fig.5(a) are for $\alpha < \alpha_c$. At $\epsilon=0$, the energy flow in Fig.4(a) has a crossover at around $N_{0}^{\ast} \approx 16$ (solid circle), from the critical fixed point to the weak-coupling one. As $\epsilon$ increases from zero, the flow does not change for $\epsilon < 10^{-5}$ and the crossover $N^{\ast}$ (empty squares) begins to decreases only for $\epsilon > 10^{-4}$. The corresponding evolution of $C(\omega)$ is shown in Fig.5(a). For $\epsilon < 10^{-5}$, $C(\omega) \propto \omega^{-s}$ for $\omega \gg \omega_{0}^{\ast}= T_{0}^{\ast}$ and $C(\omega) \propto \omega^{s}$ for $\omega \ll \omega_{0}^{\ast}$. The crossover scale of the symmetric SBM is given by $\omega_{0}^{\ast} = \Lambda^{-N_{0}^{\ast}} \approx 10^{-5}$. With increasing $\epsilon$, the crossover frequency $\omega^{\ast}$ increases and $\omega^{\ast} > \omega_{0}^{\ast}$ occurs only when $\epsilon > 10^{-4}$, corresponding to the occurrence of $N^{\ast} < N_0^{\ast}$ in the flow diagram. A crossover scale $\epsilon^{\ast}$ can be defined as such that for $\epsilon > \epsilon^{\ast}$, $\omega^{\ast}$ becomes significantly larger than $\omega_{0}^{\ast}$, or equivalently, $N^{\ast} < N_0^{\ast}$. For $\epsilon \ll \epsilon^{\ast}$, $C(\omega)$ stays same as the symmetric case ($\epsilon=0$) and $\omega^{\ast} = T_{0}^{\ast}$. For $\epsilon \gg \epsilon^{\ast}$, $C(\omega)$ is suppressed in the low frequency regime and $\omega^{\ast} > T_{0}^{\ast}$ is set by $\epsilon$.
Fig.4(b) and Fig.5(b) show the influence of $\epsilon$ in the localized phase. In Fig.4(b), the excitation energy level at $\epsilon=0$ flows from the critical fixed point towards a two-fold degenerate fixed point, with the critical-to-localize crossover around $N^{\ast} = N_{0}^{\ast}=15$ (solid circle). Now we study the change of $N^{\ast}$ under bias (empty squares). With a vanishingly small $\epsilon$, the degeneracy is lifted and the excited energy level has an upturn at an arbitrarily large $N^{\ast} \gg N_{0}^{\ast}$, showing a crossover from the localized symmetric state to the biased state. We find that the crossover energy scale $T^{\ast} \equiv \Lambda^{-N^{\ast}} \propto \epsilon$. With further increasing $\epsilon$, the upturn moves to the left and for $\epsilon > \epsilon^{\ast}$, $N^{\ast} < N_{0}^{\ast}$ occurs, which means that $N^{\ast}$ is now the crossover from the critical fixed point to the biased fixed point. NRG data give the critical-to-biased crossover $T^{\ast} \propto \epsilon^{\theta}$. The critical exponent $\theta$ will be discussed with Fig.5(c).
In Fig.5(b), the evolution of $C(\omega)$ with $\epsilon$ is shown, which looks similar to the case $\alpha < \alpha_c$. That is, $C(\omega)$ does not change much with $\epsilon$ for $\epsilon \ll \epsilon^{\ast}$ and begins to be suppressed for $\epsilon \gg \epsilon^{\ast}$. It is noted that for a given $\epsilon < \epsilon^{\ast}$, the degenerate-to-biased crossover at $N^{\ast} > N_{0}^{\ast}$ in the energy flow has no correspondence in $C(\omega)$: the latter has a perfect $\omega^{s}$ behavior at $\omega \sim \omega^{\ast} = \Lambda^{-N^{\ast}}$. The crossover in the energy flow will only show up in the temperature dependence of $C(\omega)$.
![(color online) Checking the Shiba relation at $s=0.8$ and $\Delta=0.1$. $C(\omega)$ (empty symbols) is compared to $I(\omega) \equiv \alpha \chi^{2} \omega^{s} / 4$ (solid symbols) for various $\alpha$ and $\epsilon$ values. Here $\omega = 1.01561 \times 10^{-8}$. NRG parameters are $\Lambda=2.0$, $M_{s}=200$, and $N_b=16$. []{data-label="Fig6"}](fig6_shiba.eps){width="400pt" height="300pt"}
Fig.4(c) and Fig.5(c) show the the influence of $\epsilon$ on the flow and $C(\omega)$ in the critical regime $\alpha \approx \alpha_c$. Since $T_{0}^{\ast}=0$ at this point, $\epsilon$ is the only energy scale that controls the critical-to-biased crossover in the energy flow and $C(\omega)$. Especially, Fig.5(c) shows that $C(\omega) \propto \omega^{-s}$ for $\omega \gg \omega^{\ast}$ and $C(\omega) \propto \omega^{s}$ for $\omega \ll \omega^{\ast}$. We define an critical exponent $\theta$ as $$\omega^{\ast}(\alpha = \alpha_c, \epsilon) \propto \epsilon^{\theta}.$$ For $s=0.8$, the fitted exponent from Fig.5(c) is $\theta = 1.12$. Note that the same dependence of $\omega^{\ast}$ on $\epsilon$ applies to the $\alpha > \alpha_c$ and $\alpha < \alpha_c$ cases in the regime $\epsilon \gg \epsilon^{\ast}$, which are shown in (a) and (b) of Figs.4 and 5.
The NRG result in Fig.5(c) shows that for $\alpha = \alpha_c$, $C(\omega \gg \omega^{\ast}) \propto c \omega^{-s}$ with an $\epsilon$-independent factor $c$. Combining this observation with the assumption that the Shiba relation Eq.(8) holds at $\alpha = \alpha_c$ and $\epsilon >0$ (which will be discussed in Fig.6 below), we can derive $\theta$ by equating the small and the large frequency expression at $\omega = \omega^{\ast}$, giving $$\left( \omega^{\ast} \right)^{-s} \sim \alpha_c [\chi(\alpha_c, \epsilon)]^{2} \left( \omega^{\ast} \right)^{s}.$$ Employing the critical behavior $\chi(\alpha_c) \propto \epsilon^{1/\delta -1}$, one obtains $\omega^{\ast} \propto \epsilon^{(\delta-1)/(\delta s)}$, giving $\theta = (\delta-1)/(\delta s)$. From the exact expression $\delta=3$ for $0 < s < 1/2$ and $\delta = (1+s)/(1-s)$ for $1/2 \leqslant s < 1$, we obtain $$\begin{aligned}
\theta = \left\{
\begin{array}{lll}
\frac{2}{3s}, \,\, \,\,\, \,\, & (0 < s \leqslant \frac{1}{2} ); \\
& \\
\frac{2}{1+s}, \,\,\,\,\, \,\, & ( \frac{1}{2} < s <1).
\end{array} \right. \end{aligned}$$ For $s=0.8$, this expression gives $\theta=1.111$, which agree well with the NRG result $1.12$.
Having shown that $C(\omega) \propto \omega^{s}$ ($\omega \to 0$) for general $\alpha$ and $\epsilon$ values in the previous section, we now check the Shiba relation Eq.(8) at $s=0.8$ using a fixed small frequency $\omega = 1.01561 \times 10^{-8}$. Fig.6 shows $C(\omega)$ and the right-hand side of Eq.(8) $I(\omega) \equiv \alpha \chi^{2} \omega^{s}/4$ as functions of $\epsilon$, for the same $\alpha$ values as in Figs.4 and 5. We find qualitative agreement between them for a wide range of $\epsilon$. For $\alpha > \alpha_c$ ($\alpha < \alpha_c$), $C(\omega)$ shows the crossover from critical (power law dependence on $\epsilon$) for $\epsilon \gg \epsilon^{\ast}$ to the localized-like (delocalized-like) behavior (being constant) for $\epsilon \ll \epsilon^{\ast}$, as expected. For $\alpha=\alpha_c$, a power-law dependence is observed, with the fitted exponent $-1.88$ (for $C(\omega)$) and $-1.84$ (for $I(\omega)$), respectively, in reasonable agreement with the exact $2(1/\delta -1) = -1.78$. It is notable that although $C(\omega)$ and $I(\omega)$ have more than $5$ decades of variations in the range $\epsilon \in \left[10^{-7} , 10^{-3}\right]$, their ratio $I(\omega)/C(\omega) = 2 \sim 3$ does not change much. This result is a support to the relation $C(\omega) \propto \chi^{2} \omega^{s}$. The ratio also depends weakly on $\omega$ due to the slight inaccuracy in the exponent of NRG-produced $C(\omega)$. The uniform deviation from Eq.(8) observed here is more likely due to the error of NRG data than due to the invalidity of Shiba relation. After all, a factor of $3$ is a reasonable level of error in the NRG calculation of dynamical quantities [@Bulla1], considering the logarithmic error, truncation errors, as well as the approximation used to calculate $C(\omega)$ and $\chi$ (Ref. ). It is expected that the agreement can be improved if we increase $M_{s}$ and $N_{b}$, and extrapolate $\Lambda$ to unity, which will not be pursued here. The good agreement of the NRG value of the exponent $\theta$ and Eq.(11) is also consistent with $C(\omega) \propto \chi^{s}\omega^{s}$, since Eq.(11) is derived from the this assumption.
![ (color online) (a) $C(\omega)$ at $\epsilon=10^{-5}$ and for a series of $\alpha$’s. From top to bottom, $\alpha=0.39$, $0.385$, $0.38$, $0.375$, and $0.37$ (solid lines), and $\alpha=0.395$, $0.40$, $0.405$, and $0.41$ (dashed lines). (b) The maximum value of $C(\omega)$ as functions of $\alpha$ for a series of $\epsilon$. From top to bottom, $\epsilon=0.0$, $2.0 \times 10^{-7}$, $1.0 \times 10^{-6}$, $3.0 \times 10^{-5}$, $9.0 \times 10^{-6}$, $1.8 \times 10^{-5}$, and $2.7 \times 10^{-5}$. $s=0.8$, $\Delta=0.1$. NRG parameters are $\Lambda=2.0$, $M_{s}=100$, and $N_b=12$. []{data-label="Fig7"}](fig7_region2.eps){width="320pt" height="250pt"}
In Fig.7(a), we fix $\epsilon = 10^{-5}$ and plot $C(\omega)$ for different $\alpha$ values. For $\alpha \ll \alpha_c$, $C(\omega)$ increases with increasing $\alpha$ and the height of the peak reaches a maximum at certain $\alpha < \alpha_c$. With further increasing $\alpha$, $C(\omega)$ begins to decrease uniformly, with the peak height suppressed. In Fig.7(b), we show the maximum of $C(\omega)$ as functions of $\alpha$ for various $\epsilon$’s. From this figure we can extract the $\alpha$ value at which the height of the peak in $C(\omega)$ reaches the maximum.
![(color online) Phase diagram of sub-Ohmic SBM near QCP extracted from $C(\omega)$. Region I, II, and III are the delocalized, quantum critical, and the localized phases, respectively. They are separated by a crossover line (solid squares with eye-guiding line) at which the peak height of $C(\omega)$ drops to $90\%$ of its $\epsilon=0$ value. The peak height of $C(\omega)$ reaches maximum at dashed line with empty circles. Inset: log-log plot of $\epsilon^{\ast}$ versus $|\alpha-\alpha_c|$ for $\alpha < \alpha_c$ (solid squares) and $\alpha > \alpha_c$ (solid circles). The fitted exponents are $1.989$ and $1.985$, respectively. []{data-label="Fig8"}](fig8_epi_alpha.eps){width="250pt" height="180pt"}
The phase diagram on $\alpha-\epsilon$ plane shown in Fig.8 summarizes our results for $C(\omega)$ near the QCP. Close to $\alpha_c$ and with very small $\epsilon$, three regions I, II, and III of different nature are separated by a crossover line (squares with eye-guiding line). Region I and III are continuously connected to the delocalized ($\alpha < \alpha_c$) and localized phase ($\alpha > \alpha_c$) of the symmetric SBM, respectively. They are characterized by $\epsilon^{\ast} \ll T_{0}^{\ast}$. In region I, $\langle \sigma_z \rangle \propto \chi \epsilon$ with $\chi$ being the magnetic susceptibility. In region III, $\langle \sigma_z \rangle \sim -1$ close to the saturate value. For each $\alpha$, the crossover $\epsilon^{\ast}$ in Fig.8 is defined as the $\epsilon$ value at which the maximum of $C(\omega)$ decreases to $90\%$ of its value at $\epsilon=0$.
In region II, $T_{0}^{\ast}$ vanishes near $\alpha_c$ and $\epsilon \gg T_{0}^{\ast}$ becomes the only characteristic energy scale. In this regime, the magnetization $\langle \sigma_z \rangle$ shows the quantum critical behavior as $\langle \sigma_z \rangle \propto \epsilon^{1/\delta}$, with $\delta$ being an critical exponent. The empty circles with eye-guiding line marks out the parameter at which $C(\omega)$ has a highest peak. Both the crossover line $\epsilon^{\ast}(\alpha)$ and the peak position can be extracted from the data in Fig.7(b). It is observed that the location of the maximum peak resembles that of the maximum of entanglement entropy [@Hur1]. This is not a pure coincident since the crossover peak of $C(\omega)$ reflects a strong fluctuation near the critical point and it is therefore naturally related to the entanglement maximum.
In the inset of Fig.8, we fit the crossover line close to the QCP in a power law $$\epsilon^{\ast} \propto c|\alpha - \alpha_c|^{\eta},$$ with the obtained critical exponent $\eta=1.99$ for $s=0.8$. Note that the asymmetry of $\epsilon^{\ast}$ in $\alpha > \alpha_c$ and $\alpha < \alpha_c$ regimes means different pre-factors $c$. Besides using $C(\omega)$ here, the critical behavior of $\epsilon^{\ast}$ can also be determined by the scaling form of $\langle \sigma_z \rangle \left( \alpha, \epsilon, \Delta \right)$ obtained from the $N_b$ scaling analysis of the NRG data [@Tong2]. These two approaches give consistent result. For the second approach, the $N_b$ scaling analysis of NRG data gives [@Note1] $$\langle \sigma_z \rangle(\alpha, \epsilon, \Delta) = m \left[ \left( \frac{\tau}{\Delta^{1-s}} \right)^{\beta}, \left( \frac{\epsilon}{\Delta} \right)^{1/\delta} \right].$$ Here $\tau \equiv \alpha - \alpha_c$ and $m(x, y)$ is a two-variable scaling function. By comparing the magnitude of the two variables in Eq.(13), one obtains $\epsilon^{\ast}(\alpha) \propto |\alpha - \alpha_c|^{\beta \delta}$, giving $\eta = \beta \delta$. Here, $\beta$ is the critical exponent of the order parameter. It is known that for $0 \leqslant s \leqslant 1/2$, $\beta = 1/2$ and $\delta=3$. For $1/2 < s < 1$, $\beta$ is a function of $s$ whose explicit expression is unknown yet, and $\delta = (1+s)/(1-s)$ (Refs. ). The NRG result $\beta \delta = 1.93$ is in reasonable agreement with the fitted exponent $\eta=1.99$ in the inset of Fig.8, showing the consistency of the static and dynamical approaches.
For $0 \leqslant s < 1/2$, NRG calculation is hindered by the slow convergence with $N_b$ and it is difficult to obtain the quantitative data. However, we note that the scaling form Eq.(13) is obtained from the NRG calculation combined with the $N_b$ scaling analysis for the full sub-Ohmic regime $0 \leqslant s < 1$. It is stronger than the scaling ansatz of free energy which applies only to the regime $1/2 < s < 1$ (Ref. ). The two exponents introduced above, $\theta$ and $\eta$, are not independent. We have $\epsilon^{\ast} \propto |\alpha - \alpha_c|^{ \beta \delta} \sim \left( T^{\ast} \right)^{\beta \delta / (z\nu) }$. This gives $T^{\ast} \propto \epsilon^{z\nu / (\beta \delta) }$, meaning $\theta = z \nu / \eta$. This is a relation independent of the validity of hyperscaling relation.
[ Summary ]{} In summary, we use the bosonic NRG method to study the equilibrium dynamics of the sub-Ohmic SBM under a bias. We found that the small $\omega$ behvior $C(\omega) \propto \omega^s$ holds for any bias $\epsilon$ and coupling strength $\alpha$ except exactly at the critical point $\alpha = \alpha_c$, $\epsilon=0$, where $C(\omega) \propto \omega^{-s}$. Our results strongly supports that $C(\omega) \propto \chi^{2} \omega^{s}$ in all parameter regimes, including the biased and strong coupling regime. This is in favour of the validity of the generalized Shiba relation in this regime. Close to the QCP, the competition between the energy scale $T^{\ast}_{0}$ and bias $\epsilon$ determines a crossover scale $\epsilon^{\ast}$ which separate the weak-biased regime ($\epsilon \ll \epsilon^{\ast}$) from a strong-biased regime ($\epsilon \gg \epsilon^{\ast}$). Near the QCP, $\epsilon^{\ast} \propto |\alpha-\alpha_c|^{\eta}$ with $\eta = \beta \delta$. These results are summarized in the crossover phase diagram on the $\alpha-\epsilon$ plane.
[Acknowledgments ]{}
This work is supported by 973 Program of China (2012CB921704), NSFC grant (11374362), Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China 15XNLQ03.
[99]{}
A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. [**59**]{}, 1 (1987).
U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993).
A. Shnirman, Y. Makhlin, and G. Schon, Phys. Scr. [**T102**]{}, 147 (2002).
N. H. Tong and M. Vojta, Phys. Rev. Lett. [**97**]{}, 016802 (2006).
D. Rosenberg, P. Nalbach, and D. D. Osheroff, Phys. Rev. Lett. [**90**]{}, 195501 (2003).
Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature (London) [**413**]{}, 804 (2001); P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel, F. Steglich, E. Abrahams, and Q. Si, Science [**315**]{}, 969 (2007).
C. Seoanez, F. Guinea, and A.H. Castro Neto, Europhys. Lett. [**78**]{}, 60002 (2007).
L. Mühlbacher and R. Egger, Chem. Phys. [**296**]{}, 193 (2004).
P. Borri [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 157401 (2001).
S. Kehrein and A. Mielke, Phys. Lett. A [**219**]{}, 313 (1996).
R. Bulla, N. H. Tong, and M. Vojta, Phys. Rev. Lett. [**91**]{}, 170601 (2003); R. Bulla, H. J. Lee, N. H. Tong, and M. Vojta, Phys. Rev. B [**71**]{}, 045122 (2005).
C. Guo, A. Weichselbaum, J. von Delft, and M. Vojta, Phys. Rev. Lett. [**108**]{}, 160401 (2012).
A. Winter, H. Rieger, M. Vojta, and R. Bulla, Phys. Rev. Lett. [**102**]{}, 030601 (2009).
A. Alvermann and H. Fehske, Phys. Rev. Lett. [**102**]{}, 150601 (2009).
A. Chin and M. Turlakov, Phys. Rev. B [**73**]{}, 075311 (2006).
F. B. Anders, R. Bulla, and M. Vojta, Phys. Rev. Lett. [**98**]{}, 210402(2007).
Z. G. Lü and H. Zheng, Phys. Rev. B [**75**]{}, 054302 (2007).
S. Florens, A. Freyn, D. Venturelli, and R. Narayanan, Phys. Rev. B [**84**]{}, 155110 (2011).
K. Le Hur, P. Doucet-Beauprè, and W. Hofstetter, Phys. Rev. Lett. [**99**]{}, 126801 (2007).
Y. Yao, L. Duan, Z. Lü, C. Q. Wu, and Y. Zhao, Phys. Rev. E [**88**]{}, 023303 (2013).
H. B. Liu, J. H. An, C. Chen, Q. J. Tong, H. G. Luo, and C. H. Oh, Phys. Rev. A [**87**]{}, 052139 (2013).
D. Kast and J. Ankerhold, Phys. Rev. Lett. [**110**]{}, 010402 (2013); D. Kast and J. Ankerhold, Phys. Rev. B [**87**]{}, 134301 (2013).
P. Nalbach and M. Torwart, Phys. Rev. B [**81**]{}, 054308 (2010).
K. Le Hur, Phys. Rev. B [**85**]{}, 140506 (2012).
M. Goldstein, M. H. Devoret, M. Houzet, and L. I. Glazman, Phys. Rev. Lett. [**110**]{}, 017002 (2013).
D. J. Egger and F. K. Wilhelm, Phys. Rev. Lett. [**111**]{}, 163601 (2013).
R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. [**80**]{}, 395 (2008).
K. G. Wilson, Rev. Mod. Phys. [**47**]{}, 773 (1975).
Y. H. Hou and N. H. Tong, Eur. Phys. J. B [**78**]{}, 127 (2010).
N. H. Tong and Y. H. Hou, Phys. Rev. B [**85**]{}, 144425 (2012).
R. Bulla, T. A. Costi, and D. Vollhardt, Phys. Rev. B [**64**]{}, 045103 (2001).
T. Liu and Z. X. Cao, arXiv:1312.4606 (unpublished).
M. Sassetti and U. Weiss, Phys. Rev. Lett. [**65**]{}, 2262 (1990).
R. Egger, H. Grabert, and U. Weiss, Phys. Rev. E [**55**]{}, R3809 (1997).
K. Volker, Phys. Rev. B [**58**]{}, 1862 (1998).
T. A. Costi and C. Kieffer, Phys. Rev. Lett. [**76**]{}, 1683 (1996); T. A. Costi, [*ibid.*]{} [**80**]{}, 1038 (1998).
M. Keil and H. Schoeller, Phys. Rev. B [**63**]{}, 180302 (2001).
S. Florens, D. Venturelli, and R. Narayanan, Lect. Notes Phys. [**802**]{}, 145 (2010).
C. Gan and H. Zheng, Phys. Rev. E [**80**]{}, 041106 (2009); C. Zhao, Z. Lu, and H. Zheng, Phys. Rev. E [**84**]{}, 011114 (2011).
Eq.(31), (33), (35), and (36) in Ref. .
M. Vojta, N. H. Tong, and R. Bulla, Phys. Rev. Lett. [**94**]{}, 070604 (2005); Phys. Rev. Lett. [**102**]{}, 249904(E) (2009).
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[ **Solvable lattice models for metals with Z2 topological order** ]{}
Brin Verheijden^1^, Yuhao Zhao^2^, Matthias Punk^3,4\*^
[**1**]{} Physics Department, Ludwig-Maximilians-University Munich, D-80333 München\
[**2**]{} Department of Physics, ETH Zurich, 8093 Zürich, Switzerland\
[**3**]{} Physics Department, Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-University Munich, D-80333 München\
[**4**]{} Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany\
\* matthias.punk@lmu.de
Abstract {#abstract .unnumbered}
========
[**We present quantum dimer models in two dimensions which realize metallic ground states with Z2 topological order. Our models are generalizations of a dimer model introduced in \[PNAS 112, 9552-9557 (2015)\] to provide an effective description of unconventional metallic states in hole-doped Mott insulators. We construct exact ground state wave functions in a specific parameter regime and show that the ground state realizes a fractionalized Fermi liquid. Due to the presence of Z2 topological order the Luttinger count is modified and the volume enclosed by the Fermi surface is proportional to the density of doped holes away from half filling. We also comment on possible applications to magic-angle twisted bilayer graphene.** ]{}
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Introduction {#sec:intro}
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Landau’s Fermi liquid theory is one of the cornerstones of condensed matter physics and is remarkably successful in describing conventional metallic phases of interacting electrons. Despite its wide success, various strongly correlated electron materials show unconventional metallic behavior which does not fit into the Fermi liquid framework. One prime example are the cuprate high-$T_c$ superconductors, which exhibit a distinct non-Fermi liquid or “strange metal” phase around optimal doping [@Gurvitch1987; @Legros2019], as well as a metallic “pseudogap” phase at low hole-doping featuring Fermi-liquid like transport properties, but an anomalously low charge carrier concentration [@Mirazei2013; @Chan2014; @Badoux2016]. Interestingly, phases with a similar non-Fermi liquid phenomenology have been recently observed in magic-angle twisted bilayer graphene [@Cao2018; @Cao2019] .
The theoretical description of unconventional metallic phases in dimensions $d\geq2$ largely focuses on two broad classes of non-Fermi liquids. The first are metals without well-defined electronic quasiparticle excitations and appear e.g. in the phenomenological theory of marginal Fermi liquids [@Varma1989], in the vicinity of metallic quantum critical points [@Lohneysen], or have recently been discussed in the context of SYK models [@Sachdev1993; @Sachdev2015]. A second class of models with non-Fermi liquid phenomenology is based on the concepts of topological order and fractionalization, where electronic degrees of freedom fractionalize into partons, each carrying some of the quantum numbers of an electron [@Anderson1987; @Kotliar1986; @Senthil2000; @Florens2004; @Nandkishore2012; @SachdevZ2]. One striking consequence of topological order in such models is the possibility to modify Luttinger’s theorem, which states that the volume enclosed by the Fermi surface is proportional to the density of electrons in the conduction band for ordinary Fermi liquids [@Oshikawa2000]. By contrast, so-called fractionalized Fermi liquids, which have been introduced originally in the context of heavy Fermion systems [@Senthil2003], provide a prime example for metallic phases with a modified Luttinger count and feature a reconstructed, small Fermi surface in the absence of broken translational symmetries.
In this work we present exctly solvable, two-dimensional lattice models that exhibit metallic ground states with Z2 topological order. These models are defined on the non-bipartite triangular and kagome lattices and are generalizations of a quantum dimer model introduced in Ref. [@Punk2015]. The latter is defined on the square lattice and has been argued to capture some of the unusual electronic properties of the pseudogap phase in underdoped cuprates and is itself a generalization of the well-known Rokhsar-Kivelson (RK) model [@Rokhsar1988]. Its Hilbert space is spanned by hard-core configurations of bosonic spin-singlet dimers, as well as fermionic spin-1/2 dimers carrying charge $q=+e$, both living on nearest neighbor bonds. Fermionic dimers represent a hole in a bonding orbital between two neighboring lattice sites and can be viewed as bound states of a spinon and a holon in a doped resonating valence bond liquid. The density of fermionic dimers is equal to the density of doped holes away from half filling and the model reduces to the RK model at half filling. Subsequent numerical studies of the square lattice model computed single-electron spectral functions and revealed the presence of an anti-nodal pseudogap as well as Fermi arc-like features in the electron spectral function [@Huber]. Furthermore, exact ground-state wave functions were constucted for a specific parameter regime in Ref. [@Feldmeier].
Due to the fact that the model in Ref. [@Punk2015] is defined on the bipartite square lattice and only allows for nearest neighbor dimers, a fractionalized Fermi liquid ground state without broken symmetries only appears by fine-tuning to the special RK-point, where the bosonic dimers form a $U(1)$ spin liquid. This is not a stable phase of matter, however, as the $U(1)$ spin liquid is considered to be confining at large length scales [@Polyakov1977; @Hermele2004]. By contrast, the analogous quantum dimer models on the non-bipartite triangular and kagome lattices constructed in this work feature a stable fractionalized Fermi liquid (FL\*) phase and don’t require fine-tuning. At half-filling, i.e. at a vanishing density of fermionic dimers, these models exhibit an extended Z2 spin liquid phase [@Moessner2001] and realize a metallic Z2-FL\* ground state without broken symmetries upon doping with holes [@SachdevZ2].
Lastly we also note that a metallic phase with an unusually low charge carrier density has been observed in magic-angle twisted bilayer graphene (TBG) [@Cao2018]. It appears upon hole-doping the Mott-like insulating phase at half filling of the lower Moiré mini-band close to charge neutrality, in remarkable analogy with the pseudogap phase in underdoped cuprates. In this work we argue that the triangular lattice quantum dimer model presented here could provide a toy model for the description of this unconventional metallic phase and we point out specific signatures which can be checked in future experiments. In particular we argue that the Fermi surface in the Z2-FL\* phase of the triangular lattice dimer model consists of small hole pockets centered at the M points of the Brillouin zone (i.e. the Moiré mini-Brillouin zone in the case of twisted bilayer graphene). Angle resolved photoemission (ARPES) experiments with a sufficiently high momentum resolution or quasi-particle interference in scanning tunnneling microscopy (STM) experiments on TBG should be able to test this prediction.
The rest of the paper is outlined as follows: in Sec. \[sec:triang\] we introduce the two-species quantum dimer model on the triangular lattice and construct exact ground states at a specific line in parameter space in Sec. \[sec:exact\]. In Sec. \[sec:perturb\] we consider perturbations away from the exactly solvable line and compare the perturbative results with numerical exact diagonalization data. Moreover, we discuss the emergence of a Z2-FL\* ground state and compute the single electron spectral function. Finally, in Sec. \[sec:TBG\] we discuss possible applications of this model to magic-angle twisted bilayer graphene. In the appendix we briefely present a construction of exact ground states for analogous dimer models on the kagome lattice.
Triangular lattice dimer model {#sec:triang}
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As in Ref. [@Punk2015] the Hilbert space of our model is spanned by hard-core coverings of the triangular lattice with two kinds of dimers living on nearest neighbor bonds. A bosonic dimer which represents two electrons in a spin-singlet configuration as in the usual Rokhsar-Kivelson model, as well as a fermionic spin-1/2 dimer carrying electric charge $+e$ with respect to a bosonic dimer background. The latter represents an electron in a bonding orbital delocalized between two neighboring lattice sites and can also be viewed as a tightly bound spinon-holon pair in a doped resonating valence bond liquid. An example of a dimer configuration is shown in Fig. \[fig1\].
Dynamics on this Hilbert space is generated by a Hamiltonian including various local dimer resonance terms which can change the dimer configuration on each elementary plaquette consisting of an up- and a down-facing triangle as shown in Fig. \[fig2\], as well as potential energy terms for two parallel dimers on a plaquette. The terms which we consider here are sketched in Fig. \[fig3\]. We define the canonical bosonic (fermionic) creation and annihilation operators $D^\dagger_{j,\eta}$ and $D_{j,\eta}$ ($F^\dagger_{j,\eta,\sigma}$ and $F_{j,\eta,\sigma}$) of a bosonic (fermionic) dimer emanating from lattice site $j$ in one of the three directions $\mathbf{e}_\eta$ with $\eta \in \{1,2,3\}$, as depicted in Fig. \[fig2\]. For fermionic dimers the index $\sigma$ labels the two spin components. In order to make a connection to the electronic Hilbert space of a Hubbard- or t-J model, the dimer operators can be expressed in terms of electron creation and annihilation operators $c^\dagger_{j\sigma}$ and $c_{j\sigma}$ as $$\begin{aligned}
D^\dagger_{j,\eta} &\sim& \frac{1}{\sqrt{2}} \left( c^\dagger_{j \uparrow} c^\dagger_{j+\mathbf{e}_\eta \downarrow} - c^\dagger_{j \downarrow} c^\dagger_{j+\mathbf{e}_\eta \uparrow} \right) \label{Dop} \ , \\
F^\dagger_{j,\eta,\sigma} &\sim& \frac{1}{\sqrt{2}} \left( c^\dagger_{j \sigma} + c^\dagger_{j+\mathbf{e}_\eta \sigma} \right) \label{Fop} \ ,\end{aligned}$$ up to a phase factor which depends on a gauge choice [@Punk2015]. Using these dimer operators the Hamiltonian takes the form $$\begin{aligned}
H &=& -J \sum_{j} D^\dagger_{j,1} D^\dagger_{j+\mathbf{e}_2,1} D^{\ }_{j,2} D^{\ }_{j+\mathbf{e}_1,2} + \dots \notag \\
&& + V \sum_{j} D^\dagger_{j,1} D^\dagger_{j+\mathbf{e}_2,1} D^{\ }_{j+\mathbf{e}_2,1} D^{\ }_{j,1} + \dots \notag \\
&& - t_1 \sum_{j,\sigma} F^\dagger_{j,1,\sigma} D^\dagger_{j+\mathbf{e}_2,1} F^{\ }_{j+\mathbf{e}_2,1,\sigma} D^{\ }_{j,1} + \dots \notag \\
&& + v_1 \sum_{j,\sigma} F^\dagger_{j,1,\sigma} D^\dagger_{j+\mathbf{e}_2,1} D^{\ }_{j+\mathbf{e}_2,1} F^{\ }_{j,1,\sigma} + \dots \notag \\
&& - t_2 \sum_{j,\sigma} F^\dagger_{j,1,\sigma} D^\dagger_{j+\mathbf{e}_2,1} D^{\ }_{j,2} F^{\ }_{j+\mathbf{e}_1,2,\sigma} + \dots \ ,
\label{ham0}\end{aligned}$$ where the dots indicate hermitean conjugate terms and analogous terms defined on the other two elementary plaquettes shown in Fig. \[fig2\], as well as symmetry related $t_1$, $t_2$ and $v_1$ terms, where bosonic and fermionic dimers are interchanged and/or rotated in the initial or final state. As already mentioned, the density of fermionic dimers is equal to the density of doped holes away from the half-filled electron band. At half-filling, i.e. without any fermionic dimers, only the terms in the first two lines of Eq. remain and the Hamiltonian reduces to the Rokhsar-Kivelson model, which is exactly solvable at the so-called RK-point $J=V$. At this special point the Hamiltonian can be written as a sum of projectors on each elementary plaquette and the ground state is an equal weight superposition of all bosonic dimer coverings. It is important to note that the RK-point in the triangular lattice model is part of an extended Z2 spin liquid phase [@Moessner2001]. This is in contrast to the analogous model on the square lattice, where the RK-point is a singular point in the phase diagram exhibiting a $U(1)$ spin liquid ground state.
Since the Hamiltonian is local, the Hilbert space splits into different topological sectors depending on the lattice topology, in precise analogy to the RK-model on the triangular lattice. Using periodic boundary conditions e.g. in x-direction, the parity associated with the number of dimers crossing an arbitrary closed path around the lattice in x-direction, as indicated by the dashed line in Fig. \[fig1\], is conserved under arbitrary local operations on the dimer Hilbert space, which thus splits into two topological superselection sectors. This Z2 topological order is an important property of metallic phases realized in this model and allows for a modification of the conventional Luttinger count [@Sachdev2016]. As we will see below, the model in Eq. exhibits a metallic ground state where the Fermi volume is proportional to the density of fermionic dimers, i.e. proportional to the density of holes ($p$) away from half-filling. This is in contrast to the conventional Luttinger count for an ordinary Fermi-liquid metal, where the Fermi volume is proportional to the total density of holes measured from the fully filled band (i.e. $1+p$). In the following we are going to construct exact ground states for the full model including fermionic dimers. Note that we did not include purely fermionic plaquette resonance terms in Eq. , as this model is intended to describe systems at small hole doping slightly below half filling, where the density of fermionic dimers is small and such terms are not expected to play a prominent role.
Exact ground state solution {#sec:exact}
===========================
We proceed by constructing the exact ground state of a single fermionic dimer interacting with a background of bosonic dimers at a specific line in parameter space. A generalization to an arbitrary number of fermions (also with spin) is then straightforward. Our notation and construction follows closely Ref. [@Feldmeier].
Exact ground states can be constructed for the special choice of parameters $J=V$ and $v_1=t_2=-t_1$, in which case the Hamiltonian reduces to a sum of projectors $$H=H_\text{RK} + v_1 \sum_{j,\alpha} P_{j,\eta} \ ,
\label{HRK}$$ where $H_\text{RK}$ is the standard Rokhsar-Kivelson Hamiltonian (i.e. the first two lines of Eq. ) at the special RK-point $J=V$ [@Rokhsar1988] and we defined the three different plaquette projectors $P_{j,\eta} = |\phi_{j,\eta}\rangle \langle \phi_{j,\eta}|$ with $\eta \in \{ 1,2,3 \}$ for every lattice site $j$ and plaquette $\eta$ as defined in Fig. \[fig2\]. Furthermore, $$\begin{aligned}
|\phi_{j,1} \rangle &=& \big| {\mathord{\vcenter{\hbox{\includegraphics[height=5ex]{./elem3.pdf}}}}}\big\rangle + \big| {\mathord{\vcenter{\hbox{\includegraphics[height=5ex]{./elem4.pdf}}}}}\big\rangle - \big| {\mathord{\vcenter{\hbox{\includegraphics[height=5ex]{./elem1.pdf}}}}}\big\rangle - \big| {\mathord{\vcenter{\hbox{\includegraphics[height=5ex]{./elem2.pdf}}}}}\big\rangle \ ,\\
|\phi_{j,2} \rangle &=& \big| {\mathord{\vcenter{\hbox{\includegraphics[height=2.5ex]{./elem7.pdf}}}}}\big\rangle + \big| {\mathord{\vcenter{\hbox{\includegraphics[height=2.5ex]{./elem8.pdf}}}}}\big\rangle - \big| {\mathord{\vcenter{\hbox{\includegraphics[height=3ex]{./elem5.pdf}}}}}\big\rangle - \big| {\mathord{\vcenter{\hbox{\includegraphics[height=3ex]{./elem6.pdf}}}}}\big\rangle \ , \\
|\phi_{j,3} \rangle &=& \big| {\mathord{\vcenter{\hbox{\includegraphics[height=2.5ex]{./elem11.pdf}}}}}\big\rangle + \big| {\mathord{\vcenter{\hbox{\includegraphics[height=2.5ex]{./elem12.pdf}}}}}\big\rangle - \big| {\mathord{\vcenter{\hbox{\includegraphics[height=3ex]{./elem9.pdf}}}}}\big\rangle - \big| {\mathord{\vcenter{\hbox{\includegraphics[height=3ex]{./elem10.pdf}}}}}\big\rangle \ .\end{aligned}$$ Here white ellipses denote bosonic spin-singlet dimers and black ellipses represent fermionic dimers. Since we focus on a single fermionic dimer here, the spin index is suppressed. Also note that the Hamiltonian in Eq. is positive semi-definite and thus all states with zero energy are automatically ground states.
We define basis states with one fermionic dimer fixed at a position specified by the lattice site $j$ and the direction $\eta \in \left\{1,2,3\right\}$ (see Fig. \[fig2\] for a definition) $$|(j,\eta)\rangle = F^\dagger_{j,\eta} D^{ \ }_{j,\eta} | RK \rangle \ ,$$ where $|RK \rangle$ is the usual Rokhsar-Kivelson wave function, i.e. the equal weight superposition of all hard-core coverings with bosonic spin-singlet dimers in a given topological sector. It is important to note that these basis states are ground states of the RK Hamiltonian $H_\text{RK}$ at the exactly solvable RK point $J=V$ per construction, i.e. $H_\text{RK} |(j,\eta)\rangle = 0$, because applying the RK Hamiltonian to a plaquette with a fermionic dimer gives zero. Moreover, applying the plaquette projectors $P_{\ell,\alpha} $ to these basis states gives $$\begin{aligned}
P_{\ell,1} |(j,\eta)\rangle &=& \left[ \delta_{\eta,1} (\delta_{j,\ell}+\delta_{j,\ell+e_2}) - \delta_{\eta,2} (\delta_{j,\ell}+\delta_{j,\ell+e_1})\right] |\phi_{\ell,1} \rangle \ , \ \ \ \ \ \ \ \label{proj1} \\
P_{\ell,2} |(j,\eta)\rangle &=& \left[ \delta_{\eta,2} (\delta_{j,\ell}+\delta_{j,\ell+e_3}) - \delta_{\eta,3} (\delta_{j,\ell}+\delta_{j,\ell+e_2})\right] |\phi_{\ell,2} \rangle \ , \ \ \ \ \ \ \ \label{proj2} \\
P_{\ell,3} |(j,\eta)\rangle &=& \left[ \delta_{\eta,1} (\delta_{j,\ell}+\delta_{j,\ell+e_3}) - \delta_{\eta,3} (\delta_{j,\ell}+\delta_{j,\ell+e_1})\right] |\phi_{\ell,3} \rangle \ . \ \ \ \ \ \ \ \label{proj3} \end{aligned}$$ At the exactly solvable line the ground state turns out to be highly degenerate and different ground states $|\mathbf{p}\rangle$ can be parametrized by a lattice momentum $\mathbf{p}$. We make the ansatz $$|\mathbf{p}\rangle = \sum_{j,\eta} e^{i \mathbf{p} \cdot \mathbf{R}_j} c_\eta(\mathbf{p}) |(j,\eta)\rangle \ ,
\label{GSansatz}$$ where $\mathbf{R}_j$ is the lattice position of site $j$ and determine the complex coefficients $c_\eta(\mathbf{p})$ by demanding that $H | \mathbf{p} \rangle = 0$. Since $H_\text{RK} |\mathbf{p}\rangle = 0$ per construction, this immediately leads to the three equations $$\sum_{\ell} P_{\ell,\eta} |\mathbf{p}\rangle = 0$$ for each $\eta \in \{1,2,3\}$. Using Eqs. , , these equations reduce to $$\begin{aligned}
c_1(\mathbf{p}) \left( 1+ e^{i p_2} \right) - c_2(\mathbf{p}) \left( 1+ e^{i p_1} \right) & = & 0 \ , \\
c_2(\mathbf{p}) \left( 1+ e^{i p_3} \right) - c_3(\mathbf{p}) \left( 1+ e^{i p_2} \right) & = & 0 \ , \\
c_1(\mathbf{p}) \left( 1+ e^{i p_3} \right) - c_3(\mathbf{p}) \left( 1+ e^{i p_1} \right) & = & 0 \ ,\end{aligned}$$ where we defined $p_\eta \equiv \mathbf{p} \cdot \mathbf{e}_\eta$. Together with the normalization condition $$\langle \mathbf{p} | \mathbf{p} \rangle = N Q_c[(j,\eta)] \sum_\eta |c_\eta(\mathbf{p})|^2 = 1 \ ,
\label{normaliz}$$ with $N$ the number of lattice sites and $Q_c[(j,\eta)] = \langle (j,\eta) | (j,\eta) \rangle =1/6$ as the average dimer density per bond, these equations are solved by $$c_\eta(\mathbf{p}) = \sqrt{\frac{6}{N}} \, \frac{1+e^{i p_\eta}}{\sqrt{\sum_\zeta | 1+e^{i p_\zeta}|^2}} \ .
\label{coeff}$$ The above construction can be straightforwardly generalized to an arbitrary number $N_f$ of fermionic dimers, since any of the projectors in the Hamiltonian applied to a plaquette with more than one fermionic dimer gives zero. Accordingly the states $$|\mathbf{p}_1, \dots \mathbf{p}_{N_f} \rangle = \sum_{j_1, \eta_1, \dots j_{N_f}, \eta_{N_f}} e^{i \sum_j \mathbf{p}_j \cdot \mathbf{R}_j} c_{\eta_1}(\mathbf{p}_1) \dots c_{\eta_{N_f}}(\mathbf{p}_{N_f}) \, | (j_1,\eta_1),\dots,(j_{N_f}, \eta_{N_f})\rangle \ ,
\label{exactGSmulti}$$ with the coefficients $c_\eta(\mathbf{p})$ given in Eq. are exact, zero-energy ground states of the Hamiltonian with $N_f$ fermionic dimers. Also note that these states are anti-symmetric as required, since the basis states $| (j_1,\eta_1),\dots,(j_{N_f}, \eta_{N_f})\rangle = F^\dagger_{j_1,\eta_1} D^{\ }_{j_1,\eta_1} \dots F^\dagger_{j_{N_f},\eta_{N_f}} D^{\ }_{j_{N_f},\eta_{N_f}} |\text{RK}\rangle$ are anti-symmetric under the exchange of two fermionic dimers.
Perturbing away from the exactly solvable line: Z2-FL\* {#sec:perturb}
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The massive ground state degeneracy is immediately lifted away from the exactly solvable line. Within first order perturbation theory the ground states $|\mathbf{p}\rangle$ obtain an energy shift $\varepsilon(\mathbf{p})$, which can be interpreted as a fermion dispersion. Accordingly, at a finite density of fermionic dimers the ground state corresponds to a Fermi sea with the lowest energy states $|\mathbf{p}\rangle$ occupied up to the Fermi energy. This state realizes a fractionalized Fermi liquid with Z2 topological order ( Z2-FL\*) and a small Fermi surface, the volume of which is determined by the density of fermionic dimers, i.e. the density of holes ($p$) away from half filling, rather than the total density of holes ($1+p$) measured from the completely filled band.
Here we compute the dispersion $\varepsilon(\mathbf{p})$ for small deviations of the amplitude $t_1 \to t_1+\delta t_1$ in the Hamiltonian away from the exactly solvable line $v_1=t_2=-t_1$ using first order perturbation theory. A straightforward computation gives $$\begin{aligned}
\varepsilon(\mathbf{p}) &=& \langle \mathbf{p} | \Delta H | \mathbf{p} \rangle \notag \\
&=& - \delta t_1 \sum_{i, \eta_i,j,\eta_j.\ell} e^{-i \mathbf{p} \cdot (\mathbf{R}_i-\mathbf{R}_j)} c^*_{\eta_i}(\mathbf{p}) c_{\eta_j}(\mathbf{p}) \, \langle (i,\eta_i)| F^\dagger_{\ell,1,\sigma} D^\dagger_{\ell+e_2,1} F^{\ }_{\ell+e_2,1,\sigma} D^{\ }_{\ell,1} + \dots | (j,\eta_j) \rangle \notag \\
&=& - \frac{2 \, \delta t_1}{3} \frac{\epsilon^2_{\mu \nu \lambda} (1+\cos p_\mu) ( \cos p_\nu + \cos p_\lambda )}{\sum_\zeta | 1+e^{i p_\zeta}|^2} \ ,
\label{dispexact}\end{aligned}$$ where $\epsilon_{\mu\nu\lambda}$ is the antisymmetric tensor and sums over repeated greek indices $\mu,\nu,\lambda \in \{1,2,3\}$ are implied. In the last line we used that $\langle (\ell,1) | (\ell+\mathbf{e}_2, 1) \rangle = Q_c[(\ell,1),(\ell+\mathbf{e}_2, 1)] =1/18$ is the classical dimer correlation function of two parallel dimers on a plaquette, which can be computed using standard Grassmann techniques [@Fendley2002].
In Fig. \[fig4\] we check the validity of the perturbative expansion by comparing Equ. with numerical results. Shown is the dispersion along the high symmetry line $\Gamma-M-K-\Gamma$ for $\delta t_1=-0.01$. The numerical results were obtained using Lanczos exact diagonalization of the Hamiltonian $\eqref{ham0}$, where we computed the ground state energy of a single fermionic dimer with fixed momentum $\mathbf{p}$ interacting with a background of bosonic dimers on lattices of size $4\times4$ and $6\times6$ with twisted boundary conditions to increase the momentum resolution. The numerical results show reasonably good agreement with the perturbative result in Eq. and the small discrepancies can be attributed to finite size effects.
Fig. \[fig5\] shows a contour plot of the perturbative dispersion $\varepsilon(\mathbf{p})$ for $\delta t_1 = -0.25$. Note that the dispersion minima are located at the $M$ points on the edges of the Brillouin zone. This feature persists away from the exactly solvable line and is a generic feature of the fermion dispersion for realistic parameter choices. Indeed, the dimer resonance amplitudes can be estimated by computing matrix elements of a simple nearest-neighbor tight binding Hamiltonian $H_t = -t \sum_{\langle i,j \rangle} c^\dagger_i c_j$ of electrons on the triangular lattice of the form $$\begin{aligned}
t_1 &\simeq& - \langle {\mathord{\vcenter{\hbox{\includegraphics[height=2.5ex]{./elem11.pdf}}}}}| H_t | {\mathord{\vcenter{\hbox{\includegraphics[height=2.5ex]{./elem12.pdf}}}}}\rangle = - \frac{3 t}{4} \ , \\
t_2 &\simeq& - \langle {\mathord{\vcenter{\hbox{\includegraphics[height=2.5ex]{./elem11.pdf}}}}}| H_t | {\mathord{\vcenter{\hbox{\includegraphics[height=3ex]{./elem10.pdf}}}}}\rangle = \frac{t}{2} \ , \end{aligned}$$ where $t$ is the nearest-neighbor electron hopping amplitude. Note in particular that $t_1$ is negative and $50\%$ larger in magnitude than $t_2$. This puts the parameters in the vicinity of the exactly solvable line, where $t_1=-t_2$. Accordingly we might suspect that the exact solution captures physical properties in the realistic parameter regime. In Fig. \[fig6\] we show a plot of the numerically obtained fermion dispersion for the choice of parameters $J=V=1$, $t_1=-3$, $t_2=2$ and $v_1=2$, corresponding to a nearest neighbor electron hopping amplitude $t=4J$. Again, the data was obtained using Lanczos exact diagonalization of the Hamiltonian $\eqref{ham0}$ for a single fermionic dimer with fixed momentum on a lattice of size $4\times4$ with twisted boundary conditions. Note that the minima of the fermion dispersion remain at the $M$ points for this choice of parameters.
It is also important to note that the Fermi surface of the fermionic dimers as obtained from the dispersion $\varepsilon(\mathbf{p})$ in Eq. has a direct imprint on the electronic Fermi surface. The argument is analogous to Ref. [@Feldmeier] and here we focus on the main idea. Defining the operators $$f^\dagger_{\mathbf{p},\sigma} = \sum_{j,\eta} e^{i \mathbf{p} \cdot \mathbf{R}_j} c_\eta(\mathbf{p}) F^\dagger_{j,\eta,\sigma} D^{\ }_{j,\eta}$$ and acting with them on the bosonic RK ground state $|RK\rangle$ generates the exact ground states in Equ. . Note that we’ve reintroduced the spin label here. The operators $f^\dagger_{\mathbf{p},\sigma}$ and $f_{\mathbf{p},\sigma}$ obey canonical anti-commutation relations $\big\{ f^{\ }_{\mathbf{p},\sigma} , f^\dagger_{\mathbf{q},\sigma'} \big\} = \delta_{\mathbf{p},\mathbf{q}} \delta_{\sigma,\sigma'}$ in the thermodynamic limit on the Hilbert space spanned by the states in Equ. . This can be shown by computing $\big| \big\{ f^{\ }_{\mathbf{p},\sigma} , f^\dagger_{\mathbf{q},\sigma'} \big\} |RK\rangle \big|^2$ and realizing that it is proportional to the Fourier transform of the classical dimer correlation function. Within first order perturbation theory the Hamiltonian thus takes the form of a free fermion Hamiltonian $H = \sum_{\mathbf{p},\sigma} \varepsilon(\mathbf{p}) f^\dagger_{\mathbf{p},\sigma} f^{\ }_{\mathbf{p},\sigma}$ in the vicinity of the exactly solvable line.
Lastly, the operators $f^\dagger_{\mathbf{p},\sigma}$ can be directly related to electron annihilation operators $c_{\mathbf{p},\sigma}$, which can be seen as follows. On the dimer Hilbert space the electron destruction operator $c_{j,\sigma}$ on lattice site $j$ can be uniquely written in terms of the dimer operators in Eqs. , as [@Punk2015] $$c_{j,\sigma} =\frac{\epsilon_{\sigma \sigma'}}{2} \sum_\eta \left( F^\dagger_{j,\eta,\sigma'} D^{\ }_{j,\eta} + F^\dagger_{j-\mathbf{e}_\eta,\eta,\sigma'} D^{\ }_{j-\mathbf{e}_\eta,\eta} \right) \ ,$$ which can be checked by computing matrix elements on both sides of the equation in the dimer Hilbert space. Fourier transforming this relation it is straightforward to show that $$c_{\mathbf{p},\sigma} = \epsilon_{\sigma,\sigma'} \sqrt{Z_\mathbf{p}} \, f^\dagger_{-\mathbf{p},\sigma'} \ ,$$ where $Z_\mathbf{p}=\sum_\eta | 1+ e^{-i p_\eta} |^2 / 24$ is the electronic quasiparticle residue. In the vicinity of the exactly solvable line the electron spectral function thus takes the form $$A_e(\mathbf{p},\omega) = Z_\mathbf{p} \, \delta\big(\omega-\varepsilon(\mathbf{p})+\mu\big) \ ,$$ where we introduced the chemical potential $\mu$ to fix the average density of fermionic dimers. This demonstrates that the small Fermi surface of fermionic dimers is directly imprinted on the electronic Fermi surface and proves that the ground state is indeed a fractionalized Fermi liquid with Z2 topological order. It features a small Fermi surface with an enclosed volume proportional to the density of doped holes away from half filling in the absence of any broken symmetries.
Application to Twisted Bilayer Graphene {#sec:TBG}
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In this section we comment on a possible application of the previously introduced triangular lattice quantum dimer model in the context of magic-angle twisted bilayer graphene (TBG). Recent experiments on TBG showed correlated Mott-like insulating behavior at half filling of the lower Moiré mini-band near the charge neutrality point [@Cao2018; @Cao2018b; @Lu2019]. Moreover, TBG becomes superconducting upon doping the correlated insulator with electons or holes. Interestingly, Shubnikov - de Haas as well as Hall resistivity measurements above the superconducting transition temperature on the hole-doped side of the correlated insulator indicate an anomalously small charge carrier density, which is equal to the density of doped carriers measured from the correlated insulator at half filling [@Cao2018]. This is in remarkable analogy to Hall measurements in the pseudogap phase of underdoped cuprates [@Badoux2016]. One possible explanation of a small carrier density involves translational symmetry breaking orders, which enlarge the size of the unit cell and reconstruct the Fermi surface into small pockets. A different possibility is the presence of topological order, which can give rise to small pocket Fermi surfaces and a small carrier density as well, as discussed in this work. So far no symmetry breaking orders have been observed in TBG which could explain the small carrier density, consequently the Z2-FL\* scenario discussed here could provide a viable explanation.
The microscopic details of TBG are rather complex and a lot of effort has been put into developing realistic tight binding models [@Koshino2018; @Kang2018; @Po2018; @Po2019]. In the following we argue that the triangular lattice dimer model studied here can be viewed as a very simplistic toy model to gain intuition about the unconventional metallic state near the Mott-like insulator at half filling of the lower Moiré mini-band. Indeed, electric charge in TBG is localized at the AA stacked regions of the two twisted graphene layers, which form a triangular lattice [@Lucian2011]. Even though symmetry constraints preclude the definition of a triangular lattice tight binding model for TBG and microscopic models have a lower rotational symmetry, it is expected that the six-fold rotational symmetry of the triangular lattice emerges in effective low-energy theories. As such, triangular lattice models like the one discussed here should be able to capture effects of local correlations on the low-energy properties of TBG [@Thomson].
An additional important complication is the two-fold valley degeneracy in TBG. In our simplistic picture this would lead to two degenerate orbitals per triangular lattice site due to the valley degree of freedom, i.e. a four fold degeneracy per lattice site in total, if spin is included. This fourfold degeneracy is indeed observed in the Landau level structure emanating from the charge neutrality point [@Cao2018]. However, the Landau level degeneracy in the metallic state emanating from the Mott-like insulator at half filling is only two-fold. It has been argued that this could be due to the presence of so-called inter-valley coherent order, where the valley quantum number is lost [@Po2018]. The spin quantum number is likely to remain unaffected, since the superconductor, which develops out of this unconventional metallic state, can be suppressed by a parallel magnetic field and thus spin-singlet pairing is probable. Using this scenario as a basis, our triangular quantum dimer model can be viewed as a simple toy-model for the remaining two-fold degenerate spin-states per triangular lattice site in a conjectured inter-valley coherent phase and offers a possible explanation for the low-charge carrier density upon hole-doping the Mott-like insulator. As discussed in the main part of this manuscript, a prominent signature of the Z2-FL\* state are the small hole-pockets centered at the M points of the Moiré mini-Brillouin zone in the absence of translational symmetry breaking orders. However, the Fermi-pockets at the three distinct M points should give rise to an additional three-fold degeneracy of the Landau-levels, which is not observed in experiments. This could be due to two reasons: either the Fermi surface undergoes an additional reconstruction in the presence of a strong magnetic field, or our simple estimate of the dimer hopping amplitudes from a single-band, nearest-neighbor triangular tight-binding model does not hold for TBG and the dispersion minimum is in fact at the $\Gamma$ point. In any case, the location of the small Fermi pockets in TBG could be determined in principle using either ARPES measurements with very high energy and momentum resolution, which is currently out of reach, or using quasi-particle interference in STM experiments. We also note here that a different route towards a description of metallic states with Z2 topological order in TBG is discussed in Ref. [@Thomson].
Discussion and Conclusions {#sec:conclusions}
==========================
We presented a generalized, two-species quantum dimer model on the triangular lattice which features a fractionalized Fermi liquid ground state with Z2 topological order (Z2-FL\*). An exact ground state solution was constructed for a specific line in parameter space. At this exactly solvable line the ground state is highly degenerate and can be interpreted as a flat fermionic band. Using perturbation theory we computed the fermion dispersion away from the exactly solvable line and showed that the ground state is indeed a Z2-FL\* with a modified Luttinger count, where the electronic Fermi surface encloses a volume proportional to the density of doped holes away from half filling. In particular we showed that the Fermi surface consists of small hole pockets in the vicinity of the M points on the edges of the Brillouin zone at low doping. This feature is quite robust for realistic parameter choices derived from a single-band tight-binding model of electrons on a triangular lattice and can be used as a signature of the Z2-FL\* state on the triangular lattice.
Even though we focused on the unconventional metallic Z2-FL\* state in this work, the complete phase diagram of this model also contains symmetry broken states that we did not consider here. In the undoped case, transitions to symmetry broken valence bond solid (VBS) phases can be described as confinement transitions of the effective Z2 gauge theory for the resonating valence bond phase [@Wegner1971; @Huh2011]. It is an interesting open problem how such confinement transitions are affected by the presence of a finite density of fermionic dimers. Related questions on the square lattice have been studied recently using sign-problem free quantum Monte-Carlo [@Gazit2017; @Gazit2018] and it would be interesting to extend this approach to the model studied here.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Johannes Feldmeier and Sebastian Huber for helpful discussions.
#### Funding information
This research was supported by the German Excellence Initiative via the Nano Initiative Munich (NIM) as well as by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868.
Kagome lattice dimer model
==========================
The construction presented in Sec. \[sec:exact\] can be straightforwardly generalized to different lattice geometries, such as the kagome lattice, which we will briefly discuss in this appendix. A general dimer Hamiltonian again consists of various plaquette resonance terms, in analogy to Eq. \[ham0\] on the triangular lattice. We consider terms on the four different plaquettes shown in Fig. \[fig7\], namely the elementary hexagon, as well as the three diamond shaped loops of length eight. Instead of specifying the various resonance terms in the Hamiltonian in detail, we directly define an exactly solvable Hamiltonian in terms of plaquette projectors, as in Eq. . Again, we keep an RK-like term $H_\text{RK}$, which ensures that the ground state without fermionic dimers is an equal weight superposition of all bosonic dimer coverings. Moreover, we define the four plaquette projectors $P_{j,\eta} = | \phi_{j,\eta} \rangle \langle \phi_{j,\eta} |$ involving a fermionic dimer, where $$\begin{aligned}
|\phi_{j,1} \rangle &=& {\mathord{\vcenter{\hbox{\includegraphics[height=4ex]{./elemKagome1.pdf}}}}}\ , \ \ \ \ \ \\
|\phi_{j,2} \rangle &=& {\mathord{\vcenter{\hbox{\includegraphics[height=6ex]{./elemKagome2.pdf}}}}}\ \ \ \ \ \ \ \end{aligned}$$ and analogous terms $|\phi_{j,3} \rangle$ and $|\phi_{j,4} \rangle$ for the other two diamond shaped plaquettes. Note that we have six independent bonds per kagome unit cell (see Fig. \[fig7\]) and thus we need to determine six coefficients $c_\eta(\mathbf{p})$ with $\eta \in \{ 1,\dots,6 \}$ in our ansatz for the ground state wave function in Eq. . Consequently we would need five linearly independent plaquette projectors together with the normalization condition in order to uniquely specify the coefficients $c_\eta(\mathbf{p})$. With our four plaquette operators the coefficients are underdetermined, but we can define exact ground states nonetheless.
Applying the four projectors to our ansatz and requiring that the eigenenergy is zero leads to the four equations $$\begin{aligned}
c_1(\mathbf{p}) + c_2(\mathbf{p}) + c_3(\mathbf{p}) + c_4(\mathbf{p}) + c_5(\mathbf{p}) + c_6(\mathbf{p}) &=& 0 \ , \\
c_1(\mathbf{p}) \left( 1+ e^{i 2 p_1} \right) + c_2(\mathbf{p}) \left( 1+ e^{i 2 p_3} \right) + c_4(\mathbf{p}) \left( 1+ e^{-i 2 p_1} \right) + c_5(\mathbf{p}) \left( 1+ e^{-i 2 p_3} \right) &=& 0 \ , \\
c_2(\mathbf{p}) \left( 1+ e^{-i 2 p_2} \right) + c_3(\mathbf{p}) \left( 1+ e^{-i 2 p_1} \right) + c_5(\mathbf{p}) \left( 1+ e^{i 2 p_2} \right) + c_6(\mathbf{p}) \left( 1+ e^{i 2 p_1} \right) &=& 0 \ , \\
c_1(\mathbf{p}) \left( 1+ e^{-i 2 p_2} \right) + c_3(\mathbf{p}) \left( 1+ e^{i 2 p_3} \right) + c_4(\mathbf{p}) \left( 1+ e^{i 2 p_2} \right) + c_6(\mathbf{p}) \left( 1+ e^{-i 2 p_3} \right) &=& 0 \ .\end{aligned}$$ Solving these equations for $c_{2,3,5,6}$ together with the normalization condition Eq. , where the mean dimer density per bond on the kagome lattice is $Q_c[(i,\eta)] = 1/6$, and setting $c_4 = - c_1$ gives a particularly simple solution. We finally obtain $$\begin{aligned}
c_1(\mathbf{p}) & = & - c_4(\mathbf{p}) \ \ = \ \ \sqrt{\frac{6}{N}} \frac{|\sin (2 p_3) |}{\sqrt{3-\cos (4 p_1) - \cos (4 p_2) - \cos (4 p_3)}} \ , \\
c_2(\mathbf{p}) & = & - c_5(\mathbf{p}) \ \ = \ \ -c_1(\mathbf{p}) \ \frac{e^{i 2 (p_1-p_3)} \big( 1-e^{-i 4 p_1}\big)}{1-e^{-i 4 p_3}} \ , \\
c_3(\mathbf{p}) & = & - c_6(\mathbf{p}) \ \ = \ \ c_1(\mathbf{p}) \ \frac{e^{i 2 (p_2-p_3)} \big( 1-e^{-i 4 p_2}\big)}{1-e^{-i 4 p_3}} \ .\end{aligned}$$ Again, this solution can be straightforwardly extended to an arbitrary number of fermionic dimers.
[99]{} M. Gurvitch and A. T. Fiory, [*Resistivity of La$_{1.825}$Sr$_{0.175}$CuO$_4$ and YBa$_2$Cu$_3$O$_7$ to 1100 K: Absence of saturation and its implications*]{}, Phys. Rev. Lett. **59**, 1337 (1987), . A. Legros, S. Benhabib, W. Tabis, F. Laliberté, M. Dion, M. Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Z. Li, P. Auban-Senzier, N. Doiron-Leyraud, P. Fournier, D. Colson, L. Taillefer, C. Proust, [*Universal T-linear resistivity and Planckian dissipation in overdoped cuprates*]{}, Nature Physics **15**, 142?147 (2019), . S. I. Mirzaei, D. Stricker, J. N. Hancock, C. Berthod, A. Georges, E. van Heumen, M. K. Chan, X. Zhao, Y. Li, M. Greven, N. Barisić, D. van der Marel, [*Spectroscopic evidence for Fermi liquid-like energy and temperature dependence of the relaxation rate in the pseudogap phase of the cuprates*]{}, Proc. Natl. Acad. Sci. **110**, 5774-5778 (2013), . M. K. Chan, M. J. Veit, C. J. Dorow, Y. Ge, Y. Li, W. Tabis, Y. Tang, X. Zhao, N. Barisić, and M. Greven, [*In-Plane Magnetoresistance Obeys Kohler’s Rule in the Pseudogap Phase of Cuprate Superconductors*]{}, Phys. Rev. Lett. **113**, 177005 (2014), . S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, C. Proust, [*Change of carrier density at the pseudogap critical point of a cuprate superconductor*]{}, Nature **531**, 210 (2016), . Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, P. Jarillo-Herrero, [*Magic-angle graphene superlattices: a new platform for unconventional superconductivity*]{}, Nature **556**, 43-50 (2018), . Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, P. Jarillo-Herrero, [*Strange metal in magic-angle graphene with near Planckian dissipation*]{}, <http://arxiv.org/abs/1901.03710>. C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, [*Phenomenology of the normal state of Cu-O high-temperature superconductors*]{}, Phys. Rev. Lett. **63**, 1996 (1989), . H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, [*Fermi-liquid instabilities at magnetic quantum phase transitions*]{}, Rev. Mod. Phys. **79**, 1015 (2007), . S. Sachdev and J. Ye, [*Gapless spin-fluid ground state in a random quantum Heisenberg magnet*]{}, Phys. Rev. Lett. **70**, 3339 (1993), . S. Sachdev, [*Bekenstein-Hawking Entropy and Strange Metals*]{}, Phys. Rev. X **5**, 041025 (2015), . P. W. Anderson, [*The Resonating Valence Bond State in $La_2CuO_4$ and Superconductivity*]{}, Science **235**, 1196-1198 (1987), . G. Kotliar, A. E. Ruckenstein, [*New Functional Integral Approach to Strongly Correlated Fermi Systems: The Gutzwiller Approximation as a Saddle Point*]{}, Phys. Rev. Lett. **57**, 1362 (1986), . T. Senthil, M. P. A. Fisher, [*$Z_2$ gauge theory of electron fractionalization in strongly correlated systems*]{}, Phys. Rev. B **62**, 7850 (2000), . S. Florens, A. Georges, [*Slave-rotor mean-field theories of strongly correlated systems and the Mott transition in finite dimensions*]{}, Phys. Rev. B **70**, 035114 (2004), . R. Nandkishore, M. A. Metlitski, T. Senthil, [*Orthogonal metals: The simplest non-Fermi liquids*]{}, Phys. Rev. B 86, 045128 (2012), . S. Sachdev, [*Topological order, emergent gauge fields, and Fermi surface reconstruction*]{}, Reports on Progress in Physics **82**, 014001 (2019), . M. Oshikawa, [*Topological Approach to Luttinger’s Theorem and the Fermi Surface of a Kondo Lattice*]{}, Phys. Rev. Lett. **84**, 3370 (2000), . T. Senthil, S. Sachdev, M. Vojta, [*Fractionalized Fermi Liquids*]{}, Phys. Rev. Lett. **90**, 216403 (2003), . M. Punk, A. Allais, S. Sachdev, [*Quantum dimer model for the pseudogap metal*]{}, Proc. Natl. Acad. Sci. **112** 9552-9557 (2015), . D. S. Rokhsar, S. A. Kivelson, [*Superconductivity and the Quantum Hard-Core Dimer Gas*]{}, Phys. Rev. Lett. **61**, 2376 (1988), . S. Huber, J. Feldmeier, M. Punk, [*Electron spectral functions in a quantum dimer model for topological metals*]{}, Phys. Rev. B **97**, 075144 (2018), . J. Feldmeier, S. Huber, M. Punk, [*Exact solution of a two-species quantum dimer model for pseudogap metals*]{}, Phys. Rev. Lett. **120**, 187001 (2018), . A. M. Polyakov, [*Quark confinement and topology of gauge theories*]{}, Nucl. Phys. B **120**, 429 (1977), . M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, X.-G. Wen, [*Stability of U(1) spin liquids in two dimensions*]{}, Phys. Rev. B **70**, 214437 (2004), . R. Moessner , S. L. Sondhi, [*Resonating Valence Bond Phase in the Triangular Lattice Quantum Dimer Model*]{}, Phys. Rev. Lett. **86**, 1881 (2001), . S. Sachdev, D. Chowdhury, Progress of Theoretical and Experimental Physics 12C102 (2016), . P. Fendley, R. Moessner, S. L. Sondhi, [*Classical dimers on the triangular lattice*]{}, Phys. Rev. B **66**, 214513 (2002), . Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, P. Jarillo-Herrero, [*Correlated Insulator Behaviour at Half-Filling in Magic Angle Graphene Superlattices*]{}, Nature **556**, 80-84 (2018), . X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold, A. H. MacDonald, D. K. Efetov, [*Superconductors, Orbital Magnets, and Correlated States in Magic Angle Bilayer Graphene* ]{}, <http://arxiv.org/abs/1901.03710>. M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, L. Fu, [*Maximally-localized Wannier orbitals and the extended Hubbard model for the twisted bilayer graphene*]{}, Phys. Rev. X **8**, 031087 (2018), . J. Kang, O. Vafek, [*Symmetry, maximally localized Wannier states, and low energy model for the twisted bilayer graphene narrow bands*]{}, Phys. Rev. X **8**, 031088 (2018), . H. C. Po, L. Zou, A. Vishwanath, T. Senthil, [*Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene*]{}, Phys. Rev. X **8**, 031089 (2018), . H. C. Po, L. Zou, T. Senthil, A. Vishwanath, [*Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene*]{}, Phys. Rev. B **99**, 195455 (2019), . A. Luican, G. Li, A. Reina, J. Kong, R. R. Nair, K. S. Novoselov, A. K. Geim, and E. Y. Andrei, [*Single-Layer Behavior and Its Breakdown in Twisted Graphene Layers*]{}, Phys. Rev. Lett. **106**, 126802 (2011), . A. Thomson, S. Chatterjee, S. Sachdev, and M. S. Scheurer, [*Triangular antiferromagnetism on the honeycomb lattice of twisted bilayer graphene*]{}, Phys. Rev. B **98**, 075109 (2018), . F. Wegner, [*Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter*]{}, J. Math. Phys. **12**, 2259 (1971), . Y. Huh, M. Punk, S. Sachdev, [*Vison states and confinement transitions of Z2 spin liquids on the kagome lattice*]{}, Phys. Rev. B **94**, 094419 (2011), . S. Gazit, M. Randeria, A. Vishwanath, [*Emergent Dirac fermions and broken symmetries in confined and deconfined phases of Z2 gauge theories*]{}, Nature Physics **13**, 484 (2017), . S. Gazit, F. F. Assaad, S. Sachdev, A. Vishwanath, C. Wang, [*Confinement transition of Z2 gauge theories coupled to massless fermions: emergent QCD3 and SO(5) symmetry*]{}, Proc. Natl. Acad. Sci. **115**, E6987 (2018), .
|
---
abstract: |
First searches for the coherent dissociation of relativistic oxygen nuclei into four a particles are reported. It is shown that reactions of this type are characterized by a significantly lower decay temperature than the conventional multifragmentation of residual projectile nuclei. The momentum spectra and correlations of a panicles are not reproduced by the simple statistical model of direct fragmentation. The possibility that the oxygen nucleus undergoing fragmentation acquires a nonzero angular momentum in the collision process is discussed.
DOI: 1063-7788/96/5901-0102S10.00
author:
- 'N. P. Andreeva'
- 'Z. V. Anzon'
- 'V. I. Bubnov'
- 'A. Sh. Gaitinov'
- 'G. Zh. Eligbaeva'
- 'L. E. Eremenko'
- 'G. S. Kalyachkina'
- 'E. K. Kanygina'
- 'A. M. Seitimbetov'
- 'I. Ya. Chasnikov'
- 'Ts. I. Shakhova'
- 'M. Haiduk'
- 'S. A. Krasnov'
- 'T. N. Maksimkina'
- 'K. D. Tolstov'
- 'G. M. Chernov'
- 'N. A. Salmanova'
- 'D. A. Salomov'
- 'A. Khushvaktova'
- 'F. A. Avetyan'
- 'N. A. Marutyan'
- 'L. T. Sarkisova'
- 'V. F. Sarkisyan'
- 'M. I. Adamovich'
- 'Yu. A. Bashmakov'
- 'V. G. Larionova'
- 'G. I. Orlova'
- 'N. G. Peresadko'
- 'M. I. Tretyakova'
- 'S. P. Kharlamov'
- 'M. M. Chemyavsky'
- 'V. G. Bogdanov'
- 'V. A. Plyushchev'
- 'Z. I. Soloveva'
- 'V. V. Belaga'
- 'A. I. Bondarenko'
- 'Sh. A. Rustamova'
- 'A. G. Chemov'
- 'N. N. Kostanashvili'
- 'S. Vokál'
title: 'Coherent Dissociation $^{16}$O $\rightarrow$ 4$\alpha$ in Photoemulsion at an Incident Momentum of 4.5 GeV/$c$ per Nucleon'
---
Multifragmentation of relativistic projectile nuclei provides unique insight into their structure: as charged secondaries are detected without any thresholds, the corresponding transitions can be observed at the lowest possible values of 4-momentum transfer. Extremely peripheral, coherent, inelastic nuclear collisions in which the target nucleus acts on the dissociating projectile nucleus as a discrete particle - that is, the former does not change charge, does not undergo breakup, and is not excited - are especially promising in this respect. Investigations of such collisions between high-energy particles were initiated largely by the milestone theoretical results by Pomeranchuk and Feinberg [@Pomeranchuk]. Note, however, that these results equally apply to collisions in which high-energy nuclei take the place of particles as projectiles, while the products of nuclear fragmentation appear as final states instead of elementary particles produced [@Chernov].
As in coherent particle production on nuclei, processes of two types - a diffractive transition mediated by a Pomeron and a Coulomb transition mediated by a virtual photon - occur in the above nuclear reactions. The latter is expected to be dominant for colliding nuclei with high electric charges. Typically, inelastic coherent reactions have rather high energy thresholds. For example, the threshold incident momentum for coherent dissociation of a projectile nucleus $A$ with mass $M_0$ into $n$ fragments with masses $m_i(i~=~1,...,n)$ is estimated as [@Chernov] $$p^{min}_0~\approx~(M_0B^{1/3}/\mu)\Delta~~~~~(1)$$ where $\mu$ is the pion mass, $B$ is the mass number of the target nucleus that coherently interacts with the projectile $A$, and $\Delta~=~\sum^n_{i=1}m_i~-~M_0$ is the mass defect with respect to the dissociation channel considered. In particular, for the reaction $^{16}$O $\rightarrow$ 4$\alpha$, the momentum threshold amounts to several hundred MeV/$c$ per nucleon, approaching the relativistic region for heavy targets.
Only a few studies devoted to the coherent dissociation of relativistic projectile nuclei have been performed thus far. This paper reports the results of first searches for the coherent dissociation process $$^{16}O~\stackrel{Em}{\longrightarrow}~4\alpha~~~~~(2)$$ at $p_0$ = 4.5 GeV/$c$ per nucleon in nuclear emulsion (Em). Earlier, the dissociation of oxygen nuclei into four $\alpha$ particles was investigated at nonrelativistic energies (below 90 MeV per nucleon; see [@Pouliot; @Badala] and references therein), which probably fall short of the coherent threshold at least for the heavy target nuclei.
Stacks of nuclear emulsion BR-2 irradiated by a beam of $^{16}$O ions with a momentum of 4.5 GeV/$c$ per nucleon from the JINR synchrophasotron were used in searches for events satisfying criteria necessary to select reaction (2).
To estimate the mean range $\lambda$ of $^{16}$O nuclei in emulsion for reaction (2) without additional charged secondaries, a slow-fast scan along the track was carried out in a part of emulsion sheets. Over the primary track of total length 375.2 m, we found 12 pure events of reaction (2). This corresponds to a $\lambda$ value of 31.3$^{+12.6}_{-7.0}$ m. From this, the cross section for the so-called average emulsion nucleus $(\langle A\rangle\approx45)$ was estimated at 10 mb/nucleus.
To maximize the statistics of events we are interested in, we made use of a special procedure based on a fast area scan in a large number of emulsion sheets. The scan was performed over bands normal to the incident oxygen beam. The distance between the bands was 0.5 cm. In the scan, we selected close groups comprising 3 to 4 gray tracks of similar ionization. Once such a group has been located, search along the group axis was carried out in the reverse direction. In comparison with scanning along the track, this strategy resulted in more than a tenfold increase in the scanning efficiency for reaction (2).
In this way, we found and measured as many as 641 events that involve four well-identified relativistic fragments with $z$ = 2 in the final state and which show evidence neither for the emergence of additional charged particles, nor for the excitation of the target nucleus. These events are further analyzed in the present study.
![\[Fig:1\] Distribution of $^{16}$O $\rightarrow$ 4$\alpha$ events in $q^2_T=-t'$. A fit to the functional form (6) is shown by a curve.](fig1.pdf){width="4in"}
Figure 1 shows the distribution in the square $q^2_T$ of the transverse-momentum transfer to the $^{16}$O nucleus through interaction with an emulsion nucleus. This quantity can be used to estimate the square $t$ of the 4-momentum transfer to the oxygen nucleus splitting into $n$ fragments. This is done with the aid of the relation $$|t'|~=~|t-t^{min}(\sum_{i=1}^{n}m_i)|~\simeq~q^2_T.~~~~~(3)$$ Indeed, in the coherent diffractive production of the fragments (as in elastic diffractive scattering), the distribution in $t'$ [and in $t$ as well, because the factor exp$[t^{min}(\sum_{i=1}^nm_i)]$ that takes into account the minimum 4-momentum transfer squared $t^{min}(\sum_{i=1}^nm_i)$ required for the production of $n$ free fragments is insignificant here]{} must have the simple exponential form $$d\sigma/dt'~\varpropto~exp(-a|t'|),~~~~~(4)$$ where the slope of the diffractive peak $a$ is expressed in terms of the radii of the projectile and target nuclei, $R_A$ and $R_B$, as $a~\approx~(R_A~+~R_B)^2/4$. On average, the resulting longitudinal component of the 3-momentum transfer $\mathbf q$ is much less than the transverse component. It follows that $-t'~\approx~q^2_T$; hence, the distribution in the transverse momentum squared must have the Rayleigh form $$d\sigma/dq^2_T~\varpropto~exp(-aq^2_T)~~~~~(5)$$ with $\langle q_T\rangle~\approx~\pi^{1/2}/(R_A~+~R_B)$. We assumed that there are no neutral fragments in the events under study. In this case, we have $\mathbf q_T~=~\sum_{i=1}^4\mathbf p_T$, where $\mathbf p_T$ is the $\alpha$-particle transverse momentum. Its absolute value $p_T$ was estimated by the formula $p_T~=~4p_0sin\theta$, where $p_0~=~4.5$ GeV/$c$ and $\theta$ is the polar emission angle. A straight line corresponds to distribution (5) on the exponential scale used in Fig. 1.
It can be seen from Fig. 1 that the experimental distribution in $q^2_T$ is not consistent with the functional form (5) ($\chi^2$/NDF = 3.2). However, it can be described by the sum of two Rayleigh distributions, $$d\sigma/dq^2_T~=~\alpha exp(-a_1q^2_T)~+~(1~-~\alpha)exp(-a_2q^2_T),~~~~~(6)$$ with two different slope parameters $a_1$ and $a_2$. A maximum-likelihood fit of (6) to experimental data yields ($\chi^2$/NDF = 0.9) $$a_1~=~19~\pm~2,~a_2~=~4.2~\pm~0.4,~\alpha~=~0.66~\pm~0.06.~~~~~(7)$$ Once the nuclear composition of emulsion and experimental uncertainties in $q_T$, which effectively broaden the distribution in $q_T$, are taken into account, the resulting values of $a_1$ and $a_2$ are compatible with the assumption that the first and second terms in (6) correspond, respectively, to the diffractive dissociation $^{16}$O $\rightarrow$ 4$\alpha$ on an emulsion nucleus as a discrete particle and dissociation on a nucleon. At the same, any attempts at estimating the relative contributions of these two dissociation channels (that is, the slope parameters $\alpha$) and their cross sections are seriously impeded by the fact that events with a recoil proton are not observed in our experiment, by the possible contribution of Coulomb dissociation, by the area scan used, etc.
Along with the complete sample of 641 pure events of the type $^{16}$O $\rightarrow$ 4$\alpha$, we will henceforth consider the subsample of 428 ($\sim67\%$ of the complete sample) events that satisfy the condition $q_T~<~0.32$ GeV/$c$, which is equivalent to $葉'~<~0.1~($GeV/$c)^2$. This value of $t'$ (see Fig. 1) may be taken for an approximate boundary between the coherent and incoherent events in the selected sample. Provided that the coherent reaction (2) is indeed responsible for the first term in (6), this separation enables us to constrain from below the characteristics of the two reactions. In other words, we may hope to obtain some reliable estimates by comparing the characteristics of the complete sample and of the subsample with $-t'~<~0.1$ (GeV/$c)^2$.
![\[Fig:2\] Inclusive $p^2_T$ distribution of $\alpha$ particles for ($a$) the complete event sample and ($b$) events with $-t'~<~0.1$ (GeV/$c$). The straight lines are the Rayleigh distributions with $\langle p^2_T\rangle$ corresponding to the observed values.](fig2.pdf){width="3in"}
Figure 2 shows the transverse-momentum-squared distributions of $\alpha$ particles from reaction (2) for the complete sample and for the subsample with $-t'~<~0.1$ (GeV/$c)^2$. The corresponding root-mean-square values $\langle p^2_T\rangle^{1/2}$ are 167 $\pm$ 4 and 145 $\pm$ 4 MeV/$c$. The two distributions display deviations from the Rayleigh form, showing high-energy tails that are due to particles with high $p_T$. These deviations cannot be explained by the complex composition of emulsion, in view of the well-known result from mathematical statistics that an arbitrary superposition of any number of Rayleigh distributions with different a values is a Rayleigh distribution as well.
![\[Fig:3\] Distribution in the $\alpha\alpha$ azimuthal angle $\varepsilon_{ij}$ for the complete sample of $^{16}$O $\rightarrow$ 4$\alpha$ events.](fig3.pdf){width="3in"}
It is also well known that the laboratory transverse momenta of fragments are effectively overestimated because of the transverse motion of the nucleus underｬgoing fragmentation (the so-called bounce-off effect discussed, for example, in [@Bengus]). In our data, this effect manifests itself in the distribution in the azimuthal angle $\varepsilon_{ij}$ = arccos($\mathbf p_{T_i}\mathbf p_{T_j}/p_{T_i}p_{T_j}$) between the transverse momenta of two $\alpha$ particles produced in the same event of dissociation (2) (see Fig. 3). This distribution contradicts the phase-space prediction (the curve in the same figure). In particular, for the observed azimuthal asymmetry $$A~=~(N_{\varepsilon_{ij}\ge\pi/2}~-~N_{\varepsilon_{ij}<\pi/2})/N_{0\le\varepsilon_{ij}\le\pi}~~~~~(8)$$ of the distribution $d\sigma/d\varepsilon_{ij}$, we have -0.01 $\pm$ 0.02, while the value predicted by the law of transverse-momentum conservation is $1/(N_{\alpha}$ - 1) = 0.33 [@Bondarenko]. This means that the fragmentation of the oxygen nucleus must be described in terms of the quantities defined in its rest frame.
For an exclusive reaction like (2), the recipe for going over to the rest frame of the nucleus undergoing fragmentation is straightforward. (In the following, all the quantities that refer to this frame are labeled with asterisks.) Under the assumption that the projectile nucleus is scattered at a small angle, a high-precision approximation to the transverse momenta of $\alpha$ particles is given by the formula $$\mathbf p^*_{T_i}~\cong~\mathbf p_{T_i}~-~\sum_{i=1}^4\mathbf p_{T_i}/4~~~~~(9)$$ Figures 4 and 5 show, respectively, the $\mathbf p^*_{T_i}$ and $\varepsilon^*_{ij}$ distributions of $\alpha$ particles for ($a$) the complete sample of events (2) and for ($b$) the subsample with $葉'~<~0.1$ (GeV/$c$). The main quantitative characteristics of these distributions are given in the table. These are the mean values $\langle p^*_T\rangle$ and $\langle p^{*2}_T\rangle^{1/2}$, the azimuthal asymmetry $A^*$ \[see formula (8) for $\varepsilon_{ij}$\], and the azimuthal collinearity $B^*$ defined as $$B^*~=~(N_{\varepsilon^*_{ij}\le\pi/4}~+~N_{\varepsilon^*_{ij}\ge3\pi/4}~-~N_{\pi/4<\varepsilon^*_{ij}<3\pi/4})/N_{0\le\varepsilon^*_{ij}\le\pi}.~~~~~(10)$$ The data presented in Figs. 4 and 5 and in the table lead to the following conclusions.
![\[Fig:4\] As in Fig. 2, but for the $\alpha$ transverse momentum defined in the $^{16}$O rest frame.](fig4.pdf){width="3in"}
![\[Fig:5\] Distributions in $\varepsilon^*_{ij}$ for ($a$) the complete event sample and ($b$) events with $-t'~<~0.1$ (GeV/$c)^2$ The curves represent distribution (11) with coefficients (12).](fig5.pdf){width="3in"}
(1) As might have been expected, the mean values $\langle p^*_T\rangle$ are substantially smaller than $\langle p_T\rangle$ (by some 30% for the complete sample of $^{16}$O $\rightarrow$ 4$\alpha$). It is noteworthy that the $\langle p^*_T\rangle$ values for all events and for those with $-t'~<~0.1$ (GeV/$c)^2$ differ only slightly.
(2) As in the laboratory frame, the $p^{*2}_T$ distributions show deviations from the Rayleigh form for both sets. It should be emphasized that within errors, the relative contribution of the high-$p^*_T$ tail for the complete sample does not differ from that for the subsample with $-t'~<~0.1$ (GeV/$c)^2$ (23 $\pm$ 4 and 21 $\pm$ 4%, respectively).
(3) The $\varepsilon^*_{ij}$ distributions also disagree with the functional form $$d\sigma/d\varepsilon^*~\simeq~\frac{1}{\pi}(1~+~c_1cos\varepsilon^*~+~c_2cos2\varepsilon^*),~~~~~(11)$$ which follows from the statistical picture of the direct nuclear decay into the observed $\alpha$ particles. Assuming that in the c.m.s., each component of the $\alpha$ 3-momentum obeys the normal distribution $n(0,\sigma)$ (this automatically leads to the Rayleigh form of the distribution $d\sigma/dp^{*2}_T)$ and taking into account energy-momentum conservation in the decay, we find that the coefficients $c_1$ and $c_2$ in (11) are related to the azimuthal asymmetry $A^*$ and the azimuthal collinearity $B^*$ by the equations $$c_1~=~-(\pi/2)A^*~=~-(\pi/2)(N_{\alpha}~-~1)^{-1},$$ $$c_2~=~(\pi/2)B^*~=~(8\pi/25)(N_{\alpha}~-~1)^{-2},~~~~~(12)$$ where (in our case) $N_{\alpha}$ = 4 \[6\]. The predicted forms (11) with coefficients (12) are represented by the curves in Fig. 5, and the values of $A^*$ and $B^*$ are given in the table. The discrepancy between the predictions and experimental data is clearly seen.
(4) In relation to the predictions based on transverse-momentum conservation, the experimental distribution $d\sigma/d\varepsilon^*_{ij}$ shows a higher degree of collinearity between the $\alpha$-particle transverse momenta $\mathbf p_T$ in the transverse plane (see the observed excess of experimental values over theoretical results for $\varepsilon_{ij}~\approx~0$ and $\varepsilon_{ij}~\approx~\pi$ in Fig. 5; the values of $B^*$ are given in the table).
[l|c|c|c|c|c|c]{} &\
Event sample & $\langle p^*_T\rangle$, & $\langle p^{*2}_T\rangle^{1/2}$, & $A^*$ & $B^*$ & $\alpha^*$ & $\beta^*$\
& MeV/$c$ & VeV/$c$ & & & &\
Experiment, all events&$121\pm2$&$145\pm3$&$-0.28\pm0.02$&$0.27\pm0.02$&$-0.23\pm0.01$&$0.22\pm0.02$\
Experiment, &$115\pm2$&$134\pm4$&$-0.27\pm0.02$&$0.30\pm0.02$&$-0.23\pm0.01$&$0.25\pm0.02$\
events with $t'<0.1$ (MeV/$c)^2$&&&&&&\
Statistical model, $^{16}$O$\rightarrow4\alpha$&120&-&-0.33&0.07&-0.26&0.06\
Statistical model, &120&-&-0.33&0.11&-0.27&0.09\
$^{16}$O$\rightarrow^8$Be+$\alpha+\alpha\rightarrow4\alpha$&&&&&&\
Statistical model, &120&-&-0.33&0.10&-0.27&0.08\
$^{16}$O$\rightarrow^8$Be+$^8$Be$\rightarrow4\alpha$&&&&&&\
Statistical model, &120&-&-0.33&0.07&-0.27&0.06\
$^{16}$O$\rightarrow^{12}$C$^*+\alpha\rightarrow4\alpha$&&&&&&\
Statistical model, &120&-&-0.33&0.08&-0.27&0.06\
$^{16}$O$\rightarrow^{12}$C$^*+\alpha\rightarrow^8$Be+$\alpha+\alpha\rightarrow4\alpha$&&&&&&\
Let us now discuss the results. We will use the Feshbach-Huang-Goldhaber statistical theory of prompt fragmentation [@Feshbach; @Goldhaber]. In this theory (which is widely employed in analyses of fragmentation of excited nuclei), the measured values of $\langle p^{*2}_T\rangle$ can be related to the temperature of the oxygen nucleus decaying through the channel (2). In energy units, the temperature is estimated as $$KT~=~\frac{A}{A-1}(\sigma^2_N/m_N),~~~~~(13)$$ where $m_N$ is the nucleon mass, and $\sigma^2_N$ is the variance of the momentum distribution of intranuclear nucleons; the latter governs the variance of the momentum distribution for arbitrary fragments through the so-called parabolic law $$\sigma^2_F~=~\sigma^2_NA_F(A-A_F)/(A-1).~~~~~(14)$$ Here, $A$ and $A_F$ are the mass numbers of the nucleus undergoing fragmentation and the fragment under consideration, respectively. In our case, we have $A~=~16$, $A_F~\equiv~A_{\alpha}~=~4$, and $\sigma^2_{\alpha}~=~\langle p^{*2}_T\rangle/2$. Using the experimental values of $\langle p^{*2}_T\rangle$ (see table) for the complete sample of events (2), we obtain $kT~=~3.7$ MeV. For the events with $葉'~<~0.1$ (GeV/$c)^2$ and with $葉'~>~0.1$ (GeV/$c)^2$, we arrive at $kT~=~3.2$ and 4.8 MeV, respectively.
The above estimates indicate that the coherent dissociation $^{16}$O $\rightarrow~4\alpha$ is characterized by a lower value of $kT~(kT~\simeq~3~-~3.5$ MeV) than conventional multifragmentation in reactions of the type $A~+~B~\rightarrow~\alpha~+~X$ that are investigated in inclusive experiments (see, for example, [@Bengus; @Bhanjo; @Bondarenko2; @Adamovich]). The last statement remains valid even if we consider that in many experimental studies, the decay temperature was overestimated because of erroneously using the laboratory momentum characteristics of $\alpha$ particles without taking into account the transverse motion of the system undergoing fragmentation. The value that we obtained for $kT$ is also significantly lower than the nucleon binding energy in the oxygen nucleus. On the other hand, such a low value of temperature corresponds to a small energy-momentum transfer to the nucleus undergoing fragmentation, which is typical of coherent processes. If nonstatistical tails in the distributions shown in Fig. 4 are due to the decays of massive intermediate states (the cascade decays of $^{16}$O are discussed below), the true $kT$ value is even lower than that quoted above.
Let us consider this issue in more detail. First, we address the question of whether there is $\mathbf p_T$ collinearity in individual events. For this purpose, we supplemented the analysis of the inclusive characteristics of azimuthal asymmetry and collinearity ($A^*$ and $B^*$) with the calculation of the asymmetry $\alpha^*$ and collinearity $\beta^*$ in individual events by the formulas [@Bondarenko] $$\alpha^*~=~\langle cos\varepsilon^*_{ij}\rangle~=~\sum_{i,j=1}^{4}cos\varepsilon^*_{ij}/N_{\alpha}(N_{\alpha}-1) (i~\neq~j),~~~~~(15)$$ $$\alpha^*~=~\langle cos2\varepsilon^*_{ij}\rangle~=~\sum_{i,j=1}^{4}cos2\varepsilon^*_{ij}/N_{\alpha}(N_{\alpha}-1) (i~\neq~j),~~~~~(16)$$ $(-(N-1)^{-1}=-(1/3)\le\alpha^*, \beta^*\le1)$. Assuming that the components of $\mathbf p^*_T$ are normally distributed and that the transverse momentum is conserved in the decay, we find that the mean values $\langle\alpha^*\rangle$ and $\langle\beta^*\rangle$ of random variables $\alpha^*$ and $\beta^*$ are given by [@Bondarenko] $$\langle\alpha^*\rangle~=~-(\pi/4)(N_{\alpha}-1)^{-1}~=~-0.26,$$ $$\langle\beta^*\rangle~=~(4\pi/25)(N_{\alpha}-1)^{-2}~=~0.056.~~~~~(17)$$
The distribution in the collinearity $\beta^*$ is plotted in Fig. 6 for the 428 $^{16}$O $\rightarrow~4\alpha$ events with $-t'~<~0.1$ (GeV/$c)^2$. The observed mean values $\langle\alpha^*\rangle$ and $\langle\beta^*\rangle$ are presented in the last two columns of the table.
![\[Fig:6\] Distribution in the azimuthal collinearity $\beta^*$ for an individual event. The solid and dashed curves represent the results of calculations made in the statistical theory for reacｬtion channels (18) and (19). respectively.](fig6.pdf){width="3in"}
To evaluate the contributions of various $^{16}$O cascade decays to the observed tendency of the product $\alpha$ particles to diverge collinearly in the transverse plane, we performed a simulation of all possible cascade channels of dissociation according to the simplest statistical model [@Goldhaber]. Thus, we considered the transitions $$^{16}O~\rightarrow~4\alpha~(the~~direct~~channel),~~~~~(18)$$ $$^{16}O~\rightarrow~^8Be~+~\alpha~+~\alpha~\rightarrow~4\alpha,~~~~~(19)$$ $$^{16}O~\rightarrow~^8Be~+~^8Be~\rightarrow~4\alpha,~~~~~(20)$$ $$^{16}O~\rightarrow~^{12}C~+~\alpha~\rightarrow~4\alpha,~~~~~(21)$$ $$^{16}O~\rightarrow~^{12}C^+~\alpha~\rightarrow~^8Be~+~\alpha~+~\alpha~\rightarrow~4\alpha.~~~~~(22)$$ For each decay fragment from reactions (18)-(22), we assumed that each 3-momentum component defined in the rest frame of the decaying nucleus or of the unstable intermediates state obeys a normal distribution $n(0,~\sigma^2)$, the variance $\sigma^2$ being dependent on the fragment mass according to the parabolic law (14). After the transformation from the rest frames of intermediate states to the c.m.s. of the $^{16}$O nucleus, we were left with the single model parameter $\sigma^2_N$ \[see (14)\]. This parameter was determined by imposing the requirement that the mean transverse momentum $\langle p^{*2}_T\rangle$ for the final-state $\alpha$ particles in each channel (18)-(22) be equal to its measured value $\approx$ 120 MeV/$c$ (see table). For each of the reactions (18)-(22) simulated according to the Monte Carlo method, the values of $A^*$, $B^*$, $\langle\alpha^*\rangle$, and $\langle\beta^*\rangle$ are shown in the table. The computed distributions $dN/d\beta^*$ are illustrated in Fig. 6 for channels (18) (the direct reaction) and (19) (under the above assumptions, the latter leads to the maximum collinearity of the vectors $\mathbf p^*_T$ in the final state).
The results shown in the table and in Fig. 6 demonstrate that the observed effect of $\mathbf p^*_T$ collinearity cannot be attributed entirely to the cascade decays of $^{16}$O into $\alpha$ particles. If we go beyond the model from [@Goldhaber] by assuming that the temperature of primary $^{16}$O decays is substantially higher than that of the decays of intermediate fragments, a high degree of collinearity of the $\alpha$-particle transverse momenta in the final state can be obtained. In this case, however, theoretical calculations fail to describe other experimental characteristics (mean values $\langle p^*_T\rangle$, azimuthal asymmetries, etc.) under any assumptions about the relative contributions of channels (18) - (22).
Therefore, the only plausible way to explain the high values of $B^*$ and $\langle\beta^*\rangle$ is to assume that a nonzero angular momentum is transferred to the $^{16}$O nucleus in the collision. Needless to say, a comprehensive analysis of conclusions that can be drawn on basis of this assumption is beyond the scope of the present study. We look forward to testing it with increased statistics that will be accumulated in the near future.
We are grateful to the administration and staff of the High Energy Division of JINR for the aid in performing our experiment at the synchrophasotron. We also acknowledge the efforts of all operators and technicians involved in scanning and measurements.
Pomeranchuk, I.Ya. and Feinberg, E.L., Dokl. Akad. Nauk SSSR, 1953, vol. 93, p. 439; Feinberg, E.L. and Pomeranchuk, I.Ya., Suppl. Nuovo Cimento, 1956, vol. 3, p. 652. Chernov, G.M., Preprint of Inst, of Nuclear Physics, Tashkent, 1993, no. R-7-584. Pouliot, J. et al., Phvs. Lett. B, 1991, vol. 263, p. 18; 1993, vol. 299, p. 210. Badala, A. et al., Phys. Lett. B, 1993, vol. 299, p. 11. Bengus, L.E. et al., Pis知a Zh. Eksp. Teor. Fiz., 1983, vol. 38, p. 353. Bondarenko, A.I. et al., Uprugie i Neuprugie Soudareniya Chastits Bol痴hoi Energii s Nuklonami i Yadrami (Elastic and Inelastic Collisions of High-Energy Particles with Nucleons and Nuclei), Tashkent: FAN, 1975, p. 119. Feshbach, H. and Huang, K., Phys. Lett. B, 1973, vol. 47, p. 300. Goldhaber, A.S., Phys. Lett. B, 1974, vol. 53, p. 306. Bhanja, R. et al., Nucl. Phys. A, 1985, vol. 438, p. 740. Bondarenko, A.I. et al., Yad. Fiz., 1992, vol. 55, p. 137. Adamovich, M.I. et al,. Mod. Phys. Lett. A, 1993, vol. 8, p. 21.
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abstract: 'In [@Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are closely related to the generalized Ramanujan problem as posted by Sarnak in [@Sar05].'
address:
- |
School of Mathematics\
University of Minnesota\
Minneapolis, MN 55455, USA
- |
School of Mathematics\
Institute for Advanced Study\
Einstein Drive\
Princeton, New Jersey 08540 USA
author:
- Dihua Jiang
- Baiying Liu
title: On Cuspidality of Global Arthur Packets for Symplectic Groups
---
[^1]
Introduction
============
Let $F$ be a number field and ${{\mathbb {A}}}$ be the ring of adeles of $F$. For an $F$-split classical group ${{\mathrm {G}}}$, ${{\mathcal {A}}}_2({{\mathrm {G}}})$ denotes the set of equivalence classes of all automorphic representations of ${{\mathrm {G}}}({{\mathbb {A}}})$ that occur in the discrete spectrum of the space of all square-integrable automorphic forms on ${{\mathrm {G}}}({{\mathbb {A}}})$. The automorphic representations $\pi$ in the set ${{\mathcal {A}}}_2({{\mathrm {G}}})$ have been classified, up to global Arthur packets, in the fundamental work of J. Arthur ([@Ar13]), based of the theory of endoscopy. More precisely, for any $\pi\in{{\mathcal {A}}}_2({{\mathrm {G}}})$, there exists a global Arthur packet, denoted by ${\widetilde}{\Pi}_\psi(G)$, such that $\pi\in{\widetilde}{\Pi}_\psi({{\mathrm {G}}})$ for some global Arthur parameter $\psi\in{\widetilde}{\Psi}_2({{\mathrm {G}}})$. Following [@Ar13], a global Arthur parameter $\psi\in{\widetilde}{\Psi}_2({{\mathrm {G}}})$ can be written formally as $$\label{ap}
\psi=(\tau_1,b_1)\boxplus(\tau_2,b_2)\boxplus\cdots\boxplus(\tau_r,b_r)$$ where $\tau_j\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_j})$ and $b_j\geq 1$ are integers. We refer to Section 2 for more details. A global Arthur parameter $\psi$ is called [*generic*]{}, following [@Ar13], if the integers $b_j$ are one, i.e. a generic global Arthur parameter $\psi$ can be written as $$\label{gap}
\psi=\phi=(\tau_1,1)\boxplus(\tau_2,1)\boxplus\cdots\boxplus(\tau_r,1).$$
For a generic global Arthur parameter $\phi$ as in , the global Arthur packet ${\widetilde}{\Pi}_\phi({{\mathrm {G}}})$ contains at least one member $\pi$ from the set ${{\mathcal {A}}}_2({{\mathrm {G}}})$. More precisely, this $\pi$ must belong to the subset ${{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}})$, i.e. it is cuspidal. This assertion follows essential from the theory of automorphic descents of Ginzburg-Rallis-Soudry ([@GRS11]), as discussed in [@JL15a]. In fact, as in [@JL15a Section 3.1], one can show that a global Arthur parameter $\psi$ is generic if and only if the global Arthur packet ${\widetilde}{\Pi}_\psi({{\mathrm {G}}})$ contains a member $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}})$ that has a nonzero Whittaker-Fourier coefficient (Theorem 3.4 in [@JL15a]). It is not hard to show that when a global Arthur parameter $\psi=\phi$ is generic, the following holds: $${\widetilde}{\Pi}_\phi({{\mathrm {G}}})\cap{{\mathcal {A}}}_2({{\mathrm {G}}})\subset{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}}).$$ All members in ${\widetilde}{\Pi}_\phi({{\mathrm {G}}})\cap{{\mathcal {A}}}_2({{\mathrm {G}}})$ may be constructed via the [*twisted automorphic descents*]{} as developed in [@JLXZ] and more generally in [@JZ].
In [@M08] and [@M11], C. Mœglin investigates the following problem: for a global Arthur parameter $\psi\in{\widetilde}{\Psi}_2(G)$, when does the global Arthur packet ${\widetilde}{\Pi}_\psi({{\mathrm {G}}})$ contain a non-cuspidal member in ${{\mathcal {A}}}_2({{\mathrm {G}}})$ and how to construct such non-cuspidal members if exist? Mœglin states her results in terms of her local and global conjectures in the papers. We refer to [@M08] and [@M11] for detailed discussions on those problems.
The objective of this paper is to investigate the following question: For a global Arthur parameter $\psi\in{\widetilde}{\Psi}_2({{\mathrm {G}}})$, when does the global Arthur packet ${\widetilde}{\Pi}_\psi({{\mathrm {G}}})$ contain no cuspidal members, i.e. when is the intersection $${\widetilde}{\Pi}_\psi({{\mathrm {G}}})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}})$$ an empty set? The approach that we are taking to investigate this problem is based on our understanding of the structure of Fourier coefficients of automoprhic forms associated to nilpotent orbits or partitions, following the discussions and conjectures in [@J14 Section 4] and [@JL15a]. This study can be regarded as an extension of the fundamental work of R. Howe on the theory of singular automorphic forms using his notion of ranks for unitary representations ([@H81]).
In this paper, we consider mainly the case that $G={{\mathrm{Sp}}}_{2n}$, the symplectic groups. The method is applicable to other classical groups. Due to technical reasons, we leave the discussion for other classical groups to our future work.
We start the discussion with a global Arthur parameter $$\psi=(\tau,2e)\boxplus(1,1)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{4e})$$ with $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ of symplectic type. When $e=1$, the well-known example of Saito-Kurokawa provides irreducible cuspidal automorphic representations in the global packet ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_4)$, as constructed by Piatetski-Shapiro in [@PS83] using global theta correspondences. This is the first known counter-example to the generalized Ramanujan conjecture, which is not of unipotent cuspidal type. Of course, the counter-examples of unipotent cuspidal type were constructed in 1979 by Howe and Piatetski-Shapiro in [@HPS79], also using global theta correspondences. It was desirable to find such non-tempered cuspidal automorphic representations for general ${{\mathrm{Sp}}}_{2n}$ or even for general reductive groups. In 1996, W. Duke and Ö. Imamoglu made a conjecture in [@DI96] that when $F={{\mathbb {Q}}}$, there exists the analogy of the Saito-Kurokawa type cuspidal automorphic forms on ${{\mathrm{Sp}}}_{4e}$ for all integers $e\geq 1$. In terms of the endoscopic classification theory ([@Ar13]), the Duke-Imamoglu conjecture asserts that when $F={{\mathbb {Q}}}$, the intersection $${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{4e})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{4e})$$ is non-empty if the global Arthur parameter $\psi=(\tau,2e)\boxplus(1,1)$. This conjecture was confirmed positively by T. Ikeda in his 2001 Annals paper ([@Ik01]) and an extension to the case that $F$ is totally real in [@Ik]. The questions remain to ask:
1. What happens to the symplectic groups ${{\mathrm{Sp}}}_{4e+2}$?
2. What happens if $F$ is not totally real?
For a general number field $F$, the authors joint with L. Zhang proved in [@JLZ13] that the intersection $${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$$ is non-empty for a family of global Arthur parameters $\psi$, including the case that $\psi=(\tau,2e)\boxplus(1,1)$. We explicitly constructed non-zero square-integrable residual representations in the global Arthur packets ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ for a family of global Arthur parameters and hence confirmed the conjecture of Mœglin in [@M08] and [@M11] for those cases. Our main motivation in [@JLZ13] is to find [*automorphic kernel functions*]{} for the automorphic integral transforms that explicitly produce endoscopy correspondences as explained in [@J14].
One of the main results in this paper confirms that when $F$ is [*totally imaginary*]{} and $n\geq 5$, the intersection $${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$$ is empty for the global Arthur parameters $\psi=(\tau,2e)\boxplus(1,1)$ if $n=2e$ and $\psi=(\tau,2e+1)\boxplus(\omega_\tau,1)$ if $n=2e+1$, where $\omega_\tau$ is the central character of $\tau$, and $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ is self-dual. Note that when $n=2e$, $\tau$ is of symplectic type; and when $n=2e+1$, $\tau$ is of orthogonal type. This conclusion is a consequence of more general results obtained in Section 4, where various versions of criteria for global Arthur packets containing no cuspidal members are given in Theorems \[ncmain1\], \[ncmain2\], \[ncmain3\], and \[ncmain4\]; and explicit examples are also discussed in Section 4.2.
On the other hand, we discuss the characterization of cuspidal automorphic representations with smallest possible Fourier coefficients, which are called [*small*]{} cuspidal representations in Section 2. We first explain how to re-interpret the result of Li that cuspidal automorphic representations of classical groups are non-singular, in terms of the Fourier coefficients associated to partitions or nilpotent orbits. This leads to a question about the smallest possible Fourier coefficients for the cuspidal spectrum of classical groups, which is closely related to the generalized Ramanujan problem as posted by P. Sarnak in 2005 ([@Sar05]). As a consequence of the discussion in Section 3, we find simple criterion for ${{\mathrm{Sp}}}_{4n}$ that determines families of global Arthur parameters of unipotent type, with which the global Arthur packets contains no cuspidal members (Theorem \[unip\]). Examples and the relation of Theorem \[unip\] with the work of S. Kudla and S. Rallis ([@KR94]) are also discussed briefly in Section 3.
Generally speaking, by the endoscopic classification of the discrete spectrum of Arthur ([@Ar13]), the global Arthur parameters provide the bounds for the Hecke eigenvalues or the exponents of the Satake parameters at the unramified local places for automorphic representations occurring in the discrete spectrum. Since it is not clear how to deduce directly from the endoscopic classification which global Arthur packets contains no cuspidal members, we apply the method of Fourier coefficients associated to unipotent orbits. Hence it is expected that our discussion improves those bounds for the exponents of the Satake parameters of cuspidal spectrum if we find more global Arthur packets containing no cuspidal members. In Section 5, we obtain a preliminary result towards the generalized Ramanujan problem. For general number fields, we show in Proposition 5.1 that when $n=2e$ is even, the cuspidal automorphic representations of ${{\mathrm{Sp}}}_{4e}$ constructed by Piatetski-Shapiro and Rallis ([@PSR88]) achieve the worst bound, which is $\frac{n}{2}=e$, for the exponents of the Satake parameters of the cuspidal spectrum. While in Proposition 5.2, we assume that $F$ is totally imaginary and $n=2e+1\geq 5$ is odd, $\frac{n-1}{2}=e$ is an upper bound for the exponents of the Satake parameters of the cuspidal spectrum. It needs more work to understand if the bound $\frac{n-1}{2}=e$ is [*sharp*]{} when $F$ is totally imaginary and $n=2e+1\geq 5$ is odd. It is also not clear that how to construct cuspidal representations with the worst bound for the exponents of the Satake parameters. We will come back to those issues in our future work.
In the last section (Section 6), we characterize the small cuspidal automorphic representations of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ by means of Fourier coefficients of Fourier-Jacobi type, and by the notion of hyper-cuspidal automorphic representations in the sense of Piatetski-Shapiro ([@PS83]). As consequence, we prove (Theorem \[nohc\]) that when $F$ is totally imaginary and $n\geq 5$, there does not exist any hyper-cuspidal automorphic representation of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$.
The basic facts on the endoscopic classification of the discrete spectrum and the basic conjecture on the relations between the Fourier coefficients of automorphic forms and their global Arthur parameters are recalled in Section 2. Here we also recall the recent, relevant results of the authors, which are used in the rest of this paper.
Finally, we would like to thank J. Arthur, L. Clozel, J. Cogdell, R. Howe, R. Langlands, C. M[œ]{}glin, P. Sarnak, F. Shahidi, R. Taylor, D. Vogan, and J.-L. Waldspurger for their interest in the problems discussed in this paper and for their encouragement.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Fourier Coefficients and Global Arthur Packets {#secfcgapk}
==============================================
Fourier coefficients attached to nilpotent orbits
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In this section, we recall Fourier coefficients of automorphic forms attached to nilpotent orbits, following the formulation in [@GGS15], which is slightly more general and easier to use than the one taken in [@J14] and [@JL15a]. Let ${{\mathrm {G}}}$ be a reductive group defined over $F$, or a central extension of finite degree. Fix a nontrivial additive character $\psi$ of $F {\backslash}{{\mathbb {A}}}$. Let $\frak{g}$ be the Lie algebra of ${{\mathrm {G}}}(F)$ and $f$ be a nilpotent element in $\frak{g}$. The element $f$ defines a function on $\frak{g}$: $$\psi_f: \frak{g} \rightarrow {{\mathbb {C}}}^{\times}$$ by $\psi_f(x) = \psi(\kappa(f,x))$, where $\kappa$ is the killing form on $\frak{g}$.
Given any semi-simple element $h \in \frak{g}$, under the adjoint action, $\frak{g}$ is decomposed to a direct sum of eigenspaces $\frak{g}^h_i$ of $h$ corresponding to eigenvalues $i$. For any rational number $r \in {{\mathbb {Q}}}$, let $\frak{g}^h_{\geq r} = \oplus_{r' \geq r} \frak{g}^h_{r'}$. The element $h$ is called [*rational semi-simple*]{} if all its eigenvalues are in ${{\mathbb {Q}}}$. Given a nilpotent element $f$, a [*Whittaker pair*]{} is a pair $(h,f)$ with $h \in \frak{g}$ being a rational semi-simple element, and $f \in \frak{g}^h_{-2}$. The element $h$ in a Whittaker pair $(h,f)$ is called a [*neutral element*]{} for $f$ if $f \in \frak{g}^h_{-2}$ and the map $\frak{g}^h_0 \rightarrow \frak{g}^h_{-2}$ via $X \mapsto [X,f]$ is surjective. For any nilpotent element $f \in \frak{g}$, by the Jacobson-Morozov Theorem, there is an $\frak{sl}_2$-triple $(e,h,f)$ such that $[h,f]=-2f$. In this case, $h$ is a neutral element for $f$. By [@GGS15 Lemma 2.2.1], a Whittaker pair $(h,f)$ comes from an $\frak{sl}_2$-triple $(e,h,f)$ if and only if $h$ is a neutral element for $f$. For any $X \in \frak{g}$, let $\frak{g}_X$ be the centralizer of $X$ in $\frak{g}$.
Given any Whittaker pair $(h,f)$, define an anti-symmetric form $\omega_f$ on $\frak{g}$ by $\omega_f(X,Y):=\kappa(f,[X,Y])$, where $\kappa$ is the killing form. We denote by $\omega = \omega_f$ when there is no confusion. Let $\frak{u}_h= \frak{g}^h_{\geq 1}$ and let $\frak{n}_h=\ker(\omega)$ be the radial of $\omega |_{\frak{u}_h}$. Then $[\frak{u}_h, \frak{u}_h] \subset \frak{g}^h_{\geq 2} \subset \frak{n}_h$. By [@GGS15 Lemma 3.2.6], $\frak{n}_h = \frak{g}^h_{\geq 2} + \frak{g}^h_1 \cap \frak{g}_f$. Note that if the Whittaker pair $(h,f)$ comes from an $\frak{sl}_2$-triple $(e,h,f)$, then $\frak{n}_h=\frak{g}^h_{\geq 2}$. Let $U_{h}=\exp(\frak{u}_h)$ and $N_h=\exp(\frak{n}_h)$ be the corresponding unipotent subgroups of ${{\mathrm {G}}}$. Define a character of $N_h$ by $\psi_f(n)=\psi(\kappa(f,\log(n)))$. Let $N_h' = N_h \cap \ker (\psi_f)$. Then $U_h/N_h'$ is a Heisenberg group with center $N_h/N_h'$. It follows that for each Whittaker pair $(h,f)$, $\psi_f$ defines a character of $N_h({{\mathbb {A}}})$ which is trivial on $N_h(F)$.
Assume that $\pi$ be an automorphic representation of ${{\mathrm {G}}}({{\mathbb {A}}})$. Define a [*degenerate Whittaker-Fourier coefficient*]{} of $\varphi \in \pi$ by $$\label{fc}
{{\mathcal {F}}}_{h,f}(\varphi)(g) = \int_{N_{h}(F) {\backslash}N_{h}({{\mathbb {A}}})} \varphi(ng){\overline}{\psi}_f(n)dn, g \in {{\mathrm {G}}}({{\mathbb {A}}}).$$ Let ${{\mathcal {F}}}_{h,f}(\pi)=\{{{\mathcal {F}}}_{h,f}(\varphi) | \varphi\in \pi\}$. If furthermore, $h$ is a neutral element for $f$, then ${{\mathcal {F}}}_{h,f}(\varphi)$ is also called a [*generalized Whittaker-Fourier coefficient*]{} of $\varphi$. The (global) [*wave-front set*]{} $\frak{n}(\pi)$ of $\pi$ is defined to the set of nilpotent orbits ${{\mathcal {O}}}$ such that ${{\mathcal {F}}}_{h,f}(\pi)$ is nonzero, for some Whittaker pair $(h,f)$ with $f \in {{\mathcal {O}}}$ and $h$ being a neutral element for $f$. Note that if ${{\mathcal {F}}}_{h,f}(\pi)$ is nonzero for some Whittaker pair $(h,f)$ with $f \in {{\mathcal {O}}}$ and $h$ being a neutral element for $f$, then it is nonzero for any such Whittaker pair $(h,f)$, since the non-vanishing property of such Fourier coefficients does not depends on the choices of representatives of ${{\mathcal {O}}}$. Let $\frak{n}^m(\pi)$ be the set of maximal elements in $\frak{n}(\pi)$ under the natural order of nilpotent orbits. We recall [@GGS15 Theorem C] as follows.
\[ggsglobal\] Let $\pi$ be an automorphic representation of ${{\mathrm {G}}}({{\mathbb {A}}})$. Given two Whittaker pairs $(h,f)$ and $(h',f')$, with $h$ being a neutral element for $f$, if $f \in {\overline}{{{\mathrm {G}}}_{h'}(F) f'}$, where ${{\mathrm {G}}}_{h'}$ is the centralizer of $h'$ in ${{\mathrm {G}}}$, and ${{\mathcal {F}}}_{h',f'}(\pi)$ is nonzero, then ${{\mathcal {F}}}_{h,f}(\pi)$ is nonzero.
Note that a particular case of Theorem \[ggsglobal\] is that $f=f'$. In this case, the condition $f \in {\overline}{{{\mathrm {G}}}_{h'}(F) f'}$ is automatically satisfied, and hence Theorem \[ggsglobal\] asserts in this case that if ${{\mathcal {F}}}_{h',f}(\pi)$ is nonzero, for some Whittaker pair $(h',f)$, then ${{\mathcal {F}}}_{h,f}(\pi)$ is nonzero, for any Whittaker pair $(h,f)$ with $h$ being a neutral element for $f$.
When ${{\mathrm {G}}}$ is a quasi-split classical group, it is known that the nilpotent orbits are parametrized by pairs $({\underline}{p}, {\underline}{q})$, where ${\underline}{p}$ is a partition and ${\underline}{q}$ is a set of non-degenerate quadratic forms (see [@W01]). When ${{\mathrm {G}}}= {{\mathrm{Sp}}}_{2n}$, then ${\underline}{p}$ is symplectic partition, namely, odd parts occur with even multiplicities. When ${{\mathrm {G}}}= {{\mathrm{SO}}}^{\alpha}_{2n}, {{\mathrm{SO}}}_{2n+1}$, then ${\underline}{p}$ is orthogonal partition, namely, even parts occur with even multiplicities. In these cases, let $\frak{p}^m(\pi)$ be the partitions corresponding to nilpotent orbits in $\frak{n}^m(\pi)$, that is, the maximal nilpotent orbits in the wave-front set $\frak{n}(\pi)$ of the automorphic representation $\pi$.
**Convention**. [*When ${{\mathrm {G}}}$ is a quasi-split classical group, $\pi$ is an automorphic representation of ${{\mathrm {G}}}({{\mathbb {A}}})$, for any symplectic/orthogonal partition ${\underline}{p}$, by a Fourier coefficient attached to ${\underline}{p}$, we mean a generalized Whittaker-Fourier coefficient attached to an orbit ${{\mathcal {O}}}$ parametrized by a pair $({\underline}{p}, {\underline}{q})$ for some ${\underline}{q}$, that is, ${{\mathcal {F}}}_{h,f}(\varphi)$, where $\varphi \in \pi$, $u\in {{\mathcal {O}}}$ and $h$ is a neutral element for $f$. Sometimes, for convenience, we also write a Fourier coefficient attached to ${\underline}{p}$ as ${{\mathcal {F}}}^{\psi_{{\underline}{p}}}(\varphi)$ without specifying the $F$-rational orbit ${{\mathcal {O}}}$ and Whittaker pairs.*]{}
Next, we recall the following result of [@JL15], which is one of the main ingredients of this paper.
\[ti\] Let $F$ be a totally imaginary number field. And let $\pi$ be a cuspidal automorphic representation of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$. Then there exists an even partition (that is, consists of only even parts) in $\frak{p}^m(\pi)$, constructed in [@GRS03], of the form $${\underline}{p}_{\pi}:=[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}],$$ with $2n_1 > 2n_2 > \cdots > 2n_r$ and $s_i \leq 4$ holds for $1 \leq i \leq r$.
In this paper, we will consider two orders of partitions as follows. Given a partition ${\underline}{p}=[p_1p_2\cdots p_r]$, let $\lvert {\underline}{p} \rvert=\sum_{i=1}^r p_i$.
\[orders\] $(1)$. [**Lexicographical order**]{}. Given two partitions ${\underline}{p}=[p_1p_2 \cdots p_r]$ with $p_1 \geq p_2 \geq \cdots \geq p_r$, and ${\underline}{q}=[q_1q_2 \cdots q_r]$ with $q_1 \geq q_2 \geq \cdots \geq q_r$, (add zeros at the end if needed) which may not be partitions of the same positive integer, i.e., $\lvert {\underline}{p} \rvert$ and $\lvert {\underline}{q} \rvert$ may not be equal. If there exists $1 \leq i \leq r$ such that $p_j = q_j$ for $1 \leq j \leq i-1$, and $p_i < q_i$, then we say that ${\underline}{p} < {\underline}{q}$ under the lexicographical order of partitions. Lexicographical order is a total order.
$(2)$. [**Dominance order**]{}. Given two partitions ${\underline}{p}=[p_1p_2 \cdots p_r]$ with $p_1 \geq p_2 \geq \cdots \geq p_r$, and ${\underline}{q}=[q_1q_2 \cdots q_r]$ with $q_1 \geq q_2 \geq \cdots \geq q_r$ (add zeros at the end if needed), which again may not be partitions of the same positive integer, i.e., $\lvert {\underline}{p} \rvert$ and $\lvert {\underline}{q} \rvert$ may not be equal. If for any $1 \leq i \leq r$, $\sum_{j=1} p_j \leq \sum_{j=1}^i q_j$, then we say that ${\underline}{p} \leq {\underline}{q}$ under the dominance order of partitions. Dominance order is a partial order.
Given two partitions ${\underline}{p}$ and ${\underline}{q}$, if we do not specify which order of partitions, by ${\underline}{p} \leq {\underline}{q}$, we mean that it is under the dominance order of partitions.
Automorphic discrete spectrum and Fourier coefficients
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In this paper, we consider mainly the symplectic groups. Although the methods are expected to work for all quasi-split classical groups, due to the state of art in the current development of the theory, one knows much less when the classical groups are not of symplectic type. Hence we will be concentrated on symplectic groups here and leave the discussion for other classical groups in future.
For symplectic group ${{\mathrm{Sp}}}_{2n}$, the endoscopic classification of the discrete spectrum was obtained by Arthur in [@Ar13]. A preliminary statement of the endoscopic classification is recalled below.
For any $\pi\in{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$, there exists a global Arthur parameter $$\psi = \psi_1 \boxplus \cdots \boxplus \psi_r,$$ such that $\pi\in{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$, the global Arthur packet associated to $\psi$.
The notation used in this theorem can be explained as follows. Each $\psi_i=(\tau_i, b_i)$ is called a [*simple Arthur parameter*]{}, where $\tau_i$ is an irreducible self-dual unitary cuspidal automorphic representation of ${{\mathrm{GL}}}_{a_i}({{\mathbb {A}}})$ with central character $\omega_{\tau_i}$, $b_i \in {{\mathbb {Z}}}_{\geq 1}$. Every simple Arthur parameter $\psi_i$ is of orthogonal type. This means that if $\tau_i$ is of symplectic type, that is, $L(s, \tau_i, \wedge^2)$ has a pole at $s=1$, then $b_i$ must be even; and if $\tau_i$ is of orthogonal type, that is, $L(s, \tau_i, {{\mathrm{Sym}}}^2)$ has a pole at $s=1$, then $b_i$ must be odd. In order for the formal sum $\psi = \psi_1 \boxplus \cdots \boxplus \psi_r$ to be a global Arthur parameter in ${\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n})$, one requires that $2n+1 = \sum_{i=1}^r a_ib_i$, $\prod_{i=1}^r \omega_{\tau_i}^{b_i}=1$, and the simple parameters $\psi_i$’s are pair-wise different.
A global Arthur parameter $\psi$ is called [*generic*]{}, following [@Ar13], if the integers $b_i$ are one. The set of generic global Arthur parameters is denoted by ${\widetilde}{\Phi}_2({{\mathrm{Sp}}}_{2n})$. A generic global Arthur parameter $\phi$ can be written as $\phi=(\tau_1,1)\boxplus(\tau_2,1)\boxplus\cdots\boxplus(\tau_r,1)$.
\[Shahidi\] For any generic global Arthur parameter $\phi=\boxplus_{i=1}^r(\tau_i,1) \in {\widetilde}{\Phi}_2({{\mathrm{Sp}}}_{2n})$, there is an irreducible generic cuspidal automorphic representation $\pi$ of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ belonging to ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})$, and hence $\frak{p}^m(\pi)=\{[(2n)]\}$.
This conjecture has been proved in [@JL15a Theorem 3.3]，using the automorphic descent of Ginzburg, Rallis and Soudry. Note that by analyzing constant terms of residual representations, M[œ]{}glin ([@M08 Proposition 1.2.1]) shows that if there is a residual representation occurring in ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})$, then the Arthur parameter is never generic. Hence we have $${\widetilde}{\Pi}_\phi({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})\subset{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$$ for all generic global Arthur parameters $\phi\in{\widetilde}{\Phi}_2({{\mathrm{Sp}}}_{2n})$. For general Arthur parameters $\psi=\boxplus_{i=1}^r(\tau_i,b_i)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n})$, the following conjecture made in [@J14] extends Conjecture \[Shahidi\] naturally.
\[J14\] Let ${{\mathrm {G}}}$ be a quasi-split classical group. For a given global Arthur parameter $\psi=\boxplus_{i=1}^r(\tau_i,b_i)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n})$, the partition $\eta({\underline}{p}_{\psi})$, which is the Barbasch-Vogan dual of the partition ${\underline}{p}_{\psi}=[b_1^{a_1}b_2^{a_2} \cdots b_r^{a_r}]$ associated to the parameter $\psi$, has the following properties:
1. $\eta({\underline}{p}_{\psi})$ is bigger than or equal to any ${\underline}{p} \in \frak{p}^m(\pi)$ for all $\pi \in {\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$, under the dominance order of partitions as in Definition \[orders\]; and
2. there exists a $\pi \in {\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ with $\eta({\underline}{p}_{\psi}) \in \frak{p}^m(\pi)$.
\[rmkbv\] The Barbasch-Vogan duality can be explained as follows. Given a partition ${\underline}{p}=[p_1p_2 \cdots p_r]$ of $2n+1$, with $p_1 \geq p_2 \geq \cdots \geq p_r$, by [@BV85 Definition A1] and [@Ac03 Section 3.5], the Barbasch-Vogan dual $\eta({\underline}{p})$ is defined to be $(({\underline}{p}^{-})_{{{\mathrm{Sp}}}})^t$. More precisely, one has that ${\underline}{p}^{-}=[p_1p_2\cdots (p_r-1)]$ and $({\underline}{p}^{-})_{{{\mathrm{Sp}}}}$ is the biggest symplectic partition that is smaller than or equal to ${\underline}{p}^{-}$. We refer to [@CM93 Lemma 6.3.8] for the recipe of obtaining $({\underline}{p}^{-})_{{{\mathrm{Sp}}}}$ from ${\underline}{p}^{-}$. $({\underline}{p}^{-})_{{{\mathrm{Sp}}}}$ is called the symplectic collapse of ${\underline}{p}^{-}$. Finally, $(({\underline}{p}^{-})_{{{\mathrm{Sp}}}})^t$ is the transpose of $({\underline}{p}^{-})_{{{\mathrm{Sp}}}}$. By [@Ac03 Lemma 3.3], one has that $\eta({\underline}{p})=(({\underline}{p}^t)^-)_{{{\mathrm{Sp}}}}$.
We recall the following result from [@JL15b], which is also a main ingredient of this paper.
\[ub\] For a given global Arthur parameter $\psi=\boxplus_{i=1}^r(\tau_i,b_i) \in \widetilde{\Psi}_2({{\mathrm{Sp}}}_{2n})$, the Barbasch-Vogan dual $\eta({\underline}{p}_{\psi})$ is bigger than or equal to any ${\underline}{p} \in \frak{p}^m(\pi)$ for every $\pi \in {\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$, under the lexicographical order of partitions as in Definition \[orders\].
It is clear that when the global Arthur parameter $\psi=\phi$ is generic, the partition $\underline{p}_\phi=[1^{2n+1}]$, and hence the partition $\eta(\underline{p}_\phi)=[(2n)]$, which corresponds to the regular nilpotent orbit in $\frak{sp}_{2n}$. Since any symplectic partition is smaller than or equal to $[(2n)]$, it follows that Conjecture \[J14\] holds for all generic Arthur parameters $\phi\in{\widetilde}{\Phi}_2({{\mathrm{Sp}}}_{2n})$. Hence, it is more delicate to understand the lower bound for partitions ${\underline}{p}\in{{\mathfrak {p}}}^m(\pi)$ for all $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$. It is even harder to understand the lower bound for partitions ${\underline}{p}\in{{\mathfrak {p}}}^m(\pi)$ when $\pi\in{\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ for a given global Arthur parameter $\psi\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n})$.
\[splb\] Find symplectic partitions ${\underline}{p}_0$ of $2n$ with the property that
1. there exists a $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ such that ${\underline}{p}_0\in{{\mathfrak {p}}}^m(\pi)$, but
2. for any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$, there does not exist a partition ${\underline}{p}\in{{\mathfrak {p}}}^m(\pi)$ such that ${\underline}{p}<{\underline}{p}_0$, under the dominance order of partitions as in Definition \[orders\].
This problem was motivated by the theory of singular automorphic representations of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$, which is briefly recalled in the following section.
Singular automorphic representations {#sar}
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In this section, consider ${{\mathrm {G}}}_n={{\mathrm{Sp}}}_{2n}, {{\mathrm{SO}}}_{2n+1}$ or ${{\mathrm{SO}}}_{2n}$ to be split classical groups. The theory of singular automorphic representations of ${{\mathrm {G}}}_n({{\mathbb {A}}})$ has been developed based on the notion of ranks for unitary representations of Howe ([@H81]) and by the fundamental work of Li ([@Li92]).
When ${{\mathrm {G}}}_n={{\mathrm{Sp}}}_{2n}$ is the symplectic group, defined by the skew-symmetric matrix $J_n=\begin{pmatrix}0&w\\-w&0\end{pmatrix}$, with $w=(w_{ij})_{n\times n}$ anti-diagonal. Take $P_n=M_nU_n$ to be the Siegel parabolic subgroup of ${{\mathrm{Sp}}}_{2n}$. Hence $M_n\cong{{\mathrm{GL}}}_n$ and the elements of $U_n$ are of form $$u(X)=\begin{pmatrix}I_n&X\\ 0&I_n\end{pmatrix}.$$ The Pontryagin duality tells us that the group of unitary characters $U_n({{\mathbb {A}}})$ which are trivial on $U_n(F)$ is isomorphic to ${{\mathrm{Sym}}}^2(F^n)$, i.e. $${\widehat}{U_n(F)\bks U_n({{\mathbb {A}}})}\cong{{\mathrm{Sym}}}^2(F^n).$$ The explicit isomorphism is given as follows. Take $\psi_F$ to be a nontrivial additive character of $F\bks{{\mathbb {A}}}$. For any $T\in{{\mathrm{Sym}}}^2(F^n)$, i.e. any $n\times n$ symmetric matrix $T$, the corresponding character $\psi_T$ is given by $$\psi_T(u(X)):=\psi_F({{\mathrm{tr}}}(TwX)).$$ The adjoint action of the Levi subgroup ${{\mathrm{GL}}}_n$ on $U_n$ induces an action of ${{\mathrm{GL}}}_n$ on ${{\mathrm{Sym}}}^2(F^n)$. For an automorphic form $\varphi$ on ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$, the $\psi_T$-Fourier coefficient is defined by $$\label{Tfc}
{{\mathcal {F}}}^{\psi_T}(\varphi)(g)
:=
\int_{U_m(F)\bks U_n({{\mathbb {A}}})}\varphi(u(X)g)\psi_T^{-1}(u(X))du(X).$$ An automorphic form $\varphi$ on ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ is called [*non-singular*]{} if $\varphi$ has a nonzero $\psi_T$-Fourier coefficient with the $F$-rank of $T$ maximal, which is $n$. In other words, an automorphic form $\varphi$ on ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ is called [*singular*]{} if $\varphi$ has the property that if a $\psi_T$-Fourier coefficient ${{\mathcal {F}}}^{\psi_T}(\varphi)$ is nonzero, then $\det(T)=0$.
Based on his notion of ranks for unitary representations, Howe shows in [@H81] that if an automorphic form $\varphi$ on ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ is singular, then $\varphi$ can be expressed as a linear combination of certain theta functions. Li in [@Li89] shows that a cuspidal automorphic form of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ with $n$ even is distinguished, i.e. $\varphi$ has nonzero $\psi_T$-Fourier coefficient with only one ${{\mathrm{GL}}}_n$-orbit of non-degenerate $T$ if and only if $\varphi$ is in the space of theta lifting from the orthogonal group ${{\mathrm {O}}}_T$ defined by $T$. A family of explicit examples of such distinguished cuspidal automorphic representations of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ with $n$ even was constructed by Piatetski-Shapiro and Rallis in [@PSR88]. Furthermore, Li proves in [@Li92] the following theorem.
\[li\] For any classical group ${{\mathrm {G}}}_n$, cuspidal automorphic forms on ${{\mathrm {G}}}_n({{\mathbb {A}}})$ are non-singular.
For orthogonal groups ${{\mathrm {G}}}_n$, the singularity of automorphic forms can be defined as follows, following [@Li92]. Let $(V,q)$ be a non-degenerate quadratic space defined over $F$ of dimension $m$ with the Witt index $n=[\frac{m}{2}]$. Let $X^+$ be a maximal totally isotropic subspace of $V$, which has dimension $n$, and let $X^-$ be the maximal totally isotropic subspace of $V$ dual to $X^+$ with respect to $q$. Hence we have the polar decomposition $$V=X^-+V_0+X^+$$ with $V_0$ being the orthogonal complement of $X^-+X^+$, which has dimension less than or equal to one. The generalized flag $$\{0\}\subset X^+\subset V$$ which defines a maximal parabolic subgroup $P_{X^+}$, whose Levi part $M_{X^+}$ is isomorphic to ${{\mathrm{GL}}}_{n}$ and whose unipotent radical $N_{X^+}$ is abelian if $m$ is even; and is a two-step unipotent subgroup with its center $Z_{X^+}$ if $m$ is odd. When $m$ is even, we set $Z_{X^+}=N_{X^+}$. Again, by the Pontryagin duality, we have $${\widehat}{Z_{X^+}(F)\bks Z_{X^+}({{\mathbb {A}}})}\cong\wedge^2(F^{n}),$$ which is given explicitly, as in the case ${{\mathrm{Sp}}}_{2n}$, by the following formula: For any $T\in\wedge^2(F^{[\frac{m}{2}]})$, $$\psi_T(z(X)):=\psi_F({{\mathrm{tr}}}(TwX)).$$ The adjoint action of the Levi subgroup ${{\mathrm{GL}}}_{n}$ on $Z_{X^+}$ induces an action of ${{\mathrm{GL}}}_{n}$ on the space $\wedge^2(F^{n})$. For an automorphic form $\varphi$ on ${{\mathrm {G}}}({{\mathbb {A}}})$, the $\psi_T$-Fourier coefficient is defined by $$\label{Tfc2}
{{\mathcal {F}}}^{\psi_T}(\varphi)(g)
:=
\int_{Z_{X^+}(F)\bks Z_{X^+}({{\mathbb {A}}})}\varphi(z(X)g)\psi_T^{-1}(z(X))dz(X).$$ An automorphic form $\varphi$ on ${{\mathrm {G}}}({{\mathbb {A}}})$ is called [*non-singular*]{} if $\varphi$ has a non-zero $\psi_T$-Fourier coefficient with $T\in\wedge^2(F^{n})$ of maximal rank.
Following Section 2.1, we may reformulate the maximal rank Fourier coefficients of automorphic forms in terms of partitions, and denote by ${\underline}{p}_{{\mathrm{ns}}}$ the partition corresponding to the non-singular Fourier coefficients. It is easy to figure out the following:
1. When ${{\mathrm {G}}}_n={{\mathrm{Sp}}}_{2n}$, one has ${\underline}{p}_{{\mathrm{ns}}}=[2^n]$. This is a special partition for ${{\mathrm{Sp}}}_{2n}$.
2. When ${{\mathrm {G}}}_n={{\mathrm{SO}}}_{2n+1}$, one has $${\underline}{p}_{{\mathrm{ns}}}=
\begin{cases}
[2^{2e}1]&\ \text{if}\ n=2e;\\
[2^{2e}1^3]&\ \text{if}\ n=2e+1.
\end{cases}$$ This is not a special partition of ${{\mathrm{SO}}}_{2n+1}$.
3. When ${{\mathrm {G}}}_n={{\mathrm{SO}}}_{2n}$, one has $${\underline}{p}_{{\mathrm{ns}}}=
\begin{cases}
[2^{2e}]&\ \text{if}\ n=2e;\\
[2^{2e}1^2]&\ \text{if}\ n=2e+1.
\end{cases}$$ This is a special partition of ${{\mathrm{SO}}}_{2n}$.
According to [@JLS15], for any automorphic representation $\pi$, the set $\frak{p}^m(\pi)$ contains only special partitions. Since the non-singular partition ${\underline}{p}_{{\mathrm{ns}}}$ is not special when ${{\mathrm {G}}}_n={{\mathrm{SO}}}_{2n+1}$, the partitions contained in $\frak{p}^m(\pi)$ as $\pi$ runs in the cuspidal spectrum of ${{\mathrm {G}}}_n$ should be bigger than or equal to the following partition $${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm {G}}}_n}=
\begin{cases}
[32^{2e-2}1^2]&\ \text{if}\ n=2e;\\
[32^{2e-2}1^4]&\ \text{if}\ n=2e+1.
\end{cases}$$ Following [@CM93], ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm {G}}}_n}$ denotes the ${{\mathrm {G}}}_n$-expansion of the partition ${\underline}{p}_{{\mathrm{ns}}}$, i.e., the smallest special partition which is bigger than or equal to ${\underline}{p}_{{\mathrm{ns}}}$. Of course, when ${{\mathrm {G}}}_n={{\mathrm{Sp}}}_{2n}$ or ${{\mathrm{SO}}}_{2n}$, one has that ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm {G}}}_n}={\underline}{p}_{{\mathrm{ns}}}$.
\[lbpt\] For split classical group ${{\mathrm {G}}}_n$, the ${{\mathrm {G}}}_n$-expansion of the non-singular partition, ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm {G}}}_n}$, is a lower bound for partitions in the set ${{\mathfrak {p}}}^m(\pi)$ as $\pi$ runs in the cuspidal spectrum of ${{\mathrm {G}}}_n$.
It is natural to ask whether the lower bound ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm {G}}}_n}$ is sharp. This is to construct or find an irreducible cuspidal automorphic representation $\pi$ of ${{\mathrm {G}}}_n({{\mathbb {A}}})$ with the property that ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm {G}}}_n}\in{{\mathfrak {p}}}^m(\pi)$.
When ${{\mathrm {G}}}_n={{\mathrm{Sp}}}_{4e}$ with $n=2e$ even, and when $F$ is totally real, the examples constructed by T. Ikeda ([@Ik01] and [@Ik]) are irreducible cuspidal automorphic representations $\pi$ of ${{\mathrm{Sp}}}_{4e}({{\mathbb {A}}})$ with the global Arthur parameter $\psi=(\tau,2e)\boxplus(1,1)$, where $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ is of symplectic type. By Theorem \[ub\], for any partition ${\underline}{p}\in{{\mathfrak {p}}}^m(\pi)$, we should have $${\underline}{p}\leq \eta({\underline}{p}_\psi)=[2^{2e}]={\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm{Sp}}}_{4e}},$$ under the lexicographical order of partitions, which automatically implies that ${\underline}{p}\leq[2^{2e}]={\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm{Sp}}}_{4e}}$ under the dominance order of partitions. On the other hand, by Theorem \[li\], for any partition ${\underline}{p}\in{{\mathfrak {p}}}^m(\pi)$, we must have $${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm{Sp}}}_{4e}}=[2^{2e}]\leq{\underline}{p},$$ under the dominance order of partitions. It follows that ${{\mathfrak {p}}}^m(\pi)=\{[2^{2e}]={\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm{Sp}}}_{4e}}\}$.
\[ik\] When $F$ is totally real, the non-singular partition ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm{Sp}}}_{4e}}={\underline}{p}_{{\mathrm{ns}}}=[2^{2e}]$ is the sharp lower bound in the sense that for all $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{4e})$, the partition ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm{Sp}}}_{4e}}\in{{\mathfrak {p}}}(\pi)$ and there exists a $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{4e})$, as constructed in [@Ik01] and [@Ik], such that ${\underline}{p}_{{\mathrm{ns}}}^{{{\mathrm{Sp}}}_{4e}}\in{{\mathfrak {p}}}^m(\pi)$.
It is clear that the assumption that $F$ must be totally real is substantial in the construction of Ikeda in [@Ik01] and [@Ik]. However, there is no known approach to proceed the similar construction when $F$ is not totally real. We are going to discuss the situation in the following sections when $F$ is totally imaginary, which leads to a totally different conclusion.
Also the situation is different when we consider orthogonal groups. For ${{\mathrm {G}}}_n$ to ${{\mathrm{SO}}}_{2n+1}$ or ${{\mathrm{SO}}}_{2n}$, in spirit of a conjecture of Ginzburg ([@G06]), any partition ${\underline}{p}$ in ${{\mathfrak {p}}}(\pi)$ with $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}}_n)$ should contain only odd parts. Hence it is reasonable to conjecture the existence of a lower bound which is better than the one determined by non-singularity of cuspidal automorphic representations.
\[lbso\] For ${{\mathrm {G}}}_n$ to be $F$-split ${{\mathrm{SO}}}_{2n+1}$ or ${{\mathrm{SO}}}_{2n}$, the sharp lower bound partition ${\underline}{p}_0^{{{\mathrm {G}}}_n}$ for ${\underline}{p} \in \frak{p}(\pi)$, as $\pi$ runs in ${{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}}_n)$, is given as follows:
1. When ${{\mathrm {G}}}_n={{\mathrm{SO}}}_{2n+1}$, $${\underline}{p}_0^{{{\mathrm{SO}}}_{2n+1}}
=
\begin{cases}
[3^e1^{e+1}]&\ \text{if}\ n=2e;\\
[3^{e+1}1^e]&\ \text{if}\ n=2e+1.
\end{cases}$$
2. When ${{\mathrm {G}}}_n={{\mathrm{SO}}}_{2n}$, $${\underline}{p}_0^{{{\mathrm{SO}}}_{2n}}
=
\begin{cases}
[3^e1^e]&\ \text{if}\ n=2e;\\
[53^{e-1}1^e]&\ \text{if}\ n=2e+1.
\end{cases}$$
We note that a shape lower bound partition for the Fourier coefficients of all irreducible cuspidal representations of ${{\mathrm {G}}}_n({{\mathbb {A}}})$ involves deep arithmetic of the base field $F$, which is one of the main concerns in our investigation. Following the line of ideas in [@H81] and [@Li92], we define the following set of [*small partitions*]{} for the cuspidal spectrum of ${{\mathrm {G}}}_n({{\mathbb {A}}})$: $$\label{smallpartitions}
{{\mathfrak {p}}}_{{{\mathrm{sm}}}}^{{{\mathrm {G}}}_n, F} := \min \{{\underline}{p} \in {{\mathfrak {p}}}^m(\pi)\ |\ \text{ for some}\ \pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}}_n) \}.$$ We call a $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}}_n)$ [*small*]{} if $\frak{p}^m(\pi) \cap \frak{p}_{{{\mathrm{sm}}}}^{{{\mathrm {G}}}_n, F}$ is not empty. Our discussion for small cuspidal automorphic representations will resume in Section \[SCAR\].
On Cuspidality for General Number Fields {#CGNF}
========================================
In this section, we assume that $F$ is a general number field. We mainly consider the [*cuspidality problem*]{} for the global Arthur packets with a family of global Arthur parameters of form: $$\psi=(\chi,b) \oplus (\tau_2, b_2) \oplus \cdots \oplus (\tau_r,b_r)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n}).$$ When $b$ is large, it is most likely that the corresponding global Arthur packet ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members.
Recall from Section 2.2 that by Conjecture \[J14\] for ${{\mathrm {G}}}_n={{\mathrm{Sp}}}_{2n}$, for any $\pi\in{\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$, it is expected that for any partition ${\underline}{p}\in{{\mathfrak {p}}}^m(\pi)$, one should have $$\label{upbfc2}
{\underline}{p}\leq \eta({\underline}{p}_{\psi}),$$ under the dominance order of partitions. We will take this as an [*assumption*]{} for the discussion in this section.
For $\psi=(\chi,b) \oplus (\tau_2, b_2) \oplus \cdots \oplus (\tau_r,b_r) \in \widetilde{\Psi}_2({{\mathrm{Sp}}}_{2n})$, with $\chi$ a quadratic character, the partition associated to $\psi$ is $${\underline}{p}_{\psi}=[(b)^1 (b_2)^{a_2} \cdots (b_r)^{a_r}].$$ By the definition of Arthur parameters for ${{\mathrm{Sp}}}_{2n}$, $b$ is automatically odd. As explained in Remark \[rmkbv\], $\eta({\underline}{p}_{\psi})
=(({\underline}{p}_{\psi}^t)^{-})_{{{\mathrm{Sp}}}}$. Assume that $b >b_0:=\max(b_2, \ldots, b_r)$, then $${\underline}{p}_{\psi}^t=[(1)^b]+[(a_2)^{b_2}]+\cdots + [(a_r)^{b_r}]$$ has the form $[(1+\sum_{i=2}^r a_i)p_2 \cdots p_{b_0} (1)^{b-b_0}]$, and $$({\underline}{p}_{\psi}^t)^{-}=[(1+\sum_{i=2}^r a_i)p_2 \cdots p_{b_0} (1)^{b-b_0-1}].$$ After taking the symplectic collapse, $\eta({\underline}{p}_{\psi})
= (({\underline}{p}_{\psi}^t)^{-})_{{{\mathrm{Sp}}}}$ has the form $$[q_1 q_2\cdots q_k (1)^m],$$ with $m \leq b-1-\sum_{i=2}^r b_i$, and $k+m=b-1$.
If there is a $\pi\in{\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$, by Theorem \[li\], $\pi$ has a nonzero Fourier coefficient attached to the partition $[2^n]$. It is clear that $b>n+1$ if and only if $[2^n]$ is either bigger than or not related to the above partition $[q_1 q_2\cdots q_k (1)^m]$. Hence, we have the following result.
\[unip\] Assume that $\eqref{upbfc2}$ holds. For $$\psi=(\chi,b) \oplus (\tau_2, b_2) \oplus \cdots \oplus (\tau_r,b_r)\in \widetilde{\Psi}_2({{\mathrm{Sp}}}_{2n})$$ with $\chi$ a quadratic character, if $b>n+1$, then the intersection $${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$$ is empty.
Here is an example illustrating the theorem.
Consider $\psi = (\chi, 7) \oplus (\tau,2)\in\widetilde{\Psi}_2({{\mathrm{Sp}}}_{10})$, where $\chi=1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}$, and $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ with $L(s, \tau, \wedge^2)$ having a pole at $s=1$. ${\underline}{p}_{\psi}=[72^2]$ and $\eta({\underline}{p}_{\psi})=[3^21^4]$, which is not related to $[2^5]$. Hence, by the assumption that $\eqref{upbfc2}$ holds, there is no cuspidal members in the global Arthur packet ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{10})$.
\[rmkKR\] In [@KR94 Theorem 7.2.5], Kudla and Rallis show that for a given $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ and a quadratic character $\chi$, the $L$-function $L(s, \pi \times \chi)$ has the right-most possible pole at $s=1+[\frac{n}{2}]$. This implies that the simple global Arthur parameter of type $(\chi,b)$ occurring in the global Arthur parameter of $\pi$ must have the condition that $b$ is at most $1+2[\frac{n}{2}]$. Because $b$ has to be odd in this case, it follows that $b$ is at most $n+1$ if $n$ is even, and $b$ is at most $n$ if $n$ is odd. In any case, one obtains that if $b > n+1$, then the simple global Arthur parameter of type $(\chi,b)$ can not occur in the global Arthur parameter of $\pi$ for any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$. This matches the result in the above theorem.
On Cuspidality for Totally Imaginary Fields {#CTIF}
===========================================
In this section, we assume that $F$ is a totally imaginary number field. We show that there are more global Arthur packets contain no cuspidal members. It is an interesting question to discover the significance of such a difference depending on the arithmetic of the ground field $F$.
On criteria for cuspidality
---------------------------
For any ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$, define a set $B_{{\underline}{a}}$, depending only on ${\underline}{a}$, to be the subset of ${{\mathbb {Z}}}_{\geq 1}^r$ that consists of all $r$-tuples ${\underline}{b}=(b_1, b_2, \ldots, b_r)$ with the property: There are some self-dual $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ for $1 \leq i \leq r$, such that $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)$$ belongs to ${\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n})$ for some $n\geq 1$ with $2n+1=\sum_{i=1}^ra_ib_i$. We define an integer $N_{{\underline}{a}}$ depending only on ${\underline}{a}$ by $$\label{Na}
N_{{\underline}{a}}=\begin{cases}
(\sum_{i=1}^r a_i)^2+2(\sum_{i=1}^r a_i) &\textit{if}\ \sum_{i=1}^r a_i \textit{is even};\\
(\sum_{i=1}^r a_i)^2-1 & \textit{otherwise}.
\end{cases}$$
\[ncmain1\] Assume that $F$ is a totally imaginary number field. Given an ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$ that defines the set $B_{{\underline}{a}}$ and the integer $N_{{\underline}{a}}$ as above. For any ${\underline}{b}=(b_1, b_2, \ldots, b_r) \in B_{{\underline}{a}}$, write $2n+1=\sum_{i=1}^ra_ib_i$. If the condition $$2n=(\sum_{i=1}^r a_ib_i)-1 > N_{{\underline}{a}}$$ holds, then for any global Arthur parameter $\psi$ of the form $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n}),$$ with $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ for $i=1,2,\cdots,r$, ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members.
By assumption, $\psi=\boxplus_{i=1}^r (\tau_i, b_i)$ belongs to ${\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n})$. Recall that $${\underline}{p}_{\psi}=[(b_1)^{a_1} (b_2)^{a_2} \cdots (b_r)^{a_r}]$$ is the partition of $2n+1$ attached to $\psi$. By Remark \[rmkbv\], $\eta({\underline}{p}_{\psi})=(({\underline}{p}_{\psi}^t)^{-})_{{{\mathrm{Sp}}}}$. Then the Barbasch-Vogan dual $\eta({\underline}{p}_{\psi})$ has the following form $$\label{bvsp}
[(\sum_{i=1}^r a_i)p_2 \cdots p_s]_{{{\mathrm{Sp}}}},$$ where $\sum_{i=1}^r a_i \geq p_2 \geq \cdots \geq p_s$. After taking the symplectic collapse of the partition in , one obtains that $\eta({\underline}{p}_{\psi})$ must be one of the following three possible forms:
1. It equals $[(\sum_{i=1}^r a_i)p_2 \cdots p_s]$ if $\sum_{i=1}^r a_i$ is even and $$\sum_{i=1}^r a_i \geq p_2 \geq \cdots \geq p_s.$$
2. It equals $[(\sum_{i=1}^r a_i)p_2 \cdots p_s]$ if $\sum_{i=1}^r a_i$ is odd and $$\sum_{i=1}^r a_i \geq p_2 \geq \cdots \geq p_s.$$
3. It equals $[((\sum_{i=1}^r a_i)-1)p_2 \cdots p_s]$ if $(\sum_{i=1}^r a_i)$ is odd and $$(\sum_{i=1}^r a_i)-1 \geq p_2 \geq \cdots \geq p_s.$$
Assume that $\pi$ belongs to ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$. By Theorem \[ti\], one may assume that $${\underline}{p}_{\pi} = [(2n_1)^{s_1} (2n_2)^{s_2} \cdots (2n_r)^{s_r}]\in\frak{p}^m(\pi)$$ with $n_1 > n_2 > \cdots > n_k \geq 1$ and with the property that $1 \leq s_i \leq 4$ holds for $1 \leq i \leq r$.
[**Case 1:**]{} By Theorem \[ub\], we have $2n_1 \leq \sum_{i=1}^r a_i$. It follows that $$\begin{aligned}
2n
&=&\sum_{i=1}^r 2n_is_i \\
&\leq& 4(2 + 4 + 6 + \cdots + \sum_{i=1}^r a_i)\\
&=&(\sum_{i=1}^r a_i)^2+2(\sum_{i=1}^r a_i)=N_{{\underline}{a}}.\end{aligned}$$
[**Cases 2 and 3:**]{} By Theorem \[ub\], we have $2n_1 \leq (\sum_{i=1}^r a_i)-1$. It follows that $$\begin{aligned}
2n
&=&\sum_{i=1}^r 2n_is_i \\
&\leq& 4(2 + 4 + 6 + \cdots + (\sum_{i=1}^r a_i)-1)\\
&=&(\sum_{i=1}^r a_i)^2-1=N_{{\underline}{a}}.\end{aligned}$$
Now it is easy to check that for any $r$-tuple ${\underline}{b}=(b_1, b_2, \ldots, b_r) \in B_{{\underline}{a}}$, if $2n=(\sum_{i=1}^r a_ib_i)-1 > N_{{\underline}{a}}$, then the global Arthur packets ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ associated to any global Arthur parameters of the form $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)$$ contain no cuspidal members. This completes the proof of the theorem.
Note that in Theorem \[ncmain1\], for a given ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$, the integer $n$ defining the group ${{\mathrm{Sp}}}_{2n}$ depends on the choice of ${\underline}{b}=(b_1, b_2, \ldots, b_r) \in B_{{\underline}{a}}$. We may reformulate the result for a given group ${{\mathrm{Sp}}}_{2n}$ as follows.
For any $r$-tuple ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$, define $B_{{\underline}{a}}^{2n}$ to be the subset of ${{\mathbb {Z}}}_{\geq 1}^r$, consisting of $r$-tuples ${\underline}{b}=(b_1, b_2, \ldots, b_r)$ such that $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n})$$ for some self-dual $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ with $1 \leq i \leq r$. Note that this set $B_{{\underline}{a}}^{2n}$ could be empty in this formulation. The integer $N_{{\underline}{a}}$ is defined to be the same as in . Theorem \[ncmain1\] can be reformulated as follows.
\[ncmain2\] Assume that $F$ is a totally imaginary number field and that ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$ has a non-empty $B_{{\underline}{a}}^{2n}$. If $2n > N_{{\underline}{a}}$, then for any global Arthur parameter $\psi$ of the form $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n}),$$ with $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ for $i=1,2,\cdots,r$, ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members.
On one hand, the integer $N_{{\underline}{a}}$ is not hard to calculate. This makes Theorems \[ncmain1\] and \[ncmain2\] easy to use. On the other hand, the integer $N_{{\underline}{a}}$ depends only on ${\underline}{a}$, and hence may not carry enough information for some applications. Next, we try to improve the above bound $N_{{\underline}{a}}$, by defining a new bound $N_{{\underline}{a},{\underline}{b}}^{(1)}$, depending on both ${\underline}{a}$ and ${\underline}{b}$.
For a partition ${\underline}{p}=[p_1p_2\cdots p_r]$, set $\lvert {\underline}{p} \rvert=\sum_{i=1}^r p_i$. Given an ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$ that defines the set $B_{{\underline}{a}}$. For any ${\underline}{b}=(b_1, b_2, \ldots, b_r)$ belonging to the set $B_{{\underline}{a}}$ that defines the integer $n$ with $$2n+1=\sum_{i=1}^ra_ib_i,$$ the new bound $N_{{\underline}{a},{\underline}{b}}^{(1)}$ is defined to be maximal value of $\lvert {\underline}{p} \rvert$ for all symplectic partitions ${\underline}{p}$, which may not be a partition of $2n$, satisfying the following conditions:
1. ${\underline}{p} \leq \eta({\underline}{p}_{\psi})$ under the [*lexicographical order*]{} of partitions as in Definition \[orders\], and
2. ${\underline}{p}$ has the form $[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}]$ with $2n_1 > 2n_2 > \cdots > 2n_r$ and $s_i \leq 4$ for $1 \leq i \leq r$.
Note that the integer $N_{{\underline}{a},{\underline}{b}}^{(1)}$ depends on ${\underline}{b}$ through Condition (1) above. For this new bound, we have the following result.
\[ncmain3\] Assume that $F$ is a totally imaginary number field. Given an ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$ that defines the set $B_{{\underline}{a}}$. For any ${\underline}{b}=(b_1, b_2, \ldots, b_r) \in B_{{\underline}{a}}$, if $2n=(\sum_{i=1}^r a_ib_i)-1 > N_{{\underline}{a},{\underline}{b}}^{(1)}$, then for any global Arthur parameter $\psi$ of the form $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n}),$$ with $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ for $i=1,2,\cdots,r$, ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members.
Assume that there is a $\pi \in {\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$. By Theorem \[ub\], for any ${\underline}{p} \in \frak{p}^m(\pi)$, which is a partition of $2n$, we must have that ${\underline}{p} \leq \eta({\underline}{p}_{\psi})$ under the lexicographical order of partitions. In particular, the even partition ${\underline}{p}_{\pi} \in \frak{p}^m(\pi)$, constructed in [@GRS03], enjoys this property. On the other hand, since $F$ is totally imaginary, by Theorem \[ti\], ${\underline}{p}_{\pi}$ has the form $[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}]$ with $2n_1 > 2n_2 > \cdots > 2n_r$ and $s_i \leq 4$ for $1 \leq i \leq r$. Hence, ${\underline}{p}_{\pi}$ satisfies the above two conditions defining the bound $N_{{\underline}{a},{\underline}{b}}^{(1)}$. It follows that $N_{{\underline}{a},{\underline}{b}}^{(1)} \geq 2n=\lvert {\underline}{p}_\pi \rvert$. This contradicts the assumption that $2n > N_{{\underline}{a},{\underline}{b}}^{(1)}$.
If we assume that Part (1) of Conjecture \[J14\] holds, namely, $\eta({\underline}{p}_{\psi})$ is bigger than or equal to any ${\underline}{p} \in \frak{p}^m(\pi)$, under the dominance order of partitions, for all $\pi \in {\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$, we may replace the bound $N_{{\underline}{a},{\underline}{b}}^{(1)}$ by an even better bound $N_{{\underline}{a},{\underline}{b}}^{(2)}$ as follows.
Given an ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$ that defines the set $B_{{\underline}{a}}$. For any ${\underline}{b}=(b_1, b_2, \ldots, b_r) \in B_{{\underline}{a}}$ that defines the integer $n$ with $$2n+1=\sum_{i=1}^ra_ib_i,$$ the new bound $N_{{\underline}{a},{\underline}{b}}^{(2)}$ is defined to be the maximal value of $\lvert {\underline}{p} \rvert$ for all symplectic partitions ${\underline}{p}$, which may not be a partition of $2n$, satisfying the following conditions:
1. ${\underline}{p} \leq \eta({\underline}{p}_{\psi})$ under the [*dominance order*]{} of partitions, as in Definition \[orders\], and
2. ${\underline}{p}$ has the form $[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}]$ with $2n_1 > 2n_2 > \cdots > 2n_r$ and $s_i \leq 4$ holds for $1 \leq i \leq r$.
It is clear that the integer $N_{{\underline}{a},{\underline}{b}}^{(2)}$ depends on ${\underline}{b}$ through Condition (1) above. By assuming Part (1) of Conjecture \[J14\], we can prove the following with this new bound.
\[ncmain4\] Assume that $F$ is a totally imaginary number field, and that Part (1) of Conjecture \[J14\] is true. Given an ${\underline}{a}=(a_1, a_2, \ldots, a_r) \in {{\mathbb {Z}}}_{\geq 1}^r$ that defines the set $B_{{\underline}{a}}$. For any ${\underline}{b}=(b_1, b_2, \ldots, b_r) \in B_{{\underline}{a}}$, if $2n=(\sum_{i=1}^r a_ib_i)-1 > N_{{\underline}{a},{\underline}{b}}^{(2)}$, then for any global Arthur parameter $\psi$ of the form $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{2n}),$$ with $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ for $i=1,2,\cdots,r$, ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members.
Assume that there is a $\pi \in {\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$. By Part (1) of Conjecture \[J14\], for any ${\underline}{p} \in \frak{p}^m(\pi)$, which is a partition of $2n$, we must have that ${\underline}{p} \leq \eta({\underline}{p}_{\psi})$ under the dominance order of partitions. In particular, the even partition ${\underline}{p}_{\pi} \in \frak{p}^m(\pi)$, constructed in [@GRS03], enjoys this property. On the other hand, since $F$ is totally imaginary, by Theorem \[ti\], ${\underline}{p}_{\pi}$ has the form $[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}]$ with $2n_1 > 2n_2 > \cdots > 2n_r$ and $s_i \leq 4$ for $1 \leq i \leq r$. Hence, ${\underline}{p}_{\pi}$ satisfies the above two conditions defining the bound $N_{{\underline}{a},{\underline}{b}}^{(2)}$. It follows that $N_{{\underline}{a},{\underline}{b}}^{(2)} \geq 2n=\lvert {\underline}{p}_\pi \rvert$. This contradicts the assumption that $2n > N_{{\underline}{a},{\underline}{b}}^{(2)}$.
First, it is clear that $N_{{\underline}{a}} \geq N_{{\underline}{a},{\underline}{b}}^{(1)} \geq N_{{\underline}{a},{\underline}{b}}^{(2)}$. We expect that the bound $N_{{\underline}{a},{\underline}{b}}^{(2)}$ is sharp. Namely, for any ${\underline}{b}=(b_1, b_2, \ldots, b_r) \in B_{{\underline}{a}}$ with $\sum_{i=1}^r a_ib_i = N_{{\underline}{a},{\underline}{b}}^{(2)}+1$, we expect that any global packet ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{N_{{\underline}{a},{\underline}{b}}^{(2)}})$ associated to any global Arthur parameter $\psi$ of the form $$\psi=(\tau_1, b_1) \boxplus (\tau_2, b_2) \boxplus \cdots \boxplus (\tau_r, b_r)\in{\widetilde}{\Psi}_2({{\mathrm{Sp}}}_{N_{{\underline}{a},{\underline}{b}}^{(2)}}),$$ with $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ for $i=1,2,\cdots,r$, contains a cuspidal member. An interesting problem is to figure out the explicit formula of the bounds $N_{{\underline}{a},{\underline}{b}}^{(1)}$ and $N_{{\underline}{a},{\underline}{b}}^{(2)}$ as functions of ${\underline}{a}$ and ${\underline}{b}$. Secondly, one may easily write down the corresponding analogues of Theorem \[ncmain2\] for bounds $N_{{\underline}{a},{\underline}{b}}^{(1)}$ and $N_{{\underline}{a},{\underline}{b}}^{(2)}$, we omit them here. Finally, we give examples to indicate that $N_{{\underline}{a}} > N_{{\underline}{a},{\underline}{b}}^{(1)} > N_{{\underline}{a},{\underline}{b}}^{(2)}$.
Consider $\psi=(\tau_1, 1) \boxplus (\tau_2, 8)$, where $\tau_1\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_5)$ of orthogonal type, and $\tau_2\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ of symplectic type. By Remark \[rmkbv\], $$\eta({\underline}{p}_{\psi})=(([1^58^2]^t)^-)_{{{\mathrm{Sp}}}}=[72^61]_{{{\mathrm{Sp}}}}=[62^7].$$ In this case, one has that $N_{{\underline}{a}}=(5+2)^2-1=48$. On the other hand, one has that $N_{{\underline}{a},{\underline}{b}}^{(1)}=24$ and $N_{{\underline}{a},{\underline}{b}}^{(2)}=16$.
In fact, $[4^42^4]$ is the only partition ${\underline}{p}$ that gives maximal $\lvert {\underline}{p} \rvert$, and satisfies the conditions: ${\underline}{p} \leq \eta({\underline}{p}_{\psi})$ under the lexicographical order of partitions, and ${\underline}{p}$ has the form $[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}]$ with $2n_1 > 2n_2 > \cdots 2n_r$ and $s_i \leq 4$ for $1 \leq i \leq r$. This shows that $N_{{\underline}{a},{\underline}{b}}^{(1)}=24$.
Also, $[4^22^4]$ is the only partition ${\underline}{p}$ that gives maximal $\lvert {\underline}{p} \rvert$, and satisfies the conditions: ${\underline}{p} \leq \eta({\underline}{p}_{\psi})$ under the dominance order of partitions, and ${\underline}{p}$ has the form $[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}]$ with $2n_1 > 2n_2 > \cdots 2n_r$ and $s_i \leq 4$ for $1 \leq i \leq r$. This shows that $N_{{\underline}{a},{\underline}{b}}^{(2)}=16$.
Note that the bound $N_{{\underline}{a},{\underline}{b}}^{(1)}$ uses Theorem \[ub\], while the bound $N_{{\underline}{a},{\underline}{b}}^{(2)}$ needs the assumption that Part (1) of Conjecture \[J14\] holds.
Examples
--------
We give examples of Arthur parameters $\psi$ such that ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members.
[**Example 1:**]{}Let $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{2l})$ be such that $L(s, \tau, \wedge^2)$ has a pole at $s=1$. Consider the Arthur parameter $\psi=(\tau, 2m) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 1)$. In this case, we have that ${\underline}{a}=(2l,1)$ and ${\underline}{b}=(2m,1)$. Since $a_1+a_2=2l+1$ is odd, we have that $$N_{{\underline}{a}}=(a_1+a_2)^2-1=(2l+1)^2-1.$$ If $m > l+1$, then we have $$4ml=a_1b_1+a_2b_2-1=2l(2m)+1-1 > (2l+1)^2-1=N_{{\underline}{a}},$$ and hence, by Theorem \[ncmain1\] or Theorem \[ncmain2\], ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4ml}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{4ml})$ contains no cuspidal members.
But, if in addition, $L(\frac{1}{2}, \tau) \neq 0$, we can construct a residual representation in ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4ml}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{4ml})$ as follows. Let $P_{2ml} = M_{2ml}N_{2ml}$ be the parabolic subgroup of ${{\mathrm{Sp}}}_{4ml}$ with Levi subgroup $M_{2ml} \cong {{\mathrm{GL}}}_{2l}^{\times m}$. For any $$\phi \in A(N_{2ml}({{\mathbb {A}}})M_{2ml}(F) {\backslash}{{\mathrm{Sp}}}_{4ml}({{\mathbb {A}}}))_{\Delta(\tau,m)},$$ following [@L76] and [@MW95], a residual Eisenstein series can be defined by $$E(\phi,s)(g)=\sum_{\gamma\in P_{2ml}(F)\bks {{\mathrm{Sp}}}_{4ml}(F)}\lambda_s \phi(\gamma g).$$ It converges absolutely for real part of $s$ large and has meromorphic continuation to the whole complex plane ${{\mathbb {C}}}$. Since $L(\frac{1}{2}, \tau) \neq 0$, by [@JLZ13], this Eisenstein series has a simple pole at $\frac{m}{2}$, which is the right-most one. Denote the representation generated by these residues at $s=\frac{m}{2}$ by ${{\mathcal {E}}}_{\Delta(\tau, m)}$, which is square-integrable. By [@JLZ13 Section 6.2], ${{\mathcal {E}}}_{\Delta(\tau,m)}$ has the global Arthur parameter $\psi=(\tau, 2m) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 1)$, and hence belongs to ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4ml}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{4ml})$.
[**Example 2:**]{}Consider a family of Arthur parameters of symplectic groups of the form $$\psi=(1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, b_1) \boxplus (\tau, b_2),$$ where $b_1\geq 1$ is odd, $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ is of symplectic type and $b_2\geq 1$ is even. By definition, ${\underline}{p}_{\psi} = [b_1b_2^2]$, and $$\eta({\underline}{p}_{\psi})=(({\underline}{p}_{\psi}^{-})_{{{\mathrm{Sp}}}})^t = (({\underline}{p}_{\psi}^t)^{-})_{{{\mathrm{Sp}}}}=(([1^{b_1}]+[2^{b_2}])^{-})_{{{\mathrm{Sp}}}}.$$ It is clear that the biggest part occurring in the partition $\eta({\underline}{p}_{\psi})$ is at most $3$. Note that $2n=a_1b_1+a_2b_2-1=b_1+2b_2-1$.
Assume that $\pi$ belongs to ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ with the above given global Arthur parameter $\psi$. By Theorem \[ub\], for any ${\underline}{p} \in \frak{p}^m(\pi)$, its biggest part is smaller than or equal to 3. On the other hand, the partition ${\underline}{p}_{\pi} \in \frak{p}^m(\pi)$ constructed in [@GRS03] is even. Hence, ${\underline}{p}_{\pi} = \{[2^n]\}$. Since $F$ is totally imaginary, by Theorem \[ti\], we must have that $n \leq 4$. Hence, one can see that $N_{{\underline}{a}}=N_{{\underline}{a},{\underline}{b}}^{(1)}=N_{{\underline}{a},{\underline}{b}}^{(2)}=8$, where ${\underline}{a}=\{1,2\}$, ${\underline}{b}=\{b_1,b_2\}$. It follows from Theorems \[ncmain1\]–\[ncmain4\] that ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members except probably the following cases (see Figure 1 below) $$(b_1,b_2)=(1,2), (1,4), (3,2), (5,2).$$ In particular, the global Arthur packet ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members if $n\geq 5$.
(7,2) circle(4pt); (3,4) circle(4pt); (5,4) circle(4pt); (7,4) circle(4pt); (1,6) circle(4pt); (3,6) circle(4pt); (5,6) circle(4pt); (7,6) circle(4pt); (0, 0) – (8, 0); (0, 0) – (0, 7); at (8,-0.4) [$b_1$]{}; at (-0.4, 7) [$b_2$]{}; at (1,0) [$+$]{}; at (3,0) [$+$]{}; at (5,0) [$+$]{}; at (2,0) [$+$]{}; at (4,0) [$+$]{}; at (6,0) [$+$]{}; at (7,0) [$+$]{}; at (0,2) [$+$]{}; at (0,4) [$+$]{}; at (0,6) [$+$]{}; at (0,1) [$+$]{}; at (0,3) [$+$]{}; at (0,5) [$+$]{}; at (1,2) [$*$]{}; at (3,2) [$*$]{}; at (5,2) [$*$]{}; at (1,4) [$*$]{}; (4,-1)
As we mentioned before that for generic global Arthur parameters $\phi\in{\widetilde}{\Phi}_2({{\mathrm {G}}})$, one must have $${\widetilde}{\Pi}_\phi({{\mathrm {G}}})\cap{{\mathcal {A}}}_2({{\mathrm {G}}})\subset{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm {G}}}).$$ In [@M08] and [@M11], M[œ]{}glin considers the problem on which non-generic global Arthur packets contains non-cuspidal members, i.e. the square-integrable residual representations of ${{\mathrm {G}}}({{\mathbb {A}}})$. She gives a conjecture on necessary and sufficient conditions for this problem and proves the conjecture when the square-integral representations with cohomology at infinity. Moreover, in [@M08 Section 4.6], M[œ]{}glin predicts that her conjecture implies that for a given global Arthur parameter $\psi=\boxplus_{i=1}^r (\tau_i, b_i)$ of a symplectic group ${{\mathrm{Sp}}}_{2n}$, where $\tau_i\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_{a_i})$ is self-dual, if there exist $1 \leq j_1 \leq r$ such that $b_{j_1} \geq a_{j_1}+a_{j_2}+b_{j_2}$, for any $1 \leq j_2 \neq j_1 \leq r$, then ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members. Comparing to our discussions and examples above, one may easily find that [**Example 1**]{} gives examples that ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members, which matches her prediction. But, our [**Example 2**]{} contains many more cases that ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$ contains no cuspidal members, which can not be determined by the condition suggested by M[œ]{}glin. We remark that [**Example 2**]{} also includes cases that can not be decided by the discussion in Section 3. One of such cases is that given by $(b_1,b_2)=(5,6)$.
On generalized Ramanujan problem
================================
The generalized Ramanujan problem as proposed by P. Sarnak in [@Sar05 Section 2] is to understand the behavior of the local components of irreducible cuspidal automorphic representations of ${{\mathrm {G}}}({{\mathbb {A}}})$ for general reductive algebraic group ${{\mathrm {G}}}$ defined over a number field $F$. The generalized Ramanujan conjecture asserts that all local components of irreducible generic cuspidal representations are tempered. When the group ${{\mathrm {G}}}$ is not a general linear group, an irreducible cuspidal automorphic representation $\pi$ of ${{\mathrm {G}}}({{\mathbb {A}}})$ may have non-tempered local components. Examples are those cuspidal members in a global Arthur packet with a non-generic global Arthur parameters. Hence it is important also from this prospective to determine which non-generic global Arthur packets have no cuspidal members.
More precisely, the endoscopic classification of Arthur provides certain bounds for the exponents of the unramified local components of the irreducible automorphic representations occurring in the discrete spectrum. It is clear that if one is able to determine which non-generic global packets have no cuspidal members, the bounds of the exponents of the unramified local components of the cuspidal spectrum would be much improved, which definitely helps us to the understanding of the generalized Ramanujan problem.
In this section, we take a preliminary step to understand the bounds of exponents of the unramified local components of the cuspidal spectrum of ${{\mathrm{Sp}}}_{2n}$ based on the results obtained in Section 4.
For $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ and $\theta \in {{\mathbb {R}}}_{\geq 0}$, we say that $\pi$ satisfies $R(\theta)$ if each of its unramified components $\pi_v$ is the unique unramified component of the induced representation $${{\mathrm{Ind}}}_{B(F_v)}^{{{\mathrm{Sp}}}_{2n}(F_v)} \chi_1 \lvert \cdot \rvert^{\alpha_1} \otimes \chi_2 \lvert \cdot \rvert^{\alpha_2}
\otimes \cdots \otimes \chi_n \lvert \cdot \rvert^{\alpha_n},$$ where $B$ is the standard Borel subgroup of ${{\mathrm{Sp}}}_{2n}$, with the property that for $1 \leq i \leq n$, $\chi_i$ are unitary unramified characters of $F_v^*$, such that $0 \leq \alpha_i \leq \theta$.
By the discussion in Remark \[rmkKR\], if there is a simple global Arthur parameter $(\chi, b)$ occurring as a formal summand in the global Arthur parameter $\psi$ of $\pi$, one must have that $b \leq n+1$ if $n$ is even, and that $b \leq n$ if $n$ is odd, where $\chi$ is a quadratic automorphic character of ${{\mathrm{GL}}}_1({{\mathbb {A}}})$. In order to figure out an upper bound $\theta$ for every $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ to satisfy $R(\theta)$, one only needs to consider simple global Arthur parameters $(\tau,b)$ that may occur in the global Arthur parameter $\psi$ of $\pi$, where $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ being self-dual.
First, assume that $n$ is even. Consider a global Arthur parameter of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$, $\psi=(1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})},1) \boxplus (\tau, n)$, with $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ of symplectic type. By using the bound of Kim-Sarnak ([@KS03]) and Blomer-Brumley ([@BB11]) towards the Ramanujan conjecture for ${{\mathrm{GL}}}_2$, which is $R(\frac{7}{64})$, one may easily figure out that any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})\cap{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ satisfies $R(\frac{7}{64}+\frac{n-1}{2})$. By the result of Kudla and Rallis ([@KR94]), for any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})\cap{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ (with $n$ even), if a simple global Arthur parameter $(\chi,b)$ occurs in the global Arthur parameter $\psi$ of $\pi$, one must have that $b$ is at most $n+1$, and hence satisfies $R(\frac{n}{2})$. Note that $\frac{7}{64}+\frac{n-1}{2} < \frac{n}{2}$. It follows that $\frac{n}{2}$ is a possible upper bound for all $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$. On the other hand, Piatetski-Shapiro and Rallis ([@PSR88]) construct a cuspidal member $\pi\in{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ (with $n$ even) that has the simple global Arthur parameter $(\chi, n+1)$ occurring in the $\psi$. Therefore, we obtain that $\frac{n}{2}$ is the sharp upper bound for all $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ when $n$ is even. We state the conclusion of the above discussion as
\[ubeven\] Let $F$ be a number field. When $n$ is an even integer, all $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ satisfy $R(\frac{n}{2})$, and the bound $\frac{n}{2}$ is achieved by the $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ constructed by Piatetski-Shapiro and Rallis in [@PSR88].
Next, assume that $n$ is odd. Consider a global Arthur parameter of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$, $\psi=(\omega_{\tau},1) \boxplus (\tau, n)$, with $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ of orthogonal type and $\omega_{\tau}$ the central character of $\tau$. By the same reason, one has that all $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})\cap{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ satisfy $R(\frac{7}{64}+\frac{n-1}{2})$. Again by [@KR94], for any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})\cap{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$ (with $n$ odd), if a simple global Arthur parameter $(\chi,b)$ occurs in the global Arthur parameter $\psi$ of $\pi$, one must have that $b$ is at most $n$, and hence satisfies $R(\frac{n-1}{2})$. Because $\frac{n-1}{2} < \frac{7}{64}+\frac{n-1}{2}$, we obtain that $\frac{7}{64}+\frac{n-1}{2}$ is a possible upper bound for any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$.
However, by Theorem \[ncmain1\], if we assume that $F$ is totally imaginary and $n \geq 5$, then for the Arthur parameters $\psi=(\omega_{\tau},1) \boxplus (\tau, n)$ given above, there does not exist any cuspidal member in ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n}) \cap {{\mathcal {A}}}_2({{\mathrm{Sp}}}_{2n})$. Hence, we obtain the following conclusion.
\[ubodd\] Assume that $F$ is totally imaginary and $n \geq 5$ is odd. Any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$ satisfies $R(\frac{n-1}{2})$.
We may expect that a simple global Arthur parameter $(\tau,n-1)$ with $n$ odd and $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_2)$ of symplectic type could have cuspidal members in the global Arthur packet ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{2n})$, although we do not know how to construct them for the moment. However, in the case the bound is $\frac{7}{64}+\frac{n-2}{2}$, which is less than $\frac{n-1}{2}$. Also, for $\tau\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{GL}}}_a)$ (self-dual) with $a\geq 3$, the simple global Arthur parameters of type $(\tau, b)$ produce naturally a bound better than what obtained above, and hence are omitted for further consideration.
It is a very interesting problem to determine the sharp upper bound $\theta$ for the cuspidal spectrum of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ when $n$ is odd. This would involve a generalization or extension of the constructions by Piatetski-Shapiro and Rallis ([@PSR88]) and by Ikeda ([@Ik01] and [@Ik]). We will get back to this issue in our future work.
Small Cuspidal Automorphic Representations {#SCAR}
==========================================
In this section, we discuss some criteria on the [*smallness*]{} of cuspidal automorphic representations of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ and gives examples of small cuspidal automorphic representations, in addition to the examples constructed by Ikeda in [@Ik01]. From now on, we assume that $F$ is a number field.
Characterization of small cuspidal representations
--------------------------------------------------
The characterization of small cuspidal automorphic representations will be given in terms of a vanishing condition on Fourier coefficients related to the automorphic descent method ([@GRS11]), and also in terms of the notion of hyper-cuspidality in the sense of Piatetski-Shapiro ([@PS83]). Also, our discussions cover the case of symplectic group ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ and the case of the metaplectic double cover ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$ of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ together.
\[spclt\] Assume that $\pi$ is an irreducible cuspidal automorphic representation of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$. Then $\frak{p}^m(\pi) = \{[2^n]\}$ if and only if $\pi$ has no nonzero Fourier coefficients attached to the partition $[41^{2n-4}]$.
First, assume that $\frak{p}^m(\pi) = \{[2^n]\}$. In this case, the even partition ${\underline}{p}_{\pi}$ constructed in [@GRS03] is exactly $[2^n]$. Note that ${\underline}{p}_{\pi}$ has the property that “maximal at every stage" (see the proof of [@GRS03 Theorem 2.7] or [@JL15 Remark 5.1]), which implies directly that $\pi$ has no nonzero Fourier coefficients attached to the partition $[41^{2n-4}]$.
Next, assume that $\pi$ has no nonzero Fourier coefficients attached to the partition $[41^{2n-4}]$. By Lemma \[spvandescent\] below, $\pi$ has no nonzero Fourier coefficients attached to the partition $[(2k)1^{2n-2k}]$, for any $2 \leq k \leq n$. Assume that ${\underline}{p}=[p_1p_2\cdots p_s] \in \frak{p}^m(\pi)$, with $p_1 \geq p _2 \geq \cdots \geq p_s$. If $p_1$ is odd, then one must have that $p_1 \geq 3$. By [@JL15 Lemma 3.3], $\pi$ has a nonzero Fourier coefficient attached to the partition $[(p_1)^21^{2n-2p_1}]$. Then [@GRS03 Lemma 2.4] shows that $\pi$ must have a nonzero Fourier coefficient attached to the partition $[(2r)1^{2n-2r}]$ for some $2r > 2p_1 \geq 6$, which contradicts the assumption of the theorem. Now, if $p_1$ is even, then by [@GRS03 Lemma 2.6] or [@JL15 Lemma 3.1], $\pi$ has a nonzero Fourier coefficient attached to the partition $[(p_1)1^{2n-p_1}]$. By assumption of the theorem, we must have that $p_1=2$. Hence we obtain that $2 = p_1 \geq p_2 \geq \cdots \geq p_s$, which implies that ${\underline}{p} \leq [2^n]$. On the other hand, by Theorem \[li\], the cuspidal $\pi$ must have a nonzero Fourier coefficient attached to the partition $[2^n]$. It follows that for any ${\underline}{p}\in\frak{p}^m(\pi)$, the case that ${\underline}{p} < [2^n]$ can not happen. Therefore, we conclude that ${\underline}{p} = [2^n]$, and hence $\frak{p}^m(\pi) = \{[2^n]\}$. This completes the proof of the theorem.
Let $\alpha = e_1 - e_{2n}$ be the longest positive root of ${{\mathrm{Sp}}}_{2n}$ and let $X_{\alpha}$ be the corresponding one-dimensional root subgroup. Recall from [@PS83 Section 6] that an automorphic function $\varphi$ is called [*hypercuspidal*]{} if $$\int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} \varphi(xg) dx \equiv 0.$$ It is clear that any hypercupsidal function is automatically cuspidal. An automorpic representation $\pi$ of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$ is called [*hypercuspidal*]{} if every $\varphi \in \pi$ is hypercuspidal.
For $0 \leq i \leq n-1$, let $P_i=M_iN_i$ be the parabolic subgroup of ${{\mathrm{Sp}}}_{2n}$ with Levi subgroup $M \cong {{\mathrm{GL}}}_1^i \times {{\mathrm{Sp}}}_{2n-2i}$. Define a character of $N_i$ by $\psi_i(n)=\psi(\sum_{j=1}^{i} n_{j,j+1})$. Let $\pi$ be an automorphic representation of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$. For any $\varphi \in \pi$, let $${{\mathcal {F}}}_i(\varphi)(g)=\int_{N_i(F) {\backslash}N_i({{\mathbb {A}}})} \varphi(ng) \psi_i^{-1}(n) dn.$$
\[spfe\] Let $\pi$ be a cuspidal automorphic representation of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$. For any $\varphi \in \pi$, ${{\mathcal {F}}}_i(\varphi)$ is a linear combination of ${{\mathcal {F}}}_{i+1}(\varphi)$ and Fourier coefficients attached to the partition $[(2i+2)1^{2n-2i-2}]$.
Let $\alpha$ be the root $e_{i+1}-e_{2n-i}$ and let $X_{\alpha}$ be the corresponding one-dimensional root subgroup. Since $X_{\alpha}$ normalizes $N_i$ and preserves the character $\psi_i$, one can take the Fourier expansion of ${{\mathcal {F}}}_i(\varphi)$ along $X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})$. The non-constant terms give us exactly Fourier coefficients attached to the partition $[(2i+2)1^{2n-2i-2}]$. Now consider the constant term, that is $\int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} {{\mathcal {F}}}_i(\varphi)(xg)dx$.
For $i+2 \leq j \leq 2n-i-1$, let $\alpha_j$ be the root $e_{i+1}-e_j$, and let $X_{\alpha_j}$ be the corresponding one-dimensional root subgroup. Let $X=\prod_{j=i+2}^{2n-i-1} X_{\alpha_j}$. Then, one can see that $X$ normalizes $N_iX_{\alpha}$ and preserves the character $\psi_i$. Here $\psi_i$ is extended trivially to $N_iX_{\alpha}$. Hence, one can take the Fourier expansion of $\int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} {{\mathcal {F}}}_i(\varphi)(xg)dx$ along $X(F) {\backslash}X({{\mathbb {A}}})$, and obtain that $$\begin{aligned}
\ & \int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} {{\mathcal {F}}}_i(\varphi)(xg)dx\\
= \ & \sum_{\xi \in X(F)} \int_{X(F) {\backslash}X({{\mathbb {A}}})} \int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} {{\mathcal {F}}}_i(\varphi)(xx'g)\psi_{\xi}^{-1}(x')dxdx'.\end{aligned}$$
Note that the constant term corresponding to $\xi=0$ is identically zero, since $\varphi \in \pi$ is cuspidal. Also note that ${{\mathrm{Sp}}}_{2n-2i-2}(F)$ acts on $X(F) {\backslash}\{0\}$ transitively, and one can take a representative $\xi_0 =(1,0,\ldots,0)$. Denote the stabilizer of $\xi_0$ in ${{\mathrm{Sp}}}_{2n-2i-2}(F)$ by $H(F)$, which is a Jacobi group ${{\mathcal {H}}}_{2n-2i-4}(F) \rtimes {{\mathrm{Sp}}}_{2n-2i-4}(F)$. Embed ${{\mathrm{Sp}}}_{2n-2i-2}$ into ${{\mathrm{Sp}}}_{2n}$ via $g \rightarrow \begin{pmatrix}
I_{i+1} & 0 & 0\\
0& g & 0\\
0 & 0 & I_{i+1}
\end{pmatrix}$, and identify it with its image under this embedding. Then the above Fourier expansion can be rewritten as $$\begin{aligned}
\ & \int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} {{\mathcal {F}}}_i(\varphi)(xg)dx\\
= \ & \sum_{\gamma \in H(F) {\backslash}{{\mathrm{Sp}}}_{2n-2i-2}(F)} \int_{X(F) {\backslash}X({{\mathbb {A}}})} \int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} {{\mathcal {F}}}_i(\varphi)(xx'\gamma g)\psi_{\xi_0}^{-1}(x')dxdx',\end{aligned}$$ which is exactly $$\sum_{\gamma \in H(F) {\backslash}{{\mathrm{Sp}}}_{2n-2i-2}(F)} {{\mathcal {F}}}_{i+1}(\varphi)(\gamma g).$$ Therefore, ${{\mathcal {F}}}_i(\varphi)$ is a linear combination of ${{\mathcal {F}}}_{i+1}(\varphi)$ and Fourier coefficients attached to the partition $[(2i+2)1^{2n-2i-2}]$. This completes the proof of the lemma.
The following lemma is [@GRS05 Key Lemma 3.3], which has been used in the proof of Theorem \[spclt\]. We state it here and give a shorter proof, using Theorem \[ggsglobal\].
\[spvandescent\] Let $\pi$ be any automorphic representation of ${{\mathrm {G}}}({{\mathbb {A}}})={{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$. If $\pi$ has no nonzero Fourier coefficients attached to the partition $[(2k)1^{2n-2k}]$, then $\pi$ has no nonzero Fourier coefficients attached to the partition $[(2k+2)1^{2n-2k-2}]$.
We recall that the $F$-rational nilpotent orbits ${{\mathcal {O}}}$ of $\frak{sp}_{2n}(F)$ corresponding to the partition $[(2k+2)1^{2n-2k-2}]$ are parameterized by square classes $\beta \in F^* / (F^*)^2$. For $1 \leq j \leq k$, define the root $\alpha_j=e_{j+1}-e_j$. Let $\alpha_{k+1}=e_{2n+1-k-1}-e_{k+1}$. For $1 \leq j \leq k+1$, let $x_{\alpha_j}$ be the corresponding root subspace in the Lie algebra. A representative of ${{\mathcal {O}}}$ can be chosen to be $f=\sum_{j=1}^{k} x_{\alpha_j}(\frac{1}{2})+x_{\alpha_{k+1}}(\beta)$. It is clear that $f$ can be decomposed as $f_1 +f_2$, where $f_1 =
x_{\alpha_1}(\frac{1}{2})$, and $f_2=\sum_{j=2}^{k} x_{\alpha_j}(\frac{1}{2})+x_{\alpha_{k+1}}(\beta)$. Note that the nilpotent orbit containing $f_2$ corresponds to the partition $[(2k)1^{2n-2k}]$.
For $f$ and $f_i$, $i=1,2$, By Jacobson-Morozov Theorem, there exist $\frak{sl}_2$-triples $(e,h,f)$, $(e_i,h_i,f_i)$, such that $[h,u]=-2u$, $[h_i, u_i]=-2u_i$. Let ${{\mathrm {G}}}({{\mathbb {A}}})_h$ be the centralizer of $h$ in ${{\mathrm {G}}}({{\mathbb {A}}})$, which contains the maximal split torus ${{\mathrm {T}}}({{\mathbb {A}}})$ of ${{\mathrm {G}}}({{\mathbb {A}}})$. It is clear that $f_2 \in {\overline}{{{\mathrm {G}}}({{\mathbb {A}}})_h f}$. Indeed, take $t = \diag(t_1, 1, \ldots, 1, t_1^{-1})$ with $t_1^{-1} \rightarrow 0$, then $t \cdot f \rightarrow f_2$. By Theorem \[ggsglobal\], if $\pi$ has a nonzero Fourier coefficient attached to $f$, then it has a nonzero Fourier coefficient attached to $f_2$. Since by assumption, $\pi$ has no nonzero Fourier coefficients attached to $[(2k)1^{2n-2k}]$, we can conclude that $\pi$ also has no nonzero Fourier coefficients attached to the partition $[(2k+2)1^{2n-2k-2}]$.
\[spclt2\] For an irreducible cuspidal automorphic representation $\pi$ of ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$, $\frak{p}^m(\pi) = \{[2^n]\}$ if and only if $\pi$ is hypercuspidal.
By Theorem \[spclt\], we just need to show that $\pi$ is hypercuspidal if and only if $\pi$ has no nonzero Fourier coefficients attached to partition $[41^{2n-4}]$. First, it is clear that if $\pi$ is hypercuspidal, then $\pi$ has no nonzero Fourier coefficients attached to partition $[41^{2n-4}]$, since $X_{\alpha}$， for the longest root $\alpha$, is the center of the standard maximal unipotent subgroup of ${{\mathrm{Sp}}}_{2n}$. Now assume that $\pi$ has no nonzero Fourier coefficients attached to partition $[41^{2n-4}]$. By Lemma \[spvandescent\], $\pi$ has no nonzero Fourier coefficients attached to partition $[(2k)1^{2n-2k}]$, for any $2 \leq k \leq n$.
Let $Y$ be the unipotent subgroup of ${{\mathrm{Sp}}}_{2n}$ consisting of elements $y=\begin{pmatrix}
1 & x & *\\
0 & I_{2n-2} & x^*\\
0 & 0 & 1
\end{pmatrix}$, where $x \in {{\mathrm{Mat}}}_{1 \times (2n-2)}$. It is clear that $Y$ normalizes $X_{\alpha}$. Hence, $f(g):=\int_{X_{\alpha}(F) {\backslash}X_{\alpha}({{\mathbb {A}}})} \phi(xg) dx$ can be viewed as an automorphic function over $Y(F) {\backslash}Y({{\mathbb {A}}})$. After taking Fourier expansion along $Y(F) {\backslash}Y({{\mathbb {A}}})$, $$\label{spclt2equ1}
f(g)=\sum_{\xi \in F^{2n-2}{\backslash}\{0\}}
\int_{Y(F) {\backslash}Y({{\mathbb {A}}})} f(yg)\psi_{\xi}^{-1}(y) dy,$$ since $\pi$ is a cuspidal.
Note that the action of ${{\mathrm{Sp}}}_{2n-2}(F)$ on $F^{2n-2}{\backslash}\{0\}$ via conjugation is transitive. Take a representative $\xi_0=(1,0,\ldots,0)$. Then its stabilizer in ${{\mathrm{Sp}}}_{2n-2}(F)$ is a subgroup (denoted by $H$) consisting of elements $\begin{pmatrix}
1 & x & y \\
0 & g' & x^*\\
0 & 0 & 1
\end{pmatrix}$, where $x \in {{\mathrm{Mat}}}_{1 \times {2n-4}}$, $y \in F$, $g' \in {{\mathrm{Sp}}}_{2n-4}$. Embed ${{\mathrm{Sp}}}_{2n-2}$ into ${{\mathrm{Sp}}}_{2n}$ via the map $g \rightarrow \begin{pmatrix}
1 & 0 & 0 \\
0 & g & 0\\
0 & 0 & 1
\end{pmatrix}$, and identify ${{\mathrm{Sp}}}_{2n-2}$ with its image under this embedding. Then, after changing of variables, the Fourier expansion in can be rewritten as $$\label{spclt2equ2}
f(g)=\sum_{\gamma \in H {\backslash}{{\mathrm{Sp}}}_{2n-2}(F)}
\int_{Y(F) {\backslash}Y({{\mathbb {A}}})} f(y\gamma g)\psi_{\xi_0}^{-1}(y) dy,$$ which is exactly $\sum_{\gamma \in H {\backslash}{{\mathrm{Sp}}}_{2n-2}(F)} {{\mathcal {F}}}_1(f)(\gamma g)$. Hence, to show that $f$ is identically zero, it is enough to show that ${{\mathcal {F}}}_1(f)$ is identically zero.
Applying Lemma \[spfe\] repeatedly, ${{\mathcal {F}}}_1(f)$ is a linear combination of Fourier coefficients attached to the partitions $[(2k)1^{2n-2k}]$, $2 \leq k \leq n$, which are all identically zero, by the above discussion. Therefore, $f$ is identically zero, i.e., $\pi$ is hypercuspidal.
This completes the proof of the theorem.
Combining Theorems \[spclt\], \[spclt2\] with Theorem \[ti\], we have the following corollary.
\[nohc\] Assume that $F$ is a totally imaginary number field and $n \geq 5$. Then ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ and ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$ have no cuspidal representations having nonzero Fourier coefficients attached to the partitions $[41^{4n-4}]$, and equivalently, have no nonzero hypercuspidal representations.
Assume that ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ and ${\widetilde}{{{\mathrm{Sp}}}}_{2n}({{\mathbb {A}}})$ has a nonzero cuspidal representation $\pi$ having nonzero Fourier coefficients attached to the partitions $[41^{4n-4}]$, equivalently, $\pi$ is hypercuspidal. Then, by Theorems \[spclt\], \[spclt2\], $\frak{p}^m(\pi) = \{[2^n]\}$. In particular, the even partitions ${\underline}{p}_{\pi}$ constructed in [@GRS03] is exactly $[2^n]$. On the other hand, since $F$ is totally imaginary, by Theorem \[ti\], ${\underline}{p}_{\pi}$ can not be $[2^n]$ because of $n \geq 5$. Contradiction.
Examples of small cuspidal representations
------------------------------------------
In this section, we assume that $F$ is not totally imaginary number field if $n \geq 5$. In order to provide examples of global Arthur packets of ${{\mathrm{Sp}}}_{2n}$ whose cuspidal automorphic members $\pi$ have the property that ${{\mathfrak {p}}}^m(\pi) = \{[2^n]\}$, we separate the discussion according the parity of the integer $n$.
### [**Case of $n=2e$.**]{}
\[speven2\] Any $\pi\in{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{4e}) \cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{4e})$ with $$\psi=(\tau, 2i) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 4e-4i+1), e \leq 2i \leq 2e,$$ and $\tau \in {{\mathcal {A}}}_{{{\mathrm{cusp}}}}({{\mathrm{GL}}}_2)$ of symplectic type, has the property that $\frak{p}^m(\pi) = \{[2^{2e}]\}$, and hence is small.
For $\psi=(\tau, 2i) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 4e-4i+1)$, with $e \leq 2i \leq 2e$, we must have that ${\underline}{p}_{\psi}=[(2i)^2 (4e-4i+1)]$ and $\eta({\underline}{p}_{\psi})$ has largest part at most $3$. Any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{4e})\cap{\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{4e})$, by Theorem \[ub\], any partition ${\underline}{p} \in \frak{p}^m(\pi)$ satisfies the property that ${\underline}{p} \leq {\underline}{p}_{\psi}$ under the lexicographical order of partitions. Hence, any partition ${\underline}{p}=[p_1p_2 \cdots p_r] \in \frak{p}^m(\pi)$ has largest part $p_1 \leq 3$.
If $p_1=3$, then by [@JL15 Lemma 3.3], $\pi$ has a nonzero Fourier coefficient attached to the partition $[(p_1)^2 1^{4e-2p_1}]$. Furthermore, by [@GRS03 Lemma 2.4], $\pi$ has a nonzero Fourier coefficient attached to the partition $[(2r)1^{4e-2r}]$ for some $2r > p_1 = 3$, which contradicts Theorem \[ub\]. Hence we may have to take that $p_1=2$ and ${\underline}{p} \leq [2^{2e}]$ under the dominance order of partitions. In this case, by Theorem \[li\], $\pi$ is non-singular. It follows again that any ${\underline}{p} \in \frak{p}^m(\pi)$ satisfies the property that ${\underline}{p} \geq [2^{2e}]$ under the dominance order of partitions. Therefore, we must have that $\frak{p}^m(\pi) = \{[2^{2e}]\}$.
Note that if $2i < e$, then $4e-4i+1 > 2e+1$. By Remark \[rmkKR\], the global Arthur packet ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{4e})$ corresponding to the global Arthur parameter $$\psi=(\tau, 2i) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 4e-4i+1)$$ contains no cuspidal automorphic representations.
In the case of $2i=2e$, $\psi=(\tau, 2e) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 1)$, where $\tau \in {{\mathcal {A}}}_{{{\mathrm{cusp}}}}({{\mathrm{GL}}}_2)$ is of symplectic type. If in addition $L(\frac{1}{2}, \tau) \neq 0$, then we can construct a residual representation in ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4e})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{4e})$ as follows.
Let $\Delta(\tau, e)$ be a Speh residual representation in the discrete spectrum of ${{\mathrm{GL}}}_{2e}({{\mathbb {A}}})$. For more information about the Speh residual representations, we refer to [@MW89], or [@JLZ13 Section 1.2]. Let $P_r=M_rN_r$ be the maximal parabolic subgroup of ${{\mathrm{Sp}}}_{2l}$ with Levi subgroup $M_r$ isomorphic to ${{\mathrm{GL}}}_r \times {{\mathrm{Sp}}}_{2l-2r}$. Using the normalization in [@Sh10], the group $X_{M_{r}}^{{{\mathrm{Sp}}}_{2l}}$ of all continues homomorphisms from $M_{r}({{\mathbb {A}}})$ to ${{\mathbb {C}}}^{\times}$, which is trivial on $M_{r}({{\mathbb {A}}})^1$ (see [@MW95]), will be identified with ${{\mathbb {C}}}$ by $s \rightarrow \lambda_s$.
For any $\phi \in A(N_{2e}({{\mathbb {A}}})M_{2e}(F) {\backslash}{{\mathrm{Sp}}}_{4e}({{\mathbb {A}}}))_{\Delta(\tau,e)}$, following [@L76] and [@MW95], a residual Eisenstein series can be defined by $$E(\phi,s)(g)=\sum_{\gamma\in P_{2e}(F)\bks {{\mathrm{Sp}}}_{4e}(F)}\lambda_s \phi(\gamma g).$$ It converges absolutely for real part of $s$ large and has meromorphic continuation to the whole complex plane ${{\mathbb {C}}}$. Since $L(\frac{1}{2}, \tau) \neq 0$, by [@JLZ13], this Eisenstein series has a simple pole at $\frac{e}{2}$, which is the right-most one. Denote by ${{\mathcal {E}}}_{\Delta(\tau, e)}$ the representation generated by these residues at $s=\frac{e}{2}$. This residual representation is square-integrable. By [@JLZ13 Section 6.2], the global Arthur parameter of ${{\mathcal {E}}}_{\Delta(\tau,e)}$ is $\psi=(\tau, 2e) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 1)$. Hence ${{\mathcal {E}}}_{\Delta(\tau,e)} \in {\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4e})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{4e})$.
By [@L13 Theorem 1.3], $\frak{p}^m({{\mathcal {E}}}_{\Delta(\tau,e)}) = \{[2^{2e}]\}$. For $\psi$ above, ${\underline}{p}_{\psi}= [(2e)^21]$ and $\eta({\underline}{p}_{\psi})=[2^{2e}]$. Hence, as mentioned in [@L13], combining with Theorem \[ub\], all parts of Conjecture \[J14\] have been proved for the Arthur parameter $\psi=(\tau, 2e) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 1)$ above.
### [**Case of $n=2e+1$.**]{}
\[spodd2\] Any $\pi\in{\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4e+2})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{4e+2})$ with $\psi=(\tau, 2i+1) \boxplus (\omega_{\tau}, 4e-4i+1)$, $e \leq 2i \leq 2e$, and $\tau \in {{\mathcal {A}}}_{{{\mathrm{cusp}}}}({{\mathrm{GL}}}_2)$ of orthogonal type, has the property that $\frak{p}^m(\pi) = \{[2^{2e+1}]\}$, and hence is small.
The proof of this proposition is similar to that of Proposition \[speven2\], and is omitted here. Note that if $2i < e$, then $4e-4i+1 > 2e+1$. By Remark \[rmkKR\], the global Arthur packet ${\widetilde}{\Pi}_\psi({{\mathrm{Sp}}}_{4e+2})$ associated to the global Arthur parameter $$\psi=(\tau, 2i+1) \boxplus (1_{{{\mathrm{GL}}}_1({{\mathbb {A}}})}, 4e-4i+1)$$ contains no cuspidal automorphic representations.
In the case of $2i=2e$, we can also construct a residual representation in ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4e+2})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{4e+2})$ as follows.
Since $\tau \in {{\mathcal {A}}}_{{{\mathrm{cusp}}}}({{\mathrm{GL}}}_2)$ is of orthogonal type, by the theory of automorphic descent of Ginzburg, Rallis and Soudry, there is a cuspidal representation $\pi'$ of ${{\mathrm{SO}}}_2^{\alpha}({{\mathbb {A}}})$ which is anisotropic, such that $\pi'$ lifts to $\tau$ by the automorphic induction. Assume that there is an irreducible generic cuspidal representation $\pi$ of ${{\mathrm{Sp}}}_2({{\mathbb {A}}})$ corresponding to $\pi'$ under the theta correspondence. Then the global Langlands functorial transfer from ${{\mathrm{Sp}}}_2$ to ${{\mathrm{GL}}}_3$ takes $\pi$ to $\tau\boxplus 1$.
For any $\phi \in A(N_{2e}({{\mathbb {A}}})M_{2e}(F) {\backslash}{{\mathrm{Sp}}}_{4e+2}({{\mathbb {A}}}))_{\Delta(\tau,e) \otimes \pi}$, a residual Eisenstein series can be defined as before by $$E(\phi,s)(g)=\sum_{\gamma\in P_{2e}(F)\bks {{\mathrm{Sp}}}_{4e+2}(F)}\lambda_s \phi(\gamma g).$$ It converges absolutely for real part of $s$ large and has meromorphic continuation to the whole complex plane ${{\mathbb {C}}}$. By [@JLZ13], this Eisenstein series has a simple pole at $\frac{e+1}{2}$, which is the right-most one. Denote by ${{\mathcal {E}}}_{\Delta(\tau, e) \otimes \pi}$ the representation generated by these residues at $s=\frac{e+1}{2}$. This residual representation is square-integrable. By [@JLZ13 Section 6.2], the global Arthur parameter of ${{\mathcal {E}}}_{\Delta(\tau,e) \otimes \pi}$ is $\psi=(\tau, 2e+1) \boxplus (\omega_{\tau}, 1)$. Hence ${{\mathcal {E}}}_{\Delta(\tau,e) \otimes \pi} \in {\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{4e+2})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{4e+2})$.
By [@JL15c Theorem 2.1], $\frak{p}^m({{\mathcal {E}}}_{\Delta(\tau,e) \otimes \pi}) = \{[2^{2e+1}]\}$. For $\psi=(\tau, 2e+1) \boxplus (\omega_{\tau}, 1)$ above, ${\underline}{p}_{\psi}=[(2e+1)^21]$ and $\eta({\underline}{p}_{\psi})=[2^{2e+1}]$. Hence, combining with Theorem \[ub\], all parts of Conjecture \[J14\] have been proved for the Arthur parameter $\psi=(\tau, 2e+1) \boxplus (\omega_{\tau}, 1)$ above.
### [**Case $n=3e+1$.**]{}
\[spodd3\] Any $\pi\in{\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{6e+2})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{6e+2})$ with $\psi=(\tau, 2e+1)$, and $\tau \in {{\mathcal {A}}}_{{{\mathrm{cusp}}}}({{\mathrm{GL}}}_3)$ of orthogonal type and with trivial central character, has the property that $\frak{p}^m(\pi) = \{[2^{3e+1}]\}$, and hence is small.
For $\psi=(\tau, 2e+1)$, we must have that ${\underline}{p}_{\psi}=[(2e+1)^3]$ and $\eta({\underline}{p}_{\psi})=[3^{2e}2]$. Take any $\pi\in{\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{6e+2})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{6e+2})$. By Theorem \[ub\], for any ${\underline}{p}=[p_1 p_2 \cdots p_r] \in \frak{p}^m(\pi)$, we have that ${\underline}{p} \leq [3^{2e}2]$ under the lexicographical order of partitions. It follows that $$3 \geq p_1 \geq \cdots \geq p_r.$$ If $p_1=3$, then by [@JL15 Lemma 3.3], $\pi$ has a nonzero Fourier coefficient attached to the partition $[(p_1)^2 1^{6e+2-2p_1}]$. Then, by [@GRS03 Lemma 2.4], $\pi$ has a nonzero Fourier coefficient attached to the partition $[(2r)1^{6e+2-2r}]$ for some $2r > p_1 = 3$, which contradicts Theorem \[ub\]. Hence $p_1=2$, and ${\underline}{p} \leq [2^{3e+1}]$ under the dominance order of partitions. On the other hand, by Theorem \[li\], $\pi$ is non-singular. Hence, any ${\underline}{p} \in \frak{p}^m(\pi)$ also satisfies the property that ${\underline}{p} \geq [2^{3e+1}]$ under the dominance order of partitions. Therefore, we have proved that $\frak{p}^m(\pi) = \{[2^{3e+1}]\}$.
This completes the proof of the proposition.
We can also construct a residual representation in ${\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{6e+2})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{6e+2})$ as follows. Since $\tau \in {{\mathcal {A}}}_{{{\mathrm{cusp}}}}({{\mathrm{GL}}}_3)$ has trivial central character, and $L(s, \tau, {{\mathrm{Sym}}}^2)$ has a pole at $s=1$, by the theory of automorphic descent ([@GRS11]), there is an irreducible generic cuspidal automorphic representation $\pi$ of ${{\mathrm{Sp}}}_2({{\mathbb {A}}})$ that lifts to $\tau$.
For any $\phi \in A(N_{3e}({{\mathbb {A}}})M_{3e}(F) {\backslash}{{\mathrm{Sp}}}_{6e+2}({{\mathbb {A}}}))_{\Delta(\tau,e) \otimes \pi}$, a residual Eisenstein series can also be defined by $$E(\phi,s)(g)=\sum_{\gamma\in P_{3e}(F)\bks {{\mathrm{Sp}}}_{6e+2}(F)}\lambda_s \phi(\gamma g).$$ It converges absolutely for real part of $s$ large and has meromorphic continuation to the whole complex plane ${{\mathbb {C}}}$. By [@JLZ13], this Eisenstein series has a simple pole at $\frac{e+1}{2}$, which is the right-most one. Denote by ${{\mathcal {E}}}_{\Delta(\tau, e) \otimes \pi}$ the representation generated by these residues at $s=\frac{e+1}{2}$. This residual representation is square-integrable. By [@JLZ13 Section 6.2], the global Arthur parameter of ${{\mathcal {E}}}_{\Delta(\tau,e) \otimes \pi}$ is $\psi=(\tau, 2e+1)$. Hence ${{\mathcal {E}}}_{\Delta(\tau,e) \otimes \pi} \in {\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{6e+2})\cap{{\mathcal {A}}}_2({{\mathrm{Sp}}}_{6e+2})$.
For $\psi=(\tau, 2e+1)$ as above, ${\underline}{p}_{\psi}=[(2e+1)^3]$, and $\eta({\underline}{p}_{\psi})=[3^{2e}2]$. Hence, by Theorem \[ub\], for any $\pi\in{\widetilde}{\Pi}_{\psi}({{\mathrm{Sp}}}_{6e+2})\cap{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{6e+2})$, we have that for any ${\underline}{p}=[p_1 p_2 \cdots p_r] \in \frak{p}^m(\pi)$, ${\underline}{p} \leq [3^{2e}2]$ under the lexicographical order of partitions, and hence, ${\underline}{p} \leq [3^{2e}2]$ under the dominance order of partitions also. By [@JL15c Theorem 2.1], $\frak{p}^m({{\mathcal {E}}}_{\Delta(\tau,e) \otimes \pi}) = \{[3^{2e}2]\}$. Therefore, all parts of Conjecture \[J14\] have been proved for the global Arthur parameter $\psi=(\tau, 2e+1)$ as above.
Small cuspidal representations over totally imaginary number fields
-------------------------------------------------------------------
In this section, let $F$ be a totally imaginary number field. Assume that $\frak{p}^m(\pi)$ is a singleton. Then $\frak{p}^m(\pi)$ consists of exactly the partition ${\underline}{p}_{\pi}$ constructed in [@GRS03].
Let ${\underline}{p}({{\mathrm{Sp}}}_{2n}, F)$ be the smallest even partition of $2n$ of the form $$[(2n_1)^{s_1}(2n_2)^{s_2} \cdots (2n_r)^{s_r}],$$ with $2n_1 > 2n_2 > \cdots > 2n_r$ and $s_i \leq 4$ for $1 \leq i \leq r$.
${\underline}{p}({{\mathrm{Sp}}}_{8}, F) = [2^4]$, ${\underline}{p}({{\mathrm{Sp}}}_{10}, F) = [42^3]$, ${\underline}{p}({{\mathrm{Sp}}}_{12}, F) = [42^4]$, ${\underline}{p}({{\mathrm{Sp}}}_{14}, F) = [4^22^3]$ and ${\underline}{p}({{\mathrm{Sp}}}_{26}, F) = [64^32^4]$.
\[lowerbound\] Let $F$ be a totally imaginary number field. Assume that $\frak{p}^m(\pi)$ is a singleton for any $\pi\in{{\mathcal {A}}}_{{\mathrm{cusp}}}({{\mathrm{Sp}}}_{2n})$. Then any partition ${\underline}{p} \in \frak{p}_{sm}^{{{\mathrm{Sp}}}_{2n}, F}$ is bigger than or equal to ${\underline}{p}({{\mathrm{Sp}}}_{2n}, F)$.
[abcdefg]{} P. Achar, [*An order-reversing duality map for conjugacy classes in Lusztig’s canonical quotient.*]{} Transform. Groups **8** (2003), no. 2, 107–145.
J. Arthur, [*The endoscopic classification of representations: Orthogonal and Symplectic groups*]{}. Colloquium Publication Vol. **61**, 2013, American Mathematical Society.
D. Barbasch and D. Vogan, [*Unipotent representations of complex semisimple groups.*]{} Ann. of Math. (2) **121** (1985), no. 1, 41–110.
V. Blomer and F. Brumley, *On the Ramanujan conjecture over number fields*. Ann. of Math. (2) **174** (2011), no. 1, 581–605.
D. Collingwood and W. McGovern, [*Nilpotent orbits in semisimple Lie algebras.*]{} Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. xiv+186 pp.
W. Duke and Ö. Imamoglu, [*A converse theorem and the Saito-Kurokawa lift*]{}, Internat. Math. Res. Notices **7** (1996), 347–355.
D. Ginzburg, [*Certain conjectures relating unipotent orbits to automorphic representations.*]{} Israel J. Math. [**151**]{} (2006), 323–355.
D. Ginzburg, S. Rallis and D. Soudry, [*On Fourier coefficients of automorphic forms of symplectic groups.*]{} Manuscripta Math. **111** (2003), no. 1, 1–16.
D. Ginzburg, S. Rallis and D. Soudry, [*Contruction of CAP representations for Symplectic groups using the descent method.*]{} Automorphic representations, L-functions and applications: progress and prospects, 193–224, Ohio State Univ. Math. Res. Inst. Publ., [**11**]{}, de Gruyter, Berlin, 2005.
D. Ginzburg, S. Rallis and D. Soudry, [*The descent map from automorphic representations of [${\rm GL}(n)$]{} to classical groups.*]{} World Scientific, Singapore, 2011. v+339 pp.
R. Gomez, D. Gourevitch and S. Sahi, [*Generalized and degenerate Whittaker models.*]{} Preprint. 2015.
R. Howe, [*Automorphic forms of low rank*]{}. Noncommutative harmonic analysis and Lie groups, 211–248, Lecture Notes in Math., [**880**]{}, Springer, 1981.
R. Howe and I. Piatetski-Shapiro, [*A counterexample to the "generalized Ramanujan conjecture” for (quasi-) split groups*]{}. Automorphic forms, representations and L-functions, Part 1, pp. 315–322, Proc. Sympos. Pure Math., [**33**]{}, Amer. Math. Soc., Providence, R.I., 1979.
T. Ikeda, [*On the lifting of elliptic cusp forms to Siegel cusp forms of degree $2n$*]{}. Ann. of Math. (2) [**154**]{} (2001), no. 3, 641–681.
T. Ikeda, [*On the lifting of automorphic representations of ${{\mathrm{PGL}}}_2({{\mathbb {A}}})$ to ${{\mathrm{Sp}}}_{2n}({{\mathbb {A}}})$ or ${\widetilde}{{{\mathrm{Sp}}}}_{2n+1}({{\mathbb {A}}})$ over a totally real field*]{}. Preprint.
D. Jiang, [*Automorphic Integral transforms for classical groups I: endoscopy correspondences*]{}. Automorphic Forms: L-functions and related geometry: assessing the legacy of I.I. Piatetski-Shapiro. Comtemp. Math. **614**, 2014, AMS.
D. Jiang and B. Liu, [*On Fourier coefficients of automorphic forms of ${\rm GL}(n)$*]{}. Int. Math. Res. Not. 2013 (17): 4029–4071.
D. Jiang and B. Liu, [*On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups*]{}. J. Number Theory [**146**]{} (2015), 343–389.
Jiang, Dihua; Liu, Baiying [*Fourier coefficients for automorphic forms on quasisplit classical groups*]{}. To appear in Cogdell volume. 2015.
D. Jiang and B. Liu, [*Arthur parameters and Fourier coefficients for automorphic forms on symplectic groups.*]{} To appear in Annales de l’Institut Fourier. 2015.
D. Jiang and B. Liu, [*On Fourier coefficients of certain residual representations of symplectic groups.*]{} To appear in Pacific J. of Math. 2015.
D. Jiang, B. Liu, and G. Savin, [*Raising nilpotent orbits for classical groups.*]{} Submitted. 2015.
D. Jiang, B. Liu, B. Xu, and L. Zhang, [*The Jacquet-Langlands correspondence via twisted descent*]{}. accepted by IMRN 2015.
D. Jiang, B. Liu and L. Zhang, [*Poles of certain residual Eisenstein series of classical groups*]{}. Pacific J. of Math. Vol. **264** (2013), No. 1, 83–123
D. Jiang, L. Zhang, [*Arthur parameters and cuspidal automorphic modules of classical groups*]{}. submitted, 2015.
H. Kim and P. Sarnak, *Refined estimates towards the Ramanujan and Selberg conjectures*. Journal of AMS. **16** (2003), 175–181.
S. Kudla and S. Rallis, [*A regularized Siegel-Weil formula: the first term identity.*]{} Ann. of Math. (2) [**140**]{} (1994), no. 1, 1–80.
R. Langlands, [*On the functional equations satisfied by Eisenstein series*]{}. Springer Lecture Notes in Math. **544**. 1976.
J.-S. Li, [*Distinguished cusp forms are theta series*]{}. Duke Math. J. [**59**]{} (1989), no. 1, 175–189.
J.-S. Li, [*Nonexistence of singular cusp forms*]{}. Compositio Math. [**83**]{} (1992), no. 1, 43–51.
B. Liu, [*Fourier coefficients of automorphic forms and Arthur classification.*]{} Thesis (Ph.D.) University of Minnesota. 2013. 127 pp. ISBN: 978-1303-19255-5.
W. Luo, Z. Rudnick and P. Sarnak, [*On the generalized Ramanujan conjecture for ${{\mathrm{GL}}}(n)$.*]{} Proc. Sympos. Pure Math., [**66**]{}, part 2, (1999), 301–310.
C. Mœglin, [*Formes automorphes de carré intégrable noncuspidales*]{}. Manuscripta Math. **127** (2008), no. 4, 411–467.
C. Mœglin, [*Image des opérateurs d’entrelacements normalisés et pôles des séries d’Eisenstein*]{}. Adv. Math. **228** (2011), no. 2, 1068–1134.
C. Mœglin and J.-L. Waldspurger, [*Le spectre residuel de [${\rm GL}(n)$.]{}*]{} Ann. Sci. École Norm. Sup. (4) **22** (1989), no. 4, 605–674.
C. Mœglin and J.-L. Waldspurger, [*Spectral decomposition and Eisenstein series.*]{} Cambridge Tracts in Mathematics, [**113**]{}. Cambridge University Press, Cambridge, 1995.
O. Paniagua-Taboada, [*Some consequences of Arthur’s conjectures for special orthogonal even groups.*]{} J. Reine Angew. Math. **661** (2011), 37–84.
I. Piatetski-Shapiro, [*On the Saito-Kurokawa lifting.*]{} Invent. Math. **71** (1983), no. 2, 309–338.
I. Piatetski-Shapiro and S. Rallis, [*A new way to get Euler products.*]{} J. reine angew. Math. [**392**]{} (1988), 110–124.
P. Sarnak, [*Notes on the generalized Ramanujan conjectures.*]{} Harmonic analysis, the trace formula, and Shimura varieties, 659–685, Clay Math. Proc., [**4**]{}, Amer. Math. Soc., Providence, RI, 2005.
F. Shahidi, [*Eisenstein series and automorphic L-functions,*]{} volume [**58**]{} of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2010. ISBN 978-0-8218- 4989-7.
J.-L. Waldspurger, [*Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés*]{}. Astérisque [**269**]{}, 2001.
[^1]: The research of the first named author is supported in part by the NSF Grants DMS–1301567, and that of the second named author is supported in part by NSF Grants DMS–1302122, and in part by a postdoc research fund from Department of Mathematics, University of Utah
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abstract: 'We present simple coding strategies, which are variants of the Schalkwijk-Kailath scheme, for communicating reliably over additive white noise channels in the presence of corrupted feedback. More specifically, we consider a framework comprising an additive white forward channel and a backward link which is used for feedback. We consider two types of corruption mechanisms in the backward link. The first is quantization noise, i.e., the encoder receives the quantized values of the past outputs of the forward channel. The quantization is uniform, memoryless and time invariant (that is, symbol-by-symbol scalar quantization), with bounded quantization error. The second corruption mechanism is an arbitrarily distributed additive bounded noise in the backward link. Here we allow symbol-by-symbol encoding at the input to the backward channel. We propose simple explicit schemes that guarantee positive information rate, in bits per channel use, with positive error exponent. If the forward channel is additive white Gaussian then our schemes achieve capacity, in the limit of diminishing amplitude of the noise components at the backward link, while guaranteeing that the probability of error converges to zero as a doubly exponential function of the block length. Furthermore, if the forward channel is additive white Gaussian and the backward link consists of an additive bounded noise channel, with signal-to-noise ratio (SNR) constrained symbol-by-symbol encoding, then our schemes are also capacity-achieving in the limit of high SNR.'
author:
- 'Nuno C Martins and Tsachy Weissman[^1]'
title: '**Coding for Additive White Noise Channels with Feedback Corrupted by Uniform Quantization or Bounded Noise** '
---
\[section\] \[theorem\][Corollary]{} \[theorem\][Conjecture]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[section\] \[section\] \[section\] \[section\] \[section\] \[theorem\][Corollary A]{}
Introduction {#sec:Introduction}
============
That noiseless feedback does not increase the capacity of memoryless channels, but can dramatically enhance the reliability and simplicity of the schemes that achieve it, is well known since Shannon’s work [@Shannon56]. The assumption of noiseless feedback is an idealization often meant to capture communication scenarios where the noise in the backward link is significantly smaller than in the forward channel. However, all the known simple schemes for reliable communication in the presence of feedback rely heavily on the assumption that the feedback is completely noise-free, and break down when noise is introduced into the backward link.
As a case in point, it was recently shown in [@KimLapidothWeissman] that *any* feedback scheme with linear encoding (of which the Schalkwijk-Kailath scheme and its variants are special cases) breaks down completely in the presence of additive white noise of arbitrarily small variance in the backward link: not only is it impossible to achieve capacity, but, with such schemes it is impossible to communicate reliably at any positive information rate.
It is therefore of primary importance, from both the theoretical and the practical viewpoints, to develop channel coding schemes that, by making use of *noisy* feedback, maintain the simplicity of noiseless feedback schemes while achieving a positive rate of reliable communication. It is the quest for such schemes that motivates this paper.
Our main contribution is the derivation of simple coding strategies, which are variants of the Schalkwijk-Kailath scheme, for communicating over additive white channels in the presence of corrupted feedback. More specifically, we consider two types of corruption mechanisms in the backward link:
- Quantization noise: the encoder receives the quantized values of the past outputs of the forward channel. The quantization is uniform, memoryless and time invariant (that is, symbol-by-symbol scalar quantization), with bounded quantization error.
- Additive bounded noise: the noise in the backward link is additive, and has bounded components, but is otherwise arbitrarily distributed. Here we allow symbol-by-symbol encoding at the input to the backward channel.
The coding schemes that we present achieve positive information rate with positive error exponent. In addition, if the forward channel is additive white Gaussian then our schemes are capacity-achieving, in the limit of diminishing amplitude of the noise components in the backward link. Furthermore, if the backward link consists of an additive bounded noise channel, with instantaneous encoding, then our schemes are also capacity-achieving in the limit of high SNR (in the backward link). We note that the diminishing of the gap to capacity with vanishing noise in the backward link is a desired property, not to be taken for granted in light of the negative results in [@KimLapidothWeissman]. In addition, the probability of error of our coding schemes converges to zero as a doubly exponential function of the block length, provided that the forward channel is additive, white and Gaussian. As will be seen in subsequent sections, our analysis of the performance of the suggested schemes is based on elementary linear systems theory.
To our knowledge, the impact of noise in the feedback link on fundamental performance limits and on explicit schemes that attain them has hitherto received little attention. Exceptions are the papers [@SahaiSimsek; @DraperSahai] which study the trade-off between reliability and delay in coding for discrete memoryless channels with noisy feedback, and suggest concrete coding schemes for this scenario. Another exception is the recent [@Permuter], which considers the capacity of discrete finite-state channels in the presence of non-invertible maps in the feedback link, such as quantization. Yet another paper is the aforementioned [@KimLapidothWeissman], which is primarily concerned with the impact of noise in the backward link on the error exponents.
The remainder of this paper is structured as follows. Section \[sec:726pm17jun06\] presents preliminary results and definitions, while Section \[sec:231pm05june06\] specifies and analyzes a coding scheme in the presence of feedback corrupted by bounded additive noise, under the assumption that the noise is observable at the decoder. The main results of the paper are presented in Sections \[sec:447pm15jun06\] and \[sec:639pm17jun06\], where we describe and analyze coding schemes for the cases where the backward link features uniform quantization or bounded additive noise, respectively. The paper ends with conclusions in Section \[sec:conclusions\].
**Notation:**
- Random variables are represented in large caps, such as $Z$.
- Stochastic processes are indexed by the discrete time variable $t$, like in $X_t$. We also use $X^t$ to represent $(X_0, \ldots, X_t)$, provided that $t \geq 0$. If $t$ is a negative integer then we adopt the convention that $X^t$ is the empty set.
- A realization of a random variable $Z$ is represented in small caps, such as $z$.
Preliminary Results and Definitions {#sec:726pm17jun06}
===================================
In this Section, we define and analyze a feedback system whose structure is described by the diagram of Fig \[Fig:904pm30may06\]. The aforementioned system will be present in the coding schemes proposed in subsequent Sections.
For the remainder of this paper, we consider that $W_t$ is a zero mean and white stochastic process of variance $\sigma_W^2$ and that $Z$ is a real random variable taking values in $[0,1]$. In addition, $Z$ and $W^t$ are assumed independent for all $t$. The feedback noise $V_t$ is a bounded real stochastic process whose amplitude has a least upper-bound given by: $$\bar{\sigma}_V \overset{def}{=} \inf \{ \alpha \in \mathbb{R}_{\geq 0} : Prob(|V_t|>\alpha)=0, t \geq 0\}$$ meaning that the following holds: $$Prob(|V_t| \leq \bar{\sigma}_V)=1, \text{ } t \geq 0$$ The remaining signals $U_t$, $Y_t$ and $\hat{Z}_t$ are also real stochastic processes. The block represented in Fig \[Fig:904pm30may06\] by $\phi_{\bar{r}}$ is an operator that maps $Z$ and $U^{t-1}$ into $X_t$ for all $t$. Similarly, $\hat{\phi}_{\bar{r}}$ maps $Y^t$ and $V^t$ into $\hat{Z}_t$. The description of the maps $\phi_{\bar{r}}$ and $\hat{\phi}_{\bar{r}}$ is given in the following definition.
\[def:1011pm30may06\] Given a positive real constant $\bar{r}$, the operators $\phi_{\bar{r}}: \left( t,Z,U^{t-1} \right) \mapsto X_t$ and $\hat{\phi}_{\bar{r}}: \left( t,Y^t,V^{t} \right) \mapsto \hat{Z}_t$, represented in Fig \[Fig:904pm30may06\], are defined as follows: $$\label{eq:701pm31may06}
X_{t}=\phi_{\bar{r}}\left(t, Z,U^{t-1} \right) \overset{def}{=} \begin{cases} (2^{-\bar{r}}-2^{\bar{r}}) \left( \sum_{i=0}^{t-1} 2^{\bar{r}(t-i-1)}U_i+2^{\bar{r}t}Z \right) & \text{if $t \geq 1$} \\ (2^{-\bar{r}}-2^{\bar{r}})Z & \text{if $t=0$} \end{cases}$$ $$\label{eq:702pm31may06}
\hat{Z}_{t}= \hat{\phi}_{\bar{r}}\left(t, Y^t,V^t \right) \overset{def}{=} \begin{cases} -\sum_{i=0}^{t-1} 2^{-\bar{r}(i+1)} (V_{i}+Y_{i}) & \text{ if $t \geq 1$} \\ 0 & \text{if $t=0$} \end{cases}$$
Notice that (\[eq:701pm31may06\]) has a term, given by $2^{\bar{r}t}Z$, that grows exponentially. However, it should be observed that if the feedback loop is closed (see Fig \[Fig:904pm30may06\]) by using $U_t=X_t+V_t+W_t$ then $X_t$ is given by: $$\label{eq:12368sep06}
X_t=(2^{-\bar{r}}-2^{\bar{r}})\left( \sum_{i=0}^{t-1} 2^{-\bar{r}(t-i-1)}(W_i+V_i)+2^{-\bar{r}t}Z\right), \text{ } t \geq 1$$ which describes a system that is stable, in the bounded input implies bounded output sense. In the absence of backward link noise, i.e. $V_t=0$, (\[eq:701pm31may06\]) and (\[eq:702pm31may06\]) are equivalent to the equations used in the original work by Schalkwijk-Kailath [@SchalkwijkKailath]. An alternative minimum variance control interpretation to (\[eq:701pm31may06\]) and (\[eq:702pm31may06\]), in the presence of *perfect* feedback, is given in [@Elia04]. In addition, the work by [@Elia04] extends Schalkwijk-Kailath’s algorithm, with *perfect* feedback, to the multi-user case. A general control theoretic framework to feedback capacity is given in [@Tatikonda]. The following lemma states a few properties of (\[eq:701pm31may06\]) and (\[eq:702pm31may06\]) which motivate their use in the construction of coding schemes.
Let $\sigma_W^2$, $\bar{\sigma}_V$ and $\bar{r}$ be given positive real constants. Consider the feedback system of Fig \[Fig:904pm30may06\], which is described by (\[eq:701pm31may06\])-(\[eq:702pm31may06\]) in conjunction with the following equations: $$Y_t=X_t+W_t$$ $$U_t=X_t+V_t+W_t$$ The following holds: $$\label{eq:932pm31may06}
X_t=2^{\bar{r}t}(2^{\bar{r}}-2^{-\bar{r}})(\hat{Z}_t - Z), \text{ } t \geq 0$$ $$\label{eq:1019pm30may06}
E[X_t^2] \leq \left( \sigma_W \sqrt{2^{2\bar{r}}-1}+\bar{\sigma}_V (2^{\bar{r}}+1) + 2^{-\bar{r}t} (2^{\bar{r}}-2^{-\bar{r}}) \right)^2, \text{ } t \geq 0$$ If $W_t$ is zero-mean, white and Gaussian, with variance $\sigma_W^2$, then the following holds: $$\label{eq:257pm4sept06}
Prob \left( |X_t| \geq \alpha \right) \leq e^{-\frac{\left( \alpha-\gamma\right)^2}{2 \beta^2}}, \text{ } \alpha > 0, \text{ } t \geq 0$$ where $\gamma$ and $\beta$ are the following positive real constants:
$$\label{eq:258pm4sept06}
\gamma \overset{def}{=} \left( 2^{\bar{r}} + 1 \right) \bar{\sigma}_V + 2^{\bar{r}}-2^{-\bar{r}}$$
$$\label{eq:259pm4sept06}
\beta^2 \overset{def}{=} \left( 2^{2\bar{r}}-1 \right) \sigma_W^2$$
**Proof:** In order to derive (\[eq:932pm31may06\]), we substitute $U_t=V_t+Y_t$ in (\[eq:702pm31may06\]). We now proceed to proving the validity of (\[eq:1019pm30may06\]). Since the operators $\phi_{\bar{r}}$ and $\hat{\phi}_{\bar{r}}$ are linear, we can bound the variance of $X_t$ by separately quantifying the contribution of the external *inputs* $Z$, $W_t$ and $V_t$. By making use of the triangular inequality, we arrive at the following bound: $$\label{eq:1036pm30may06}
\left( E[X_t^2] \right)^{1/2} \leq \left( \sigma_W^2 \frac{1}{2 \pi} \int_{-\pi}^{\pi} \left|T \left( e^{j\omega} \right) \right|^2 d \omega \right)^{1/2}+ \bar{\sigma}_V \max_{\omega \in (-\pi,\pi]} \left|T \left( e^{j\omega} \right) \right|+2^{-\bar{r}t}(2^{\bar{r}}-2^{-\bar{r}})$$ where $T\left( e^{j\omega} \right)$ is the following transfer function: $$T\left( e^{j\omega} \right) = \frac{2^{-\bar{r}}-2^{\bar{r}}}{e^{j\omega}-2^{-\bar{r}}}$$ The transfer function $T\left( e^{j\omega} \right)$ describes the input-output behavior of the feedback loop from $V_t$ to $X_t$ and from $W_t$ to $X_t$. The first term in the right hand side of (\[eq:1036pm30may06\]) quantifies the contribution from the white process $W_t$, while the second term is an upper-bound to the contribution of $V_t$ and the last term comes from the *initial condition* determined by $Z$. Standard computations lead to the following results: $$\label{eq:1044pm30may06}
\frac{1}{2 \pi} \int_{-\pi}^{\pi} \left|T \left( e^{j\omega} \right) \right|^2 d \omega = 2^{2\bar{r}}-1$$ $$\label{eq:1045pm30may06}
\max_{\omega \in (-\pi,\pi]} \left|T \left( e^{j\omega} \right) \right| = \frac{2^{\bar{r}}-2^{-\bar{r}}}{1-2^{-\bar{r}}} = 2^{\bar{r}}+1$$ After substituting (\[eq:1044pm30may06\]) and (\[eq:1045pm30may06\]) in (\[eq:1036pm30may06\]), we arrive at (\[eq:1019pm30may06\]). In order to prove (\[eq:257pm4sept06\])-(\[eq:259pm4sept06\]), under the assumption that $W_t$ is zero mean white Gaussian, we define the following auxiliary Gaussian process: $$\tilde{X}_t= \begin{cases} 0 & \text{if $t=0$} \\ (2^{-\bar{r}}-2^{\bar{r}}) \sum_{i=0}^{t-1} 2^{-\bar{r}(t-i-1)} W_i & \text{if $t \geq 1$} \end{cases}$$ After simple manipulations, similar to the ones leading to (\[eq:1044pm30may06\])-(\[eq:1045pm30may06\]), we get the following properties of $\tilde{X}_t$: $$E[\tilde{X}_t^2] = \left( 2^{2\bar{r}}-1 \right) \left( 1-2^{-2\bar{r}t} \right) \sigma_W^2 \leq \beta^2$$ $$|\tilde{X}_t -X_t| \leq \left( 2^{\bar{r}}-2^{-\bar{r}} \right) \left( \bar{\sigma}_V \frac{1-2^{-\bar{r}t}}{1-2^{-\bar{r}}}+2^{-\bar{r}t} \right) \leq \gamma$$ where we used the definitions (\[eq:258pm4sept06\]) and (\[eq:259pm4sept06\]) along with (\[eq:12368sep06\]). Consequently, we arrive at: $$Prob \left( |X_t| \geq \alpha \right) \leq Prob \left( |\tilde{X}_t| \geq \alpha-\gamma \right) \leq \sqrt{\frac{2}{\pi \beta^2}} \int_{\alpha-\gamma}^{\infty} e^{-\frac{\mu^2}{2 \beta^2}} d \mu, \text{ } \alpha > 0$$ where we used the facts that, by definition, $|\tilde{X}_t -X_t| \leq \gamma$, that $E[\tilde{X}_t^2] \leq \beta^2$ and that $\tilde{X}_t$ is normally distributed. The derivation of (\[eq:257pm4sept06\]) is complete once we use the following upper-bound [@Proakis page 220 eq. (5.1.8)]: $$\sqrt{\frac{2}{\pi \beta^2}} \int_{\alpha-\gamma}^{\infty} e^{-\frac{\mu^2}{2 \beta^2}} d \mu \leq e^{-\frac{\left( \alpha-\gamma\right)^2}{2 \beta^2}}$$ $\square$
A coding scheme with feedback {#sec:231pm05june06}
=============================
In this Section, we describe a coding scheme in the presence of feedback according to the framework of Fig \[Fig:1012pm31may06\], where $\phi_{\bar{r}}$ and $\hat{\phi}_{\bar{r}}$ are defined by (\[eq:701pm31may06\])-(\[eq:702pm31may06\]), while the maps $\theta_{n,r}$ and $\hat{\theta}_{n,r}$ will be defined below. Notice that the scheme of Fig \[Fig:1012pm31may06\] assumes that $\hat{\phi}_{\bar{r}}$ has direct access to the feedback noise $V_t$. Under such an assumption, in this Section we construct an efficient and simple coding and decoding scheme which will be used as a basic building block in the rest of the paper. In Section \[sec:447pm15jun06\] we use the fact that if the backward link is corrupted by uniform quantization then, in fact, $V_t$ is the quantization error which can be recovered from the output of the forward channel and used as an input to $\hat{\phi}_{\bar{r}}$. Finally, in Section \[sec:639pm17jun06\] we show that bounded noise in the feedback link can be dealt with by using a modification of the quantized feedback framework of Section \[sec:447pm15jun06\]. It should be noted that in the schemes presented in Sections \[sec:447pm15jun06\] and \[sec:639pm17jun06\], the decoder relies solely on the output of the forward channel.
The main result of this Section is stated in Theorem \[th:313pm03may06\], where we compute a rate of reliable[^2] transmission, in bits per channel use, which is achievable by the scheme of Fig \[Fig:1012pm31may06\], in the presence of a power constraint at the input of the forward channel. Such a transmission rate is a function of the parameters $\sigma_W^2$, $\bar{\sigma}_V$ and it also depends on the forward channel’s input power constraint, which we denote as $P_X^2$. Theorem \[th:313pm03may06\] also provides a lower bound on the error exponent of the resulting scheme. If the forward channel is additive, white and Gaussian then Theorem \[th:313pm03may06\] shows that the probability of error of the scheme of Fig \[Fig:1012pm31may06\] decreases as a doubly exponential function of the block length.
We start with the following definitions of the *ceiling* and *floor* functions denoted by $\bar{\Theta}$ and $\Theta$, respectively. $$\bar{\Theta}(a)\overset{def}{=} \min \{n \in \mathbb{N}: a \leq n \}, \text{ } a \in \mathbb{R}$$ $$\label{eq:353pm08june06}
\Theta(a) \overset{def}{=} \max \{n \in \mathbb{N}: a \geq n \}, \text{ } a \in \mathbb{R}$$
The following definition specifies the maps $\theta_{n,r}$ and $\hat{\theta}_{n,r}$ represented in Fig \[Fig:1012pm31may06\].
Given a positive integer $n$, a positive real constant $r$, a random variable $M$ taking values in the set $\{1, \ldots,
2^{\Theta(rn)}\}$ and a real stochastic process $\hat{Z}_t$, the following is the definition of the maps $\theta_{n,r}:M \mapsto Z$ and $\hat{\theta}_{n,r}:\hat{Z}_t \mapsto \hat{M}_t$: $$\label{eq:1028pm31may06} Z= \theta_{n,r} (M) \overset{def}{=} \left(M-\frac{1}{2}
\right) 2^{-\Theta(rn)}$$ $$\label{eq:1029pm31may06} \hat{M}_t = \hat{\theta}_{n,r} (\hat{Z}_t) \overset{def}{=}
\bar{\Theta}\left( 2^{\Theta(rt)} \hat{Z}_t \right), \text{ } t \in \{0, \ldots,n \}$$
For the remainder of this paper, $n$ denotes the block length of the coding schemes and $r$ represents a design parameter that quantifies the desired information rate, in bits per channel use. The following equations, describing the coding scheme of Fig \[Fig:1012pm31may06\], will be used in the statement of Lemma \[lem:948pm03may06\] and Theorem \[th:313pm03may06\].
$$\label{eq:1129pm6sept06}
\hat{M}_t=\hat{\theta}_{n,r} \left(\hat{\phi}_{\bar{r}}(t,Y^t,V^t)\right)$$
$$Y_t=W_t+\underbrace{\phi_{\bar{r}} \left( t,\theta_{n,r}(M),U^{t-1} \right)}_{X_t}$$
$$\label{eq:1130pm6sept06}
U_t=Y_t+V_t$$
\[lem:948pm03may06\] Let $\sigma_W^2$, $\bar{\sigma}_V$ and $\bar{r}$ be given positive real parameters. Consider that the block length is given by a positive integer $n$, that the desired transmission rate is a positive real number $r$ strictly less than $\bar{r}$ and that $M$ is a random variable arbitrarily distributed in the set $\{1,\ldots,2^{\Theta(rn)}\}$. If we adopt the scheme of Fig \[Fig:1012pm31may06\], alternatively described by (\[eq:1129pm6sept06\])-(\[eq:1130pm6sept06\]), then the following holds: $$\label{eq:1136pm31may06}
Prob \left(M \neq \hat{M}_n \right) \leq \frac{ 2^{-2(\bar{r}-r)n } E[X_n^2] }{4 (2^{\bar{r}}-2^{-\bar{r}})^{2}}$$ If $W_t$ is zero mean, white and Gaussian with variance $\sigma_W^2$ then the following doubly exponential decay, with increasing block size $n$, of the probability of error holds:
$$\label{eq:517pm4sept06}
Prob \left(M \neq \hat{M}_n \right) \leq e^{-\frac{1}{2 \beta^2} \left( 2 (2^{\bar{r}}-2^{-\bar{r}}) 2^{(\bar{r}-r)n} - \gamma\right)^2}$$
where $\gamma$ and $\beta$ are positive real constants given by (\[eq:258pm4sept06\]) and (\[eq:259pm4sept06\]), respectively.
**Proof:** We start by using (\[eq:1028pm31may06\])-(\[eq:1029pm31may06\]) and the fact that $2^{\Theta(rn)}Z$ is in the set $ \{\frac{1}{2}, \ldots,2^{\Theta(rn)}-\frac{1}{2} \}$ to conclude the following: $$\left| 2^{\Theta(rn)}Z - 2^{\Theta(rn)}\hat{Z}_n \right| < \frac{1}{2} \implies M=\hat{M}_n$$ leading to: $$\label{eq:1137pm31may06}
Prob\left(M \neq \hat{M}_n \right) \leq Prob \left( \left|Z -\hat{Z}_n \right| \geq 2^{-(\Theta(rn)+1)} \right)$$
Using (\[eq:932pm31may06\]), (\[eq:1137pm31may06\]) and the fact that $\Theta(rn) \leq rn$, we get:
$$\label{eq:505pm4sept06}
Prob\left(M \neq \hat{M}_n \right) \leq Prob \left( |X_n| \geq 2 (2^{\bar{r}}-2^{-\bar{r}}) 2^{(\bar{r}-r)n} \right)$$
The inequality (\[eq:1136pm31may06\]) follows from Markov’s inequality applied to (\[eq:505pm4sept06\]). Finally, the inequality (\[eq:517pm4sept06\]) follows from (\[eq:505pm4sept06\]) and (\[eq:257pm4sept06\]). $\square$
Lower-bounds on the achievable rate of reliable transmission in the presence of a power constraint at the input of the forward channel
--------------------------------------------------------------------------------------------------------------------------------------
Below, we define a function that quantifies an achievable rate of reliable transmission for the scheme of Fig \[Fig:1012pm31may06\], in the presence of a power constraint at the input of the forward channel.
\[defvarrho\] For every choice of positive real parameters $\sigma_W^2$, $P_X^2$ and $\bar{\sigma}_V$ satisfying $ 4 \bar{\sigma}_V^2 \leq P_X^2$, define a function $\varrho:(\sigma_W^2,P_X^2,\bar{\sigma}_V) \mapsto \mathbb{R}_{\geq 0}$ as the non-negative real solution $\varrho$ of the following equation: $$\label{eq:249pm03june06}
\sigma_W \sqrt{2^{2\varrho}-1}=P_X-\bar{\sigma}_V \left( 1+2^{\varrho } \right)$$ If, instead, $ 4 \bar{\sigma}_V^2 > P_X^2$ then $\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)\overset{\Delta}{=}0$.
It is readily verifiable that a non-negative real solution of (\[eq:249pm03june06\]), in terms of $\varrho$, exists and is unique, provided that $\sigma_W^2$ and $P_X^2$ are strictly positive and that $4 \bar{\sigma}_V^2$ is less or equal than $P_X^2$.
\[th:313pm03may06\] Let $\sigma_W^2$, $P_X^2$ and $\bar{\sigma}_V$ be given positive real parameters satisfying $4\bar{\sigma}_V^2 < P_X^2$. In addition, select a positive transmission rate $r$ and a positive real constant $\bar{r}$ satisfying $r<\bar{r}<\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)$. For every positive integer block length $n$ the coding scheme of Fig \[Fig:1012pm31may06\], alternatively described by (\[eq:1129pm6sept06\])-(\[eq:1130pm6sept06\]), leads to:
$$\label{eq:832pm16jun06}
E[X_t^2]\leq \left( P_X + \underbrace{2^{-\bar{r}t} (2^{\bar{r}}-2^{-\bar{r}}) }_{\text{vanishes with increasing $t$}} \right)^2, 0 \leq t \leq n$$
$$\label{eq:931pm4sept06}
Prob \left(M \neq \hat{M}_n \right) \leq \frac{ 2^{-2 \left( \bar{r} - r \right) n } E[X_n^2]}{ 4 (2^{\bar{r}}-2^{-\bar{r}})^{2} }$$
where $M$ is a random variable arbitrarily distributed in the set $\{1,\ldots,2^{\Theta(nr)}\}$. If $W_t$ is zero mean, white and Gaussian with variance $\sigma_W^2$ then the following doubly exponential decay, with increasing block size $n$, of the probability of error holds:
$$\label{eq:902pm4sept06}
Prob \left(M \neq \hat{M}_n \right) \leq e^{-\frac{1}{2 \beta^2} \left( 2 (2^{\bar{r}}-2^{-\bar{r}}) 2^{(\bar{r}-r)n} - \gamma \right)^2}, \text{ } \alpha > 0$$
where $\gamma$ and $\beta$ are positive real constants given by (\[eq:258pm4sept06\]) and (\[eq:259pm4sept06\]), respectively.
Theorem \[th:313pm03may06\] shows that the scheme of Fig \[Fig:1012pm31may06\], under the constraint that the time average of the second moment of $X_t$ is less or equal[^3] than $P_X^2$, allows for reliable transmission at any rate $r$ strictly less than $\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)$. In addition, Theorem \[th:313pm03may06\] shows that any rate of transmission $r$, if strictly less than $\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)$, leads to an achievable error exponent arbitrarily close to $2\left[r-\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)\right]$. In addition, Theorem \[th:313pm03may06\] shows that if the forward channel is additive, white and Gaussian then the probability of error decreases with the block length $n$ at a doubly exponential rate (see (\[eq:902pm4sept06\])).
**Proof of Theorem \[th:313pm03may06\]:** The inequalities (\[eq:931pm4sept06\]) and (\[eq:902pm4sept06\]) follow directly from Lemma \[lem:948pm03may06\]. The derivation of (\[eq:832pm16jun06\]) follows from (\[eq:1019pm30may06\]) and from the fact that, from Definition \[defvarrho\], $\bar{r} < \varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)$ implies that $\sigma_W \sqrt{2^{2\bar{r}}-1}+\bar{\sigma}_V(2^{\bar{r}}+1) < P_X$. $\square$
It follows from its definition, as the solution to (\[eq:249pm03june06\]), that $\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)$ also satisfies the following 3 properties: $$\label{eq:424pm03june06}
\lim_{\bar{\sigma}_V \rightarrow 0^+}
\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V) = \frac{1}{2} \log_2 \left(1+\frac{P_X^2}{\sigma_W^2}\right),
\text{ } \sigma_W^2>0, \text{ } P_X^2 > 0$$
$$\label{eq:1059pm06may06}
\varrho\left(\sigma_W^2,P_X^2,\frac{P_X^2}{4}\right) =0, \text{ } \sigma_W^2>0, \text{ } P_X^2 > 0$$
$$\label{eq: large P behavior}
\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V) \simeq \log_2
\left(\frac{P_X}{\sigma_W+\bar{\sigma}_V}\right), \text{ } P_X^2 >>
\max \{\sigma_W^2,\bar{\sigma}_V^2\},$$
where $\simeq$ indicates that the ratio between the left and right hand sides of (\[eq: large P behavior\]) tends to $1$ as $P_X
\rightarrow \infty$. If $W_t$ is white Gaussian then (\[eq:424pm03june06\]) indicates that in the limit, as the second moment of feedback noise goes to zero, the scheme of Fig \[Fig:1012pm31may06\] approaches capacity[^4]. We have computed $\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)$ for $\sigma_W^2=1$, $P_X^2=4$ and one thousand equally spaced values of $\bar{\sigma}_V$, ranging from zero to one and the results are plotted in Fig \[Fig:1112pm3june06\]. The plot illustrates a graceful (continuous) degradation of $\varrho(1,4,\bar{\sigma}_V)$ as a function of $\bar{\sigma}_V$, going from the highest rate of $\frac{1}{2}\log_2 5$, achieving capacity when $W_t$ is Gaussian, down to zero when $\bar{\sigma}_V=1$, which is consistent with (\[eq:424pm03june06\]) and (\[eq:1059pm06may06\]), respectively.
Specification of a coding scheme using uniformly quantized feedback {#sec:447pm15jun06}
===================================================================
In this Section, we consider the scheme of Fig \[Fig:658pm07june06\], where $\Phi_{\bar{\sigma}_V}$ represents a memoryless uniform quantizer with sensitivity $\bar{\sigma}_V$ and $\Delta_{\bar{\sigma}_V}$ gives the associated quantization error. The main result of this Section is Corollary \[cor:140pm13jun06\], where we indicate that the results of Section \[sec:231pm05june06\] hold in the presence of uniformly quantized feedback. Notice that the diagram of Fig \[Fig:658pm07june06\] follows from Fig \[Fig:1012pm31may06\] by adopting $V_t$ as the quantization error, which the decoder re-constructs by making use of $\Delta_{\bar{\sigma}_V}$ applied to the output of the forward channel. The precise definitions of the uniform quantizer $\Phi_{\bar{\sigma}_V}$ and of the quantization error function $\Delta_{\bar{\sigma}_V}$ are given below:
Given a positive real parameter $b$, a uniform quantizer with sensitivity $b$ is a function $\Phi_b:\mathbb{R} \rightarrow \mathbb{R}$ defined as: $$\Phi_b(y)=2b \Theta\left( \frac{y+b}{2b}\right)$$ where $\Theta$ is the *floor* function specified in (\[eq:353pm08june06\]). Similarly, the quantization error is given by the following function: $$\Delta_b(y)=\Phi_b(y) - y, y \in \mathbb{R}$$ which satisfies the following bound: $$\label{eq:216pm13june06}
| \Delta_b(y) | \leq b, y \in \mathbb{R}$$
The coding scheme of Fig \[Fig:658pm07june06\] can be equivalently expressed by the following equations[^5]: $$\label{eq:1220pm7sept06}
\hat{M}_t=\hat{\theta}_{n,r} \left( \hat{\phi}_{\bar{r}}(t,Y^t,V^t) \right)$$ $$Y_t=W_t+\underbrace{\phi_{\bar{r}} \left( t,\theta_{n,r}(M),U^{t-1} \right)}_{X_t}$$ $$U_t=\Phi_{\bar{\sigma}_V}(Y_t)=Y_t+V_t$$ $$\label{eq:1221pm7sept06}
V_t=\Delta_{\bar{\sigma}_V}(Y_t)$$
The Corollary below follows directly from Theorem \[th:313pm03may06\] applied to the scheme of Fig \[Fig:658pm07june06\], along with the upper-bound (\[eq:216pm13june06\]).
\[cor:140pm13jun06\] Let $\sigma_W^2$, $P_X^2$ and $\bar{\sigma}_V$ be positive real constants satisfying $4\bar{\sigma}_V^2 < P_X^2$, where $\bar{\sigma}_V$ represents the sensitivity of the quantizer. In addition, select a positive transmission rate $r$ and a positive real constant $\bar{r}$ satisfying $r<\bar{r}<\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V^2)$. For every positive integer block length $n$, the coding scheme specified by (\[eq:1220pm7sept06\])-(\[eq:1221pm7sept06\]) (see Fig \[Fig:658pm07june06\]) leads to:
$$E[X_t^2] \leq \left( P_X + \underbrace{ 2^{-\bar{r}t} (2^{\bar{r}}-2^{-\bar{r}}) }_{\text{vanishes with increasing $t$}} \right)^2, 0 \leq t \leq n$$
$$Prob \left(M \neq \hat{M}_n \right) \leq \frac{ 2^{- 2 \left( \bar{r} - r \right) n } E[X_n^2]}{4 (2^{\bar{r}}-2^{-\bar{r}})^{2} }$$
where $M$ is a random variable arbitrarily distributed in the set $\{1,\ldots,2^{\Theta(nr)}\}$. If $W_t$ is zero mean, white and Gaussian with variance $\sigma_W^2$ then the following doubly exponential decay, with increasing block size $n$, of the probability of error holds: $$\label{eq:903pm4sept06}
Prob \left(M \neq \hat{M}_n \right) \leq e^{-\frac{1}{2 \beta^2} \left(2 (2^{\bar{r}}-2^{-\bar{r}}) 2^{(\bar{r}-r)n} - \gamma\right)^2}$$ where $\gamma$ and $\beta$ are positive real constants given by (\[eq:258pm4sept06\]) and (\[eq:259pm4sept06\]), respectively.
Notice that Corollary \[cor:140pm13jun06\] shows that, in the presence of uniformly quantized feedback with sensitivity $\bar{\sigma}_V$, any rate $r$ strictly less than $\varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)$ allows for reliable transmission. This implies that the properties (\[eq:424pm03june06\])-(\[eq:1059pm06may06\]), along with the conclusions derived in Section \[sec:231pm05june06\], hold for uniformly quantized feedback. In particular, the achievable rate of reliable transmission of the coding scheme of Fig \[Fig:658pm07june06\] degrades gracefully as a continuous function of the quantizer sensitivity $\bar{\sigma}_V$ (see the numerical example portrayed in Fig \[Fig:1112pm3june06\]).
Coding and decoding in the presence of feedback corrupted by bounded noise. {#sec:639pm17jun06}
===========================================================================
From Corollary \[cor:140pm13jun06\], we conclude that there exist simple explicit coding strategies based on Schalkwijk-Kailath’s framework that, even in the presence of *uniformly quantized* feedback, provide positive rates with positive error exponents. In this Section, we aim at designing coding schemes in the presence of feedback corrupted by bounded noise. The main result of this Section is discussed in Section \[sec:712pm7jul06\], where we describe a communication scheme whose structure is that of Fig \[Fig:124am16june06\]. In addition, we analyze the performance of such a scheme in the presence of power constraints at the input of the forward and backward channels. The proposed scheme retains the simplicity of the Schalkwijk-Kailath scheme [@SchalkwijkKailath], but, in contrast to the original scheme (which breaks down in the presence of noise in the backward link [@SchalkwijkKailath Section III.D]), achieves a positive rate of reliable communication and is in fact capacity achieving in the limit of high SNR in the backward link (assuming white Gaussian noise in the forward channel). The scheme proposed in Section \[sec:712pm7jul06\] also guarantees that, if the forward channel is additive, white and Gaussian, then the probability of error converges to zero as a doubly exponential function of the block length. The main results of this Section are stated in Theorem \[thm:234pm5jul06\].
Performance in the presence of a power constraint at the input of the backward channel. {#sec:712pm7jul06}
---------------------------------------------------------------------------------------
For the remainder of this Section, we will define a coding scheme whose structure is that of Fig \[Fig:124am16june06\]. The additive noise $S_t$ in the feedback link is arbitrarily distributed, bounded and the tightest upper-bound to its amplitude is defined below:$$\bar{\sigma}_S \overset{def}{=} \inf \{ \alpha \in \mathbb{R}_{\geq 0} : Prob(|S_t|>\alpha)=0, t \geq 0\}$$ meaning that the following holds: $$Prob(|S_t| \leq \bar{\sigma}_S)=1, \text{ } t \geq 0$$
The following remark will be used in the construction of a coding scheme with the structure of Fig \[Fig:124am16june06\].
\[rem:830pm15jun06\] Let $\bar{\sigma}_S$ be a positive real constant and $S_t$ be a real valued stochastic process satisfying $|S_t| \leq \bar{\sigma}_S$ with probability one. Given a positive real parameter $\bar{\sigma}_V$, the following holds with probability one: $$\frac{\bar{\sigma}_V}{\bar{\sigma}_S} \Phi_{\bar{\sigma}_S} \left( S_t + Q_t \right) = \Phi_{\bar{\sigma}_V}(Y_t)$$ where $Q_t$ is given by: $$\label{eq:1205pm16jun06}
Q_t=\Phi_{\bar{\sigma}_S} \left( \frac{\bar{\sigma}_S}{\bar{\sigma}_V} Y_t \right)$$
The schematic representation of the equivalence expressed in Remark \[rem:830pm15jun06\] is displayed in Fig \[Fig:829pm15june06\]. In such a scheme, $S_t$ is the bounded additive noise at the backward channel with input $Q_t$.
Aiming at constructing a coding scheme according to the structure of Fig \[Fig:124am16june06\], we use Remark \[rem:830pm15jun06\] to obtain a new coding strategy by substituting the feedback quantizer $\Phi_{\bar{\sigma}_V}$ of Fig \[Fig:658pm07june06\] with the equivalent additive noise channel diagram of Fig \[Fig:829pm15june06\]. The resulting scheme, along with the encoding and decoding strategy of Section \[sec:447pm15jun06\], provides a solution to the problem of designing encoders and decoders in the presence of an additive (bounded) noise backward channel (see Fig \[Fig:118pm17aug06\]). Under such a design strategy, $\bar{\sigma}_V$ becomes a design parameter. Notice that viewing $\bar{\sigma}_V$ as a design knob is in contrast with the framework of Section \[sec:447pm15jun06\], where $\bar{\sigma}_V$ was a given constant.
Regarding the role of $\bar{\sigma}_V$, we have shown in (\[eq:424pm03june06\]) that as $\bar{\sigma}_V$ approaches zero the achievable rate of reliable transmission converges to a positive value, which, in the case where $W_t$ is white Gaussian, coincides with capacity. However, for any given positive real $\bar{\sigma}_S$, the smaller $\bar{\sigma}_V$ the larger the scaling constant $\frac{\bar{\sigma}_S}{\bar{\sigma}_V}$ in (\[eq:1205pm16jun06\]) and that may lead to $Q_t$ having an arbitrarily large second moment. In Theorem \[thm:234pm5jul06\], we show that the function defined below solves the aforementioned problem by providing a suitable choice for $\sigma_V$, in the presence of power constraints at the input of the forward and backward channels.
Let $\sigma_W$, $\bar{\sigma}_S$, $P_X$ and $P_Q$ be given positive real constants, where $P_Q^2$ symbolizes a power constraint at the input of the backward channel $Q_t$. Below, we define the function $\Gamma:\mathbb{R}_{\geq 0}^4 \rightarrow \mathbb{R}_{\geq 0}$, which we will use as a selection for the design parameter $\bar{\sigma}_V$: $$\Gamma\left(\sigma_W,\bar{\sigma}_S,P_X,P_Q \right)=
\left(P_X+\sigma_W \right)\frac{\bar{\sigma}_S}{P_Q-\bar{\sigma}_S}, \text{ } P_Q>\bar{\sigma}_S$$
The following Theorem is one of the main results of this paper.
\[thm:234pm5jul06\] Let $\sigma_W^2$, $P_X^2$, $P_Q^2$ and $\bar{\sigma}_S$ be positive constants satisfying $4\Gamma\left(\sigma_W,\bar{\sigma}_S,P_X,P_Q \right)^2 < P_X^2$ and $\bar{\sigma}_S < P_Q$. In addition, select a positive transmission rate $r$ and a positive real constant $\bar{r}$ satisfying $r<\bar{r}<\left. \varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V) \right|_{\bar{\sigma}_V=\Gamma\left(\sigma_W,\bar{\sigma}_S,P_X,P_Q \right)}$. For every positive integer block length $n$, the coding scheme of Fig \[Fig:118pm17aug06\], alternatively described by (\[eq:1220pm7sept06\])-(\[eq:1221pm7sept06\]) and (\[eq:1205pm16jun06\]), leads to:
$$\label{eq:1237pm5jul06} E[X_t^2] \leq \left( P_X +
\underbrace{2^{-\bar{r}t} \left(
2^{\bar{r}}-2^{-\bar{r}}\right)}_{\text{vanishes with increasing
$t$}} \right)^2, \text{ } 0 \leq t \leq n$$
$$\label{eq:1238pm5jul06}
E[Q_t^2] \leq \left( P_Q+\underbrace{ 2^{-\bar{r} t} \frac{P_Q-\bar{\sigma}_S}{ P_X+\sigma_W} \left( 2^{\bar{r}}-2^{-\bar{r}}\right) }_{
\text{vanishes with increasing $t$}} \right)^2, \text{ } 0 \leq t \leq n$$
$$\label{eq:1239pm5jul06}
Prob \left(M \neq \hat{M}_n \right) \leq \frac{ 2^{-2 \left( \bar{r} - r \right) n } E[X_n^2] }{4 (2^{\bar{r}}-2^{-\bar{r}})^{2} }$$
where $M$ is a random variable arbitrarily distributed in the set $\{1,\ldots,2^{\Theta(nr)}\}$. If $W_t$ is zero mean, white and Gaussian with variance $\sigma_W^2$ then the following doubly exponential decay, with increasing block size $n$, of the probability of error holds:
$$\label{eq:905pm4sept06}
Prob \left(M \neq \hat{M}_n \right) \leq e^{-\frac{1}{2 \beta^2} \left( 2 (2^{\bar{r}}-2^{-\bar{r}}) 2^{(\bar{r}-r)n} - \gamma \right)^2}$$
where $\gamma$ and $\beta$ are positive real constants given by (\[eq:258pm4sept06\]) and (\[eq:259pm4sept06\]), respectively, where $\bar{\sigma}_V $ is given by the assumed selection $\bar{\sigma}_V = \Gamma\left(\sigma_W,\bar{\sigma}_S,P_X,P_Q \right)$.
**Proof:** The inequalities (\[eq:1237pm5jul06\]), (\[eq:1239pm5jul06\]) and (\[eq:905pm4sept06\]) follow directly from Corollary \[cor:140pm13jun06\]. In order to arrive at (\[eq:1238pm5jul06\]), we start by noticing that we can use the triangular inequality to find the following inequalities: $$\label{eq:808pm16jun06}
\left( E[Y_t^2] \right)^{\frac{1}{2}} \leq \left( E[X_t^2]\right)^{\frac{1}{2}} + \sigma_W$$
$$\label{eq:809pm16jun06}
\left( E[Q_t^2] \right)^{\frac{1}{2}} \leq \frac{\bar{\sigma}_S}{\bar{\sigma}_V}\left( E[Y_t^2]\right)^{\frac{1}{2}} + \bar{\sigma}_S$$
In addition, substitution of (\[eq:808pm16jun06\]) in (\[eq:809pm16jun06\]), leads to:
$$\label{eq:802pm16jun06}
E[Q_t^2] \leq \left( \frac{\bar{\sigma}_S}{\bar{\sigma}_V} \left( E[X_t^2]^{\frac{1}{2}}+\sigma_W\right)+\bar{\sigma}_S \right)^2$$
which, from (\[eq:1237pm5jul06\]), implies the following: $$\label{eq:632pm5sept06}
E[Q_t^2] \leq \left( \frac{\bar{\sigma}_S}{\bar{\sigma}_V} \left( P_X +
2^{-\bar{r}t} \left(2^{\bar{r}}-2^{-\bar{r}}\right)+\sigma_W\right)+\bar{\sigma}_S \right)^2$$
The proof is complete since (\[eq:1238pm5jul06\]) follows by substituting our choice $\bar{\sigma}_V=\Gamma\left(\sigma_W,\bar{\sigma}_S,P_X,P_Q \right)$ in (\[eq:632pm5sept06\]). $\square$
Under the conditions of Theorem \[thm:234pm5jul06\], including our choice of the design parameter $\bar{\sigma}_V$, the following limit holds:
$$\label{eq:651pm17jun06}
\lim_{\bar{\sigma}_S \rightarrow 0^+} \left. \varrho(\sigma_W^2,P_X^2,\bar{\sigma}_V)\right|_{\bar{\sigma}_V=\Gamma\left(\sigma_W,\bar{\sigma}_S,P_X,P_Q \right)} = \frac{1}{2} \log_2 \left( 1+ \frac{P_X^2}{\sigma_W^2} \right), \text{ } \sigma_W>0, P_X >0, P_Q>0$$
Notice that (\[eq:651pm17jun06\]) leads to the conclusion that, under our choice of $\bar{\sigma}_V$, the performance of the scheme of Theorem \[thm:234pm5jul06\] (see Fig \[Fig:118pm17aug06\]) degrades gracefully as a function of $\bar{\sigma}_S$, in terms of both the rate and the error exponent. If $W_t$ is white Gaussian then (\[eq:651pm17jun06\]) indicates that as $\bar{\sigma}_S$ tends to zero, the scheme of Theorem \[thm:234pm5jul06\] can be used to reliably communicate at a rate arbitrarily close to capacity. Moreover, such a conclusion holds in the presence of an arbitrarily low power constraint at the backward channel. The plot of Fig \[Fig:315pm5jul06\] displays how the achievable rate changes as a function of $\bar{\sigma}_S$, under the choice $\bar{\sigma}_V=\Gamma\left(\sigma_W,\bar{\sigma}_S,P_X,P_Q
\right)$. Such a plot also illustrates that by increasing $P_Q$ we can reduce the sensitivity of the achievable rate, of reliable transmission, relative to variations in $\bar{\sigma}_S$.
Further comments on the location of the one-step feedback delay
---------------------------------------------------------------
In the framework of Fig \[Fig:118pm17aug06\], the one-step delay block is located *after* the feedback decoder. However, we should stress that, since the feedback decoder is time-invariant, our coding scheme would be unaltered if we had placed the delay block *before* as indicated in Fig \[Fig:742pm5sept06\]. Indeed, the diagrams of Fig \[Fig:118pm17aug06\] and \[Fig:742pm5sept06\] are equivalent, implying that Theorem \[thm:234pm5jul06\] holds also for the coding scheme of Fig \[Fig:742pm5sept06\].
Conclusions {#sec:conclusions}
===========
We derived simple schemes for reliable communication over a white noise forward channel, in the presence of corrupted feedback. Both the case of uniform quantization noise and the case of additive bounded noise in the backward link were considered, where, in the latter case, encoding at the input to the backward channel is allowed. The schemes were seen to achieve a positive rate of reliable communication, and in fact be capacity-achieving in the presence of an additive white Gaussian forward channel, in the limit of small noise (or high SNR when encoding is allowed) in the backward link. In addition, still under the assumption that the forward channel is additive white Gaussian, the proposed schemes guarantee that the probability of error converges to zero as a doubly exponential function of the block length.
We believe that our approach to the construction and analysis of coding schemes carries over naturally to the case where the noise in the forward channel is non-white. In this case, we expect to obtain variations on the schemes in [@Kim] that are analogous to those in the present work and whose gap to capacity behaves similarly.
[99]{}
T. M. Cover and J. A. Thomas; *“Elements of Information Theory,”* Wiley-Iterscience Publication, 1991
S. C. Draper and A. Sahai, *“Noisy feedback improves communication reliability,”* Proceedings of the *International Symposium of Information Theory,* Seattle, Washington, July 2006
N. Elia, *“When Bode Meets Shannon: Control-Oriented Feedback Communication Schemes,”* *IEEE Transactions on Automatic Control*, Vol. 49, No. 9, pp. 1477-1488, Sept. 2004
Y. H. Kim, *“Feedback capacity of stationary Gaussian channels,”* submitted to *IEEE Transactions on Information Theory*. Available at “ arxiv.org/abs/cs.IT/0602091”
Y. H. Kim, A. Lapidoth and T. Weissman, *“Upper bounds on error exponents of channels with feedback,”* to appear in *IEEE 24th Convention of Electrical & Electronics Engineers in Israel*, Eilat, Israel, November 2006
H. Permuter, T. Weissman and A. Goldsmith, *“Finite State Channels with Time-Invariant Deterministic Feedback,”* Available at “http://arxiv.org/abs/cs.IT/0608070”
J. G. Proakis, M. Salehi, *“Fundamentals of communication systems,”* Prentice Hall, 2005
A. Sahai and T. Şimşek, *“On the variable-delay reliability function of discrete memoryless channels with access to noisy feedback,”* Proceedings of the *IEEE Information Theory Workshop*, San Antonio, Texas, 2004.
J. P. M. Schalkwijk and T. Kailath, *“A coding scheme for additive noise channels with feedback [I]{}: no bandwidth constraint,”* *IEEE Transactions on Information Theory*, vol. 12, pp. 172-182, April 1966
C. E. Shannon, *“The zero-error capacity of a noisy channel,”* *IRE Transactions on Information Theory*, vol. IT-2, pp. 8–19, September 1956.
S. Yang, A. Kavcic and S. Tatikonda,*“Feedback capacity of finite-state machine channels,”* *IEEE Transactions on Information Theory*, Volume 51, Issue 3, pp. 799 –810, March 2005
[^1]: ([nmartins@umd.edu]{}) Nuno C. Martins is with the Electrical and Computer Engineering Department and the Institute for Systems Research at the University of Maryland, College Park. ([tsachy@stanford.edu]{}) Tsachy Weissman is with the Department of Electrical Engineering and the Information Systems Laboratory at Stanford University. Note: An abridged version of this work was presented at Stanford University on July 7th of 2006, in the Colloquium on Feedback Communications.
[^2]: By reliable transmission we mean that the probability of error converges to zero with increasing block length $n$.
[^3]: See inequality (\[eq:832pm16jun06\]).
[^4]: It is a standard fact [@Thomas] that the capacity in bits per channel use of an additive Gaussian channel, with noise variance $\sigma_W^2$ and input power constraint $P_X^2$, is given by $\frac{1}{2} \log_2\left(1+\frac{P_X^2}{\sigma_W^2}\right)$.
[^5]: Some of these equations have been used before, but we repeat them here for convenience.
|
---
abstract: 'With the online proliferation of hate speech, organizations and governments are trying to tackle this issue, without upsetting the ‘freedom of speech’. In this paper, we try to understand the temporal effects of allowing hate speech on a platform (Gab) as a norm (protected as freedom of speech). We observe that the amount of hate speech is steadily increasing, with new users being exposed to hate speech at a faster rate. We also observe that the language used by the Gab users are aligning more with the hateful users with time. We believe that the hate community is evolving a new language culture of their own in this unmoderated environment and the rest of the (benign) population is slowly getting adapted to this new language culture. Our work provides empirical observations to the HCI questions regarding the freedom of hate speech.'
author:
- |
Binny Mathew, Anurag Illendula, Punyajoy Saha, Soumya Sarkar,\
**Pawan Goyal, Animesh Mukherjee**\
Indian Institute of Technology(IIT), Kharagpur\
binnymathew@iitkgp.ac.in, {aianurag09, punyajoysaha1998}@gmail.com, soumya015@iitkgp.ac.in\
{pawang, animeshm}@cse.iitkgp.ac.in
bibliography:
- 'Main.bib'
title: Temporal effects of Unmoderated Hate speech in Gab
---
Introduction
============
The question about where is the borderline or whether there is indeed any borderline between ‘free speech’ and ‘hate speech’ is an ongoing subject of debate which has recently gained a lot of attention. With crimes related to hate speech increasing in the recent times[^1], hate speech is considered to be one of the fundamental problems that plague the Internet. The online dissemination of hate speech has even lead to real-life tragic events such as genocide of the Rohingya community in Myanmar, anti-Muslim mob violence in Sri Lanka, and the Pittsburg shooting. The big tech giants are also unable to control the massive dissemination of hate speech[^2].
Recently, there have been a lot of research concerning multiple aspects of hate speech such as *detection* [@davidson2017automated; @Badjatiya:2017:DLH:3041021.3054223; @zhang2018detecting], *analysis* [@Chandrasekharan2017YouCS; @Olteanu2018TheEO], *target identification* [@silva2016analyzing; @mondal2017measurement; @elsherief2018hate], *counter-hate speech* [@gagliardone2015countering; @mathew2018thou; @benesch2016countertwitter] etc. However, very little is known about the temporal effects of hate speech in online social media, especially if it is considered as normative. In order to have a clear understanding on this, we would need to see the effects on a platform which allows free flow of hate speech. To understand the true nature of the hateful users, we need to study them in an environment that would not stop them from following/enacting on their beliefs. This led us to focus our study on Gab ($Gab.com$). Gab is a social media site that calls itself the *‘champion of free speech’*. The site does not prohibit a user from posting any hateful content. This natural environment in which the only moderation is what the community members impose on themselves provides a rich platform for our study. Using a large dataset of $\sim 21M$ posts spanning around two years since the inception of the site, we develop a data pipeline which allows us to study the temporal effects of hate speech in an unmoderated environment. Our work adds the temporal dimension to the existing literature on hate speech and tries to study and characterize hate in unmoderated online social media.
Despite the importance of understanding hate speech in the current socio-political environment, there is little HCI work which looks into the temporal aspects of these issues. This paper fills an important research gap in understanding how hate speech evolves in an environment where it is protected under the umbrella of free speech. This paper also opens up questions on how new HCI design policies of online platforms should be regulated to minimize/mitigate the problem of the temporal growth of hate speech. We posit that HCI research, acknowledging the far-reaching mal consequences of this problem, should factor it into the ongoing popular initiative of *platform governance*[^3].
Outline of the work
-------------------
To understand the temporal characteristics, we needed data from consecutive time points in Gab. As a first step, using a heuristic, we generate successive graphs which capture the different time snapshots of Gab at one month intervals. Then, using the DeGroot model, we assign a hate intensity score to every user in the temporal snapshot and categorize them based on their degrees of hate. We then perform several *linguistic* and *network* studies on these users across the different time snapshots.
Research questions
------------------
**RQ1**: How can we characterize the growth of hate speech in Gab?
**RQ2**: How have the hate speakers affected the Gab community as a whole?
RQ1 attempts to investigate the general growth of hate speech in Gab. Previous research on Gab [@zannettou2018gab] states that the hateful content is 2.4x as compared to Twitter. RQ2, on the other hand, attempts to identify how these hateful users have affected the Gab community. We study this from two different perspectives: language and network characteristics.
Key observations
----------------
For RQ1, we found that the amount of hate speech in Gab is consistently increasing. This is true for the new users joining as well. We found that the recently joining new users take much less time to become hateful as compared those that joined at earlier time periods. Further, the fraction of users getting exposed to hate speech is increasing as well.
For RQ2, we found that the language used by the community is aligning more with the hateful users as compared to the non-hateful ones. The hateful users also seem to be playing a pivotal role from the network point of view. We found that the hateful users reach the core of the community faster and in larger sizes.
Prior Work
==========
The hate speech research has a substantial literature and it has recently gained a lot of attention from the Computer Science perspective. In the following sections, we will examine the various aspects of research on hate speech. Interested readers can follow Fortuna et al. [@fortuna2018survey] and Schmidt et al. [@schmidt2017survey] for a comprehensive survey of this subject.
Definition of hate speech
-------------------------
Hate speech lies in a complex confluence of freedom of expression, individual, group and minority rights, as well as concepts of dignity, liberty and equality [@gagliardone2015countering]. Owing to the subjective nature of this issue, deciding if a given piece of text contains hate speech is onerous. In this paper, we use the hate speech definition outlined in the work done by Elsherief et al. [@elsherief2018hate]. The author defines hate speech as a “direct and serious attack on any protected category of people based on their race, ethnicity, national origin, religion, sex, gender, sexual orientation, disability or disease.”. Others have a slightly different definition for hate speech but the spirit is mostly the same. In our work we shall mostly go by this definition unless otherwise explicitly mentioned.
Related concepts
----------------
Hate speech is a complex phenomenon, intrinsically associated to relationships among groups, and also relying on linguistic nuances [@fortuna2018survey]. It is related to some of the concepts in social science such as incivility [@maity2018opinion], radicalization [@agarwal2015using], cyberbullying [@chen2011detecting], abusive language [@chandrasekharan2017bag; @nobata2016abusive], toxicity [@gunasekara2018review; @srivastava2018identifying], profanity [@sood2012profanity] and extremism [@mcnamee2010call]. Owing to the overlap between hate speech and these concepts, sometimes it becomes hard to differentiate between them [@davidson2017automated]. Teh et al. [@teh2018identifying] obtained a list of frequently used profane words from comments in YouTube videos and categorized them into 8 different types of hate speech. The authors aim to use these profane words for automatic hate speech detection. Malmasi et al. [@malmasi2018challenges] attempts to distinguish profanity from hate speech by building models with features such as n-grams, skip-grams and clustering-based word representations.
Effects of hate speech
----------------------
Previous studies have found that public expressions of hate speech affects the devaluation of minority members [@greenberg1985effect], the exclusion of minorities from the society [@mullen2003ethnophaulisms], psychological well-being and the suicide rate among minorities [@mullen2004immigrant], and the discriminatory distribution of public resources [@fasoli2015labelling]. Frequent and repetitive exposure to hate speech has been shown to desensitize the individual to this form of speech and subsequently to lower evaluations of the victims and greater distancing, thus increasing outgroup prejudice [@soral2018exposure]. Olteanu et. al [@Olteanu2018TheEO] studied the effect of violent attacks on the volume and type of hateful speech on two social media platforms, Twitter and Reddit. They found that extremist violence tends to lead to an increase in online hate speech, in particular, on messages directly advocating violence.
Computational approaches
------------------------
The research interest in hate speech, from a computer science perspective, is gaining interest. Larger datasets [@davidson2017automated; @founta2018large; @de2018hate] and different approaches have been devised by researchers to detect hateful social media comments. These methods include techniques such as dictionary-based [@guermazi2007using], distributional semantics [@djuric2015hate], multi-feature [@salminen2018anatomy] and neural networks [@Badjatiya:2017:DLH:3041021.3054223].
Burnap et al. [@burnap2016us] used a bag of words approach combined with hate lexicons to build machine learning classifiers. Gitari et al. [@gitari2015lexicon] used sentiment analysis along with subjectivity detection to generate a set of words related to hate speech for hate speech classification. Chau et. al [@chau2007mining] used analysis of hyperlinks among web pages to identify hate group communities. Zhou et al. [@zhou2005us] used multidimensional scaling (MDS) algorithm to represent the proximity of hate websites and thus capture their level of similarity. Lie et al. [@liu2015new] incorporated LDA topic modelling for improving the performance of the hate speech detection task. Saleem et al. [@saleem2017web] proposed an approach to detecting hateful speech using self-identifying hate communities as training data for hate speech classifiers. Davidson et al. [@davidson2017automated] used crowd-sourcing to label tweets into three categories: hate speech, only offensive language, and those with neither. Waseem et al. [@waseem2016hateful] presented a list of criteria based on critical race theory to identify racist and sexist slurs.
More recently, researchers have started using deep learning methods [@Badjatiya:2017:DLH:3041021.3054223; @zhang2018detecting] and graph embedding techniques [@ribeiro2018characterizing] to detect hate speech. Badjatiya et al. [@Badjatiya:2017:DLH:3041021.3054223] applied several deep learning architectures and improved the benchmark score by $\sim$18 F1 points. Zhang et al. [@zhang2018detecting] used deep neural network, combining convolutional and gated recurrent networks to improve the results on 6 out of 7 datasets. Gao et al. [@gao2017detecting] utilized the context information accompanied with the text to develop hate speech detection models. Grondahl et al. [@grondahl2018all] found that several of the existing state-of-the-art hate speech detection models work well only when tested on the same type of data they were trained on.
While most of the computational approaches focus on detecting if a given text contains hate speech, very few works focus on the user account level detection. Gian et al. [@qian2018leveraging] proposed a model that leverages intra-user and inter-user representation learning for hate speech detection.
Gibson [@gibson2017safe] studied the moderation policies on Reddit communities and observed that ‘safe space’ have higher levels of censorship and is directly related to the politeness in the community. Studying the effects of hate speech in online social media remains an understudied area in HCI research. By employing our data processing pipeline, we study the temporal effects of hate speech on Gab.
Dataset
=======
The Gab social network
----------------------
Gab is a social media platform launched in August 2016 known for promoting itself as the “Champion of free speech”. However, it has been criticized for being a shield for alt-right users [@zannettou2018gab]. The site is very similar to Twitter but has very loose moderation policies. According to the Gab guidelines, the site does not restrain users from using hateful speech[^4]. The site allows users to read and write posts of up to 3,000 characters. The site employs an upvoting and downvoting mechanism for posts and categorizes posts into different topics such as News, Sports, Politics, etc.
Dataset collection
------------------
We use the dataset developed by Mathew et al. [@mathew2018spread] for our analysis. For the sake of completeness of the paper, we present the general statistics of the dataset in Table \[tab: dataset-details\]. The dataset contains information from August 2016 to July 2018. We do not use the data for the initial two months (August-September 2016) and the last month (July 2018) as they had less posts.
Property Value
---------------------------------- ------------
Number of posts 21,207,961
Number of reply posts 6,601,521
Number of quote posts 2,085,828
Number of reposts 5,850,331
Number of posts with attachments 9,669,374
Number of user accounts 341,332
Average follower per account 62.56
Average following per account 60.93
: Description of the dataset.[]{data-label="tab: dataset-details"}
Methodology
===========
To address our research questions, we need to have a temporal overview of the activity of each user. So, our first task involves generating temporal snapshots to capture the month-wise activity of the users. We develop a pipeline to generate the hate vectors of each users for this purpose. A hate vector is a representation used to capture the activity of each user. Higher value in the hate vector is an indication of the hatefulness of a user, whereas a lower value means that the user potentially did not indulge in any hateful activity.
In this section, we will explain the pipeline we used to study the temporal properties of hate. The pipeline mainly consists of the following three tasks:
**Generating temporal snapshots**: We divide the data such that a snapshot will represent the activities of a particular month.
**Hate intensity calculation**: We calculate the hate intensity score for each user, which represent the hateful activity of a user based on his/her posts, reposts, and network connections.
**User profiling**: We profile users based on his/her temporal activity of hate speech, which is represented by a vector of his/her hate intensity score.
![Our overall methodology to generate the hate vector for a user.[]{data-label="fig:DataProcessing"}](Figures/Data_Processing){width="\linewidth"}
Figure \[fig:DataProcessing\] shows our overall data processing pipeline.
Generating Temporal Snapshots
=============================
In order to study the temporal nature of hate speech, we need a temporal sequence of posts, reposts, users being followed, and users following the account. Thus, for each snapshot, we should have a list of the new posts, reposts, followers, and following of each user. This would allow us to have a better picture of the user stance/opinion in each snapshot. The Gab dataset gives us the information regarding the post creation date, but it does not provide any information about when a particular user started following another user. Using various data points we have, we come up with a technique in the following section to approximate the month in which a user started following another user.
New followers in each snapshot
------------------------------
While the post creation date is available in the dataset, the Gab API does not provide us with the information regarding when a particular user started following another user. Hence, we apply a heuristic by Meeder et al. [@meeder2011we], which was used in previous works [@lang2011anti; @antonakaki2015evolving]; to get a lower bound on the *following* link creation date. The heuristic is based on the fact that the API returns the list of followers/friends of a user ordered by the link creation time. We can thus obtain a lower bound on the follow time using the account creation date of a followers. For instance, if a user $U_A$ is followed by users $\{U_0, U_1, \ldots, U_n\}$ (in this order through time) and the users joined Gab on dates $\{D_0, D_1, \ldots, D_n\}$, then we can know for certain that $U_1$ was not following $U_A$ before $\max(D_0, D_1)$. We applied this heuristic on our dataset and ordered all of the *following* relationships according to this. The authors [@meeder2011we] proved that this heuristic is pretty accurate (within several minutes) specially on time periods where there are high follow rates. Since in our case we have considered a much larger window of one month, it would provide a fairly accurate estimate about the link creation time.
Hence the above heuristic helps to get the list of followers/friends each month for a particular user. This information, combined with the creation dates of his posts allows us to construct a temporal snapshot of his/her activity each month.
Dynamic graph generation {#sec:dynamic graph}
------------------------
We consider the Gab graph ($\mathcal{G}$) as a dynamic graph with no parallel edges. We represent the *dynamic graph* $\mathcal{G}$ as a set of successive time step graphs $\{G_0, \ldots, G_{t_\text{max}}\}$, where $G_s = (V_s, E_s)$ denotes the graph at snapshot $s$, where the set of nodes is $V_s$ (=$\{\bigcup_{i=0}^{s}V_i\}$) nodes and the set of edges is $E_s$ (=$\{\bigcup_{i=0}^{s}E_s\}$). In this paper, we consider the time duration between each successive snapshot as **one month**. An example of this dynamic graph is provided in figure \[fig:DynamicGraph\].
Each snapshot, $G_s$ is a weighted directed graph with the users as the nodes and the edges representing the following relationship. The edge weight is calculated based on the user’s posting and reposting activity. We shall explain the exact mechanism of calculation of this weight in the following section.
![An example dynamic graph. The nodes represent user accounts and the edges represent the ‘follows’ relationship.[]{data-label="fig:DynamicGraph"}](Figures/Dynamic_Graph.pdf){width="\linewidth"}
Hate Intensity calculation
==========================
We make use of the temporal snapshots to calculate the *hate intensity* of a user. The notion of *hate intensity* allows us to capture the overall hatefulness of a user. A user with a high value of *hate intensity* would be considered to be a potential hateful user as compared to another with lower value. The *hate intensity* value ranges from 0 to 1, with 1 representing highly hateful user and values close to zero representing non-hateful user.
We use the DeGroot model [@DeGroot1974reaching; @golub2010naive; @ribeiro2018characterizing; @mathew2018spread] to calculate the *hate intensity* of a user at each snapshot. Similar to Mathew et al. [@mathew2018spread], our purpose of using DeGroot model is to capture users who did not use these hate keywords explicitly, yet have a high potential to spread hate. We later perform manual evaluation to ensure the quality of the model.
DeGroot model
-------------
In the DeGroot opinion dynamics model [@DeGroot1974reaching], each individual has a fixed set of neighbours, and the local interaction is captured by taking the convex combination of his/her own opinion and the opinions of his/her neighbours at each time step [@xu2015modified]. The DeGroot model describes how each user repeatedly updates its opinion to the average of those of its neighbours. Since this model reflects the fundamental human cognitive capability of taking convex combinations when integrating related information [@anderson1981foundations], it has been studied extensively in the past decades [@chamley2013models]. We will now briefly explain the DeGroot model and how we modify it to calculate the *hate intensity* of a user account.
In the DeGroot model, each user starts with an initial belief. In each time step, the user interacts with its neighbours and updates his/her belief based on the neighbour’s beliefs. The readers should remember that each snapshot is a directed graph, $G_s=(V_s,E_s)$ with $V_s$ representing the set of vertices and $E_s$ representing the set of edges at snapshot $s$. Let $N(i)$ denote the set of neighbours of node $i$ and $z_i(t)$ denote the belief of the node $i$ at iteration $t$. The update rule in this model is the following: $\mathbf{z}_{i}(t+1)= \frac{w_{ii}\mathbf{z}_{i}(t) + \sum_{j\in N(i)}w_{ij}\mathbf{z}_{j}(t)} {w_{ii}+\sum_{j\in N(i)}w_{ij}}$ where $(i,j)\in E_s$.
For each snapshot, we assign the initial edge weights based on the following criteria:
$$\begin{aligned}
\small\label{weight_update}
w_{ij} =
\begin{cases}
\mathrm{e}^\text{ $R_{ij}$ } & \quad \text{if $F_{ij}=1$}\\
\mathrm{e}^\text{ $R_{ij}$ } & \quad \text{if $F_{ij}=0$ and $R_{ij}>0$}\\
0 & \quad \text{if $F_{ij}=0$ and $R_{ij}=0$}\\
1+ \text{ $P_i$} & \quad \text{if $i$ = $j$}\\
\end{cases}\end{aligned}$$
where $R_{ij}$ denotes the number of reposts done by user $i$, where the original post was made user $j$. $F_{ij}$ represents the following relationship, where $F_{ij}=1$ means that user $i$ is following user $j$, and $F_{ij}=0$ means that user $i$ is not following user $j$. Similarly, $P_i$ denotes the number of posts by user $i$.
We then run the DeGroot model on each snapshot graph for 5 iterations, similar to Mathew et al. [@mathew2018spread], to obtain the hate score for each of the users.
Hate lexicon
------------
We initially started with the lexicon set available in Mathew et al. [@mathew2018spread]. These high-precision keywords were selected from Hatebase[^5] and Urban dictionary[^6]. To further enhance the quality of the lexicon, we adopt the word embedding method, skip-gram [@mikolov2013distributed], to learn distributed representation of the words from our Gab dataset in an unsupervised manner. This would allow us to enhance the hate lexicon with words that are specific to the dataset as well as spelling variations used by the Gab users. For example, we found more than five variants for the derogatory term *ni\*\*er* in the dataset used by hateful users. We manually went through the words and carefully selected only those words which could be used in a hateful context. This resulted in a final set of 187 phrases which we have made public[^7] for the use of future researchers. In figure \[fig:hate\_post\_increase\], we plot the % of posts that have at least one of the words from these hate lexicon. We can observe from these initial results that the volume of hateful posts on Gab is increasing over time. Further, in order to establish the quality of this lexicon, we collected three posts randomly for each of the words in the hate lexicon. Two of the authors independently annotated these posts for the presence of hate speech, which yielded 88.5% agreement where both the annotators found the posts to be hateful. The value indicates that the lexicons developed are of high quality. The annotators were instructed to follow the definition of hate speech used in Elsherief et al. [@elsherief2018hate].
![The percentage of posts over time that have at least one of the hate words. The shows the increasing trend of posting such messages on Gab.[]{data-label="fig:hate_post_increase"}](Figures/hate_post_increase){width="\linewidth"}
Calculating the hate score
--------------------------
Using the high precision hate lexicon directly to assign a hate score to a user should be problematic because of two reasons: first, we might miss out on a large set of users who might not use any of the words in the hate lexicon directly or use spelling variations, thereby, getting a much lower score. Second, many of the users share hateful messages via images, videos and external links. Using the hate lexicon for these users will not work. Instead, we use a variant of the methodology used in Riberio et al. [@ribeiro2018characterizing] to assign each user in each snapshot a value in the range $[0, 1]$ which indicates the users’ propensity to be hateful.
We enumerate the steps of our methodology below. We apply this procedure for each snapshot to get the hate score for each user.
We identify the initial set of potential hateful users as those who have used the words from the hate lexicon in at least two posts. Rest of the users are identified as non-hateful users.
Using the snapshot graph, we assign the edge weight according to equation \[weight\_update\]. We convert this graph into a belief graph by reversing the edges in the original graph and normalizing the edge weights between $0$ and $1$.
We then run a diffusion process based on the DeGroot’s learning model on the belief network. We assign an initial belief value of $1$ to the set of potential hateful users identified earlier and $0$ to all the other users.
We observe the belief values of all the users in the network after *five* iterations of the diffusion process.
Threshold selection
-------------------
The DeGroot’s model will assign each user a hate score in the range $[0,1]$ with $0$ implying the least hateful and $1$ implying highly hateful. In order to draw the boundary between the hateful and non-hateful users, we need a threshold value, above which we might be able to call a user is hateful. The same argument goes for the non-hateful users as well: a threshold value below which the user can be considered to be non-hateful.
In order to select such threshold values, we used $k$-means [@macqueen1967some; @jain1999data] as a clustering algorithm on the scalar values of the hate score. Briefly, $k$-means selects $k$ points in space to be the initial guess of the $k$ centroids. Remaining points are then allocated to the nearest centroid. The whole procedure is repeated until no points switch cluster assignment or a number of iterations is performed. In our case, we assign $k=3$ which would give us three regions in the range $[0,1]$ represented by three centroids $C_L$, $C_M$, and $C_H$ denoting ‘low hate’, ‘medium hate’ and ‘high hate’, respectively. The purpose of having medium hate category is to capture the ambiguous users. These will be the users who will have values that are neither high enough to be considered hateful nor low enough to be considered non-hateful. We apply $k$-means algorithm on the list of hate scores from all the snapshots. Figure \[fig:hate\_distribution\] shows the fraction of users in each category of hate in each snapshot. The DeGroot model is biased toward non-hate users as in every snapshot, a substantial fraction of users are initially assigned a value of zero. As shown in figure \[fig:hate\_score\_kmeans\], the centroid values are 0.0421 ($C_L$), 0.2111 ($C_M$), 0.5778 ($C_H$) for the low, mid, and high hate score users, respectively.
![The distribution of hate scores and the centroid values based on the $k$-means algorithm.[]{data-label="fig:hate_score_kmeans"}](Figures/hate_score_kmeans){width="\linewidth"}
![The number of accounts that are labelled as low, mid, and high hate in each of the snapshots.[]{data-label="fig:hate_distribution"}](Figures/hate_distribution){width="\linewidth"}
User profiling
==============
Using the centroid values ($C_L$, $C_M$, and $C_H$), we transform the activities of a user into a sequence of low, medium, and high hate over time. We denote this sequence by a vector $V_{hate}$. Each entry in $V_{hate}$ consists of one of the three values of low, mid, and high hate. This would allow us to find the changes in the perspective of a user at multiple time points.
Consider the example given in figure \[fig:hateful\_user\_vector\]. The vector represents a user who had high hate score for most of the time period with intermittent low and medium hate score. Similarly, figure \[fig:non\_hateful\_user\_vector\] shows a user who had low hate score for most of the time period. For the purpose of this study, we mainly focus on only two types of user profiles: consistently hateful users and the consistently non-hateful users.
![The hate vector consisting of sequence of low (L), mid (M), and high (H) hate. (a) An example of a hateful user. (b) An example of a non-hateful user.[]{data-label="fig:hate_sequence_example"}](Figures/Hate_Vector_hate.pdf){width="\textwidth"}
![The hate vector consisting of sequence of low (L), mid (M), and high (H) hate. (a) An example of a hateful user. (b) An example of a non-hateful user.[]{data-label="fig:hate_sequence_example"}](Figures/Hate_Vector_nonhate.pdf){width="\textwidth"}
We would like to point out here that other types of variations could also be possible. Like a user’s hate score might change from one category to other multiples times, but we have not considered such cases here.
In order to find these users, we adopt a conservative approach and categorize the users based on the following criteria:
**Hateful**: We would call a user as hateful if at least 75% of his/her $V_{hate}$ entries contain an ‘H’.
**Non-hateful**: We would call a user as non-hateful user if at least 75% of his/her $V_{hate}$ entries contain an ‘L’.
In addition, we used the following filters on the user accounts as well:
The user should have posted at least **five** times.
The account should be created before February 2018 so that there are at least **six** snapshots available for the user.
We have not considered users with hate score in the mid-region as they are ambiguous. After the filtering, the number of users in the two different categories are noted in Table \[tab:account-characteristics\]. In the following section, we will perform textual and network analysis on these types of users and try to characterize them.
Type \#users
------------- ---------
Hateful 1,019
Non-hateful 19,814
: Number of user accounts in each type of hate[]{data-label="tab:account-characteristics"}
Sampling the appropriate set of non-hateful users
-------------------------------------------------
We use the non-hateful users as the candidates in the control group. Our idea of the control group is to find non-hate users with similar characteristics as the hateful users. For sanity check purpose, we identify users who have (nearly) the same activity rate as the users in the hate set. We define the activity rate of a user as the sum of the number of posts and reposts done by the user, divided by the account age as of June 2018. For each hateful user, we identify a user from the non-hateful set with the nearest activity rate. We repeat this process for all the users in the hate list. We then performed Mann-Whitney $U$-test [@depuy2005w] to measure the goodness. We found the value of $U = 517,208$ and $p$-value = 0.441. This indicates that the hate and non-hate users have nearly the same distribution. By using this subset of non-hate users, we aim to capture any general trend in Gab. Our final set consists of 1,019 hateful users and the corresponding 1,019 non-hateful users who have very similar activity profile.
Evaluation of user profiles
---------------------------
We evaluate the quality of the final dataset of hateful and non-hateful accounts through human judgment. We ask two annotators to determine if a given account is hateful or non-hateful as per their perception. Since Gab does not have any policy for hate speech, we use the definition of hate speech provided by Elsherief et al. [@elsherief2018hate] for this task. We provide the annotators with a class balanced random sample of 200 user accounts.
Each account was evaluated by two independent annotators. After the labelling was complete, we only took those annotations where both the annotators agreed. This gave us a final set of 258 user accounts, where 135 accounts were hateful and 123 accounts were non-hateful. We compared these ground truth annotations with our model predictions and found that they were in almost 100% agreement.
RQ1: How can we characterize the growth of hate speech in Gab?
==============================================================
The volume of hate speech is increasing
---------------------------------------
As a first step to measure the growth of hate speech in Gab, we use the hate lexicon that we generated to find the number of posts which contain them in each month. We can observe from figure \[fig:hate\_post\_increase\] that the amount of hate speech in Gab is indeed increasing.
More number of new users are becoming hateful
---------------------------------------------
Another important aspect about the growth that we considered was the fraction of new users who are getting exposed to hate. In this scenario, we say that a user $A$ has become hateful, if his/her hate vector has the entry ‘H’ at least $\mathcal{N}$ times within $\mathcal{T}$ months from the account creation. In figure \[fig:hate\_score\_from\_creation\_month\], we plot for $\mathcal{T}=3$ and $\mathcal{N}=1, 2, 3$ ‘H’ entries, to observe the fraction of users for each month who are becoming hateful. As we can observe, the fraction of users being exposed to hate speech is increasing over time.
![The figure shows the fraction of users in each month who got assigned at least **(1, 2, 3)** ‘H’ within the first **three** months of their joining.[]{data-label="fig:hate_score_from_creation_month"}](Figures/Hate_Fraction_3_months){width="\linewidth"}
![The figure shows the amount of time (in months) that each user requires to achieve $\mathcal{N}$ ‘H’ entries from his/her month of joining.[]{data-label="fig:user_change_to_hate_time"}](Figures/user_change_to_hate_time){width="\linewidth"}
New users are becoming hateful at a faster rate
-----------------------------------------------
In figure \[fig:user\_change\_to\_hate\_time\], we show how much time does a user take to have the first, second and third ‘H’ entry[^8]. We observe that with time the time required for a user to get his/her first exposure to hate decreases in Gab.
[0.45]{} {width="\textwidth"}
[0.45]{} {width="\textwidth"}
RQ2: What was the impact of the hateful users on Gab?
=====================================================
Hate users receive replies much faster
--------------------------------------
In order to understand the user engagement, we define *‘first reply time’(FRT)* which tries to measure the time taken to get the first comment/reply to a post by a user.
We define the *‘first reply time’* (FRT) for a set of users $U$ as $ FRT_{U}= \frac{1}{|U|}\sum_{(u)\in U} T_u $, where $T_u$ represents the average time taken to get the first reply for the posts written by a user $u$.
We calculated the FRT values for the set of hateful and non-hateful users and found that the average time for the first reply is 51.32 minutes for non-hate users, whereas it is 44.38 minutes for the hate users ($p$-value $<0.001$). The indicates that the community is engaging with the hateful users at a faster speed as compared to the non-hateful users.
Hateful users: lone wolf or clans
---------------------------------
In this section, we study the hateful and non-hateful users from a network-centric perspective by leveraging user-level dynamic graph. This approach has been shown to be effective in extracting anomalous patterns in microblogging platforms such as Twitter and Weibo [@zhang2015influenced; @shin2016corescope]. In similar lines, we conduct an unique experiment, where we track the influence of hateful and non-hateful users across successive temporal snapshots.
We utilize the node metric – [*k-core or coreness*]{} to identify influential users in the network [@shin2016corescope]. Nodes with high coreness are embedded in major information pathways. Hence they have been shown to be influential spreaders, that can diffuse information to a large portion of the network [@malliaros2016locating; @kitsak2010identification]. For further details about [*coreness*]{} and its several applications in functional role identification, we refer to Malliaros et al. [@malliaros2019core]. We first calculate coreness of the undirected [*follower/followee*]{} graph for each temporal snapshot using [*k-core decomposition*]{} [@malliaros2016locating]. In each snapshot, we subdivide all the nodes into $10$ buckets where consecutive buckets comprise increasing order of influential nodes, i.e., the bottom $10$ percentile nodes to the top $10$ percentile nodes in the network. We calculate the proportion of each category of users in all the proposed buckets across multiple dynamic graphs. We further estimate the proportion of migration from different buckets in consecutive snapshots. We illustrate results as a flow diagram in figure \[Fig:Alluvial\]. The innermost core is labeled 0, the next one labeled 1 and so on. The bars that have been annotated with a label $NA$ denote the proportion of users who have eventually been detected to be in a particular category but have not yet entered in the network at that time point (Account is not yet created).
**Position of hateful users**: We demonstrate the flow hateful users in figure \[fig:h-h\]. The leftmost bar denotes the entire group strength. The following bars indicate consecutive time points, each showcasing the evolution of the network.
We could observe several interesting patterns in figure \[fig:h-h\]. In the initial three time points, we observe that a large proportion of users are confined to the outer shells of the network. This forms a network-centric validation of the hypothesis that newly joined users tend to familiarize themselves with the norms of the community and do not exert considerable influence [@danescu2013no]. However, in the final time points we observe that the hateful users rapidly rise in ranks and the majority of them assimilate in the inner cores. This trend among Gab users has been found consistent with other microblogging sites like Twitter [@ribeiro2018characterizing] where hate mongers have been found with higher eigenvector and betweenness centrality compared to normal accounts. There are also surprising cases where a fraction of users who have just joined the network, become part of the inner core very quickly. We believe that this is by their virtue of already knowing a lot of ‘inner core’ Gab users even before they join the platform.
**Position of non-hateful users**: Focusing on figure \[fig:nh-nh\], which illustrates the case of non-hateful users, we see a contrasting trend. The flow diagram shows that users already in influential buckets continue to stay there over consecutive time periods. The increase in core size at a time point can be mostly attributed to the nodes of the nearby cores in the previous time point. We also observe that in the final snapshot of the graph all the cores tend to have a similar number of nodes. These results are in sharp contrast with those observed for the hateful users (figure \[fig:h-h\]).
![The ratio of hateful users to non-hateful users for each month in a specific core of the network[]{data-label="Fig:Core_transition"}](Figures/core_transition.pdf){width="\linewidth"}
**Acceleration toward the core**: We were also interested in understanding the rate at which the users were accelerating toward the core. To this end, we calculated the time it took for the users to reach bucket 0 from their account creation time. We found that a hateful user takes only 3.36 month to do this, whereas a non-hateful users requires 6.94 months to reach an inner core in the network. We test the significance of this result with the Mann-Whitney $U$-test and found $U = 35203.5$ and $p$-value$=2.68e-06$.
To further understand the volume of users transitioning in-between the cores of the network, we compute the ratio of the hateful to the non-hateful users in a given core for each month. Figure \[Fig:Core\_transition\] plots the ratio values. A value of 1.0 means that an equal number of hateful and non-hateful users occupy the same core in a particular month. A value less than one means that there were more non-hateful users in a particular core than there were hateful users. We observe that in the initial time periods (October 2016 - July 2017), the non-hateful users were occupying the inner core of the network more. However, after this, the fraction of hateful users in the innermost started increasing, and around August 2017 the fraction of hateful users surpassed the non-hateful ones. We observe similar trends in all the four innermost cores (0, 1, 2, and 3).
![Month wise entropy of the predictions obtained from the H-SLM and N-SLM when the full data for that month is used as the test set.[]{data-label="fig:entropy_month_wise"}](Figures/entropy_month_wise){width="\linewidth"}
Gab community is increasingly using language similar to users with high hate scores
-----------------------------------------------------------------------------------
Gab started in August 2016 with the intent to become the ‘champion of free speech’. Since its inception, it has attracted several types of users. As the community evolves, so does the members in the community. To understand the temporal nature of the language of the users and the community, we utilize the framework developed by Danescu et al. [@danescu2013no]. In their work, the authors use language models to track the linguistic change in communities.
We use kenLM [@Heafield-estimate] to generate language models for each snapshot. These ‘Snapshot Language Models’ (SLM) are generated for each month, and they capture the linguistic state of a community at one point of time. The SLMs allow us to capture how close a particular utterance is to a community. The ‘Hate Snapshot Language Model’ (H-SLM) is generated using the posts written by the users with high hate score in a snapshot as the training data. Similarly, we generate the ‘Non-hate Snapshot Language Model’ (N-SLM), which uses the posts written by users with low hate score in a snapshot for the training data. Note that unlike in the previous sections where we were building hate vectors aggregated over different time snapshots to call a user hateful/non-hateful, here we consider posts of users with high/low hate scores for a given snapshot to build the snapshot wise training data for the language models[^9]. For a given snapshot, we use the full data for testing. Using these two models, we test them on all the posts of the month and report the average cross entropy $$H(d, {\textrm{SLM}}_{c_t}) = \frac{1}{|d|}\sum_{b_i \in d}{\textrm{SLM}}_ {c_t}(b_i)$$ where $H(.)$ represents the cross-entropy, ${\textrm{SLM}}_{c_t}(b_i)$ is the probability assigned to bigram $b_i$ from comment $d$ in community-month $c_t$. Here, the community can be hate (H-SLM) or non-hate (N-SLM)[^10].
A higher value of cross-entropy indicates that the posts of the month deviate from the respective type of the community (hate/ non-hate). We plot the entropy values in figure \[fig:entropy\_month\_wise\]. As is observed, the whole Gab community seems to be more aligned with the language model built using the posts of the users with high hate scores. A remarkable observation is that from around May 2017, the language used by the Gab community started getting closer to the language of users with high hate scores. This may indicate that the hate community is evolving its own linguistic culture and the other users on the platform are slowly (and possibly unknowingly) aligning themselves to this culture thus making it the normative. If left unmoderated, one might notice in the near future that there would be only one language on this platform – *the language of hate*.
Discussion
==========
Ethical considerations and implications
---------------------------------------
The ongoing discussion of whether to include hate speech under the umbrella of free speech is a subject of great significance. A major argument used by the supporters of hate speech is that any form of censorship of speech is a violation of freedom of speech. Our work provides a peek into the hate ecosystem developed on a platform which does not have any sort of moderation, apart form ‘self-censorship’.
We caution against our work being perceived as a means for full-scale censorship. Our work is not indented to be perceived as a support for full-scale censorship. We simply argue that the ‘free flow’ of hate speech should be stopped. We leave it to the platform or government to implement a system that would reduce such speech in the online space.
Moderation
----------
Although the intent of Gab was to provide support for free speech, it is acting as a fertile ground for the fringe communities such as alt-right, neo-Nazis etc. Freedom of speech which harms others is no longer considered freedom. While banning of users and/or comments is not democratic, platforms/governments would still need to curb such hateful contents for proper functioning of the ‘online’ society. Recently, some of the works have started looking into alternatives to the banning approach. One of the contenders for this is counterspeech [@benesch2014countering; @mathew2018thou; @mathew2018analyzing]. The basic idea is that instead of banning hate speech, we should use crowdsourced responses to reply to these messages. The main advantage of such an approach is that it does not violate the freedom of speech. However, there are some doubts on how much applicable/practical this approach is. Large scale studies would need to be done to observe the benefits and costs of such an approach. Understanding the effects of such moderation is an area of future work.
We suggest that social media platforms could start incentive programs for counterspeech. They could also provide interface to group moderators to identify hateful activities and take precautionary steps. This would allow platforms to stop the spread of hateful messages in an early stage itself. The effect of hate speech from influential users is also much more as compared to others and thus targeted campaigns are required to overcome such adverse effects.
Monitoring the growth of hate speech
------------------------------------
The platform should have interface which allows moderators to monitor the growth of hate speech in the community. This could be a crowdsourced effort which could help identify users who are attempting to spread hate speech.
As we have seen that the new users are gravitating toward the hateful community at a faster rate and quantity, there is a need for methods that could detect and prevent such movement. There could exist radicalization pipelines [@ribeiro2019auditing] which could navigate a user toward hateful contents. Platforms should make sure that their user feed and recommendation algorithms are free from such issues. Exposure to such content could also lead to desensitization toward the victim community [@soral2018exposure]. We would need methods which would take the user network into consideration as well. Instead of waiting for a user to post his/her hateful post after the indoctrination, the platforms will need to be proactive instead of reactive. Some simple methods such as nudge [@thaler2009nudge], or changing the user feed to reduce polarization [@celis2019controlling] could be an initial step. Further research is required in this area to study these points more carefully.
Platform governance – the rising role of HCI
--------------------------------------------
All the points that we had discussed above related to moderation and monitoring can be aptly summarized as initiatives toward platform governance. We believe that within this initiative the HCI design principles of the social media platforms need to completely overhauled. In February 2019, the United Kingdom’s Digital, Media, Culture, and Sport (DCMS) committee issued a verdict that social media platforms can no longer hide themselves behind the claim that they are merely a ‘platform’ and therefore have no responsibility of regulating the content of their sites[^11]. In fact, the European Union now has the ‘EU Code of Conduct on Terror and Hate Content’ (CoT) that applies to the entire EU region. Despite the increase in toxic content and harassment, Twitter did not have a policy of their own to mitigate these issues until the company created a new organisation – ‘Twitter Trust and Safety Council’ in 2015. A common way deployed by the EU to combat such online hate content involves creation of working groups that combine voices from different avenues including academia, industry and civil society. For instance, in January 2018, 39 experts met to frame the ‘Code of Practice on Online Disinformation’ which was signed by tech giants like Facebook, Google etc. We believe that HCI practitioners have a lead role to play in such committees and any code of conduct cannot materialize unless the HCI design policies of these platforms are reexamined from scratch.
Limitations and future works
============================
There are several limitations of our work. We are well aware that studies conducted on only one social media such as Gab have certain limitations and drawbacks. Especially, since other social media sites delete/suspend hateful posts and/or users, it becomes hard to conduct similar studies on those platforms. The initial keywords selected for the hate users were in English. This would bias the initial belief value assignment as users who use non-English hate speech would not be detected directly. But, since these users follow similar hate users and repost several of their content, we would still detect many of them. We plan to take up the multilingual aspect as an immediate future work. Another major limitation of our work is the high-precision focus of the work which would leave out several users who could have been hateful.
As part of the future work, we also plan to use the discourse structure of these hateful users for better understanding of the tactics used by these users in spreading hate speech [@phadke2018framing]. This would allow us to break down the hate speech discourse into multiple components and study them in detail.
Conclusion
==========
In this paper, we perform the first temporal analysis of hate speech in online social media. Using an extensive dataset of 21M posts by 314K users on Gab, we divide the dataset into multiple snapshots and assign a hate score to each user at every snapshot. We then check for variations in the hate score of the user. We characterize these account types on the basis of text and network structure. We observe that a large fraction of hateful users occupy the core of the Gab network and they reach the core at a much faster rate as compared to non-hateful users. The language of the hate users seem to pervade across the whole network and convert even benign users unknowingly to speak the language of hate. Our work would be extremely useful to platform designers to detect the hateful users at an early stage and introduce appropriate measure to change the users’ stance.
[^1]: <https://www.justice.gov/hatecrimes/hate-crime-statistics>
[^2]: <https://tinyurl.com/facebook-leaked-moderation>
[^3]: <https://www.tandfonline.com/eprint/KxDwNEpqTY86MNpRDHE9/full>
[^4]: <https://gab.com/about/tos>
[^5]: <https://hatebase.org>
[^6]: <https://www.urbandictionary.com>
[^7]: <https://www.dropbox.com/sh/spidpraeln0qrtj/AACyFRPAWURXT05dbHwH9-Kta?dl=0>
[^8]: The reason for the initial dip in the plot is that some of the users who have 1 ‘H’ do not further account for 2 ‘H’ and 3 ‘H’ since they were never assigned a ‘H’ after the first one.
[^9]: It is not possible to extend the hate vector concept here as we are building language models snapshot by snapshot.
[^10]: We controlled for the spurious length effect by considering only the initial 30 words only [@danescu2013no]; the same controls are used in the cross-entropy calculations.
[^11]: https://policyreview.info/articles/analysis/platform-governance-triangle-conceptualising-informal-regulation-online-content
|
---
abstract: |
In [@X] Xia introduced a simple dynamical density basis for partially hyperbolic sets of volume preserving diffeomorphisms. We apply the density basis to the study of the topological structure of partially hyperbolic sets. We show that if $\Lambda$ is a strongly partially hyperbolic set with positive volume, then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and the global unstable manifolds over ${\omega}(\Lambda^d)$.
We give several applications of the dynamical density to partially hyperbolic maps that preserve some $acip$. We show that if $f$ is essentially accessible and $\mu$ is an $acip$ of $f$, then $\text{supp}(\mu)=M$, the map $f$ is transitive, and $\mu$-a.e. $x\in M$ has a dense orbit in $M$. Moreover if $f$ is accessible and center bunched, then either $f$ preserves a smooth measure or there is no $acip$ of $f$.
address: |
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China\
and CEMA, Central University of Finance and Economics, Beijing 100081, China
author:
- Pengfei Zhang
title: partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems
---
Introduction
============
Let $M$ be a $n$-dimensional connected, closed manifold, $r>1$ and $f\in\mathrm{Diff}^r(M)$ be a $C^r$ diffeomorphism on $M$. A compact $f$-invariant subset $\Lambda\subset M$ is said to be [*partially hyperbolic*]{} if there are a nontrivial $Tf$-invariant splitting of $T_xM = E^s_x\oplus E^{c}_x\oplus E^u_x$ for every $x\in\Lambda$, a smooth Riemannian metric $g$ on $M$ for which we can choose continuous positive functions $\nu,\tilde{\nu},\gamma$ and $\tilde{\gamma}$ on $\Lambda$ with $\nu,\tilde{\nu}<1$ and $\nu<\gamma\le\tilde{\gamma}^{-1}<\tilde{\nu}^{-1}$ such that, for all $x\in\Lambda$ and for all unit vectors $v\in E^s_x$, $w\in
E^c_x$ and $v'\in E^u$, $$\label{partial}
\|Tf(v)\|\le\nu(x)<\gamma(x)\le\|Tf(w)\|\le\tilde{\gamma}(x)^{-1}
<\tilde{\nu}^{-1}(x)\le\|Tf(v')\|.$$ The notation here is taken from [@BW]. Such a metric is called [*adapted*]{} (see [@G]). If both $E^s$ and $E^u$ are nontrivial, then we say $\Lambda$ is strongly partially hyperbolic. In particular the map $f$ is called a (strongly) partially hyperbolic diffeomorphism if $M$ itself is a (strongly) partially hyperbolic set. It is well known that $E^s$ and $E^u$ are uniquely integrable and tangent to the stable lamination $\mathcal{W}^s$ and the unstable lamination $\mathcal{W}^u$ respectively.
In [@X] Xia introduced a simple dynamical density basis for general partially hyperbolic sets. Namely let $\delta>0$ small, $W^s(x,\delta)$ be the local stable manifold over $x\in\Lambda$. Let $B^s_n(p)=f^nW^s(f^{-n}p,\delta)$ for each $p\in\Lambda$ and $n\ge0$. The collection of sets $\mathcal{S}=\{B^s_n(p):n\ge0,p\in\Lambda\}$ is called the [*stable basis*]{} (see [@X]). Let $A\subset\Lambda$ be a measurable subset. A point $p\in A$ is said to be a $\mathcal{S}$-density point of $A$ if $$\lim_{n\to\infty}\frac{m_{W^s(p)}(B^s_n(p)\cap
A)}{m_{W^s(p)}(B^s_n(p))}=1.$$ where $m_{W^s(p)}$ is the leaf-volume induced by the Riemannian metric. Let $A^d$ be the set of $\mathcal{S}$-density points of $A$. Following [@X] we have:
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**Proposition.** [*Let $r>1$, $f\in\mathrm{Diff}^r(M)$ and $\Lambda$ be a partially hyperbolic set with positive volume. For each measurable subset $A\subset\Lambda$, $m$-a.e. point in $A$ is $\mathcal{S}$-density point of $A$, that is, $m(A\backslash A^d)=0$. In words, $\mathcal{S}$ forms a density basis.*]{}
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This simply defined density basis turns out to be useful in the study of the topological structure of (partially) hyperbolic sets. There is an extensive literature discussing the topology of (partially) hyperbolic sets. We just name a few that are close related to the results here. Bowen showed in [@B], there exists $C^1$ horseshoe of positive volume (It is also showed in [@B1] that this [*fat*]{} horseshoe can not exist among the $C^2$ diffeomorphisms). In [@AP] Alves and Pinheiro showed that for a diffeomorphism $f\in\mathrm{Diff}^r(M)$, if $\Lambda$ is a partially hyperbolic set that attracts a positive volume set, then $\Lambda$ contains some local unstable disk (hence $\Lambda$ cannot be a horseshoe-like set). Under a much stronger setting, we can get a useful characterization that serves well for later applications. More precisely let $\alpha(x)$ be the set of accumulating points of $\{f^nx:n\le0\}$ of $x\in M$. For $E\subset M$, let ${\alpha}(E)$ be the closure of $\bigcup_{x\in E}\alpha(x)$. Similarly we can define ${\omega}(x)$ and ${\omega}(E)$. Then we have
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**Theorem A.**
*Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$ and $\Lambda$ be a partially hyperbolic set with positive volume. Then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$, that is, $W^s(x)\subset\Lambda$ for each $x\in {\alpha}(\Lambda^d)$.*
In particular if $\Lambda$ is a strongly partially hyperbolic set with positive volume, then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and the global unstable manifolds over ${\omega}(\Lambda^d)$.
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The argument here relies on the bounded distortion estimates and the absolute continuity of stable and unstable laminations, which fail for $C^1$ maps. See [@B; @RY]. Although ${\alpha}(\Lambda^d)$ is nonempty, the volume of ${\alpha}(\Lambda^d)$ could be zero (even in the hyperbolic case). In fact Fisher [@F] constructed several hyperbolic sets $\Lambda$ with nonempty interior such that $\alpha(\Lambda^d)$ are repellers and $\omega(\Lambda^d)$ are attractors (hence their volume must be zero).
A point $x$ is said to be backward recurrent if $x\in\alpha(x)$, and to be recurrent if $x\in\alpha(x)\cap\omega(x)$. An interesting case is when most points are recurrent. This will hold in particular if $\mu(\Lambda)>0$ for some [*absolutely continuous invariant probability*]{} measure ([*acip*]{} for short) $\mu\ll m$. For simplicity we assume that $\Lambda=\text{supp}\mu$.
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**Corollary B.** [*Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$ and $\Lambda$ a strongly partially hyperbolic set supporting some $acip$ $\mu$. Then $\Lambda$ is bi-saturated, that is, for each point $p\in\Lambda$, the global stable manifolds and the global unstable manifolds over $p$ lie in $\Lambda$.* ]{}
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In particular we give a dichotomy for maps $f\in\mathrm{Diff}^{r}(M)$: either $f$ is a transitive Anosov diffeomorphism, or each $f$-invariant hyperbolic set $\Lambda$ is $acip$-null, that is, $\mu(\Lambda)=0$ for every $acip$ $\mu$.
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**Theorem C.** [*Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$, $\mu$ be an $acip$ and $\Lambda$ be a hyperbolic set with positive $\mu$-measure. Then $\Lambda=M$ and $f$ is a transitive Anosov diffeomorphism on $M$.* ]{}
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The similar result has been proved if the $acip$ $\mu$ assumed to be equivalent to $m$ (see [@BocV; @X]). Moreover it is proved in [@B1] that for $C^r$ transitive Anosov diffeomorphism, the $acip$ must have Hölder continuous density with respect to the volume and be an ergodic (indeed $Bernoulli$) measure, see Remark \[bernoulli\]. Also note that the condition that $\Lambda$ has positive $\mu$-measure for some $acip$ is nontrivial and see [@F] for counter-examples.
Theorem C motivates the analogous generalizations from hyperbolic systems to accessible partially hyperbolic systems. Recall that an $f$-invariant measure $\mu$ is said to be [*weakly ergodic*]{} if for $\mu$-a.e. $x$, $\mathcal{O}(x)$ is dense in $\text{supp}(\mu)$. Following generalizes the well known result of Brin [@Br] to the presence of $acip$.
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**Theorem D.** [*Let $f\in\mathrm{SPH}^r(M)$ for some $r>1$ be essentially accessible. If there exists some $acip$ $\mu$ of $f$, then $\mathrm{supp}(\mu)=M$, the map $f$ is transitive, and the $acip$ $\mu$ is weakly ergodic. In particular $\mathcal{O}(x)$ is dense in $M$ for $\mu$-a.e. $x\in M$.* ]{}
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In the following we assume $r=2$ for simplicity. Burns and Wilkinson proved in [@BW] that if a map $f\in\mathrm{SPH}^2(M)$ is center bunched, then every measurable bi-essential saturated set is essential bi-saturated. Applying to $acip$ we have
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**Proposition.** [*Let $f\in \mathrm{SPH}^2(M)$ be essentially accessible and center bunched. If there exists some $acip$ $\mu$, then $\mu$ must be equivalent to the volume. In particular $\mu$ is ergodic.* ]{}
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Note that the arguments in [@BW] still work if $f\in
\mathrm{SPH}^{r}(M)$ for $r>1$, as long as we assume [*strong center bunching*]{} (see [@BW Theorem 0.3]). So our results also extend to this setting. Then applying the [*cohomologous theory*]{} developed in [@W], we show that the $acip$ is a [*smooth measure*]{}, that is, the density $\frac{d\mu}{dm}$ of $\mu$ with respect to $m$ is Hölder continuous on $M$, bounded and bounded away from zero.
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**Theorem E.** [*Let $f\in \mathrm{SPH}^2(M)$ be accessible and center bunched. If there exists some $acip$, then the $acip$ must have Hölder continuous density with respect to the volume of $M$. In words, either $f$ preserves some smooth measure or there is no $acip$ for $f$.* ]{}
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Combining the results in [@DW] we have the following direct corollary:
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**Corollary F.** [*The set of maps that admit no $acip$ contains a $C^1$ open and dense subset of $C^2$ strongly partially hyperbolic and center bunched diffeomorphisms. In particular the set of maps that admit no $acip$ contains a $C^1$ open and dense subset of $C^2$ strongly partially hyperbolic diffeomorphisms with $\dim(E^c)=1$.* ]{}
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Finally we remark that although the volume measure need not be $f$-invariant, there always exists some $f$-invariant measures. The density argument combines the dynamics of $acip$ and the dynamics of volume on $M$. This is why most results of volume-preserving partially hyperbolic systems have parallel generalizations to the systems with $acip$.
Dynamical density basis for partially hyperbolic set
====================================================
In this section we will consider a $C^r$ diffeomorphism for some $r>1$ and a partially hyperbolic invariant set with positive volume. More precisely let $M$ be a closed and connected smooth manifold. Each Riemannian metric $g$ on $M$ induces a (geodesic) distance $d$ on $M$ and a normalized volume measure $m$ on $M$. Let $\mathcal{B}$ be the Borel $\sigma$-algebra of $M$. A Borel probability measure $\mu$ on $M$ is said to be [*absolutely continuous*]{} with respect to $m$, denoting $\mu\ll m$, if $\mu(A)=0$ for each set $A\in
\mathcal{B}$ with $m(A)=0$, and to be [*equivalent to*]{} $m$ if $\mu\ll m$ and $m\ll \mu$. It is evident for any other Riemannian metric $g'$ compatible with $g$, the induced volume of $g'$ is equivalent to $m$.
Let $f\in\mathrm{Diff}^r(M)$ for $r>1$ and $\Lambda$ be a compact partially hyperbolic invariant set with positive volume. In the following we always assume that the stable subbundle $E^s$ is nontrivial on $\Lambda$ and $m$ is the normalized volume measure on $M$ induced by some Riemannian metric adapted to the invariant splitting (see [@G]).
Since $r>1$, it is well known that the stable bundle $E^s$ is Hölder continuous over $\Lambda$ (the Hölder exponent may be much smaller than $r-1$, see [@BS]) and is tangent to the stable lamination $\mathcal{W}^s$ over $\Lambda$. (A lamination over $\Lambda$ is a partial foliation which may not foliate an open neighborhood of $\Lambda$, see [@HPS]. In case that $\Lambda=M$, $\mathcal{W}^s$ turns out to be a foliation.) For $\delta>0$ small we use $W^s(x,\delta)$ to denote the local stable manifold over $x\in\Lambda$. Note that $W^s(x,\delta)$ varies uniformly $\alpha$-Hölder continuously with respect to the base point $x\in\Lambda$. For simplicity we also write $E_y^s=T_yW^s(x)$ for all $y\in W^s(x,\delta)$ and $x\in\Lambda$. The invariance of $\mathcal{W}^s$ implies that the extended distribution $E^s$ is also invariant. The Hölder continuity of $E^s$ ensures that the family $\mathcal{W}^s$ is absolutely continuous. By slightly increasing $\nu$ and decreasing $\delta$ if necessary, we can assume that for each $x\in\Lambda$ the following holds: $$\label{stable}
\text{if }p,p'\in W^s(x,\delta),\text{ then
}d(fp,fp')\le\nu(p)d(p,p').$$ In particular we have $fW^s(x,\delta)\subseteq
W^s(fx,\delta\!\cdot\!\nu(x))$ for all $x\in \Lambda$.
Before moving on, let’s fix some notations as in [@BW]. Let $S\subset M$ be a submanifold of $M$, $m_S$ be the volume measure on $S$ induced by the restricted Riemannian metric $g|_{S}$ on $S$. In particular if $S=W^s(x)$, we abbreviate the induced measure as $m_{s,x}$. Denote $m_{s,x}(A)$ the restricted submanifold measure for a measurable subset $A\subseteq W^s(x)$. This should not be confused with conditional measures. Let $\eta=\min\{\|Tf(v)\|:v\in
TM\text{ with }\|v\|=1\}$ and $\overline{\nu}=\sup_{p\in\Lambda}\nu(p)$. Clearly $0<\eta\le\nu(p)\le\overline{\nu}<1$ by compactness. For each $p\in
\Lambda$ we let $p_i=f^ip$ for $i\in\mathbb{Z}$, $\nu_0(p)=1$ and $\nu_n(p)=\nu(p_{n-1})\cdots\nu(p_0)$ for $n\ge1$. Let $B^s_n(p)=f^nW^s(p_{-n},\delta)$. Since $\Lambda$ is $f$-invariant, we have $B^s_n(p)\subset W^s(p,\delta\!\cdot\! \nu_n(p_{-n}))$.
Since each stable manifold is a smooth submanifold of the Riemannian manifold $M$ and $f$ is $C^r$ for $r>1$, the [*stable Jacobian*]{} $J^s(f,x)$ of the restricted map $Tf:E^s_x\rightarrow E^s_{x_1}$ is well defined and Hölder continuous with uniform Hölder exponent and Hölder constant. That is, there exist $\alpha>0$ and $C_0>0$ such that for any $p\in\Lambda$ and $x,y\in W^s(p,\delta)$ we have $|J^s(f,x)-J^s(f,y)|\le C_0 d(x,y)^\alpha$. Also there exists $J^*\ge1$ such that $1/J^*\le J^s(f,x)\le J^*$ for all $x\in
W^s(p,\delta)$ and $p\in\Lambda$. Decreasing $\delta$ again if necessary we assume $C_1=\prod_{k=0}^{\infty}\frac{(1+J^*C_0
\delta^\alpha \overline{\nu}^{k\alpha})}{(1-J^*C_0 \delta^\alpha
\overline{\nu}^{k\alpha})}<\infty$.
Let $B^s_n(p)=f^nW^s(f^{-n}p,\delta)$ for each $p\in\Lambda$, $n\ge0$ and $\mathcal{S}=\{B^s_n(p):n\ge0,p\in\Lambda\}$ be the stable basis. It is easy to see that $\{B^s_n(p):n\ge0\}$ forms a nesting sequence of neighborhoods of $p\in\Lambda$ relative to $W^s(p,\delta)$ and $B^s_n(p)$ shrinks to $p$ as $n\rightarrow\infty$. Note that the basis here is in leafwise sense and may has infinite eccentricity. The proposition below states that the stable basis $\mathcal{S}$ behaves well in the sense of [@PS]:
\[basis\] The following properties hold for stable basis $\mathcal{S}$:
1. For any $p\in\Lambda$, $m_{s,p}(B^s_n(p))\rightarrow0$ if and only if $n\rightarrow\infty$.
2. For any $k\ge0$, there exists $c_k\ge1$ such that $\frac{m_{s,p}(B^s_n(p))} {m_{s,p}(B^s_{n+k}(p))}\le c_k$ for all $p\in\Lambda$, $n\ge0$.
3. There exists $L\in\mathbb{N}$ such that for any $p,q\in\Lambda$, $n\ge0$, if $B^s_{n+L}(p)\cap B^s_{n+L}(q)\neq\emptyset$, then $B^s_{n+L}(p)\cup B^s_{n+L}(q)\subseteq B^s_{n}(p)\cap B^s_{n}(q)$.
The properties listed above appeared in [@PS] (in a general setting) and is named to be [*volumetrically engulfing*]{} (also see [@X] for example). The proof mainly uses distortion estimates.
Let $A\in\mathcal{B}_{\Lambda}$ be a measurable subset of $\Lambda$. Recall that a point $x\in A$ is said to be a $\mathcal{S}$-density point of $A$ if $$\label{density}
\lim_{n\to\infty}\frac{m_{s,p}(B^s_n(p)\cap
A)}{m_{s,p}(B^s_n(p))}=1.$$ For different $\delta$’s, the induced stable bases are [*internested*]{} (see [@BW Lemma 2.1] for details). So the definition of $\mathcal{S}$-density point is independent of the choice of the size of stable manifolds and the choice of adapted Riemannian metric on $M$. Let $A^d$ be the set of $\mathcal{S}$-density points of $A$.
For each $A\in\mathcal{B}_{\Lambda}$ and each $p\in\Lambda$, $A\cap
W^s(p,\delta)$, the intersection of two Borel measurable subsets, is a Borel measurable subset of the submanifold $W^s(p,\delta)$. (Note that if $A$ is Lebesgue measurable, above relation will hold for $m$-a.e. $p\in\Lambda$ by Fubini Theorem.) Let us denote $A^d_p$ the set of $\mathcal{S}$-density points of $A\cap W^s(p,\delta)$. Clearly we have $A^d=\bigcup_{p\in\Lambda}A^d_p$.
\[dup\] Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$ and $\Lambda$ be a partially hyperbolic set with positive measure. For each subset $A\in\mathcal{B}_{\Lambda}$, we have
1. for each $p\in\Lambda$, $m_{s,p}$-a.e. point in $W^s(p,\delta)\cap A$ is a $\mathcal{S}$-density point of $A$: $m_{s,p}(W^s(p,\delta)\cap A\backslash A^d_p)=0$.
2. $m$-a.e. point of $A$ is a $\mathcal{S}$-density point of $A$: $m(A\backslash A^d)=0$.
Moreover if $A\in\mathcal{B}_{\Lambda}$ is $f$-invariant, so is $A^d$.
The first item follows by applying Theorem 3.1 in [@PS] to stable basis $\mathcal{S}$ to each intersection $A\cap
W^s(p,\delta)$. Proposition \[basis\] ensures that $\mathcal{S}$ forms a density basis in this leafwise sense.
Using the absolute continuity of the stable foliation $\mathcal{W}^s$ and the relation $A^d=\bigcup_{p\in\Lambda}A^d_p$, we have $m(A\backslash A^d)=0$. Hence $\mathcal{S}$ also forms a density basis in the ambient sense.
For the last item, we note that each local leaf $W^s(s,\delta)$ is a $C^r$ submanifold of $M$ and the restriction of $f$ between local stable manifolds is diffeomorphic onto its image. So $p\in\Lambda$ is a $\mathcal{S}$-density point of $A\cap W^s(p,\delta)$ (or equally, of $A$) if and only if $fp$ is a $\mathcal{S}$-density point of $A\cap W^s(fp,\delta)$.
Topological structure of partially hyperbolic sets
==================================================
In this section we give some descriptions of the topological structure of partially hyperbolic sets with positive volume. As in Section $2$ we let $M$ be a connected closed manifold, $f\in\mathrm{Diff}^r(M)$ for some $r>1$ and $\Lambda$ a partially hyperbolic set with positive volume.
Given a Borel subset $A\subset \Lambda$, we consider a family of functions $\eta_n$ on $\Lambda$ as $$\eta_n(x)=m_{s,x}(B^s_n(x)\backslash A)/m_{s,x}(B^s_n(x)).$$ The following result shows the increasing occupation of an invariant set $A$ in the local stable manifolds along the backward iterates of an $\mathcal{S}$-density point of $A$.
\[distortion\] There exists a constant $C\ge1$ such that given an $f$-invariant subset $A\in\mathcal{B}_{\Lambda}$, $m_{s,x_{-n}}(W^s(x_{-n},\delta)\backslash A) \le C_2\cdot\eta_n(x)$ for each $x\in\Lambda$ and $n\ge0$.
We only need to adapt the notations in [@X Lemma 3.2], since the proof is essentially the same. Let $A$ be an invariant set and $x\in\Lambda$ be fixed. Let $B^k_n=f^kW^s(x_{-n},\delta)$ and $D_n^k=B^k_n\backslash A$ for each $0\le k\le n$. Note that $B^0_n=W^s(x_{-n},\delta)$ is a local stable leaf and $B^n_n=B^s_n(x)$ is an element in the stable basis $\mathcal{S}$. Then using the constant $C_5$ given by [@X Page 816], we have $m_{s,x_{-n}}(D_n^{0}) \le C_5\cdot\eta_n(x)\cdot
m_{s,x_{-n}}(B_n^{0})$. Applying $B_n^{0}=W^s(x_{-n},\delta)$ and $D_n^{0}=W^s(x_{-n},\delta)\backslash A$, we finish the proof with a uniform constant $C_2=C_5\cdot\max_{p\in
\Lambda}m_{s,p}(W^s(p,\delta))$.
Recall that $\alpha(x)$, the $\alpha$-set of $x$, is the set of accumulating points along the backward orbit $\{x,f^{-1}x,\cdots\}$. Let ${\alpha}(E)$ be the closure of $\bigcup_{x\in E}\alpha(x)$. Note that for each point $x\in\Lambda$ and each subset $E\subset
\Lambda$, the sets $\alpha(x)$ and ${\alpha}(E)$ are compact $f$-invariant subsets of $\Lambda$.
Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$ and $\Lambda$ a partially hyperbolic set with positive volume. Then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$.
First let us consider $y\in\alpha(x)$ for some $x\in\Lambda^d$. Pick a sequence of times $n_i\to+\infty$ such that $x_{-n_i}\to y$ (clearly all these points are in $\Lambda$). By Lemma \[distortion\] we have $m_{s,x_{-n}}(W^s(x_{-n},\delta)\backslash
\Lambda)\le C_2\eta_n(x)$. (Note that $\eta_n(x)\to 0$ as $n\to\infty$.) Passing to a subsequence if necessary, we can assume that $W^s(x_{-n_i},\delta)\cap\Lambda$ contains a $\frac{1}{i}$-dense subset $E_{x_{-n_i},i}$ of $W^s(x_{-n_i},\delta)$. Let $E=\limsup_{i\to\infty}E_{x_{-n_i},i}=\bigcap_{k\ge1}\overline{\bigcup_{i\ge
k}E_{x_{-n_i},i}}$. It is clear that $E\subset\Lambda$ since $\Lambda$ is compact. By continuity of the stable manifolds, $E$ contains a dense subset of $W^s(y,\delta)$, and hence $W^s(y,\delta)\subset E$. So $W^s(y,\delta)\subset \Lambda$ for each $y\in\alpha(x)$ and each $x\in\Lambda^d$.
Still by the compactness of $\Lambda$, $W^s(y,\delta)\subset\Lambda$ for each $y\in {\alpha}(\Lambda^d)$. By the invariance of $\Lambda$ and ${\alpha}(\Lambda^d)$, the global stable manifolds $W^s(y)\subset\Lambda$ for each $y\in {\alpha}(\Lambda^d)$.
Similarly we consider the $\omega$-set and define ${\omega}(\Lambda^d)$. For a strongly partially hyperbolic set we have
\[main\] Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$ and $\Lambda$ a strongly partially hyperbolic set with positive volume. Then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and the global unstable manifolds over ${\omega}(\Lambda^d)$.
So every partially hyperbolic set with positive volume is far from being a topological horseshoe-like set. Although the sets ${\alpha}(\Lambda^d)$ and ${\omega}(\Lambda^d)$ are always nonempty, we do not know how large they could be and when they could intersect with each other. This can be improved if we require that $\Lambda$ admits some recurrence.
A point $x$ is said to be [*backward recurrent*]{} if $x\in\alpha(x)$. The definition of [*forward recurrent*]{} is analogous. A point is said to be [*recurrent*]{} if it is both backward and forward recurrent.
Let $E$ be a measurable subset of $\Lambda$. Then $E$ is said to be [*$s$-saturated*]{} if for each $x\in E$, $W^s(x)\subset E$. Similarly we can define [*$u$-saturated*]{} sets. Then the set $E$ is [*bi-saturated*]{} if it is $s$-saturated and $u$-saturated.
\[sph\] Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$ and $\Lambda$ be a strongly partially hyperbolic set supporting some $acip$ $\mu$. Then $\Lambda$ is bi-saturated.
By Poincaré recurrence theorem, we have that $\mu$-a.e. $x\in\Lambda$ is recurrent, that is, $\mu(\mathrm{Rec}_{\Lambda})=1$ where $\mathrm{Rec}_{\Lambda}$ is the set of recurrent points in $\Lambda$. Also we have $\mu(\Lambda\backslash \Lambda^d)=0$ since $\mu\ll m$ and $m(\Lambda\backslash \Lambda^d)=0$. So $\mu(\Lambda^d\cap\mathrm{Rec}_{\Lambda})=1$ and the closed set ${\alpha}(\Lambda^d)$ contains $\Lambda^d\cap\mathrm{Rec}_{\Lambda}$, which is a subset of full $\mu$-measure in $\Lambda$ and hence dense in $\text{supp}\mu=\Lambda$. So ${\alpha}(\Lambda^d)=\Lambda$ and the set $\Lambda$ is $s$-saturated by Theorem \[main\]. Similarly we can show $\Lambda$ is $u$-saturated. This completes the proof.
Regularity of $acip$: hyperbolic case. {#acip1}
======================================
In this section we consider the hyperbolic sets. We show that if a hyperbolic set has positive $acip$-measure, then the map is a transitive Anosov diffeomorphism. Then it is well known that the $acip$ is not only equivalent to the volume, but also has smooth density with respect to the volume. This motivates the generalization to partial hyperbolic systems in next section.
\[anosov\] Let $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$, $\mu$ be an $acip$ and $\Lambda$ be a hyperbolic set with positive $\mu$-measure. Then $\Lambda=M$ and $f$ is an transitive Anosov diffeomorphism on $M$.
By considering $\Lambda_\mu=\Lambda\cap\text{supp}(\mu)$ and $\mu|_{\Lambda_\mu}$ if necessary, we can assume that $\Lambda=\text{supp}(\mu)$. By Corollary \[sph\], we have that $\Lambda$ is bi-saturated. By the uniform hyperbolicity of $\Lambda$, there exists $\epsilon>0$ such that $B(x,\epsilon)\subset\bigcup_{y\in
W^u(x,\delta)}W^s(y,\delta)\subset\Lambda$ for each $x\in\Lambda$. So the set $\Lambda$ is both close and open, hence coincides with the whole manifold $M$. Since the recurrent set is a dense subset of $\text{supp}(\mu)=\Lambda=M$ and is contained in the nonwandering set $\Omega(f)$, we have that $\Omega(f)=M$ and $f$ is a transitive Anosov diffeomorphism on $M$ (by spectrum decomposition theorem, see [@B1]).
\[bernoulli\] Spectrum decomposition theorem actually implies that $f$ is mixing. Moreover by Corollary 4.13 and Theorem 4.14 in [@B1], $\mu$ coincides with the equilibrium state $\mu_{\phi^u}$ of the potential $\phi^u(x)=-\log(J^u(f,x))$, and has Hölder continuous density with respect to $m$. Furthermore the smooth measure $\mu$ is ergodic and $Bernoulli$.
The regularity of $f\in\mathrm{Diff}^{r}(M)$ for some $r>1$ is an essential assumption in a two-fold sense. In [@RY] Robinson and Young constructed a $C^1$ Anosov diffeomorphism with non-absolutely continuous stable and unstable foliations, which does have some closed invariant set with positive volume. In [@B] Bowen constructed a $C^1$ horseshoe with positive volume and absolutely continuous local stable and unstable laminations, where the bounded distortion property in Lemma \[distortion\] fails.
Regularity of $acip$: partially hyperbolic case {#acip2}
===============================================
In this section we show analogous results in Section \[acip1\] hold for accessible strongly partially hyperbolic systems. Namely, let $f\in\mathrm{SPH}^r(M)$ for some $r>1$ be a $C^r$ strongly partially hyperbolic diffeomorphism and $m$ be the volume measure associated to some Riemannian metric adapted to the partially hyperbolic splitting. Let $\mathcal{W}^s$ be the stable foliation tangent to the stable bundle and $\mathcal{W}^u$ the unstable foliation tangent to the unstable bundle.
Let $E$ be a measurable subset of $M$. Then $E$ is said to be [*essentially $s$-saturated*]{} if there exists an $s$-saturated set $\widehat{E}^s$ with $m(E\triangle \widehat{E}^s)=0$. Similarly we can define [*essentially $u$-saturated*]{} sets. The set $E$ is [*essentially bi-saturated*]{} if there exists a bi-saturated set $\widehat{E}^{su}$ with $m(E\triangle \widehat{E}^{su})=0$, and [*bi-essentially saturated*]{} if $E$ is essentially $s$-saturated and essentially $u$-saturated.
A strongly partially hyperbolic diffeomorphism $f: M\to M$ is said to be [*accessible*]{} if each nonempty bi-saturated set is the whole manifold $M$. The map $f$ is [*essentially accessible*]{} if every measurable bi-saturated set has either full or zero volume.
\[tran\] Let $f\in\mathrm{SPH}^r(M)$ be essentially accessible. If there exists some $acip$ for $f$, then the support of the $acip$ is the whole manifold and the map $f$ is transitive.
Before the proof, we mention that there exists a $C^1$ open set of accessible but non-transitive diffeomorphisms (see [@NT]).
Let $\mu$ be an $acip$ of $f$. Then the support $\mathrm{supp}(\mu)$ of $\mu$ is a strongly partially hyperbolic set supporting $\mu$, hence is a bi-saturated set by Corollary \[sph\]. Essential accessibility of $f$ implies that $m(\mathrm{supp}(\mu))=1$. Hence $\mathrm{supp}(\mu)=M$ since $\mathrm{supp}(\mu)$ is closed.
Suppose on the contrary that $f$ is not transitive. That is, there exists an $f$-invariant nonempty open set $U$ such that $M\backslash
\overline{U}\neq\emptyset$. So the set $\Lambda=M\backslash U$ is $f$-invariant, closed with nonempty interior. Hence $\mu(\Lambda)>0$ and $\mu|_{\Lambda}$ is again an $acip$. Corollary \[sph\] implies that $\Lambda$ is bi-saturated. Since $f$ is essentially accessible, we have $m(\Lambda)=1$ and $m(U)=0$. This contradicts the openness of $U$.
Generally for a transitive map $f$, the set $\mathrm{Tran}_f$ of points with dense orbit is measure-theoretic meagre (although topological residual). In [@RHRHU Section 5.7] they extracted following property which can be viewed as a stronger form of transitivity (or a weak form of ergodicity).
An $f$-invariant measure $\mu$ is said to be [*weakly ergodic*]{} if the set of points with dense orbit in $\text{supp}(\mu)$ has full $\mu$-measure.
Clearly that ergodicity implies weak ergodicity, and weak ergodicity implies the transitivity of the subsystem $(f,\text{supp}(\mu))$. In the following we show some analogous results in [@Br; @BDP; @RHRHU] hold for $acip$. To this end let us introduce some necessary notations. Let $\mu$ be an $acip$ of $f\in\mathrm{SPH}^r(M)$ for some $r>1$ and $\phi=\frac{d\mu}{dm}$ be the [*Radon-Nikodym density*]{} of $\mu$ relative to $m$. Note that the [*Jacobian*]{} $J_f:M\to\mathbb{R},x\mapsto \mathrm{Jac}(Df:T_xM\to T_{fx}M)$ is a Hölder continuous function, bounded and bounded away from $0$ on $M$. Now for each measurable subset $A\subset M$ we have: $$\int_{A}\phi(x)dm(x)=\mu(A)
=\mu(fA)=\int_{fA}\phi(y) dm(y)=\int_{A}\phi(fx) J_f(x) dm(x).$$ So the following holds: $$\label{coho1}
\phi(fx) J_f(x)=\phi(x)\text{ for }m-\text{a.e. }x\in M.$$ Let us consider the set $E=\{x\in M:\phi(x)>0\}$. Clearly $E$ is measurable and $m(E)>0$. By we see that $E$ is also $f$-invariant. Restricted to the set $E$, the measure $m|_{E}$ is equivalent to $\mu$. So ‘(P) for $m$-a.e. $x\in E$’ is the same as ‘(P) for $\mu$-a.e. $x\in E$’. In this case we will say ‘(P) a.e. $x\in E$’ for short.
\[biess\] Let $f\in\mathrm{SPH}^r(M)$, $\mu$ be an $acip$ with density $\phi$ and $E=\{x\in M:\phi(x)>0\}$. Then $E$ is bi-essentially saturated.
Note that all essential saturations are defined with respect the volume. If $f$ is volume preserving, then every invariant set is always bi-essentially saturated by Hopf argument. See [@BS Lemma 6.3.2] and [@RHRHU Theorem 5.5].
It suffices to prove that $E$ is essentially $s$-saturated. Let $B^s_n(x)=f^{n}W^s(x_{-n},\delta)$ and $E^d$ be the set of $\mathcal{S}$-density points of $E$. By Proposition \[dup\] we have $m(E\backslash E^d)=0$.
Consider the functions $\eta_n(x)=m_{W^s(x)}(B^s_n(x)\backslash
E)/m_{W^s(x)}(B^s_n(x))$ for $n\ge1$. So $\eta_n(x)\to 0$ as $n\to+\infty$ for a.e. $x\in E$. For each $\epsilon>0$, there exists a subset $E_\epsilon\subset E$ with $m(E\backslash
E_\epsilon)<\epsilon$ on which $\eta_n$ converges uniformly to zero. For a recurrent point $x\in E_\epsilon$, let $n_i$ be the forward recurrent times of $x$ with respect to $E_\epsilon$, that is, $f^{n_i}x\in E_\epsilon$. Note that a.e. $x\in E_\epsilon$ is recurrent.
By Lemma \[distortion\], there exists a uniform constant $C_2$ such that for the point $y=f^{n_i}x$ and $n=n_i$ the following holds: $$m_{W^s(x)}(W^s(x,\delta)\backslash E) \le C_2\cdot
\eta_{n_i}(f^{n_i}x).$$ Passing $n_i$ to $\infty$ we have $m_{W^s(x)}(W^s(x,\delta)\backslash E)=0$ for a.e. $x\in
E_\epsilon$.
Since $\epsilon$ can be arbitrary small, we have $m_{W^s(x)}(W^s(x,\delta)\backslash E)=0$ for a.e. $x\in E$. Since $E$ is $f$-invariant and $f$ is smooth between leaves of $\mathcal{W}^s$, $m_{W^s(x)}(f^{-n}W^s(f^{n}x,\delta)\backslash
E)=0$ for each $n\ge1$ and a.e. $x\in E$. Hence $m_{W^s(x)}(W^s(x)\backslash E)=0$ for a.e. $x\in E$. It follows from the absolute continuity of $\mathcal{W}^s$ that $E$ is essentially $s$-saturated. Similarly we can show $E$ is essentially $u$-saturated. This completes the proof.
Let $f\in\mathrm{SPH}^r(M)$ be essentially accessible. Then every $acip$ is weakly ergodic. In particular if $\mu$ is an $acip$, then the orbit $\mathcal{O}(x)$ is dense in $M$ for $\mu$-a.e. $x\in M$.
This result is well known if the system is volume preserving (see [@Br; @BDP; @RHRHU]). The idea of the proof is similar to Lemma 5 in [@BDP]. Also see Proposition 5.17 in [@RHRHU].
Let $\phi$ be the density of $\mu$ with respect to $m$ and $E=\{x\in
M:\phi(x)>0\}$. By Proposition \[biess\], we have $E$ is bi-essentially saturated. Hence $\overline{E}=\mathrm{supp}(\mu)=M$ by Theorem \[tran\] since $f$ is essentially accessible.
[*Step 1.*]{} We will show that for each open ball $B$, $\mathcal{O}(x)\cap B\neq\emptyset$ for $m$-a.e. point $x\in E$. To this end we first consider $G(B)$, the subset of points $x$ which has a neighborhood $U$ of $x$ such that $\mathcal{O}(y)\cap
B\neq\emptyset$ for $m$-a.e. $y\in U\cap E$. Evidently $G(B)$ is a nonempty open subset (and $f$-invariant).
[**Claim.**]{} $G(B)$ is bi-saturated. So $m(G(B))=1$ since $f$ is essentially accessible.
[*Proof of Claim*]{}. Let us prove $G(B)$ is $s$-saturated. It suffices to show that $q\in G(B)$ for each $q\in W^s(z,\delta)$ and each $p\in G(B)$, where the size $\delta$ is fixed. So the justification lies in a local foliation box $X$ of $\mathcal{W}^s$ around $p$. Note that we can replace $E$ by its saturate $\widehat{E}^s$ in the definition of $G(B)$ since $E$ is essentially $s$-saturated. For a point $x\in X$, denote $W^s_X(x)$ the component of $W^s(x)\cap X$ that contains $x$. Let $U$ be a small neighborhood of $p$ with $\mathcal{O}(y)\cap B\neq\emptyset$ for $m$-a.e. $y\in
U\cap \widehat{E}^s$. Let $R$ be the set of recurrent points $z\in
U\cap \widehat{E}^s$ whose orbits enter $B$. Note that $m(U\cap
\widehat{E}^s\backslash R)=0$ since $m|_{E}$ is equivalent to the invariant measure $\mu$ and $m(E\triangle\widehat{E}^s)=0$. So we can pick a smooth transverse $T$ of $\mathcal{W}^s_{X}$ in $U$ such that $T\cap W^s_U(p)\neq\emptyset$ and $m_T(\widehat{E}^s\backslash
R)=0$, where $m_T$ is the induced volume on $T$ (It is helpful to keep in mind that $\widehat{E}^s$ is not only essentially $s$-saturated, but $s$-saturated). Now we have
1. For each $y\in W^s_X(x)$ and $x\in R$, we have $\mathcal{O}(y)\cap
B\neq\emptyset$. This follows from that $d(f^nx,f^ny)\to0$ and the recurrence of $x$: the orbit of $x$ will enters $B$ infinite many times.
2. The set $\bigcup_{x\in T\cap R}W^s_X(x)$ has full $m$-measure in the set $\bigcup_{x\in T\cap \widehat{E}^s}W^s_X(x)$. This follows from that both sets are measurable and $\mathcal{W}^s_X$-saturated, $\mathcal{W}^s_X$ is an absolutely continuous lamination of $X$ and $m_T(\widehat{E}^s\backslash R)=0$.
3. The set $\bigcup_{x\in T}W^s_X(x)$ contains an open neighborhood $V$ of $q$. This follows from that the holonomy maps along $\mathcal{W}^s_X$ are homeomorphisms.
Also note that $\bigcup_{x\in T\cap
\widehat{E}^s}W^s_X(x)=\left(\bigcup_{x\in T}W^s_X(x)\right)\cap
\widehat{E}^s$. So $\mathcal{O}(y)\cap B\neq\emptyset$ for $m$-a.e. $y\in V\cap \widehat{E}^s$. This implies $q\in G(B)$ and hence $G(B)$ is $s$-saturated. Similarly we have $G(B)$ is also $u$-saturated and hence $m(G(B))=1$ by the essential accessibility of $f$. This completes the proof of Claim.
Now let $F(B)=\{x\in E: \mathcal{O}(x)\cap B\neq\emptyset\}$. We need to show that $m(E\backslash F(B))=0$. To derive a contradiction we assume $m(E\backslash F(B))>0$ and $p\in G(B)$ be a Lebesgue density point of $E\backslash F(B)$ (here we use $m(G(B))=1$). So there exists an open neighborhood $U$ of $p$ such that $\mathcal{O}(x)\cap B\neq\emptyset$ for a.e. $x\in U\cap E$. Then we have $m(U\cap E\backslash F(B))=0$. But this is impossible since we choose $p$ as a Lebesgue density point of $E\backslash F(B)$. So we have $m(E\backslash F(B))=0$ for each open ball $B$.
[*Step 2.*]{} Since $M$ is compact, there exists a countable collection of open balls $\{B_n:n\ge1\}$ which forms a basis of the topology on $M$. Let $F(B_n)$ be given by Step 1. We have $m(E\backslash F)=0$ where $F=\bigcap_{n\ge1}F(B_n)$. Now for each $x\in F$, $\mathcal{O}(x)\cap B_n\neq\emptyset$ for each $n\ge1$. So the orbit $\mathcal{O}(x)$ is dense in $M$ for each point $x\in F$. Equivalently we see $\mu$-a.e. $x\in M$ has a dense orbit. So the $acip$ $\mu$ is weakly ergodic. This completes the proof.
A natural question is, if $f\in\mathrm{SPH}^r(M)$ is essentially accessible and preserves some $acip$ $\mu$, is $\mu$ an ergodic measure? This is related to the uniqueness of $acip$. Clearly uniqueness of $acip$ forces the ergodicity of $acip$. On the other hand, let us assume that exists two $acip$’s: $\mu=\phi m$ and $\nu=\psi m$. Let $E=\{x:\phi(x)>0\}$ and $F=\{x:\psi(x)>0\}$. If $m(E\triangle F)>0$ we can further assume $E$ and $F$ are disjoint. Proposition \[biess\] implies that both $E$ and $F$ are bi-essentially saturated (and nontrivial). In particular none of them can be essentially bi-saturated.
We do not know whether such example can exist, or a bi-essentially saturated set is automatically essentially bi-saturated. A sufficient condition for this property is center bunching. From now on we assume $r=2$ for simplicity.
\[centerb\] A strongly partially hyperbolic diffeomorphism $f$ is [*center bunched*]{} if the functions $\nu$, $\tilde{\nu}$ and $\gamma$, $\tilde{\gamma}$ given in can be chosen so that: $\nu<\gamma\tilde{\gamma}$ and $\tilde{\nu}<\gamma\tilde{\gamma}$.
Let $f\in \mathrm{SPH}^2(M)$ be center bunched. Then every measurable bi-essentially saturated subset is essentially bi-saturated.
\[equi\] Let $f\in \mathrm{SPH}^2(M)$ be essentially accessible and center bunched. If there exists some $acip$, then the $acip$ must be equivalent to the volume.
Let $\mu$ be an $acip$ and $\phi$ be the density of $\mu$ with respect to $m$. We showed that $E=\{x\in M:\phi(x)>0\}$ is bi-essentially saturated. Center bunching implies that $E$ is also essentially bi-saturated. Since $f$ is essentially accessible and $m(E)>0$, $m(E)=1$ and hence $\mu$ is equivalent to the volume $m$.
In [@BW], a map $f$ is said to be [*volume preserving*]{} if $f$ preserves some invariant measure $\mu$ that is equivalent to the volume. They proved that if $f\in \mathrm{SPH}^2(M)$ is essentially accessible, center bunched and preserves some $\mu$ equivalent to the volume, then the measure $\mu$ is ergodic (and $Kolmogorov$). It is well known that ergodic measures either coincide or absolutely singular with respect to each other. So by Corollary \[equi\], if $f\in \mathrm{SPH}^2(M)$ is essentially accessible and center bunched, then either $f$ is volume preserving in the board sense, or there exists no $acip$ at all.
Followed by Corollary \[equi\] we get that the density $\phi=\frac{d\mu}{dm}$ of an $acip$ is positive a.e. on $M$. Now we use [*Cohomologous Theory*]{} developed in [@W] to show the smoothness of the density of $\mu$. Namely let $\psi:M\to\mathbb{R}$ be a potential on $M$ and consider the cohomologous equation on $M$: $$\label{coho2}
\psi=\Psi\circ f-\Psi.$$
\[smooth\] Let $f\in \mathrm{SPH}^2(M)$ be accessible, center bunched, and volume-preserving. Let $\psi:M\to\mathbb{R}$ be a Hölder continuous potential. If there exists a measurable solution $\Psi$ such that holds for a.e. $x\in M$, then there is a Hölder continuous solution $\Phi$ of with $\Phi=\Psi$ a.e. $x\in M$.
Let $\psi=-\log J_f$ and $\Psi=\log\phi$. Now $\psi$ is a $C^1$ function and $\Psi$ is a well defined measurable function. Corollary \[equi\] implies that $\Psi$ is a measurable solution of the cohomologous equation . Then applying Proposition \[smooth\] we get a Hölder continuous solution $\Phi$ of which coincides with $\Psi$ a.e.. It is evident that $\mu=e^{\Phi}m$ and the density $e^{\Phi}$ is bounded and bounded away from zero on $M$. Such a measure $\mu$ is called a [*smooth measure*]{}. So we have
\[dich\] Let $f\in \mathrm{SPH}^2(M)$ be accessible and center bunched. If there exists some $acip$, then the $acip$ must have a Hölder continuous density with respect to the volume of $M$ which is also bounded and bounded away from $0$. In words, either $f$ preserves a smooth measure or there is no $acip$ of $f$.
In particular center bunching holds whenever $E^c$ is one-dimensional. As a corollary, we obtain:
Let $f\in \mathrm{SPH}^2(M)$ be accessible and $\dim(E^c)=1$. Then either $f$ preserves a smooth measure or there is no $acip$ of $f$.
Let $\mathrm{CB}^2(M)\subset\mathrm{SPH}^2(M)$ be the collection of $C^2$ strongly partially hyperbolic diffeomorphisms that are center bunched. Clearly $\mathrm{CB}^2(M)$ forms an open subset of $\mathrm{SPH}^2(M)$. Applying Theorem \[dich\] and the result in [@DW] we have
\[noacip\] The set of maps that admit no $acip$ contains a $C^1$ open and dense subset of $\mathrm{CB}^2(M)$. In particular the set of maps that admit no $acip$ contains a $C^1$ open and dense subset of $C^2$ strongly partially hyperbolic diffeomorphisms with $\dim(E^c)=1$.
The main obstruction for $C^2$ density in Theorem \[noacip\] is that we do not know whether stable accessibility is $C^2$ dense in $\mathrm{SPH}^2(M)$.
Dolgopyat and Wilkinson proved in [@DW] that there is a $C^1$ dense subset of stable accessible diffeomorphisms in $\mathrm{SPH}^2(M)$ (also $C^1$ dense in $\mathrm{CB}^2(M)$). Starting with arbitrary $f\in\mathrm{CB}^2(M)$, we first perturb it to a stable accessible one, say $f_1$. By $C^1$ closing lemma, there exists $f_2\in\mathrm{CB}^2(M)$ close to $f_1$ that has some periodic point. We can assume that $f_2$ is also stable accessible since we can make it arbitrary close to $f_1$. By Franks’ Lemma [@Fr] we can assume that the periodic point $p$ is hyperbolic with period $k$ and the Jacobian of $Df_2^k:T_xM \to T_xM$ has absolute value different from $1$. These properties hold robustly for all maps in a small neighborhood $\mathcal{U}\subset\mathrm{CB}^2(M)$ of $f_2$.
Let $g\in\mathcal{U}$ and $p_g$ be the continuation of $p$. By the choice of $\mathcal{U}$, we know that $g$ is accessible and center bunched. If $g$ admits some $acip$ $\mu$, then by Theorem \[dich\] $\mu=\phi m$ for some Hölder continuous function $\phi$ which is bounded and bounded away from zero. By Equation we have $\phi(p_g)=J_{g^k}(p_g)\phi(g^k p_g)=J_{g^k}(p_g)\phi(p_g)$. This is impossible sice $|J_{g^k}(p_g)|\neq1$ and $\phi(p_g)\neq0$. So each $g\in\mathcal{U}$ admits no $acip$. Hence there exists an open set $\mathcal{U}$ close to $f$ in which each map admits no $acip$. This finishes the proof.
It is well known that among $C^2$ Anosov diffeomorphisms the ones that admits no $acip$ are open and dense, see [@B1 Corollary 4.15]. This is due to the fact that there are many periodic points for every Anosov diffeomorphisms. Recently Avila and Bochi [@AB] proved that a $C^1$-generic map in $C^1(M,M)$ has no $acip$. In particular a $C^1$-generic map in $\mathrm{Diff}^1(M)$ has no $acip$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Wenxiang Sun, Lan Wen, Amie Wilkinson and Zhihong Xia for discussions and suggestions. We are grateful to Shaobo Gan for useful comments and corrections to the original manuscript. Especially we thank Amie Wilkinson for explaining her results in [@W].
[99]{}
(MR2415085) J. Alves and V. Pinheiro, *Topological structure of (partially) hyperbolic sets with positive volume*, Trans. Amer. Math. Soc. **360** (2008), no. 10, 5551–5569.
(MR2267725) A. Avila and J. Bochi, *A generic $C^1$ map has no absolutely continuous invariant measure*, Nonlinearity **19** (2006), 2717–2725.
(MR2090775) J. Bochi and M. Viana, *Lyapunov exponents: how frequently are dynamical systems hyperbolic?*, in Modern dynamical systems and applications, 271–297, Cambridge Univ. Press, Cambridge, 2004.
(MR0380890) R. Bowen, *A horseshoe with positive measure*, Invent. Math. **29** (1975), 203–204.
(MR0442989) R. Bowen, *Equilibrium states and the ergodic theory of Axiom A diffeomorphisms*, Lecture Notes in Mathematics, Vol. **470**. Springer-Verlag, Berlin-New York, 1975.
M. Brin, *Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature*, Functional Anal. Appl. **9** (1975), 8–16.
(MR1963683) M. Brin and G. Stuck, *Introduction to dynamical systems*, Cambridge University Press, 2002.
K. Burns and A. Wilkinson, *On the ergodicity of partially hyperbolic systems*, Annals of Math. **171** (2010) 451–489.
(MR1933439) K. Burns, D. Dolgopyat and Ya. Pesin, *Partial hyperbolicity, Lyapunov exponents and stable ergodicity*, J. Statist. Phys. **108** (2002), no. 5-6, 927–942.
(MR2039999) D. Dolgopyat and A Wilkinson, *Stable accessibility is $C^1$ dense*, Geometric methods in dynamics. II. Astérisque No. **287** (2003), xvii, 33–60.
T. Fisher, “On the structure of hyperbolic sets", Ph.D thesis, Northwestern University, 2004.
(MR0283812) J. Franks, *Necessary conditions for stability of diffeomorphisms*, Trans. A.M.S. **158** (1971), 301–308.
(MR2371598) N. Gourmelon, *Adapted metrics for dominated splittings*, Ergod. Th. Dynam. Sys. **27** (2007), 1839–1849.
(MR0501173) M. Hirsch, C. Pugh and M. Shub, *Invariant manifolds*, Lecture Notes in Mathematics, Vol. **583**, Springer-Verlag, Berlin-New York, 1977.
(MR1808220) V. Niţică and A. Török, *An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one*, Topology **40** (2001), no. 2, 259–278.
(MR1750453) C. Pugh and M. Shub, *Stable ergodicity and julienne quasiconformality*, J. Eur. Math. Soc. **2** (2000), no. **1**, 1–52.
(MR0590160) C. Robinson, L. S. Young, *Nonabsolutely continuous foliations for an Anosov diffeomorphism*, Invent. Math. **61** (1980), no. 2, 159–176.
(MR2388690) F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, *A survey of partially hyperbolic dynamics*, Partially hyperbolic dynamics, laminations, and Teichmüller flow, Fields Inst. Commun., vol. **51**, Amer. Math. Soc., 2007, 35–87.
A. Wilkinson, *The cohomological equation for partially hyperbolic diffeomorphisms*, arXiv:[0809.4862]{}.
(MR2220749) Z. Xia, *Hyperbolic invariant sets with positive measures*, Discrete Contin. Dyn. Syst. **15** (2006), no. 3, 811–818.
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---
abstract: 'We prove that every finite set of homothetic copies of a given compact and convex body in the plane can be colored with four colors so that any point covered by at least two copies is covered by two copies with distinct colors. This generalizes a previous result from Smorodinsky (SIAM J. Disc. Math. 2007). Then we show that for any $k\geq 2$, every three-dimensional hypergraph can be colored with $6(k-1)$ colors so that every hyperedge $e$ contains $\min\{ |e|,k \}$ vertices with mutually distinct colors. This refines a previous result from Aloupis [*et al.*]{} (Disc. & Comp. Geom. 2009). As corollaries, we improve on previous results for conflict-free coloring, $k$-strong conflict-free coloring, and choosability. Proofs of the upper bounds are constructive and yield simple, polynomial-time algorithms.'
author:
- Jean Cardinal
- Matias Korman
bibliography:
- 'fourCol.bib'
title: 'Coloring Planar Homothets and Three-Dimensional Hypergraphs'
---
Introduction
============
The well-known graph coloring problem has several natural generalizations to hypergraphs. A rich literature exists on these topics; in particular, the two-colorability of hypergraphs (also known as property B), has been studied since the sixties. In this paper, we concentrate on coloring geometric hypergraphs, defined by simple objects in the plane. Those hypergraphs serve as models for wireless sensor networks, and associated coloring problems have been investigated recently. This includes conflict-free colorings [@shakharcf; @HPS05], and covering decomposition problems [@pachtoth; @pachindecomp; @GV09].
Smorodinsky [@Smo07] investigated the chromatic number of geometric hypergraphs, defined as the minimum number of colors required to make every hyperedge non-monochromatic. He considered hypergraphs induced by a collection $S$ of regions in the plane, whose vertex set is $S$, and the hyperedges are all subsets $S'\subseteq S$ for which there exists a point $p$ such that $S'= \{ R \in S: p\in R \}$. He proved the following result.
\[thm\_4col\]
- Any hypergraph induced by a family of n simple Jordan regions in the plane such that the union complexity of any $m$ of them is given by $u(m)$ and $u(m)/m$ is non-decreasing is $O(u(n)/n)$-colorable so that no hyperedge is monochromatic. In particular, any finite family of pseudodisks can be colored with $O(1)$ colors.
- Any hypergraph induced by a finite family of disks is 4-colorable
Later, Aloupis, [*et al.*]{} [@ACCLS09] considered the quantity $c(k)$, defined as the minimum number of colors required to color a given hypergraph, such that every hyperedge of size $r$ has at least $\min \{ r, k\}$ vertices with distinct colors. For hypergraphs induced by a collection of regions in the plane, such that no point is covered more than $k$ times (a $k$-fold packing), this number corresponds to the minimum number of (1-fold) packings into which we can decompose this collection. It generalizes the usual chromatic number, equal to $c(2)$. They proved the following.
\[thm\_ck\] Any finite family of pseudodisks in the plane can be colored with $24k+1$ colors in a way that any point covered by $r$ pseudodisks is covered by $\min \{r, k\}$ pseudodisks with distinct colors.
#### Our results.
We show in Section \[sec\_dual\] that the second statement of Theorem \[thm\_4col\] actually holds for homothets of any compact and convex body in the plane. The proof uses a lifting transformation that allows us to identify a planar graph, such that every hyperedge of the initial hypergraph contains an edge of the graph. The result then follows from the Four Color Theorem.
We actually give two definitions of this graph: one is based on a weighted Voronoi diagram construction, while the other relates to Schnyder’s characterization of planar graphs. Schnyder showed that a graph is planar if and only if its vertex-edge incidence poset has dimension at most $3$ [@schnyder]. In Section \[sec\_3D\], we show that the chromatic number $c(k)$ of three-dimensional hypergraphs is at most $6(k-1)$. This improves the constant of Theorem \[thm\_ck\] for this special case, which includes hypergraphs induced by homothets of a triangle.
In Section \[sec\_lb\], we give a lower bound for all the above problems. Finally, in Section \[sec\_appl\], we give some corollaries of these results involving other types of colorings, namely conflict-free and $k$-strong conflict-free colorings, and choosability.
#### Definitions.
We consider hypergraphs defined by [*ranges*]{}, which are compact and convex bodies ${Q}\subset {\mathbb{R}}^d$ containing the origin. The [*scaling*]{} of ${Q}$ by a factor $ \lambda\in{\mathbb{R}}$ is the set $\{\lambda x : x\in {Q}\}$. Note that, the scaling of ${Q}$ with $\lambda=-1$ is the reflection of ${Q}$ around the origin. The [*translate*]{} of ${Q}$ by a vector $t\in{\mathbb{R}}^d$ is the set $\{x+t : x\in {Q}\}$. The [*homothet*]{} of ${Q}$ of [*center*]{} $t$ and [*scaling*]{} $\lambda$ is the set $\{\lambda x + t : x\in {Q}\}$ and is denoted by ${Q}(t,\lambda)$. Given a finite collection $S$ of points in ${\mathbb{R}}^d$, the [*primal hypergraph*]{} defined by these points and a range ${Q}$ has $S$ as vertex set, and $\{ S\cap {Q}' : {Q}'\mathrm{\ homothet\ of\ }{Q}\}$ as hyperedge set. Similarly, the [*dual hypergraph*]{} defined by a finite set $S$ of homothets of ${Q}$ has $S$ as vertex set, and the hyperedges are all subsets $S'\subseteq S$ for which there exists a point $p\in{\mathbb{R}}^d$ such that $S'= \{ R \in S: p\in R \}$ (i.e., the set of ranges of $S$ that contain $p$). While we give these definitions for an arbitrary dimension $d$, we will be mostly concerned by the case $d=2$.
For a given range ${Q}$, the chromatic number $c_{{Q}} (k)$ is the minimum number $c$ such that every primal hypergraph (induced by a set of points) can be colored with $c$ colors, so that every hyperedge of size $r$ contains $\min \{r, k\}$ vertices with mutually distinct colors. Similarly, the chromatic number $\bar{c}_{{Q}} (k)$ is the smallest number $c$ such that every dual hypergraph (induced by a set of homothets of ${Q}$) can be $c$-colored so that every hyperedge of size $r$ contains $\min \{r, k\}$ vertices with mutually distinct colors. In what follows, we refer to these two coloring problems as [*primal*]{} and [*dual*]{}, respectively. Such colorings are called [*polychromatic*]{}[^1].
Coloring Primal Hypergraphs {#sec_primal}
===========================
As a warm-up, we consider the primal version of the problem for $k=2$. Given a set of points $S\subset\mathbb{R}^2$ and a two-dimensional range ${Q}$, the [*Delaunay graph*]{} of $S$ induced by ${Q}$ is the graph $G_Q(S)=(S,E)$ with $S$ as vertex set [@fortune]. For any two points $p,q\in S$, $pq\in E$ if and only if there exists a homothet ${Q'}$ of ${Q}$ such that ${Q'}\cap S =\{p,q\}$. Note that, the Delaunay graph induced by disks in the plane corresponds to the ordinary Delaunay triangulation, which is planar. In fact, planarity holds for many ranges.
[@BCCS08c; @sarioz]\[lem\_planar\] For any convex range ${Q}\subseteq {\mathbb{R}}^2$ and set of points $S$, $G_Q(S)$ is planar.
Previously published versions of this result required that the points of $S$ are in general position (that is, no four points of $S$ are on the boundary of a range). The generalization to any point set was done by Bose [*et al.*]{} [@BCCS08c]. Whenever a homothet ${Q'}$ contains more than $3$ points on its boundary, the edges $uv,uw$ and $vw$ are added to $G_Q(S)$, where $u,v$, and $w$ are the three lexicographically smallest points of $S\cap {Q'}$. With this definition, they showed that planarity holds for any compact and convex range. The compactness requirement was afterwards removed by Sarioz [@sarioz].
![Proof of Lemma \[lem\_edge\]: given a homothet ${Q'}$ (light grey), we shrink it until further shrinking will have less than two points (dark grey). Afterwards we keep shrinking while remaining tangent to a point $q$ on the boundary until the point in the interior of the range (if any) reaches the boundary. The resulting range ${Q''}$ is depicted in white. []{data-label="fig_shrink"}](fig_shrink){width="40.00000%"}
\[lem\_edge\] For any homothet ${Q'}$ containing two or more points of $S$, there exist $p,q\in S \cap {Q'}$ such that $pq\in E$.
Let ${Q'}$ be a homothet of center $c_0$ and scaling $\lambda_0$ that contains two or more points of $S$. We shrink it continuously keeping the same center; let $\lambda_{\min}$ be the smallest scaling such that ${Q}(c_0,\lambda_{\min})$ has two (or more) points of $S$. If ${Q}(c_0, \lambda_{\min})$ contains exactly two points $p,q\in S$, we have $pq\in E$ by definition of $G_Q(S)$. However, we might have some kind of degeneracy in which ${Q}(c_0,\lambda_{\min})$ contains three (or more) points of $S$. Observe that this can only happen if there are two or more points on the boundary and possibly an interior point.
First consider the case in which there exists a point $p\in S$ in the interior of $\lambda_{\min}$. Pick any point $q\in S$ on the boundary of ${Q}(c_0,\lambda_{\min})$ and shrink the homothet remaining tangent to $q$ until $p$ reaches the boundary (see Figure \[fig\_shrink\]). After this shrinking process, both $p$ and $q$ will be on the boundary of the new homothet. Moreover, any other point that was previously in ${Q}(c_0,\lambda_{\min})$ either remains on the boundary or is not in the range anymore.
That is, we can always shrink a range ${Q'}$ to another range ${Q''}\subseteq {Q'}$ that contains two or more points on its boundary and none in the interior. Hence, by the result of [@BCCS08c] we know that there will be an edge between the two lexicographically smallest points of $S \cap {Q''}$.
\[theo\_primal\] For any two-dimensional range ${Q}$, we have $c_{Q}(2)\leq4$.
By Lemma \[lem\_planar\], $G_Q(S)$ is planar, hence 4-colorable. By Lemma \[lem\_edge\], any homothet ${Q'}$ containing two or more points of $S$ must contain $p,q\in S\cap {Q'}$ such that $pq\in E$. In particular, these points cannot have the same color assigned, hence ${Q'}$ cannot be monochromatic.
The proof yields an $O(n^2)$-time algorithm. The bound of Theorem \[theo\_primal\] is tight for a wide class of ranges (see Section \[sec\_lb\]).
Coloring Dual Hypergraphs {#sec_dual}
=========================
In this section we describe a similar approach for the dual variant of the problem in the plane. Recall that in the dual problem, we are given a set $S$ of compact and convex homothets of ${Q}$, and we are interested in coloring the elements of $S$ so that any point of the plane covered at least twice is covered by two homothets of different colors. For simplicity, we first suppose that no range of $S$ is contained in another; we will show how to remove this assumption afterwards.
In order to solve this problem, we lift the two-dimensional ranges to a three dimensional space. We map the homothet of ${Q'}$ of center $(x,y)$ and scaling $\lambda$ to the three dimensional point $\rho ({Q'})=(x, y, \lambda)\in {\mathbb{R}}^3$. Given a set $S$ of homothets of ${Q}$, we define $\rho (S) = \{\rho ({Q'}) : {Q'}\in S\}$ as the set containing the images of the ranges in $S$. For any point $p=(x,y,d)$, we associate the three dimensional range $\pi(p)$ as the cone with apex at $(x,y,d)$ such that the intersection with the horizontal plane of height $z$ is $Q(0, z-d)$ (if $z\geq d$) or empty (if $z<d$), where ${Q}^*={Q}(0,-1)$ is the reflection of ${Q}$ about its center. Note that the cone $\pi (p)$ so defined is convex (see Figure \[fig\_trans\]). We define the [*downward cone*]{} $\pi^* (p)$ as the centrally symmetric image of $\pi(p)$ through $p$. By symmetry, we observe the following:
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\[lem\_dual\] For any $p,q\in{\mathbb{R}}^3$, we have $q\in\pi (p)\Leftrightarrow p\in\pi^*(q)$. Moreover, for any point $(x, y)\in {\mathbb{R}}^2$ and range ${Q'}$, $(x, y)\in {Q'}\Leftrightarrow \rho({Q'}) \in \pi((x, y, 0))$.
It follows that any coloring of $\rho(S)$ with respect to the conic ranges $\pi$ is a valid coloring of $S$. Let $G_\pi(\rho(S))$ be the Delaunay graph in ${\mathbb{R}}^3$ with cones as ranges. That is, the vertex set of $G_\pi(\rho(S))$ is $S$ and two ranges ${Q'},{Q''}$ of $S$ are adjacent if and only if there exists a point $p\in{\mathbb{R}}^3$ such that $\pi (p) \cap \rho (S) = \{ \rho ({Q'}), \rho ({Q''})\}$. We claim that $G_\pi(\rho(S))$ satisfies properties similar to those of Lemmas \[lem\_planar\] and \[lem\_edge\]. In order to prove so, we first introduce some inclusion properties.
\[lem\_inclus\] For any $p\in{\mathbb{R}}^3$, $q\in \pi (p)$ and $m$ on the line segment $pq$, we have $\pi (q) \subseteq \pi (p)$ and $q \in \pi (m)$.
Observe that the projections of the cones $\pi (p)$ and $\pi (q)$ on any vertical plane (i.e., any plane of equation $ax+by+c=0$) are two-dimensional cones; that is, the set of points above two halflines with a common origin. Moreover, the slope of the halflines only depends on $a,b$ and ${Q}$. Let $z_q$ be the $z$-coordinate of $q$, and consider the intersections of the cones $\pi (p)$ and $\pi (q)$ with a horizontal plane $\Pi$ of $z$-coordinate $t \geq z_q$. We get two homothets of ${Q}^*$, say ${Q}^*_p$ and ${Q}^*_q$. We have to show that ${Q}^*_q \subseteq {Q}^*_p$ for any $t$.
Suppose otherwise. There exists a vertical plane $\Pi'$ for which the projection of ${Q}^*_q$ on $\Pi'$ is not included in the projection of ${Q}^*_p$. To see this, we can find a common tangent to ${Q}^*_p$ and ${Q}^*_q$ in $\Pi$, slightly rotate it so that it is tangent to ${Q}^*_q$ only, then pick a plane that is orthogonal to that line. But the projections of $\pi (p)$ and $\pi (q)$ on $\Pi'$ are two cones with parallel bounding halflines, thus the projection of the apex of $\pi (q)$ cannot be in that of $\pi (p)$, a contradiction.
The proof of the second claim is similar. We know that $q\in \pi (p)$, hence from Lemma \[lem\_dual\], $p\in\pi^* (q)$. Now from the convexity of $\pi^* (q)$, we have that $m\in \pi^*(q)$. Using again Lemma \[lem\_dual\], we obtain $q\in\pi (m)$.
\[lem\_dualplanar\] The graph $G_\pi(\rho(S))$ is planar.
By definition of $E(S)$, we know that for every edge ${Q'}{Q''}\in E$ there exists $p\in{\mathbb{R}}^3$ such that $\pi (p) \cap \rho (S) = \{ \rho ({Q'}), \rho ({Q''})\}$. We draw the edge ${Q'}{Q''}$ as the projection (on the horizontal plane $z=0$) of the two line segments connecting respectively $\rho ({Q'})$ and $\rho ({Q''})$ to $p$.
First note that crossings involving two edges with a common endpoint can be eliminated by rerouting the two polygonal lines at their intersection point. Thus it suffices to show that this embedding has no crossing involving vertex-disjoint edges. Consider two edges $uu'$ and $vv'$, and their corresponding witness cones $\pi_1\ni u, u'$ and $\pi_2\ni v, v'$. By definition of witness, each cone must contain exactly two points. In particular, we have $u\not\in\pi_2$ and $v\not\in\pi_1$.
Suppose that the projections of the segments connecting $u$ with the apex of $\pi_1$ and $v$ with the apex of $\pi_2$ cross at a point $x$ (other than the endpoints). Consider the vertical line $\ell$ that passes through $x$: by construction, this line must intersect with both segments at points $a$ and $b$, respectively. Without loss of generality we assume that $a$ has lower $z$ coordinate than $b$ (see Figure \[fig\_planar\]).
From the convexity of $\pi_1$, we have $a\in \pi_1$. From Lemma \[lem\_inclus\], we have $v\in \pi (b)$, $b\in \pi (a)$, and $\pi (a)\subseteq \pi (b)$. In particular, we have $v\in \pi (b) \subseteq \pi (a) \subseteq\pi_1$, which contradicts $v\not\in \pi_1$.
Alternative construction via weighted Voronoi diagrams {#sec:wVD}
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We introduce an alternative definition of $G_\pi(\rho(S))$ so as to prove its planarity. For any point $p$, we define its distance to $q$ as $d(p,q)=\min\{\lambda \geq 0 | q\in {Q}(p,\lambda)\}$. That is, the smallest possible scaling so that a range of center $p$ contains $q$. This distance is called the [*convex distance function*]{} from $p$ to $q$ (with respect to ${Q}$). Given a set $S=\{{Q}_1, \ldots, {Q}_n\}$ of homothets of ${Q}$, we construct an additively weighted Voronoi diagram $V_{Q}(S)$ with respect to the convex distance function [@fortune]. Let $c_i$ and $\lambda_i$ be the center and scaling of ${Q}_i$. Then $V_{Q}(S)$ has $\{c_1,\ldots c_n\}$ as the set of sites, and each site $c_i$ is given the weight $-\lambda_i$. The additively weighted Voronoi diagram for this set of sites has a cell for each site $c_i$, defined as the locus of the points $p$ of the plane whose weighted distance $d(p,c_i)-\lambda_i$ to $c_i$ is the smallest among all sites. The dual graph for this Voronoi diagram has an edge between any two sites whose cells share a boundary.
In the following we show that the dual of $V_{Q}(S)$ is $G_\pi(\rho(S))$. Let $p = (x,y)\in{\mathbb{R}}^2$ be any point covered by one or more ranges of $S$. We denote by $(p,z)$ the point of ${\mathbb{R}}^3$ of coordinates $(x,y,z)$, for any $z\in{\mathbb{R}}$. From Lemma \[lem\_dual\], the points of $\rho (S)$ contained in $\pi(p,0)$ are exactly the ranges of $S$ that contain $p$. We translate the cone $\pi (p,0)$ vertically upward; in this lifting process, the points of $\rho(S)\cap \pi(p,0)$ will leave the cone. For any homothet ${Q'}\in S$ such that $p\in {Q'}$, we define $z_{{Q'}}(p)$ as the height in which $\rho({Q'})$ is on the boundary of the cone $\pi(p,z_{{Q'}}(p))$.
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\[lem\_height\] For any homothet ${Q'}$ of center $c$ and scaling $\lambda$ and $p\in{Q'}$, $z_{{Q'}}(p)=\lambda - d(c,p)$.
Consider the cone $\pi(p,z_{{Q'}}(p))$ and the halfplane $z=z_{{Q'}}(p)$. By Lemma \[lem\_dual\], the fact that $\rho({Q'})$ is on the boundary of $\pi(p,z_{{Q'}}(p))$ is equivalent to $(p,z_{{Q'}}(p))$ being on the boundary of the downwards cone $\pi^*(\rho({Q'}))=(c,\lambda)$. Observe that the distance between points $(c,z_{{Q'}}(p))$ and $(p,z_{{Q'}}(p))$ is exactly $d(c,p)$. Consider the plane that passes through the points $(c,0)$, $(p,0)$, and $(c,\lambda)$: we have two similar triangles whose bases have lengths $\lambda$ and $d(c,p)$, respectively. Since the height of the large triangle is $\lambda$, we conclude that the height of the smaller one must be $d(c,p)$ (see Figure \[fig\_vorono\]). That is, the difference in the $z$ coordinates between the points $(c,z_{{Q'}}(p))$ and $(c, \lambda)$ is $d(c,p)$. Since, by definition of $\rho$, the difference in $z$ coordinates between $(c,0)$ and $\rho({Q'})$ is exactly $\lambda$, we obtain the equality $\lambda=z_{{Q'}}(p)+d(c,p)$ and the Lemma follows.
This shows the duality between the weighted Voronoi diagram and the graph $G_\pi(\rho(S))$: let $p\in {\mathbb{R}}^2$ be any point in the plane covered by at least one range of ${Q'}$. Consider the cone $\pi(p,0)$ and lift it continuously upward. The last point of $\rho(S)\cap \pi(p,0)$ to leave the cone will be one with highest $z_{{Q'}}(p)$. By Lemma \[lem\_height\], it will be the homothet ${Q'}$ of center $c'$ and scaling $\lambda'$ that has the [*smallest*]{} $d(c',p)-\lambda'$. Observe that this is exactly how we defined the weights of the sites, hence ${Q'}$ being the last range in the cone is equivalent to $c'$ being the [*closest*]{} site of $p$ in $V_{Q'}(S)$. This can be interpreted as shrinking simultaneously all ranges until $p$ is only covered by its closest homothet ${Q'}$. This shrinking process is simulated in our construction through the $z$ coordinate.
\[lem\_dualVorDel\] The dual graph of $V_{Q'}(S)$ is $G_\pi(\rho(S))$.
Let $p=(x,y)\in {\mathbb{R}}^2$ be a point on a bisector of $V_{Q}(S)$ between sites $c_1$ and $c_2$ (corresponding to ranges ${Q}_1$ and ${Q}_2$ of scaling $\lambda_1$ and $\lambda_2$, respectively). By definition, we have that $d(c_1,p)-\lambda_1=d(c_2,p)-\lambda_2$ and $d(c',p)-\lambda' > d(c_1,p)-\lambda_1$ for all other homothets ${Q'}\in S$ of center $c'$ and scaling $\lambda'$.
Let $w_{\min}=d(c_1,p)-\lambda_1$ and consider the cone $\pi(x,y,-w_{\min})$: by Lemma \[lem\_height\], both $\rho({Q}_1)$ and $\rho({Q}_2)$ are on the boundary. Moreover, any other homothet ${Q'}\in S$ will satisfy ${Q'}\not\in \pi(x,y, -w_{\min})$ (since other ranges have larger weighted distance, which is equivalent to having smaller $z_{{Q'}}(p)$). That is, the cone $\pi(x,y,-w({Q}_1))$ contains exactly points $\rho({Q}_1)$ and $\rho({Q}_2)$. Moreover, no other point of $\rho(S)$ will be in the cone, hence ${Q}_1{Q}_2\in E$.
The other inclusion is shown analogously: let ${Q}_1{Q}_2$ be two ranges such that ${Q}_1{Q}_2\in E$. Let $(x,y,z)\in {\mathbb{R}}^3$ be the apex of the minimal cone (with respect to inclusions) such that $\pi(x,y,z)\cap \rho(S)=\{\rho({Q}_1),\rho({Q}_2)\}$. Since $\pi(x,y,z)$ is minimal, both $\rho({Q}_1)$ and $\rho({Q}_2)$ must be on the boundary of the cone. In particular, $z_{{Q}_1}(x,y)=z_{{Q}_2}(x,y)$ and other ranges satisfy $z_{{Q'}}(x,y)<z_{{Q}_1}(x,y)$ (for all other ${Q'}\in S$). Using again Lemma \[lem\_height\], this is equivalent to the fact that $p=(x,y)$ is equidistant to sites $c_1$, $c_2$, and all other sites have strictly larger distance.
#### Coloring.
As an application of the above construction, we show how to solve the dual coloring problem. By Lemma \[lem\_dualplanar\], we already know that $G_\pi(\rho(S))$ is 4-colorable. For any point $p\in{\mathbb{R}}^2$, let $S_p$ be the set of ranges containing $p$ (i.e., $S_p=\{{Q'}\in S : p\in {Q'}\}$).
\[lem\_enough\] For any $p\in {\mathbb{R}}^2$ such that $|S_p| \geq 2$, there exist ${Q}_1,{Q}_2 \in S_p$ such that ${Q}_1{Q}_2\in E(S)$.
From the second property of Lemma \[lem\_dual\], the number of points of $\rho (S)$ contained in the cone $\pi(p,0)$ is the number of ranges of $S$ containing $p$. The proof is now analogous to Lemma \[lem\_edge\], where the shrinking operation is replaced by a vertical lifting of the cone. Let $z_0\geq 0$ be the largest value such that the cone $\pi(p,z_0)$ has two or more points of $\rho(S)$. If $\pi(p,z_0)$ contains exactly two points we are done, hence it only remains to treat the degeneracies. Remember that in such a case, there must be at least two points on the boundary of $\pi(p,z_0)$ (and possibly a point in its interior).
If there is a point $\rho({Q}_1)$ in the interior, we select a second point $\rho({Q}_2)$ on the boundary of $\pi(p,z_0)$ and translate the apex of the cone towards $\rho({Q}_2)$. Note that when the apex is located at $\rho({Q}_2)$, the point $\rho({Q}_1)$ cannot be in the cone (since it would imply that ${Q}_1 \subseteq {Q}_2$, and we assumed otherwise). Thus, at some point in the translation $\rho({Q}_1)$ reaches the boundary of the cone. During this translation process, points that were on the boundary of $\pi(p,z_0)$ have either remained on the boundary or have left the cone. In either case, we obtain a cone $\pi(p',z')\subseteq \pi(z,0)$ with two (or more) vertices of $\rho(S)$ on its boundary and none in its interior.
If it contains exactly two points of $\rho(S)$ we are done. Otherwise, by duality of Lemma \[lem\_dualVorDel\], $p'$ is a vertex of the weighted Voronoi diagram. We pick a point $p''$ in an edge $e$ of the Voronoi diagram incident to $p'$. Since $p'$ is in an edge of the Voronoi diagram, it is equidistant to two ranges ${Q}_1,{Q}_2 \in S_p$. Hence, by Lemma \[lem\_dualVorDel\], when we do the lifting operation on $p'$ we will obtain a cone that exactly contains ${Q}_1,{Q}_2 \in S_p$ on its boundary and no other point in its interior. It suffices to show the claim for points $p\in{\mathbb{R}}^2$ that are $3$ or more deep. We translate the cone vertically upward; points of $\rho(S)$ will be leaving the cone. If at some height exactly two points $\rho({Q}_1),\rho({Q}_2)$ remain, we have ${Q}_1{Q}_2\in E(S)$. Otherwise, we have that, at some height $z_0\geq 0$, the cone $\pi(p,z_0)$ has three or more points of $\rho(S)$ and any vertical lifting (even by an arbitrarily small amount) gives a cone with less than two points. If there is a point of $\rho({Q}_1)\in \rho(S)$ in the interior of $\pi(p,z_0)$ we select a second point $\rho({Q}_2)$ on the boundary of $\pi(p,z_0)$ and translate the apex of the cone towards $\rho({Q}_2)$. During this translation process, point $\rho({Q}_2)$ will always stay in the cone. Moreover, for a sufficiently small translation $\rho({Q}_1)$ will also be in the interior of the cone. Observe that the translated cone is be tangent to $\pi(p,z_0)$ only at the line that ray emanating from $(p,z_0)$ and passing through $\rho({Q}_2)$. Hence, no other point $\rho({Q'})$ can be in the ray (since then we would have a point of $\rho(S)$ dominating another, which is equivalent to a range being included in another range). That is, after a small translation, we have that only ${Q}_1$ and ${Q}_2$ will be in the cone. We now study the case in which there is no interior point. That is, the cone $\pi(p,z_0)$ has three or more points of $\rho(S)\cap \pi(p,z_0)$ on its boundary and none in the interior. By Lemma \[lem\_dualVorDel\], this case corresponds to the case in which $p$ is a vertex of the weighted Voronoi diagram. Hence, $p$ is equidistant to ranges ${Q}_1, {Q}_2, \ldots {Q}_k$ (for some $k\geq 3$). Let $e$ be any edge of the Voronoi diagram incident to $p$, and let $p'$ be any point of $e$. Without loss of generality, we can assume that $e$ is the bisector between ranges ${Q}_1$ and ${Q}_2$. By Lemma \[lem\_dualVorDel\], when we do the lifting operation on $p'$ we will obtain a cone that only contains $\rho({Q}_1)$ and $\rho({Q}_2)$ on its boundary and no other point of $\rho(S)$. As a result, we have ${Q}_1{Q}_2\in E$ completing the proof of the Lemma.
\[theo\_dual\] $\bar{c}_{Q}(2) \leq 4$.
Let $I\subset S$ be the set of homothets included in other homothets of $S$. Recall that we initially assumed that $I$ was empty. Thus, to finish the proof it only remains to study the $I\neq \emptyset$ case. First we color $S\setminus I$ with 4 colors, using a 4-coloring of $G_\pi(\rho(S))$. This is possible from Lemma \[lem\_dualplanar\]. Then, for each homothet ${Q'}$ of $I$ there exists one homothet ${Q''}$ in $S \setminus I$ that contains it. We assign to ${Q'}$ any color different than the one assigned to ${Q''}$. Any point $p\in {Q'}$ will also satisfy $p\in{Q''}$, since ${Q'}\subseteq{Q''}$, hence $p$ will be covered by two ranges of different colors.
Coloring Three-dimensional Hypergraphs {#sec_3D}
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Lemma \[lem\_dualplanar\] actually generalizes the “easy" direction of Schnyder’s characterization of planar graphs. We first give a brief overview of this fundamental result. The [*vertex-edge incidence poset*]{} of a hypergraph $G=(V,E)$ is a bipartite poset $P=(V\cup E,\preceq_P)$, such that $e\preceq_P v$ if and only if $e\in E$, $v\in V$, and $v\in e$. The [*dimension*]{} of a poset $P=(S,\preceq_P)$ is the smallest $d$ such that there exists an injective mapping $f:S\to{\mathbb{R}}^d$, such that $u\preceq_P v$ if and only if $f(u)\leq f(v)$, where the order $\leq$ is the componentwise partial order on $d$-dimensional vectors. When $P$ is the vertex-edge incidence poset of a hypergraph $G$, we will refer to this mapping as a [*realizer*]{} of $G$, and to $d$ as its [*dimension*]{}.
There exists a relation between the dimension of a graph and its chromatic number. For example, the graphs of dimension $2$ or less are subgraphs of the path, hence are 2-colorable. Schnyder pointed out that all 4-colorable graphs have dimension at most $4$ [@schnyder], and completely characterized the graphs whose incidence poset has dimension $3$:
A graph is planar if and only if its dimension is at most three.
The “easy" direction of Schnyder’s theorem states that every graph of dimension at most three is planar. The non-crossing drawing that is considered in one of the proofs is similar to ours, and simply consists of, for every edge $e=uv$, projecting the two line segments $f(e)f(u)$, and $f(e)f(v)$ onto the plane $x+y+z=0$ [@trotter; @BD81]. It is easy to see that octants in ${\mathbb{R}}^3$ satisfy the equivalent of our Lemma \[lem\_enough\] (by translating the apex of the octant with vector $(1,1,1)$ for example). Combining this result with the Four Color Theorem gives the following result.
\[lem\_3dim\] Every hypergraph of dimension at most three is 4-colorable.
Upper bounds for three-dimensional hypergraphs {#sec_3dim}
----------------------------------------------
We now adapt the above result for higher values of $k$. That is, we are given a three-dimensional hypergraph $G=(V,H)$ and a constant $k\geq 2$. We would like to color the vertices of $G$ such that any hyperedge $e\in H$ contains at least $\min\{|e|,k\}$ vertices with different colors. We denote by $c_3(k)$ the minimum number of colors so that any three-dimensional hypergraph can be suitably colored. Note that the problem is self-dual: any instance of the dual coloring problem can be transformed into a primal coloring problem by symmetry.
For simplicity, we assume that no two vertices of $V$ in the realizer share an $x$, $y$ or $z$ coordinate. This can be obtained by making a symbolic perturbation of the point set in ${\mathbb{R}}^3$. Recall that, from the definition of the realizer, the point $q_e$ dominates $u\in S$ if and only if $u\in e$. For any hyperedge $e\in H$, there exist many points in ${\mathbb{R}}^3$ that dominate the points of $e$. We also assume that hyperedge $e$ is mapped to the minimal point $q_e\in{\mathbb{R}}^3$, obtained by translating $q_e$ in each of the three coordinates until a point of hits the boundary of the upper octant whose apex is $q_e$.
For any hyperedge $e\in H$, we define the $x$-extreme of $e$ as the point $x(e)\in e$ whose image has smallest $x$-coordinate. Analogously we define the $y$ and $z$-extremes and denote them $y(e)$ and $z(e)$, respectively. We say that a hyperedge $e$ is [*extreme*]{} if two extremes of $e$ are equal.
\[lem\_boundeg\] For any $k\geq 2$, $G$ has up to $3n$ extreme hyperedges of size exactly $k$.
We charge any extreme hyperedge to the point that is repeated. By the pigeonhole principle, if a point is charged more than three times, there exist two extreme hyperedges $e_1,e_2$ of size exactly $k$ that charge on the same coordinates. Without loss of generality, we have $x(e_1)=x(e_2)$ and $y(e_1)=y(e_2)$. Let $q_1$ and $q_2$ be the mappings of $e_1$ and $e_2$, respectively. By hypothesis, the $x$ and $y$ coordinates of $q_1$ and $q_2$ are equal. Without loss of generality, we assume that $q_1$ has higher $z$ coordinate than $q_2$. In particular, we have $q_1\subset q_2$. Since both have size $k$, we obtain $e_1=e_2$.
Let $S$ be the the 3-dimensional realizer of the vertices of $G$. For simplicity, we assume that $G$ is maximal. That is, for any $e\subseteq S$, we have $e\in H$ if and only if there exists a point $q_e\in {\mathbb{R}}^3$ dominating exactly $e$. Since we are only adding hyperedges to $G$, any coloring of this graph is a valid coloring of $G$.
For any $2\leq k\leq n$, we define the graph $G_k(S)=(S,E_k)$, where for any $u,v\in S$ we have $uv\in E_k$ if and only if there exists a point $q\in {\mathbb{R}}^3$ that dominates $u,v$ and at most $k-2$ other points of $S$ (that is, we replace hyperedges of $G$ whose size is at most $k$ by cliques). The main property of this graph is that any proper coloring of $G_k(S)$ induces a polychromatic coloring of $G$. Using Lemma \[lem\_boundeg\], we can bound the number of edges of $G_k(S)$.
\[lem\_edges\] For any set $S$ of points and $2\leq k\leq n$, graph $G_k(S)$ has at most $3(k-1)n-6$ edges.
The claim is true for $k=2$ from Schnyder’s characterization. Notice that $E_{k-1}\subseteq E_k$, thus it suffices to bound the total number of edges $uv\in E_k\setminus E_{k-1}$. By definition of $G_k$ and $G_{k-1}$, there must exist a hyperedge $e$ of size exactly $k$ such that $u,v\in e$. In the three-dimensional realizer, this corresponds to a point $q_e\in {\mathbb{R}}^3$ that dominates $u,v$ and $k-2>0$ other points of $S$.
We translate the point $q_e$ upward on the $x$ coordinate until it dominates only $k-1$ points. By definition, the first point to leave must be the $x$-extreme point $x(e)$. After this translation we obtain point $q'_e$ that dominates $k-1$ points. All these points will form a clique in $E_{k-1}$. Since $uv\not\in E_{k-1}$, we either have $u=x(e)$ or $v=x(e)$. We repeat the same reasoning translating in the $y$ and $z$ coordinates instead and, combined with the fact that a point cannot be extreme in the three directions, either $uv\in E_{k-1}$ or $u$ and $v$ are the only two extremes of $e$. In particular, the hyperedge $e$ is extreme. From Lemma \[lem\_boundeg\] we know that this case can occur at most $3n$ times, hence we obtain the recurrence $|E_k|\leq |E_{k-1}|+3n$.
\[theo\_colz\] For any $k\geq 2$, we have $c_3(k)\leq 6(k-1)$.
From Lemma \[lem\_edges\] and the handshake lemma, the average degree of $G$ is strictly smaller than $6(k-1)$. In particular, there must exist a vertex whose degree is at most $6(k-1)-1$. Moreover, this property is also satisfied by any induced subgraph, as any edge $(u,v)\in E_k$ is an edge of $G_k(S\setminus\{w\})$, $\forall w\neq u,v$. Hence, for any $S'\subseteq S$, the induced subgraph $G_k(S) \setminus S'$ is a subgraph of $G_k(S\setminus\{S'\})$. In particular, the graph $G_k(S)$ is $(6(k-1)-1)$-degenerate, and can therefore be colored with $6(k-1)$ colors.
Note that dual hypergraphs induced by collections of homothetic triangles have dimension at most 3, so our result directly applies.
\[cor\_triangle\] For any $k\geq 3$, any set $S$ of homothets of a triangle can be colored with $6(k-1)$ colors so that any point $p\in{\mathbb{R}}^2$ covered by $r$ homothets is covered by $\min\{r,k\}$ homothets with distinct colors.
Lower Bounds {#sec_lb}
============
We now give a lower bound on $c_{Q}(k)$. The normal vector of ${Q}$ at the boundary point $p$ is the unique unit vector orthogonal to the halfplane tangent to ${Q}$ at $p$, if it is well-defined. We say that a range has $m$ distinct directions if there exist $m$ different points with defined, pairwise linearly independent normal vectors.
\[lem\_lowerprim\] Any range ${Q}$ with at least three distinct directions satisfies $c_{Q}(k)\geq 4\lfloor k/2 \rfloor$ and $\bar{c}_{Q}(k)\geq 4\lfloor k/2 \rfloor$.
We first show that $c_{Q}(2)\geq 4$. Scale ${Q}$ by a large enough value so that it essentially becomes a halfplane. By hypothesis, we can obtain halfplane ranges with three different orientations. By making an affine transformation to the problem instance, we can assume that the halfplanes are of the form $x\geq c$, $y\geq \sqrt{3}x+c$ or $y\leq \sqrt{3}x+c$ for any constant $c\in {\mathbb{R}}$ (i.e. the directions of the equilateral triangle). Let $\Delta$ be the largest equilateral triangle with a side parallel to the abscissa that can be circumscribed in ${Q}$. Let $p_1,p_2, p_3$ and $p_4$ be the vertices and the incenter of $\Delta$, respectively (see Figure \[fig\_ranges\]). Note that any two points of $\{p_1,p_2,p_3,p_4\}$ can be selected with the appropriate halfplane range, hence any valid coloring must assign different colors to the four points. The proof of the dual bound is analogous: it suffices to consider the ranges that contain exactly two points of $\{p_1,p_2,p_3,p_4\}$.
For higher values of $k$ it suffices to replace each point $p_i$ for a cluster of $\lfloor k/2 \rfloor$ points. That is, we have $4\lfloor k/2 \rfloor$ points clustered into four groups so that any two groups can be covered by one range. By the pigeonhole principle, any coloring that uses strictly less than $4\lfloor k/2 \rfloor$ colors must have two points with the same color. The range containing them (and any other $k-2$ points) will have at most $k-1$ colors, hence will not be polychromatic.
Observe that parallelograms are the only ranges that do not have three or more distinct normal directions (in this case, we can show a weaker $3\lfloor k/2 \rfloor$ lower bound). In particular, the results of Sections \[sec\_primal\] and \[sec\_dual\] are tight for any range other than a parallelogram. Also notice that, since triangle containment posets are 3-dimensional, the lower bound also applies to $c_3(k)$.
Applications to other coloring problems {#sec_appl}
=======================================
#### Conflict-free colorings.
A coloring of a hypergraph is said to be [*conflict-free*]{} if, for every hyperedge $e$ there is a vertex $v\in e$ whose color is distinct from all other vertices of $e$. Even [*et al.*]{} [@shakharcf] gave an algorithm for finding such a coloring. Their method repeatedly colors (in the polychromatic sense) the input hypergraph with few colors, and removes the largest color class. By repeating this process iteratively a conflict-free coloring is obtained. The number of colors is at most $\log_{\frac c{c-1}} n$, where $n$ is the number of vertices, and $c$ is the maximum number of colors used at each iteration. Our 4-colorability proof of Theorem \[theo\_dual\] is constructive and can be computed in $O(n^2)$ time. Hence, by combining both results we obtain the following corollary.
Any dual hypergraph induced by a finite set of $n$ homothets of a compact and convex body in the plane has a conflict-free coloring using at most $\log_{4/3} n \leq 2.41\log_2 n$ colors. Furthermore, such a coloring can be found in $O(n^2\log n )$ time.
#### $k$-strong conflict-free colorings.
Abellanas [*et al.*]{} [@ABGHNR09] introduced the notion of $k$-strong conflict free colorings, in which every hyperedge $e$ has $\min\{|e|,k\}$ vertices with a unique color. Conflict-free colorings are $k$-strong conflict-free colorings for $k=1$. Recently, Horev, Krakovski, and Smorodinsky [@HKS10] showed how to find $k$-strong conflict-free colorings by iteratively removing the largest color class of a polychromatic coloring with $c(k)$ colors. Again, combining this result with Theorem \[theo\_colz\] yields the following corollary.
Any dual hypergraph induced by a finite set of $n$ homothets of a compact and convex body in the plane has a $k$-strong conflict-free coloring using at most $\log_{(1+\frac{1}{6(k-1)})}n$ colors.
#### Choosability.
Cheilaris and Smorodinsky [@CS10] introduced the notion of choosability in geometric hypergraphs. A hypergraph with vertex set $V$ is said to be $k$-choosable whenever for any collection $\{L_v\}_{v\in V}$ of subsets of positive integers of size at least $k$, the hypergraph admits a proper coloring, where the color of vertex $v$ is chosen from $L_v$. Our construction of Section \[sec\_dual\] provides a planar graph, and planar graphs are known to be 5-choosable. This directly yields the following result.
Any dual hypergraph induced by a finite set of homothets of a convex body in the plane is 5-choosable.
[^1]: The term [*$k$-colorful*]{} is also used in the literature [@Smo07].
|
---
abstract: 'In this paper we study the effect of a torus topology on Loop Quantum Cosmology. We first derive the Teichmüller space parametrizing all possible tori using Thurston’s theorem and construct a Hamiltonian describing the dynamics of these torus universes. We then compute the Ashtekar variables for a slightly simplified torus such that the Gauss constraint can be solved easily. We perform a canonical transformation so that the holomies along the edges of the torus reduce to a product between almost and strictly periodic functions of the new variables. The drawback of this transformation is that the components of the densitized triad become complicated functions of these variables. Nevertheless we find two ways of quantizing these components, which in both cases leads surprisingly to a continuous spectrum.'
address: |
Institut für Theoretische Physik, Universität Ulm\
Albert-Einstein-Allee 11\
D-89069 Ulm, Germany
author:
- Raphael Lamon
title: Loop Quantum Cosmology on a Torus
---
Introduction {#sec:intro}
============
The Einstein field equations are local equations in the sense that they only describe the local geometry of the spacetime. For example the Robertson-Walker metric explicitly contains the parameter $k$ which gives an account of the intrinsic spatial curvature. Using the Friedmann equations this parameter can be determined experimentally since it is directly related to the density parameter $\Omega_{\mathrm{tot}}$ and the Hubble parameter $h$. Recent measurements of the energy density of the universe tend to slightly favor a positively curved universe [@Komatsu:08], yet a flat curvature lies within the 1-$\sigma$ range. The most direct conclusion is that the spatial topology of the universe is just $\mathds{R}^3$ which is the assumption of the $\Lambda$CDM model. Nevertheless in the mathematical literature it is well known that a flat space does not mean that its topology is necessarily $\mathds{R}^3$, in fact there are 18 possible flat topologies. Since the Einstein field equations are not sensitive to topology every possibility has to be considered as a possible candidate for the global geometry of our universe until it is ruled out by experiment. In order to do so we first note that the spectrum of the Laplace operator sensitively depends on the topology, i.e. it is discrete if the eigenstates are normalizable and continuous if not. In the first case the solution for e.g. a torus is given by plane waves with a wave vector $\vec k_n$ taking only discrete values $n\in\mathds{N}$ while in the second case the (weak) solution to the eigenvalue equation is given by the (distributional) plane waves with a continuous wave vector $\vec k$. For example, the eigenvalue problem for $\triangle$ on $\mathds{T}^3$ is given by $(\triangle+E_{\vec n})\Psi_{\vec n}=0,\; \vec n\in\mathds{N}^3$, and on $S^3$ by $\triangle\Psi_{\beta,l,m}=(\beta^2-1)\Psi_{\beta,l,m},$ where $\beta\in\mathds{N}$, $0\leq l\leq\beta-1$ and $|m|\leq l$. The implication of a solution of the form $\Psi_{\vec n}$ is the existence of a wave function $\Psi_n$ with a maximum length corresponding to e.g. the length of the edges of the torus. Since the departure from a continuous solution is biggest for large wavelengths we have to look for large-scale structures of the universe in order to distinguish between cosmic topologies. The best way to do so is to measure the inhomogeneities of the cosmic microwave background (CMB), expand these in multipole moments and compare the low multipoles with the predictions from theory. It can be shown that in certain closed topologies a suppression in the power spectrum of the low multipoles is expected because of the existence of a largest wavelength. Since such a suppression is present in the CMB several studies compared the theoretical predictions for various topologies with the data. While most analyzed topologies can already be ruled out three of them describe the data even better than the infinite $\Lambda$CDM model, namely the torus [@Aurich:08; @Aurich:08:2; @Aurich:09], the dodecahedron[@Aurich:05; @Caillerie:07] and the binary octahedron[@Aurich:05:2] (see also references therein). While the last two topologies are spherical the torus is the simplest model of a closed flat topology.
However, we know that standard cosmology cannot be the final answer as its predictability breaks down at the big bang. A quantization of the Friedmann equations a la Wheeler-DeWitt does not improve this behavior either. This situation has changed thanks to a new model called loop quantum cosmology (LQC) developed over the last few years which removes the initial singularity. LQC [@Bojowald:08; @Bojowald:00; @Bojowald:00:2; @Bojowald:00:3; @Bojowald:02; @Ashtekar:03; @Bojowald:03; @Ashtekar:06; @Ashtekar:06:2] is the approach motivated by loop quantum gravity (LQG) [@Ashtekar:04; @Rovelli:04; @Thiemann:07] to the quantization of symmetric cosmological models. The usual procedure is to reduce the classical phase space of the full theory to a phase space with a finite number of degrees of freedom. The quantization of these reduced models uses the tools of the LQG and is therefore called LQC but it does not correspond to the cosmological sector of LQG. The results of LQC not only provide new insights into the quantum structure of spacetime near the Big-Band singularity but also remove this singularity by extending the time evolution to negative times.
In sum, on the one hand we have hints from observation that our universe may have a closed topology, on the other hand we have a very successful loop quantization of various cosmologies. Thus, starting from these two motivations, we would like to study LQC with a torus topology. But contrary to the works on the CMB we don’t want to restrict the analysis to a cubical torus. To do so we construct a torus using Thurston’s theorem and find that the most general torus has six degrees of freedom which consist of e.g. three lengths and three angles. We will study its dynamics by numerically solving the Hamiltonian coupled to a scalar field. After rewritting this Hamiltonian in terms of Ashtekar variables we will see that the quantization of such a torus leads to a product between the standard Hilbert spaces of LQC and the Hilbert spaces over the circle. Moreover, we will find two ways to quantize the components of the triad and show that both (generalized) eigenfunctions are not normalizable in this Hilbert space.
As a side remark we would like to point out that the consequences of putting a non-abelian gauge theory into a box with periodic boundary conditions have been studied in e.g. [@thooft:79]. The motivation behind this idea is an attempt to explain the quark confinement in QCD without explicitely breaking gauge invariance. To simplify the analysis the $su(N)$-valued gauge field is chosen to be pure gauge, i.e. $A=U^{-1}dU$ with $U\in SU(N)$, such that the holonomy around a closed curve $C$ only depends on the topological property of $C$. Since general relativity written in terms of Ashtekar variables is also a (constrained) Yang-Mills theory it may be tentalizing to use the methods developed for QCD in a box to LQC of a torus universe. However we will derive an Ashtekar connection for the homogeneous torus which is not pure gauge so that the holonomies along $C$ also depend on the length of $C$. This may not be surprising in view of the fact that the Hilbert space of LQC on $\mathds{R}^3$ is spanned by almost periodic functions with an arbitrary length parameter $\mu$.
This paper is organized as follows: in we first introduce the classical dynamics of a torus universe and numerically solve the Friedmann equations with a massless scalar field. In we introduce the Ashtekar variables for a torus and also explain the complications that arise because of a closed topology. The loop quantization and the construction of a Hilbert space are explained in and provides a summary and directions for future works. \[sec:fundamentaldomain\] gives a short review of the fundamental domain of the 3-torus and \[sec:TorusIwasawa\] describes the dynamics of the torus in terms of Iwasawa coordinates.
Compact Homogeneous Universes and their Dynamics {#sec:CHU}
================================================
The purpose of this section is to study models in which the spatial section has a compact topology. The compactness of a locally homogeneous space brings new degrees of freedom of deformations, known as Teichmüller deformations. This leads to the conclusion that cosmology on a torus is simply cosmology on $\mathds{R}^3$ restricted to a cube may be too naive a point of view, especially since the space of solutions of a torus gets nine additional degrees of freedom, as already mentioned in [@Ashtekar:91]. We will introduce Teichmüller spaces with an emphasis on a Thurston geometry admitting a Bianchi I geometry as its subgeometry [@Wolf:74; @Koike:94; @Tanimoto:97; @Tanimoto:97:02; @Yasuno:01] and derive the vacuum Friedmann equations using the Hamiltonian formalism.
Compact Homogeneous Spaces {#sec:CHS}
--------------------------
Let $\Sigma$ be a three-dimensional, arcwise connected Riemannian manifold.
A metric on a manifold $\Sigma$ is locally homogeneous if $\forall p,q\in \Sigma$ there exist neighborhoods $U,V$ of $p$ resp. $q$ and an isometry $(U,p)\rightarrow (V,q)$. The manifold is globally homogeneous if the isometry group acts transitively on the whole manifold $\Sigma$.
Since $\Sigma$ is arcwise connected we know that there is a unique universal covering manifold $\tilde \Sigma$ up to diffeomorphisms with a metric given by the pullback of the metric on $\Sigma$ by the covering map $$\label{coveringmap}
\pi:\tilde \Sigma\rightarrow \Sigma.$$ Singer [@Singer:60] proved that the metric on $\tilde \Sigma$ is then globally homogeneous and $\tilde \Sigma$ is given by $\tilde \Sigma\cong\tilde S/F$, where $\tilde S$ is the orientation preserving isometry group of $\tilde \Sigma$ and $F$ its isotropy subgroup.
On the other hand, we can also start from a three-dimensional, simply connected Riemannian manifold $\tilde \Sigma$ which admits a compact quotient $\Sigma$. In order to construct this compact manifold consider the covering group $\Gamma\subset \tilde S$ which is isomorphic to the fundamental group $\pi_1(\Sigma)$ of $\Sigma$. This implies that $$\Sigma=\tilde \Sigma/\Gamma,$$ which is Hausdorff iff $\Gamma$ is a discrete subgroup of $\tilde S$ and a Riemannian manifold iff $\Gamma$ acts freely on $\tilde \Sigma$.
A geometry is the pair $(\tilde\Sigma,S)$ where $\tilde S$ a group acting transitively on $\tilde \Sigma$ with compact isotropy subgroup. A geometry $(\tilde \Sigma,\tilde S')$ is a subgeometry of $(\tilde \Sigma,\tilde S)$ if $\tilde S'$ is a subgroup of $\tilde S$. A geometry $(\tilde \Sigma,\tilde S)$ is called maximal if it is not a subgeometry of any geometry and minimal if it does not have any subgeometry.
We will need the following important theorem:
Any maximal, simply connected 3-dimensional geometry which admits a compact quotient is equivalent to the geometry $(\tilde \Sigma,\tilde S)$ where $\tilde \Sigma$ is one of $E^3$ (Euclidean), $H^3$ (hyperbolic), $S_p^3$ (3-sphere), $S_p^2\times\mathds{R}$, $H^2\times\mathds{R}$, $\widetilde{SL}(2,\mathds{R})$, Nil or Sol.
If $(\tilde \Sigma,\tilde S')$ is not a maximal geometry but is simply connected and admits a compact quotient as well we can find a discrete subgroub $\Gamma'$ of $\tilde S'$ acting freely so as to make $\tilde \Sigma/\Gamma'$ compact. Define $(\tilde \Sigma,\tilde S)$ as the maximal geometry with $(\tilde \Sigma,\tilde S')$ as its subgeometry, i.e. $\tilde S'\subset \tilde S$. By Thurston’s Theorem $(\tilde \Sigma,\tilde S)$ is one of the eight Thurston geometries, which implies that $(\tilde \Sigma,\tilde S')$ is a subgeometry of one of the eight Thurston geometries.
Any minimal, simply connected three-dimensional geometry is equivalent to $(\tilde \Sigma,\tilde S)$, where $\tilde \Sigma=\mathds{R}^3$, $\tilde S=$Bianchi I-VIII; $\tilde \Sigma=S_p^3$, $\tilde S=$Bianchi IX; or $\tilde \Sigma=S_p^2\times\mathds{R}$, $\tilde S=SO(3)\times\mathds{R}$, where $S_p^3$ is the three-sphere and $S_p^2$ the two-sphere.
Let Rep$(\Sigma)$ denote the space of all discrete and faithful representations $\rho:\pi_1(\Sigma)\rightarrow \tilde S$ and the diffeomorphism $\phi:\tilde \Sigma\rightarrow \tilde \Sigma$ a [*global conformal isometry*]{} if $\phi_*\tilde h_{ab}=\mathrm{const}\cdot \tilde h_{ab}$, where $\tilde h_{ab}$ is the spatial metric of the universal covering manifold $\tilde \Sigma$. This allows us to define a relation $\rho\sim\rho'$ in Rep$(\Sigma)$ if there exists a conformal isometry $\phi$ of $\tilde \Sigma$ connected to the identity with $\rho'(a)=\phi\circ\rho(a)\circ\phi'$.
\[def:Teichmueller\] The Teichmüller space is defined as $$\mathrm{Teich}(\Sigma)=\mathrm{Rep}(\Sigma)/\sim$$ with elements called Teichmüller deformations, which are smooth and nonisometric deformations of the spatial metric $h_{ab}$ of $\Sigma$, leaving the universal cover $(\tilde \Sigma,\tilde h_{ab})$ globally conformally isometric.
The situation gets more complicated when we try to extend the previous construction to four-dimensional Lorentzian manifolds. The reason is that the action of the covering group $\Gamma$ needs to preserve both the extrinsic curvature and the spatial metric of $\tilde \Sigma$. Thus we cannot construct a homogeneous compact manifold by the action of a discrete subgroup of $\tilde S$ on the spatial three-section $\tilde \Sigma$. Instead we need the isometry group of the four-dimensional manifold $\tilde M$. Let $M=\mathds{R}\times \Sigma$ be a compact homogeneous Lorentzian manifold with metric $g_{\mu\nu}$ and $\tilde M=\mathds{R}\times\tilde\Sigma$ its covering with metric $\tilde g_{\mu\nu}$ $(\mu,\nu=0,\ldots,4)$.
Let $(\tilde \Sigma,\tilde h_{ab})$ be a spatial section of $(\tilde M,\tilde g_{\mu\nu})$. An extendible isometry is defined by the restriction of an isometry of $(\tilde M,\tilde g_{ab})$ on $\tilde \Sigma$ which preserves $\tilde \Sigma$ and forms a subgroup $\mathrm{Esom}(\tilde \Sigma)$ of $\tilde S$.
Thus, in order to get a compact homogeneous manifold from $\tilde M$ the covering group $\Gamma$ must be a subgroup of Esom$(\tilde \Sigma)$, i.e. $$\Gamma\subset\mathrm{Esom}(\tilde \Sigma).$$ The line element of $M=\mathds{R}\times \Sigma$ is given by $$ds^2=-dt^2+h_{ab}(t)\sigma^a\sigma^b,$$ where $\sigma^a$ are the invariant one-forms.
Therefore the Teichmüller parameters enlarge the parameter space by bringing new degrees of freedom from the deformations defined in Definition \[def:Teichmueller\]. In fact, the set of all possible universal covers $(\tilde M,\tilde g_{ab})$ carries the degrees of freedom of the local geometry and the covering maps $\Gamma$ the degrees of freedom of the global geometry which are parameterized by the Teichmüller parameters.
The Torus Universe {#sec:torusuniverse}
------------------
In this section we restrict the above analysis to the case of a flat torus and give only the main results. Further details can be found in [@Wolf:74; @Koike:94; @Tanimoto:97; @Tanimoto:97:02; @Yasuno:01]. Let $\tilde M=\mathds{R}\times\tilde \Sigma$ be the universal cover of $M$ and $\tilde \Sigma$ the Thurston geometry $(E^3,ISO(3,\mathds{R}))$. The isometry group $ISO(3)$ is expressed as $g(\mathbf{x})=\mathbf{Rx}+\mathbf{a}$, where $\mathbf{a}$ is a constant vector and $\mathbf{R}\in SO(3)$ in order that the orientation be preserved. The Killing vectors of $E^3$ are $$\begin{aligned}
\xi_1=\partial_x,\quad \xi_2=\partial_y,\quad \xi_3=\partial_z,\nonumber \\
\xi_4=-z\partial_y+y\partial_z,\quad \xi_5=-x\partial_z+z\partial_x,\quad \xi_6=-y\partial_x+x\partial_y.\end{aligned}$$ The line element of $\tilde M$ is thus given by $$ds^2=-dt^2+\tilde h_{ab}dx^adx^b=-dt^2+a^2(t)\,^0\tilde h_{ab}dx^adx^b,$$ where $\,^0\tilde h_{ab}$ is called the fiducial metric in the LQC literature and $dx^a$ are the invariant one-forms of the group $ISO(3,\mathds{R})$ [^1].
![The vectors $a_1$, $a_2$ and $a_3$ span the torus with six Teichmüller parameters. The global conformal invariance was used in order to align $a_1$ with $\xi_1$ and $a_2$ with span$\{\xi_1,\xi_2\}$.[]{data-label="fig:torus"}](torus.eps){width="8cm"}
The covering group $\Gamma\subset\mathrm{Esom}(\tilde\Sigma)\equiv \mathrm{Esom}(E^3)$ allows us to construct a torus via $M=\tilde M/\Gamma$, where $M=\mathds{R}\times\mathds{T}^3$. The freedom of global conformal transformations allows us to choose the coordinate system of $\tilde \Sigma$ such that the generators of the torus have a simple representation. We thus require one of the generators to be aligned with $\xi_1$ and one to lie in the $\xi_1\xi_2$-plane. The Teichmüller space is then generated by six Teichmüller parameters in three vectors $$\label{Teichmuellervectors}
a_1=\left(
\begin{array}{c}
a_1\,^1 \\ 0 \\ 0
\end{array}
\right),
\quad
a_2=\left(
\begin{array}{c}
a_2\,^1 \\ a_2\,^2 \\ 0
\end{array}
\right),
\quad
a_3=\left(
\begin{array}{c}
a_3\,^1 \\ a_3\,^2 \\ a_3\,^3
\end{array}
\right),$$ where all $a_i\,^j$ only depend on the coordinate time $t$. The configuration space $\mathcal{C}$ is therefore spanned by the six Teichmüller parameters such that $\mathcal{C}\subset\mathds{R}^6$ (see \[sec:fundamentaldomain\]). The [*flat*]{} spatial metric on ${\mathds{T}^3}$ is then given by $(a,b=1,2,3)$ $$\label{ds2Teich}
ds^2=h_{ab}dx^{a}dx^{b},\quad h_{ab}=\sum_c a_a\,^ca_b\,^c$$ where $$\label{metrich}
(h_{ab})=\left(
\begin{array}{ccc}
(a_1\,^1)^2 & a_1\,^1a_2\,^1 & a_1\,^1a_3\,^1 \\
& (a_2\,^1)^2+(a_2\,^2)^2 & a_2\,^1a_3\,^1+a_2\,^2a_3\,^2 \\
\mathrm{(sym)} & & (a_3\,^1)^2+(a_3\,^2)^2+(a_3\,^3)^2
\end{array}
\right).$$ This metric is invariant under transformations in $SL(3,\mathds{Z})$. For example it is left invariant by $(a_1\,^1\rightarrow -a_1\,^1,a_2\,^1\rightarrow -a_2\,^1,a_3\,^1\rightarrow -a_3\,^1)$ (see \[sec:fundamentaldomain\] for more details). From we can make a Legendre transform of the Einstein-Hilbert action $$\label{SEH}
S_{\mathrm{E-H}}[g]=\frac{1}{2\kappa}\int_{\mathds{R}\times{\mathds{T}^3}}*R[g], \quad \kappa=8\pi G$$ to obtain a Hamiltonian, where $*$ is the Hodge star operator. After a partial integration of $\ddot a_i\,^i$ (which also cancels the surface term we omitted in ) we find the following Lagrangian: =&&\
&We introduce the momenta $$\label{defp}
p^a\,_b:=\frac{\partial\mathcal{L}}{\partial \dot a_a\,^b}$$ conjugate to the configuration variables $a_a\,^b$ such that the phase space $\mathcal{P}=T^*\mathcal{C}\subset\mathds{R}^{12}$ is the cotangent bundle over $\mathcal{C}$ with $$\label{bracketap}
\{a_a\,^b,p^c\,_d\}=\delta_a^c\delta^b_d, \quad \{a_a\,^b,a_c\,^d\}=0, \quad \{p^a\,_b,p^c\,_d\}=0,$$ where the Poisson brackets are defined as $$\{f,g\}=\sum_{a,b=1}^{3}\frac{\partial f}{\partial a_a\,^b}\frac{\partial g}{\partial p^a\,_b}-\frac{\partial g}{\partial a_a\,^b}\frac{\partial f}{\partial p^a\,_b}$$ for any smooth functions on the phase space. We insert $\dot a_a\,^b=\dot a_a\,^b(p^c\,_d)$ into the Legendre transform of and get the Hamiltonian \[Hamiltonian\] \_g=&&\
&The Hamiltonian constraint $\mathcal{H}_g\approx0$ reduces the dynamical degrees of freedom from dim $\mathcal{P}=12$ to dim $\mathcal{P}=10$, which agrees with [@Ashtekar:91]. To compare this Hamiltonian with the usual Bianchi type I models we set all offdiagonal elements to zero and $a_i\,^i=a_i$, $p^i\,_i=p^i$ (no summation), and get \_g=(++-2-2-2), which agrees with the result given in [@Chiou:07] up to a factor 2 in the definition of the action. To get the isotropic case[^2] we further set $a_i=a$, $p^i=p/3$ and find that the Hamiltonian reduces to the usual first Friedmann equation $$\mathcal{H}_g=-\frac{\kappa p^2}{12a}$$ and the Hamiltonian equation $\dot p^i\,_j=-\partial \mathcal{H}_g/\partial a_i\,^j$ to the usual second Friedmann equation $$\dot p=-\frac{\partial \mathcal{H}_g}{\partial a}=\frac{\kappa p^2}{12a^2}.$$ The second Hamiltonian equation is given by $\dot a=\partial \mathcal{H}_g/\partial p=-\kappa p/(6a)$ and allows us to recast the first Friedmann equation into the usual form $$\mathcal{H}_g=-3a\dot a^2/\kappa.$$ Furthermore, notice that all $a_i\,^j$ and $p^i\,_j$, $i\neq j$, have to vanish in order for the torus to remain aligned with the Killing fields $\xi_I$.
![[*Left panel*]{}: Solutions corresponding to the Hamiltonian with the initial conditions $a_i\,^i(1)=1$, $p^i\,_i(1)=-1$ (no summation), $a_i\,^j(1)=0$ $(i\neq j)$, $p^3\,_1(1)=p^3\,_2(1)=0$, $p^2\,_1(1)=0.2$, $\phi(1)=10^{-3}$, $\pi(1)=1.2$. The diagonal momenta $p^i\,_i$ are chosen to be negative such that all sides of the torus expand. The solid black line is $a_1\,^1$, the dashed one $a_2\,^2$, the dotted one $a_3\,^3$ and the gray one the off-diagonal $a_2\,^1$. The time $t$ parametrizes the coordinate time in natural units ($c=\kappa=\hbar=1$). [*Right panel*]{}: Solution corresponding to the Hamiltonian at two different times. The initial condition is a cubic universe with $a_i\,^i\equiv a_0$, $p^i\,_i\equiv p_0$, $a_i\,^j=0$ $(i\neq j)$, $p^i\,_j\neq 0$ $(i\neq j)$. For both panels the mass and the potential of the scalar field have been set to zero.[]{data-label="fig:torus_sim"}](aoft.eps "fig:"){width="7.4cm"}![[*Left panel*]{}: Solutions corresponding to the Hamiltonian with the initial conditions $a_i\,^i(1)=1$, $p^i\,_i(1)=-1$ (no summation), $a_i\,^j(1)=0$ $(i\neq j)$, $p^3\,_1(1)=p^3\,_2(1)=0$, $p^2\,_1(1)=0.2$, $\phi(1)=10^{-3}$, $\pi(1)=1.2$. The diagonal momenta $p^i\,_i$ are chosen to be negative such that all sides of the torus expand. The solid black line is $a_1\,^1$, the dashed one $a_2\,^2$, the dotted one $a_3\,^3$ and the gray one the off-diagonal $a_2\,^1$. The time $t$ parametrizes the coordinate time in natural units ($c=\kappa=\hbar=1$). [*Right panel*]{}: Solution corresponding to the Hamiltonian at two different times. The initial condition is a cubic universe with $a_i\,^i\equiv a_0$, $p^i\,_i\equiv p_0$, $a_i\,^j=0$ $(i\neq j)$, $p^i\,_j\neq 0$ $(i\neq j)$. For both panels the mass and the potential of the scalar field have been set to zero.[]{data-label="fig:torus_sim"}](torus_sim.eps "fig:"){width="7.9cm"}
We add a matter term consisting of a homogeneous massive scalar field[^3] to to obtain the Hamiltonian $$\label{Hamiltonianmatter}
\mathcal{H}=\mathcal{H}_g+\mathcal{H}_{\phi}=\mathcal{H}_g+\frac{1}{2\sqrt{h}}\pi^2+\frac{\sqrt{h}}{2}m_{\phi}^2\phi^2+\sqrt{h}V(\phi),$$ where $\pi$ is the momentum of the scalar field, $m_{\phi}$ its mass, $h=(a_1\,^1)^2(a_2\,^2)^2(a_3\,^3)^2$ the determinant of the spatial metric and $V(\phi)$ the potential which we set to zero in the sequel. From this equation we calculate the Friedmann equations and compute the shape of the universe for a special choice of initial conditions, which is shown in . All classical solutions have the limit $\lim_{t\rightarrow 0}a_i\,^j=0$ and grow with $a\propto t^{1/3}$ for a massless scalar field with zero potential. Furthermore, note the convergence of $a_1\,^1$ and $a_2\,^2$, which is explained in \[sec:TorusIwasawa\].
Symmetry Reduction and Classical Phase Space for Ashtekar Variables {#sec:symmetryreduction}
===================================================================
In this section we shall repeat the complete analysis introduced in [@Kobayashi:63; @Bojowald:00; @Bojowald:00:2] in order to see the role of a compact topology on a connection. Our strategy is to find an invariant connection on the covering space $\tilde M$ and then restrict it to the compact space $M$ by means of the covering map . In the following section, when referring to the covering space, we shall use a tilde.
Invariant Connections {#sec:invariantconnections}
---------------------
Let $\tilde P(\tilde M,SU(2),\pi)$ be a principal fiber bundle over $\tilde M$ with structure group $SU(2)$ and projection $\pi:\tilde P\rightarrow \tilde M$. We require that there be a symmetry group $\tilde S\subset \mathrm{Aut}(\tilde P)$ of bundle automorphisms which acts transitively. Furthermore, for Bianchi I models $\tilde S$ does not have a non-trivial isotropy subgroup $\tilde F$ so that the base manifold is isomorphic to the symmetry group $\tilde S$, i.e. $\tilde M/\tilde S=\{x_0\}$ is represented by a single point that can be chosen arbitrarily in $\tilde M$. Since the isotropy group $\tilde F$ is trivial the coset space $\tilde S/\tilde F\cong \tilde S$ is reductive with a decomposition of the Lie algebra of $\tilde S$ according to ${\mathcal{L}}\tilde S={\mathcal{L}}\tilde F\oplus{\mathcal{L}}\tilde F_{\perp}={\mathcal{L}}\tilde F_{\perp}$ together with the trivial condition $\mathrm{Ad}_{\tilde F}{\mathcal{L}}\tilde F_{\perp}\subset{\mathcal{L}}\tilde F_{\perp}$. This allows us to use the general framework described in [@Bojowald:00; @Bojowald:00:2; @Kobayashi:63].
Since the isotropy group plays an important role in classifying symmetric bundles and invariant connections we describe the general case of a general isotropy group $\tilde F$. Fixing a point $x\in\tilde M$, the action of $\tilde F$ yields a map $\tilde F:\pi^{-1}(x)\rightarrow\pi^{-1}(x)$ of the fiber over $x$. To each point $p\in\pi^{-1}(x)$ in the fiber we assign a group homomorphism $\lambda_p:\tilde F\rightarrow G$ defined by $f(p)=:p\cdot\lambda_p(f)$, $\forall f\in \tilde F$. As this homomorphism transforms by conjugation $\lambda_{p\cdot g}=\mathrm{Ad}_{g^{-1}}\circ\lambda_p$ only the conjugacy class $[\lambda]$ of a given homomorphism matters. In fact, it can be shown [@Kobayashi:63] that an $\tilde S$-symmetric principal bundle $P(\tilde M,G,\pi)$ with isotropy subgroup $\tilde F\subseteq \tilde S$ is uniquely characterized by a conjugacy class $[\lambda]$ of homomorphisms $\lambda:\tilde F\rightarrow G$ together with a reduced bundle $Q(\tilde M/\tilde S,Z_G(\lambda(\tilde F)),\pi_Q)$, where $Z_G(\lambda(\tilde F))$ is the centralizer of $\lambda(\tilde F)$ in $G$. In our case, since $\tilde F=\{1\}$ all homomorphisms $\lambda:\tilde F\rightarrow G=SU(2)$ are given by $1\mapsto 1_G$.
After having classified the $\tilde S$-symmetric fiber bundle $\tilde P$ we seek a $[\lambda]$-invariant connection on $\tilde P$. We use the following general result [@Brodbeck:96]:
Let $\tilde P$ be an $\tilde S$-symmetric principal bundle classified by $([\lambda],Q)$ and let $\tilde \omega$ be a connection in $\tilde P$ which is invariant under the action of $\tilde S$. Then $\tilde \omega$ is classified by a connection $\tilde \omega_Q$ in $Q$ and a scalar field (usually called the Higgs field) $\phi:Q\times{\mathcal{L}}\tilde F_{\perp}\rightarrow{\mathcal{L}}G$ obeying the condition $$\label{condonphi}
\phi(\mathrm{Ad}_f(X))=\mathrm{Ad}_{\lambda(f)}\phi(X)$, $\forall f\in \tilde F,\; X\in{\mathcal{L}}\tilde F_{\perp}.$$
The connection $\tilde \omega$ can be reconstructed from its classsifying structure as follows. According to the decomposition $\tilde M\cong\tilde M/\tilde S\times \tilde S/\tilde F$ we have $\tilde \omega=\tilde{\omega}_Q+\tilde \omega_{\tilde S/\tilde F}$ with $\tilde \omega_{\tilde S/\tilde F}=\phi\circ\iota^*\tilde \theta_{\mathrm{MC}}$, where $\iota:\tilde S/\tilde F\hookrightarrow \tilde S$ is a local embedding and $\tilde \theta_{\mathrm{MC}}$ is the Maurer-Cartan form on $\tilde S$. The structure group $G$ acts on $\phi$ by conjugation, whereas the solution space of is only invariant with respect to the reduced structure group $Z_G(\lambda(\tilde F))$. This fact leads to a partial gauge fixing since the connection form $\tilde \omega_{\tilde S/\tilde F}$ is a $Z_G(\lambda(\tilde F))$-connection which explicitly depends on $\lambda$. We then break down the structure group from $G$ to $Z_G(\lambda(\tilde F))$ by fixing a $\lambda\in[\lambda]$.
In our case, the embedding $\iota:\tilde S\rightarrow \tilde S$ is the identity and the base manifold $\tilde M/\tilde S=\{x_0\}$ of the orbit bundle is represented by a single point so that the invariant connection is given by $$\tilde A=\phi\circ\tilde \theta_{\mathrm{MC}}.$$ The three generators of ${\mathcal{L}}\tilde S$ are given by $T_I$, $1\leq I\leq 3$, with the relation $[T_I,T_J]=0$ for Bianchi I models. The Maurer-Cartan form is given by $\tilde \theta_{\mathrm{MC}}=\tilde \omega^I T_I$ where $\tilde \omega^I$ are the left invariant one-forms on $\tilde S$. The condition is empty so that the Higgs field is given by $\phi:{\mathcal{L}}\tilde S\rightarrow{\mathcal{L}}G,\;T_I\mapsto \phi(T_I)=:\phi_I\,^i\tau_i$, where the matrices $\tau_j=-i\sigma_j/2$, $1\leq j\leq 3$, generate ${\mathcal{L}}G$, where $\sigma_j$ are the standard Pauli matrices[^4]. In summary the invariant connection is given by $$\label{Anondiag}
\tilde A=\phi_I\,^i\tau_id\tilde \omega^I.$$
In order to restrict this invariant connection we define the invariant connection $A$ on $\mathds{T}^3$ with the pullback given by the covering map . The generators of the Teichmüller space (see ) allow us to write $A$ as: $$\label{defofphigen}
A_a^i:=\bar \phi_I\,^i\omega_a^I,\quad (\bar\phi_I\,^i)=\left(
\begin{array}{ccc}
\bar\phi_1\,^1 & \bar\phi_2\,^1 & \bar\phi_3\,^1 \\
0 & \bar\phi_2\,^2 & \bar\phi_3\,^2 \\
0 & 0 & \bar\phi_3\,^3
\end{array}
\right).$$
### Simplified Model {#sec:simplmodel}
In the sequel we shall concentrate on a simpler model for which we can also easily satisfy the Gauss constraint. We choose a torus generated by the vectors $a_1=(a_1\,^1,0,0)^T$, $a_2=(0,a_2\,^2,a_2\,^3)^T$ and $a_3=(0,a_3\,^2,a_3\,^3)^T$ (see ) such that \[defofphi\] (|\_I\^i)=(
[ccc]{} |\_1\^1 & 0 & 0\
0 & |\_2\^2 & |\_3\^2\
0 & |\_2\^3 & |\_3\^3
),(\_a\^I)=(
[ccc]{} a\_1\^1 & 0 & 0\
0 & a\_2\^2 & a\_3\^2\
0 & a\_2\^3 & a\_3\^3
) and (A\_a\^i)=(
[ccc]{} a\_1\^1|\_1\^1 & 0 & 0\
0 & a\_2\^2|\_2\^2+a\_2\^3|\_3\^2 & a\_3\^2|\_2\^2+a\_3\^3|\_3\^2\
0 & a\_2\^2|\_2\^3+a\_2\^3|\_3\^3 & a\_3\^2|\_2\^3+a\_3\^3|\_3\^3
) The vectors $X_I$ dual to $\omega^I$ are given by X\_1=(
[c]{}\
0\
0
), X\_2=(
[c]{} 0\
a\_3\^3\
-a\_2\^3
), X\_3=(
[c]{} 0\
-a\_3\^2\
a\_2\^2
), where we defined $\mathfrak{h}=a_2\,^2a_3\,^3-a_2\,^3a_3\,^2$.
![The vectors $a_1$, $a_2$ and $a_3$ span the torus with five Teichmüller parameters. The vectors $a_2$ and $a_3$ lie in the $\xi_2\xi_3$-plane while $a_1$ is aligned with $\xi_1$[]{data-label="fig:torus_simpl"}](torus_simpl.eps){width="6cm"}
Classical Phase Space for Ashtekar Variables {#sec:classicalphasespace}
--------------------------------------------
The phase space of full general relativity in the Ashtekar representation is spanned by the $SU(2)$-connection $A_a^i=\Gamma_a^i+\gamma K_a^i$ and the densitized triad $E_i^a=|\det e|e_i^a$, where $\Gamma_a^i$ is the spin connection, $K_a^i$ the extrinsic curvature, $e_i^a$ the triad and $\gamma>0$ the Immirzi parameter [@Ashtekar:04; @Rovelli:04; @Thiemann:07]. The symplectic stucture of full general relativity is given by the Poisson bracket $$\label{AEvariables}
\{A^i_a(y),E^b_j(x)\}=\kappa\delta_a^b\delta_j^i\delta(x,y).$$ The connection between the metric and the densitized triad is given by \[EEeqh\] hh\^[ab]{}=\^[ij]{}E\_i\^aE\_j\^b,where $h^{ab}$ is the inverse of the metric $h_{ab}$.
We can now use the results obtained in to construct the phase space $\mathcal{P}$ in this representation. In the preceding subsection we have already found the configuration space is spanned by $\bar \phi_I\,^i$ (see ). On the other hand, the densitized triad dual to the connection is given by $$\label{defofp}
(E^a_i)=\sqrt{h}\bar p^I\,_i X^a_I=\sqrt{h}\left(
\begin{array}{ccc}
\frac{\bar p^1\,_1}{a_1\,^1} & 0 & 0 \\
0 & \frac{a_3\,^3\bar p^2\,_2-a_3\,^2\bar p^3\,_2}{\mathfrak{h}} & \frac{a_3\,^3\bar p^2\,_3-a_3\,^2\bar p^3\,_3}{\mathfrak{h}} \\
0 & \frac{a_2\,^2 \bar p^3\,_2-a_2\,^3\bar p^2\,_2}{\mathfrak{h}} & \frac{a_2\,^2\bar p^3\,_3-a_2\,^3\bar p^2\,_3}{\mathfrak{h}}
\end{array}
\right),$$ where (|p\^I\_i)=(
[ccc]{} |p\^1\_1 & 0 & 0\
0 & |p\^2\_2 & |p\^2\_3\
0 & |p\^3\_2 & |p\^3\_3
), together with $\omega_a^JX^a_I=\delta^J_I$ and $h=(a_1\,^1)^2(a_2\,^3a_3\,^2-a_2\,^2a_3\,^3)^2$ is the determinant of the spatial metric constructed from the vectors $a_i$ and $\bar p^I\,_i$ the momentum dual to $\bar\phi_I\,^i$ satisfying the Poisson bracket $$\label{bracketphipV0}
\{\bar\phi_I\,^i,\bar p^J\,_j\}=\frac{\kappa\gamma}{V_0}\delta_I^J\delta_j^i$$ with the volume $V_0=\int_{{\mathds{T}^3}}d^3x\sqrt{h}$ of ${\mathds{T}^3}$ as measured by the metric $h$. For later purpose we define new variables $$\label{pphi}
\phi_I\,^i = L_I\bar\phi_I\,^i,\quad p^I\,_i=\frac{V_0}{L_I}\bar p^I\,_i,$$ such that $$\label{bracketphip}
\{\phi_I\,^i, p^J\,_j\}=\kappa\gamma\delta_I^J\delta_j^i,$$ where $$\begin{aligned}
L_1=a_1\,^1, \; L_2=\sqrt{(a_2\,^2)^2+(a_2\,^3)^2},\; L_3=\sqrt{(a_3\,^2)^2+(a_3\,^3)^2}.\end{aligned}$$ Thus we conclude that
The classical configuration space $\mathcal{A}_S=\mathds{R}^{5}$ is spanned by the five configuration variables $\phi_I\,^i$. The phase space $\mathcal{P}=\mathds{R}^{10}$ is spanned by $\phi_I\,^i$ and the five momenta $p^J\,_j$ satisfying the Poisson bracket .
Furthermore, note that the determinant of the densitized triad is given by $$\label{detE}
\det E_i^a=\mathfrak{k}\, p^1\,_1( p^2\,_3 p^3\,_2- p^2\,_2 p^3\,_3),$$ where we defined :=.
The relation between the new variables $(\phi_I\,^i,p^J\,_j)$ and the ’scale factors’ $a_a\,^b$ and their respective momenta $p^a\,_b$ can be found by using and the Poisson brackets and . A closed form could only be found for $p^1\,_1$ and is given by $$|p^1\,_1|=|a_2\,^2a_3\,^3-a_2\,^3a_3\,^2|.$$
Constraints in Ashtekar Variables on the Torus
----------------------------------------------
In the canonical variables the Legendre transform of the Einstein-Hilbert action results in a fully constrained system [@Ashtekar:04; @Rovelli:04; @Thiemann:07] $$\label{SAE}
S=\frac{1}{2\kappa}\int_{\mathds{R}}dt\int_{{\mathds{T}^3}}d^3x\left(2\dot A_a^iE^a_i-[\Lambda^jG_j+N^aH_a+N\mathcal{H}]\right),$$ where $G_j$ is the Gauss constraint, $H_a$ the diffeomorphism (or vector) constraint, $\mathcal{H}$ the Hamiltonian and $\Lambda^j$, $N^a$, $N$ are Lagrange multipliers. The Hamiltonian constraint simplifies to $$\label{Cgravdef}
C_{\mathrm{grav}}=-\frac{1}{2\kappa}\int_{{\mathds{T}^3}}d^3xN\epsilon_{ijk}F^i_{ab}\frac{E^{aj}E^{bk}}{\sqrt{|\mathrm{det}E|}}$$ due to spatial flatness, where the curvature of the Ashtekar connection is given by F\_[ab]{}\^i=\_aA\_b\^i-\_bA\_a\^i+\^i\_[jk]{}A\^j\_aA\^k\_b= \^i\_[jk]{}A\^j\_aA\^k\_b. Homogeneity further requires that $N\neq N(x)$. Inserting and into we get \[Cgrav\] C\_=&&-\
&, where we defined $N=\sqrt{L_1 L_2 L_3/V_0}$ in order to simplify the Hamiltonian. Using the Hamiltonian we can compute the time evolution of the basic variables $\phi_i\,^j$ and $p^i\,_j$ (see ). Setting all off-diagonal terms to zero we see that matches with Eq. (3.20) in [@Chiou:07]. If we further set $\phi_{(i)}\,^i=c$ and $p^{(i)}\,_i=p$ we get $$C_{\mathrm{grav}}=-\frac{3}{\kappa\gamma^2}c^2\sqrt{|p|},$$ which is exactly the same result as the homogeneous and isotropic case [@Bojowald:08].
![Solutions corresponding to the Hamiltonian coupled to a massless scalar field with vanishing potential. [*Left panel*]{}: the black thick solid shows the evolution of $\phi_1\,^1$, $\phi_2\,^2$ is the black dashed line, $\phi_3\,^3$ the dotted line, $\phi_2\,^3$ the gray dashdotted one and $\phi_3\,^2$ the solid gray one. [*Right panel*]{}: the black thick solid shows the evolution of $p^1\,_1$, $p^2\,_2$ is the black dashed line, $p^3\,_3$ the dotted line, $p^2\,_3$ the gray dashdotted one and $p^3\,_2$ the solid gray one. In both cases the initial conditions are $\phi_1\,^1=1.0$, $\phi_2\,^2=0.2$, $\phi_3\,^3=0.4$, $\phi_2\,^3=0.6$, $\phi_3\,^2=0.7$, $p^1\,_1=1.0$, $p^2\,_2=0.3$, $p^3\,_3=0.5$, $p^2\,_30.5$, $p^3\,_2=1.4$, $\phi=0.01$ and $p_{\phi}=8.1$. The time $t$ parametrizes the coordinate time in natural units ($c=\kappa=\hbar=1$).[]{data-label="fig:phipoft"}](phioft.eps "fig:"){width="7.0cm"}![Solutions corresponding to the Hamiltonian coupled to a massless scalar field with vanishing potential. [*Left panel*]{}: the black thick solid shows the evolution of $\phi_1\,^1$, $\phi_2\,^2$ is the black dashed line, $\phi_3\,^3$ the dotted line, $\phi_2\,^3$ the gray dashdotted one and $\phi_3\,^2$ the solid gray one. [*Right panel*]{}: the black thick solid shows the evolution of $p^1\,_1$, $p^2\,_2$ is the black dashed line, $p^3\,_3$ the dotted line, $p^2\,_3$ the gray dashdotted one and $p^3\,_2$ the solid gray one. In both cases the initial conditions are $\phi_1\,^1=1.0$, $\phi_2\,^2=0.2$, $\phi_3\,^3=0.4$, $\phi_2\,^3=0.6$, $\phi_3\,^2=0.7$, $p^1\,_1=1.0$, $p^2\,_2=0.3$, $p^3\,_3=0.5$, $p^2\,_30.5$, $p^3\,_2=1.4$, $\phi=0.01$ and $p_{\phi}=8.1$. The time $t$ parametrizes the coordinate time in natural units ($c=\kappa=\hbar=1$).[]{data-label="fig:phipoft"}](poft.eps "fig:"){width="7.0cm"}
Diffeomeorphism and Gauss Constraints {#sec:diffgauss}
-------------------------------------
The Gauss constraint stems from the fact that we chose the densitized triads $E_i^a$ as the momenta conjugated to the connections $A_a^i$. In fact, the spacial metric can be directly obtained from the densitized triads through and is invariant under rotations given by $E_i^a\mapsto O_i^jE_j^a$. In order that the theory be invariant under such rotations the Gauss constraint \[Gaussconstraint\] G\_i=\_aE\^a\_i+\_[ijk]{}A\^j\_aE\^a\_k0 must be satisfied. The diffeomorphism constraint modulo Gauss constraint originates from the requirement of independence from any spatial coordinate system or background and is given by \[diffeoconstraint\] H\_a=F\^i\_[ab]{}E\^b\_i0.
However, as mentioned in [@Ashtekar:91] we have to be careful when dealing with these constraints in the case where the topology is closed. We thus divide this subsection into two parts, starting with the general case of open models.
### Open Models {#sec:openmodels}
Due to spatial homogeneity of Bianchi type I models the basic variables can be diagonalized to [@Bojowald:00; @Bojowald:03] A\^[’i]{}\_a=c’\_[(K)]{}\^[’i]{}\_K\^[’K]{}\_a,E\^[’a]{}\_i=p\^[’(K)]{}\^[’K]{}\_iX\^[’a]{}\_K, where $\omega'$ is the left-invariant 1-form, $X'$ the densitized left-invariant vector field dual to $\omega'$ and $\Lambda'\in SO(3)$[^5]. This choice of variables automatically satisfies the vector and Gauss constraints, thus reducing the analysis of to the Hamiltonian constraint . The homogeneous, anisotropic vacuum solution to the Einstein field equations is called the Kasner solution and is given by the following metric: ds\^2=-d\^2+\^[2\_1]{}dx\_1\^2+\^[2\_2]{}dx\_2\^2+\^[2\_3]{}dx\_3\^2 where the two constraints $\alpha_i\in\mathds{R}$, $\sum \alpha_i=\sum \alpha_i^2=1$ have to be fulfilled. These imply that not all Kasner exponents can be equal, i.e. isotropic expansion or contraction of space is not allowed. By contrast the RW metric is able to expand or contract isotropically because of the presence of matter. At the end, from the twelve-dimensional phase space only two degrees of freedom remains.
An infinitesimal diffeomorphism generated by a vector field $V$ induces the following action on the left-invariant 1-form $\omega'$: \[HPD\] \_a’\_a+\_V’\_a, where $\mathcal{L}_V$ is the Lie derivative along $V$. Such transformations leave the metric homogeneous provided the vector fields satisfy \[condHPD\] V\^a=-(f\_j\^iy\^j)X\^[’a]{}\_i for some constants $f_j^i$ and functions $y^i$ given by $\mathcal{L}_{K_j}y^i=\delta_j^i$ [@Ashtekar:91]. The last equation for $y^i$ relies on the fact that the 3-surface is topologically $\mathds{R}^3$ and the Killing vectors $K_i$ commute. As we shall see below this will not be the case in the closed models.
In the case of rotational symmetry the diffeomorphism constraint is once again satisfied by the choice of variables whereas the Gauss constraint is not. However, in such a case the triad components can be rotated until the Gauss constraint is also satisfied. Further details can be found in [@Ashtekar:06:3].
### The Torus as a Closed Model {#sec:closedmodels}
As we have seen in it is not possible to align the Killing fields with the left-invariant vectors, whence the metric takes the non-diagonal form and the Ashtekar connection the form . In the previous subsection we saw that a diffeomorphism preserves homogeneity provided it satisfies the condition . In the closed model the analysis goes through as well and we find that $V_i$ has to satisfy the same condition . However, since such fields lack the required periodicity in $x^i$ we are led to the conclusion that there are no globally defined, non-trivial homogeneity preserving diffeomorphisms (HPDs) and there is no analog of . Thus, instead of one degree of freedom we get additional degrees of freedom.
The Gauss constraint for a Bianchi type I model is given by \[GaussBianchi\] G\_i=\_[ijk]{}\_I\^ip\^I\_k. With our choice of variables two Gauss constraints are automatically satisfied, namely $G_2=G_3\equiv0$. However, we can still perform a global $SU(2)$ transformation along $\tau_1$ which is implemented in the nonvanishing Gauss constraint \[G1\] G\_1=\_2\^2p\^2\_3+\_3\^2p\^3\_3-\_2\^3p\^2\_2-\_3\^3p\^3\_20 generating simultaneous rotations of the pairs $(\phi_2\,^2,\phi_2\,^3)$, $(p^2\,_2,p^2\,_3)$ resp. $(\phi_3\,^2,\phi_3\,^3)$, $(p^3\,_2,p^3\,_3)$. Thus the norms of these vectors and the scalar products between them are gauge invariant. The Gauss constraint allows us to get rid of e.g. the pair $(\phi_3\,^2,p^3\,_2)$ by fixing the gauge in the following way: we rotate the connection components such that $\phi_3\,^2=0$. Because the length $\|\phi_3\|=\sqrt{(\phi_3\,^2)^2+(\phi_3\,^3)^2}$ is preserved we know that $\phi_3\,^3\neq0$. The Gauss constraint then implies that $p^3\,_2=(\phi_2\,^2p^2\,_3-\phi_2\,^3p^2\,_2)/\phi_3\,^3$. This gauge fixing reduces the degrees of freedom by two units.
The diffeomorphism constraint is given by and since $F_{ab}^i=\epsilon^i\,_{jk}A_a^jA_b^k$ ($\partial_aA_b^i=0$ thanks to homogeneity) we find that \[diffeopropGauss\] H\_a=\^i\_[jk]{}A\_a\^jA\_b\^kE\^b\_iA\^i\_aG\_i. The gauge fixing we just performed ensures that the diffeomorphism constraint also vanishes.
Canonical Transformation {#sec:cantransf}
------------------------
In this subsection we introduce a set of new variables which will greatly simplify the analysis of the kinematical Hilbert space. We first perform a canonical transformation on the unreduced phase space: \[cantransf\] Q\_1=\_1\^1,&& P\^1=p\^1\_1,\
Q\_2=,&& P\^2=\
Q\_3=,&& P\^3=\
\_1=\_k(),&& P\_[\_1]{}=p\^2\_3\_2\^2-p\^2\_2\_2\^3\
\_2=\_k(),&& P\_[\_2]{}=-p\^3\_3\_3\^2+p\^3\_2\_3\^3such that the variables are mutually conjugate: {Q\_I,P\^J}=\_I\^J,{\_,P\_[\_]{}}=\_[,]{}. We choose the convention that the diagonal limit can be recovered by setting $\theta_1=\theta_2=0$. The inverse of this canonical transformation will be important in the sequel and is given by: \[invcantransf\] \_2\^2=Q\_2(\_1),&&\_2\^3= Q\_2(\_1),\
p\^2\_2=P\^2(\_1)-,&&p\^2\_3=+P\^2(\_1),\
\_3\^2= Q\_3(\_2),&&\_3\^3= Q\_3(\_2),\
p\^3\_2= P\^3(\_2)+,&&p\^3\_3=-+ P\^3(\_2).It is important to note that $Q_2,Q_3\in\mathds{R}_+$ and $\theta_1,\theta_2\in [k\pi,(k+1)\pi]$ where we restrict the values of $k$ to be either $k=0$ if $\mathrm{sgn}(\phi_2\,^3)>0$ or $k=1$ if $\mathrm{sgn}(\phi_2\,^3)<0$. If $\mathrm{sgn}(\phi_2\,^3)=0$ then we have the case $k=0$ if $\mathrm{sgn}(\phi_2\,^2)>0$ or $k=1$ if $\mathrm{sgn}(\phi_2\,^2)<0$. The function arc$_1$cos$(x)$ is related to the principal value via arc$_1$cos$(x)=2\pi-$arccos$(x)$. With this convention we can recover unambiguiously from .
The Hamiltonian constraint is given in terms of the new variables by \[Cgravnew\] C\_=\
{2P\^1Q\_1\
+P\^2Q\_2\
+P\_[\_1]{}}Using this Hamiltonian we can compute the time evolution of the basic variables $Q_i$, $\theta_{\alpha}$, $P^i$ and $P_{\theta_{\alpha}}$ (see ). We choose the initial conditions so that they correspond to the values of the old variables (see caption of ). By doing so we are able to check whether the solutions to are equivalent to the solutions to by performing the canonical transformation . The different solutions do indeed match up to a very good accuracy.
![Solutions corresponding to the Hamiltonian coupled to a massless scalar field with vanishing potential. [*Left panel*]{}: the black thick solid shows the evolution of $Q_1$, $Q_2$ is the black dashed line, $Q_3$ the dotted line, $\theta_1$ the gray dashdotted one and $\theta_2$ the solid gray one. [*Right panel*]{}: the black thick solid shows the evolution of $P^1$, $P^2$ is the black dashed line, $P^3$ the dotted line, $P_{\theta_1}$ the gray dashdotted one and $P_{\theta_2}$ the solid gray one. In both cases the initial conditions are $Q_1=1$, $Q_2=0.63$, $Q_3=0.81$, $\theta_1=1.25$, $\theta_2=1.05$, $P^1=1$, $P^2=0.57$, $P^3=1.46$, $P_{\theta_1}=-0.08$, $P_{\theta_2}=0.21$, $\phi=0.01$ and $p_{\phi}=8.1$. The time $t$ parametrizes the coordinate time in natural units ($c=\kappa=\hbar=1$).[]{data-label="fig:QPoft"}](Qoft.eps "fig:"){width="7.0cm"}![Solutions corresponding to the Hamiltonian coupled to a massless scalar field with vanishing potential. [*Left panel*]{}: the black thick solid shows the evolution of $Q_1$, $Q_2$ is the black dashed line, $Q_3$ the dotted line, $\theta_1$ the gray dashdotted one and $\theta_2$ the solid gray one. [*Right panel*]{}: the black thick solid shows the evolution of $P^1$, $P^2$ is the black dashed line, $P^3$ the dotted line, $P_{\theta_1}$ the gray dashdotted one and $P_{\theta_2}$ the solid gray one. In both cases the initial conditions are $Q_1=1$, $Q_2=0.63$, $Q_3=0.81$, $\theta_1=1.25$, $\theta_2=1.05$, $P^1=1$, $P^2=0.57$, $P^3=1.46$, $P_{\theta_1}=-0.08$, $P_{\theta_2}=0.21$, $\phi=0.01$ and $p_{\phi}=8.1$. The time $t$ parametrizes the coordinate time in natural units ($c=\kappa=\hbar=1$).[]{data-label="fig:QPoft"}](Poft.eps "fig:"){width="7.0cm"}
The only nontrivial Gauss constraint is then given by \[G1inPQ\] G\_1=P\_[\_1]{}-P\_[\_2]{}, which vanishes only when $P_{\theta_2}=P_{\theta_1}$. We are free to fix the gauge by setting $\theta_2=0$. The same result can be obtained from the gauge fixing performed in so that $$Q_3=\phi_3\,^3,\quad P^3=p^3\,_3,\quad \theta_2=0\quad\mathrm{and}\quad P_{\theta_2}=P_{\theta_1}.$$ The symplectic structure of the reduced 8-dimensional phase space is given by $$\mathbf{\Omega}=\frac{V_0}{\kappa\gamma}(dQ_1\wedge dP^1+dQ_2\wedge dP^2+dQ_3\wedge dP^3+d\theta_1\wedge dP_{\theta_1}).$$
Kinematical Hilbert Space {#sec:Hkin}
=========================
Holonomies {#sec:holonomyfluxalgebra}
----------
In the last section we parametrized the classical phase space and gave the Hamiltonian in terms of the Ashtekar variables. To quantize the theory we have to select a set of elementary observables which have unambiguous operator analogs. In order to do so we have first to find elementary variables of the 5-dimensional configuration space.
According to [@Bojowald:00:2] the configuration space on the covering space $\tilde \Sigma$ is given by Higgs fields in a single point $x_0$ which is the only point in the reduced manifold $\tilde \Sigma/\tilde S$. In quantum theory these fields are represented as point holonomies associated to the point $x_0$ [@Thiemann:98:4]. On $\tilde S$ we can take the three edges $\xi_1$, $\xi_2$ and $\xi_3$ in order to regularize the point holonomies. However, we would like to apply this construction to a closed manifold. First we note that $\Sigma\cong \tilde S/\Gamma$ such that two elements $g,g'\in \tilde S$ are equivalent if there is an element $\gamma\in\Gamma$ such that $g'=g+\gamma$. We can thus restrict the regularization of the point holomonies to the three edges $X_1$, $X_2$ and $X_3$ meeting at $x_0$ without losing information. Our elementary configuration variables are then the holonomies along straight lines $\gamma_I:[0,1]\rightarrow \Sigma$ defined by the connection $A \propto (\phi_I\,^i)$ [@Bojowald:00:2; @Bojowald:00:3; @Bojowald:02; @Ashtekar:03; @Bojowald:03]. Now the holonomies along $X_1$, $X_2$ resp. $X_3$ are given by $$\begin{aligned}
\label{holonomiesalonga}
h_1^{(\lambda_1)}&=&\exp(\lambda_1\phi_1\,^1\tau_1)=\cos(\lambda_1\phi_1\,^1)+2\tau_1\sin(\lambda_1\phi_1\,^1)\nonumber\\
h_2^{(\lambda_2)}&=&\exp(\lambda_2\phi_2\,^2\tau_2+\lambda_2\phi_2\,^3\tau_3)\\
h_3^{(\lambda_3)}&=&\exp(\lambda_3\phi_3\,^2\tau_2+\lambda_3\phi_3\,^3\tau_3), \nonumber\end{aligned}$$ where $\lambda_I\in(-\infty,\infty)$ and $\lambda_IL_I$ is the length of the edge $I$ with respect to the spatial metric $h_{ab}$. The auxilary Hilbert space is then generated by spin networks associated with graphs consisting of the three edges $\gamma_I$ meeting at the vertex $x_0$.
In open Bianchi type I models the gauge invariant information of the connection can be separated from the gauge degrees of freedom via the relation $\phi_I^i=c_{(I)}\Lambda_I^i$ with $\Lambda\in SO(3)$ so that the holonomies become simple trigonometric functions. In our case the situation is more complicated because the holonomies $h_2$ and $h_3$ cannot be reduced to such functions since h\_\^[(\_)]{}=(\_\_/2)+2(\_\_/2)(), where $\alpha=2,3$ and \_:=.The problem is that this expression cannot be used in this form since there is no well defined operator $\hat \phi_I\,^i$ on the kinematical Hilbert space. Using the canonical transformation we can re-express the holonomies such that \[holinQP\] h\_1\^[(\_1)]{}=(\_1Q\_1/2)+2\_1(\_1Q\_1/2),\
h\_2\^[(\_2)]{}=(\_2Q\_2/2)+2(\_2\_1+\_3\_1)(\_2Q\_2/2),\
h\_3\^[(\_3)]{}=(\_3Q\_3/2)+2(\_2\_2+\_3\_2)(\_3Q\_3/2).Since $\lambda_I\in\mathds{R}$ matrix elements of the exponentials of $Q_1$, $Q_2$ and $Q_3$ form a $C^*$-algebra of almost periodic functions. On the other hand the variables $\theta_{1,2}$ are periodic angles such that only strictly periodic functions $\exp(ik_{\alpha}\theta_{\alpha})\in U(1)$ with $k_{\alpha}\in\mathds{Z}$ are allowed. Thus, any function generated by this set can be written as \[gofQP\] g(Q\_1,Q\_2,Q\_3,\_1,\_2)=\_[\_1,\_2,\_3,k\_1,k\_2]{}\_[\_1,\_2,\_3,k\_1,k\_2]{}\
(i\_1Q\_1+i\_2Q\_2+i\_3Q\_3+ik\_1\_1+ik\_2\_2)with coefficients $\xi_{\lambda_1,\lambda_2,\lambda_3,k_1,k_2}\in\mathds{C}$, generating the $C^*$-algebra $\mathcal{A}_S$. Note that this function is almost periodic in $Q_1$,$ Q_2$ and $Q_3$ and strictly periodic in $\theta_1$ and $\theta_2$. The spectrum of the algebra of the almost periodic functions is called the Bohr compactification $\bar \mathds{R}_B:=\Delta({\mathrm{Cyl}_S})$ of the real line and can be seen as the space of generalized connections [@Ashtekar:03; @Velhinho:07]. Thus the functions provide us a complete set of continuous functions on $\bar\mathds{R}_B\times\bar\mathds{R}_B\times\bar\mathds{R}_B\times S^1\times S^1$. Moreover the Gel’fand theory guarantees that the space $\bar\mathds{R}_B$ is compact and Hausdorff [@Bratteli:79] with a unique normalized Haar measure $d\mu(c)$ such that f(c)d(c):=\_[T]{}\_[-T]{}\^Tf(c)dc.
A Cauchy completion leads to a Hilbert space ${\mathcal{H}^S}$ defined by the tensor product ${\mathcal{H}^S}=\mathcal{H}_B^{\otimes3}\otimes\mathcal{H}_{S^1}^{\otimes2}$ with the Hilbert spaces $\mathcal{H}_B=L^2(\bar\mathds{R}_B,d\mu(c))$ and $\mathcal{H}_{S^1}=L^2(S^1,d\phi)$ of square integrable functions on $\bar\mathds{R}_B$ and the circle respectively, where $d\phi$ is the Haar measure for $S^1$. An orthonormal basis for $\mathcal{H}_B$ is given by the almost periodic functions $\langle Q_I|\mu_I\rangle=\exp(i\mu_IQ_I/2)$ (no summation) with $\mu_I\in\mathds{R}$ with $\langle\mu_I|\mu_I'\rangle=\delta_{\mu_I,\mu_I'}$. Analogously a basis for $\mathcal{H}_{S^1}$ is given by the strictly periodic functions $\langle\theta_{\alpha}|k_{\alpha}\rangle=\exp(ik_{\alpha}\theta_{\alpha})$ with $\langle k_{\alpha}|k'_{\alpha}\rangle=\delta_{k_{\alpha},k'_{\alpha}}$.
We choose a representation where the configuration variables, now promoted to operators, act by multiplication via: (g\_1g\_2)(Q,)=g\_1(Q,)g\_2(Q,).The momentum operators act by derivation in the following way: \[Pop\] P\^I=-i[l\_]{}\^2,P\_[\_]{}=-i[l\_]{}\^2.The eigenstates of all momentum operators are given by |,k&:=&|\_1,\_2,\_3,k\_1,k\_2\
&:=&|\_1|\_2|\_3|k\_1|k\_2with \[eigenvaluesofP\] P\^I|,k=[l\_]{}\^2\_I|,k,P\_[\_]{}|,k=k\_|,k.
The simple form of the momentum operators may suggest that the Hilbert space of LQC on a torus is simply expanded from $L^2(\bar\mathds{R}_B^3)$ to $L^2(\bar\mathds{R}_B^3)\times L^2(U(1)^2)$. However the situation is far more complicated because the important variables for the Gauss and Hamiltonian constraints are not the new momenta $P^I$ and $P_{\theta_{\alpha}}$ but the components $p^I\,_i$ of the triad. In terms of the new canonical variables they are complicated functions of both the configuration and momentum variables, as can be seen from . These expressions cannot be quantized directly since the operators $\hat Q_{2,3}$ fail to be well defined on the Hilbert space. The solution is to consider the momentum operators of the full theory given by a sum of left and right invariant vector fields. In [@Bojowald:00:3] the same strategy was used to show that the triad components $p^I\,_i$ act by derivation. In our case the situation is more complicated since the triad components contain both configuration and momentum variables. The triad operators act on functions in ${\mathcal{H}^S}$ and are given by \[defofpX\] p\^I\_i=-i(X\^[(R)]{}\_i(h\_I)+X\^[(L)]{}\_i(h\_I)),where $X^{(R)}_i(h_I)$ and $X^{(L)}_i(h_I)$ are the right and left invariant vector fields acting on the copy of $SU(2)$ associated with the edge $e_I$ of length 1 and are given by X\^[(R)]{}\_i(h\_I)=,X\^[(L)]{}\_i(h\_I)=.Applying the operators $\hat p^2\,_2$ and $\hat p^2\,_3$ on the function $\tr(h_2)$ we get \[operatorp\] p\^2\_2(h\_2)=2p\^2\_2(\_2Q\_2/2)=i[l\_]{}\^2\_2(\_2Q\_2/2)(\_1),\
p\^2\_3(h\_2)=2p\^2\_3(\_2Q\_2/2)=i[l\_]{}\^2\_2(\_2Q\_2/2)(\_1). We see that the usual expressions for an open topology can be recovered by simply setting $\theta_1=0$. Applying these operators once again we get the expressions: (p\^2\_2)\^2(h\_2)=\^2[l\_]{}\^4\_2\^2(\_2Q\_2/2)=(p\^2\_3)\^2(h\_2), which means that $\cos(\lambda_2Q_2/2)$ is an eigenfunction of both $(\hat p^2\,_2)^2$ and $(\hat p^2\,_3)^2$ with eigenvalue $\gamma^2{l_{\mathrm{Pl}}}^4/2\lambda_2^2$. On the other hand we have p\^2\_2 p\^2\_3(h\_2)=p\^2\_3 p\^2\_2(h\_2)=0.
Quantization: 1. Possibility {#sec:loopquantization}
----------------------------
As previously mentioned we cannot directly quantize the expressions because $\hat Q_I$ does not exist as multiplication operator on ${\mathcal{H}^S}$. In a loop quantization only holonomies of the connections are represented as well-defined operators on ${\mathcal{H}^S}$. Thus we replace every configuration variable $Q_I$ in by $\sin(\delta_I Q_I/2)/\delta_I$ [@Bojowald:04], where $\delta_I\in\mathds{R}\backslash\{0\}$ plays the role of a regulator, and compare it with the results just obtained in terms of left and right invariant vector fields. For later purpose we order the operators in a symmetrical way get the following operators acting on functions of ${\mathcal{H}^S}$: \[operators\] \_2\^2=\_1,&&p\^2\_2=\_1P\^2-P\_[\_1]{},\
\_2\^3= \_1,&&p\^2\_3=\_1P\^2+P\_[\_1]{},\
\_3\^2= \_2,&&p\^3\_2= \_2P\^3+P\_[\_2]{},\
\_3\^3= \_2,&&p\^3\_3= \_2P\^3-P\_[\_2]{}.Applying e.g. the operator $\hat p^2\,_2$ on $\cos(\lambda_2Q_2/2)$ with the definitions we see that we obtain the same result as for $\delta=1$. This is not surprising in view of the fact that we defined the operator $\hat p^I\,_i$ in with holonomies along edges $e_I$ of length 1.
This substitution is problematic since the configuration variables $Q_{2,3}$ are by definition positive (see ). Therefore, for $Q_{2,3}\rightarrow \sin(\delta_{2,3}Q_{2,3})/\delta_{2,3}$ to be valid we restrict the analysis to the domain $0<Q_{2,3}<\pi$. In the diagonal case the situation is less problematic because the configuration variable $c$ is arbitrary such that $\sin(\delta c)$ is also allowed to be negative.
Classically, since the change of variables is a canonical transformation the symplectic structure is conserved, i.e. the Poisson bracket between $p^2\,_2$ and $p^2\,_3$ vanishes: $$\left\{p^2\,_2,p^2\,_3\right\}_{Q,P}=0$$ A quantization of the above expression is obtained with the substitution $\{,\}\rightarrow -i[,]\hbar$ such that the commutator between $\hat p^2\,_2$ and $\hat p^2\,_3$ should also vanish. However, the consequence of the substitution of $1/Q_I$ by $\delta_I/\sin(\delta_I Q_I)$ is that commutator between these two variables doesn’t vanish anymore: \[commp22p23\]f(Q\_2,\_1)=-\^2[l\_]{}\^4\_2Formally we can recover the classical limit by taking the limit $$\lim_{\delta_2\rightarrow0}[\hat p^2\,_2,\hat p^2\,_3]f(Q_2,\theta_1)=0,$$ which however fails to exist on ${\mathcal{H}^S}$.
The operators $\hat p^I\,_i$ are partial differential operators with periodic coefficients in both $\theta$ and $Q$. In spherically symmetric quantum geometry a similar situation arises when considering the quantization of a nondiagonal triad component [@Bojowald:04]. However the expression of this component reduces to a Hamiltonian whose eigenvalues are discrete. In our case the situation is more complicated.
### Quantization of $p^2\,_2$
In order to find eigenfunctions of the triad operators let us consider an operator of the form A\_:=-i+i.A substitution $\xi=\delta Q$ shows that $\hat A_{\delta}=\delta \hat A_1\equiv\delta \hat A$ so that it is sufficient to determine the spectrum for $\delta=1$. This operator is symmetric on $\mathcal{H}_A:=L^2(\bar{\mathds{R}}_B,d\mu_B)\otimes L^2(U(1))$: $$\langle f,\hat A g\rangle=\langle\hat Af,g\rangle,\quad\forall f,g\in\mathcal{D}(\hat A),$$ where $\mathcal{D}(\hat A)\subset\mathcal{H}_A$ is the domain of $\hat A$. The eigenfunctions of $\hat A$ are obtained by solving $\hat A f_{\lambda}(\xi,\theta)=\lambda f_{\lambda}(\xi,\theta)$, i.e. \[PDE\] -i+i+f\_(,)=f\_(,),where we constrain $\xi$ to be in the interval $[0,\pi]$ in order to avoid negative values of $\sin\xi$. We look for a solution of the form $w=w(\xi,\theta)$ [@Kamke:79] satisfying -i+i=(-)f\_such that the characteristic functions are given by \[chareq\] =-i(t),=i\_=(-) f\_(t),where the dot is the time derivative. Combining the first two equations gives after integration \[C1\] =C\_1,meaning that every $C^1$-function $\Omega_1(\sin\theta\tan(\xi/2))$ solves the left-hand side of . In order to solve for $\lambda\neq0$ we first note that \[cosoft\](t)=((C\_1(/2)))=i.An integration of this equation gives the result \[solfort\] t=-i,where $$a=\sqrt{1+C_1^2} \quad\mathrm{and}\quad b=\sqrt{-1+C_1^2+\cos\xi(1+C_1^2)}.$$ The last characteristic equation in can be written as $$\dot f_{\lambda}=\frac{\partial f_{\lambda}}{\partial\xi}\dot\xi=\left(\lambda-\frac{i}{2}\frac{\cos\theta}{\sin\xi}\right)f_{\lambda}$$ such that $$\frac{\partial f_{\lambda}}{\partial\xi}=\left(i\frac{\lambda}{\cos\theta}+\frac{1}{2\sin\xi}\right)f_{\lambda}.$$ can be inserted into the last equation such that after an integration we get the result $$\log f_{\lambda}=\lambda t+\log\left(\sqrt{\tan(\xi/2)}\right)+C,$$ where $t$ is given by and $C$ is an integration constant. The final solution to the PDE is thus given by \[flambda\] f\_(,)=&&\_1\
&&((/2)\_1+\_1)\^[-i]{}\_1((/2)),where \_1(,)&=&\
\_1(,)&=&. The $C^1$-function $\Omega_1(\sin\theta\tan(\xi/2))$ can be determined by e.g. boundary conditions. For simplicity we set $\Omega_1(\sin\theta\tan(\xi/2))\equiv1$ subsequently. As a cross-check we see that the first line of solves $$-i\cos\theta\frac{\partial\sqrt{\tan(\xi/2)}}{\partial\xi}+\frac{i}{2}\frac{\cos\theta}{\sin\xi}\sqrt{\tan(\xi/2)}=0$$ and the second one the eigenvalue problem of the operator $\hat A$. The scalar product on $\mathcal{H}_A$ is given by \[scalarproductH\]f\_,f\_[’]{}=\_[T]{}\_[-T]{}\^[T]{}d\_0\^[2]{}d|f\_f\_[’]{}.The integral of $|\sqrt{\tan(\xi/2)}|^2$ over one period is not finite and since the second line of never vanishes the eigenfunctions $f_{\lambda}$ are not normalizable in $\mathcal{H}_A$. We could choose the function $\Omega_1\propto(\tan(\xi/2))^{-1/2}$ but we would automatically get the factor $(\sin\theta)^{-1/2}$ which is also not normalizable. The surprising implication is that the spectrum of $\hat A$ is continuous. Note that the function $\alpha_1$ is always real while $\beta_1$ is always purely imaginary ($\lim_{\xi\rightarrow\pi/2}\beta_1=i\cos\theta)$. The exponent of $f_{\lambda}$ is thus always real, implying that $f_{\lambda}$ is uniquely determined.
### Self-adjointness of $\hat A$ {#SAofA}
In the previous section we constructed a symmetric operator $\hat A$ with respect to the scalar product of $\mathcal{H}_A$, i.e. $\hat A=\hat A^+$ with domain $\mathcal{D}(\hat A)\subset\mathcal{D}(\hat A^+)$. In this subsection we give a possible domain for $\hat A$ and check if there exists a self-adjoint extension of $\hat A$.
In analogy with [@Shubin:74; @Shubin:78] define the space CAP$(\mathds{R})$ of the (uniform) almost periodic functions[^6] such that its completion is the Hilbert space $L^2(\bar{\mathds{R}}_B)$. The Sobolev space $H^1(\bar{\mathds{R}}_B)$ is given by the completion of the space of trigonometric polynomials Trig$(\mathds{R})$ in the Sobolev norm $\|f\|_{H^1}^2=\|f\|^2_{L^2(\bar{\mathds{R}}_B)}+\|f'\|^2_{L^2(\bar{\mathds{R}}_B)}$, i.e. $H^1(\bar{\mathds{R}}_B)$ consists of all almost periodic functions $f\in \mathrm{CAP}(\mathds{R})$ such that $f'\in \mathrm{CAP}(\mathds{R})$.
Let the differential operator $\hat p:=-i\frac{d}{d\xi}$ on $L^2(\bar{\mathds{R}}_B)$ have the domain of definition Trig$(\mathds{R})$. Then its closure has the domain $H^1(\bar{\mathds{R}}_B)$. The adjoint operator to $\hat p$ on $L^2(\bar{\mathds{R}}_B)$ has also the domain $H^1(\bar{\mathds{R}}_B)$ and coincides with $\hat p^+$ on it. Since $\hat p=\hat p^+$, $\hat p$ is essentially self-adjoint on Trig$(\mathds{R})$ [@Shubin:74; @Shubin:78].
Since every almost periodic function $f(x)$ is bounded a necessary condition for the inverse $f^{-1}(x)$ to be almost periodic is that $\min_x |f(x)|\neq0$. It follows that $\sin^{-1}\xi$ is not an almost periodic function. We thus define the domain \[domainofA\](A):={H\^1(|\_B)H\^1(U(1))|(k,)=0=’(k,),k},which, according to [@Roberts:66; @Roberts:66:2], is dense. Any function $\varphi\in\mathcal{D}(\hat A)$ removes the pole caused by $\sin^{-1}\xi$, i.e. we require that $\lim_{\xi\rightarrow k\pi} \varphi(\xi)(\sin\xi)^{-1}=0$ and $\lim_{\xi\rightarrow k\pi} \varphi'(\xi)(\sin\xi)^{-1}=0$. $k\in\mathds{Z}$. On the other hand, thanks to $\sin\theta$ in front of the differential operator $i\partial/\partial\theta$, the boundary term of an integration by part is automatically annihilated so that no boundary conditions on $\theta$ have to be imposed. Moreover the deficiency indices $n_{\pm}$ for $\hat A$ are defined by $$n_{\pm}:=\mathrm{dim}\,\mathrm{ker}(\hat A^+\mp i).$$ The solutions to this equation do not lie in $\mathcal{D}(\hat A^+)$ such that $n_{\pm}=0$. It follows that the operator $\hat A$ is essentially self-adjoint.
### Quantization of $p^2\,_3$
The eigenfunctions of $\hat p^2\,_3$ can be obtained by applying the same procedure on the symmetrized operator $$\hat B:=-i\sin\theta\frac{\partial}{\partial \xi}-i\frac{\sqrt{\cos\theta}}{\sin\xi}\frac{\partial}{\partial\theta}\sqrt{\cos\theta}$$ The eigenfunctions $g_{\lambda}(\xi,\theta)$ are given by \[glambda\] g\_(,)=\
((/2)\_2+\_2)\^[-i]{}\_2((/2)),where \_2&=&\
\_2&=& and $\Omega_2$ is any $C^1$-function that can be determined by boundary conditions. While the function $\beta_2$ is always purely imaginary the function $\alpha_2$ is only real when $\cot^2\xi/2<\cos^2\theta$. This means that $g_{\lambda}$ is not uniquely determined. We can write $g_{\lambda}$ as $$g_{\lambda}(\xi,\theta)=\frac{k_2}{\sqrt{\tan(\xi/2)}}e^{F_1(\xi,\theta)\ln F_2(\xi,\theta)}$$ with the logarithm is defined by $\ln F_2=\mathrm{Ln} F_2+2\pi in$, where $n\in \mathds{Z}$ and $\mathrm{Ln}$ is the principal value of the logarithm. Inserting this solution into the eigenvalue problem $\hat B g_{\lambda}=\lambda g_{\lambda}$ it can be shown that there is only a solution for $n=0$. The eigenfunctions $g_{\lambda}$ are not normalizable since the integral of $1/|\tan(\xi/2)|$ over one period is not finite.
As for $f_{\lambda}$ we are led to the conclusion that the spectrum of $\hat B$ is continuous. We can construct a dense subspace along the lines described in , the only difference being that $g_{\lambda}$ has poles at $\xi=2k\pi$ and $\theta=(2k+1)\pi/2$ whereas $f_{\lambda}$ has poles at $\xi=(2k+1)\pi$, $k\in\mathds{Z}$.
![Absolute value of the eigenfunctions $f_{\lambda}(\xi,\theta)$ (left panel) and $g_{\lambda}(\xi,\theta)$ (right panel). The black thick line is the eigenfunction for $\lambda=1$, $\theta=1$ and the black dashed line for $\lambda=2$, $\theta=1$.[]{data-label="fig:EF"}](EF.eps "fig:"){width="7.2cm"}![Absolute value of the eigenfunctions $f_{\lambda}(\xi,\theta)$ (left panel) and $g_{\lambda}(\xi,\theta)$ (right panel). The black thick line is the eigenfunction for $\lambda=1$, $\theta=1$ and the black dashed line for $\lambda=2$, $\theta=1$.[]{data-label="fig:EF"}](EFg.eps "fig:"){width="7.2cm"}
![Absolute value of the eigenfunctions $f_{\lambda}(\xi,\theta)$ (left panel) and $g_{\lambda}(\xi,\theta)$ (right panel). The black thick line is the eigenfunction for $\lambda=1$, $\xi=2$ and the black dashed line for $\lambda=2$, $\xi=2$.[]{data-label="fig:EFth"}](EFth.eps "fig:"){width="7.2cm"}![Absolute value of the eigenfunctions $f_{\lambda}(\xi,\theta)$ (left panel) and $g_{\lambda}(\xi,\theta)$ (right panel). The black thick line is the eigenfunction for $\lambda=1$, $\xi=2$ and the black dashed line for $\lambda=2$, $\xi=2$.[]{data-label="fig:EFth"}](EFthg.eps "fig:"){width="7.2cm"}
Quantization: 2. Possibility {#sec:Schrquantization}
----------------------------
In order to quantize the triad components $\hat p^I\,_i$ we replaced the configuration variables $Q_I$ with $\sin(\delta _IQ_I)/\delta_I$. The question we may ask is to what extend this substitution changes the eigenfunctions. We define the symmetrized operator $\hat A_2$ quantized without the substition of $Q_I$ as $$\hat A_2=-i\cos\theta\frac{\partial}{\partial \xi}+i\frac{\sin\theta}{\xi}\frac{\partial}{\partial\theta}+\frac{i}{2}\frac{\cos\theta}{\xi}.$$ The solution to the eigenvalue problem $\hat A_2f_{\lambda}(\xi,\theta)=\lambda f_{\lambda}(\xi,\theta)$ is given by $$f_{\lambda}(\xi,\theta)=\exp\left(i\lambda\xi\cos\theta\right)\sqrt{\xi}\Gamma(\log(\xi\sin\theta)),$$ We see that the eigenfunctions are not almost periodic in $\xi$. However we can choose the function $\Gamma$ such that $\sqrt{\xi}$ disappears, i.e. we set $$\Gamma=\mathcal{N}_1\exp\left(-\frac{1}{2}\log(\xi\sin\theta)\right),$$ where $\mathcal{N}_1$ is a constant, such that the eigenfunctions to $\hat A_2$ are given by f\_(,)=\_1 . The above eigenfunction is almost periodic in $\xi$ but fails to be normalizable on $L^2(\bar\mathds{R}_B)\otimes L^2(U(1))$. As in the preceding section the spectrum of $\hat A_2$ is thus continuous. Note that the eigenfunction is constant in the non-diagonal limit $\theta\rightarrow\pi/2$.
Similarly the eigenfunctions of the symmetrized operator $$\hat B_2=-i\sin\theta\frac{\partial}{\partial \xi}-i\frac{\cos\theta}{\xi}\frac{\partial}{\partial\theta}+\frac{i}{2}\frac{\sin\theta}{\xi}$$ are given by g\_(,)=\_2.The diagonal limit $\theta\rightarrow0$ of $g_{\lambda}$ is just a constant function such that $\hat p^2\,_3$, i.e. the expectation value $\langle \hat p^2\,_2\rangle$ measures the ’diagonality’ of the torus and $\langle \hat p^2\,_3\rangle$ its departure. Once again, the eigenfunctions $g_{\lambda}$ fail to be normalizable on the Hilbert space such that the spectrum of $\hat B_2$ is continuous. Also note that contrary to the first method the commutator between both operators vanishes: $$[\hat A_2,\hat B_2]=0.$$
Volume Operator {#sec:kinop}
---------------
The classical expression for the volume of $V$ is given by $$\mathbf{\mathcal{V}}(V)=\int_V\sqrt{\left|\frac{1}{6}\epsilon_{abc}\epsilon_{ijk}E^{ai}E^{bj}E^{ck}\right|}d^3x.$$ Inserting the definition of the homogeneous densitized triad we get: $$\label{volume}
\mathbf{\mathcal{V}}(V)=\sqrt{\mathfrak{k}\left|p^1\,_1(p^2\,_2p^3\,_3-p^2\,_3p^3\,_2)\right|}$$ The factor $\mathfrak{k}$ depends on the specific form of the torus and is equal to one if the torus is cubic such that we recover the usual expression in this limit (see e.g. Eq. (4.5) in [@Chiou:07]). Using the classical solution of the Gauss constraint we get the following expression for the physical volume of the torus: $$\mathbf{\mathcal{V}}(V)=\sqrt{\mathfrak{k}\left|p^1\,_1\left[p^2\,_2p^3\,_3-p^2\,_3\frac{\phi_2\,^2p^2\,_3-\phi_2\,^3p^2\,_2}{\phi_3\,^3}\right]\right|}$$ or in terms of the new variables $$\begin{aligned}
\label{volumeGauss}
\mathbf{\mathcal{V}}(V)&=&\sqrt{\mathfrak{k}\left|\frac{P^1}{Q_2Q_3}\right|}\times\Bigl|\left((P_{\Theta})^2-P^2P^3Q_2Q_3\right)\cos\Theta+\nonumber\\
&&\quad\quad\quad\quad\quad\quad\quad+P_{\Theta}(P^2Q_2+P^3Q_3)\sin\Theta\Bigr|^{1/2}.\end{aligned}$$
### Quantization of the Volume Operator according to 1. Method {#sec:loopvolop}
To perform a quantization of the volume operator we insert the definitions into $\hat \mathbf{\mathcal{V}}(V)$. Despite the fact that we know the eigenfunctions of the operators $\hat p^I\,_i$ it is not straightforward to give the eigenfunctions of the volume operator $\hat\mathcal{V}$ because, as explained in , they do not necessarily commute. Thus the difficult task is to determine the spectrum of the operator \[volop\]:=p\^2\_2p\^3\_3-p\^3\_2p\^2\_3&=&-\
&&++.However, this operator is not symmetric on ${\mathcal{H}^S}$. Let us define the symmetric operator $$\hat \mathfrak{V}^S:=\frac{1}{2}\left(\hat \mathfrak{V}^++\hat\mathfrak{V}\right).$$ A calculation shows that $\hat\mathfrak{V}^S$ is given by $$\hat\mathfrak{V}^S=\hat\mathfrak{V}+\frac{1}{2}\left(-\frac{\cos\Theta}{\sin Q_2\sin Q_3}-2\frac{\sin \Theta}{\sin Q_2\sin Q_3}\frac{\partial}{\partial\Theta}+\frac{\cos\Theta}{\sin Q_3}\frac{\partial}{\partial Q_2}+\frac{\cos\Theta}{\sin Q_2}\frac{\partial}{\partial Q_3}\right)$$ This operator is rather complicated and no analytic solutions to the eigenvalue problem could be found.
### Quantization of the Volume Operator according to 2. Method {#sec:Schrvolop}
In this subsection we consider the quantization of $\mathcal{V}$ as described in where the commutator between $\hat p^I\,_i$ and $\hat p^J\,_j$ vanishes. This fact simplifies dramatically the analysis because the (generalized) eigenvalue problem can now be written in terms of products and sums of the eigenfunctions of the $\hat p^I\,_i$. Let us define $$T_{\lambda_1,\lambda_{22},\lambda_{23},\gamma_{33},\gamma_{32}}:=\mathcal{N}_{\lambda_1}\otimes(f_{\lambda_{22}}g_{\lambda_{23}})\otimes (f'_{\gamma_{33}}g'_{\gamma_{32}}),$$ where $f_{\gamma}(Q_2,\theta_1)$, $g_{\gamma}(Q_2,\theta_1)$, $f'_{\gamma}(Q_3,\theta_2)$ and $g'_{\gamma}(Q_3,\theta_2)$ are the (generalized) eigenfunctions of $\hat p^2\,_2$, $\hat p^2\,_3$, $\hat p^3\,_3$ and $\hat p^3\,_2$ respectively given in . Furthermore we denoted the eigenfunctions of $\hat p^1\,_1$ by $\mathcal{N}_{\lambda_1}:=\langle Q_1|\lambda_1\rangle$. Since we have $$(f_{\lambda_{22}}g_{\lambda_{23}})(f'_{\gamma_{33}}g'_{\gamma_{32}})\propto\frac{\exp\left(iQ_2(\lambda_{22}\cos\theta_1-\lambda_{23}\sin\theta_1)\right)}{\sqrt{\sin\theta_1\cos\theta_1}}\frac{\exp\left(iQ_3(\gamma_{33}\cos\theta_2-\gamma_{32}\sin\theta_2)\right)}{\sqrt{\sin\theta_2\cos\theta_2}}$$ we see that $T_{\lambda_1,\lambda_{22},\lambda_{23},\gamma_{33},\gamma_{32}}$ is not normalizable in $\mathcal{H}^S$. The generalized eigenvalue problem is thus given by T\_[\_1,\_[22]{},\_[23]{},\_[33]{},\_[32]{}]{}\[\]=T\_[\_1,\_[22]{},\_[23]{},\_[33]{},\_[32]{}]{}\[\]\
=\^[3/2]{}[l\_]{}\^3T\_[\_1,\_[22]{},\_[23]{},\_[33]{},\_[32]{}]{}\[\]for $\varphi\in\mathcal{D}(\hat\mathcal{V})$.
Quantum Gauss Constraint {#sec:Gaussconstraint}
------------------------
In we computed the classical Gauss constraint for a Bianchi type I model. In the open case the elementary variables can always be diagonalized such that both the diffeomorphism and Gauss constraints are automatically satisfied. In the closed model this is not the case anymore so that a quantization of the constraints is mandatory. Since in Bianchi type I models the diffeomorphism constraint is proportional to the Gauss constraint we only need to quantize and solve the latter. However, contrary to the diffeomorphism constraint the Gauss constraint can be quantized infinitesimally.
A gauge transformation of an $su(2)$-connection is given by AA’=\^[-1]{}A+\^[-1]{}dwhere $\lambda:\Sigma\mapsto SU(2)$. Infinitesimally we can write this equation as A\_a\^iA\^[’i]{}\_a=A\_a\^i+\_a\^i+\^i\_[jk]{}\^jA\_a\^k+(\^2).The classical Gauss constraint ensuring $SU(2)$-invariance is given by G()=-\_[[\^3]{}]{}d\^3xE\^a\_jD\_a\^jwhere $D_a\Lambda^j=\partial_a\Lambda^j+\epsilon^j\,_{kl}A_a^k\Lambda^l$ is the covariant derivative of the smearing field $\Lambda^j$. The infinitesimal quantization of this expression yields an operator containing a sum of right and left invariant vector fields over all vertices and edges of a given graph $\alpha$. This operator is essentially self-adjoint and can, by Stone’s theorem, be exponentiated to a unitary operator $U_{\phi}$ defining a strongly continuous one-parameter group in $\phi$. Usually, in order to find the kernel of the Gauss constraint operator one restrict the scalar product on $\mathcal{H}_{\mathrm{aux}}$ to the gauge-invariant scalar product on $\mathcal{H}^G_{\mathrm{inv}}$. This Hilbert space is a true subspace of $\mathcal{H}_{\mathrm{aux}}$ since zero is in the discrete part of the spectrum of the Gauss constraint operator.
We saw in that thanks to the symmetry reduction two of the Gauss constraints are automatically satisfied. While the nonvanishing Gauss constraint is still a complicated function in $\phi_I\,^i$ and $p^J\,_j$ it simplifies to after the canonical transformation. A quantization of this expression is then given by G\_1=P\_[\_1]{}-P\_[\_2]{}.Since the eigenstates of the momentum operators $\hat P_{\theta_{\alpha}}$ are the strict periodic functions satisfying the action of the Gauss constraint on ${|\mathbf{\vec\mu,\vec k}\rangle}$ is given by G\_1[|]{}=(k\_1-k\_2)[|]{}which vanishes if k\_1=k\_2.We can thus introduce a new variable $\Theta:=\theta_1+\theta_2$ such that the algebra $\mathcal{A}_S$ given by reduces to the invariant algebra $\mathcal{A}_S^{\mathrm{inv}}$ generated by the functions \[ginv\] g(Q\_1,Q\_2,Q\_3,)&=&\_[\_1,\_2,\_3,k]{}\_[\_1,\_2,\_3,k]{}\
&&(i\_1Q\_1+i\_2Q\_2+i\_3Q\_3+ik).A Cauchy completion leads to the invariant Hilbert space $\mathcal{H}^S_{\mathrm{inv}}=\mathcal{H}_B^{\otimes 3}\times\mathcal{H}_{S^1}$. A comparison with ${\mathcal{H}^S}$ shows that we ’lost’ one Hilbert space $\mathcal{H}_{S^1}$ by solving the quantum Gauss constraint. Furthermore, instead of two momentum operators conjugated to $\theta_1$ and $\theta_2$ we have just one momentum operator conjugated to $\Theta$ defined by $$\hat P_{\Theta}=-i\gamma{l_{\mathrm{Pl}}}^2\frac{\partial}{\partial\Theta}.$$ The eigenstates of all momentum operators are given by $$|\vec\mu,k\rangle:=|\mu_1,\mu_2,\mu_3,k\rangle,$$ where $k\in\mathds{Z}$ defines the representation of $U(1)$.
Conclusions and Outlook {#sec:conclusions}
=======================
In this paper we studied how a torus universe affects the results of LQC. To do so we first introduced the most general tori using Thurston’s theorem and found that six Teichmüller parameters are needed. We construted a metric describing a flat space but respecting the periodicity of the covering group used to construct the torus and used it to derive a gravitational Hamiltonian. We studied the dynamics of a torus universe driven by a homogeneous scalar field by numerically solving the full Hamiltonian and saw that its form only remains cubic if all off-diagonal terms vanish. The Ashtekar connection and the densitized triad for a torus were then derived for both the most general and a slighty simplified torus. The reason for this simplification was that a simple solution to the Gauss constraint could be given. We also derived the Hamiltonian constraint in these new variables and showed that it reduces to the standard constraint of isotropic LQC in case of a cubical torus.
The passage to the quantum theory required a canonical transformation so as to be able to write the holonomies as a product of strictly and almost periodic functions. A Cauchy completion then led to a Hilbert space given by square integrable functions over both $\mathds{R}_B$ and $U(1)$. However the drawback of the canonical transformation is a much more complicated expression for the components of the densitized triad containing both the momentum and the configuration variables. Following the standard procedure of LQC we substituted these configuration variables with the sine thereof and were able to solve the eigenvalue problem analytically. Surprisingly it turned out that the (generalized) eigenfunctions of the triad operators do not lie in the Hilbert space, i.e. the spectrum is continuous. On the other hand we were also able to find almost periodic solutions to the eigenvalue problem of the triad operators without performing the substitution just described, but once again these eigenfunctions do not lie in the Hilbert space. The reason why both ways lead to a continuous spectrum is the non-cubical form of the torus, for if we set the angles $\theta_{1,2}=0$ in the triads correspond to the ones obtained in isotropic models. Furthermore we were able to find the spectrum of the volume operator for the second case because, contrary to the first case, it is a product of commutating triad operators.
Although we gave a couple of numerical solutions to the classical Hamiltonian we didn’t consider its quantization. The constraint describing quantum dynamics of a torus is given by inserting the holonomies into Thiemann’s expression for the quantum Hamiltonian [@Thiemann:98] C\_\^[ijk]{}(h\_i\^[(\^0\_i)]{}h\_j\^[(\^0\_j)]{}(h\_i\^[(\^0\_i)]{})\^[-1]{}(h\_j\^[(\^0\_j)]{})\^[-1]{}h\_k\^[(\^0\_k)]{}\[(h\_k\^[(\^0\_k)]{})\^[-1]{},V\]). Contrary to LQG and LQC we saw that the spectrum of the volume operator of a torus is continuous. It would thus be very interesting to know how far $\hat C_{\mathrm{grav}}$ departs from the usual difference operator of LQC. Furthermore, whether a quantization of the torus a la LQG removes the Big Band singularity needs also to be addressed, especially since we saw that many characteristics of both LQG and isotropic LQC are not present in this particular topology.
In this work we only considered the simplest closed flat topology but there are many other closed topologies. As we saw there are eight geometries admitting at least one compact quotient. For example there are six different compact quotients with covering $\mathds{E}^3$, namely $\mathds{T}^3$, $\mathds{T}^3/\mathds{Z}_2$, $\mathds{T}^3/\mathds{Z}_3$, $\mathds{T}^3/\mathds{Z}_4$, $\mathds{T}^3/\mathds{Z}_6$ and a space where all generators are screw motions with rotation angle $\pi/2$. It would be interesting to know how these discrete groups $\mathds{Z}_{2,3,4,6}$ affect the results of this work, especially since the last five spaces are inhomogeneous (observer dependent) [@Fagundes:92].
I would like to thank Martin Bojowald, Frank Steiner and Jan Eric Sträng for many useful comments and corrections of previous versions of this manuscript.
Fundamental Domain of the Torus {#sec:fundamentaldomain}
===============================
In two dimensions the upper half-plane $H$ is the set of complex numbers $H=\{x+iy\,|\,y>0;\,x,y\in\mathds{R}\}$. When endowed with the Poincaré metric $ds^2=(dx^2+dy^2)/y^2$ this half-plane is called the Poincaré upper half-plane and is a two-dimensional hyperbolic geometry. The special linear group $SL(2,\mathds{R})$ acts on $H$ by linear fractional transformations $z\mapsto (az+b)/(cz+d)$, $a,b,c,d\in\mathds{R}$, and is an isometry group of $H$ since it leaves the Poincaré metric invariant . The modular group $SL(2,\mathds{Z})\subset SL(2,\mathds{R})$ defines a fundamental domain by means of the quotient space $H/SL(2,\mathds{Z})$. This fundamental domain parametrizes inequivalent families of 2-tori and can thus be identified as the configuration space of the two-dimensional torus. Since we consider a three-dimensional torus with six independent Teichmüller parameters (see ) we need a generalization of the Poincaré upper half-plane [@Gordon:87; @Terras:88] to a six-dimensional upper half-space.
A fundamental domain $D$ for $SL(3,\mathds{Z})$ is a subset of the space $\mathscr{P}_3:=\{A\in Mn(3,\mathds{R})\,|\,A^T=A,\;A\;\mathrm{positive} \; \mathrm{definite}\}$ which is described by the quotient space $\mathscr{P}_3/SL(3,\mathds{Z})$. In other words, if both $A\in\mathscr{P}_3$ and $A[g]:=g^TAg$, $g\in SL(3,\mathds{Z})$, are in the fundamental domain then either $A$ and $A[g]$ are on the boundary of the fundamental domain or $g=id$.
Since $\mathscr{P}_3$ is a subspace of the six-dimensional Euclidean space (there are six independent matrix elements for $A\in\mathscr{P}_3$), the generalization of the Poincaré upper half-plane is now a six-dimensional upper half-space $U^6:=\{(a_1,\ldots,a_6)\in \mathds{E}^6\,|\,a_6>0\}$ upon which the group $SL(3,\mathds{R})$ acts transitively. To identify $\mathscr{P}_3$ with an upper half-space we introduce the Iwasawa coordinates such that $\forall \, A\in\mathscr{P}_3$ there is the unique decomposition: $$A=\left(
\begin{array}{ccc}
y_1 & 0&0\\
0&y_2&0\\
0&0&y_3
\end{array}\right)\left(
\begin{array}{ccc}
1&x_1&x_2\\
0&1&x_3\\
0&0&1
\end{array}\right),$$ with $x_i,y_j\in\mathds{R}$ with $\prod y_i=1$. The geometry of the upper half-space $U^6$ is given by the $GL(3,\mathds{R})$-invariant line element $$\begin{aligned}
\label{dsuhs}
ds^2=\mathrm{tr}((A^{-1}dA)^2)=&&\frac{dy_1\,^2}{y_1\,^2}+\frac{dy_2\,^2}{y_2\,^2}+\frac{dy_3\,^2}{y_3\,^2}.\end{aligned}$$ Note that the Ricci scalar of the metric is constant and negative.
In order to give a parametrization of the fundamental region we use Minkowski’s reduction theory [@Minkowski:05], which tells us that for a metric $h_{i,j}$ the following inequalities must be satisfied: $$\begin{aligned}
h_{i,i}&\leq&h_{i+1,i+1},\quad i=1,2,3\nonumber\\
h_{i,j}&\leq&\frac{1}{2}h_{i,i},\quad i<j.\end{aligned}$$ Since the metric is invariant under the map $a_3\,^3\rightarrow -a_3\,^3$ we can define the upper half-space as $U^6=\{(a_1\,^1,a_2\,^1,a_2\,^2,a_3\,^1,a_3\,^2,a_3\,^3)\in\mathds{E}^6\,|\,a_3\,^3>0\}$, where we have identified the element $a_6$ with $a_3\,^3$. In our parametrization we therefore obtain the fundamental domain $D$ delimited by the inequalities: $$\begin{aligned}
(a_1\,^1)^2\leq (a_2\,^1)^2+(a_2\,^2)^2\leq (a_3\,^1)^2+(a_3\,^2)^2+(a_3\,^3)^2\nonumber \\
a_2\,^1\leq\frac{1}{2}a_1\,^1\nonumber \\
a_3\,^1\leq\frac{1}{2}a_1\,^1 \nonumber \\
a_2\,^1a_3\,^1+a_2\,^2a_3\,^2\leq \frac{1}{2}\left((a_2\,^1)^2+(a_2\,^2)^2\right)\end{aligned}$$
The first inequality tells us that the length of the generators of the torus are ordered: $\|a_1\|\leq\|a_2\|\leq\|a_3\|$. However, starting with such an ordered triplet does not necessarily imply that the order is preserved by dynamics. Thus we think that it may be more appropriate to choose the equivalent representation of the configuration space given by $\mathcal{C}=\mathds{R}^6$. Otherwise, we would have to rotate the coordinate system every time the torus leaves the fundamental domain. Note that similar results have also been obtained in M-theory, where one considers a compactification of the extra dimensions on $\mathds{T}^n$ (see e.g. [@McGuigan:90; @McGuigan:03]). However, the situation is different in string theory where one really integrates only over the fundamental domain, e.g. $Z(\mathds{T}^n)=\int_D d\bm{\tau} Z(\bm{\tau})$.
The Torus Universe in Iwasawa Coordinates {#sec:TorusIwasawa}
=========================================
In this appendix we use a parametrization of the torus using the Iwasawa coordinates which are more apt to describe the asymptotic behavior of the metric [@Damour:03]. It is important to understand the role of the off-diagonal terms in the metric and to know what happens near the singularity and at late times. The metric can be decomposed as follows: $$\label{metricI}
h=\mathcal{N}^T\mathcal{D}^2\mathcal{N},$$ where $$\mathcal{D}=\left(
\begin{array}{ccc}
e^{-z_1} & 0&0\\
0&e^{-z_2}&0\\
0&0&e^{-z_3}
\end{array}\right), \quad
\mathcal{N}=\left(
\begin{array}{ccc}
1 & n_1&n_2\\
0&1&n_3\\
0&0&1
\end{array}\right).$$ An easy calculation shows that can be transformed into with $n_1=a_2\,^1/a_1\,^1$, $n_2=a_3\,^1/a_1\,^1$, $n_3=a_3\,^2/a_2\,^2$, $z_i=-\ln a_i\,^i$ (no summation)[^7]. The analogue to is now given by $$\label{habI}
h_{ab}=\sum_{i=1}^3e^{-2z_i}\mathcal{N}_a\,^i\mathcal{N}_b\,^i.$$ The Iwasawa decomposition can also be viewed as the Gram-Schmidt orthogonalization of the forms $dx^a$: $$h_{ab}dx^a\otimes dx^b=\sum_{i=1}^3e^{-2z_i}\theta^i\otimes \theta^i,$$ where the coframes $\theta^i$ are given by $$\theta^i=\mathcal{N}_a\,^idx^a.$$ Analogously, the frames $e_i$ dual to the coframes $\theta^i$ are given by the inverse of $\mathcal{N}_a\,^i$: $$e_i=\mathcal{N}^a\,_i\frac{\partial}{\partial x^a}.$$ Since the determinant of the matrix $\mathcal{N}$ is equal to one the basis given by the coframe is orthonormal. Note that this is different from the construction in .
In order to determine the asymptotic behavior of the off-diagonal terms we follow the analysis in [@Damour:03]. The metric $h$ being symmetric, we automatically know that its eigenvalues are real. We call these eigenvalues $t^{2\alpha_i}$, $1\leq i \leq3$ and $\alpha_1<\alpha_2<\alpha_3$, in analogy to the diagonal Kasner solution (see ) and construct a metric $h_K(t)=\mathrm{diag}(t^{2\alpha_i})$ by means of a constant matrix $L$ $$h(t)=L^Th_K(t)L, \quad L=\left(
\begin{array}{ccc}
l_1 & l_2 & l_3\\
m_1 & m_2 & m_3\\
r_1 & r_2 & r_3
\end{array}\right).$$ With these relations we can obtain the evolution of the Iwasawa variables. For example, we have $$n_1(t)=\frac{t^{2\alpha_1}l_1l_2+t^{2\alpha_2}m_1m_2+t^{2\alpha_3}r_1r_2}{t^{2\alpha_1}l_1^2+t^{2\alpha_2}m_1^2+t^{2\alpha_3}r_1^2}.$$ In [@Damour:03] it was shown that the asymptotic behavior $t\rightarrow 0^+$ of the off-diagonal terms of the Iwasawa variables is given by $$n_1\rightarrow \frac{l_2}{l_1},\quad n_2\rightarrow \frac{l_3}{l_1},\quad n_3\rightarrow\frac{l_1m_3-l_3m_1}{l_1m_2-l_2m_1}, \quad (t\rightarrow 0^+),$$ which means that the off-diagonal terms freeze in as we approach the singularity. We can also calculate the other limit $t\rightarrow\infty$ and obtain e.g. $$n_1\rightarrow \frac{r_2}{r_1},\quad n_2\rightarrow\frac{r_3}{r_1},\quad n_3\rightarrow\frac{m_1r_3-m_3r_1}{m_1r_2-m_2r_1},\quad(t\rightarrow \infty).$$ We have checked this result numerically, which can be seen in where the gray line $(a_2\,^1)$ converges for $t\rightarrow\infty$ toward the solid line ($a_1\,^1)$, i.e. $n_1\rightarrow \mathrm{const}$. However, the limit $t\rightarrow0^+$ could not be checked due to the numerical instability of the solutions when approaching the singularity.
References {#references .unnumbered}
==========
[99]{} Komatsu E [*et al*]{}, [*Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation*]{}, 2009 [*Astrophys.J.Suppl.*]{} [**180**]{} 330 Aurich R, Janzer H S, Lustig S and Steiner F, [*Do we live in a ’small universe’*]{}, 2008 [*Class. Quantum Grav.*]{} [**25**]{} 125006 Aurich R, [*A spatial correlation analysis for a toroidal universe*]{}, 2008 [*Class. Qautnum Grav.*]{} [**25**]{} 225017 Aurich R, Lustig S and Steiner F, [*Hot pixel contamination in the CMB correlation function*]{}, 2009 [`a`stro-ph.CO/0903.3133]{} Aurich R, Lustig S and Steiner F, [*CMB Anisotropy of the Poincaré dodecahedron*]{}, 2005 [*Class. Quantum Grav.*]{} [**22**]{} 2061 Caillerie S [*et al*]{}, [*A New Analysis of the Poincaré Dodecahedral Space Model*]{}, 2007 [*A & A*]{} [**476**]{} 691C Aurich R, Lustig S and Steiner F, [*CMB Anisotropy of Spherical Spaces*]{}, 2005 [*Class. Quantum Grav.*]{} [**22**]{} 3443 Bojowald M, [*Loop Quantum Cosmology*]{}, 2008 [*Living Re. Relativity*]{} [**11**]{} 4 Bojowald M and Kastrup H A, [*Symmetry Reduction for Quantized Diffeomorphism-invariant Theories of Connections*]{}, 2000 [*Class. Quantum Grav.*]{} [**17**]{} 3009 Bojowald M, [*Loop Quantum Cosmology I: Kinematics*]{}, 2000 [*Class. Quantum Grav.*]{} [**17**]{} 1489 Bojowald M, [*Loop Quantum Cosmology II: Volume Operator*]{}, 2000 [*Class. Quantum Grav.*]{} [**17**]{} 1509 Bojowald M, [*Isotropic Loop Quantum Cosmology*]{}, 2002 [*Class. Quantum Grav.*]{} [**19**]{} 2717 Ashtekar A, Bojowald M and Lewandowski J, [*Mathematical Structure of Loop Quantum Cosmology*]{}, 2003 [*Adv. Theor. Math. Phys.*]{} [**7**]{} 233 Bojowald M, [*Homogeneous Loop Quantum Cosmology*]{}, 2003 [*Class. Quantum Grav.*]{} [**20**]{} 2595 Ashtekar A, Pawlowski T and Singh P, [*Quantum Nature of the Big Bang: An Analytical and Numerical Investigation*]{}, 2006 [*Phys. Rev. D*]{} [**73**]{} 124038 Ashtekar A, Pawlowski T and Singh P, [*Quantum Nature of the Big Bang: Improved Dynamics*]{}, 2006 [*Phys.Rev. D*]{} [**74**]{} 084003 Ashtekar A and Lewandowski J, [*Background Independent Quantum Gravity: A Status Report*]{}, 2004 [*Class. Quantum Grav.*]{} [**21**]{} R53 Rovelli C, [*Quantum Gravity*]{}, 2004 (Cambridge: Cambridge University Press) Thiemann T, [*Modern Canonical Quantum General Relativity*]{}, 2007 (Cambridge: Cambridge University Press) ’t Hooft G, [*A property of electric and magnetic flux in non-abelian gauge theories*]{}, 1979 [*Nucl. Phys.*]{} B [**153**]{} 141 Ashtekar A and Samuel J, [*Bianchi Cosmologies: the Role of Spatial Topology*]{}, 1991 [*Class. Quantum Grav.*]{} [**8**]{} 2191 Wolf J A, [*Spaces of Constant Curvature*]{}, 1974 (Boston: Publish Or Perish) Koike T, Tanimoto M and Hosoya A, [*Compact Homogeneous Universes*]{}, 1994 [*J. Math. Phys.*]{} [**35**]{} 4855 Tanimoto M, Koike T and Hosoya A, [*Dynamics of Compact Homogeneous Universes*]{}, 1997 [*J. Math. Phys.*]{} [**38**]{} 350 Tanimoto M, Koike T and Hosoya A, [*Hamiltonian Structures for Compact Homogeneous Universes*]{}, 1997 [*J. Math. Phys.*]{} [**38**]{} 6560 Yasuno K, Koike T and Siino M, [*Thurston’s Geometrization Conjecture and Cosmological Models*]{}, 2001 [*Class. Quantum Grav.*]{} [**18**]{} 1405 Singer I M, [*Infinitesimally Homogeneous Spaces*]{}, 1969 [*Comm. Pure Appl. Math.*]{} [**13**]{} 685 Thurston W, [*Three-dimensional geometry and topology*]{}, 1997 Vol. 1. (Princeton: Princeton University Press) Kobayashi S and Nomizu K, [*Foundations of Differential Geometry*]{}, volume 1 (John Wiley & Sons, New York 1963); volume 2 (New York 1969) Brodbeck O, [*On Symmetric Gauge Fields for Arbitrary Gauge and Symmetry Groups*]{}, 1996 [*Helv. Phys. Acta*]{} [**69**]{} 321 Chiou D-W, [*Loop Quantum Cosmology in Bianchi Type I Models: Analytical Investigation*]{}, 2007 [*Phys.Rev.*]{} D[**75**]{} 024029 Ashtekar A and Bojowald M, [*Quantum Geometry and the Schwarzschild Singularity*]{}, 2006 [*Class. Quantum Grav.*]{} [**23**]{} 391 Thiemann T, [*Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories*]{}, 1998 [*Class. Quantum Grav.*]{} [**15**]{} 1487 Velhinho J M, [*The Quantum Configuration Space of Loop Quantum Cosmology*]{}, 2007 [*Class. Quantum Grav.*]{} [**24**]{} 3745 Bratelli O and Robinson D W, [*Operator Algebras and Quantum Statistical Mechanics*]{}, 1979 (New York: Springer) Bojowald M and Swiderski R, [*The Volume Operator in Spherically Symmetric Quantum Geometry*]{}, 2004 [*Class. Quantum Grav.*]{} [**21**]{} 4881 Kamke E, [*Diffenrentialgleichungen, Lösungsmethoden und Lösungen II*]{}, 1979 (Stuttgart: B.G. Teubner) Shubin M A, [*Differential and pseudodifferential operators in spaces of almost periodic functions*]{}, 1974 [*Math. USSR Sbornik*]{} [**24**]{} 547 Shubin M A, [*Almost periodic functions and partial differential operators*]{}, 1978 [*Russian Math. Surveys*]{} [**33**]{} 1 Bohr H, [*Almost Periodic Functions*]{}, 1947 Chelsea Publishing Company Roberts J E, [*The [D]{}irac bra and ket formalism*]{}, 1966 [*J. Math. Phys.*]{} [**7**]{} 1097 Roberts J E, [*Rigged [H]{}ilbert Space in quantum mechanics*]{}, 1966 [*Commun. Math. Phys.*]{} [**3**]{} 98 Thiemann T, [*Anomaly-free Formulation of Non-perturbative, Four-dimensional Lorentzian Quantum Gravity*]{}, 1998 [*Phys. Lett.*]{} B [**380**]{} 257 Fagundes H V, [*Closed Spaces in Cosmology*]{}, 1992 [*Gen. Rel. Grav.*]{} [**24**]{} 199 Gordon D, Grenier D and Terras A, [*Hecke Operators and the Fundamental Domain of $SL(3,\mathds{Z})$*]{}, 1987 [*Math. Comp.*]{} [**48**]{} 159 Terras A, [*Harmonic Analysis on Symmetric Spaces and Applications II*]{}, 1988 Vol. II (New York, Springer Verlag) Minkowski H, [*Diskontinuitätsbereich für arithmetische Äquivalenz*]{}, 1905 [*J. Reine Angew. Math.*]{} [**129**]{} 220 McGuigan M, [*Fundamental Regions of Superspace*]{}, 1990 [*Phys. Rev.*]{} D [**41**]{} 1844 McGuigan M, [*Three Dimensional Gravity and M-Theory*]{}, 2003 [`h`ep-th/0312327]{} Damour T, Henneaux M and Nicolai H, [*Cosmological Billards*]{}, 2003 [*Class. Quantum Grav.*]{} [**20**]{} R145-R200
[^1]: When dealing with the open case $\mathds{R}^3$ one has to distinguish between the fiducial volume $V_0$ of a cell as measured by the fiducial metric $\,^0\tilde h_{ab}$ and the physical volume $V$ as measured by the physical metric $\tilde h_{ab}$. Since we shall deal with a closed universe we have the “preferred fiducial cell” $\mathds{T}^3$ at our disposal. Furthermore, in the open case the spatial integrals have to be restricted to this fiducial cell whereas in the closed case these integrals are naturally restricted to the physical cell $\mathds{T}^3$.
[^2]: At this point care has to be taken because there is no homogeneous and isotropic vacuum solution to the Einstein equation (see ). Only a nonvanishing energy-momentum tensor allows for the isotropic limit of $\mathcal{H}_g$, which corresponds then to the usual Friedmann solutions.
[^3]: Notice that since every scalar field lives in the trivial representation of the rotation group it is not possible to construct a scalar field which is homogeneous but anisotropic.
[^4]: We use the convention $\tau_i\tau_j=\frac{1}{2}\epsilon_{ij}\,^k\tau_k-\frac{1}{4}\delta_{ij}\mathds{1}_{2\times 2}$
[^5]: In order to avoid confusion with the rest of this work we tag every variable with a $"'"$ when dealing with the open case.
[^6]: An almost periodic function $f(x)$ is uniformly continuous for $x\in\mathds{R}$ and bounded [@Bohr:47].
[^7]: For simplicity we assume that all diagonal scale factors $a_i\,^i$ are strictly positive. However the nondiagonal scale factors can be negative or zero.
|
---
abstract: 'In this note we study the limiting behaviour of the symbolic generic initial system $\{ \text{gin}(I^{(m)}) \}$ of an ideal $I \subseteq K[x,y,z]$ corresponding to an arrangement of $r$ points of $\mathbb{P}^2$ lying on an irreducible conic. In particular, we show that the *limiting shape* of this system is the subset of $\mathbb{R}^2_{\geq 0}$ such consisting of all points above the line through $(\text{min} \{ \frac{r}{2}, 2 \},0)$ and $(0, \text{max} \{ \frac{r}{2}, 2 \})$.'
author:
- Sarah Mayes
bibliography:
- 'ConicBib.bib'
nocite: '[@*]'
title: The Symbolic Generic Initial System of Points on an Irreducible Conic
---
The general research trend looking at the asymptotic behaviour of collections of algebraic objects is motivated by the idea that there is often a structure revealed in the limit that is difficult to see when studying individual objects (see, for example, [@ELS01], [@Siu01], [@Huneke92], and [@ES09]). The asymptotic behaviour of a collection of monomial ideals $\mathrm{a}_{\bullet}$ such that $\mathrm{a}_i \cdot \mathrm{a}_j \subseteq \mathrm{a}_{i+j}$ (a *graded system of monomial ideals*) can be described by its *limiting shape* $P$. If $P_{\mathrm{a}_i}$ denotes the Newton polytope of $\mathrm{a}_i$, then the limiting shape $P$ is defined to be the limit $\lim_{m \rightarrow \infty} \frac{1}{m} P_{\mathrm{a}_m}$ ([@Mayes12a]). In addition to giving a simple geometric interpretation of the limiting behaviour, $P$ completely determines the asymptotic multiplier ideals of $\mathrm{a}_{\bullet}$ (see [@Howald01]).
Generic initial ideals have a nice combinatorial structure, but are often difficult to compute and usually have complicated sets of generators (see [@Green98] for a survey or [@Cimpoeas06] and [@Mayes12c] for examples). This motivates a series of work describing the limiting shape of generic initial systems, $\{ \text{gin}(I^m) \}_m$, and of symbolic generic initial systems, $\{ \text{gin}(I^{(m)}) \}_m$ ([@Mayes12a], [@Mayes12b], [@Mayes12c], [@Mayes12e]). The goal of this paper is to describe the limiting shape of the symbolic generic initial system of the ideal of $r$ points in $\mathbb{P}^2$ lying on an irreducible conic.
We will see that when $I$ is the ideal of points in $\mathbb{P}^2$, each of the polytopes $P_{\text{gin}(I^{(m)})}$, and thus $P$ itself, can be thought of as a subset of $\mathbb{R}^2$. The following theorem describes $P$ in the case we are interested in.
\[thm:mainthm\] Let $I \subseteq R = K[x,y,z]$ be the ideal of $r >1$ distinct points $p_1, \dots, p_r$ of $\mathbb{P}^2$ lying on an irreducible conic and let $P \subseteq \mathbb{R}^2_{\geq 0}$ be the limiting shape of the reverse lexicographic symbolic generic initial system $\{ \textnormal{gin}(I^{(m)})\}_m$. If $ r \geq 4$, then $P$ has a boundary defined by the line through the points $(2,0)$ and $(0, \frac{r}{2})$ (see Figure \[fig:limitingshape\]). If $r =2$ or $r=3$, then $P$ has a boundary defined by the line through the points $(\frac{r}{2}, 0)$ and $(0, 2)$.
![The limiting shape $P$ of $\{ \text{gin}(I^{(m)})\}_m$ where $I$ is the ideal of $r \geq 4$ points lying on an irreducible conic.[]{data-label="fig:limitingshape"}](ConicPolytope.png){width="4cm"}
The proof of this theorem is an application of ideas that have been described elsewhere. Rather than repeating arguments here, we refer the reader elsewhere for details where necessary.
The following result describes the structure of the individual ideals $\text{gin}(I^{(m)})$ that make up the generic initial system.
\[thm:formofgin\] Suppose that $I \subseteq K[x,y,z]$ be the ideal of a set of distinct points of $\mathbb{P}^2$. Then the minimal generators of $\textnormal{gin}(I^{(m)})$ are $$\big\{ x^{\alpha(m)}, x^{\alpha(m)-1}y^{\lambda_{\alpha(m)-1}(m)}, \dots, xy^{\lambda_1(m)}, y^{\lambda_0(m)}\big \}$$ for some positive integers $\lambda_0(m), \dots, \lambda_{\alpha(m)-1}$ such that $\lambda_0(m)> \lambda_1(m) > \cdots >\lambda_{\alpha(m)-1}(m)$. Further, if the minimal free resolution of $I^{(m)}$ is of the form $$0 \rightarrow F_1=\bigoplus_{i=1}^{\psi} R(-u_i) \rightarrow F_1=\bigoplus_{i=1}^{\mu} R(-d_i) \rightarrow I^{(m)} \rightarrow 0$$ with $U(m)= \textnormal{max}\{ u_i\}$ and $D(m) = \textnormal{min} \{ d_i\}$, then $$\alpha(m) = D(m)$$ and $$\lambda_0(m) = U(m)-1.$$
The first part of the theorem is Corollary 2.9 of [@Mayes12c] and follows from results in [@BS87] and [@HS98]. The second statement follows the a result of Hilbert-Burch, which says that, with the notation in the theorem, the minimal free resolution of $\text{gin}(I^{(m)})$ is of the form $$0 \rightarrow G_1 \rightarrow G_0 \rightarrow \text{gin}(I^{(m)}) \rightarrow 0$$ where $G_1 = \bigoplus_{i=0}^{\alpha(m)} R(-\lambda_i(m)-i-1)$ and $G_0 = \big[\bigoplus_{i=0}^{\alpha(m)} R(-\lambda_i(m)-i)\big] \oplus R(-\alpha(m))$ (Corollary 4.15 of [@Green98]). A *consecutive cancellation* takes a sequence $\{ \beta_{i,j} \}$ to a new sequence by replacing $\beta_{i,j}$ by $\beta_{i,j}-1$ and $\beta_{i+1,j}$ by $\beta_{i+1,j}-1$. The ‘Cancellation Principle’ says that the graded Betti numbers of $J$ can be obtained by the graded Betti numbers of $\text{gin}(J)$ by making a series of consecutive cancellations (see Corollary 1.21 of [@Green98]). Since $\lambda_0(m) +1 > \lambda_i(m)+i$ for all $i$, $\beta_{1,\lambda_0+1}\geq 1$ does not change with any such consecutive cancellation; thus, $R(-\lambda_0(m)-1)$ is the summand of $F_1$ with the largest shift. Likewise, $\alpha(m)<\lambda_i(m)+i+1$ for all $i$ so $\beta_{0, \alpha(m)} \geq 1$ does not change with a consecutive cancellation; thus, $R(-\alpha(m))$ is the summand of $F_0$ with the smallest shift.
In the case where $m$ is even and $I$ is the ideal of $r\geq 3$ points lying on an irreducible conic in $\mathbb{P}^2$, we will see in Proposition \[prop:resolutions\] that we can write down the entire minimal free resolution of $I^{(m)}$. This will give us $D(m)$ and $U(m)$ when $m$ is even so that we can find the powers of $x$ and $y$, $x^{\alpha(m)} = x^{D(m)}$ and $y^{\lambda_0(m)}=y^{U(m)-1}$, that appear in a minimal generating set of $\text{gin}(I^{(m)})$. In particular, Proposition \[prop:resolutions\] implies the following.
\[lem:highlowshifts\] Suppose that $I$ is the ideal of $r \geq 3$ points in $\mathbb{P}^2$ lying on an irreducible conic and use the notation of the previous theorem.
- If $r \geq 4$ is even, $D(m)=2m$ and $U(m) = \frac{rm}{2}+2$.
- If $r>4$ is odd and $m$ is even, then $D(m) = 2m$ and $U(m) = \frac{rm}{2}+2$.
- If $r = 3$ and $m$ is even, then $D(m) =\frac{3m}{2}$ and $U(m) =2m+1$.
By Lemma \[thm:formofgin\], each of the generic initial ideals $\text{gin}(I^{(m)})$ is generated in the variables of $x$ and $y$, so we can think of each Newton polytope $P_{\text{gin}(I^{(m)})}$, and thus the limiting shape $P$ itself, as a subset of $\mathbb{R}^2$.
The following result is the key for proving the main theorem: it describes when the limiting polytope $P$ of the symbolic generic initial system in $\mathbb{P}^2$ is defined by a single boundary line. The proof is contained in [@Mayes12c].
\[prop:maxvolumeattained\] Let $I \subseteq K[x,y,z]$ be the ideal of $r$ distinct points in $\mathbb{P}^2$ and let $P$ be the limiting shape of the symbolic generic initial system $\{ \textnormal{gin}(I^{(m)})\}_m$. Suppose that the $x$-intercept $\gamma_1$ and the $y$-intercept $\gamma_2$ of the boundary of $P$ are such that $\gamma_1 \cdot \gamma_2 = r$. Then the limiting polytope $P$ has a boundary defined by the line passing through $(\gamma_1, 0)$ and $(0, \gamma_2)$.
With these results in mind, we can now prove the main theorem.
Suppose first that $r \geq 4$ and that $m$ is even if $r$ is odd. By Theorem \[thm:formofgin\] and Lemma \[lem:highlowshifts\], $x^{D(m)} = x^{2m}$ and $y^{U(m)-1} = y^{\frac{rm}{2}+1}$ are the smallest powers of $x$ and $y$ contained in $\text{gin}(I^{(m)})$. This means that the intercepts of the boundary of $P_{\text{gin}(I^{(m)})}$ are $(2m,0)$ and $(0, \frac{rm}{2}+1)$. Thus, the intercepts of the boundary of the limiting polytope $P$ of the entire symbolic generic initial system are $(\lim_{m \rightarrow \infty} \frac{2m}{m},0) = (2, 0)$ and $(0, \lim_{m \rightarrow \infty} \frac{rm/2}{m}+1) = (0,\frac{r}{2})$.[^1] By Proposition \[prop:maxvolumeattained\], the fact that $\frac{r}{2} \cdot 2 = r$ implies that the limiting polytope $P$ is as claimed.
Now suppose that $r=3$ and that $m$ is even. By the same argument as above, the intercepts of the boundary of the limiting polytope $P$ are $(\lim_{m \rightarrow \infty} \frac{3m/2}{m},0) = (\frac{3}{2},0)$ and $(0,\lim_{m\rightarrow \infty} \frac{2m}{m}) = (0,2)$. Since $\frac{3}{2} \cdot 2 = 3$, the limiting polytope is as claimed.
The case where $r=2$ follows from the main theorem of [@Mayes12a] since $I$ is a type (1,2) complete intersection in this case.
It remains to prove Lemma \[lem:highlowshifts\] which follows immediately from the next proposition. In particular, we will write the minimal free resolutions of the ideals $I^{(m)}$ when $I$ is the ideal of $r \geq 3$ points on an irreducible conic and $m$ is even if $r$ is odd.
\[prop:resolutions\] Let $I$ be the ideal of $r \geq 3$ points of $\mathbb{P}^2$ lying on an irreducible conic and suppose that that the minimal free resolution of $I^{(m)}$ is of the form $$0 \rightarrow G_1 \rightarrow G_0 \rightarrow I^{(m)} \rightarrow 0.$$
- If $r$ is even, $$G_0 = \bigoplus_{j=0}^m R\big(-2(m-j)-\frac{rj}{2}\big)$$ and $$G_1 = \bigoplus_{j=1}^m R\big(-2(m-j) - \frac{rj}{2}-2\big).$$
- If $r\geq 5$ is odd and $m$ is even, $$G_0 = \Bigg[ \bigoplus_{j=0}^{m/2} R(-2m-j(r-4)) \Bigg] \oplus \Bigg[ \bigoplus_{j=0}^{m/2-1} R^2\big(-2m-j(r-4)-\frac{r-1}{2}+1\big)\Bigg]$$ and $$G_1 = \Bigg[ \bigoplus_{j=1}^{m/2} R(-2m-j(r-4)-2) \Bigg] \oplus \Bigg[ \bigoplus_{j=0}^{m/2-1} R^2\big(-2m-j(r-4)-\frac{r-1}{2}\big) \Bigg].$$
- If $r=3$ and $m$ is even, $$G_0 = R\Big(-\frac{3m}{2}\Big) \oplus \Bigg[ \bigoplus_{j=0}^{m/2-1} R^3\big(-\frac{3m}{2} - j-1\big) \Bigg]$$ and $$G_1 = \bigoplus_{j=0}^{m/2-1} R^3\Big(-\frac{3m}{2} -j-2\Big).$$
To prove this proposition we will follow the results of Catalisano described in [@Catalisano91] that can be used to compute the minimal free resolution of *any* fat point ideal $$I_{(m_1, \dots, m_r)} = I^{m_1}_{p_1} \cap I^{m_2}_{p_2} \cap \cdots \cap I^{m_r}_{p_r}$$ as long as the points $p_1, \dots, p_r$ lie on an irreducible conic. The following is a specialization of Catalisano’s work to the case where $r\geq4$ and $m_i = m$ for all $i$ (that is, when $I$ is the ideal of a uniform fat point subscheme).
\[prop:algorithm\] Let $I$ be the ideal of $r \geq 4$ points of $\mathbb{P}^2$ lying on an irreducible conic. Suppose that the minimal free resolution of $I^{(t-1)}$ is of the form $$0 \rightarrow F'_1 \rightarrow F'_0 \rightarrow I^{(t-1)} \rightarrow 0$$ where $F'_1 = \oplus_{i=1}^{\mu-1} R(-u_i)$ and $F'_0 = \oplus_{i=1}^{\mu} R(-d_i)$ and that the minimal free resolution of $I^{(t)}$ is of the form $$0 \rightarrow F_1 \rightarrow F_0 \rightarrow I^{(t)} \rightarrow 0.$$ If $rt$ is even
- $F_1 = [\oplus_{i=1}^{\mu-1} R(-u_i-2)] \oplus R(-\frac{rt}{2} - 2)$ and $F_0 = [\oplus_{i=1}^{\mu} R(-d_i-2)] \oplus R(-\frac{rt}{2})$
while if $rt$ is odd
- $F_1 = [\oplus_{i=1}^{\mu-1} R(-u_i-2)] \oplus R^2(-\frac{rt+1}{2} - 1)$ and $F_0 = [\oplus_{i=1}^{\mu} R(-d_i-2)] \oplus R^2(-\frac{rt+1}{2})$.
Therefore, one can apply this result $m$ times to find the minimal free resolution of $I^{(m)}$. That is, first find the minimal free resolution of $I^{(1)}=I$ from $0 \rightarrow 0 \rightarrow R(0) \rightarrow I^{(0)} \rightarrow 0$, then find the minimal free resolution of $I^{(2)}$ from that of $I$, and so on.
We will give an idea of how to find the minimal free resolutions of the ideals $I^{(m)}$ using the algorithm in Proposition \[prop:algorithm\].
If we are in case (a) where $r$ is even, $rt$ is even for all $t$, so to find the resolution of $I^{(t)}$ from the minimal free resolution of $I^{(t-1)}$ we follow the first case of Proposition \[prop:algorithm\]. In particular, to find the resolution of $I^{(m)}$ from the resolution of $I^{(0)}$, we apply part (1) exactly $m$ times for $t = 1, \dots, m$.
If we are in case (b) where $r$ is odd, $rt$ is odd for odd $t$ and $rt$ is even for even $t$. Thus, we need to apply both cases of Proposition \[prop:algorithm\] to find the resolution of $I^{(m)}$. To obtain the resolution $$0 \rightarrow F_1 \rightarrow F_0 \rightarrow I^{(t)} \rightarrow 0$$ of $I^{(t)}$ from the resolution $$0 \rightarrow F''_1 \rightarrow F''_0 \rightarrow I^{(t-2)} \rightarrow 0$$ of $I^{(t-2)}$ when $t$ is even, one needs to:
- shift each summand of $F_0$ and $F_1$ by $-4$;
- add $R^2(-\frac{r(t-1)+1}{2}-3)$ and $R(-\frac{rt}{2}-2)$ to $F''_1$; and
- add $R^2(-\frac{r(t-1)+1}{2}-3)$ and $R(-\frac{rt}{2})$ to $F''_0$.
If $m$ is even, we can follow this procedure $\frac{m}{2}$ times with $t = 2, 4, 6, \dots, m$ to find the resolution of $I^{(m)}$ from that of $I^{(0)}$.
For case (c) when $r=3$ and $m$ is even, one needs to use other results of [@Catalisano91] beyond those stated in Proposition \[prop:algorithm\]. The general idea is the same as above: use a sequence of fat point schemes $Z_0 = m(p_1+p_2+p_3), Z_1, Z_2, \dots, Z_H = 0(p_1+p_2+p_3)$ and find the minimal free resolution of $I_{Z_{H-1}}$ from that of $I_{Z_H}$, then find the minimal free resolution of $I_{Z_{H-2}}$ from that of $I_{Z_{H_1}}$, and so on, until we can find the minimal free resolution of $I^{(m)} = I_{Z_0}$ from that of $I_{Z_1}$. However, when $r=3$ not all of the $Z_i$ will be uniform fat point subschemes. In particular, subsequences of the form $Z_l = t(p_1+p_2+p_3)$, $Z_{l+1} = (t-1)p_1+(t-1)p_2+tp_3$, $Z_{l+2} = (t-1)p_1+(t-2)p_2+(t-1)p_3$, $Z_{l+3} = (t-2)(p_1+p_2+p_3)$ come together to form the sequence $Z_0, \dots, Z_H$. See [@Catalisano91] for further details.
[^1]: In the case where $r$ is odd we take the limits over even $m$.
|
---
author:
- 'Tushar Athawale, Dan Maljovec, Chris R. Johnson, Valerio Pascucci, Bei Wang'
title: Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps
---
Acknowledgments {#acknowledgments .unnumbered}
===============
This project is supported in part by NSF IIS-1513616, DBI-1661375, and IIS-1910733; National Institute of General Medical Sciences of NIH under grant P41 GM103545-18; and the Intel Parallel Computing Centers Program.
|
---
abstract: 'The observation of several neutron stars in the center of supernova remnants and with significantly lower values of the dipolar magnetic field than the average radio-pulsar population has motivated a lively debate about their formation and origin, with controversial interpretations. A possible explanation requires the slow rotation of the proto-neutron star at birth, which is unable to amplify its magnetic field to typical pulsar levels. An alternative possibility, the hidden magnetic field scenario, considers the accretion of the fallback of the supernova debris onto the neutron star as responsible for the submergence (or screening) of the field and its apparently low value. In this paper we study under which conditions the magnetic field of a neutron star can be buried into the crust due to an accreting, conducting fluid. For this purpose, we consider a spherically symmetric calculation in general relativity to estimate the balance between the incoming accretion flow and the magnetosphere. Our study analyses several models with different specific entropy, composition, and neutron star masses. The main conclusion of our work is that typical magnetic fields of a few times $10^{12}$ G can be buried by accreting only $10^{-3}-10^{-2} {M}_\odot$, a relatively modest amount of mass. In view of this result, the Central Compact Object scenario should not be considered unusual, and we predict that anomalously weak magnetic fields should be common in very young ($<$ few kyr) neutron stars.'
author:
- |
Alejandro Torres-Forné$^{1}$[^1], Pablo Cerdá-Durán$^1$, José A. Pons$^2$ and José A. Font$^{1,3}$\
$^{1}$Departamento de Astronomía y Astrofísica, Universitat de València, Dr. Moliner 50, 46100, Burjassot (València), Spain\
$^{2}$Departament de Física Aplicada, Universitat d’Alacant, Ap. Correus 99, 03080 Alacant, Spain\
$^{3}$Observatori Astronòmic, Universitat de València, Catedrático José Beltrán 2, 46980, Paterna (València), Spain
title: 'Are pulsars born with a hidden magnetic field?'
---
\[firstpage\]
stars: magnetic field – stars: neutron – pulsars: general
Introduction
============
Central Compact Objects (CCOs) are isolated, young neutron stars (NSs) which show no radio emission and are located near the center of young supernova remnants (SNRs). Three such NSs, PSR E1207.4-5209, PSR J0821.0-4300, and PSR J1852.3-0040, show an inferred magnetic field significantly lower than the standard values for radio-pulsars (i.e. $10^{12}$ G). The main properties of these sources are summarized in Table \[Table1\]. In all cases, the difference between the characteristic age of the neutron star $\tau_c=P/\dot{P}$ and the age of the SNR indicates that these NSs were born spinning at nearly their present periods ($P\sim 0.1-0.4$ s). This discovery has challenged theoretical models of magnetic field generation, that need to be modified to account for their peculiar properties.
-------------------- ------- ------- ------- ------------------- --------------------- ---------------- ---------- ------------------
Age d P $\rm{B}_{\rm{s}}$ $\rm{L}_{x,bol}$ SNR $\tau_c$ References
(kyr) (kpc) (s) $10^{11} $G (erg $\rm{s}^{-1})$ (Myr)
J0822.0-4300 3.7 2.2 0.112 $0.65$ $6.5\times10 ^{33}$ Puppis A $190$ 1, 2
1E 1207.4-5209 7 2.2 0.424 $2$ $2.5\times10^{33}$ PKS 1209-51/52 $ 310$ 2, 3, 4, 5, 6, 7
J185238.6 + 004020 7 7 0.105 0.61 $5.3\times10^{33}$ Kes 79 $ 190$ 8, 9, 10, 11
\[Table1\]
-------------------- ------- ------- ------- ------------------- --------------------- ---------------- ---------- ------------------
[**References:**]{} (1) [@Hui:2006], (2) [@Gotthelf:2013], (3) [@Zavlin:2000], (4) [@Mereghetti:2002], (5) [@Bignami:2003], (6) [@DeLuca:2004], (7) [@Gottfeld:2007], (8) [@Seward:2003], (9) [@Gottfeld:2005], (10) [@Halpern:2007], (11) [@Halpern:2010]
The first possible explanation for the unusual magnetic field found in these objects simply assumes that these NSs are born with a magnetic field much lower than that of their classmates. This value can be amplified by turbulent dynamo action during the proto-neutron star (PNS) phase [@Thompson:1993; @Bonanno:2005] . In this model, the final low values of the magnetic field would reflect the fact that the slow rotation of the neutron star at birth does not suffice to effectively amplify the magnetic field through dynamo effects. However, recent studies have shown that, even in the absence of rapid rotation, magnetic fields in PNS can be amplified by other mechanisms such as convection and the standing accretion shock instability (SASI) [@Endeve:2012; @Obergaulinger:2014].
An alternative explanation is the hidden magnetic field scenario [@Young:1995; @Muslimov:1995; @Geppert:1999; @Shabaltas:2012]. Following the supernova explosion and the neutron star birth, the supernova shock travels outwards through the external layers of the star. When this shock crosses a discontinuity in density, it is partially reflected and moves backwards (reverse shock). The total mass accreted by the reverse shock in this process is in the range from $\sim 10^{-4} M_\odot$ to a few solar masses on a typical timescale of hours to days [@Ugliano:2012]. Such a high accretion rate can compress the magnetic field of the NS which can eventually be buried into the neutron star crust. As a result, the value of the external magnetic field would be significantly lower than the internal ‘hidden’ magnetic field. [@Bernal:2010] performed 1D and 2D numerical simulations of a single column of material falling onto a magnetized neutron star and showed how the magnetic field can be buried into the neutron star crust.
Once the accretion process stops, the magnetic field might eventually reemerge. The initial studies investigated the process of reemergence using simplified 1D models and dipolar fields [@Young:1995; @Muslimov:1995; @Geppert:1999] and established that the timescale for the magnetic field reemergence is $\sim1-10^{7}$ kyr, critically depending on the depth at which the magnetic field is buried. More recent investigations have confirmed this result. [@Ho:2011] observed similar timescales for the reemergence using a 1D cooling code. [@Vigano:2012] carried out simulations of the evolution of the interior magnetic field during the accretion phase and the magnetic field submergence phase.
In the present work we study the feasibility of the hidden magnetic field scenario using a novel numerical approach based on the solutions of 1D Riemann problems (discontinuous initial value problems) to model the compression of the magnetic field of the NS. The two initial states for the Riemann problem are defined by the magnetosphere and by the accreting fluid, at either sides of a moving, discontinuous interface. Following the notation defined in [@Michel:1977], the NS magnetosphere refers to the area surrounding the star where the magnetic pressure dominates over the thermal pressure of the accreting fluid. The magnetopause is the interface between the magnetically dominated area and the thermally dominated area. The equilibrium point is defined as the radius at which the velocity of the contact discontinuity is zero.
The paper is organized as follows. In Sections \[Model\] to \[sec:magnetosphere\] we present the model we use to perform our study. We describe in these sections the equation of state (EoS) of the accreting fluid, the spherically symmetric Michel solution characterizing the accreting fluid, and all the expressions needed to compute the potential solution for the magnetic field in the magnetosphere. Section \[Results\] contains the main results of this work. After establishing a reference model, we vary the remaining parameters, namely entropy, composition and the initial distribution of the magnetic field, and study their influence on the fate of the magnetic field. Finally, in Section \[Summary\] we summarize the main results of our study and present our conclusions and plans for future work. If not explicitly stated otherwise we use units of $G=c=1$. Greek indices ($\mu,\nu \dots$) run from $0$ to $3$ and latin indices ($i, j \dots$) form $1$ to $3$.
The reverse shock and the fallback scenario. {#Model}
============================================
At the end of their lives, massive stars ($M_{\rm star} \gtrsim 8 M_{\odot}$) possess an onion-shell structure as a result of successive stages of nuclear burning. An inner core, typically formed by iron, with a mass of $\sim 1.4M_{\odot}$ and $\sim 1000$ km radius develops at the centre, balancing gravity through the pressure generated by a relativistic, degenerate, $\gamma=4/3$, fermion gas. The iron core is unstable due to photo-disintegration of nuclei and electron captures, which result in a deleptonization of the core and a significant pressure reduction ($\gamma<4/3$). As a result, the core shrinks and collapses gravitationally to nuclear matter densities on dynamical timescales ($\sim 100$ ms). As the center of the star reaches nuclear saturation density ($\sim 2\times 10^{14}$ g cm$^{-3}$), the EoS stiffens and an outward moving (prompt) shock is produced. As it propagates out the shock suffers severe energy losses dissociating Fe nuclei into free nucleons ($\sim 1.7 \times 10^{51}$ erg$/0.1 M_{\odot}$), consuming its entire kinetic energy inside the iron core (it stalls at $\sim 100-200$ km), becoming a standing accretion shock in a few ms. There is still debate about the exact mechanism and conditions for a successful explosion, but it is commonly accepted that the standing shock has to be revived on a timescale of $\lesssim 1$ s by the energy deposition of neutrinos streaming out of the innermost regions, and some form of convective transport for the shock to carry sufficient energy to disrupt the whole star [see @Janka:2007 for a review on the topic].
Even if the shock is sufficiently strong to power the supernova, part of the material between the nascent neutron star and the propagating shock may fall back into the neutron star [@Colgate:1971; @Chevalier:1989]. Determining the amount of fallback material depends not only on the energy of the shock but also on the radial structure of the progenitor star [@Fryer:2006]. Most of the fallback accretion is the result of the formation of an inward moving reverse shock produced as the main supernova-driving shock crosses the discontinuity between the helium shell and the hydrogen envelope [@Chevalier:1989]. For typical supernova progenitors ($10-30\,M_\odot$) the base of the hydrogen envelope is at $r_{\rm H} \sim 10^{11}$ cm to $3\times 10^{12}$ cm [@Woosley:2002], which is reached by the main shock on a timescale of a few hours. The reverse shock travels inwards carrying mass that accretes onto the NS. It reaches the vicinity of the NS on a timescale of hours, about the same time at which the main supernova shock reaches the surface of the star [@Chevalier:1989]. By the time the reverse shock reaches the NS, the initially hot proto-neutron star has cooled down significantly. In its first minute of life the PNS contracts, cools down to $T < 10^{10}$ K and becomes transparent to neutrinos [@BL:1986; @Pons:1999]. In the next few hours the inner crust ($\rho \in [2\times 10^{11},2\times 10^{14}]$ g cm$^{-3}$) solidifies but the low density envelope ($\rho < 2\times10^{11}$ g cm$^{-3}$), which will form the outer crust on a timescale of $1-100$ yr, remains fluid [@Page:2004; @Aguilera:2008].
Understanding the processes generating the magnetic field observed in NSs, in the range from $\sim 10^{10}$ G to $\sim 10^{15}$ G, is still a open issue. Most likely, convection, rotation and turbulence during the PNS phase play a crucial role in field amplification [@Thompson:1993]. However, at the time in the evolution that we are considering (hours after birth), none of these processes can be active anymore and the electric current distribution generating the magnetic field will be frozen in the interior of the NS. These currents evolve now on the characteristic Hall and Ohmic timescales of $10^4$-$10^6$ yr [@Pons:2007; @Pons:2009; @Vigano:2013], much longer than the timescale $t_{\rm acc}$ during which fallback is significant, which can be estimated as the free-fall time from the base of the hydrogen envelope $$\begin{aligned}
t_{\rm acc} \sim
\frac{1}{2}\left(\frac{r_{\rm H}^3}{GM}\right)^{1/2}\,.\end{aligned}$$ This ranges from 30 minutes to several days for the typical values of $r_{\rm H}$ and a $M=1.4 M_{\odot}$.
The total mass accreted during this phase is more uncertain. Detailed 1D numerical simulations of the shock propagation and fallback estimate that typical values range from $10^{-4} M_\odot$ to a few solar masses [@Woosley:1995; @Zhang:2008; @Ugliano:2012]. If more than a solar mass is accreted, the final outcome would be the delayed formation of a black hole, hours to days after core bounce. [@Chevalier:1989] and [@Zhang:2008] showed that the accretion rate is expected to be maximum when the reverse shock reaches the NS and decreases as $t^{-5/3}$ at later times. Therefore, the total amount of accreted mass is dominated by the fallback during the first few hours. Given the theoretical uncertainties, we assume for the rest of this work that a total mass of $\delta M \in [10^{-5} M_\odot,\delta M_{\rm max}]$ is accreted during a typical timescale of $t_{\rm acc} \in [10^3,10^4]$ s, being $\delta M_{\rm max}\sim 1 M_\odot$ the amount of mass necessary to add to the NS to form a black hole. Therefore, the typical accretion rate during fallback is $\dot M \in [10^{-9}, 10^{-3}] M_\odot / \rm{s}$, which, for practical purposes, we assume to stay constant during the accretion phase. This accretion rate, even at its lowest value, exceeds by far the Eddington luminosity $$\begin{aligned}
\frac{{\dot M} c^2}{L_{\rm Edd}} = 5\times 10^6 \left ( \frac{\dot M}{10^{-9} M_\odot / \rm{s}} \right ),\end{aligned}$$ with $L_{\rm Edd} = 3.5\times 10^{38}$ erg s$^{-1}$ the Eddington luminosity for electron scattering.
In the hypercritical accretion regime, the optical depth is so large that photons are advected inwards with the flow faster than they can diffuse outwards [@Blondin:1986; @Chevalier:1989; @Houck:1991]. As a result the accreting material cannot cool down resulting in an adiabatic compression of the fluid. The dominant process cooling down the accreting fluid and releasing the energy stored in the infalling fluid is neutrino emission [@Houck:1991]. At temperatures above the pair creation threshold, $T_{\rm pair}\approx 10^{10}$ K, pair annihilation can produce neutrino-antineutrino pairs, for which the infalling material is essentially transparent and are able to cool down very efficiently the material as it is decelerated at the surface of the NS or at the magnetopause. Therefore, the specific entropy, $s$, of the fallback material remains constant all through the accretion phase until it decelerates in the vicinity of the NS.
The value for $s$ is set at the time of the reverse shock formation. Detailed 2D numerical simulations of the propagation of the shock through the star [@Scheck:2006; @Kifonidis:2003; @Kifonidis:2006] show that typical values of $s\sim20~ k_{\rm B}/{\rm nuc}$ are found at the reverse shock. At this stage of the explosion the flow is highly anisotropic due to the Rayleigh-Taylor instability present in the expanding material and the Richtmyer-Meshkov instability at the He/H interface. Those instabilities generate substantial mixing between hydrogen and helium and even clumps of high-entropy heavier elements (from C to Ni) rising from the innermost parts of the star. Therefore, the fallback material has entropy in the range $s \sim 1-100 ~k_{\rm B}/{\rm nuc}$ and its composition, although it is mostly helium, can contain almost any element present in the explosion. 3D simulations show qualitatively similar results regarding the entropy values and mixing [@Hammer:2010; @Joggerst:2010; @Wongwathanarat:2015].
Outside the NS, the expanding supernova explosion leaves behind a low density rarefaction wave which is rapidly filled by the NS magnetic field, forming the magnetosphere. For the small magnetospheric densities, the inertia of the fluid can be neglected, and the magnetosphere can be considered force-free. The fallback reverse shock propagates inwards compressing this magnetosphere. The boundary between the unmagnetized material falling back and the force-free magnetosphere, i.e. the magnetopause, can be easily compressed at long distances ($r \gtrsim 10^8$ cm ) due to the large difference of the pressure of the infalling material with respect to the magnetic pressure. The dynamical effect of the magnetosphere only plays a role at $r \lesssim 10^8$ cm, i.e. inside the light cylinder for most cases. The precise radius where the magnetic field becomes dynamically relevant is estimated later in Section \[sec:riemann\]. Only in the case of magnetar-like magnetic fields and fast initial spin ($P\lesssim 10$ ms) this consideration is not valid, although this is not the case for CCOs.
To conclude this scenario overview, we note that the magnetospheric torques will spin-down the NS on a characteristic timescale [@Shapiro:2004] given by $$\begin{aligned}
\tau_c = \frac{P}{2 \dot P} \sim 180 \left( \frac{B_p}{10^{15} {\rm G}} \right)^{-2} \left ( \frac{P}{1 {\rm s}}\right )^2 {\rm yr},\end{aligned}$$ for a typical NS with radius $10$ km and mass $1.4 M_\odot$. $B_p$ is the value of the magnetic field at the pole of the NS. The value of the moment of inertia is $1.4\times10^{45}~\rm{g}~\rm{cm}^2$. At birth, the spin period of a NS is limited by the mass-shedding limit to be $P>1$ ms [@Goussard:1998]. If all NSs were born with millisecond periods, purely magneto-dipolar spin-down would limit the observed period of young NSs ($10^4$ yr) to $$\begin{aligned}
P_{{\rm obs}, 10^4 {\rm yr}}\lesssim 5.5 \left ( \frac{B}{10^{15} {\rm G}}\right )^2 {\rm s}.\end{aligned}$$ For magnetic fields $B\lesssim 1.4\times 10^{13}$ G this criterion fails for the vast majority of pulsars and all CCOs ($P\gtrsim0.1$ s) and therefore the measured spin period must be now very close to that hours after the onset of the supernova explosion. Detailed population synthesis studies of the radio-pulsar population clearly favor a broad initial period distribution in the range 0.1-0.5 s [@Faucher:2006; @Gullon:2014], rather than fast millisecond pulsars. Therefore, from observational constraints, it is reasonable to assume that progenitors of pulsars (including CCOs) have spin periods of $P\sim 0.1-0.5$ s at the moment of fallback. For such low rotation rates, the NS can be safely considered as a spherically symmetric body and its structure can thus be computed by solving the Tolman-Oppenheimer-Volkoff (TOV) equation.
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Stationary spherical accretion {#sec:michel}
==============================
We model the fallback of the reverse shock as the spherically symmetric accretion of an unmagnetized relativistic fluid. The stationary solutions for this system were first obtained by [@Michel:1972] for the case of a polytropic EoS. Here, we extend this work to account for a general (microphysically motivated) EoS. The equations that describe the motion of matter captured by a compact object, i.e. a NS or black hole, can be derived directly from the equations of relativistic hydrodynamics , namely the conservation of rest mass, $$\begin{aligned}
\nabla_{\mu}J^\mu=0\,,
\end{aligned}$$ and the conservation of energy-momentum, $$\begin{aligned}
\nabla_\mu T^{\mu \nu}=0\,,
\end{aligned}$$ where we use the notation $\nabla_{\mu}$ for the covariant derivative and the density current $J^{\mu}$ and the (perfect fluid) energy-momentum tensor $T^{\mu\nu}$ are given by $$\begin{aligned}
\label{eq:density_current}
J^{\mu} &=& \rho u^{\mu}\,,
\\
T_{\mu\nu} &=& \rho h u_{\mu}u_{\nu} + p g_{\mu \nu}\,.\end{aligned}$$ In the above equations $\rho$ is the rest-mass density, $p$ is the pressure and $h$ is the specific enthalpy, defined by $h = 1 + \varepsilon + p/\rho$, where $\varepsilon$ is the specific internal energy, $u^{\mu}$ is the four-velocity of the fluid and $g_{\mu \nu}$ defines the metric of the general spacetime where the fluid evolves. Assuming spherical symmetry and a steady state we have $$\begin{aligned}
\frac{d}{dr}(J^1\sqrt{-g} )&=& 0\,,
\label{eq:der_r_mass}
\\
\frac{d}{dr}(T_0^1\sqrt{-g}) &=& 0\,,
\label{eq:der_r_mom}\end{aligned}$$ where $g\equiv \det(g_{\mu\nu})$. The exterior metric of a non-rotating compact object is given by the Schwarzschild metric $$\begin{aligned}
ds^2 &=& -\left(1-\frac{2{M}}{r}\right)dt^2+\left(1-\frac{2{M}}{r}\right)^{-1}dr^2
\nonumber \\
&+& r^2 (d\theta^2+ \sin^2\theta \,d\varphi^2)\,.\label{eq:schwarzschild}\end{aligned}$$ In Schwarzschild coordinates Eqs. (\[eq:der\_r\_mass\]) and (\[eq:der\_r\_mom\]) can be easily integrated to obtain [cf. @Michel:1972] $$\begin{aligned}
\rho \, u \, r^2 &=& C_1\,, \label{eq:michel1}
\\
h^2 \left(1-\frac{2{M}}{r}+u^2\right)&=&C_2\, \label{eq:michel2}\end{aligned}$$ where $C_1$ and $C_2$ are integration constants and $u\equiv u^r$. To obtain an adiabatic solution for the accreting fluid, we differentiate Eqs. (\[eq:michel1\]) and (\[eq:michel2\]) at constant entropy and eliminate $d\rho$ $$\begin{aligned}
\label{eq:crit_point}
&&\frac{du}{u}\left[ V^2-u^2 \left(1-\frac{2{M}}{r}+u^2 \right)^{-1}\right]
\nonumber \\
&+& \frac{dr}{r}\left[ 2V^2-\frac{{M}}{r} \left(1-\frac{2{M}}{r}+u^2 \right)^{-1}\right] =0\,,\end{aligned}$$ where $$\begin{aligned}
V^2\equiv\frac{\rho}{h}\left. \frac{\partial h}{\partial \rho} \right|_s\,.\end{aligned}$$ The solutions of this equation are those passing through a critical point where both terms in brackets in equation (\[eq:crit\_point\]) are zero, i.e. those fulfilling $$\begin{aligned}
2u_c^2 &=& \frac{{M}}{r_c}\,,
\nonumber \\
V_c^2 &=& u_c^2(1-3u_c^2)^{-1}\,, \label{eq:crit_point2}\end{aligned}$$ where sub-index $c$ indicates quantities evaluated at the critical point. The critical point can be identified as the sonic point, i.e. the point where the velocity of the fluid equals its own sound speed. After some algebra, it can be shown that the constant $C_1$ in equation (\[eq:michel1\]) is related to the accretion rate $\dot{M}$ by $$\begin{aligned}
\dot{M}=-4\pi C_1.\end{aligned}$$ Thereby we can obtain the accretion solution by simply selecting the mass accretion rate and the specific entropy of the fluid, which fixes the two constants $C_1$ and $C_2$. We note that, for each pair of values, the system (\[eq:crit\_point2\]) has two solutions, although only one represents a physical accretion solution ($|u|\to 0$ at $r \to \infty$). In this case the fluid is supersonic for radii below the critical radius and subsonic above. Figure \[fig:michel\_solution\] displays one illustrative accretion solution for a mass accretion rate $\dot{M}=10^{-5} M_{\odot}/\rm{s}$ and entropy per baryon $s=80k_{\rm B}/{\rm nuc}$. For the accreting material, we use the tabulated Helmholtz EoS [@Timmes:2000], which is an accurate interpolation of the Helmholtz free-energy of the Timmes EoS [@Timmes_Arnett:1999]. Timmes EoS, and Helmholtz EoS by extension, include the contributions from ionized nuclei, electrons, positrons and radiation. By default, Timmes EoS uses the rest mass density $\rho\ [\rm{g}/\rm{cm}^3]$, temperature $T\ [\rm{K}]$ and composition as input. For convenience, we have developed a search algorithm that allows to call the EoS with different thermodynamical variables as input (e.g. $\rho$, $s$ and composition as inputs for the adiabatic flow of accreting material). Helmholtz EoS also requires the mean mass number $\bar{A}$ and the mean atomic number $\bar{Z}$.
At low densities, $\rho < 6\times 10^7$ g cm$^{-3}$, and temperatures, $T\lesssim 2\times 10^9$ K, nuclear reactions proceed much slower than the accretion timescale and the composition remains frozen during the accretion. We fix the composition to that at the reverse shock formation point. Given the uncertainties, we consider two possibilities in this regime, either pure helium or pure carbon. At temperatures $T\gtrsim 2\times 10^9$ K nuclear burning becomes fast enough to change the composition. For $T\gtrsim 4\times 10^9$ K the fluid reaches nuclear statistical equilibrium (NSE) on a significantly shorter timescale than the accretion timescale [see e.g. @Woosley:2002]. To deal with the high temperature regime, $T\ge 2\times 10^9$ K, we have tried three different approaches: 1) unchanged composition of the accreting material, 2) compute the NSE composition at a given temperature and density using a thermonuclear reaction network with 47 isotopes [@Timmes:1999; @Seitenzahl:2008] and 3) simplified burning with four transitions: $^{4}\rm{He}$ for $T\le 2\times 10^9$ K, $^{56}\rm{Ni}$ for $2\times 10^9 > T\ge 5\times 10^9$ K, $^{4}\rm{He}$ for $5\times 10^9 > T\ge 2\times 10^{10}$ K and protons and neutrons for $T> 2\times 10^{10}$ K. We use the publicly available routines of the Hemlholtz EoS and the NSE equilibrium kindly provided by the authors[^2].
Non-magnetized accretion and pile-up {#sec:non-magnetized accretion}
====================================
Before considering the case of magnetized accretion onto a NS, we study the case of non-magnetized accretion. For the span of accretion rates considered in this work, the sonic point of the accreted fluid is located at $r>23500$ km at entropy $s=10~k_{\rm{B}}/\rm{nuc}$, and hence the fallback material falls supersonically onto the NS. Inevitably an accretion shock forms at the surface of the star, which propagates outwards. The accreted fluid crossing the shock will heat up, increasing its specific entropy and will fall subsonically. The high entropy of this material ($s_{shock} \in [70 - 300]~k_{\rm B}/{\rm nuc}$) and the compression that experiments as it flows inwards raises the temperature beyond the pair creation threshold, $T_{\rm pair}\approx 10^{10}$ K, and the fluid will cool efficiently via neutrino-antineutrino annihilation. Therefore, the kinetic energy of the supersonically accreting fluid is mostly transformed into thermal energy as it crosses the accretion shock and then is dissipated to neutrinos close to the NS surface. [@Chevalier:1989] showed that the accretion shock will eventually stall at a certain radius as an energy balance is found. The radius of the stalled shock depends only on the accretion rate $\dot M$ and is located at about $R_{\rm shock} \sim 10^7 - 10^8$ km. In some estimates bellow in this work we use the values provided in table 1 in [@Houck:1991], based in a more realistic treatment of the accretion and neutrino cooling.
The final fate of the neutrino-cooled material falling steadily onto the NS surface is to pile up on top of the original NS material forming a layer of new material. In order to study the effect of the pile up we consider a NS of mass $M$ and radius $R$. If we add a mass $\delta M$ to the equilibrium model, the new NS will have a new radius $R_{\rm new}$ smaller than the original one. The original surface of the star, will now be buried at a depth $\delta R$, i.e. the new surface will be located at a distance $\delta R$ over the old surface. Although trivial, the last statement is important because most of the discussion below in this work is carried out in terms of $\delta R$ and in terms of distances with respect to the original NS surface. Therefore it makes sense to try to compute what is the dependence of the burial depth, $\delta R$, with the total accreted mass, $\delta M$. In order to compute this dependence we use the TOV equations to solve a sequence of NS equilibrium models starting with $M$ and progressively increasing to $M+\delta M$ for different values of $\delta M$. For each model in the sequence we compute $\delta R$ as the distance between the radius enclosing a mass $M$ and the surface of the star, i.e. the radius enclosing $M + \delta M$. Given the small values of $\delta M$, we integrate the TOV equations using a simple forward Euler method, with a step limited to relative variations of density of $10^{-5}$ and a maximum step of $10$ cm. We have computed the relation between $\delta R$ and $\delta M$ for four different NS masses, $M=1.2$, $1.4$, $1.6$, $1.8$ and $2.0 M_\odot$. We have used several realistic EoS in tabulated form, namely four different combinations using either EoS APR [@Akmal:1998] or EoS L [@Pandharipande:1995] for the core and EoS NV [@Negele:1973] or EoS DH [@Douchin:2001] for the crust. For each case we compute the sequence up to the maximum mass; beyond that mass, the equilibrium model is unstable and it will collapse to a black hole in dynamical timescales. All EoS allow for equilibrium solutions with maximum mass consistent with recent observations of a NS with mass close to $2 M_\odot$ [@Demorest:2010; @Antoniadis:2013]. The blue solid line in Fig. \[fig:drdm\] shows the dependence of $\delta R$ with $\delta M$ for a $1.4 M_\odot$ NS with the APRDH EoS. All other EoS and neutron star masses show similar behavior. For all EoS, any amount of accreted mass larger than $\sim 10^{-4} M_{\odot}$ will sink the original NS surface to the inner crust, and for $\delta M \sim 0.1 M_{\odot}$ the entire crust is formed by newly accreted material. The bottom line is that, if the accreted material is able to compress the magnetosphere and deposit itself on top of the NS, the magnetic field trapped with the fluid may be buried into the NS crust, and depending on the conditions (accreted mass and magnetic field strength), the burial depth could be as deep as the inner crust. We study next the impact of magnetic fields in the vicinity of the NS, namely the magnetosphere, in the burial process.
Magnetosphere {#sec:magnetosphere}
=============
Potential magnetospheric solution {#sec:forcefree}
---------------------------------
For simplicity in the following discussion we use a [*reference model*]{} with the APRDH EoS and $M=1.4 M_\odot$. This model results in a a NS with coordinate radius $R=12.25$ km. The effect of the EoS and the NS mass are discussed later in the text. Given that both the magnetosphere and the accreted material involve low energy densities compared with those inside the NS, the spacetime outside the NS can be regarded as non-self-gravitating and approximated by the Schwarzschild exterior solution.
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The magnetosphere extends between the NS surface and the magnetopause, which will be assumed to be a spherically symmetric surface at the location of the infaling reverse shock. We model this region using the force-free magnetic field approximation, $ \mathbf{J}\times\mathbf{B}=0$, $\mathbf{J}$ being the electric current and $\mathbf{B}$ the magnetic field. We neglect the currents resulting from the rotation of the star. Consequently the magnetic field has a potential solution, solution of the relativistic Grad-Shafranov equation. In spherical coordinates, the magnetic field vector components are related to the vector potencial $\mathbf{A}$ as, $$\begin{aligned}
\label{Br}
\hat{B}_r&=&\frac{1}{r^2\sin \theta}\partial_\theta A_\phi \,,
\label{Br}
\\
\hat{B}_\theta&=&\frac{-1}{r^2\sin \theta}\partial_r A_\phi \,,
\label{Btheta}
\\
\hat{B}_\phi &=& 0\,,\end{aligned}$$ where $\hat{B}_i=\sqrt{\gamma}B_i$ and $\gamma$ is the determinant of the spatial metric. If we assume axisymmetry, the unique nonzero component of the electric current is the $\phi$ component, $$\begin{aligned}
J_\phi = \sin \theta \left[ \partial_r(r \hat{B}_\theta)-\partial_\theta \hat{B}_r\right].
\label{eq:current}\end{aligned}$$ Imposing the force-free condition, we obtain, $$\begin{aligned}
-J_\phi \hat{B}_\theta &=& 0\,,
\\
J_\phi \hat{B}_r &=& 0\,.\end{aligned}$$ Since $\hat{B}_r, \hat{B}_\theta\neq0$, the only possible solution is $J_\phi=0$. As we want an expression that only depends on the vector potential, we replace Eqs. (\[Br\]) and (\[Btheta\]) in equation (\[eq:current\]) resulting in $$\begin{aligned}
\label{jphi}
J_\phi & =&\sin \theta \left[\frac{-1}{\sin \theta} \partial_r(\partial_r)A_\phi-\frac{1}{r^2}\partial_\theta \left(\frac{\partial_\theta A_\phi}{\sin\theta}\right)\right]
\nonumber \\
&=&-\partial_{rr}A_\phi-\frac{1}{r^2}\partial_{\theta\theta}A_\phi+\frac{\cot \theta}{r^2}\partial_\theta A_\phi=0\,.\end{aligned}$$ We discretize this expression using second order finite differences and solve the resulting linear system of equations using a cyclyc reduction algorithm [@Swarztrauber:1974]. We impose Dirichlet boundary conditions on $A_\phi$ at the surface of the NS to match with the interior value of the radial component of the magnetic field. Our aim is to describe a magnetosphere, which is confined within a certain radius, $R_{\rm mp}$, defining the magnetopause. Magnetic field lines at the magnetopause are parallel to this interface and they enter the NS along the axis. Therefore, they correspond to lines with $A_\phi =0$, which we use as Dirichlet boundary condition at $R_{\rm mp}$ to solve the Grad-Shafranov equation. We can obtain the field distribution after the compression by simply changing the radius where the boundary conditions are imposed. The evolution of the magnetic field geometry before and after compression is shown in Fig. \[mf\_compression\] for three illustrative cases.
For the interior magnetic field, which determines the boundary conditions at the surface of the star, we use two different magnetic field distributions, a dipolar magnetic field ([*dipole*]{} herafter) and a poloidal field generated by a circular loop of radius $r=4\times10^5$ cm [@Jackson:1962] ([*loop current*]{} hereafter). Following @Gabler:2012, it is useful to introduce the [*equivalent magnetic field*]{}, $B^*$, which we define as the magnetic field strength at the surface of a Newtonian, uniformly magnetized sphere with radius $10$ km having the same dipole magnetic moment as the configuration we want to describe. It spans the range $B^*\in[10^{10}-10^{16}]$ G.
Magnetosphere compression {#sec:riemann}
-------------------------
In the case of a fluid accreting onto a force-free magnetosphere, the magnetopause will remain spherical and will move inwards as long as the total pressure of the unmagnetized fluid, $p_{\rm tot} = p + p_{\rm ram}$ , exceeds that of the magnetic pressure, $p_{\rm mag}$, of the magnetosphere. If we approximate the magnetopause as a spherical boundary between the spherically symmetric accreting solution described in Section \[sec:michel\] and the potential solution computed in Section \[sec:forcefree\], its properties can be described as the solution of a Riemann problem at the magnetopause. Since the magnetic field of the initial state is tangential to the magnetopause, we can use the exact solution of the Riemann problem developed by [@Romero:2005]. A succinct summary of the details of the implementation of the Riemann solver can be found in Appendix \[riemann\_problem\].
For illustrative purposes the left panel of Fig. \[fig:riemann\_problem\] shows the solution of the Riemann problem for a supersonic fluid accreting from the right into a magnetically dominated region (magnetosphere) on the left. The figure displays both the density (left axis, solid lines) and the fluid velocity (right axis, dashed lines). The initial discontinuity is located at $x=0$. The right constant state of the Riemann problem corresponds to the accreting fluid with an entropy of $s=10~k_{\rm{B}}/\rm{nuc}$ and accretion rate of $\dot{M}=10^{-7} \ \rm{M}_\odot/\rm{s}$. The left constant state corresponds a state with magnetic pressure $B^2/2$. The figure plots the corresponding solutions for different values of $B$ around the equilibrium (indicated in the legend).
Looking at the left panel of Fig. \[fig:riemann\_problem\] from left to right, the first jump in density corresponds to the contact discontinuity, point at which, as expected, the velocity remains continuos. The next discontinuity is a shock wave, where both the density and velocity are discontinuous, and both decrease. For low magnetic fields, $B\le 10^{10} G$, the low magnetic pressure on the left state cannot counteract the total pressure of the accreting fluid and the contact discontinuity advances to the right at a velocity equal to that of the accreting fluid; a shock front is practically nonexistent. As the magnetic field is increased the velocity of the contact discontinuity decreases and it becomes zero at about $B=10^{13}$ G. We identify this point as the [*equilibrium point*]{}, since no net flux of matter crosses $x=0$. Around this equilibrium point, an accretion shock appears, which heats and decelerates matter coming from the right. The equilibrium point corresponds to a solution in which the matter crossing the shock has zero velocity, i.e. it piles up on top of the left state as the shock progresses to the right.
The actual accretion of matter onto a magnetically dominated magnetosphere is expected to behave in a similar way as the described Riemann problem. At large distances (low $B$) the magnetopause (contact discontinuity) is compressed at the speed of the fluid. As the magnetosphere is compressed, the magnetic field strenght raises and at some point an equilibrium point is found, beyond which the magnetosphere impedes the accretion of the fluid.
In the right panel of Fig. \[fig:riemann\_problem\] we show for the sake of completeness the solution for a subsonic accreting fluid. In accreting NS this regime is probably unrealistic, since very large specific entropy is necessary ($s=2000 k_{\rm B}/{\rm nuc}$ in the example plotted). In this case the solution is qualitatively different; instead of a shock, a rarefaction wave if formed for $B$ below the equilibrium point. For larger values of $B$, an accretion shock is formed.
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Setup
-----
Our goal is to study the conditions under which the magnetic field of a new-born NS can be buried by fallback material during a supernova. We have spanned a large range of values for both, the magnetic field strength and the accretion rate, proceeding as follows. We obtain the distance from the NS surface where the magnetosphere and the accreting fluid are in balance, i.e. the radial point where the velocity of the contact discontinuity is zero. We reduce our 2D configuration to a 1D Riemann problem by restricting the evaluation of the equilibrium point to the equatorial plane of the NS, due to the fact that the magnetic pressure is maximum at the equator. Therefore, if the magnetic field can be buried in this latitude, it will be buried in all latitudes of the NS.
The code developed by [@Romero:2005] requires as input the knowledge of the density, velocity, thermal pressure, and magnetic pressure at both left and right states of the initial discontinuity. In all cases we consider, the left state corresponds to the force-free magnetosphere while the right state is occupied by the accreting fluid. To obtain the magnetic pressure of the left state we find the solution of the Grad-Shafranov equation (see Section \[sec:forcefree\]). This allows to locate the position of the magnetopause where the Riemann problem must be solved. Since the intertia of the fluid at the magnetosphere can be neglected in front of the magnetic pressure, the value of the density on the left state is set to yield an Alfvén velocity near to one , the thermal pressure is set to be at least six orders of magnitude lower than the magnetic pressure, and the velocity is set to zero. On the other hand, the values on the right state are fixed to the corresponding values of density, pressure and velocity of the stationary spherical accretion solution (see Section \[sec:michel\]) and the magnetic pressure is set to zero.
A sketch of the different stages of the accretion process is shown in Fig. \[fig:accretion\_diagram\]. The plots depict the location of the NS (including its core and inner and outer crust), the magnetosphere, the magnetopause, and part of the region where material is falling back. Each region is shaded in a different colour for a simple identification. Note that the scale ratio of the different regions is not preserved in the figure. The upper panel in Fig. \[fig:accretion\_diagram\] shows the initial state of the process. The panels on the left column show the expected evolution for a low magnetic field case (e.g. $B\lesssim 10^{13}$ G) while those on the right column correspond to a typical high magnetic field case (e.g. $B\gtrsim 10^{13} $ G). In general, the value of $B$ separating between the two regimes depends on the accretion rate. For this figure we have chosen a value of the magnetic field that corresponds to a representative example of our results (see Section 6), for which $\dot{M}=10^{-5} M_{\odot}/$s. At the beginning of the evolution, the reverse shock falls over the magnetosphere. The magnetic field lines are confined inside the magnetosphere, which is shown in white on the diagram. Depending on the position of the sonic point, which in turn depends on the values of the specific entropy and the accretion rate, the motion of the reverse shock may be either supersonic or subsonic. We limit the qualitative description of the evolution below to the case of a supersonic reverse shock as in the subsonic case no accretion shock forms, as shown in Section \[sec:riemann\].
The middle two panels in both evolutionary tracks show only qualitative differences in the size of the resulting magnetosphere after its compression and in the amplitude of the instabilities that may arise in the magnetopause (see below). Therefore, our description can be used for either path keeping this quantitative differences in mind. The evolution on the left column shows the case where the magnetic pressure is weak compared with the ram pressure of the fluid. In this case the magnetosphere shrinks significantly until the equilibrium point is reached ($R_{\rm mp}$; zero speed contact discontinuity) close to the NS surface at $R_{\rm s}\sim 10$ km. If the infall of the reverse shock is sxupersonic an accretion shock will appear simultaneously. The location of this accretion shock is shown on the horizontal axis of the four middle panels. As a result, the velocity of the reverse shock is reduced due to the presence of a region of subsonic accretion behind the accretion shock. Nevertheless, as through the accretion shock the momentum is conserved, the compression is not affected. The evolution on the right column, where the magnetic pressure is stronger, is qualitatively similar, only the accretion shock is located further away from the NS surface and the magnetosphere is not so deeply compressed.
As we will discuss below in more detail, the compression phase is unstable against the growth of Rayleigh-Taylor instabilities and the development of convection on the dynamic timescale. Therefore, the fluid and the magnetic field lines can mix, which provides a mechanism for the infalling fluid to actually reach the star. As the fluid reaches the NS, the mass of the star grows from $M_{\rm s}^*$ to $M_{\rm s}$ and its radius increases from $R_{\rm s}^*$ to $R_{\rm s}$, encompassing the twisted magnetic field lines a short distance away. The mass accreted $\delta M$ forms part of the new crust of the NS, whose final radius will depend on the total mass accreted during the process. The bottom panels of the diagram depict a magnified view of the NS to better visualize the rearrangement the mass of the star and the magnetic field undergo. If the radius $R_{\rm mp}$ of the equilibrium point is lower than the new radius $R_{\rm s}$, all the magnetic field lines will be frozen inside the NS new crust, as shown in the bottom-left plot of Fig. \[fig:accretion\_diagram\] which corresponds to the end of the accretion process for a low magnetic field evolution. On the contrary, if the magnetic field is high, as considered on the evolutionary path on the right, the equilibrium point $R_{\rm mp}$ is far from the surface of the NS. Although part of the infalling matter may still reach the star and form a new crust, the mechanism is not as efficient as in the low magnetic field case. This is depicted in the bottom-right panel of the figure.
In our approach, that we discuss in more detail in the section on results, we compare the distance obtained by the Riemann solver for the location of $R_{\rm mp}$ (zero speed in the contact discontinuity) with the increment of the radius of the NS, $\delta R$, due to the pile up of the accreting matter. If the radial location of the equilibrium point $R_{\rm mp}$ is lower than $\delta R$ (as in the bottom-left panel of Fig. \[fig:accretion\_diagram\]) we conclude that the magnetic field is completely buried into the NS crust. On the contrary, if $R_{\rm mp}>\delta R$, our approach does not allow us to draw any conclusion. In this case, multidimensional MHD numerical simulations must be performed to obtain the final state of the magnetic field.
Results {#Results}
=======
We turn next to describe the main results of our study. In order to be as comprehensive as possible, we cover a large number of cases which are obtained from varying the physical parameters of the model, namely the composition and entropy of the accreting fluid, the mass of the NS, and the initial magnetic field distribution. For all possible combinations of these parameters the outcome of the accretion process depends both on the magnetic field strength and on the mass accretion rate. This dependence is presented in the following sections in a series of representative figures. A summary of all the combinations considered and the description of the model parameters can be found in Table \[tab:models\].
---------------- ------------- -------------------------- ---------------- ----------------- --
Composition Entropy NS Mass MF distribution
$k_{\rm{B}} / \rm{nuc}$ $\rm{M}_\odot$
Reference He + NSE 10 $1.4$ loop current
1 He + NSE 100 $1.4$ loop current
2 He + NSE 1000 $1.4$ loop current
3 He + NSE 5000 $1.4$ loop current
4 He 10 $1.4$ loop current
5 He 100 $1.4$ loop current
6 He 1000 $1.4$ loop current
7 He 5000 $1.4$ loop current
8 C + NSE 10 $1.4$ loop current
9 C + NSE 100 $1.4$ loop current
10 C + NSE 1000 $1.4$ loop current
11 C + NSE 5000 $1.4$ loop current
12 He 10 $1.4$ dipole
13 He 1000 $1.4$ dipole
14 He 10 $1.2$ loop current
15 He 10 $1.6$ loop current
16 He 10 $1.8$ loop current
17 He 10 $2.0$ loop current
\[tab:models\]
---------------- ------------- -------------------------- ---------------- ----------------- --
: Models considered in this study.
Reference model
---------------
We use as a reference model the one corresponding to an accreting fluid with $s=10~k_{\rm{B}} / \rm{nuc}$, and composed essentially by Helium. The nuclear reactions to reach nuclear statistical equilibrium are also allowed in this model. The mass of the NS is $1.4\ M_{\odot}$ and the magnetic field is generated by a [*loop current*]{} in the NS.
[![Distance above the star of the equilibrium point $\delta R$ as a function of the total mass accreted $\delta M$ for each value of the magnetic field (solid lines) for the reference model. The yellow area indicates the region where the accretion flow is supersonic. The dotted line represents the limit of the accretion shock. The red area marks the outer crust of the NS after accretion, while the green and blue areas display the inner crust and the core respectively, as shown in Fig. \[fig:accretion\_diagram\]. []{data-label="fig:alternative_reference"}](./Figures/alternative_He_0010_14_APRDH_30_07.pdf "fig:"){width="80mm"}]{}
The results are shown in Fig. \[fig:alternative\_reference\]. The solid lines in this figure represent the distance of the equilibrium point $\delta R$ (position of the magnetopause) above the NS surface as a function of the total accreted mass $\delta M$. The limit of the horizontal axis is given by the maximum mass that can be accreted without forming a black hole. Each line corresponds to a different value of the initial magnetic field, indicated in the legend of the figure. The yellow area represents the region in which the accretion of the reverse shock is supersonic and the black dotted line shows the limit of the accretion shock. The dashed red line shows the radial location of the new surface of the star due to the accretion of the infalling matter. The lines which cross the dashed red line have the equilibrium point inside the crust of the NS and, therefore, the corresponding magnetic fields will be buried into the crust. However, for the lines that are in the white area, the equilibrium point is not close enough to the NS surface and the magnetic field can not be buried. Note that for initial values of the magnetic field $B\gtrsim 10^{15}$ G, the magnetic field is never buried for all mass accretion rates considered.
An alternative view of this result is shown in Fig. \[fig:reference\]. The goal of this representation is to provide a clearer representation of the dependence of the equilibrium point with the span of values of the magnetic field and the total mass accreted we are considering. The figure shows the isocontours where the equilibrium point is equal to the increment of the radius of the NS, i.e. $R_{\rm mp}=\delta R$. The two lines plotted (dotted, $t=10^4$ s, and solid, $t=10^3$ s) correspond to the limits of the total accretion time, which relates the accretion rate $\dot{M}$ and the total mass increment $\delta M$. The black area indicates the values of the maximum mass of the NS beyond which it will form a black hole. The dark orange region represents the span of values of $\delta M$ and $B^*$ where we cannot assure that the magnetic field could be buried completely. The light orange area, on the other hand, represents the cases where the magnetic field is totally buried. The results show that for low values of the magnetic field ($B^*<10^{11} G$) the field can be buried even with the lowest accretion rates we have considered. As expected, as the accreted mass increases it is possible to bury the magnetic field for larger initial field values, up to a certain maximum. Indeed, for $B^*> 2 \times 10^{14}$ G we cannot find any accretion rate which can bury the magnetic field.
[![Outcome of the accretion depending on the total accreted mass ($\delta M$) and the initial magnetic field ($B^*$) for the reference model. For the two accretion times considered, $t=10^{3}$ s (dark brown) and $t=10^4$ s (light brown), the respective line splits the parameter space in a region where the magnetic field will be buried (left side) or not completely buried (right side). Above certain $\delta M$ a black hole will be formed. The dashed line represents the fit shown in Eq. (\[Eq:fit\]).[]{data-label="fig:reference"}](./Figures/He_0010_14_APRDH_20_20.pdf "fig:"){width="80mm"}]{}
Models with higher specific entropy
-----------------------------------
We turn next to analyze the behavior of the magnetic field compression when the accreting fluid has higher specific entropy than in the reference model, keeping the same conditions for the composition, mass and magnetic field distribution (Models 1, 2 and 3 in Table \[tab:models\]). Fig. \[fig:entropy\_variation\] shows the results for values of the specific entropy of $s=100~k_{\rm{B}} / \rm{nuc}$, $1000~k_{\rm{B}} / \rm{nuc}$ and $5000~k_{\rm{B}} / \rm{nuc}$ compared with the reference model ($s=10~k_{\rm{B}} / \rm{nuc}$). For the model with specific entropy $100~k_{\rm{B}} / \rm{nuc}$, the results are very similar to the reference model as both lines almost perfectly overlap. For larger specific entropy the difference is more noticeable; for $s=1000~k_{\rm{B}} / \rm{nuc}$ and $5000~k_{\rm{B}} / \rm{nuc}$, the burial/reemergence boundary of the parameter space is shifted toward larger magnetic fields, i.e. higher entropy material compress the magnetosphere more easily and it is possible to bury larger magnetic fields. This behavior can be understood if one considers that the equilibrium point is a balance between the total pressure of the infall material, $p_{\rm tot} = p + p_{\rm ram} \approx p + \rho v^2$, and the magnetic pressure of the magnetosphere. For low specific entropy, the total pressure is dominated by the ram pressure and changes in $s$ do not produce significant changes in the equilibrium point. Above a certain threshold, the thermal pressure $p$ dominates the total pressure and increasing $s$ induces a larger compression of the magnetosphere, shifting the equilibrium point downwards. For realistic values of the specific entropy in supernovae, $s\sim 10-100~k_{\rm{B}} / \rm{nuc}$ [@Scheck:2006; @Kifonidis:2003; @Kifonidis:2006], we expect the ram pressure to be dominant and hence the influence of $s$ to be minimal. Even for an unrealistically large value of the specific entropy, $5000~k_{\rm{B}} / \rm{nuc}$, the maximum magnetic field that can be buried increases one order of magnitude at most, and only for the largest mass accretion rates considered.
[![Similar to Fig. \[fig:reference\] but for the models with different specific entropy for the accreting fluid (namely, models 1 to 3 and reference). All cases are shown for a total accretion time of $10^3$ s. Each line ends at the maximum mass of the corresponding model.[]{data-label="fig:entropy_variation"}](./Figures/entropy_comparation.pdf "fig:"){width="85mm"}]{}\
Models with different NS mass
-----------------------------
We consider next the effect of the neutron star mass, within astrophysically relevant limits. According to observations [see @Lattimer:2012 and references therein] the lower limit for the NS mass is around $1.2~M_\odot$. The maximum achievable mass of a NS is strongly dependent on the equation of state [@Lattimer:2005]. Nowadays, there are a few observations that support the existence of pulsars and NS with masses greater than $1.5~M_\odot$, in particular an observation of a $\sim 2~M_\odot$ NS [@Demorest:2010; @Antoniadis:2013]. For this reason, we explore the results for several values of the neutron star mass between $1.2~M_\odot$ and $2~M_\odot$. The results are shown in Fig. \[fig:mass\_variation\], where each line corresponds to a model with different NS mass as indicated in the legend. The results for all masses are very similar. In general we observe that for more massive NS, a higher accreted mass is needed to bury the magnetic field. Our interpretation is that higher mass NS have lower radii and hence we have to compress more the magnetosphere to successfully bury it into the crust. Therefore, a higher accreted mass is needed to bury the field for NS with larger mass (smaller radius). Since the radius difference between a $1.2$ and a $2$ $M_\odot$ NS is small, the impact of the NS mass on the burial is minimal. The maximum value of the magnetic field which can be buried is $\sim 2\times10^{14}$ G in all cases. For smaller NS masses slightly larger values of the magnetic field can be buried due to the ability to support a larger accreted mass. We conclude that the burial of the magnetic field is not crucially sensitive to the NS mass.
[![Similar to Fig. \[fig:reference\] for the models with different NS mass: $1.2~{M}_\odot$, $1.6~{M}_\odot$ , $1.8~{M}_\odot$, $2.0~{M}_\odot$ (models 14, 15, 16 and 17 and reference). All cases are shown for a total accretion time of $10^3$ s. Each line ends at the maximum mass of the corresponding model.[]{data-label="fig:mass_variation"}](./Figures/mass_comparation.pdf "fig:"){width="85mm"}]{}
Models with different EoS
-------------------------
Fig. \[fig:eos\_variation\] shows the comparison of the results for the reference model when using the four different equations of state described in section \[sec:non-magnetized accretion\]. For $M=1.4~M_\odot$, the coordinate radius of these NS models is $12.25$ km for APRDH, $12.11$ km for APRNV, $15.77$ km for LDH and $15.37$ km for LNV. Since the maximum mass is sensitive to the EoS, each line ends at different points in the $\delta M$ vs $B^*$ plot. The use of APRDH or APRNV EoSs leads to almost indistinguishable results (the two lines lay on top of each other). This is expected since the radius of this two models differs only by about $1\%$, because the EoS are very similar and only differ at low densities (at the crust). The LDH and LNV EoSs allow the burial of a larger magnetic field for a given accreted mass, in comparison with APRDH and APRNV. The maximum magnetic field that can be buried in the LDH and LNV models is $\sim 6\times10^{14}$ G and $\sim 5\times10^{14}$ G respectively, which is about a factor $2$ larger than for the APRDH EoS. In general, for a $M=1.4~M_\odot$, EoS resulting in a larger NS radius allow to bury larger magnetic fields for a given $\delta M$. Given that the results of this work are meant to be an order-of-magnitude estimate of the location in the parameter space of the limit between burial and reemergence, a difference of a factor $2$ due to the EoS, does not change the main conclusions of this work. For practical purposes the APRDH EoS can be taken as a good estimator for this limit.
[![Similar to Fig. \[fig:reference\] for the models with different EoS, a NS of mass $1.4\rm{M}_\odot$ and specific entropy of the accreting fluid $10k_{\rm{B}} / \rm{nuc}$ All cases are shown for a total accretion time of $10^3$ s. Each line ends at the maximum mass of the corresponding model. []{data-label="fig:eos_variation"}](./Figures/eos_comparation.pdf "fig:"){width="85mm"}]{}
Remaining models
----------------
We do not observe any significant differences with respect to the reference model in the results for the models with different initial composition of the reverse shock (models 8 to 11) or the ones using the NSE calculations (models 4 to 7). As a result we do not present additional figures for these models since the limiting lines overlap with those of the reference model. The observed lack of dependence is due to the fact that the EoS only depends on the electron fraction, $Y_e$. This value is obtained from the ratio between the mean atomic mass number ($\bar{A}$) and the mean atomic number ($\bar{Z}$). For both cases of pure Helium and pure Carbon, this ratio is equal to $Y_e=0.5$ and, consequently, the values of pressure and density for the accreting fluid are almost identical, producing differences in the results below the numerical error of our method [^3]. In the case of the NSE calculation, the reason is similar. For low entropies ($s=10, 100\,k_{\rm{B}} / \rm{nuc}$) the temperature is not sufficiently high to start the nuclear reactions and the composition remains constant throughout the accretion phase. For higher entropies, although the value of the electron fraction may differ from $0.5$ during the accretion process, the differences produced in the thermodynamical variables lead to changes in the results of the Riemann problem still below the numerical error of the method.
Regarding the initial distribution of the magnetic field, we do not observe either any significant difference in the results in the two cases that we have considered, [*loop current*]{} and [*dipole*]{}. Given that we are comparing models with the same effective magnetic field, $B^*$, and thus the same magnetic dipolar moment, the magnetic field is virtually identical at long radial distances and the only differences appear close to the NS surface. In practice the magnetic field structure only changes the details of the burial in the cases in which the equilibrium point is close to the burial depth (the limiting line plotted in the Figs. \[fig:reference\] to \[fig:eos\_variation\]), but it does not change the location of the limit itself in a sensitive way. As a conclusion, we can say that the dominant ingredient affecting the burial of the magnetic field is the presence of a dipolar component of the magnetic field but, for order-of-magnitude estimations, a multipolar structure of the field is mostly irrelevant.
Summary and discussion {#Summary}
======================
We have studied the process of submergence of magnetic field in a newly born neutron star during a hypercritical accretion stage in coincidence with core collapse supernovae explosions. This is one of the possible scenarios proposed to explain the apparently low external dipolar field of CCOs. Our approach is based on 1D solutions of the relativistic Riemann problem, which provide the location of the spherical boundary (magnetopause) matching an external non-magnetised accretion solution with an internal magnetic field potential solution. For a given accretion rate and magnetic field strength, the magnetopause keeps moving inwards if the total (matter plus ram) pressure of the accreting fluid, exceeds the magnetic pressure below the magnetopause. Exploring a wide range of accretion rates and field strengths, we have found the conditions for the magnetopause to reach the equilibrium point below the NS surface, which implies the burial of the magnetic field. Our study has considered several models with different specific entropy, composition, and neutron star masses. Assuming an accretion time of 1000s, our findings can be summarised by a general condition, rather independent on the model details, relating the required total accreted mass to bury the magnetic field with the field strength. An approximate fit is (see dashed line in Fig. \[fig:reference\]) $$\label{Eq:fit}
\frac{\delta M}{M_\odot} \approx \left( \frac{B}{2.5\times10^{14}} \right)^{2/3}.$$
The most important caveat in our approach is that we are restricted to a simplistic 1D spherical geometry, which does not allow us to consistently account for the effect of different MHD instabilities that can modify the results. We also note that our scenario is quite different from the extensively studied case of X-ray binaries, in which the NS accretes matter from a companion but at much lower rates (sub-Eddington) and matter is mostly transparent to radiation during accretion. In that case, matter cools down through X-ray emission during the accretion process. [@Davidson:1973] and [@Lamb:1973] already noticed this fact and predicted that the accretion will most likely be channeled through the magnetic poles, in analogy to the Earth’s magnetosphere. In the context of X-ray binaries, [@Arons:1976] and [@Michel:1977] were able to compute equilibrium solutions with a deformed magnetosphere and a cusp like accretion region at the magnetic poles. However, as the same authors pointed out, these systems are unstable to the interchange instability [@Kruskal:1954], a Railegh-Taylor-like instabilitiy in which magnetic field flux tubes from the magnetosphere can raise, allowing the fluid to sink. This might allow for the formation of bubbles of material that fall through the magnetosphere down to the NS surface. In the case of a fluid deposited on top of a highly magnetized region, modes with any possible wavelength will be unstable [@Kruskal:1954], however, in practice these instabilities are limited to the size of the magnetosphere ($\sim R_{\rm mp}$) in the angular direction. As the bubbles of accreted material sink, magnetic flux tubes raise, as long as their magnetic pressure equilibrates the ram pressure of the unmagnetized accreting fluid [@Arons:1976]. Therefore, in a natural way, the equilibrium radius computed in Section \[sec:riemann\] roughly determines the highest value at which the magnetic field can raise.
This accretion mechanism through instabilities has been shown to work in the case of X-ray binaries in global 3D numerical simulations [e.g. @Kulkarni:2008; @Romanova:2008]. In the case of the hypercritical accretion present in the supernova fallback, Rayleigh-Taylor instabilities have been studied by [@Payne:2004; @Payne:2007; @Bernal:2010; @Bernal:2013; @Mukherjee:2013a; @Mukherjee:2013b]. The simulations of [@Bernal:2013] also show that the height of the unstable magnetic field over the NS surface decreases with increasing accretion rate, for fixed NS magnetic field strength, as expected. Using the method described in Section \[sec:riemann\] we have estimated the equilibrium height over the NS surface for the 4 models presented in Fig. 9 of [@Bernal:2013], for their lower accretion rates ($\dot{M}\le 10^{-6}\ \rm{M}_\odot/s$). Our results predict correctly the order of magnitude of the extent of the unstable magnetic field over the NS surface. Therefore, our simple 1D model for the equilibrium radius serves as a good estimator of the radius confining the magnetic field during the accretion process, although details about the magnetic field structure cannot be predicted. Another important difference with the binary scenario is the duration of the accretion process. In X-ray binaries, a low accretion rate is maintained over very long times, so that instabilities have always time to grow. In our case, hypercritical accretion can last only hundreds or thousands of seconds, and depending on the particular values of density and magnetic field, this may be too short for some instabilities to fully develop. This issue is out of the scope of this paper and deserves a more detailed study.
Our main conclusion is that a typical magnetic field of a few times $10^{12}$ G can in principle be buried by accreting only $10^{-3}-10^{-2} M_\odot$, a relatively modest amount of mass. This estimate has interesting implications: since it is likely that most neutron stars can undergo such an accretion process, and the field would only reemerge after a few thousand years [@Geppert:1999; @Vigano:2012], the CCO scenario is actually not peculiar at all and we expect that most very young NSs show actually an anomalously low value of the magnetic field. On the contrary, magnetar-like field strengths are much harder to screen and the required accreted mass is very large, in some cases so large that the neutron star would collapse to a black hole. We also stress that the concept of [*burial*]{} of the magnetic field refers only to the large scale dipolar component, responsible for the magnetospheric torque spinning down the star. Small scale structures produced by instabilities can exist in the vicinity of the star surface, and this locally strong field is likely to have a visible imprint in the star thermal spectrum, as in Kes 79 [@Shabaltas:2012], without modifying the spin-down torque. However, the high field burial scenario should not be very common because both, high field NSs are only a fraction to the entire population, and only a part of them would undergo the fallback episode with the right amount of matter. This is consistent with the recent results of [@Bogdanov:2014] who searched for the hidden population of evolved CCOs among a sample of normal pulsars with old characteristic ages but close to a supernova remnant. None of the eight sources studied was found to have a luminosity higher than $10^{33}$ erg/s, which would have been an evidence of a hidden strong field. They all show X-ray luminosities in the 0.3-3 keV band of the order of $10^{31}$ erg/s (or similar upper limits), consistent with the properties of other low field neutron stars with $B\approx 10^{12}$ G. Thus, these sample of sources are not likely to be linked to the family of descendants of Kes 79-like objects, but there is no contradiction with these being pulsars with reemerged normal fields. Finally, we note that the slow reemergence process on timescales of kyrs mimics the increase of the magnetic field strength, and it is therefore consistent with a value of the braking index smaller than 3 [@Espinoza:2011], which should be common for all young pulsars in this scenario.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to thank J.M. Martí for many fruitful discussions. This work has been supported by the Spanish MINECO grants AYA2013-40979-P and AYA2013-42184-P and by the Generalitat Valenciana (PROMETEOII-2014-069).
Riemann Problem {#riemann_problem}
===============
For an ideal magneto-fluid, the energy-momentum tensor $T^{\mu\nu}$ and Maxwell dual tensor $F^{*\mu \nu}$ are $$\begin{aligned}
T^{\mu \nu}&=&\rho\hat{h}u^{\mu}u^{\nu}+g^{\mu \nu}\hat{p}-b^\mu b^\nu\,,
\\
F^{*\mu \nu} &=& u^\mu b^\nu-u^\nu b^\mu\,,\end{aligned}$$ where $\hat{h}=1+\varepsilon+p/\rho+b^2/\rho$ is the specific enthalpy including the contribution of the magnetic field and $\hat{p}=p+b^2/2$ is the total pressure. Moreover, $b^{\mu}$ stands for the magnetic field measured by a comoving observer (see [@Anton:2006] for details and its relation with the magnetic field $B^{\mu}$ measured by a generic observer). The conservation of these two quantities, jointly with the conservation of the density current, equation (\[eq:density\_current\]), lead to the equations of ideal relativistic MHD. $$\begin{aligned}
\nabla_{\mu} J^\mu=0\,,
\\
\nabla_{\mu}T^{\mu \nu}=0\,,
\\
\nabla_{\mu}F^{*\mu\nu}=0\,.\end{aligned}$$ In the particular configuration of our Riemann problem, $u^\mu=W(1,v^x,0,v^z)$, $b^\mu=(0,0,b,0)$, so the term $\nabla_{\mu}b^\mu b^\nu$ in the conservation of the stress-energy tensor, vanish. Therefore, the conservation equations reduce to the purely hydrodynamical case.
The Riemann problem in this particular configuration is described in terms of three characteristics, one entropy wave and two fast magnetosonic waves. The initial problem with two states $L$ (left) and $R$ (right) breaks up into four states, $$\begin{aligned}
L\mathcal{W}_\leftarrow L_*\mathcal{C} R_* \mathcal{W}_\to R,\end{aligned}$$ where $\mathcal{W}$ indicate a fast magnetosonic shock wave or a rarefaction wave and $\mathcal{C}$ indicates the contact discontinuity. Solving the Riemann problem entails finding the intermediate states ($L_*,R_*$) and the position of the waves, which are determined by the pressure $\hat{p}_*$ and the flow velocity $v_*^x$. If $\hat{p}\leq\hat{p}_*$ the wave is a rarefaction wave (a self-similar continuous solution), otherwise the solution is a shock wave. In our case [see @Romero:2005] the ordinary differential equation that allows to obtain the solution for a rarefaction wave is given by $$\begin{aligned}
\label{eq:rarefaction}
\frac{{d}v^x}{1-(v^x)^2}=\pm \frac{(1+b^2/(\rho h c_s))\sqrt{\hat{h}+\hat{\mathcal{A}}^2(1-w^2)}}{\hat{h}^2+\hat{\mathcal{A}}^2}\frac{{d}p}{\rho w},\end{aligned}$$ where $\hat{\mathcal{A}}=\hat{h}Wv^z$, $w=c_s^2+v_A^2-c_s^2v_A^2$, $v_A=b^2/\rho \hat{h}$ is the Alfvén velocity and $c_s=\sqrt{\frac{1}{h}\left.\frac{\partial p}{\partial \rho}\right|_s}$ is the sound speed. The integration of equation \[eq:rarefaction\] allows to conect the states ahead (a) and behind (b) the rarefaction wave. Rarefaction waves conserve entropy, hence, all the thermodynamical variables and the differential of $p$ must be calculated at the same entropy of the initial state. From this equation, the normal velocity behind the rarefaction can be obtained directly, $$\begin{aligned}
v_b^x=\tanh \hat{\mathcal{C}},\end{aligned}$$ with $$\begin{aligned}
\hat{\mathcal{C}}&=& \frac{1}{2}\log\left(\frac{1+v_a^x}{1-v_a^x}\right)
\nonumber \\
&\pm&
\int_{\hat{p}_a}^{\hat{p}_b}\frac{(1+b^2/(\rho h c_s))\sqrt{\hat{h}+\hat{\mathcal{A}}^2(1-w^2)}}{\hat{h}^2+\hat{\mathcal{A}}^2}\frac{{d}p}{\rho w}.\end{aligned}$$ In the same way, the velocity inside the rarefaction can be obtained replacing the thermal pressure $p$ by the total pressure $\hat{p}$.
On the other hand, shock waves should fulfill the so-called Rankine-Hugoniot conditions [@Lichnerowicz:1967; @Anile:1989] $$\begin{aligned}
[\rho u^{\mu}] \,n_{\mu} &=& 0\,,
\\
{[ T^{\mu\nu}]} \,n_{\nu} &=& 0\,,
\\
{[F^{*\mu \nu}]} \,n_{\nu} &=& 0\,,\end{aligned}$$ where $n_\mu$ is the unit normal to a given surface and $[H]\equiv H_a-H_b$ being $H_a$ and $H_b$ the boundary values. The normal flow speed in the post-shock state, $v_b^x$, can be extracted from the Rankine-Hugoniot equations [see @Romero:2005 for a detailed discussion], $$\begin{aligned}
v_b^x&=&\left(\hat{h}_aW_av_a^x+\frac{W_s(\hat{p}_b-\hat{p}_a)}{j} \right)
\\
&\times&
\left(\hat{h}_aW_a +(\hat{p}_b-\hat{p}_a)\left(\frac{W_sv_a^x}{j}+\frac{1}{\rho_a W_a}\right)\right)^{-1}\,,\end{aligned}$$ where $W_s=\frac{1}{\sqrt{1-V_s^2}}$ is the Lorentz factor of the shock, $$\begin{aligned}
V_s^{\pm}=\frac{\rho_a^2W_a^2v_a^x\pm|j|\sqrt{j^2+\rho_a^2W_a^2(1-{v_a^x})^2}}{\rho_a^2W_a^2+j^2}\,,\end{aligned}$$ is the shock speed, and $$\begin{aligned}
j\equiv W_s\rho_aW_a(V_s-v_a^x)=W_s\rho_bW_b(V_s-v_b^x)\,,\end{aligned}$$ is an invariant derived directly from the Rankine-Hugoniot jump conditions. These expressions, together with the Lichnerowicz adiabat, $$\begin{aligned}
[\hat{h}^2]=\left(\frac{\hat{h}_b}{\rho_b}+\frac{\hat{h}_a}{\rho_a}\right),\end{aligned}$$ allows us to calculate the shock wave solution.
[99]{}
Aguilera D., Pons J.A., Miralles J.A, 2008, [A&A]{}, [486]{}, 255
Akmal A., Pandharipande V. R., Ravenhall D. G., 1998, [Phys. Rev. C]{}, [58]{}, 1804
Anile A.M.,1989,[*Relativistic Fluids and Magneto-Fluids; with Applications in Astrophysics and Plasma Physics.*]{} Cambridge University Press.
Antón L., Zanotti, O., Miralles, J.A., Martí, J.M., Ibáñez, J.M., Font, J.A., Pons, J.A., 2006 ,[ApJ]{}, [637]{}, 296
Antoniadis J. et al, 2013, [Science]{}, [340]{}, 1233232
Arons J. , Lea S. M., 1976, [ApJ]{}, [207]{}, 914
Bernal C.G., Lee W.H., Page D., 2010, [RMAA]{}, [46]{}, 309
Bernal C.G., Lee W.H., Page D., 2013, [ApJ]{}, [770]{}, 106
Bignami, G. F., Caraveo, P. A., De Luca, A., Mereghetti, S. 2003, Nature, 423, 725
Blondin J.M., 1986, [ApJ]{}, [308]{}, 755
Bogdanov S., Ng C.-Y. and Kaspi V. M., 2014, [ApJL]{}, [792]{}, L36
Bonanno A. [et al.]{}, 2005, [A&A]{}, [440]{}, 199
Burrows, A., Lattimer, J.M. , 1986, [ApJ]{}, [307]{}, 178
Chevalier R.A. , 1989, [ApJ]{}, [346]{}, 847
Janka, H.-T., Langanke, K., Marek, A., Martínez-Pinedo, G., Mller, B., 2007, [Phys. Rep.]{}, [442]{}, 38
Colgate S.A. , 1971, [ApJ]{}, [163]{}, 221
Davidson, K., Ostriker, J.P., 1973,[ApJ]{} [179]{}, 585
De Luca, A., Mereghetti, S., Caraveo, P. A., Moroni, M., Mignani, R. P., Bignami, G. F. 2004, [A&A]{}, 418, 625
Demorest P. B., Pennucci T., Ransom S. M., Roberts M. S. E.,Hessels J. W. T., 2010,[Nature]{}, [467]{}, 1081
Douchin F., Haensel P., 2001, [A&A]{}, [380]{}, 151
Endeve E., Cardall C.Y., Budiardja R.D., Beck S.W. and Bejnood A., 2012 [ApJ.]{}, [751]{}, 26 (28pp)
Espinoza, C. M., Lyne, A. G., Kramer, M., Manchester, R. N., & Kaspi, V. M. 2011, [ApJL]{}, [741]{}, L13
Faucher-Giguère, C.-A.; Kaspi, V. M., 2006, [ApJ]{}, [643]{}, 332
, C. L. , 2006, [New Ast. Rev.]{}, [50]{}, 492-495
Geppert U., Page D., Zannias T., 1999, [A&A]{}, [345]{}, 847
Goldreich P., Julian W.H., 1969, [ApJ]{}, [157]{}, 869
Gotthelf, E. V., Halpern J.P., 2007, [ApJL]{}, [664]{}, L35
Gotthelf E.V., Halpern J.P., Seward S.D., 2005, [ApJL]{}, [627]{}, 390-96
Gotthelf E. V., Halpern J.P., Alford, J., 2013, [ApJ]{}, [766]{}, id. 58
Goussard, J.-O.; Haensel, P.; Zdunik, J. L., 1998, [A&A]{}, [330]{}, 1005
Gullón, M.; Miralles, J. A.; Viganò, D.; Pons, J.A., 2014, [MNRAS]{}, [443]{}, 1891
Hammer N.J., Janka H.-Th., Müller E., 2010, [ApJ]{}, [714]{}, 1371
Halpern, J. P., Gotthelf, E. V., Camilo, F., Seward, F. D. 2007, ApJ, 665, 1304 (Paper 2)
Halpern, J. P., Gotthelf, E. V., Camilo, F., Seward, F. D. 2010, ApJ, 709, 436
Helfand D.J, Becker R.H., 1984, [Nature]{}, [307]{}, 215
Ho W.C.G., 2011, [MNRAS]{}, [414]{}, 2567
Houck J.C., Chevalier R.A., 1991, [ApJ]{}, [376]{}, 234
Hui, C. Y., Becker, W. 2006, [A&A]{}, 454, 543
Jackson, J. D. 1962, [*Classical electrodynamics*]{}, New York: Wiley.
Gabler M., Cerdá-Durán P., Font J.A., Müller E, Stergioulas N. , 2012, [MNRAS]{}, [430]{}, 1811
Joggerst C.C, Almgren A., Woosley S.E.,2010, [ApJ]{}, [723]{}, 353
Kifonidis K., Plewa T., Janka H.Th., Müller E., 2003, [A&A]{}, [408]{}, 621
Kifonidis K., Plewa T., Scheck L., Janka H.Th., Müller E., 2006, [A&A]{}, [453]{}, 661
Kruskal, M., Schwarzschild, M. 1954, [Proc. R. Soc. A]{}, [223]{}, 348–360
Kulkarni, A. K., Romanova, M. M., 2008, [MNRAS]{}, [386]{}, 673
Lattimer J., Prakash M., 2005, Phys. Rev. Lett. 94, 111101
Lattimer J., 2012, Annual Review of Nuclear and Particle Physics, Vol. 62, 485
Lamb, F. K., Pethick, C. J., Pines, D. 1973, Astrophysical Journal, 184, 271
Lichnerowicz A.,1967, [*Relativistic Hydrodynamics and Magnetohydrodynamics.*]{} Benjamin.
Lorimer D. R., Kramer M., 2004, [*Handbook of Pulsar Astronomy*]{}, Cambridge University Press.
Mereghetti, S., De Luca, A., Caraveo, P. A., Becker, W., Mignani, R., Bignami, G. F. 2002, ApJ, 581, 1280
Michel F.C., 1972, [Ap. Space Sci.]{}, [15]{}, 153
Michel F.C., 1977, [ApJ]{}, [214]{}, 261
Mukherjee, D., Bhattacharya, D., Mignone, A. 2013, [ApJ]{}, [430]{}, 1976
Mukherjee, D., Bhattacharya, D., Mignone, A. 2013, [ApJ]{}, [435]{}, 718
Muslimov A., Page D., 1995, [ApJ ]{}, [440]{}, L77
Negele J. W., Vautherin D., 1973, [Nuclear Physics A]{}, [207]{}, 298
Obergaulinger M., Janka H.T., Aloy-Torás M. A., 2014, MNRAS, 445, 3169
Pandharipande V. R., Smith R. A., 1975, [Physics Letters B]{}, [59]{}, 15
Page, D., Lattimer J.M., Prakash M., Steiner, A.W., 2004, [ApJS]{}, [155]{}, 623
Payne, D. J. B., Melatos, A. 2004, [ApJ]{}, [351]{}, 569–584
Payne, D. J. B., Melatos, A. 2007,[ApJ]{}, [376]{}, 609–624
Pons J.A., Geppert U., 2007, [A&A]{}, [470]{}, 303
Pons J.A., Miralles, J.A., Geppert U., 2009, [A&A]{}, [496]{}, 207
Pons J.A., Reddy S., Prakash M., Lattimer J.M., Miralles J.A., 1999, [ApJ]{}, [513]{}, 780
Romanova M. M., Kulkarni A. K., Lovelace, Richard V. E, [ApJL]{}, [673]{}, L171.
Romero R., Mart' i J.M, Pons J.A., Ibáñez J.M., Miralles J.A., 2005, [J. Fluid Mech.]{}, [544]{}, 323-338
Seitenzahl R. et al., 2008, ApJ, 685, L129
Seward, F. D., Slane, P. O., Smith, R. K., Sun, M. 2003, ApJ, 584, 414
Shabaltas N. Lai D., 2012, [ApJ]{} [748]{}, 148
Shapiro S.L., Teukolsky S.A., 2004, [*Black holes, white dwarfs and NSs*]{}, Wiley-VCH.
Scheck L., Kifonidis K., Janka H.-Th., Müller E., 2006, [A&A]{} [457]{}, 963
Swarztrauber, 1974, [SIAM J. Numer. Anal. ]{} [11]{}, 1136-1150
Thompson C., Duncan R.C., 1993, [ApJ]{}, [408]{}, 194
Timmes F.X, 1999, ApJ Supp. ser., 124, 241-263
Timmes F.X., Arnett D., 1999, [ApJ Supp. Ser.]{}, [125]{}, 277-294
Timmes F.X., Swesty D., 2000, [ApJ Supp. Ser.]{}, [126]{}, 501-516
Turolla R., Esposito P., 2013, [Int. J. Mod. Phys.]{}, D, [22]{}, 1330024
Viganò D., Pons J.A., 2012, [MNRAS]{}, 425, 2487-2492
Viganò D., Rea, N.; Pons, J. A.; Perna, R.; Aguilera, D. N.; Miralles, J. A., 2013, [MNRAS]{}, 434, 123
Ugliano M., Janka H.T., Marek A., Arcones A., 2012, [ApJ]{}, [757]{}, 69
Woosley S.E., Weaver T.A., 1995, [ApJ Supp. Ser]{}, [101]{}, 181
Woosley S.E., Heger A., Weaver T.A., 2002, [Rev. Mod. Phys.]{}, [74]{}, 1015
Wongwathanarat A., Müller E., Janka H.-Th., 2015, [A&A]{}, [577]{}, A48
Young E.J., Chanmugan G., 1995, [ApJ]{}, [442]{}, L53
Zavlin V.E., Pavlov G.G., Sanwal D., Trumper J., 2000, [ApJL]{}, [540]{}, L25
Zhang W., Woosley S.E., Heger A., 2008, [ApJ]{}, [679]{}, 639
[^1]: E-mail: Alejandro.Torres@uv.es
[^2]: http://cococubed.asu.edu/code\_pages/codes.shtml
[^3]: The numerical error is dominated by the calculation of the equilibrium point, which is computed with a relative accuracy of $10^{-4}$.
|
---
abstract: 'We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models. We find the aspect ratio of the rectangle’s length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: for long and wide channels it decreases algebraically $v_{c}\sim W^{-1/2}$ with the system’s width $W$, while in square systems it decreases logarithmically $v_{c}\sim1/\ln L$ with length $L$. Although derived for asymptotically wide rectangles, our analytical results agree with numerical data for systems as small as $W\approx10$.'
author:
- Eial Teomy
- Yair Shokef
title: 'Jamming transition of kinetically-constrained models in rectangular systems'
---
Introduction
============
Increasing the density of particles in granular matter causes them to undergo a transition from an unjammed state, where the particles can move relatively freely, to a jammed state, where almost none of the particles can move [@jamming]. Systems of interest in nature and in industrial applications typically have complicated geometries which strongly affect jamming in them [@industry; @industry2; @nature; @nature2], and it is thus important to understand how does confinement influence the jamming of granular matter. Here we investigate the effects of confinement on the jamming transition, and in particular test how does the shape of containers determine how they jam. Most theoretical work so far was done on square systems [@durian; @haxton; @ohern; @ningxu; @lerner; @barrat].
There are numerous laboratory experiments that deal with non-square systems [@exp1; @exp2; @exp3; @exp4]. For example, Daniels and Behringer conducted an experiment on polypropylene spheres in an annulus [@behringer1], which is large enough to be considered a rectangle with infinite length and finite width. A different experiment by Bi et al. [@behringer2] consists of shearing a square system such that it becomes a rectangle with the same area and particle density as the original square. In this paper we study the effects of confinement on jamming by studying these phenomena in kinetically-constrained models in rectangular domains.
The essence of jamming is captured by the various kinetically-constrained models [@review; @review2; @kronig; @fieldings; @toninelli; @sellitto; @knights; @DFOT; @sellitto2; @spiral; @jeng; @shokef; @elmatad; @driving]. For such simple models, and for other related models [@kipnis; @derrida2; @derrida; @bouchaud; @asep; @monthus; @nemeth; @scheidler; @varnik; @srebro; @zrp; @lang], it is useful to have exact solutions.
![ The difference between frozen and unfrozen particles in the Kob-Andersen model: White square are vacancies. Light-gray particles can move in this initial configuration. Dark-gray particles cannot move now, but after some other particle(s) move, they too are mobile. Black particles are permanently frozen and will never move. []{data-label="gridex"}](gridexample.eps){width="160pt"}
Kinetically-Constrained Models
------------------------------
Two types of kinetically-constrained models which describe granular and glassy materials are spin-facilitated models and lattice-gas models [@review; @review2]. In both types of models, the system is represented by a grid, such that each site on the grid can have one of two values, $1$ or $0$.
In the *lattice-gas models*, a site with a value of $1$ represents a single particle and a site with a value of $0$ represents a vacant region. In each time step, one particle and one of the possible directions are chosen randomly with equal probabilities. The chosen particle attempts to move in that direction, and if the kinetic constraint allows the move, the particle moves to the neighboring site in the chosen direction. For a given initial configuration, some of the particles can move from the start, and some can move only after (many) other particles have moved and cleared the way for them. There may also be particles that will never move, no matter how the other particles in the system move. Those that will never move are called *permanently frozen*, and those that can move eventually are called unfrozen. See Fig. \[gridex\] for an example.
In the case of *spin-facilitated models*, a site with a value of $0$ represents a region of low density and high mobility, and a site with a value of $1$ represents a region of high density and low mobility. Note that this notation is different than the common notation ($0$ for a region of high density and $1$ for a region of low density), but we use this definition in order to deal with both lattice-gas and spin-facilitated models simultaneously. In each time step of the dynamics, one of the sites is chosen randomly, and changes its value at a temperature-dependent probability if it has enough neighbors with low density (i.e., a value of $0$), with the exact geometric criteria depending on the specific model. In the limit of zero temperature, the only allowed changes are from high density to low density, i.e. from a value of $1$ to $0$. For a given initial configuration, there is a possibility that even after an infinite number of time steps, some sites will still have a value of $1$. These permanently frozen sites represent the backbone of the system which will never change.
In order to find the fraction of permanently frozen particles, one can use the bootstrap method, which iteratively removes mobile particles, until none of the remaining particles can move. Again, we have a backbone of sites which will never change. This algorithm is obviously valid for spin-facilitated models, but also for lattice gas models, since the criteria for the mobility of particles is local, and removing a mobile particle is effectively the same as moving it far enough from its neighbors. Since the algorithm for finding the backbone of both types of models is similar (but not identical), we will use the same language to describe both models, and choose the language of lattice-gas models. This means, for example, that whenever we speak of “vacancies” it should be interpreted as “sites with value 0” or “low-density regions” in the context of spin-facilitated models. For brevity, we will also use the term frozen particles interchangeably with permanently frozen particles.
We consider a two-dimensional rectangle, represented by a square lattice, such that each site either contains one particle or is vacant. The rectangle has $L$ sites in the horizontal direction, and $W$ sites in the vertical direction, such that $L\geq W$. In our numerical simulations we used hard-wall and periodic boundary conditions in both directions, but most of our analytical approximation ignores the boundary conditions. For rectangles of infinite length, hard-wall boundary conditions simulate particles inside a two dimensional channel, and periodic boundaries simulate particles on the surface of a cylindrical tube.
For the lattice-gas model we use the Kob-Andersen (KA) model [@kamodel], such that a particle can move if it has at least two neighboring vacancies before and after the move. For the spin-facilitated model we use the Fredrickson-Andersen (FA) model [@famodel], such that a site can change its state if it has at least two neighboring sites with a value of $0$. We could have chosen a different number of neighbors needed for movement, but on the square lattice the only interesting effects occur at two neighbors. If only one neighbor is needed for movement then all the particles are movable as long as there is at least one vacancy in the FA model or two adjacent vacancies in the KA model. If three neighbors are needed for movement then any closed loop is frozen, even a $2\times2$ block, which means that almost all the particles in the system are frozen. Mathematically speaking, the KA model and the FA model are very similar to each other. Also, a mobile particle in the KA model is necessarily mobile in the FA model, and a frozen particle in the FA model is necessarily frozen in the KA model, thus the fraction of frozen particles in the KA model is larger than (or at least equal to) the fraction of frozen particles in the FA model.
Toninelli, Biroli and Fisher showed [@toninelli] that for an infinite system in the KA model, none of the particles are permanently frozen as long as the lattice is not completely full with particles, which automatically means that this is also the case in the FA model. In this paper we study how many particles, on average, are permanently frozen for a given particle density, $\rho$, and given rectangle dimensions $W\times L$.
Finite Size Effects
-------------------
Numerical simulations done on square systems [@adler] $(L=W$, in our notations$)$ showed that the fraction of permanently frozen particles, $n_{PF}$ , rises rapidly from $0$ to $1$ at a certain critical density, $\rho_{c}$, which increases with system size. Holroyd [@bp] theoretically analyzed jamming in this context using the notion of *critical droplets*, which are small unjammed regions which facilitate movement throughout the system. He showed that for very large squares in the FA model the relation between the critical density and the system size is $$\begin{aligned}
&\rho^{square}_{c}=1-\frac{\lambda}{\ln L} ,\nonumber\\
&\lambda=\frac{\pi^{2}}{18}\approx0.54 . \label{lambdadef}\end{aligned}$$ Toninelli et. al. showed [@toninelli] that this value of $\lambda$ is also true for squares in the KA model. However, this result is only true for asymptotically large $L$.
We can define an effective $\lambda$ $$\begin{aligned}
\lambda^{squares}_{eff}(L)=\left[1-\rho_{c}(L)\right]\ln L ,\end{aligned}$$ which converges to $\lambda$ as the system size increases. For systems of size $L\approx10^{2}\sim10^{5}$ it was found [@aharony] that $\lambda^{squares}_{eff}\approx0.25$ for all simulated sizes. This contradiction was resolved by Holroyd’s proof [@holroydslow] that the convergence of $\lambda^{squares}_{eff}$ to $\lambda$ is very slow and may be apparent only at systems of size $L\approx10^{20}$, beyond the capabilities of modern computers, and beyond the range of physical realization ($L=10^{20}$ implies a system of $L^{2}=10^{40}$ particles).
Holroyd’s analysis considered only the size of the system, and not its shape. For long rectangular domains, this method may not be used. Instead, we find that rectangular systems may be divided into independent *sections*, in the sense that jamming in one section does not depend on the internal configuration within its neighboring sections. Within each section, Holroyd’s notion of critical droplets may be used.
Outline
-------
In this paper we show how not only the size of a system influences jamming in it, but also its shape. We demonstrate this, first by considering the limit of very large systems. When considering a square system of size $L \times L$, and taking the limit $L \rightarrow \infty$, one finds that the critical vacancy density, $v_{c}=1-\rho_{c}$, scales as $v_c \sim 1/\ln{L}$. We find that when the system’s width $W$ is fixed and the length is taken to infinity $L=\infty$, the critical density scales as $v_c \sim 1/\sqrt{W}$ when $W\rightarrow\infty$ .
The second scenario we consider is of a system of fixed area, for which we change the aspect ratio between the width and the length of the system. We find that stretching the system causes it to jam, and relate this result to recent experiments of sheared granular matter.
The paper is organized as follows. In Section \[secdiv\] of this paper we derive an approximate analytical expression for the fraction of frozen particles, $n_{PF}$, for a rectangular system of arbitrary dimensions $W\times L$. In Section \[seclarge\] we deal with large systems, $W,L\gg1$, and use our approximation to find the critical density at which the system goes from jammed to unjammed and the width of this transition. We find that the system can be considered infinite if its length is longer than the average section length, or equivalently $\ln L\gg\sqrt{4\lambda W}$. In Section \[secsmall\] we deal with narrow systems (small $W$), and improve the approximation derived in Section \[secdiv\]. We even derive an exact result for the case of very narrow systems ($L=\infty$ and $W=1,2$). In Section \[secinter\] we investigate the internal structures in the system. The Appendices contain the derivation of the lengthy expressions used in the analytical approximation.
Critical Droplets and Division into Sections {#secdiv}
============================================
Critical Droplets
-----------------
Holroyd showed that in a large enough square there is a probability of approximately $e^{-2\lambda/v}$ that a particle is part of a critical droplet, where $\lambda$ is given in Eq. (\[lambdadef\]b). Hence, the total number of critical droplets in a rectangle of size $L\times W$ is $WLe^{-2\lambda/v}$, where $v=1-\rho$ is the vacancy density. The expression for the critical density in square systems, Eq. (\[lambdadef\]a), is derived by setting $W=L$. Such a critical droplet can cause the entire system to be unfrozen, thus the critical vacancy density is when the number of critical droplets is finite, since below (above) that critical density the number of critical droplets is very small (large) when $L$ is taken to infinity.
The fraction of frozen particles in the FA and KA models is obviously different, due to the different kinetic constraints, but that difference is small. The reason that the densities of frozen particles for both the FA and the KA models are almost the same can be seen from the construction of Holroyd’s proof. Holroyd considered small critical droplets, which are unfrozen, and checked how they can be expanded to “unfreeze” the entire system. The criterion for the expansion of these droplets is the same in both models, and the only difference is in what constitutes a small critical droplet. For example, the structure $\begin{array}{cc}1&0\\0&1\end{array}$ is unfrozen in the FA model but frozen in the KA model. For large enough systems, and evidently for small ones too, the effect of this difference is negligible.
However, when $L\rightarrow\infty$ and $W$ remains constant we cannot simply set $1=WLe^{-2\lambda/v_{c}}$ to find the critical density, since the solution to this equation is $v_{c}=0$ for all $W$. Our resolution of this problem is obtained by dividing the long rectangle into finite sections, implementing the idea of critical droplets in each section, and finally averaging over all sections. Another approach, which yields the same results, is solving the equation $1=W\left\langle l\right\rangle e^{-2\lambda/v_{c}}$, where $\left\langle l\right\rangle$ is the average section length (see below).
Division into Sections
----------------------
A rectangular system may be divided into sections by noting that if there are two or more successive columns which are completely full, then all the particles in them are permanently frozen in both the KA and FA models and with either hard-wall or periodic boundary conditions. We call a pattern of $m$ successive full columns a *strip* of size $m$, where $m\geq2$. A single full column is not called a strip. These strips divide the rectangle into finite *sections*, such that the leftmost column on each section is the first not-full column after a strip, and the rightmost column is the final column of the next strip. Each section contains only one strip. For example, see Fig. \[gridex2\].
![(color online) Division into sections, represented by different colors: Columns $1-2$, $3-9$, and $10-15$. The strips are columns $1-2$, $7-9$, and $14-15$. []{data-label="gridex2"}](gridexample2.eps){width="200pt"}
Using this division, we note that the particles within one section are independent from the particles in the other sections, i.e., the state of the particle (whether it is frozen or not) depends only on the structure within its section, and not on the configuration of neighboring sections.
Each finite section of length $l+m$ ending with a strip of size $m$ and with $n+mW$ occupied sites, has many configurations for the $n$ particles in the $l$ columns not occupied by the strip. We will denote these configurations with an index $s$. Since the probability of having a strip of length $m$ is independent of the probability for a certain configuration in the rest of the strip, the probability of such a configuration occurring is $$\begin{aligned}
P(n,l,m,s)=\rho^{mW}Q(n,l,s), \end{aligned}$$ where $\rho^{mW}$ is the probability of having a strip of length $m$ containing $mW$ particles, and $Q(n,l,s)$ is the relative probability of configuration $s$ in the region with $l$ columns and $n$ occupied sites between the strips, such that there are no two adjacent full columns. The reason we exclude the possibility of two adjacent full columns is to count each type of section only once, since two (or more) adjacent full columns divide the section into smaller sections. The average fraction of frozen particles, $n_{PF}$, is the number of frozen particles divided by the number of particles, $$\begin{aligned}
n_{PF}=\frac{\sum_{n,l,m,s}P(n,l,m,s)N(n,l,m,s)}{\sum_{n,l,m,s}P(n,l,m,s)\left[mW+n\right]}, \label{npfmain}\end{aligned}$$ where $N(n,l,m,s)=N(n,l,s)+mW$ is the total number of permanently frozen particles in the section, with $N(n,l,s)$ being the number of frozen particles in the $l$ columns not occupied by the strip. The sum over $l$ and $m$ is such that $l+m\leq L$, and $m$ is greater or equal to $2$, except in the following special cases: no strip in the entire rectangle ($l=L,m=0$), and the entire section is full ($l=0,m=L$). Since the probability $P$ appears both in the nominator and the denominator in Eq. (\[npfmain\]) we need not worry about its normalization or the normalization of $Q$. However, we find that in the limit of infinite length the probability is normalized such that $\sum P=1$.
Our Approximation for Rectangular Systems
-----------------------------------------
In our case, we assume that the probability that a particle in a section of length $l+1$ is frozen is the probability that it is frozen in a section of length $l$ times the probability that the added column does not contain critical droplets, $$\begin{aligned}
\left\langle N(l+1)\right\rangle=\left\langle N(l)\right\rangle\left(1-e^{-2\lambda/v}\right)^{W} ,\end{aligned}$$ where $\left\langle..\right\rangle$ is the average over all configurations. The solution to this recursion relation is $$\begin{aligned}
&\left\langle N(n,l)\right\rangle=ne^{-kl} ,
&k=-W\ln\left(1-e^{-2\lambda/v}\right) .\label{krellam}\end{aligned}$$ This leads to very good agreement with results of numerical simulations, as shown in Fig. \[fakasec\].
![(color online) Average fraction $\frac{\left\langle N\right\rangle}{n}$ of frozen particles in sections of length $l$ in the KA model with hard-wall boundary conditions for different widths and densities: $W=4$, $\rho=0.8$ (blue squares), $W=10$, $\rho=0.89$ (purple circles), $W=20$, $\rho=0.93$ (yellow triangles), $W=40$, $\rho=0.95$ (green stars). Continuous lines are approximations (\[krellam\]), with $\lambda$ taken from simulations of long rectangles ($L=200W$). []{data-label="fakasec"}](fakasec.eps){width="\columnwidth"}
Using this assumption in Eq. (\[npfmain\]) yields $$\begin{aligned}
n_{PF}=\frac{\sum_{n,l,m}\rho^{mW}Q(n,l)\left[mW+ne^{-kl}\right]}{\sum_{n,l,m}\rho^{mW}Q(n,l)\left[mW + n\right]} , \label{npfmain0}\end{aligned}$$ where $Q(n,l)=\sum_{s}Q(n,l,s)$. The evaluation of these sums in closed form is given in Appendix A. We note here that the denominator and the first part of the nominator in Eq. (\[npfmain0\]) do not depend on our approximation relating the fraction of frozen particles with the number of critical droplets, and thus are not approximations but exact results. We further note that the ratio between the first part of the nominator and the denominator is the density of particles which are in the strips in both models and with both boundary condition. Hence, we define the density of particles which are in the strips as $$\begin{aligned}
n_{strip}=\frac{\sum_{n,l,m}\rho^{mW}Q(n,l)mW}{\sum_{n,l,m}\rho^{mW}Q(n,l)\left[mW + n\right]}, \end{aligned}$$ which in the limit of $L\rightarrow\infty$ converges to $$\begin{aligned}
n_{strip}(L\rightarrow\infty)=\rho^{2W-1}\left(2-\rho^{W}\right) .\end{aligned}$$ We find that even for $W$ as small as $10$ and for all $L\geq W$, the density of particles in the strips is very low, except in the region very near $\rho=1$, where the fraction of frozen particles, $n_{PF}$, is almost unity. This means that for wide systems, the strips hardly contribute to the total fraction of frozen particles near the critical density and below it. The only role the strips play in this regime is dividing the system into sections, which are very long since there are few strips. Using the results in Eqs. (\[denfinal\]) and (\[nom1final\]), the fraction of frozen particles at $L=\infty$ can be written as $$\begin{aligned}
n_{PF}(L=\infty)=\rho^{2W-1}\left(2-\rho^{W}\right)+\frac{\rho^{4W-1}}{W}N_{PF}(\rho,W) , \label{npfex}\end{aligned}$$ where $$\begin{aligned}
N_{PF}(\rho,W)=\sum_{l,n,s}Q(l,n,s)N(l,n,s) .\end{aligned}$$
As previously shown for square systems, the value of $\lambda_{eff}$ depends on the system’s size. We define the effective $\lambda$, $\lambda_{eff}(W,L)$, as the $\lambda$ for which the analytical approximation yields $n_{PF}(\rho_{c})=1/2$, with $\rho_{c}$ obtained by the numerical simulations. Previous simulations [@aharony] showed that for square systems in the FA model $\lambda_{eff}$ does not change much in the region $L\approx10^{2}-10^{5}$. Figure \[lameff\] shows that for constant $W$, the value of $\lambda_{eff}(W,L)$ converges to a finite value $\lambda_{eff}(W,\infty)$, which is rather close to $\lambda_{eff}(W,W)$ at large $W$. See also Fig. \[lameffinf\]. In the range of large $W$ and $L$ we see that the value of $\lambda$ depends mostly on the model, and not that much on the system’s size or shape (square or rectangle). Figure \[lameffinf\] shows the value of $\lambda$ for very long systems. We see from it that for squares it appears that $\lambda$ decreases with the width or converges to some value, but for long tunnels it is clear that the value of $\lambda$ increases with the width. Unless noted otherwise, in the rest of this paper we use the value of $\lambda$ taken from the simulations done on large squares.
![(color online) The effective $\lambda$ as a function of $W/L$ for hard-wall (a) and periodic (b) boundary conditions. In this range, $\lambda_{eff}$ does not change much with the length $L$, and converges as $L\rightarrow\infty$ to a finite value for each width, $W$. For each system size, $\lambda_{eff}$ for the KA model is higher than in the FA model, and it is higher with hard-wall boundary conditions than with periodic boundary conditions, which means that $n^{KA}_{PF}>n^{FA}_{PF}$ and $n^{hard-wall}_{PF}>n^{periodic}_{PF}$, as expected. At $L\rightarrow\infty$, $\lambda$ is almost the same for $W=20$ and $W=100$ but different for $W=10$.[]{data-label="lameff"}](lameff.eps "fig:"){width="\columnwidth"} ![(color online) The effective $\lambda$ as a function of $W/L$ for hard-wall (a) and periodic (b) boundary conditions. In this range, $\lambda_{eff}$ does not change much with the length $L$, and converges as $L\rightarrow\infty$ to a finite value for each width, $W$. For each system size, $\lambda_{eff}$ for the KA model is higher than in the FA model, and it is higher with hard-wall boundary conditions than with periodic boundary conditions, which means that $n^{KA}_{PF}>n^{FA}_{PF}$ and $n^{hard-wall}_{PF}>n^{periodic}_{PF}$, as expected. At $L\rightarrow\infty$, $\lambda$ is almost the same for $W=20$ and $W=100$ but different for $W=10$.[]{data-label="lameff"}](lameffper.eps "fig:"){width="\columnwidth"}
![(color online) The effective $\lambda$ as a function of the system’s width $W$ for hard-wall (a) and periodic boundary conditions (b). The plots show the value of $\lambda_{eff}$ at $L=200W$ (for $W\leq100$) and $L=5000W$ (for $W=200$), which is large enough to be considered infinite in this range of widths, and at $L=W$. For $L\gg W$ the effective $\lambda$ has a minimum at $W\approx50$, but there is no drastic change in its value in the range $20\leq W\leq100$. Even at $W=200$ the relative difference from the minimum is $0.08$. $\lambda_{eff}$ for long tunnels ($L\gg W$) and squares ($L=W$) is almost the same at $60\leq W\leq100$. We expect that at $W\rightarrow\infty$, $\lambda$ will converge to $\pi^{2}/18\approx0.54$.[]{data-label="lameffinf"}](lameffinf.eps "fig:"){width="\columnwidth"} ![(color online) The effective $\lambda$ as a function of the system’s width $W$ for hard-wall (a) and periodic boundary conditions (b). The plots show the value of $\lambda_{eff}$ at $L=200W$ (for $W\leq100$) and $L=5000W$ (for $W=200$), which is large enough to be considered infinite in this range of widths, and at $L=W$. For $L\gg W$ the effective $\lambda$ has a minimum at $W\approx50$, but there is no drastic change in its value in the range $20\leq W\leq100$. Even at $W=200$ the relative difference from the minimum is $0.08$. $\lambda_{eff}$ for long tunnels ($L\gg W$) and squares ($L=W$) is almost the same at $60\leq W\leq100$. We expect that at $W\rightarrow\infty$, $\lambda$ will converge to $\pi^{2}/18\approx0.54$.[]{data-label="lameffinf"}](lameffinfper.eps "fig:"){width="\columnwidth"}
![(color online) Fraction of frozen particles $n_{PF}$ vs. density $\rho$ for hard-wall (left) and periodic (right) boundary conditions, and for different widths: $W=10$ (top), $W=20$ (middle), $W=100$ (bottom). Symbols are results of numerical simulations and continuous line is analytical approximation with different values of $\lambda_{eff}$. As the system’s size increases, the approximation becomes better and the value of $\lambda_{eff}$ becomes dependent mostly on the model, and not on the system’s size. $\lambda$ was set to $\lambda_{eff}(W,L)$, as shown in Fig. \[lameff\]. []{data-label="npfw"}](npfw10.eps "fig:"){width="120pt"} ![(color online) Fraction of frozen particles $n_{PF}$ vs. density $\rho$ for hard-wall (left) and periodic (right) boundary conditions, and for different widths: $W=10$ (top), $W=20$ (middle), $W=100$ (bottom). Symbols are results of numerical simulations and continuous line is analytical approximation with different values of $\lambda_{eff}$. As the system’s size increases, the approximation becomes better and the value of $\lambda_{eff}$ becomes dependent mostly on the model, and not on the system’s size. $\lambda$ was set to $\lambda_{eff}(W,L)$, as shown in Fig. \[lameff\]. []{data-label="npfw"}](npfw10per.eps "fig:"){width="120pt"} ![(color online) Fraction of frozen particles $n_{PF}$ vs. density $\rho$ for hard-wall (left) and periodic (right) boundary conditions, and for different widths: $W=10$ (top), $W=20$ (middle), $W=100$ (bottom). Symbols are results of numerical simulations and continuous line is analytical approximation with different values of $\lambda_{eff}$. As the system’s size increases, the approximation becomes better and the value of $\lambda_{eff}$ becomes dependent mostly on the model, and not on the system’s size. $\lambda$ was set to $\lambda_{eff}(W,L)$, as shown in Fig. \[lameff\]. []{data-label="npfw"}](npfw20.eps "fig:"){width="120pt"} ![(color online) Fraction of frozen particles $n_{PF}$ vs. density $\rho$ for hard-wall (left) and periodic (right) boundary conditions, and for different widths: $W=10$ (top), $W=20$ (middle), $W=100$ (bottom). Symbols are results of numerical simulations and continuous line is analytical approximation with different values of $\lambda_{eff}$. As the system’s size increases, the approximation becomes better and the value of $\lambda_{eff}$ becomes dependent mostly on the model, and not on the system’s size. $\lambda$ was set to $\lambda_{eff}(W,L)$, as shown in Fig. \[lameff\]. []{data-label="npfw"}](npfw20per.eps "fig:"){width="120pt"} ![(color online) Fraction of frozen particles $n_{PF}$ vs. density $\rho$ for hard-wall (left) and periodic (right) boundary conditions, and for different widths: $W=10$ (top), $W=20$ (middle), $W=100$ (bottom). Symbols are results of numerical simulations and continuous line is analytical approximation with different values of $\lambda_{eff}$. As the system’s size increases, the approximation becomes better and the value of $\lambda_{eff}$ becomes dependent mostly on the model, and not on the system’s size. $\lambda$ was set to $\lambda_{eff}(W,L)$, as shown in Fig. \[lameff\]. []{data-label="npfw"}](npfw100.eps "fig:"){width="120pt"} ![(color online) Fraction of frozen particles $n_{PF}$ vs. density $\rho$ for hard-wall (left) and periodic (right) boundary conditions, and for different widths: $W=10$ (top), $W=20$ (middle), $W=100$ (bottom). Symbols are results of numerical simulations and continuous line is analytical approximation with different values of $\lambda_{eff}$. As the system’s size increases, the approximation becomes better and the value of $\lambda_{eff}$ becomes dependent mostly on the model, and not on the system’s size. $\lambda$ was set to $\lambda_{eff}(W,L)$, as shown in Fig. \[lameff\]. []{data-label="npfw"}](npfw100per.eps "fig:"){width="120pt"}
Figure \[npfw\] compares between the fraction of frozen particles, $n_{PF}$, obtained by the numerical simulations and the analytical approximation. From it we see that the approximation is roughly acceptable at $W=10$, quite good even at $W=20$, and has an excellent agreement with the numerical results at $W=100$, even for the hard-wall boundary conditions. Also, we note that the approximation is better for the periodic boundary conditions, and that the width of the transition from an unjammed state, where almost all of the particles are unfrozen, to a jammed state, where almost all the particles are frozen, is narrower with the periodic boundary conditions.
Large Systems, $W,L\gg1$ {#seclarge}
========================
In large systems, the transition from jammed ($n_{PF}\approx1$) to unjammed ($n_{PF}\approx0$) occurs in a very narrow region of densities. In what follows we find the critical density, $\rho_{c}$, at which this transition occurs and the width of the transition, $\Delta\rho$. We also show how the critical density depends on the shape of the system and not only on its size, by considering a system of fixed area and changing the aspect ratio $W/L$. Note that for finite-sized systems (and even when $L$ is infinite but $W$ is finite), there is no singularity in any physical quantity. Nonetheless we use the term critical density since permanently frozen particles exist due to the same considerations that govern jamming in the thermodynamic limit, where one may discuss the notion of critical phenomena [@durian; @haxton; @olsson].
Critical Density
----------------
### Critical Density from Fraction of Frozen Particles
We first note that as the system grows larger, the critical density grows as well and nears $1$. Therefore, we can use the known critical density for squares of size $W\times W$, Eq. (\[lambdadef\]), to find a lower bound on the critical density in rectangles of size $W\times L$ ($W\leq L$) $$\begin{aligned}
v_{c}\leq\frac{\lambda}{\ln W} .\label{boundvc}\end{aligned}$$
From Eq. (\[boundvc\]) we find that $$\begin{aligned}
&k=-W\ln\left(1-e^{-2\lambda/v_{c}}\right)\leq We^{-2\ln W}=W^{-1}\ll1 .\end{aligned}$$ Since $W$ is very large, this means that the exponent in the logarithm is very small, and thus $$\begin{aligned}
k\approx We^{-2\lambda/v} .\end{aligned}$$ We also note that the critical density is very close to $1$ but still $\left(\rho_{c}\right)^{W}\ll1$. Therefore, close to the critical density, we can use the results from Appendix \[apnpf\] and approximate $n_{PF}$ by $$\begin{aligned}
&n_{PF}\approx\nonumber\\
&\frac{1+\exp\left[-L\rho^{2W}\left(k\rho^{-2W}+1\right)\right]\left[kL\left(k\rho^{-2W}+1\right)-1\right]}{\left(k\rho^{-2W}+1\right)^{2}
\left[1-\exp\left(-L\rho^{2W}\right)\right]
} .\end{aligned}$$
In very short rectangles, such that $L\rho^{2W}\ll1$, we find that $n_{PF}$ is finite only if $k\rho^{-2W}\gg1$ and $kL$ is finite, and thus $$\begin{aligned}
n_{PF}(L\rho^{2W}\ll1)\approx e^{-kL} .\label{npfshort}\end{aligned}$$ Solving the equation $n_{PF}=1/2$ yields $$\begin{aligned}
v_{c}\approx\frac{2\lambda}{\ln\left(WL\right)} ,\label{vcshort}\end{aligned}$$ which retrieves the known result, Eq. (\[lambdadef\]), for the case $W=L\gg1$.
In very long rectangles, such that $L\rho^{2W}\gg1$, we find that $n_{PF}$ is finite only if $k\rho^{-2W}$ is finite and thus $$\begin{aligned}
n_{PF}(L\rho^{2W}\gg1)\approx\frac{1}{\left(k\rho^{-2W}+1\right)^{2}} .\label{npflong}\end{aligned}$$ Solving the equation $n_{PF}=1/2$ yields $$\begin{aligned}
v_{c}\approx\frac{\sqrt{16\lambda W+\ln^{2}\left(W\right)}-\ln\left(W\right)}{4W} .\label{vclong}\end{aligned}$$
This means that the distinction between short and long rectangles is whether $L\left(\rho_{c}\right)^{2W}$ is greater or lesser than $1$. Equating Eqs. (\[vcshort\]) and (\[vclong\]), we find that the crossover from short rectangles to long rectangles occurs at $$\begin{aligned}
\ln L_{c}=2Wv_{c}=\frac{\sqrt{16\lambda W+\ln^{2}\left(W\right)}-\ln W}{2} .\label{lcbig}\end{aligned}$$
### Critical Density from Typical Section Length
Another approach for finding the critical density is by considering only a typical section of length $\left\langle l\right\rangle$, where $$\begin{aligned}
\left\langle l\right\rangle=\frac{\sum^{L}_{l=1}Q(l)l}{\sum^{L}_{l=1}Q(l)} \end{aligned}$$ is the average section length. In this case, similarly to what was done on square systems, we need to solve the equation $$\begin{aligned}
1=W\left\langle l\right\rangle e^{-2\lambda/v_{c}} .\label{eqav}\end{aligned}$$ Using the expressions in the Appendix, and assuming that $\rho^{W}\ll1$ and $L,W\gg1$, we find that $$\begin{aligned}
\left\langle l\right\rangle\approx\rho^{-2W}
\left[1+\frac{L\rho^{2W}}{1-\exp\left(L\rho^{2W}\right)}\right]
.\label{lav}\end{aligned}$$ In the limits of $L\rho^{2W}\gg1$ and $L\rho^{2W}\ll1$, solving Eq. (\[eqav\]) yields the same results as in Eqs. (\[vcshort\]) and (\[vclong\]).
This means that when the rectangle’s length is shorter than $\left\langle l\right\rangle$ it can be considered as consisting of a single section, and that for longer rectangles we can consider only sections of average length. For this reason short, square-like rectangles can be considered to contain only one section and the critical density in them depends as a first approximation only on the system area $WL$ and not on its shape.
By considering terms of order $L\rho^{2W}=L/L_{c}$, we find that the correction to Eq. (\[vcshort\]) is $$\begin{aligned}
v_{c}\approx\frac{2\lambda}{\ln\left(WL\right)}+\frac{L}{L_{c}}\frac{\lambda}{\ln^{2}\left(WL\right)}\frac{2-\ln^{2}(2)-\ln(4)}{\ln^{3}(2)} .\label{vcshort2}\end{aligned}$$ By keeping the ratio $L/L_{c}$ constant, we see from Eq. (\[vcshort2\]) that the correction becomes less important at larger systems.
### Alternative Approach on Confinement
So far we looked at what happens when the width remains constant and the length increases, and found the crossover length $L_{c}$ between the two limiting cases described above. Another way to look at it is by starting from a square of size $L\times L$ and to generate confinement by narrowing its width. Namely, we fix $L$, and decrease $W$. In this case we find from Eq. (\[lcbig\]) a crossover width $$\begin{aligned}
W_{c}=\frac{\ln^{2}(L)}{4\lambda} ,\label{wc}\end{aligned}$$ which may be interpreted as the width below which the system can be considered infinitely long. To find the effect of the confinement on the critical density, we calculate the ratio between the critical density in squares and in rectangles. For $W>W_{c}(L)$ or equivalently for $L<L_{c}(W)$, this ratio is $$\begin{aligned}
\frac{v_{c}(W\times L)}{v_{c}(L\times L)}\approx\frac{2}{1+\frac{\ln(W)}{2\sqrt{\lambda W_{c}}}}\approx\frac{2}{1+\frac{2\ln\ln(L_{c})}{\ln L}} ,\end{aligned}$$ and for $W<W_{c}(L)$, or $L>L_{c}(W)$, the ratio is $$\begin{aligned}
\frac{v_{c}(W\times L)}{v_{c}(L\times L)}\approx2\sqrt{\frac{W_{c}}{W}}\approx\frac{2\ln(L)}{\ln(L_{c})} .\end{aligned}$$ Figure \[vcrel\] shows the ratio between the critical vacancy density in rectangles of size $W\times L$ and the critical vacancy density in squares of size $L\times L$. We see from it that the asymptotic values in Eqs. (\[vcshort\]) and (\[vclong\]) agree with the numerical results even at $W\approx W_{c}$. This means that systems really may be divided into long and short rectangles with a clear distinction between the two types.
![(color online) Ratio between critical vacancy density in rectangles of dimension $W\times L$ and critical vacancy density in squares of dimension $L\times L$ in the KA model with hard-wall boundary conditions. Symbols are results of numerical simulations and continuous line is the approximations (\[vcshort\]) and (\[vclong\]). The ratio between the critical densities significantly differs from $1$ only for $W<W_{c}(L)$. $W_{c}/L$ decreases with $L$, and the ratio between the critical densities increases with $L$. []{data-label="vcrel"}](vcrel.eps){width="\columnwidth"}
The critical density in Eq. (\[vclong\]) is a very good approximation even for $W$ as small as $3$, as shown in Fig. \[vcinf\]. This dependence of the critical density on the width of the system in long channels can be measured in experiments. The suggested value for the effective $\lambda$ is $0.257(FA),0.275(KA)$ for hard-wall boundaries and $0.249(FA),0.271(KA)$ for periodic boundaries, since this is its value for systems of infinite length and with a width of $W\approx20-100$.
![(color online) The critical vacancy density, $v_{c}$, as a function of the width for very long systems ($L=200W\gg L_{c}(W)$) with hard-wall boundaries. The symbols are the results from the numerical simulations and the continuous lines are the approximation (Eq. (\[vclong\])) with $\lambda=\lambda_{eff}(100,100)$.[]{data-label="vcinf"}](vcinf.eps){width="\columnwidth"}
Varying Aspect Ratio
--------------------
Here we consider a system of fixed volume $V=WL$, and study how changing the aspect ratio between the width and the length, $r=W/L$, effects the jamming transition. The crossover aspect ratio, $r_{c}$, is the aspect ratio which defines whether the system behaves as a square-like system or as a long system. From Eq. (\[wc\]) we find that the crossover aspect ratio satisfies the equation $$\begin{aligned}
16\lambda\sqrt{Vr_{c}}=\ln^{2}\left(r_{c}/V\right) .\label{rc}\end{aligned}$$ If the density is high enough that the system is jammed at $r=1$, then it is also jammed at any other $r<1$. If the system is not jammed at $r=1$, the density can still be the critical density at some aspect ratio smaller than $1$. This means that as the aspect ratio decreases, the system may undergo a jamming transition if the density is below the critical density at $r=1$ but above the critical density at $r\ll1$.
For example, we now show that our results may be related to recent experiments of Bi et al. [@behringer2]. In these experiments shear stress was applied on a two-dimensional system, such that its area and particle density remained constant, but the aspect ratio between its length and width changed. We consider a system of fixed area $WL=10^{4}$ and density $\rho=0.92$ with hard-wall boundaries. The density was chosen such that it is below $\rho_{c}$ for $r=1$ and above $\rho_{c}$ for $r\ll1$. By changing the aspect ratio, the fraction of frozen particles changes from almost $0$ at $r\geq0.1$ to almost $1$ at $r\leq0.01$ as seen in Fig. \[v104\]. The crossover aspect ratio from Eq. (\[rc\]) is $r_{c}=0.093$, very near the $r=0.1$ observed in the numerical results. This means that shearing the system causes it to become jammed, in agreement with the experimental results [@behringer2].
![(color online) Fraction of frozen particles vs aspect ratio $r=W/L$ at constant area $WL=10^{4}$ and particle density $\rho=0.92$ with hard-wall boundaries. Symbols are results of numerical simulations for KA (blue squares) and FA (purple circles) models. Continuous lines are analytical approximations with $\lambda_{eff}(100,100)\approx0.271$ (KA), $0.252$ (FA). []{data-label="v104"}](v104.eps){width="\columnwidth"}
Transition Width
----------------
We define the width of the transition, $\Delta\rho$ as the difference between the densities for which $n_{PF}=\delta$ and $1-\delta$, where $\delta$ is an arbitrary number much smaller than $1$. In short rectangles $[L\left(\rho_{c}\right)^{2W}\ll1]$, setting $n_{PF}$ in Eq. (\[npfshort\]) equal to $\delta\ll1$ and to $1-\delta$ yields $$\begin{aligned}
&\Delta\rho(short)=\rho_{u}-\rho_{l}\approx\nonumber\\
&\approx\frac{2\lambda}{\ln^{2}\left(WL\right)}\ln\left(\frac{\ln\delta^{-1}}{\delta}\right)\approx\frac{v^{2}_{c}\ln\delta^{-1}}{2\lambda} ,\label{dr}\end{aligned}$$ where $\rho_{l,u}$ are the values of the density at the lower $(n_{PF}=\delta)$ and the upper $(n_{PF}=1-\delta)$ bounds. For long rectangles $[L\left(\rho_{c}\right)^{2W}\gg1]$, setting $n_{PF}$ in Eq. (\[npflong\]) equal to $\delta$ and to $1-\delta$ yields $$\begin{aligned}
&\Delta\rho(long)=\rho_{u}-\rho_{l}\approx\frac{3\ln\delta^{-1}}{8W}\approx\frac{3v^{2}_{c}\ln\delta^{-1}}{8\lambda} ,\end{aligned}$$ which slightly differs from Eq. (\[dr\]) only in the numerical prefactor.
![(color online) Critical vacancy density (a) and transition width (b) with hard-wall boundaries. Symbols are the same in both panels. In panel (a), full lines are the approximations (\[vcshort\],\[vclong\]) for large $W$. In panel (b), full lines are the results from the full analytical expression and the dashed lines are the large-$W$ approximations. []{data-label="rcbig"}](rclw100.eps "fig:"){width="\columnwidth"} ![(color online) Critical vacancy density (a) and transition width (b) with hard-wall boundaries. Symbols are the same in both panels. In panel (a), full lines are the approximations (\[vcshort\],\[vclong\]) for large $W$. In panel (b), full lines are the results from the full analytical expression and the dashed lines are the large-$W$ approximations. []{data-label="rcbig"}](drw100.eps "fig:"){width="\columnwidth"}
Figure \[rcbig\] shows the critical density and width of the transition for systems of width $W=100$. We see, for example for $W=100$, that the critical density and the transition width indeed converges at $L_{c}\approx5000(KA),3000(FA)$, in agreement with Eq. (\[lcbig\]) which predicts $L_{c}=4606(KA),3152(FA)$. For the smaller width, $W=20$, there is also a convergence, but the fit is not as good as for $W=100$. The difference between the numerical results and the approximations for large $W$ is visible because $W=100$ is still not large enough for the asymptotic limit.
Systems with Small Width {#secsmall}
========================
In systems of small width we can improve our approximation, and even have exact results. In the first two subsections below we find an exact result for $n_{PF}$ for $L=\infty$ and $W=1$ or $2$ in both the KA and FA models. In the third subsection we improve our approximation for systems of width $3\leq W\leq 6$ by finding the exact number of frozen particles in small sections.
Fraction of Frozen Particles at $W=1$ and $L=\infty$
----------------------------------------------------
In systems of width $W=1$ we note that the boundary conditions are irrelevant. With hard-wall boundaries, a particle is blocked from above and below by the boundaries, and with periodic boundaries it is blocked by itself.
### KA Model
For systems with $W=1$, we denote by $f$ the number of occupied sites in the section and by $h$ the number of vacancies in the section. As there are no two adjacent occupied sites in the section, in order for a particle to be permanently frozen in the KA model, the entire section must be frozen and moreover it must be of the form $01010...01010$, i.e. $h=f+1$. The number of frozen particles in systems with $W=1$ is thus $$\begin{aligned}
N^{KA}_{PF}(\rho,1)=\sum^{\infty}_{h=1}v^{h}\rho^{h-1}\left(h-1\right)=\frac{\rho v^{2}}{\left(1-\rho v\right)^{2}} ,\end{aligned}$$ and thus, using Eq. (\[npfex\]), the fraction of frozen particles is $$\begin{aligned}
n^{KA}_{PF}(\rho,1)=2\rho-\rho^{2}+\frac{\rho^{4}v^{2}}{\left(1-\rho v\right)^{2}} .\end{aligned}$$
### FA Model
In the FA model, only the particles in the strips are frozen for $W=1$, and thus $$\begin{aligned}
n^{FA}_{PF}(\rho,1)=2\rho-\rho^{2} .\end{aligned}$$
Fraction of Frozen Particles at $W=2$ and $L=\infty$
----------------------------------------------------
For systems with $W=2$ the boundary conditions are important.
### KA Model, Hard-Wall Boundaries
Consider the following two types of patterns: First, a pattern with zigzag diagonals of occupied sites, such that the other sites are either occupied or not, and second, a pattern with a full row of occupied sites, with the sites in the other row either occupied or not. In each of these two cases, the particles in the *main part* (the full zigzag or full row) are frozen, and in the *secondary part* they are either frozen (if the main part is a zigzag) or unfrozen (if the main part is a row). A section with frozen particles can be built by dividing it into subsections with their main part either a zigzag or a row. Each two of these subsections must be divided by a *divider*, which consists of a full column and one particle in each of the adjacent columns, one at the top and one at the bottom. The following scheme shows this more clearly: $$\begin{aligned}
\begin{array}{cccccccccccccccccc}
1&0&1&0&d&d&0&0&0&2&0&0&d&d&3&3&3&3\\
0&1&0&1&0&d&d&2&2&2&2&d&d&0&3&0&3&0
\end{array}\end{aligned}$$ The sites marked with $d$ belong to a divider, and the sites marked with a number belong to one of the subsections. The first subsection is of the zigzag type. The second subsection is of the row type. Note that the site marked $2$ at the top row is not frozen. The third subsection can be of either type. We will consider it to be of a zigzag type, since all the particles in the secondary part are frozen. In order to simplify the following calculations, we will include the rightmost column of a divider in the subsection to the right of it. By denoting $d$ as the number of dividers, we note that the number of subsections is $d+1$. We account for the possibility of two adjacent dividers by considering subsections of length $0$. Also, since the left column in the divider is counted in it, we need to artificially add the rightmost column in the rightmost subsection, since it is not counted in the (non-existent) divider to the right of the last subsection. Another point to make is that the leftmost column in a subsection cannot be full. We denote each subsection by the number of vacancies, $h_{i}$, and the number of occupied sites, $f_{i}$, in the secondary part, such that the main part contains $h_{i}+f_{i}$ sites, and by its type, $t_{i}=z,r$ (zigzag or row). A particle in a section built in this way is unfrozen only if it is in the secondary part of a subsection of row type. We also need to make sure there are no two adjacent columns. Also, for each such section, there is a mirror configuration with the top and bottom rows switched, and so we can count the number of frozen particles in one such configuration (say, with the occupied site on the leftmost column in the top row) and multiply by $2$.
The number of frozen particles in such a section is thus
$$\begin{aligned}
N^{KA,hw}_{PF}(\rho,2)=2\sum^{\infty}_{d=0}\left(\rho^{3}v\right)^{d}\prod^{d+1}_{i=1}\sum_{t_{i}=z,r}\sum^{\infty}_{h_{i}=0}\sum^{h_{i}-\delta_{t_{i},r}}_{f_{i}=0}v^{h_{i}}\rho^{h_{i}+2f_{i}}\left(\begin{array}{c}h_{i}\\f_{i}\end{array}\right)\rho v\left[3d+\sum^{d+1}_{j=1}\left(h_{j}+f_{j}+f_{j}\delta_{t_{j},z}\right)+1\right] .\label{w2eq1}\end{aligned}$$
The factor of $2$ at the beginning is for the top-bottom symmetry. In the sums, we go over each subsection and check how many vacancies and occupied sites there are in the secondary part, where we note that in a subsection of row type $f_{i}<h_{i}$ (otherwise we consider it a zigzag type). The factor of $\rho v$ before the square brackets is for the rightmost column. The sum in the square brackets requires more explanations. First we add the particles in the main part ($h_{i}+f_{i}$). Next, we say that an occupied site in the secondary part is frozen only if the subsection is of zigzag type. The $1$ at the end is for the particle in the rightmost column. The final result from evaluating the sums in Eq. (\[w2eq1\]) is (see Appendix \[apka2\]) $$\begin{aligned}
&N^{KA,hw}_{PF}(\rho,2)=\frac{2\rho v\left(1-\rho^{3}v\right)\left[\left(1-\rho^{3}v\right)\left(1+4\rho^{3}v\right)+\rho v\left(2-\rho v+6\rho^{3}v\right)\right]}{\left[\left(1-\rho^{3}v\right)\left(1-2\rho^{3}v\right)-\rho v\right]^{2}} ,\label{ka2end}\end{aligned}$$ and the fraction of frozen particles is $$\begin{aligned}
&n^{KA,hw}_{PF}(\rho,2)=\rho^{3}\left(2-\rho^{2}\right)+\frac{\rho^{7}}{2}N^{KA,hw}_{PF}(\rho,2) . \label{ka2hw}\end{aligned}$$
### FA Model, Hard-Wall Boundaries
In the FA model with hard-wall boundaries, a section can be at least partially frozen only if all of its subsections are of row type, and the frozen particles are only those in the dividers and in the main part. Thus, the number of frozen particles is (see Appendix \[apfa2\]) $$\begin{aligned}
N^{FA,hw}_{PF}(\rho,2)=2\sum^{\infty}_{d=0}\left(\rho^{3}v\right)^{d}\prod^{d+1}_{i=1}\sum^{\infty}_{h_{i}=0}\sum^{h_{i}}_{f_{i}=0}v^{h_{i}}\rho^{h_{i}+2f_{i}}\left(\begin{array}{c}h_{i}\\f_{i}\end{array}\right)\rho v\left[3d+\sum^{d+1}_{j=1}\left(h_{j}+f_{j}\right)+1\right]=\frac{2\rho v\left(1+3\rho^{3}v\right)}{\left[1-\rho v\left(1+2\rho^{2}\right)\right]^{2}} . \label{fa2begin}\end{aligned}$$ The fraction of frozen particles is thus $$\begin{aligned}
n^{FA,hw}_{PF}(\rho,2)=\rho^{3}\left(2-\rho^{2}\right)+\frac{\rho^{8}v\left(1+3\rho^{3}v\right)}{\left[1-\rho v\left(1+2\rho^{2}\right)\right]^{2}} . \label{fa2hw}\end{aligned}$$
### KA Model, Periodic Boundaries
In the KA model with periodic boundaries, a section is frozen only if all of its subsections are of zigzag type, and thus the number of frozen particles is (see Appendix \[apkap2\]) $$\begin{aligned}
N^{KA,per}_{PF}(\rho,2)=2\sum^{\infty}_{d=0}\left(\rho^{3}v\right)^{d}\prod^{d+1}_{i=1}\sum^{\infty}_{h_{i}=0}\sum^{h_{i}}_{f_{i}=0}v^{h_{i}}\rho^{h_{i}+2f_{i}}\left(\begin{array}{c}h_{i}\\f_{i}\end{array}\right)\rho v\left[3d+\sum^{d+1}_{j=1}\left(h_{j}+2f_{j}\right)+1\right]=\frac{2\rho v\left(1+4\rho^{3}v\right)}{\left[1-\rho v\left(1+2\rho^{2}\right)\right]^{2}} ,\label{ka2per}\end{aligned}$$ and the fraction of frozen particles is $$\begin{aligned}
n^{KA,per}_{PF}(\rho,2)=\rho^{3}\left(2-\rho^{2}\right)+\frac{\rho^{8}v\left(1+4\rho^{3}v\right)}{\left[1-\rho v\left(1+2\rho^{2}\right)\right]^{2}} . \label{ka2p}\end{aligned}$$
The reason that Eqs. (\[fa2hw\]) and (\[ka2p\]) are very similar is that we count almost the same number of particles. The only addition to Eq. (\[ka2p\]) from Eq. (\[fa2hw\]) is the particles in the secondary part.
### FA Model, Periodic Boundaries
In this case, the only frozen particles are in the strips, and thus $$\begin{aligned}
&n^{FA,per}_{PF}(\rho,2)=\rho^{3}\left(2-\rho^{2}\right). \label{fa2p}\end{aligned}$$
Figure \[compw12\] shows how perfectly the expressions (\[ka2hw\]), (\[fa2hw\]), (\[ka2p\]), and (\[fa2p\]) fit the numerical results.
![(color online) Fraction of frozen particles vs. density for the FA (full symbols) and KA (empty symbols) models, for hard-wall (squares) and periodic (circles) boundary conditions, for $W=1,2$. Symbols are results of numerical simulations with $L=200W$, continuous line is analytical expression. For $W=2$ the results for KA with periodic boundaries and FA with hard-wall boundaries are almost the same. []{data-label="compw12"}](compw12.eps){width="\columnwidth"}
Fraction of Frozen Particles at $3\leq W\leq6$ {#sec36}
----------------------------------------------
In principle, we can find exact results also for systems with $W>2$, but as seen in the previous subsections it gets progressively more complicated with increasing $W$. However, we can improve the approximation calculated previously. For systems with small width ($3\leq W\leq6$) we can calculate exactly the average number of frozen particles in sections of length $l$, $N(l)$, by simply counting all possible configurations. As the number of possible configurations rises exponentially with the section’s length and the system’s width, we will consider this only for sections of size $Wl\leq40$, for which the number of configurations is $2^{40}\approx10^{12}$, a number which can be handled numerically. The frozen particles in the longer sections are neglected.
Figure \[compw36\] shows the fraction of frozen particles obtained by the numerical results, this improved approximation, and the previous approximation. We see that for $W=3$, it is enough to consider $l=13$, but for the wider systems we need longer sections. Since the fraction of frozen particles in a section decays roughly exponentially with the section’s length (see Fig. \[fakasec\]), we need to consider only $l\approx2\left\langle l\right\rangle$, where $\left\langle l\right\rangle$ is the average section length, see Eq. (\[lav\]). As the average section length depends on the density, we can take it at the critical density. We find that for such small widths, the average section length is $6.2$ (for $W=3$), $6.53$ ($W=4$), $7.05$ ($W=5$) and $7.6$ (for $W=6$). This explains why $l=13$ is enough for $W=3$, but $l=10$ is not enough for $W=4$. The number of configurations to scan numerically is $2^{2W\left\langle l\right\rangle}=2^{39}\approx5\times10^{11}$ ($W=3$), $2^{52}\approx4\times10^{15}$ ($W=4$), $2^{75}\approx3\times10^{22}$ ($W=5$) and $2^{96}\approx6\times10^{28}$ ($W=6$).
![ Fraction of frozen particles vs. density for $W=3,4,5,6$ with hard-wall boundaries. Dashed lines are result of the approximation of Section \[secdiv\] for the KA (higher curve) and FA (lower curve) models. Continuous lines are results from the approximation of Section \[sec36\] for the KA (higher curve) and FA (lower curve) models. Full (KA) and empty (FA) squares are numerical results. Numerical simulations were done with $L=200W$. Analytical results are for $L=\infty$. As $W$ increases, larger sections should be included. For $W=3,4$, the current approximation is better than the approximation of Section \[secdiv\], but for $W\geq5$, longer sections are required. []{data-label="compw36"}](npfw3.eps "fig:"){width="120pt"} ![ Fraction of frozen particles vs. density for $W=3,4,5,6$ with hard-wall boundaries. Dashed lines are result of the approximation of Section \[secdiv\] for the KA (higher curve) and FA (lower curve) models. Continuous lines are results from the approximation of Section \[sec36\] for the KA (higher curve) and FA (lower curve) models. Full (KA) and empty (FA) squares are numerical results. Numerical simulations were done with $L=200W$. Analytical results are for $L=\infty$. As $W$ increases, larger sections should be included. For $W=3,4$, the current approximation is better than the approximation of Section \[secdiv\], but for $W\geq5$, longer sections are required. []{data-label="compw36"}](npfw4.eps "fig:"){width="120pt"} ![ Fraction of frozen particles vs. density for $W=3,4,5,6$ with hard-wall boundaries. Dashed lines are result of the approximation of Section \[secdiv\] for the KA (higher curve) and FA (lower curve) models. Continuous lines are results from the approximation of Section \[sec36\] for the KA (higher curve) and FA (lower curve) models. Full (KA) and empty (FA) squares are numerical results. Numerical simulations were done with $L=200W$. Analytical results are for $L=\infty$. As $W$ increases, larger sections should be included. For $W=3,4$, the current approximation is better than the approximation of Section \[secdiv\], but for $W\geq5$, longer sections are required. []{data-label="compw36"}](npfw5.eps "fig:"){width="120pt"} ![ Fraction of frozen particles vs. density for $W=3,4,5,6$ with hard-wall boundaries. Dashed lines are result of the approximation of Section \[secdiv\] for the KA (higher curve) and FA (lower curve) models. Continuous lines are results from the approximation of Section \[sec36\] for the KA (higher curve) and FA (lower curve) models. Full (KA) and empty (FA) squares are numerical results. Numerical simulations were done with $L=200W$. Analytical results are for $L=\infty$. As $W$ increases, larger sections should be included. For $W=3,4$, the current approximation is better than the approximation of Section \[secdiv\], but for $W\geq5$, longer sections are required. []{data-label="compw36"}](npfw6.eps "fig:"){width="120pt"}
Internal Structures {#secinter}
===================
The main qualitative difference between systems of large widths and small widths is the internal structures within the sections. In wide systems, the vast majority of sections are either almost completely frozen or completely unfrozen, while in narrow systems there is a significant number of sections which are partially frozen. The reason for this difference is the existence of small unfrozen “islands", which are small regions that do not effect their surroundings. For example, a structure of the form $\begin{array}{cc}10\\00\end{array}$ is unfrozen (in both the KA and the FA model), but it does not necessarily cause the entire section to be unfrozen. When the system is wide these islands are not important, but in a narrow system they are. Figure \[fuf\] shows the density of completely frozen/unfrozen sections in a wide system ($W=40$) and in a narrow system ($W=7$), and Fig. \[grid100\] shows a snapshot of the system, highlighting the frozen and unfrozen particles.
A quantitative way to measure the effect of the boundary conditions is by noting that with hard-wall boundary conditions, the particles in the top and bottom rows have a slightly higher probability of being frozen than those in the middle rows. This happens because if within a section the top or bottom row is completely full, then all the particles in it are frozen, while for the middle rows this condition is not sufficient. However, as seen from Fig. \[relrows\], this difference is very small. Although at first the relative difference between the probability of being frozen at the edges and at the middle grows with $W$, this relative difference reaches a maximum at $W=12$ and then decreases for larger $W$. Hence, we can say that above $W=12$ the boundary conditions become less important.
![(color online) (a) Density of sections which are completely frozen or completely unfrozen, hard-wall boundaries. The minimum for $W=40$ is much lower than for $W=7$ because there is a larger chance of a small, inconsequential unfrozen island. (b) The distribution of the fraction of frozen particles per section for $W=7$ and $\rho=0.9$ and for $W=40$ and $\rho=0.95$ (near the minima in panel (a)). Most sections are either completely unfrozen or almost completely frozen. []{data-label="fuf"}](fuf.eps "fig:"){width="200pt"} ![(color online) (a) Density of sections which are completely frozen or completely unfrozen, hard-wall boundaries. The minimum for $W=40$ is much lower than for $W=7$ because there is a larger chance of a small, inconsequential unfrozen island. (b) The distribution of the fraction of frozen particles per section for $W=7$ and $\rho=0.9$ and for $W=40$ and $\rho=0.95$ (near the minima in panel (a)). Most sections are either completely unfrozen or almost completely frozen. []{data-label="fuf"}](fuf2.eps "fig:"){width="200pt"}
![(color online) (a) Probability that a frozen particle belongs to a certain row at $W=3$. At some width-dependent density, the probability of a frozen particle to be in the extreme rows is maximal. (b) The ratio between the maximum for the extreme rows and the minimum for the middle row(s) as a function of $W$. This is maximal at $W=12$. []{data-label="relrows"}](rows3.eps "fig:"){width="\columnwidth"} ![(color online) (a) Probability that a frozen particle belongs to a certain row at $W=3$. At some width-dependent density, the probability of a frozen particle to be in the extreme rows is maximal. (b) The ratio between the maximum for the extreme rows and the minimum for the middle row(s) as a function of $W$. This is maximal at $W=12$. []{data-label="relrows"}](relrow.eps "fig:"){width="\columnwidth"}
Summary
=======
In this paper we investigated the effect of confinement and shape of container on the jamming transition. We derived an analytical approximation for the fraction of frozen particles in rectangular systems in both the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models, by dividing the system into independent sections and using the notion of critical droplets, which was derived previously for square systems in the FA model. The number of these critical droplets is controlled, in addition to the system’s size and particle density, by a single parameter $\lambda$. We showed that the effective value of $\lambda$ does not change much when the system’s length increases, and that its value in rectangular systems is approximately the same as in square systems. Also, its value in the Kob-Andersen model is higher than in the Fredrickson-Andersen model, which means that the fraction of frozen particles in the KA model is higher than in the FA model, as expected by comparing the kinetic constraints of the two models. From both the numerical simulations and the analytical expressions, we found that the transition from an unjammed state, where most of the particles are free, to a jammed state, where most of the particles are frozen, occurs over a very narrow range of densities. Using our approximation, we found the critical density at which this transition occurs, $\rho_{c}$, which converges to $1$ as the system size increases. We also found that the width of the transition scales as $(1-\rho_{c})^{2}$.
For infinite tunnels, we derived an exact result for very narrow systems (widths $1$ and $2$) for both the KA model and the FA model. The technique we used can also be applied to wider systems. For infinite systems of width $3-6$ we showed that it is enough to explicitly count the number of frozen particles in sections shorter than twice the typical section length, since the number of frozen particles in a section decays exponentially with the section’s length. Also, using the analytical approximation for general rectangles, we found a simple expression relating the critical density with the width of an infinite tunnel, which can be verified in experiments. In particular, we found that the critical density in channels decreases algebraically with the system’s width, $1-\rho_{c}\sim 1/\sqrt{W}$, much faster than the logarithmic decrease in square systems, $1-\rho_{c}\sim1/\ln L$. These two different expressions for the critical density show that the jamming transition depends not only on the system size but also on its shape and the relation between the system’s width and length.
Our idea of dividing the system into independent sections can also be applied in three-dimensional systems, and will be addressed in future work. It will also be interesting to check the effects of confinement on the behavior of other kinetically-constrained models, such as jamming percolation models [@knights; @jeng], and to employ our approach for studying jamming in driven systems, such as granular matter flowing in a narrow tube.
We thank Svilen Kozhuharov and Vincenzo Vitelli for fruitful discussions, and Yariv Kafri and Peter Sollich for critical reading of the manuscript. This research was supported by the Israel Science Foundation grants No. $617/12$, $1730/12$.
Calculation of the Fraction of Frozen Particles in Eq. (\[npfmain0\]) {#apnpf}
=====================================================================
In this section we calculate explicitly the sums in Eq. (\[npfmain0\]). We first note that $Q(n,l)$ is proportional to $\rho^{n}v^{lW-n}$, such that $$\begin{aligned}
nQ(n,l)=\rho\partial_{\rho}\left[Q(n,l)\right] ,\end{aligned}$$ where $\partial_{\rho}\left[f(\rho)\right]$ is the partial derivative of $f(\rho)$ with respect to $\rho$ while assuming that $v$ is constant. Only after differentiating we use the relation $v+\rho=1$. Hence, $n_{PF}$ can be simplified by $$\begin{aligned}
&n_{PF}=\frac{\sum_{n,l,m}\rho\partial_{\rho}\left[\rho^{mW}\right]Q(n,l)+\rho^{mW}\rho\partial_{\rho}\left[Q(n,l)\right]e^{-kl}}{\rho\partial_{\rho}\left[\sum_{n,l,m}\rho^{mW}Q(n,l)\right]} ,\end{aligned}$$ We can now perform the sums over $n$, such that $$\begin{aligned}
n_{PF}=\frac{\sum_{l,m}\rho\partial_{\rho}\left[\rho^{mW}\right]Q(l)+\rho^{mW}\rho\partial_{\rho}\left[Q(l)\right]e^{-kl}}{\rho\partial_{\rho}\left[\sum_{l,m}\rho^{mW}Q(l)\right]} ,\end{aligned}$$ with $Q(l)=\sum_{n}Q(l,n)$. In order to have the derivative in the nominator outside the sum, we artificially change some of the $\rho$ to $\rho'$, and after differentiating set $\rho'=\rho$, and thus $$\begin{aligned}
n_{PF}=\frac{\rho\partial_{\rho}\left[\sum_{l,m}\rho^{mW}Q(l,\rho')+\rho'^{mW}Q(l,\rho)e^{-kl}\right]}{\rho\partial_{\rho}\left[\sum_{l,m}\rho^{mW}Q(l,\rho)\right]} . \label{npfmain2}\end{aligned}$$ For ease of calculation, we divide $n_{PF}$ into three parts $$\begin{aligned}
n_{PF}=\frac{A_{1}+A_{2}}{B} ,\end{aligned}$$ where $$\begin{aligned}
&A_{1}=\rho\partial_{\rho}\left[\sum_{l,m}\rho^{mW}Q(l,\rho')\right] ,\nonumber\\
&A_{2}=\rho\partial_{\rho}\left[\sum_{l,m}\rho'^{mW}Q(l,\rho)e^{-kl}\right] ,\nonumber\\
&B=\rho\partial_{\rho}\left[\sum_{l,m}\rho^{mW}Q(l,\rho)\right] ,\end{aligned}$$ and calculate each part separately.
Calculation of $Q(l)$
---------------------
For $l=1$ we denote by $0\leq f<W$ the number of particles in the column, such that $$\begin{aligned}
Q(1)=\sum^{W-1}_{f=0}\rho^{f}v^{W-f}\left(\begin{array}{c}W\\f\end{array}\right)=\left(\rho+v\right)^{W}-\rho^{W}=1-\rho^{W} .\label{ql1}\end{aligned}$$
For the longer sections, we denote by $r_{1}$ and $r_{2}$ the number of particles in the rightmost and leftmost columns, and by $n_{i}$ the number of columns with $i$ particles in them. Thus, the density of sections of length $l$ is $$\begin{aligned}
&Q(l)=\sum^{W-1}_{r_{1}=0}\sum^{W-1}_{r_{2}=0}\sum^{\left\lfloor\frac{l-1}{2}\right\rfloor}_{n_{W}=0}\prod^{W-1}_{i=1}\sum^{l-2-\sum^{W}_{j=i+1}n_{j}}_{n_{i}=0}\rho^{r_{1}+r_{2}}v^{2W-r_{1}-r_{2}}\left(\begin{array}{c}W\\r_{1}\end{array}\right)\left(\begin{array}{c}W\\r_{2}\end{array}\right)\rho^{\sum^{W}_{j=1}jn_{j}}\times\nonumber\\
&\times v^{\sum^{W}_{j=1}\left(W-j\right)n_{j}+W\left(l-2-\sum^{W}_{j=1}n_{j}\right)}\left(\begin{array}{c}l-n_{W}-1\\n_{w}\end{array}\right)\left(\begin{array}{c}l-2-\sum^{W}_{j=i+1}n_{j}\\n_{i}\end{array}\right)\left(\begin{array}{c}W\\i\end{array}\right)^{n_{i}} ,\end{aligned}$$ where $\left\lfloor x\right\rfloor$ is the integer part of $x$. The first binomial in the second line is the number of ways to arrange $n_{W}$ columns among the $l-2$ available places such that there are no two adjacent full columns. The upper limit of the sum over $n_{W}$ is such because above it the binomial is zero. The second binomial in the second line is the number of ways to arrange $n_{i}$ columns among the columns not yet taken by the already-placed columns. Note that we do not sum over $n_{0}$, since it must satisfy $n_{0}=l-2-\sum^{W}_{i=1}n_{i}$.
Summing over $r_{1}, r_{2}$ yields $$\begin{aligned}
&Q(l)=\left[\left(\rho+v\right)^{W}-\rho^{W}\right]^{2}\sum^{\left\lfloor\frac{l-1}{2}\right\rfloor}_{n_{W}=0}\left(\frac{\rho}{v}\right)^{Wn_{W}}v^{W\left(l-2\right)}\left(\begin{array}{c}l-n_{W}-1\\n_{W}\end{array}\right)\times\nonumber\\
&\times\prod^{W-1}_{i=1}\sum^{l-2-\sum^{W}_{j=i+1}n_{j}}_{n_{i}=0}\left[\left(\frac{\rho}{v}\right)^{i}\left(\begin{array}{c}W\\i\end{array}\right)\right]^{n_{i}}\left(\begin{array}{c}l-2-\sum^{W}_{j=i+1}n_{j}\\n_{i}\end{array}\right) .\end{aligned}$$
We now note that $$\begin{aligned}
\prod^{W-1}_{i=1}\sum^{N-\sum^{W-1}_{j=i+1}n_{j}}_{n_{i}=0}\left[g_{i}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W-1}_{j=i+1}n_{j}\\n_{i}\end{array}\right)=\left[\sum^{W-1}_{i=0}g_{i}\right]^{N} ,\label{proof1}\end{aligned}$$ where $g_{0}=1$. The proof for this equation is given in Appendix \[approof1\]. We can now write $$\begin{aligned}
&Q(l)=\left[\left(\rho+v\right)^{W}-\rho^{W}\right]^{2}\sum^{\left\lfloor\frac{l-1}{2}\right\rfloor}_{n_{W}=0}\left(\frac{\rho}{v}\right)^{Wn_{W}}v^{W\left(l-2\right)}\left(\begin{array}{c}l-n_{W}-1\\n_{W}\end{array}\right)\left[\sum^{W-1}_{i=0}\left(\frac{\rho}{v}\right)^{i}\left(\begin{array}{c}W\\i\end{array}\right)\right]^{l-2-n_{W}}=\nonumber\\
&=\left[\left(\rho+v\right)^{W}-\rho^{W}\right]^{2}\sum^{\left\lfloor\frac{l-1}{2}\right\rfloor}_{n_{W}=0}\left[\frac{1}{\left(1+\frac{v}{\rho}\right)^{W}-1}\right]^{n_{W}}\left(\begin{array}{c}l-n_{W}-1\\n_{W}\end{array}\right) .\end{aligned}$$ Performing the sum over $n_{W}$ yields $$\begin{aligned}
&Q(l)=\frac{\left[\left(\rho+v\right)^{W}-\rho^{W}\right]^{l}\left[\left(1+\frac{v}{\rho}\right)^{W}-1\right]}{\sqrt{\left[\left(1+\frac{v}{\rho}\right)^{W}+1\right]^{2}-4}}\times\nonumber\\
&\times\left[\left(\frac{\left(1+\frac{v}{\rho}\right)^{W}-1+\sqrt{\left[\left(1+\frac{v}{\rho}\right)^{W}+1\right]^{2}-4}}{2\left[\left(1+\frac{v}{\rho}\right)^{W}-1\right]}\right)^{l}-\left(\frac{\left(1+\frac{v}{\rho}\right)^{W}-1-\sqrt{\left[\left(1+\frac{v}{\rho}\right)^{W}+1\right]^{2}-4}}{2\left[\left(1+\frac{v}{\rho}\right)^{W}-1\right]}\right)^{l}\right]=\nonumber\\
&=\frac{1-\rho^{W}}{\sqrt{1+2\rho^{W}-3\rho^{2W}}}\left[x^{l}_{+}-x^{l}_{-}\right] ,\label{ql}\end{aligned}$$ where $$\begin{aligned}
x_{\pm}=\frac{1-\rho^{W}\pm\sqrt{\left(1+\rho^{W}\right)^{2}-4\rho^{2W}}}{2} .\end{aligned}$$ Only in the last step did we use $v+\rho=1$. Using $l=1$ in the general equation for $l>1$ yields the same result we found in Eq. (\[ql1\]) for $Q(1)$. Note also that $Q(l)$ depends on $\rho$ and $W$ only via $\rho^{W}$. An interesting point to make is that $$\begin{aligned}
Q(2)=Q(3)=\left(1-\rho^{W}\right)^{2} .\end{aligned}$$ Also, we find that $Q(l+1)\geq Q(l)$ for all values of $\rho^{W}$, and equality holds only for $l=2$.
The Final Result for $n_{PF}$
-----------------------------
Using Eq. (\[ql\]) we can now calculate the sums in Eq. (\[npfmain2\]). All the sums are such that $l+m\leq L$. Also, $m$ is greater or equal to $2$ except for the case $l=L$ where $m=0$.
### The Denominator
The denominator of $n_{PF}$ in Eq. (\[npfmain2\]) is $$\begin{aligned}
&B=\rho\partial_{\rho}\left[\sum_{l,m}\rho^{mW}Q(l,\rho)\right]=\rho\partial_{\rho}\left[Q(L,\rho)+\rho^{WL}+\sum^{L-2}_{l=1}\sum^{L-l}_{m=2}\rho^{mW}Q(l,\rho)\right] .\end{aligned}$$ Performing the sum over $m$ yields $$\begin{aligned}
&B=\rho\partial_{\rho}\left[Q(L,\rho)+\rho^{WL}+\sum^{L-2}_{l=1}\frac{\rho^{2W}-\rho^{W\left(L-l+1\right)}}{1-\rho^{W}}Q(l,\rho)\right] .\end{aligned}$$ Performing the sum over $l$, differentiating with respect to $\rho$ and finally setting $v=1-\rho$ yields $$\begin{aligned}
&B=\frac{W\rho^{1-2W}}{1-\rho^{W}}+\frac{W\rho^{W\left(L+1\right)}\left[2-2\rho^{W}-\rho^{W+1}+L\left(2-5\rho^{W}+3\rho^{2W}\right)\right]}{\left(1-\rho^{W}\right)\left(2-3\rho^{W}\right)^{2}}+LW\rho^{WL}-\nonumber\\
&-\frac{W\left[x^{L}_{+}+x^{L}_{-}\right]}{2\rho^{2W}\left(1+3\rho^{W}\right)\left(2-3\rho^{W}\right)^{2}}\times\nonumber\\
&\times\left[2L\rho^{2W}\left(2-3\rho^{W}\right)\left(1-\rho^{W}-3\rho^{2W}+3\rho^{W+1}\right)+\left(1+3\rho^{W}\right)\left(4\rho+2\rho^{3W}-8\rho^{W+1}+\rho^{2W+1}+\rho^{3W+1}\right)\right]+\nonumber\\
&+\frac{\sqrt{\left(1+\rho^{W}\right)^{2}-4\rho^{2W}}W\left[x^{L}_{+}-x^{L}_{-}\right]}{2\rho^{2W}\left(1-\rho^{W}\right)\left(2+3\rho^{W}-9\rho^{2W}\right)^{2}}\left[2\left(L+1\right)\rho^{2W}\left(2+3\rho^{W}-9\rho^{2W}\right)\left(1-3\rho^{W}+3\rho^{2W}-\rho^{W+1}\right)-\right.\nonumber\\
&\left.-\left(1-\rho^{W}\right)\left(4\rho+4\rho^{2W}+12\rho^{3W}-36\rho^{4W}+54\rho^{5W}+12\rho^{W+1}-15\rho^{2W+1}-54\rho^{3W+1}+27\rho^{4W+1}\right)\right] .\label{denfinal}\end{aligned}$$
### First Part of the Nominator
The first term in the nominator of Eq. (\[npfmain2\]) is $$\begin{aligned}
&A_{1}=\rho\partial_{\rho}\left[\sum_{l,m}\rho^{mW}Q(l,\rho')\right]=\rho\partial_{\rho}\left[Q(L,\rho')+\rho^{WL}+\sum^{L-2}_{l=1}\sum^{L-l}_{m=2}\rho^{mW}Q(l,\rho')\right]=\nonumber\\
&=\rho\partial_{\rho}\left[\rho^{WL}+\sum^{L-2}_{l=1}\sum^{L-l}_{m=2}\rho^{mW}Q(l,\rho')\right] ,\end{aligned}$$ where in the last transition we note that $\partial_{\rho}Q(l,\rho')=0$, since here $Q$ is a function of $\rho'$ and not $\rho$. Summing over $m$ yields $$\begin{aligned}
A_{1}=\rho\partial_{\rho}\left[\rho^{WL}+\sum^{L-2}_{l=1}\frac{\rho^{2W}-\rho^{\left(L-l+1\right)W}}{1-\rho^{W}}Q(l,\rho')\right] .\end{aligned}$$ Summing over $l$, differentiating with respect to $\rho$ (not $\rho'$), and lastly setting $v=1-\rho$ and $\rho'=\rho$ yields $$\begin{aligned}
&A_{1}=\frac{W\left(2-\rho^{W}\right)}{1-\rho^{W}}+\frac{W\rho^{\left(L+1\right)W}\left[L\left(2-5\rho^{W}+3\rho^{2W}\right)+3-4\rho^{W}\right]}{\left(1-\rho^{W}\right)\left(2-3\rho^{W}\right)^{2}}+WL\rho^{WL}-\nonumber\\
&-\frac{W\left[x^{L}_{+}+x^{L}_{-}\right]\left(1-\rho^{W}\right)\left(8-9\rho^{W}\right)}{2\left(2-3\rho^{W}\right)^{2}}-\frac{W\left[x^{L}_{+}-x^{L}_{-}\right]\left(1-\rho^{W}\right)\left(8-7\rho^{W}-3\rho^{2W}\right)}{2\left(2-3\rho^{W}\right)^{2}\sqrt{1+2\rho^{W}-3\rho^{2W}}} .\label{nom1final}\end{aligned}$$
### Second Part of the Nominator
The second part of the nominator is $$\begin{aligned}
&A_{2}=\rho\partial_{\rho}\left[\sum_{l,m}\rho'^{mW}Q(l,\rho)e^{-lk}\right]=\rho\partial_{\rho}\left[Q(L,\rho)e^{-Lk}+\sum^{L-2}_{l=1}\sum^{L-l}_{m=2}\rho'^{mW}Q(l,\rho)e^{-lk}\right]=\nonumber\\
&=\rho\partial_{\rho}\left[Q(L,\rho)e^{-Lk}+\sum^{L-2}_{l=1}\frac{\rho'^{2W}-\rho'^{\left(L-l+1\right)W}}{1-\rho'^{W}}Q(l,\rho)e^{-lk}\right] .\end{aligned}$$ Summing over $l$, differentiating with respect to $\rho$ and lastly setting $v=1-\rho$ and $\rho'=\rho$ yields $$\begin{aligned}
&Nom2=\frac{We^{k}\rho^{2W}\left[e^{2k}\left(\rho-\rho^{W}\right)+\rho^{W}\left(1-\rho^{W}\right)^{2}\right]y^{2}_{1}}{1-\rho^{W}}-\frac{We^{k}\rho^{W\left(L+1\right)}\left[\left(1-\rho^{W}\right)^{2}+e^{2k}\rho^{W}\left(\rho-\rho^{W}\right)\right]y^{2}_{2}}{1-\rho^{W}}-\nonumber\\
&-\frac{e^{-kL}WL\left(1-\rho\right)\left[x^{L}_{+}+x^{L}_{-}\right]}{2\left(1+3\rho^{W}\right)}+\frac{e^{-kL}\rho^{W}W\left[x^{L}_{+}+x^{L}_{-}\right]}{2\left(1-\rho^{W}\right)^{2}\left(1+3\rho^{W}\right)}\times\nonumber\\
&\times\left\{e^{3k}\left(1-\rho^{W}\right)\left[\left(1+3\rho^{W}\right)\left(1-4\rho^{W}+4\rho^{2W}-\rho^{W+1}\right)-L\left(1-\rho^{W}-4\rho^{2W}+6\rho^{3W}+\rho^{W+1}-3\rho^{2W+1}\right)\right]\times\right.\nonumber\\
&\left.\times\left[y^{2}_{1}-y^{2}_{2}\right]+e^{2k}\left(1-\rho^{W}\right)^{2}\left[2+4\rho^{W}-6\rho^{2W}-L\left(1+\rho^{W}-3\rho^{2W}+\rho^{W+1}\right)\right]\left[\rho^{W}y^{2}_{1}-y^{2}_{2}\right]-\right.\nonumber\\
&\left.-e^{4k}\left(1-\rho^{W}\right)\left[2+4\rho^{W}-6\rho^{2W}-L\left(2+\rho^{W}-6\rho^{2W}+3\rho^{W+1}\right)\right]\left[y^{2}_{1}-\rho^{W}y^{2}_{2}\right]+\right.\nonumber\\
&\left.+e^{5k}\left[1+2\rho^{W}-3\rho^{2W}-L\left(1-3\rho^{2W}+2\rho^{W+1}\right)\right]\left[y^{2}_{1}-\rho^{2W}y^{2}_{2}\right]\right\}-\nonumber\\
&-\frac{e^{-kL}W\left[x^{L}_{+}-x^{L}_{-}\right]\left[4\rho^{W}\left(1-\rho\right)-L\left(1+\rho-2\rho^{W}\right)\left(1+3\rho^{W}\right)\right]}{2\left(1+3\rho^{W}\right)\sqrt{1+2\rho^{W}-3\rho^{2W}}}+\nonumber\\
&+\frac{e^{-kL}W\rho^{W}\left[x^{L}_{+}-x^{L}_{-}\right]}{2\left(1-\rho^{W}\right)\left(1+3\rho^{W}\right)\sqrt{1+2\rho^{W}-3\rho^{2W}}}\times\nonumber\\
&\times\left\{-e^{2k}\left(1-\rho^{W}\right)\left[2+6\rho^{W}-6\rho^{2W}-6\rho^{3W}+4\rho^{2W+1}-L\left(1+3\rho^{W}\right)\left(1-\rho^{W}-\rho^{2W}+\rho^{W+1}\right)\right]\left[\rho^{W}y^{2}_{1}-y^{2}_{2}\right]-\right.\nonumber\\
&\left.-e^{3k}\left(1-\rho^{W}\right)\left[1+\rho^{W}-10\rho^{2W}+6\rho^{3W}-\rho^{W+1}+3\rho^{2W+1}-L\left(1+3\rho^{W}\right)\left(1-2\rho^{W}+\rho^{W+1}\right)\right]\left[y^{2}_{1}-y^{2}_{2}\right]+\right.\nonumber\\
&\left.+e^{4k}\left[2+6\rho^{W}-10\rho^{2W}+6\rho^{3W}-4\rho^{W+1}-L\left(1+3\rho^{W}\right)\left(2-3\rho^{W}+\rho^{W+1}\right)\right]\left[y^{2}_{1}-\rho^{W}y^{2}_{2}\right]-\right.\nonumber\\
&\left.-e^{5k}\left[1+4\rho^{W}-3\rho^{2W}-2\rho^{W+1}-L\left(1+2\rho^{W}-3\rho^{2W}\right)\right]\left[y^{2}_{1}-\rho^{2W}y^{2}_{2}\right]\right\} , \label{nom2final}\end{aligned}$$ where $$\begin{aligned}
&y_{1}=\frac{1}{\left(e^{k}+\rho^{W}\right)\left(1-\rho^{W}\right)-e^{2k}} ,\nonumber\\
&y_{2}=\frac{1}{\left(e^{k}+1\right)\left(1-\rho^{W}\right)-e^{2k}\rho^{W}} .\end{aligned}$$
Proof for Eq. (\[proof1\]) {#approof1}
==========================
Here we prove that $$\begin{aligned}
\prod^{W-1}_{i=1}\sum^{N-\sum^{W-1}_{j=i+1}n_{j}}_{n_{i}=0}
\left[g_{i}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W-1}_{j=i+1}n_{j}\\n_{i}\end{array}\right)=\left[1+\sum^{W-1}_{i=1}g_{i}\right]^{N} .\end{aligned}$$ We do this by induction on $W$. For $W=2$ it holds because $$\begin{aligned}
\sum^{N-0}_{n_{1}=0}\left[g_{1}\right]^{n_{1}}\left(\begin{array}{c}N-0\\n_{1}\end{array}\right)=\left[1+g_{1}\right]^{N} .\end{aligned}$$
For $W+1$ we first sum over $n_{1}$ $$\begin{aligned}
&\prod^{W}_{i=1}\sum^{N-\sum^{W}_{j=i+1}n_{j}}_{n_{i}=0}
\left[g_{i}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W}_{j=i+1}n_{j}\\n_{i}\end{array}\right)=\prod^{W}_{i=2}\sum^{N-\sum^{W}_{j=i+1}n_{j}}_{n_{i}=0}
\left[g_{i}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W}_{j=i+1}n_{j}\\n_{i}\end{array}\right)\left[1+g_{1}\right]^{N-\sum^{W}_{j=2}n_{j}}=\nonumber\\
&=\left[1+g_{1}\right]^{N}\prod^{W}_{i=2}\sum^{N-\sum^{W}_{j=i+1}n_{j}}_{n_{i}=0}
\left[\frac{g_{i}}{1+g_{1}}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W}_{j=i+1}n_{j}\\n_{i}\end{array}\right) .\end{aligned}$$ We now define $h_{i}$ such that $$\begin{aligned}
h_{i}=\frac{g_{i+1}}{1+g_{1}} ,\end{aligned}$$ and rewrite the sum as $$\begin{aligned}
&\prod^{W}_{i=1}\sum^{N-\sum^{W}_{j=i+1}n_{j}}_{n_{i}=0}
\left[g_{i}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W}_{j=i+1}n_{j}\\n_{i}\end{array}\right)=\left[1+g_{1}\right]^{N}\prod^{W-1}_{i=1}\sum^{N-\sum^{W-1}_{j=i+1}n_{j}}_{n_{i}=0}
\left[h_{i}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W-1}_{j=i+1}n_{j}\\n_{i}\end{array}\right) .\end{aligned}$$ Since the sum is now only up to $W-1$ we know what it is $$\begin{aligned}
&\prod^{W}_{i=1}\sum^{N-\sum^{W}_{j=i+1}n_{j}}_{n_{i}=0}
\left[g_{i}\right]^{n_{i}}\left(\begin{array}{c}N-\sum^{W}_{j=i+1}n_{j}\\n_{i}\end{array}\right)=\left[1+g_{1}\right]^{N}\left[1+\sum^{W-1}_{i=1}h_{i}\right]^{N}=\left[1+\sum^{W}_{i=1}g_{i}\right]^{N} ,\end{aligned}$$ as required.
Derivation of Eq. (\[ka2end\]) {#apka2}
==============================
The sum in Eq. (\[w2eq1\]) over each subsection is identical and independent of the others, thus we can transform Eq. (\[w2eq1\]) to $$\begin{aligned}
&N^{KA,hw}_{PF}(\rho,2)=2\rho v\sum^{\infty}_{d=0}\left(\rho^{3}v\right)^{d}\left[\left(3d+1\right)\left(\sum_{t=z,l}\sum^{\infty}_{h=0}\sum^{h-\delta_{t,r}}_{f=0}v^{h}\rho^{h+2f}\left(\begin{array}{c}h\\f\end{array}\right)\right)^{d+1}+\right.\nonumber\\
&\left.+\left(d+1\right)\left(\sum_{t=z,r}\sum^{\infty}_{h=0}\sum^{h-\delta_{t,r}}_{f=0}v^{h}\rho^{h+2f}\left(\begin{array}{c}h\\f\end{array}\right)\right)^{d}\left(\sum_{t=z,r}\sum^{\infty}_{h=0}\sum^{h-\delta_{t,r}}_{f=0}v^{h}\rho^{h+2f}\left(\begin{array}{c}h\\f\end{array}\right)\left(h+f+f\delta_{t,z}\right)\right)\right] .\end{aligned}$$ Calculating the sums over $f,h$ and $t$ yields $$\begin{aligned}
&N^{KA,hw}_{PF}(\rho,2)=2\rho v\sum^{\infty}_{d=0}\left(\rho^{3}v\right)^{d}\left[\left(3d+1\right)C^{d+1}_{2}\left(\rho\right)+\left(d+1\right)C^{d}_{2}\left(\rho\right)C_{1}\left(\rho\right)\right] ,\nonumber\\
&C_{1}\left(\rho\right)=\frac{\rho v\left[2+3\rho^{2}-6\rho^{5}v-2\rho^{4}v^{2}-2\rho^{6}v^{2}+3\rho^{8}v^{2}\right]}{\left(1-\rho^{3}v\right)^{2}\left[1-\rho v\left(1+\rho^{2}\right)\right]^{2}} ,\nonumber\\
&C_{2}\left(\rho\right)=\frac{1+\rho v\left(1-\rho^{2}\right)}{\left(1-\rho^{3}v\right)\left[1-\rho v\left(1+\rho^{2}\right)\right]} .\end{aligned}$$ Summing over $d$ yields Eq. (\[ka2end\])
Derivation of Eq. (\[fa2begin\]) {#apfa2}
================================
We rewrite some of the $\rho$ in Eq. (\[fa2begin\]) as $\rho'$, so that $N^{FA}_{PF}$ can be simplified to $$\begin{aligned}
&N^{FA,hw}_{PF}(\rho,2)=2\sum^{\infty}_{d=0}\prod^{d+1}_{i=1}\sum^{\infty}_{h_{i}=0}\sum^{h_{i}}_{f_{i}=0}v^{d}v^{h_{i}}\rho^{f_{i}}\rho'^{3d+h_{i}+f_{i}+1}\left(\begin{array}{c}h_{i}\\f_{i}\end{array}\right) v\left[3d+\sum^{d+1}_{j=1}\left(h_{j}+f_{j}\right)+1\right]=\nonumber\\
&=2\rho'\partial_{\rho'}\sum^{\infty}_{d=0}\prod^{d+1}_{i=1}\sum^{\infty}_{h_{i}=0}\sum^{h_{i}}_{f_{i}=0}v^{d}v^{h_{i}}\rho^{f_{i}}\rho'^{3d+h_{i}+f_{i}+1}\left(\begin{array}{c}h_{i}\\f_{i}\end{array}\right)v .\label{eqd1}\end{aligned}$$ As before, since each of the $d+1$ sums over $h_{i}$ and $f_{i}$ are independent, we can write $N^{FA}_{PF}$ as $$\begin{aligned}
N^{FA,hw}_{PF}(\rho,2)=2\rho'\partial_{\rho'}\sum^{\infty}_{d=0}v^{d+1}\rho'^{3d+1}\left[\sum^{\infty}_{h=0}\sum^{h}_{f=0}v^{h}\rho^{f}\rho'^{h+f}\left(\begin{array}{c}h\\f\end{array}\right)\right]^{d+1} .\end{aligned}$$ Calculating the sum over $h$ and $f$ yields $$\begin{aligned}
N^{FA,hw}_{PF}(\rho,2)=2\rho'\partial_{\rho'}\sum^{\infty}_{d=0}v^{d+1}\rho'^{3d+1}\left[\frac{1}{1-v\rho'\left(1+\rho\rho'\right)}\right]^{d+1} .\label{eqd2}\end{aligned}$$ Calculating the sum over $d$, differentiating with respect to $\rho'$, and finally setting $\rho'=\rho$, yields $$\begin{aligned}
N^{FA,hw}_{PF}(\rho,2)=\frac{2\rho v\left(1+3\rho^{3}v\right)}{\left[1-\rho v\left(1+2\rho^{2}\right)\right]^{2}} .\end{aligned}$$
Derivation of Eq. (\[ka2per\]) {#apkap2}
==============================
We rewrite Eq. (\[ka2per\]) as $$\begin{aligned}
N^{KA,per}_{PF}(\rho,2)=2\rho\partial_{\rho}\sum^{\infty}_{d=0}\left(\rho^{3}v\right)^{d}\prod^{d+1}_{i=1}\sum^{\infty}_{h_{i}=0}\sum^{h_{i}}_{f_{i}=0}v^{h_{i}}\rho^{h_{i}+2f_{i}}\left(\begin{array}{c}h_{i}\\f_{i}\end{array}\right)\rho v .\end{aligned}$$ This is exactly Eq. (\[eqd1\]) with $\rho'=\rho$, and therefore we can use Eq. (\[eqd2\]) $$\begin{aligned}
N^{KA,per}_{PF}(\rho,2)=2\rho\partial_{\rho}\sum^{\infty}_{d=0}v^{d+1}\rho^{3d+1}\left[\frac{1}{1-v\rho\left(1+\rho^{2}\right)}\right]^{d+1} .\end{aligned}$$ Summing over $d$ and differentiating with respect to $\rho$ yields $$\begin{aligned}
N^{KA,per}_{PF}(\rho,2)=\frac{2\rho v\left(1+4\rho^{3}v\right)}{\left[1-\rho v\left(1+2\rho^{2}\right)\right]^{2}} .\end{aligned}$$
[999]{}
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---
abstract: 'We consider the framework of transfer-entropy-regularized Markov Decision Process (TERMDP) in which the weighted sum of the classical state-dependent cost and the transfer entropy from the state random process to the control random process is minimized. Although TERMDP is generally a nonconvex optimization problem, we derive an analytical necessary optimality condition expressed as a finite set of nonlinear equations, based on which an iterative forward-backward computational procedure similar to the Arimoto-Blahut algorithm is proposed. Convergence of the proposed algorithm to a stationary point of the considered TERMDP is established. Applications of TERMDP are discussed in the context of networked control systems theory and non-equilibrium thermodynamics. The proposed algorithm is applied to an information-constrained maze navigation problem, whereby we study how the price of information qualitatively alters the optimal decision polices.'
author:
- 'Takashi Tanaka$^{1}$ Henrik Sandberg$^{2} $ Mikael Skoglund$^{3}$ [^1]'
bibliography:
- 'refs.bib'
title: ' Transfer-Entropy-Regularized Markov Decision Processes '
---
Introduction
============
*Transfer entropy* [@schreiber2000measuring] is an information-theoretic measure of directional information flow among interdependent random processes. It can be viewed as a variation of *directed information*, a concept that has been used in information-theory literature for the analysis of communication systems with feedback [@marko1973bidirectional; @massey1990causality; @kramer1998directed]. In recent years, these quantities have been used for causality analysis in a broad range of applications in science and engineering [@amblard2011directed]. To our knowledge, the concept equivalent to directed information first appeared in [@gourieroux1987kullback], where the quantity is termed *Kullback causality measure*. It appears that the concepts of transfer entropy [@schreiber2000measuring], directed information [@massey1990causality], and Kullback causality measure [@gourieroux1987kullback] are introduced independently in distinct literatures.
In this paper, we consider the mathematical framework of transfer-entropy-regularized Markov Decision Process (TERMDP), in which we seek a causal decision-making policy that minimizes the weighted sum of the classical state-dependent cost and the cost associated with transfer entropy from the state to the control actions. In contrast to the standard MDP formulation [@puterman2014markov], TERMDP penalizes information flow from the underlying state random process to the control random process. Consequently, TERMDP promotes “information-frugal” decision policies, thereby allowing control actions taken by the decision-maker to be statistically less dependent on underlying Markovian dynamics. This is often a favorable property in various real-time decision-making scenarios in which information acquisition, processing, and transmission are costly operations for the decision-maker.
Our own interest in TERMDP stemmed from causal [@neuhoff1982] and zero-delay [@gorbunov1973nonanticipatory; @tatikonda2004; @derpich2012; @tanaka2014semidefinite] source coding problems and their applications to networked control systems [@tatikonda2004; @silva2011framework; @extendedversion]. To our knowledge, directed information was first used in the rate-distortion problem in [@weissman2003] and [@venkataramanan2007source] in the context of competitive prediction and rate-distortion with feedforward, respectively. Directed information appeared in networked control systems theory in [@elia2004bode; @silva2011framework; @yuksel2012characterization; @zaidi2014stabilization]. A rate-distortion problem with feedback from the reproduction to the source and with minimum directed information was first formulated in [@charalambous2014optimization] and [@stavrou2015information]. The problem formulated in [@charalambous2014optimization] and [@stavrou2015information] is fairly general and includes as special cases TERMDP and other information-constrained optimal control problems considered in the past (e.g., LQG control with directed information cost [@tanaka2015lqg]). However, algorithms for solving TERMDP, which is the focus of this paper, have not been studied in the literature.
Problem formulations similar to TERMDP can be found in broader research contexts ranging from economics to neuroscience. In particular, information-theoretic costs have been commonly used in optimal control problems to account for the *bounded rationality* of decision-makers in various contexts. For instance, [@sims2003implications] considers an optimal control problem where Shannon’s mutual information is introduced as an attention cost, from which *rationally inattentive* behaviors of decision-makers in macroeconomic contexts are deduced. Comprehensive analysis of the rational inattention model can be found in [@shafieepoorfard2016rationally]. The reference [@creutzig2009past] proposes the *past-future information-bottleneck* approach to identify information-theoretic characterization of efficient decision-makers. The references [@tishby2010information] and [@rubin2012trading] consider a model of information acquisition and processing cost for decision-makers based on the idea of information-to-go, which is calculated using an appropriate Kullback-Leibler divergence function. Information-theoretic bounded rationality and its analogy to thermodynamics are discussed in [@ortega2013thermodynamics]. We will further explore this connection in Section \[secmaxwell\], where a connection between TERMDP and the generalized second law of thermodynamics [@ito2013information] will be discussed.
While we study algorithmic aspects of TERMDP, it is notable that there is a mathematical similarity between TERMDP and the so-called KL control (also known as the *linearly solvable MDP*) [@todorov2007linearly; @theodorou2010generalized; @dvijotham2012unifying], for which efficient solution algorithms are well-known [@todorov2007linearly]. However, as we will discuss shortly, there are some important differences between KL control problems and TERMDP that make TERMDP computationally more challenging. For instance, unlike the KL control, TERMDP is generally nonconvex with respect to the space of randomized policies.
To address these computational challenges, the goal of this paper is to propose an efficient iterative algorithm to find a stationary point (a locally optimal solution candidate) of the given TERMDP. As the first technical contribution, we derive a necessary optimality condition expressed as a set of nonlinear equations involving a finite number of variables. This result recovers, and partly strengthens, results obtained in prior work[@charalambous2014optimization]. As the second contribution, we propose a forward-backward iterative algorithm that can be viewed as a generalization of the Arimoto-Blahut algorithm [@blahut1972computation; @arimoto1972algorithm] to solve the optimality condition numerically. Observing that the proposed algorithm belongs to the class of block coordinate descent (BCD) algorithms, we show that the algorithm converges to a stationary point of the given TERMDP. The proposed algorithm is different from the transfer entropy *maximization* algorithm considered in [@naiss2013extension], since our algorithm is for transfer entropy *minimization*. The algorithm in [@naiss2013extension] can be viewed as a generalization of the Arimoto-Blahut “capacity algorithm” in [@blahut1972computation], while our proposed algorithm can be viewed as a generalization of the Arimoto-Blahut “rate-distortion algorithm” in [@blahut1972computation].
To demonstrate potential applications of the TERMDP framework, we discuss two different research disciplines to which the TERMDP formulation is relevant. The first is the aforementioned context of networked control systems theory, where transfer entropy has been used as a proxy for the data rate on communication channels. In particular, we show that solving TERMDP provides the fundamental trade-off between the achievable control performance and the required data rate at which the sensor data is fed back to the controller. The second discipline is non-equilibrium thermodynamics, where there has been renewed interest in generalization of the second law of thermodynamics using the transfer entropy concept [@ito2013information]. We show that TERMDP can be interpreted as the problem of operating thermal engines at a nonzero work rate near the fundamental limitation of the second law of thermodynamics.
This paper is organized as follows: in Section \[secformulation\], the TERMDP framework is formally introduced. Some mathematical preliminaries are summarized in Section \[secprelim\], including structural properties of TERMDP solutions. Section \[secmain\] summarizes the main results in our paper. Derivation of the main results are summarized in Section \[secderivation\]. Section \[secinterpret\] discusses application of the TERMDP framework. A numerical demonstration of the proposed algorithm is presented in Section \[secnum\]. Open problems and future work are summarized in Section \[secsummary\].
The following notation will be used in this paper. If $\{x_t\}$ is a sequence, a subsequence $(x_k, x_{k+1}, ... , x_l)$ is denoted by $x_k^l$. We also write $x^t\triangleq (x_1, x_2, ... , x_t)$. Upper case symbols such as $X$ are used to represent random variables, while lower case symbols such as $x$ are used to represent a specific realization. We use the natural logarithm $\log(\cdot)=\log_e (\cdot)$ throughout the paper.
Problem formulation {#secformulation}
===================
The TERMDP is formulated upon the standard Markov Decision Process (MDP) formalism [@puterman2014markov] defined by a time index $t=1, 2, ... , T$, state space $\mathcal{X}_t$, action space $\mathcal{U}_t$, transition probability $p_{t+1}(x_{t+1}|x_t, u_t)$, cost functions $c_t: \mathcal{X}_t \times \mathcal{U}_t \rightarrow \mathbb{R}$ for each $t=1, 2, ... , T$ and $c_{T+1}: \mathcal{X}_{T+1} \rightarrow \mathbb{R}$. For simplicity, we assume that both state space $\mathcal{X}_t$ and action space $\mathcal{U}_t$ are finite. The decision policy to be synthesized can be non-deterministic and history-dependent in general, and is represented by a conditional probability distribution: $$\label{eqpolicyinf}
q_t(u_t|x^t, u^{t-1}).$$ The joint distribution of the state and control trajectories is denoted by $\mu_{t+1}(x^{t+1},u^t)$, which is uniquely determined by the initial state distribution $\mu_1(x_1)$, the state transition probability $p_{t+1}(x_{t+1}|x_t, u_t)$ and the decision policy $q_t(u_t|x^t, u^{t-1})$ by a recursive formula $$\mu_{t+1}(x^{t+1}, u^{t})=p_{t+1}(x_{t+1}|x_t,u_t)q_t(u_t|x^t,u^{t-1})\mu_t(x^t,u^{t-1}). \label{eqmupq}$$ $$\begin{aligned}
&\mu_{t+1}(x^{t+1}, u^{t}) \nonumber \\
&=p_{t+1}(x_{t+1}|x_t,u_t)q_t(u_t|x^t,u^{t-1})\mu_t(x^t,u^{t-1}). \label{eqmupq}\end{aligned}$$ Introduce a stage-additive cost functional: $$\label{equsualmdp}
J(X^{T+1}, U^T) \triangleq \sum_{t=1}^T \mathbb{E} c_t(X_t, U_t)+\mathbb{E} c_{T+1}(X_{T+1}).$$ In TERMDP, we are also concerned with the transfer entropy cost defined as follows.
For nonnegative integers $m$ and $n$, the *transfer entropy of degree* $(m,n)$ is defined by $$I_{m,n}(X^T \rightarrow U^T) \triangleq \sum_{t=1}^T \mathbb{E}\log\frac{\mu_{t+1}(u_t|x_{t-m}^t, u_{t-n}^{t-1})}{\mu_{t+1}(u_t|u_{t-n}^{t-1})}. \label{eqdefte}$$
Using the notation for conditional mutual information [@CoverThomas], the transfer entropy can also be written as $$I_{m,n}(X^T \rightarrow U^T) \triangleq \sum_{t=1}^T I(X_{t-m}^t;U_t|U_{t-n}^{t-1}).$$ When $m=\infty$ and $n=\infty$, coincides with the definition of *directed information* [@massey1990causality]: $$I(X^T \rightarrow U^T) \triangleq \sum_{t=1}^T I(X^t;U_t|U^{t-1}). \label{eqdefdi}$$ The TERMDP, the main problem considered in this paper, is now formulated as follows.
(TERMDP) Let the initial state distribution $\mu_1(x_1)$ and the state transition probability $p_{t+1}(x_{t+1}|x_t, u_t)$ be given, and assume that the joint distribution $\mu_{t+1}(x^{t+1}, u^t)$ is recursively given by . For a fixed constant $\beta>0$, the *Transfer-Entropy-Regularized Markov Decision Processes* is an optimization problem $$\min_{\{q_t(u_t|x^t, u^{t-1})\}_{t=1}^T} J(X^{T+1}, U^T) + \beta I_{m,n}(X^T \rightarrow U^T). \label{eqmainprob}$$
A few remarks are in order regarding this problem formulation. First, the transfer entropy term in is interpreted as an additional cost corresponding to the information transfer from the state random process $X_t$ to the control random process $U_t$. The regularization parameter $\beta>0$ can be thought of as the unit cost incurred per *bit* of information transfer. In the limit of $\beta \rightarrow 0$, the standard MDP formulation is recovered. When $\beta>0$ is large, the optimal decision policy for tends to be more “information frugal” in the sense that the policy generates control actions that are statistically less dependent on the state of the system.
Second, in contrast to the standard MDP (i.e., $\beta=0$) which is known to admit an optimal decision policy that is deterministic and history-independent (Markovian) [@puterman2014markov Section 4.4], the optimal policy for is in general non-deterministic and history-dependent when $\beta>0$. Thus, the cardinality of the solution space we must explore to solve is drastically larger than that of the standard MDP. However, in Proposition \[theooptcond\] below, we show that one can assume without loss of performance a structure of the optimal policy of the form $$\label{eqpolicy}
q_t(u_t|x_t, u^{t-1}_{t-n})$$ rather than . In other words, it is sufficient to consider a policy that is dependent only on the most recent realization of the state and the last $n$ realizations of the control inputs.
Finally, the structure of the problem is similar to that of the KL control (linearly solvable MDP) formulation in [@todorov2007linearly]. In particular, if $(m,n)=(0,0)$, the transfer entropy cost becomes $$\sum\nolimits_{t=1}^T \mathbb{E}\log\frac{\mu_{t+1}(u_t|x_t)}{\mu_{t+1}(u_t)}.$$ In contrast, the KL control considered the KL divergence cost of the form $$\sum\nolimits_{t=1}^T \mathbb{E}\log\frac{\mu_{t+1}(u_t|x_t)}{r_{t+1}(u_t|x_t)}$$ where $r_{t+1}(u_t|x_t)$ is the conditional distribution specified by a predefined “reference” policy. Unlike the KL control, there is no need to specify a predefined reference policy in the TERMDP formulation , which is a convenient property in many applications. Unfortunately, this difference renders nonconvexity as we will observe in Section \[secnonconvex\].
Preliminaries {#secprelim}
=============
In this section, we summarize elements of preliminary results needed to derive our main results in this paper.
Structure of the optimal solution
---------------------------------
We first derive some important structural properties of the optimal decision policy. The obtained structural results will allow us to rewrite the main problem in a simpler form, which will be exploited in the later sections to develop an efficient numerical solution algorithm. The desired structural results can be obtained by applying the dynamic programming principle to . To this end, notice that can be viewed as a $T$-stage optimal control problem where the joint distribution $\mu_t$ is the “state” of the system to be controlled. The state $\mu_t$ is controlled by a multiplicative control action $q_t$ via the state evolution equation . Introduce the value function by $$\begin{aligned}
&V_k\left(\mu_k(x^k, u^{k-1})\right) \triangleq \nonumber\\
&\min_{\{q_t\}_{t=k}^T} \sum_{t=k}^T \left\{ \mathbb{E} c_t(x_t, u_t)+I(X_{t-m}^t;U_t|U_{t-n}^{t-1})\right\}. \label{eqvk}\end{aligned}$$ The value function satisfies the Bellman equation $$\begin{aligned}
&V_t\left(\mu_t(x^t, u^{t-1})\right)
=\min_{q_t} \Bigl\{ \mathbb{E}c_t(X_t, U_t) \biggr. \nonumber \\
& \hspace{5ex}\Bigl. + I(X_{t-m}^t;U_t|U_{t-n}^{t-1}) + V_{t+1}(\mu_{t+1}(x^{t+1}, u^{t}))\Bigr\} \label{eqhjb1}\end{aligned}$$ for $t=1,2, ... , T$, with the terminal condition $$V_{T+1}\left(\mu_{T+1}(x^{T+1}, u^{T})\right) =\mathbb{E}^{\mu_{T+1}} c_{T+1}(X_{T+1}). \label{eqterminal}$$ The next proposition summarizes key structural results.
\[propstructure\] For the optimization problem and its dynamic programming formulation –, the following statements hold for each $k=1, 2, ... , T$.
- For each policy sequence $\{q_t(u_t|x^t, u^{t-1})\}_{t=k}^T$, there exists a policy sequence of the form $\{q'_t(u_t|x_t, u_{t-n}^{t-1})\}_{t=k}^T$ such that the value of the objective function in the left hand side of attained by $\{q'_t\}_{t=k}^T$ is less than or equal to the cost attained by $\{q_t\}_{t=k}^T$.
- If the policy at time step $k$ is of the form $q'_k(u_k|x_k, u_{k-n}^{k-1})$, the identity $I(X_{k-m}^k;U_k|U_{k-n}^{k-1})=I(X_k;U_k|U_{k-n}^{k-1})$ holds.
- The value function $V_k(\mu_k(x^k, u^{k-1}))$ depends only on the marginal distribution $\mu_k(x_k, u_{k-n}^{k-1})$.
Appendix \[appstructure\].
An implication of Proposition \[propstructure\] (a) is that the search for the optimal policy for the original form of TERMDP can be restricted to the class of policies of the form $q_t(u_t|x_t, u_{t-n}^{t-1})$ without loss of performance. Proposition \[propstructure\] (b) implies that, as far as the policy of the form $q_t(u_t|x_t, u_{t-n}^{t-1})$ is assumed, the problem remains equivalent even after the transfer entropy term $I_{m,n}(X^T\rightarrow U^T)$ is replaced by the transfer entropy of degree $(0, n)$: $$I_{0,n}=\sum\nolimits_{t=1}^T I(X_t;U_t|U_{t-n}^{t-1}).$$ Proposition \[propstructure\] (c) implies that the distribution $\mu_t(x_t, u_{t-n}^{t-1})$, rather than the original $\mu_t(x^t, u^{t-1})$, suffices as the state of the considered problem. This “reduced” state evolves according to $$\begin{aligned}
&\mu_{t+1}(x_{t+1}, u_{t-n+1}^t)=\nonumber \\
&\sum_{\mathcal{X}_t}\sum_{\mathcal{U}_{t-n}}p_{t+1}(x_{t+1}|x_t, u_t)q_t(u_t|x_t, u_{t-n}^{t-1})\mu_t(x_t, u_{t-n}^{t-1}). \label{eqmustate}\end{aligned}$$ Based on these observations, it can be seen that Problem 1 can be solved by solving the following simplified problem:
(Simplified TERMDP) Let the initial state distribution $\mu_1(x_1)$ and the state transition probability $p_{t+1}(x_{t+1}|x_t, u_t)$ be given, and assume that the joint distribution $\mu_t(x_t, u_{t-n}^{t-1})$ is recursively given by . For a fixed constant $\beta>0$, the *simplified TERMDP* is an optimization problem $$\min_{\{q_t(u_t|x_t, u_{t-n}^{t-1})\}_{t=1}^T} J(X^{T+1}, U^T) + \beta I_{0,n}(X^T \rightarrow U^T). \label{eqsimpleprob}$$
In particular, any locally optimal solution to corresponds to a locally optimal solution to , and that a globally optimal solution to corresponds to a globally optimal solution to . For this reason, in what follows, we will develop an algorithm that solves the simplified TERMDP rather than the original TERMDP .
Proposition \[propstructure\] implies that an optimal solution to both the original and simplified TERMDP can be found by solving the Bellman equation $$\begin{aligned}
&V_t\left(\mu_t(x_t, u_{t-n}^{t-1})\right)
=\min_{q_t} \Bigl\{ \mathbb{E}c_t(X_t, U_t) \biggr. \nonumber \\
& \hspace{5ex}\Bigl. + I(X_t;U_t|U_{t-n}^{t-1}) + V_{t+1}(\mu_{t+1}(x_{t+1}, u_{t-n+1}^{t}))\Bigr\} \label{eqhjb2}\end{aligned}$$ with the state transition rule and the terminal condition $$V_{T+1}\left(\mu_{T+1}(x_{T+1}, u_{T-n+1}^{T})\right) =\mathbb{E}^{\mu_{T+1}} c_{T+1}(X_{T+1}). \label{eqterminal2}$$ Notice that the Bellman equation is simpler than the original form . However, solving remains computationally challenging as the right hand side of involves a nonconvex optimization problem. In Section \[secnonconvex\], we present a simple numerical example in which this nonconvexity is clearly observed.
Transfer entropy and directed information {#secbound}
-----------------------------------------
In some applications (e.g., networked control systems, see Section \[secncs\]), we are interested in with directed information ($m=\infty$ and $n=\infty$), even though solving such a problem is often computationally intractable. In such applications, approximating directed information with transfer entropy with finite degree is a natural idea, and we are interested in the consequence of this approximation.
\[propfinitete\] For any fixed decision policy of the form $q_t(u_t|x_t, u_{t-n}^{t-1})$, $t=1, 2, ... , T$, we have[^2] $$I_{0,n}\geq
I_{0,n+1}\geq \cdots \geq
I_{0,\infty}.$$
See Appendix \[apptemonotone\].
The following chain of inequalities shows that the optimal value of with any finite $n$ provides an upper bound on the optimal value of with $(m,n)=(\infty, \infty)$.
$$\begin{aligned}
& \min_{\{q_t(u_t|x^t, u^{t-1})\}_{t=1}^T} J(X^{T+1}, U^T) + \beta I_{\infty, \infty} \nonumber \\
&= \min_{\{q_t(u_t|x_t, u^{t-1})\}_{t=1}^T} J(X^{T+1}, U^T)+ \beta I_{\infty, \infty} \label{eqdivste1} \\
&= \min_{\{q_t(u_t|x_t, u^{t-1})\}_{t=1}^T} J(X^{T+1}, U^T) + \beta I_{0,\infty} \label{eqdivste2} \\
&\leq \min_{\{q_t(u_t|x_t, u_{t-n}^{t-1})\}_{t=1}^T} J(X^{T+1}, U^T)+ \beta I_{0,\infty} \label{eqdivste3} \\
&\leq \min_{\{q_t(u_t|x_t, u_{t-n}^{t-1})\}_{t=1}^T} J(X^{T+1}, U^T) + \beta I_{0,n}. \label{eqdivste4}\end{aligned}$$
Equalities and follows from Proposition \[propstructure\] (a) and (b), respectively. The inequality is trivial since any policy of the form $q_t(u_t|x_t, u_{t-n}^{t-1})$ is a special case of the policy of the form $q_t(u_t|x_t, u^{t-1})$. The final inequality is due to Proposition \[propfinitete\].
Rate-distortion theory and Arimoto-Blahut Algorithm {#secrd}
---------------------------------------------------
In the special case with $T=1$, $n=0$, $\beta=1$ and $c_{T+1}(\cdot)=0$, the optimization problem becomes $$\label{eqrdprob}
\min_{q(u|x)} \mathbb{E}c(x,u)+I(X;U)$$ where a probability distribution $p(x)$ on $\mathcal{X}$ is given. In this special case, it can be shown that is a convex optimization problem. The solution to this problem is well-known, and plays an important role in rate-distortion theory [@CoverThomas].
\[propext\] A conditional distribution $q^*(u|x)$ is a globally optimal solution to if and only if it satisfy the following condition $p(x)$-almost everywhere:
\[eqopt\] $$\begin{aligned}
q^*(u|x)&=\frac{\nu^*(u)\exp\left\{-c(x,u)\right\}}{\sum_{\mathcal{U}}\nu^*(u)\exp\left\{-c(x,u)\right\}} \label{eqopt1} \\
\nu^*(u)&=\sum\nolimits_{\mathcal{X}}p(x)q^*(u|x). \label{eqopt2}\end{aligned}$$
This result is standard and hence the proof is omitted. See [@petersen2012robust Appendix A] and [@theodorou2015nonlinear] for relevant discussions.
Condition is required only $p(x)$-almost everywhere since for $x$ such that $p(x)=0$, $q^*(u|x)$ can be chosen arbitrarily. Commonly, the denominator in is called the *partition function*: $$\phi^*(x)\triangleq \sum\nolimits_{\mathcal{U}}\nu^*(u)\exp\left\{-c(x,u)\right\}.$$ By substitution, it is easy to show that the optimal value of can be written in terms of $\nu^*(u)$ as $$-\sum\nolimits_{\mathcal{X}}p(x)\log \left\{\sum\nolimits_{\mathcal{U}}\nu^*(u) \exp\{-c(x,u)\}\right\},$$ or more compactly as $\mathbb{E}^{p(x)}\{-\log \phi^*(X)\}$. This quantity is often referred to as the free energy [@parrondo2015thermodynamics; @theodorou2012relative].
The Arimoto-Blahut algorithm is an iterative algorithm to find the solution $q^*(u|x)$ satisfying numerically. It is based on the alternating updates:
\[eqarimotoblahut\] $$\begin{aligned}
\nu^{(k)}(u)&=\sum\nolimits_{\mathcal{X}}p(x)q^{(k-1)}(u|x) \\
q^{(k)}(u|x)&= \frac{\nu^{(k)}(u)\exp\{-c(x,u)\}}{\sum_{\mathcal{U}}\nu^{(k)}(u) \exp\{-c(x,u)\}}.\end{aligned}$$
The algorithm is first proposed for the computation of channel capacity [@arimoto1972algorithm] and for the computation of rate-distortion functions [@blahut1972computation]. Clearly, the optimal solution $(q^*, \nu^*)$ is a fixed point of the algorithm . Under a mild assumption, convergence of the algorithm is guaranteed; see [@arimoto1972algorithm; @csiszar1974computation; @tseng2001convergence]. The main algorithm we propose in this paper to solve the simplified TERMDP can be thought of as a generalization of the standard Arimoto-Blahut algorithm.
Block Coordinate Descent Algorithm
----------------------------------
The Arimoto-Blahut algorithm can be viewed as a block Coordinate Descent (BCD) algorithm applied to a special class of objective functions. In this subsection, we summarize elements of the BCD method and a version of its convergence results that is relevant to the our analysis. Consider the problem
\[eqbcdprob\] $$\begin{aligned}
\min & \hspace{2ex} f(x) \\
\text{s.t. } & \hspace{2ex} x\in X=X_1\times X_2 \times ... \times X_N\end{aligned}$$
where the feasible set $X$ is the Cartesian product of closed, nonempty and convex subsets $X_i \subseteq \mathbb{R}^{n_i}$ for $i=1, 2, ... , N$, and the function $f: \mathbb{R}^{n_1+...+n_N}\rightarrow \mathbb{R}\cup \{\infty\}$ is continuously differentiable on the level set $\{x\in X: f(x)\leq f(x^{(0)})\}$, where $x^{(0)}\in X$ is a given initial point. We call $x^*\in X$ a *stationary point* for if it satisfies $\nabla f(x^*)^\top (y-x^*)\geq 0$ for every $y\in X$, where $\nabla f(x^*)$ is the gradient of $f$ at $x^*$. If $f$ is convex, every stationary point is a global minimizer of $f$. The BCD algorithm for is defined by the following cyclic update rule:
\[eqbcd\] $$\begin{aligned}
x_1^{(k)} &\in \operatorname*{arg\,min}_{x_1} f(x_1, x_2^{(k-1)}, ... , x_N^{(k-1)}) \\
x_2^{(k)}&\in \operatorname*{arg\,min}_{x_2} f(x_1^{(k)}, x_2, x_3^{(k-1)}, ... , x_N^{(k-1)}) \\
&\hspace{5ex}\cdots \nonumber \\
x_N^{(k)}&\in \operatorname*{arg\,min}_{x_N} f(x_1^{(k)}, x_2^{(k)}, ... , x_N).\end{aligned}$$
A number of sufficient conditions for the convergence of the BCD algorithm are known in the literature (e.g., [@bertsekas2016nonlinear; @grippo2000convergence; @tseng2001convergence] and references therein). For instance, if $f$ is pseudoconvex and has compact level sets, then every limit point of the sequence $\{x^{(k)}\}$ generated by the BCD algorithm is a global minimizer of $f$ [@grippo2000convergence Proposition 6]. This result can be applied to show the global convergence of the Arimoto-Blahut algorithm , simply by noticing that the objective function in can be written as a convex function of $\nu$ and $q$ as $$f(\nu, q)=\sum_{\mathcal{X},\mathcal{U}}p(x)q(u|x)\left(c(x,u)+\log\frac{q(u|x)}{\nu(u)}\right)$$ and that is equivalent to the BCD update rule . In the absence of the convexity assumption on $f$, it is typically required that each coordinate-wise minimization is *uniquely* attained in order to guarantee that every limit point of the BCD algorithm is a stationary point [@bertsekas2016nonlinear Proposition 2.7.1]. The counterexample by Powell [@powell1973search] with $N=3$ shows that the uniqueness of the coordinate-wise minimizer is a strict requirement in general. Unfortunately, the generalized version of the Arimoto-Blahut algorithm presented in this paper for the TERMDP is a BCD algorithm applied to a nonconvex objective function, where the uniqueness of the coordinate-wise minimizer cannot be guaranteed. Thus, none of the above results are applicable to prove its convergence.
Fortunately, the requirement of the uniqueness of the coordinate-wise minimizer can be relaxed when $N=2$ (two-block BCD algorithms). Here, we reproduce this result, which is due to [@grippo2000convergence Corollary 2] and [@tseng2001convergence Theorem 4.2 (c)].
\[lembcd\] Consider the problem with $N=2$, and suppose that the sequence $\{x^{(k)}\}$ generated by the two-block BCD algorithm has limit points. Then, every limit point $x^*$ of $\{x^{(k)}\}$ is a stationary point of the problem .
------------------------------------------------------------------------
\[eqoptcond\] $$\begin{aligned}
\mu^*_{t+1}(x_{t+1}, u^{t}_{t-n+1})&=\sum_{\mathcal{X}_t}\sum_{\mathcal{U}_{t-n}} p_{t+1}(x_{t+1}|x_t,u_t) q^*_t(u_t|x_t, u^{t-1}_{t-n})\mu^*_t(x_t, u^{t-1}_{t-n}) \label{eqoptcond1} \\
\nu^*_t(u_t|u^{t-1}_{t-n})&=\sum_{\mathcal{X}_t} q^*_t(u_t|x_t, u^{t-1}_{t-n})\mu^*_t(x_t|u_{t-n}^{t-1}), \;\; \mu_t(u^{t-1}_{t-n})\text{-almost everywhere }\label{eqoptcond2} \\
\rho^*_t(x_t, u_{t-n+1}^t)&= c_t(x_t,u_t)- \sum_{{\mathcal{X}}_{t+1}} p_{t+1}(x_{t+1}|x_t,u_t)\log \phi^*_{t+1}(x_{t+1}, u_{t-n+1}^{t}) \label{eqoptcond3} \\
\phi^*_t(x_t, u_{t-n}^{t-1}) &= \sum_{\mathcal{U}_t}\nu^*_t(u_t|u^{t-1}_{t-n})\exp \left\{-\rho^*_t(x_t, u_{t-n+1}^t)\right\} \label{eqoptcond4} \\
q_t^*(u_t|x_t, u_{t-n}^{t-1}) &=\frac{\nu^*_t(u_t|u^{t-1}_{t-n})\exp \left\{-\rho^*_t(x_t, u_{t-n}^t)\right\}}{\phi^*_t(x_t, u_{t-n}^{t-1})}, \;\; \forall (x_t, u_{t-n}^{t-1}) \text{ such that } \mu_t(x_t, u_{t-n}^{t-1})>0 \label{eqoptcond5}\end{aligned}$$
------------------------------------------------------------------------
Main Results {#secmain}
============
Necessary Optimality Condition
------------------------------
As the first part of the main technical contribution of this paper, we show that a necessary optimality condition for the simplified TERMDP is given by the nonlinear condition in terms of $(\mu^*, \nu^*, \rho^*, \phi^*, q^*)$.
\[theooptcond\] For any locally optimal solution $\{q_t^*\}_{t=1}^T$ to , there exist variables $\{\mu_{t+1}^*, \nu_t^*, \rho_t^*, \phi_t^*\}_{t=1}^T$ satisfying the set of nonlinear equations together with the initial condition $\mu_1^*(x_1)=p_1(x_1)$ and the terminal condition $\phi_{T+1}^*(x_{T+1}, u_{T-n+1}^{T})\triangleq \exp\{-c_{T+1}(x_{T+1})\}$.
Since an analytical expression is available for the the necessary optimality condition, one can apply various numerical methods to solve to find an optimal solution candidate. Unfortunately, it will soon be shown that the optimality condition is only necessary in general. Since is a nonlinear condition, it is possible that admits multiple distinct solutions, some of which may corresponds to local minima and saddle points of the simplified TERMDP . Theorem \[theooptcond\] is consistent with the previously obtained results in [@charalambous2014optimization] and [@stavrou2015information]. However, the result is refined in by reflecting the underlying Makovian structure of the simplified TERMDP .
------------------------------------------------------------------------
(Forward-Backward Arimoto-Blahut Algorithm)
\[alg1\]
------------------------------------------------------------------------
- **Initialize**
&q\_t\^[(0)]{}(u\_t|x\_[t-m]{}\^t, u\_[t-n]{}\^[t-1]{}) t=1,2, ... , T;\
&\_[T+1]{}\^[(k)]{}(x\_[T+1-m]{}\^[T+1]{}, u\_[T+1-n]{}\^[T]{}){-C\_[T+1]{}(x\_[T+1]{})} k=1,2, ..., K; &&
- **For** $k=1, 2, ... , K$ (until convergence) **do**
- **For** $t=1, 2, ... , T-1, T$ (forward path) **do**
&\_[t+1]{}\^[(k)]{}(x\_[t-m+1]{}\^[t+1]{}, u\_[t-n+1]{}\^t) =\_[\_[t-m]{}]{}\_[\_[t-n]{}]{}p\_[t+1]{}(x\_[t+1]{}|x\_t,u\_t) q\_t\^[(k-1)]{}(u\_t|x\_[t-m]{}\^t, u\_[t-n]{}\^[t-1]{})\_t\^[(k)]{}(x\_[t-m]{}\^t, u\_[t-n]{}\^[t-1]{});\
&\_t\^[(k)]{}(u\_t|u\_[t-n]{}\^[t-1]{})=\_[\_[t-m]{}\^t]{} q\_t\^[(k-1)]{}(u\_t|x\_[t-m]{}\^t, u\_[t-n]{}\^[t-1]{})\_t\^[(k)]{}(x\_[t-m]{}\^t|u\_[t-n]{}\^[t-1]{}); &&
- **End**
<!-- -->
- **For** $t=T, T-1, ... , 2, 1$ (backward path) **do**
&\^[(k)]{}\_t(x\_[t-m]{}\^t, u\_[t-n]{}\^t)= c\_t(x\_t,u\_t) - \_[\_[t+1]{}]{} p\_[t+1]{}(x\_[t+1]{}|x\_t,u\_t) \^[(k)]{}\_[t+1]{}(x\_[t-m+1]{}\^[t+1]{}, u\_[t-n+1]{}\^[t]{});\
&\^[(k)]{}\_t(x\_[t-m]{}\^t, u\_[t-n]{}\^[t-1]{}) = \_[\_t]{}\^[(k)]{}\_t(u\_t|u\^[t-1]{}\_[t-n]{}){-\^[(k)]{}\_t(x\_[t-m]{}\^t, u\_[t-n]{}\^t)};\
&q\_t\^[(k)]{}(u\_t|x\_[t-m]{}\^t, u\_[t-n]{}\^[t-1]{}) =; &&
- **End**
- **End**
- **Return** $q_t^{(K)}(u_t|x_{t-m}^t, u_{t-n}^{t-1}), t=1, 2, ... , T$;
------------------------------------------------------------------------
\[alg1\]
Initialize:
- $q_t^{(0)}(u_t|x_t, u_{t-n}^{t-1})>0$ for $t=1,2, ... , T$;
- $\phi_{T+1}^{(k)}(x_{T+1}, u_{T+1-n}^{T})\triangleq \exp\{-c_{T+1}(x_{T+1})\}$ for $k=1,2, ..., K $;
Return $q_t^{(K)}(u_t|x_t, u_{t-n}^{t-1})$
Forward-Backward Arimoto-Blahut Algorithm
-----------------------------------------
As the second part of our technical contribution, we propose an iterative algorithm to solve and show its convergence. Notice that the optimality condition is a set of coupled nonlinear equations with respect to the unknowns $(\mu^*, \nu^*, \rho^*, \phi^*, q^*)$. To solve numerically, we classify the five equations into two groups. Equations and form the first group (characterizing variables $\mu^*$ and $\nu^*$), and equations - form the second group (characterizing variables $\rho^*$, $\phi^*$, and $q^*$). Observe that if the variables $(\rho^*, \phi^*, q^*)$ are known, then the first set of equations, which can be viewed as the Kolmogorov forward equation, can be solved forward in time to compute $(\mu^*, \nu^*)$. Conversely, if the variables $(\mu^*, \nu^*)$ are known, then the second set of equations, which can be viewed as the Bellman backward equation, can be solved backward in time to compute $(\rho^*, \phi^*, q^*)$. Hence, to compute these unknowns simultaneously, the following boot-strapping method is natural: first, the forward computation is performed using the current best guess of the second set of unknowns, and then the backward computation is performed using the updated guess of the first set of unknowns. The forward-backward iteration is repeated until convergence. The proposed algorithm is summarized in Algorithm \[alg1\].
Algorithm \[alg1\] is a generalization of the standard Arimoto-Blahut algorithm , which can be recovered as special case with $T=1$, $m=n=0$ and $C_{T+1}(\cdot)=0$.
\[theoconvergefbaba\] Every limit point of the sequence $\{\mu^{(k)}, \nu^{(k)}, \rho^{(k)}, \phi^{(k)}, q^{(k)}\}$ generated by Algorithm \[alg1\] satisfies .
The following is a rough estimate of the number of arithmetic operations as a function of $T, m$ and $n$, needed to perform a single forward-backward path in Algorithm \[alg1\]. For simplicity, assume that the state and control spaces have time-invariant cardinalities of $|{\mathcal{X}}|$ and $|{\mathcal{U}}|$, respectively. It can be seen from the $\mu^{(k)}$-update rule in Algorithm \[alg1\] that each data entry of $\mu^{(k)}$ requires arithmetic operations proportional to $|{\mathcal{X}}||{\mathcal{U}}|$ to be updated. Since $\mu^{(k)}$ has $T|{\mathcal{X}}||{\mathcal{U}}|^n$ data entries, it requires $O(T|{\mathcal{X}}|^2|{\mathcal{U}}|^{n+1})$ operations in total. Similar analysis can be repeated for all variables (Table \[tab:arithmetic\]). Overall, it can be concluded that the number of arithmetic operations per iteration is $O(T|{\mathcal{X}}|^2|{\mathcal{U}}|^{n+1})$. Hence, it is linear in $T$, grows exponentially with $n$, and does not depend on $m$.
Variable Number of entries Operations per entry Operations per iteration
-------------- ----------------------------------------- ------------------------------------- ----------------------------------------------
$\mu^{(k)}$ $T|{\mathcal{X}}||{\mathcal{U}}|^n$ $O(|{\mathcal{X}}||{\mathcal{U}}|)$ $O(T|{\mathcal{X}}|^2|{\mathcal{U}}|^{n+1})$
$\nu^{(k)}$ $T|{\mathcal{U}}|^{n+1}$ $O(|{\mathcal{X}}|)$ $O(T|{\mathcal{X}}||{\mathcal{U}}|^{n+1})$
$\rho^{(k)}$ $T|{\mathcal{X}}||{\mathcal{U}}|^{n+1}$ $O(|{\mathcal{X}}|)$ $O(T|{\mathcal{X}}|^2|{\mathcal{U}}|^{n+1})$
$\phi^{(k)}$ $T|{\mathcal{X}}||{\mathcal{U}}|^{n}$ $O(|{\mathcal{U}}|)$ $O(T|{\mathcal{X}}||{\mathcal{U}}|^{n+1})$
$q^{(k)}$ $T|{\mathcal{X}}||{\mathcal{U}}|^{n+1}$ $O(1)$ $O(T|{\mathcal{X}}||{\mathcal{U}}|^{n+1})$
: Estimate of arithmetic operation counts.
\[tab:arithmetic\]
Nonconvexity {#secnonconvex}
------------
Due to the nonconvexity of the value functions in , the optimality condition is only necessary in general. In fact, depending on the initial condition, Algorithm \[alg1\] can converge to different stationary points corresponding to local minima and saddle points of the considered TERMDP . To demonstrate this, consider a simple problem instance of the TERMDP with $T=2$, $n=0$, $\mathcal{X}=\{0, 1\}$, $\mathcal{U}=\{0, 1\}$, $$c_t(x_t, u_t)=\begin{cases}
0 & \text{ if } u_t=x_t \\
1 & \text{ if } u_t\neq x_t
\end{cases}$$ for $t=1, 2$, and $c_3(x_3)\equiv 0$. The initial state distribution is assumed to be $\mu_1(x_1=0)=\mu_1(x_1=1)=0.5$, and the state transitions are deterministic in that $$p_{t+1}(x_{t+1}| x_t, u_t)=\begin{cases}
1 & \text{ if } x_{t+1}=u_t \\
0 & \text{ if } x_{t+1}\neq u_t .
\end{cases}$$ To see that this problem has multiple distinct local minima, we solve the Bellman equation numerically by griding the space of probability distributions. Specifically, at $t=2$, the value function $V_2(\mu_2(x_2))$ is computed by solving the minimization $$V_2(\mu_2(x_2))=\min_{q_2(u_2|x_2)} \{\mathbb{E}c_2(x_2, u_2)+I(X_2; U_2)\}.$$ Since $\mu_2(x_2)$ is an element of a single-dimensional probability simplex, it can be parameterized as $\mu_2(x_2=0)=\lambda$ and $\mu_2(x_2=1)=1-\lambda$ with $\lambda\in[0,1]$. For each fixed $\lambda$, the minimization above is solved by the standard Arimoto-Blahut iteration: $$\begin{aligned}
\nu_2^{(k)}(u_2)&=\sum_{x_2=0, 1}\mu_2(x_2)q_2^{(k-1)}(u_2|x_2) \\
\phi_2^{(k)}(x_2)&=\sum_{u_2=0, 1}\nu_2^{(k)}(u_2)\exp\{-c_2(x_2, u_2)\} \\
q_2^{(k)}(u_2|x_2)&=\frac{\nu_2^{(k)}(u_2)\exp\{-c_2(x_2, u_2)\}}{\phi_2^{(k)}(x_2)}.\end{aligned}$$ After the convergence, the value function is computed as $$V_2(\mu_2(x_2))=-\sum_{x_2=0,1} \mu_2(x_2)\log \phi_2^{(k)}(x_2).$$ The value function $V_2(\mu_2(x_2))$ can be plotted as a function of $\lambda$. It turns out that it is a nonconvex function shown in Fig. \[fig:contour\] (Left). After $V_2(\mu_2(x_2))$ is obtained, the Bellman equation at time $t=1$ can be evaluated as $$\label{eqvalf1}
V_1(\mu_1(x_1))=\min_{q_1(u_1|x_1)} \{\mathbb{E}c_1(x_1, u_1)+I(X_1; U_1)+V_2(\mu_2(x_2))\}.$$ Due to the nonconvexity of $V_2(\mu_2(x_2))$, the objective function in the minimization is a nonconvex function of $q_1(u_1|x_1)$. Fig. \[fig:contour\] (Right) shows the objective function in plotted as a function of $q_1$ parameterized by $\theta_0$ and $\theta_1$: $$\begin{aligned}
q_1(u_1=0|x_1=0)&=\theta_0 \\
q_1(u_1=1|x_1=0)&=1-\theta_0 \\
q_1(u_1=0|x_1=1)&=\theta_1 \\
q_1(u_1=1|x_1=1)&=1-\theta_1. \end{aligned}$$ Clearly, it is a nonconvex function, admitting two local minima (A and C) and a saddle point (B). It can be shown that each of them is a fixed point of Algorithm \[alg1\].
![Left: The value function $V_2$. Right: Contour plot of the objective function $\mathbb{E}c_1(x_1, u_1)+I(X_1; U_1)+V_2(\mu_2(x_2))$ in as a function of $\theta_0$ and $\theta_1$. Stationary points A and B are local minima, whereas C is a saddle point. Two sample trajectories of the proposed forward-backward Arimoto-Blahut algorithm (Algorithm \[alg1\]) started with different initial conditions are also shown. It can be shown that A, B and C are all fixed points of Algorithm \[alg1\].[]{data-label="fig:contour"}](contour_plot2.pdf){width="\columnwidth"}
Derivation of Main Results {#secderivation}
==========================
This section summarizes technical details to prove our main results in the previous section.
Preparation
-----------
The first step is to rewrite the objective function in as an explicit function of $q^T$. For each $t=1, 2, ... , T$, let $\mu_t(x_t|u_{t-n}^{t-1})$ be the conditional distribution obtained from $\mu_t(x_t, u_{t-n}^{t-1})$ whenever $\mu_t(u_{t-n}^{t-1})>0$. Define the conditional distribution $\nu_t$ by $$\nu_t(u_t|u_{t-n}^{t-1})=\sum\nolimits_{\mathcal{X}_t}q_t(u_t|x_t, u_{t-n}^{t-1})\mu_t(x_t|u_{t-n}^{t-1}).\label{eqdefnu}$$ When $\mu_t(u_{t-n}^{t-1})=0$, $\nu_t$ is defined to be the uniform distribution on $\mathcal{U}_t$. For each $t=1, 2, ... , T$, we consider $\nu_t$ and $q_t$ as elements of Euclidean spaces, i.e.,
\[eqnuqeuc\] $$\begin{aligned}
\nu_t(u_t|u_{t-n}^{t-1})&\in \mathbb{R}^{|{\mathcal{U}}_{t-n}|\times \cdots \times |{\mathcal{U}}_t|} \label{eqnuqeuc1} \\
q_t(u_t|x_t, u_{t-n}^{t-1})&\in \mathbb{R}^{|{\mathcal{U}}_{t-n}|\times \cdots \times |{\mathcal{U}}_t|\times |{\mathcal{X}}_t|}. \label{eqnuqeuc2}\end{aligned}$$
Since $\nu_t$ and $q_t$ are conditional probability distributions, they must satisfy entry-wise non-negativity (denoted by $\nu_t\geq 0$ and $q_t\geq 0$) and
\[eqnuqconst\] $$\begin{aligned}
&\sum_{u_t\in {\mathcal{U}}_t} \nu_t(u_t|u_{t-n}^{t-1})=1 \; \forall u_{t-n}^{t-1}\in{\mathcal{U}}_{t-n}^{t-1} \label{eqnuqconst1}\\
&\sum_{u_t\in {\mathcal{U}}_t} q_t(u_t|x_{t}, u_{t-n}^{t-1})=1 \; \forall (x_{t}, u_{t-n}^{t-1}) \in {\mathcal{X}}_{t} \times {\mathcal{U}}_{t-n}^{t-1}. \label{eqnuqconst2}\end{aligned}$$
Thus, the feasibility sets for $\nu^T$ and $q^T$ are $$\begin{aligned}
X_\nu=\{\nu^T: \text{\eqref{eqnuqeuc1}, \eqref{eqnuqconst1} and } \nu_t\geq 0 \text{ for every } t=1, 2, ... , T\}; \\
X_q=\{q^T: \text{\eqref{eqnuqeuc2}, \eqref{eqnuqconst2} and } q_t\geq 0 \text{ for every } t=1, 2, ... , T\}.\end{aligned}$$ Using $\mu_t$, $\nu_t$ and $q_t$, the stage-wise cost in can be written as $$\begin{aligned}
\ell_t(\mu_t, \nu_t, q_t) &\triangleq \mathbb{E}c_t(x_t, u_t)+I(X_t;U_t|U_{t-n}^{t-1})\\
&=\sum\nolimits_{{\mathcal{X}}_k}\sum\nolimits_{{\mathcal{U}}_{k-n}^k} \mu_k(x_k, u_{k-n}^{k-1}) q_k(u_k|x_k, u_{k-n}^{k-1}) \\
&\hspace{8ex}\times \left( \!\log \!\frac{q_k(u_k|x_k,u_{k-n}^{k-1})}{\nu_k(u_k|u_{k-n}^{k-1})} \!+\! c_k(x_k,u_k) \!\right)\end{aligned}$$ for $t=1, 2, ... , T$ and $$\begin{aligned}
\ell_{T+1}(\mu_{T+1})&\triangleq \mathbb{E} c_{T+1}(x_{T+1}) \\
&=\sum\nolimits_{\mathcal{X}_{T+1}}\mu_{T+1}(x_{T+1})c_{T+1}(x_{T+1}).\end{aligned}$$ The objective function in can be written as $$\label{eqfvuq}
f(\nu^T, q^T)=\sum\nolimits_{t=1}^T \ell_t(\mu_t, \nu_t, q_t)+\ell_{T+1}(\mu_{T+1}).$$ Although the variables $\nu^T$ and $q^T$ must satisfy the constraint , it will be convenient in the sequel to consider as a function of $\nu^T$ and $q^T$ by temporarily forgetting the equality constraint . This is possible because of the following result, which is essentially due to Blahut [@blahut1972computation Theorem 4(b)], stating that the identity will be automatically satisfied by any local minimizer $(\nu^T, q^T)$ of $f$.
\[lemblahut\] Let $q^T\in X_q$ be fixed. Then
$$\begin{aligned}
\min & \hspace{2ex} f(\nu^T, q^T) \\
\text{s.t. } & \hspace{2ex} \nu^T \in X_\nu\end{aligned}$$
is a convex optimization problem with respect to $\nu^T$, and an optimal solution is given by .
Lemma \[lemblahut\] implies that if $q^{T*}$ is a locally optimal solution to , then there exist a variable $\nu^{T*}$ such that $(\nu^{T*}, q^{T*})$ is a locally optimal solution to
\[eqsimpleprob2\] $$\begin{aligned}
\min & \hspace{2ex} f(\nu^T, q^T) \\
\text{s.t. } & \hspace{2ex} \nu^T\in X_\nu, q^T\in X_q.\end{aligned}$$
Since it turns out that the function $f$ in is coordinate-wise convex, $(\nu^{T*}, q^{T*})$ is also coordinate-wise optimal:
\[eqblockoptimal\] $$\begin{aligned}
\nu^{T*}&\in \operatorname*{arg\,min}\nolimits_{\nu^T \in X_\nu} f(\nu^T, q^{T*}) \label{eqblockoptimal1} \\
q^{T*}&\in \operatorname*{arg\,min}\nolimits_{q^T \in X_q} f(\nu^{T*}, q^T) \label{eqblockoptimal2}.\end{aligned}$$
We will obtain as a necessary condition for . To prove convergence of Algorithm \[alg1\], we will make a key observation that Algorithm \[alg1\] is equivalent to the two-block BCD algorithm applied to :
\[eqbcdfb\] $$\begin{aligned}
\nu^{T(k)}&=\operatorname*{arg\,min}\nolimits_{\nu^T \in X_\nu} f(\nu^T,q^{T(k-1)}) \label{eqbcdfb1} \\
q^{T(k)}&=\operatorname*{arg\,min}\nolimits_{q^T \in X_q} f(\nu^{T(k)}, q^T). \label{eqbcdfb2}\end{aligned}$$
Convergence of Algorithm \[alg1\] will then follows from Lemma \[lembcd\].
Analysis of Algorithm \[alg1\]
------------------------------
We first prove that the backward path is equivalent to computing a coordinate-wise minimizer . To this end, notice that for a fixed sequence $\nu^{T(k)}$, the minimization problem can be viewed as an optimal control problem with respect to the control actions $q^T=(q_1, ... , q_T)$. The next lemma essentially shows that the backward path is solving the minimization by the backward dynamic programming.
\[lemblockq\] Suppose $$\begin{aligned}
&(\nu_1^{(k)}, ... , \nu_T^{(k)}) \in X_\nu, \\
&(q_1^{(k-1)}, ... , q_{\tau-1}^{(k-1)}, q_\tau, q_{\tau+1}^{(k)}, ... , q_T^{(k)}) \in X_q,\end{aligned}$$ and $\{\mu_{t+1}\}_{t=1}^T$ is the sequence of probability measures generated by $(q_1^{(k-1)}, ... , q_{\tau-1}^{(k-1)}, q_\tau, q_{\tau+1}^{(k)}, ... , q_T^{(k)})$ via . Let $\rho_\tau^{(k)}, \phi_\tau^{(k)}$ and $q_\tau^{(k)}$ be the parameters obtained by computing backward in time for $t=T, ... , \tau$. Then, for each $\tau=T, T-1, ... ,1$, the following statements hold:
- The function $$f(\nu_1^{(k)}, ... , \nu_T^{(k)}, q_1^{(k-1)}, ... , q_{\tau-1}^{(k-1)}, q_\tau, q_{\tau+1}^{(k)}, ... , q_T^{(k)})$$ is convex in $q_\tau \geq 0$, and any global minimizer $q_\tau^\circ$ satisfies $q_\tau^\circ=q_\tau^{(k)}$ almost everywhere with respect to $\mu_t(x_t, u_{t-n}^{t-1})$.
- The cost-to-go function under the policy $\{q_t^{(k)}\}_{t=\tau}^T$ is linear in $\mu_\tau$: $$\begin{aligned}
&\sum\nolimits_{t=\tau}^T \ell_t(\mu_t, \nu_t^{(k)}, q_t^{(k)})+\ell_{T+1}(\mu_{T+1})\\
&=-\sum\nolimits_{\mathcal{X}_\tau}\sum\nolimits_{\mathcal{U}_{\tau-n}^{\tau-1}} \mu_\tau(x_\tau, u_{\tau-n}^{\tau-1})\log \phi_\tau^{(k)}(x_\tau, u_{\tau-n}^{\tau-1}).\end{aligned}$$
The proof is by backward induction. For the time step $T$, we have $$\begin{aligned}
&f(\nu_1^{(k)}, ... , \nu_T^{(k)}, q_1^{(k-1)}, ... , q_{T-1}^{(k-1)}, q_T) \nonumber \\
&= \sum\nolimits_{{\mathcal{X}}_T}\sum\nolimits_{{\mathcal{U}}_{T-n}^T} \mu_T(x_T, u_{T-n}^{T-1}) q_T(u_T|x_T, u_{T-n}^{T-1}) \nonumber\\
& \times \!\left(\log\frac{q_T(u_T|x_T, u_{T-n}^{T-1})}{\nu^{(k)}_T(u_T| u_{T-n}^{T-1})}\!+\!\rho_T^{(k)}(x_T, u_{T-n}^T) \right)+\text{const.} \label{eqqtlast}\end{aligned}$$ where “const.” is the term that does not depend on $q_T$. The fact that a minimizer is given by $q_T^{(k)}$ is a consequence of Proposition \[propext\]. This establishes (a) for the time step $\tau=T$. The statement (b) for $\tau=T$ can be directly shown by substituting the expression of $q_\tau^{(k)}$ given by with $t=\tau$ into :
\[eqind\_b\] $$\begin{aligned}
&\ell_T(\mu_T, \nu_T^{(k)}, q_T^{(k)})+\ell_{T+1}(\mu_{T+1})\nonumber \\
&= \sum\nolimits_{{\mathcal{X}}_T}\sum\nolimits_{{\mathcal{U}}_{T-n}^T} \mu_T(x_T, u_{T-n}^{T-1}) q_T^{(k)}(u_T|x_T, u_{T-n}^{T-1}) \nonumber\\
& \hspace{3ex}\times \left(\log\frac{q_T^{(k)}(u_T|x_T, u_{T-n}^{T-1})}{\nu^{(k)}_T(u_T| u_{T-n}^{T-1})}+\rho_T^{(k)}(x_T, u_{T-n}^T) \right) \\
&= \sum\nolimits_{{\mathcal{X}}_T}\sum\nolimits_{{\mathcal{U}}_{T-n}^T} \mu_T(x_T, u_{T-n}^{T-1}) q_T^{(k)}(u_T|x_T, u_{T-n}^{T-1}) \nonumber\\
& \hspace{3ex}\times \left(-\log \phi_T^{(k)}(x_T, u_{T-n}^{T-1})\right) \\
&= -\sum\nolimits_{{\mathcal{X}}_T}\sum\nolimits_{{\mathcal{U}}_{T-n}^{T-1}} \mu_T(x_T, u_{T-n}^{T-1}) \log \phi_T^{(k)}(x_T, u_{T-n}^{T-1}) \nonumber \\
& \hspace{3ex}\times \underbrace{\sum\nolimits_{{\mathcal{U}}_T}q_T^{(k)}(u_T|x_T, u_{T-n}^{T-1})}_{=1}.\end{aligned}$$
To complete the proof, we show that if (a) and (b) hold for the time step $\tau+1$, then they also hold for the time step $\tau$. Since (b) is hypothesized for $\tau+1$, using $\rho_\tau^{(k)}$, it is possible to write $$\begin{aligned}
&f(\nu_t^{(k)}, ... , \nu_T^{(k)}, q_1^{(k-1)}, ... , q_{\tau-1}^{(k-1)}, q_\tau, q_{\tau+1}^{(k)}, ... , q_T^{(k)}) \nonumber \\
&= \sum\nolimits_{{\mathcal{X}}_\tau}\sum\nolimits_{{\mathcal{U}}_{\tau-n}^\tau} \mu_\tau(x_\tau, u_{\tau-n}^{\tau-1}) q_\tau(u_\tau|x_\tau, u_{\tau-n}^{\tau-1}) \nonumber\\
& \times \left(\log\frac{q_\tau(u_\tau|x_\tau, u_{\tau-n}^{\tau-1})}{\nu^{(k)}_\tau(u_\tau| u_{\tau-n}^{\tau-1})}+\rho_\tau^{(k)}(x_\tau, u_{\tau-n}^\tau) \right)+\text{const.} \label{eqqtmid2}\end{aligned}$$ where “const.” is the term that does not depend on $q_\tau$. Proposition \[propext\] is applicable once again to conclude that a minimizer coincides with $q_\tau^{(k)}$ almost everywhere with respect to $\mu_t(x_t, u_{t-n}^{t-1})$. Hence, (a) is established for the time step $\tau$. The statement (b) for $\tau$ can be shown by the direct substitution. The details are similar to .
Proof of main results
---------------------
Based on the results in the previous two subsections, the main theorems in this paper are established as follows.
**Proof of Theorem \[theooptcond\]:** For a given $\{q_t^*\}_{t=1}^T$, we define $\{\mu_t^*\}_{t=1}^T$ via and hence is automatically satisfied. By Lemma \[lemblahut\], for any locally optimal solution $\{q_t^*\}_{t=1}^T$ to , there exist variables $\{\nu_t^*\}_{t=1}^T$ satisfying , such that $\{\nu_t^*, q_t^*\}_{t=1}^T$ is a locally optimal solution to . Since local optimality implies coordinate-wise local optimality, for each $t=1, 2, ... , T$, $q_t^*$ is a local minimizer of $$\label{eqfnuqstar}
f(\nu_1^*, ... , \nu_T^*, q_T^*, ... , q_{t+1}^*, q_t, q_{t-1}^*, ... , q_1^*).$$ However, Lemma \[lemblockq\] is applicable (with $\nu_\tau^{(k)}=\nu_\tau^*$ for $\tau=1, ... , T$, $q_\tau^{(k)}=q_\tau^*$ for $\tau=t+1, ... , T$ and $q_\tau^{(k-1)}=q_\tau^*$ for $\tau=1, ... , t-1$) to conclude that is convex in $q_t\geq 0$ and hence $$q_t^* \in \operatorname*{arg\,min}_{q_t\geq 0} f(\nu_1^*, ... , \nu_T^*, q_T^*, ... , q_{t+1}^*, q_t, q_{t-1}^*, ... , q_1^*).$$ Moreover, Lemma \[lemblockq\] (a) also implies that if the parameters $\{\rho_\tau^*, \phi_\tau^*\}_{\tau=t}^T$ are calculated by - backward in time, then any global minimizer $$q_t^\circ \in \operatorname*{arg\,min}_{q_t\geq 0} f(\nu_1^*, ... , \nu_T^*, q_T^*, ... , q_{t+1}^*, q_t, q_{t-1}^*, ... , q_1^*)$$ satisfies $$q_t^\circ=\frac{\nu^*_t(u_t|u^{t-1}_{t-n})\exp \left\{-\rho^*_t(x_t, u_{t-n}^t)\right\}}{\phi^*_t(x_t, u_{t-n}^{t-1})}$$ $\mu_t$-almost everywhere. Hence must hold.
**Proof of Theorem \[theoconvergefbaba\]:** To show that Algorithm \[alg1\] is equivalent to the block coordinate descent algorithm , observe that
- The update rule is equivalent to ; and
- The update rule is equivalent to .
The fact (a) follows from Lemma \[lemblahut\] and (b) follows from Lemma \[lemblockq\]. Now, notice that is a two-block BCD algorithm to which Lemma \[lembcd\] is applicable. Hence, it can be concluded that every limit point generated by Algorithm \[alg1\] is a stationary point. Notice that every stationary point $(\nu^{T*}, q^{T*})$ is a coordinate-wise stationary point. Since $f(\nu^{T}, q^{T})$ is coordinate-wise convex, $(\nu^{T*}, q^{T*})$ is a local minimizer. Hence, it satisfies , for which condition is necessary.
Interpretations {#secinterpret}
===============
In this section, we discuss two different application of the TERMDP , where the transfer entropy cost is provided with different physical meanings.
Networked Control Systems {#secncs}
-------------------------
The first application is the analysis of networked control systems, where sensors and controllers are placed in geographically separate locations in the control system and hence the sensor data must be transmitted to the controller over a rate-limited communication media. Fig. \[fig:channel\] shows a standard MDP with a discrete-time finite-horizon formalism, except that a decision policy must be realized by a joint design of encoder and decoder, together with an appropriate codebook for discrete noiseless channel. Most generally, assume that an encoder is a stochastic kernel $e_t(w_t|x^t, w^{t-1})$ and a decoder is a stochastic kernel $d_t(u_t|w^t,u^{t-1})$. At each time step, a codeword $w_t$ is chosen from a codebook $\mathcal{W}_t$ such that $|\mathcal{W}_t|=2^{R_t}$. We refer to $R=\sum_{t=1}^T R_t$ as the *rate* of communication in the feedback architecture in Fig. \[fig:channel\]. The next proposition claims that the rate of communication in Fig. \[fig:channel\] is fundamentally lower bounded by the directed information.
\[proplbdi\] Let an encoder and a decoder be any stochastic kernels of the form $e_t(w_t|x^t,w^{t-1})$ and $d_t(u_t|w^t, u^{t-1})$. Then $R\log 2\geq I(X^T\rightarrow U^T)$.
Note that $$\begin{aligned}
R\log 2\! &=\!\sum\nolimits_{t=1}^T R_t \log 2 \\
&\geq \!\sum\nolimits_{t=1}^T \!H(W_t) \\
&\geq \!\sum\nolimits_{t=1}^T \!H(W_t|W^{t-1},U^{t-1}) \\
&\geq \!\sum\nolimits_{t=1}^T \!H(W_t|W^{t-1}\!\!,U^{t-1})\!-\!H(W_t|X^t\!\!,W^{t-1}\!\!,U^{t-1}) \\
&=\!\sum\nolimits_{t=1}^T \!I(X^t;W_t|W^{t-1},U^{t-1}) \\
&\triangleq I(X^T\rightarrow W^T \| U^{T-1}).\end{aligned}$$ The first inequality is due to the fact that entropy of a discrete random variable cannot be greater than its log-cardinality. Notice that a factor $\log 2$ appears since we are using the natural logarithm in this paper. The second inequality holds because conditioning reduces entropy. The third inequality follows since entropy is nonnegative. The last quantity is known as the *causally conditioned directed information* [@kramer2003capacity]. The feedback data-processing inequality [@tanaka2015lqg] $$I(X^T\rightarrow U^T) \leq I(X^T \rightarrow W^T\| U^{T-1})$$ is applicable to complete the proof. $\square$
Proposition \[proplbdi\] provides a fundamental performance limitation of a communication system when both encoder and decoder have full memories of the past. However, it is also meaningful to consider restricted scenarios in which the encoder and decoder have limited memories. For instance:
- The encoder stochastic kernel is of the form $e_t(w_t|x^t_{t-m})$ and the decoder stochastic kernel is of the form $d_t(u_t|w_t, u_{t-n}^t)$; or
- The encoder stochastic kernel is $e_t(w_t|x^t_{t-m}, u_{t-n}^{t-1})$ and the decoder is a deterministic function $u_t=d_t(w_t)$. The encoder has an access to the past control inputs $u_{t-n}^{t-1}$ since they are predictable from the past $w_{t-n}^{t-1}$ because the decoder is a deterministic map.
The next proposition shows that the transfer entropy of degree $(m,n)$ provides a tighter lower bound in these cases.
\[propcom2\] Suppose that the encoder and the decoder have structures specified by (A) or (B) above. Then $$R\log 2 \geq I_{m,n}(X^T\rightarrow U^T).$$
See Appendix \[app1\]. $\square$
By solving the TERMDP with different $\beta>0$, one can draw a trade-off curve between $J(X^{T+1}, U^T)$ and $I(X^T\rightarrow U^T)$. Proposition \[propcom2\] means that this trade-off curve shows a fundamental limitation of the achievable control performance under the given data rate.
The tightness of the lower bounds provided by Propositions \[proplbdi\] and \[propcom2\] (i.e., whether it is possible to construct an encoder-decoder pair such that that data rate matches its lower bound while satisfying the desired control performance) is the natural next question. There has been much research effort on related topics, but this question has not been fully resolved in the literature. In the LQG control setting, the same question has been addressed in [@silva2013characterization; @extendedversion], based on the achievability argument for the corresponding nonanticipative rate-distortion function (see [@1701.06368] and references therein). In these references, it is shown that the conservativeness of the lower bound provided by Proposition \[proplbdi\] is no greater than a small constant.
![MDP over discrete noiseless channel.[]{data-label="fig:channel"}](channel.pdf){width="0.6\columnwidth"}
![MDP over discrete noiseless channel.[]{data-label="fig:channel"}](channel.pdf){width="0.8\columnwidth"}
Maxwell’s demon {#secmaxwell}
---------------
Maxwell’s demon is a physical device that can seemingly violate the second law of thermodynamics, which turns out to be a prototypical thought-experiment that connects statistical physics and information theory [@parrondo2015thermodynamics]. One of the simplest forms of Maxwell’s demon is a device called the Szilard engine. Below, we introduce a potential application of the TERMDP framework to analyze the efficiency of a generalized version of the Szilard engine extracting work at a non-zero rate (in contrast to the common assumption that the engine is operated infinitely slowly).
Consider a single-molecule gas trapped in a box (“engine”) that is immersed in a thermal bath of temperature $T_0$ (Fig. \[fig:szilard\]). The state of the engine at time $t$ is represented by the position and the velocity of the molecule, which is denoted by $X_t\in {\mathcal{X}}$. Assume that the state space is divided into finite cells so that ${\mathcal{X}}$ is a finite set. Also, assume that the evolution of $X_t$ is described by a discrete-time random process.
At each time step $t=0,1, ... , T-1$, suppose that one of the following three possible control actions $U_t$ can be applied: (i) insert a weight-less barrier into the middle of the engine box and move it to the left at a constant velocity $v$ for a unit time, (ii) insert a barrier into the middle of the box and move it to the right at the velocity $v$ for a unit time, or (iii) do nothing. At the end of control actions, the barrier is removed from the engine. We assume that the insertion and removal of the barrier is frictionless and as such do not consume any work. The sequence of operations is depicted in Fig. \[fig:szilard\]. Denote by $p(x_{t+1}|x_t,u_t)$ the transition probability from the state $x_t$ to another state $x_{t+1}$ when control action $u_t$ is applied. By $\mathbb{E}c(X_t,U_t)$ we denote the expected work required to apply control action $u_t$ at time $t$ when the state of the engine is $x_t$.[^3] This quantity is negative if the controller is expected to extract work from the engine. Work extraction occurs when the gas molecule collides with the barrier and “pushes” it in the direction of its movement.
Right before applying a control action $U_t$, suppose that the controller makes (a possibly noisy) observation of the engine state, and thus there is an information flow from $X^t$ to $U^t$. For our discussion, there is no need to describe what kind of sensing mechanism is involved in this step. However, notice that if an error-free observation of the engine state $X_t$ is performed, then the controller can choose a control action such that $\mathbb{E}c(X_t,U_t)$ is always non-positive. (Consider moving the barrier always to the opposite direction from the position of the gas molecule.) At first glance, this seems to imply that one can construct a device that is expected to cyclically extract work from a single thermal bath, which is a contradiction to the Kelvin-Planck statement of the second law of thermodynamics.
![Modified Szilard engine. The controller performs the following steps in a unit time. (a) The controller makes (a possibly noisy) observation of the state $X_t$ of the engine. (b) One of the three possible control actions $U_t$ is applied. (c) Barrier is removed.[]{data-label="fig:szilard"}](szilard.pdf){width="0.6\columnwidth"}
![Modified Szilard engine. The controller performs the following steps in a unit time. (a) The controller makes (a possibly noisy) observation of the state $X_t$ of the engine. (b) One of the three possible control actions $U_t$ (move the barrier to the left or to the right, or do nothing) is applied. (c) At the end of control action, the barrier is removed.[]{data-label="fig:szilard"}](szilard.pdf){width="\columnwidth"}
{width="65.00000%"}
{width="90.00000%"}
{width="\textwidth"}
It is now widely recognized that this paradox (Maxwell’s demon) can be resolved by including the “memory” of the controller into the picture. Recently, a generalized second law is proposed by [@ito2013information], which clarifies the role of transfer entropy. An entire view of the combined engine and memory system is provided by a Bayesian network comprised of $X_t$ and $U_t$ (see [@ito2013information]). Assuming that the free energy change of the engine from $t=0$ to $t=T$ is zero (which is the case when the above sequence of operations are repeated in a cyclic manner with period $T$), the generalized second law [@ito2013information equation (10)] reads $$\label{eqsecondlaw}
\sum_{t=0}^{T-1} \mathbb{E}c(X_t,U_t)+k_BT_0 I(X_0^{T-1}\rightarrow U_0^{T-1}) \geq 0$$ where $k_B$ \[J/K\] is the Boltzmann constant. The above inequality shows that a positive amount of work is extractable (i.e., the first term can be negative), but this is possible only at the expense of the transfer entropy cost (the second term must be positive).[^4] Given a fundamental law , a natural question is how efficient the considered thermal engine can be by optimally designing a control policy $q(u_t|x^t,u^{t-1})$. This can be analyzed by minimizing a term on the left hand side of while fixing the other, and the optimization algorithm considered in this paper for can be used for this purpose.
Numerical experiment {#secnum}
====================
In this section, we apply the proposed forward-backward Arimoto-Blahut algorithm (Algorithm \[alg1\]) to study how the price of information affects the level of information-frugality, which yields qualitatively different decision policies.
Consider a situation in which Alice, whose movements are described by Markovian dynamics controlled by Bob, is traveling through a maze shown in Fig. \[fig:grid\]. Suppose at any given time Alice knows her location in the maze, but she does not know the geometry of the maze. Bob, on the other hand, knows the geometry (including start and goal locations), but observing Alice’s location is costly. We model the problem as an MDP where the state $X_t$ is the cell where Alice is located at time step $t$, and $U_t$ is a navigation instruction given by Bob. The observation cost is characterized by the transfer entropy. We assume five different instructions are possible; $u=N, E, S, W$ and $R$, corresponding to *go north*, *go east*, *go south*, *go west*, and *rest*. The initial state is the cell indicated by “S” in Fig. \[fig:grid\], and the motion of Alice is described by a transition probability $p(X_{t+1}|X_t, U_t)$.
The transition probability is defined by the following rules. At each cell, a transition to the indicated direction occurs w.p. $0.8$ if there is no wall in the indicated direction, while transition to any open directions (directions without walls) occurs w.p. $0.05$ each. With the remaining probability, Alice stays in the same cell. If there is a wall in the indicated direction, or $u=R$, then transition to each open direction occurs w.p. $0.05$, while Alice stays in the same cell with the remaining probability.
At each time step $t=1, 2, ... , T$, the state-dependent cost is defined by $c_t(x_t,u_t)=0$ if $x_t$ is already the target cell indicated by “G” in Fig. \[fig:grid\], and $c_t(x_t,u_t)=1$ otherwise. The terminal cost is $0$ if $x_{T+1}=G$ and $10000$ otherwise. We also consider an information-theoretic cost proportional to the transfer entropy $I_{m,n}(X^T\rightarrow U^T)$. This term can be interpreted as the total amount of information that Bob must acquire about Alice’s location. With some positive weight $\beta$, the overall control problem can be written as .
As shown in Fig. \[fig:grid\], there are two qualitatively different paths from the origin to the target. The path A is shorter than the path B, and hence Bob will try to navigate Alice along path A when no information-theoretic cost is considered (i.e., $\beta=0$). However, navigating along the path A is risky because there are multiple alleys with dead ends. Hence, Bob needs more accurate knowledge about Alice’s location to provide appropriate navigation instructions. The path B is longer, but navigating through it is relatively simple; rough knowledge about Alice’s location is sufficient to provide correct instructions. Hence, it is expected that Bob would try to navigate Alice through A when information is relatively cheap ($\beta$ is small), while he would choose B when information is expensive ($\beta$ is large).
Fig. \[fig:maze\] shows the solutions to the considered problem. Solutions are obtained by iterating Algorithm \[alg1\] sufficiently many times in four different conditions. Each plot shows a snapshot of the state probability distribution $\mu_t(x_t)$ at time $t=25$. Fig. \[fig:maze\] (a) is obtained under the setting that the cost of information is high ($\beta=10$), the planning horizon is long ($T=55$), and the transfer entropy of degree $(m,n)=(0,0)$ is considered. Accordingly, the decision policy is of the form of $q_t(u_t|x_t)$ is considered. It can be seen that with high probability, the agent is navigated through the longer path. In Fig. \[fig:maze\] (b), the cost of information is reduced ($\beta=1$) while the other settings are kept the same. As expected, the solution chooses the shorter path. Fig. \[fig:mi\] shows the time-dependent information usage in (a) and (b); it shows that the total information usage is greater in situation (b) than in (a).
We note that this simulation result is consistent with a prior work [@rubin2012trading], where similar numerical experiments were conducted. Using Algorithm \[alg1\], we can further investigate the nature of the problem. Fig. \[fig:maze\] (c) considers the same setting as in (a) except that the planning horizon is shorter ($T=45$). This result shows that the solution becomes qualitatively different depending on how close the deadline is even if the cost of information is the same. Finally, Fig. \[fig:maze\] (d) considers the case where the transfer entropy has degree $(m,n)=(0,1)$ and the decision policy is of the form of $q_t(u_t|x_t,u_{t-1})$. Although the rest of simulation parameters are unchanged from (a), we observe that the shorter path is chosen in this case. This result demonstrates that the solution to can be qualitatively different depending on the considered degree of transfer entropy costs.
Summary and Future Work {#secsummary}
=======================
In this paper, we considered a mathematical framework of transfer-entropy-regularized Markov Decision Process (TERMDP). The considered problem was given physical interpretations in both engineering (networked control systems) and scientific (Maxwell’s demon) contexts. Based on the dynamic programming argument, we derived structural properties of the optimal solution, and recovered a necessary optimality condition written as a set of coupled nonlinear equations. As the main contribution, the forward-backward Arimoto-Blahut algorithm was proposed to solve the optimality condition numerically. Convergence of the proposed algorithm is established.
The study in this paper is currently restricted to *fully observable* MDPs where decision policies of the form $q_t(u_t|x^t, u^{t-1})$ are to be synthesized. In the future, the result will be extended to *Partially Observable Markov Decision Processes* (POMDPs), where the state process $X_t$ will be only partially observable through a dependent random process $Y_t$, the transfer entropy term will be $I_{m,n}(Y^T\rightarrow U^T)$, and policies of the form $q_t(u_t|y^t, u^{t-1})$ will be synthesized.
While a proof of convergence of Algorithm \[alg1\] was provided, further properties, such as the rate of convergence, were not studied in this paper. For instance, an application of the accelerated Arimoto-Blahut algorithm [@matz2004information] based on *natural gradients* should be considered in the future.
Proof of Proposition \[propfinitete\] {#apptemonotone}
-------------------------------------
For each $n'\geq n$, $$\begin{aligned}
&I_{0,n'}(X^T\rightarrow U^T) =\sum\nolimits_{t=1}^T I(X_t;U_t|U_{t-n'}^{t-1}) \\
&=\sum\nolimits_{t=1}^T H(U_t|U_{t-n'}^{t-1})-H(U_t|X_t,U_{t-n'}^t) \\
&=\sum\nolimits_{t=1}^T H(U_t|U_{t-n'}^{t-1})-H(U_t|X_t,U_{t-n}^t).\end{aligned}$$ In the last step, we used the fact that $H(X|Y,Z)=H(X|Y)$ holds when $X$ and $Z$ are conditionally independent given $Y$. By the structure of $q_t(u_t|x_t, u_{t-n}^{t-1})$, $U_t$ and $(X_t, U_{t-n}^{t-1})$ is conditionally independent of $U_{t-n'}^{t-n-1}$. Now, $$\begin{aligned}
&I_{0,n'}(X^T\rightarrow U^T)-I_{0,n'+1}(X^T\rightarrow U^T) \\
&=\sum\nolimits_{t=1}^T \left(H(U_t|U_{t-n'}^{t-1})-H(U_t|U_{t-n'-1}^{t-1})\right) \geq 0\end{aligned}$$ since entropy never increases by conditioning.
Proof of Proposition \[propstructure\] {#appstructure}
--------------------------------------
Our proof is based on backward induction.
: We prove (a) first. For a given policy $q_T(u_t|x^T, u^{T-1})$, let $
\lambda_T(x^T, u^T)\triangleq q_T(u_t|x^T, u^{T-1})\mu_T(x^T, u^{T-1})
$ be the joint distribution induced by $q_T$, and let $\lambda_T(x_T, u_{T-n}^T)$ and $\lambda_T(x_T, u_{T-n}^{T-1})$ be the marginals of $\lambda_T(x^T, u^T)$. Construct a new policy $q'_T$ by $
q'_T(u_T|x_T, u_{T-n}^{T-1})=\frac{\lambda_T(x_T, u_{T-n}^T)}{\lambda_T(x_T, u_{T-n}^{T-1})}.
$ Let $$\label{eqlambdapdef}
\lambda'_T(x^T, u^T)\triangleq q'_T(u_T|x^T, u^{T-1})\mu_T(x^T, u^{T-1})$$ be the joint distribution induced by $q'_T$. Then, we have $$\label{eqlambdap}
\lambda_T(x_T, u_{T-n}^T)=\lambda'_T(x_T, u_{T-n}^T)$$ by construction, which can be verified easily as $$\begin{aligned}
&\lambda'_T(x_T, u_{T-n}^T)\\
&=\sum\nolimits_{\mathcal{X}^{T-1}}\sum\nolimits_{\mathcal{U}^{T-n-1}} q'_T(u_T|x_T, u_{T-n}^{T-1})\mu_T(x^T, u^{T-1}) \\
&=q'_T(u_T|x_T, u_{T-n}^{T-1})\lambda_T(x_T, u_{T-n}^{T-1}) \\
&=\lambda_T(x_T, u_{T-n}^T).\end{aligned}$$ Now the Bellman equation at $k=T$ reads $$V_T(\mu_T(x^T,u^{T-1}))=\min_{q_T} J_T^c(\lambda_T)+J_T^I(\lambda_T)$$ where $$\begin{aligned}
J_T^c(\lambda_T)&=\mathbb{E}^{\lambda_T, p_{T+1}}\left(c_T(x_T, u_T)+c_{T+1}(x_{T+1})\right) \\
J_T^I(\lambda_T)&=I_{\lambda_T}(X_{T-m}^T;U_T|U_{T-n}^{T-1}).\end{aligned}$$ To establish (a) for $k=T$, it is sufficient to show that $J_T^c(\lambda_T)=J_T^c(\lambda'_T)$ and $J_T^I(\lambda_T)\geq J_T^I(\lambda'_T)$. The first equality holds because of . To see the second inequality,
$$\begin{aligned}
J_T^I(\lambda_T)&=I_{\lambda_T}(X_{T-m}^T;U_T|U_{T-n}^{T-1}) \\
&\geq I_{\lambda_T}(X_T;U_T|U_{T-n}^{T-1}) \\
&= I_{\lambda'_T}(X_T;U_T|U_{T-n}^{T-1}) \label{eqJI1}\\
&= I_{\lambda'_T}(X_T;U_T|U_{T-n}^{T-1}) \nonumber \\
&\hspace{10ex}+I_{\lambda'_T}(X_{T-m}^{T-1};U_T|X_T, U_{T-n}^{T-1}) \label{eqJI2} \\
&=I_{\lambda'_T}(X_{T-m}^T;U_T|U_{T-n}^{T-1}) \\
&=J_T^I(\lambda'_T).\end{aligned}$$
The equality follows from . The second term in is zero since $U_T$ is independent of $X_{T-m}^{T-1}$ given $(X_T, U_{T-n}^{T-1})$ by construction of $\lambda'_T$ in . Hence the statement (a) is established for $k=T$.
The statement (b) follows immediately from the above discussion. To establish (c), notice that due to (a), we can assume the optimal policy of the form $q_T(u_T|x_T, u_{T-n}^{T-1})$ without loss of generality. Hence, due to (b), the Bellman equation at $k=T$ can be written as $$\begin{aligned}
V_T(\mu_T(x^T, u^{T-1}))=&\min_{q_T(u_T|x_T, u_{T-n}^{T-1})} \Bigl\{I(X_T;U_T|U_{T-n}^{T-1})\Bigr. \\
&+\Bigl.\mathbb{E} c_T(x_T, u_T)+\mathbb{E} c_{T+1}(x_{T+1}) \Bigr\}.\end{aligned}$$ Since the last expression depends on $\mu_T(x^T, u^{T-1})$ only through its marginal $\mu_T(x_T, u_{T-n}^{T-1})$, we conclude that $V_T(\mu_T(x^T, u^{T-1}))$ is a function of the marginal $\mu_T(x_T, u_{T-n}^{T-1})$ only. This establishes (c) for $k=T$.
: Next, we assume (a), (b) and (c) hold for $k=t+1$. To establish (a), (b) and (c) for $k=t$, notice that under the induction hypothesis, we can assume without loss of generality that policies for $k=t+1, t+2, ... , T$ are of the form $q_k(u_k|x_k,u_{k-n}^{k-1})$, and that the value function at $k=t+1$ depends only on $\mu_{t+1}(x_{t+1}, u_{t-n+1}^t)$. With a slight abuse of notation, the latter fact is written as $$V_{t+1}(\mu_{t+1}(x^{t+1}, u^t))=V_{t+1}(\mu_{t+1}(x_{t+1}, u_{t-n+1}^t)).$$ Thus, the Bellman equation at $k=t$ can be written as $$\begin{aligned}
&V_t\left(\mu_t(x^t, u^{t-1})\right)
=\min_{q_t(u_t|x^t,u^{t-1})} \Bigl\{ \mathbb{E}c_t(x_t, u_t) \biggr. \nonumber \\
& \hspace{1ex}\Bigl. + I(X_{t-m}^t;U_t|U_{t-n}^{t-1}) + V_{t+1}(\mu_{t+1}(x_{t+1}, u_{t-n+1}^{t}))\Bigr\}. \label{eqvfindt}\end{aligned}$$ Now, using the similar construction to the case for $k=T$, one can show that for every $q_t(u_t|x^t,u^{t-1})$, there exists a policy of the form $q'_t(u_t|x_t,u_{t-n}^{t-1})$ such that the value of the objective function in the right hand side of attained by $q'_t$ is less than or equal to the value attained by $q_t$. This observation establishes (a) for $k=t$. Statements (b) and (c) for $k=t$ follows similarly.
Proof of Proposition \[propcom2\]
=================================
Proof of Proposition \[propcom2\]
---------------------------------
\[app1\]
For each $t=1,2, ... , T$, we have $$\begin{aligned}
I(X_{t-m}^t; W_t|U_{t-n}^{t-1}) &=I(X_{t-m}^t; W_t, U_t|U_{t-n}^{t-1}) \\
&=I(X_{t-m}^t; U_t|U_{t-n}^{t-1})+I(X_{t-m}^t;W_t|U_{t-n}^t) \\
&\geq I(X_{t-m}^t; U_t|U_{t-m}^{t-1}).\end{aligned}$$ $$\begin{aligned}
&I(X_{t-m}^t; W_t|U_{t-n}^{t-1}) \\
&=I(X_{t-m}^t; W_t, U_t|U_{t-n}^{t-1}) \\
&=I(X_{t-m}^t; U_t|U_{t-n}^{t-1})+I(X_{t-m}^t;W_t|U_{t-n}^t) \\
&\geq I(X_{t-m}^t; U_t|U_{t-m}^{t-1}).\end{aligned}$$ The first equality is due to the particular structure of the decoder specified by (A) or (B). Thus $$\sum\nolimits_{t=1}^T I(X_{t-m}^t; W_t|U_{t-n}^{t-1}) \geq I_{m,n}(X^T\rightarrow U^T).$$ The proof of Proposition \[propcom2\] is complete by noticing the following chain of inequalities. $$\begin{aligned}
R\log 2\! &=\!\sum\nolimits_{t=1}^T R_t \log 2 \\
&\geq \!\sum\nolimits_{t=1}^T \!H(W_t) \\
&\geq \!\sum\nolimits_{t=1}^T \!H(W_t|W^{t-1},U_{t-n}^{t-1}) \\
&\geq \!\sum\nolimits_{t=1}^T \!H(W_t|U_{t-n}^{t-1})-H(W_t|X_{t-m}^t, U_{t-n}^{t-1}) \\
&=\!\sum\nolimits_{t=1}^T \!I(X_{t-m}^t;W_t|U_{t-n}^{t-1}).\end{aligned}$$
[^1]: $^{1}$University of Texas at Austin, USA, [ttanaka@utexas.edu]{}; $^{2}$KTH Royal Institute of Technology, Sweden, [hsan@kth.se]{}; $^{3}$KTH Royal Institute of Technology, Sweden, [skoglund@kth.se]{}.
[^2]: $I_{m,n}$ is a short-hand notation for $I_{m,n}(X^T \rightarrow U^T)$.
[^3]: Here, we do not provide a detailed model of the function $c(x_t,u_t)$. See for instance [@horowitz2014second] for a model of work extraction based on the Langevin equation.
[^4]: The consistency with the classical second law is maintained if one accepts Landauer’s principle, which asserts that erasure of one bit of information from any sort of memory device in an environment at temperature $T_0$ \[K\] requires at least $k_B T_0 \log 2$ \[J\] of work. See [@bennett1982thermodynamics] for the further discussions.
|
---
abstract: 'We study the dissipative dynamics and the formation of entangled states in driven cascaded quantum networks, where multiple systems are coupled to a common unidirectional bath. Specifically, we identify the conditions under which emission and coherent reabsorption of radiation drives the whole network into a pure stationary state with non-trivial quantum correlations between the individual nodes. We illustrate this effect in more detail for the example of cascaded two-level systems, where we present an explicit preparation scheme that allows one to tune the whole network through “bright" and “dark" states associated with different multi-partite entanglement patterns. In a complementary setting consisting of cascaded non-linear cavities, we find that two cavity modes can be driven into a non-Gaussian entangled dark state. Potential realizations of such cascaded networks with optical and microwave photons are discussed.'
address:
- '$^1$Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria'
- '$^2$Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria'
author:
- 'K Stannigel$^{1,2}$, P Rabl$^1$, and P Zoller$^{1,2}$'
bibliography:
- 'CascadedQSexport.bib'
title: 'Driven-dissipative preparation of entangled states in cascaded quantum-optical networks'
---
Introduction
============
The coupling of a quantum system to an environment is often associated with decoherence. This is exemplified by the field of quantum computation, where the presence of additional reservoirs degrades the performance of quantum algorithms based on unitary operations executed on large many body systems [@Nielsen2000]. On the other hand, in many quantum optical settings the environment is actually useful for realizing certain applications. Prominent examples are laser cooling or optical pumping, where quantum systems are prepared in highly pure states with the help of a reservoir [@Metcalf1999], or continuous measurement schemes which enable the conditioned preparation of quantum states [@Wiseman2010]. While quantum control schemes employing engineered unitary evolution are by now standard, the last decade has witnessed an increasing interest in alternative methods based on the concept of “quantum reservoir engineering", i.e., on controlling a quantum system by tailoring its coupling to an environment [@Mueller2012]. Efforts along these lines have lead, e.g., to a broad range of proposals for the dissipative preparation of entangled few body quantum states [@Plenio1999; @Parkins2000; @Clark2003; @Kraus2004; @Paternostro2004; @Parkins2006; @Kastoryano2011]. However, in recent years attention has in particular been devoted to the study of dissipative *many body* systems. In this context, it has been realized that quantum reservoir engineering allows one to dissipatively prepare interesting many body states [@Diehl2008; @Kraus2008; @Verstraete2009; @Weimer2010; @Cho2011], perform universal quantum computation [@Verstraete2009], realize a dissipative quantum repeater [@Vollbrecht2011], or a dissipatively protected quantum memory [@Pastawski2011]. Meanwhile, first experiments demonstrating the dissipative preparation of GHZ states in systems of trapped ions [@Barreiro2011] and EPR entangled states of two atomic ensembles [@Krauter2011; @Muschik2011] have been reported. In these experiments the underlying principle has been to carefully design and implement a many-particle master equation, where a fully dissipative dynamics drives the system into a unique [*pure*]{} steady state representing the entangled state of interest.
In this work, we study entanglement formation in driven few- and many-particle [*cascaded quantum networks*]{} as introduced by Gardiner and Carmichael [@Gardiner1993; @Carmichael1993], where the unconventional coupling of multiple systems to a common unidirectional bath offers remarkable new opportunities for dissipative preparation of highly correlated states. As illustrated in , a cascaded quantum network consists of $N$ systems coupled to a 1D reservoir that has the unique feature that excitations can only propagate along a *single* direction, thereby driving successive systems in the network in a unidirectional way. While such a scenario is reminiscent of edge modes in quantum Hall systems [@Kane1997; @Haldane2008; @Wang2008; @Wang2009; @Hafezi2011], various artificial and more controlled realizations based on (integrated) non-reciprocal devices for optical [@Wang2005; @Feng2011] and microwave [@Koch2010; @Kamal2011] photons are currently developed. Our goal below is to identify situations where cascaded quantum networks are driven by classical fields in such a way that they exhibit pure and entangled steady states. As shown in , such *dark states* emerge if the continuous stream of photons emitted by the first part (“A") of the network is coherently reabsorbed by the second part (“B"), such that no photons escape from the system and the output remains dark. The system hence acts as its own [*coherent quantum absorber*]{} while the constant stream of photons maintains entanglement between its two parts.
![A driven cascaded quantum network which is realized by a set of two-level systems coupled to a unidirectional bath. The two-level systems are driven by classical fields $\sim \Omega$ and the continuously emitted radiation propagates along the waveguide and excites successive nodes. Under specific conditions, all photons emitted in a subsystem A are coherently reabsorbed in subsystem B. In this case the system relaxes into a *dark state* where no radiation escapes from the network, but a constant stream of photons running from A to B establishes entanglement between the two subsystems.[]{data-label="fig:figure1"}](Figure1_NJP){width="80.00000%"}
We will first discuss the construction of coherent quantum absorbers on general grounds. Given the first part A of the network, we show how to choose the second part B such that the whole system evolves into a dark state. In doing so, we make use of the unidirectionality of the reservoir, which allows us to solve the first part independently of the second. These formal developments are then illustrated by two settings in which the coherent quantum absorber scenario can be realized. The first is a many body cascaded network, where each of the $N$ nodes consists of a driven two-level system (TLS, “spin"). We show that this system exhibits a whole class of multi-partite entangled dark states, whose entanglement structure can be adjusted by tuning local parameters. As a second, complementary system we consider a network consisting of two non-linear cavities described by bosonic mode operators. By choosing appropriate laser drives and Kerr-type non-linearities one can ensure that this system evolves into a non-Gaussian dark state in which the two modes are entangled. In both examples, the coherent quantum absorber scenario thus leads to a dissipative state preparation scheme for non-trivial entangled states. In a more general context, these networks realize a novel type of non-equilibrium (many body) quantum system, which by changing the system parameters can be tuned between “dark" or “passive" phases (with no scattered photons emerging) and “bright" or “active" phases (with light scattered), while the nodes are driven into pure entangled or mixed states, respectively.
The remainder of this paper is structured as follows. In section \[sec:absorbers\] we present the general model for an $N$-node cascaded network and show how to construct the coherent quantum absorber subsystem B for some given subsystem A (cf. ). Complementing these rather general developments, we discuss the two mentioned realizations of the coherent quantum absorber scenario in subsequent sections. Section \[sec:spin\] presents the driven cascaded spin-system for which we derive and discuss a multi-partite entangled class of dark states and also comment on the influence of various imperfections. Subsequently, section \[sec:kerr\] discusses the setup based on cascaded cavities with Kerr-type non-linearity. In the latter case, we also obtain a purification of the steady state density matrix of the well-known dispersive optical bi-stability problem [@Drummond1980] as an interesting by-product. Implementations of the proposed cascaded networks are discussed in and concluding remarks can be found in .
Coherent quantum absorbers {#sec:absorbers}
==========================
Cascaded quantum networks {#sec:cascadedSystems}
-------------------------
We consider the general setting of a cascaded quantum network as shown in . Here, $N\geq 2$ subsystems located at positions $x_i$ are coupled to a 1D continuum of right-propagating bosonic modes $b_{\omega}$, which represent, for example, photons in an optical or microwave waveguide. The whole network can be modeled by a Hamiltonian ($\hbar=1$) $$\begin{aligned}
\label{eq:Htot}
H=
\sum_i H_i + H_{\rm bath}
+ \sum_i \int \rmd{\omega}\, g_\omega \left( c_i^\dag b_{{\omega}} \rme^{\rmi {\omega}x_i/v} + {\textrm{H.c.}}\right),\end{aligned}$$ where $H_{\rm bath}=\int\rmd{\omega}\,{\omega}\, b_{\omega}^\dag b_{\omega}$ is the free Hamiltonian of the bath modes and the integrals run over a broad bandwidth $\Delta \omega$ around the characteristic system frequency ${\omega}_0$. In this frequency range the bath modes are assumed to exhibit a linear dispersion relation with speed of light $v$. In [equation ]{} the $H_i$ describe the dynamics of the individual systems and include classical driving fields (e.g. lasers or microwave fields). The system-bath coupling is determined by the “jump operators" $c_i$ and coupling constants $g_{{\omega}}$, which we assume to be approximately constant over the frequency range $\Delta \omega$. Implementations of an effective model of the form given in [equation ]{} can be achieved with atoms or solid state TLSs coupled to 1D optical or microwave waveguides and will be discussed in more detail in below.
The system-bath interaction in [equation ]{} breaks time reversal symmetry and while photons can be emitted to the right, drive successive subsystems and eventually leave the network, the reverse processes cannot occur. To study the effects of this unconventional coupling, we assume that ${\omega}_0$ as well as the bandwidth $\Delta \omega$ are large compared to the other relevant frequency scales and eliminate the bath modes in a Born-Markov approximation. This yields a generalized cascaded master equation (ME) for the reduced system density operator $\rho$ [@Gardiner1993; @Carmichael1993; @Stannigel2011], $$\label{eq:ME1}
\dot \rho= \sum_i \mathcal{L}_i \rho
-\gamma \sum_{j>i}\left([c_j^\dag,c_i \rho] + [\rho c_i^\dag,c_j] \right).
$$ Here the first part describes the uncoupled evolution of each subsystem $\mathcal{L}_i \rho= -i[H_i,\rho] +\gamma \mathcal{D}[c_i]\rho$, where the Lindblad terms ${\mathcal{D}}[x]\rho=x \rho x^\dag-\{x^\dag x,\rho\}/2$ model dissipation due to emission of photons into the waveguide with a rate $\gamma=2\pi g_{{\omega}_0}^2$. The unidirectionality of the bath is reflected by the last term in [equation ]{}, which accounts for the possibility to reabsorb photons emitted at system $i$ by all successive nodes located at $x_j>x_i$. The explicit Lindblad form of the ME reads $$\label{eq:ME2}
\dot \rho = -\rmi [H_{\rm casc}, \rho] +\gamma \mathcal{D}[c]\rho\,,$$ where $H_{\rm casc}= \sum_i H_i - \rmi\frac{\gamma}{2}\sum_{j>i} (c_j^\dag c_i - c_i^\dag c_j)$ now includes the non-local coherent part of the environment-mediated coupling, while the only decay channel with collective jump operator $c=\sum_ic_i$ is associated with a photon leaving the system to the right.[^1] Note that equations and are understood in a rotating frame in order to account for the explicit time-dependence of the driving fields.
Coherent quantum absorbers {#sec:constructionAbsorbers}
--------------------------
In the following we are interested in steady state situations where every photon emitted within the system is perfectly reabsorbed by successive nodes in the network, such that there is no spontaneous emission via the waveguide output and the system relaxes to a pure steady state $\rho_0=|\psi_0\rangle\langle \psi_0|$. To identify the general conditions for the existence of such states, we partition the network into two subsystems A and B as indicated in , with local Hamiltonians $H_A$ and $H_B$, and jump operators $c_A$ and $c_B$, respectively. Specifically, $H_A= \sum^\prime_i H_i - \rmi\frac{\gamma}{2}\sum^\prime_{j>i} (c_j^\dag c_i - c_i^\dag c_j)$ and $c_A= \sum^\prime_i c_i$, where the primed sums run over the nodes of part A, and corresponding expressions hold for B. Then, in [equation ]{}, $H_{\rm casc}= H_A+H_B- \rmi \frac{\gamma}{2}(c_A c_B^\dag - c^\dag_A c_B )$ and $c=c_A+c_B$, and the conditions for the existence of a pure stationary state are (see Ref.[@Kraus2008] and further comments in \[sec:app:absorber\]): $$\label{eq:conditions}
{\bf (I)}\,\, \, (c_A+c_B)|\psi_0\rangle =0\,, \qquad
{\bf (II)}\,\,\, [H_{\rm casc},\rho_0]=0\,.$$ The first condition implies that the waveguide output is dark, i.e., ${\langle c^\dag c \rangle}=0$, and the second one ensures stationarity. Within a quantum trajectory picture, condition ${\bf (I)}$ means that there are no stochastic quantum jumps, which would lead to a mixed state. In situations where $[H_{A,B},\rho_0]=0$ and $c_{A,B}|\psi_0\rangle=0$ for each subsystem separately, the network can simply be devided into two smaller parts which are then treated independently. In the following, we thus focus on situations where such a division is not possible and where the steady state possesses non-trivial correlations between A and B, as characterized for example by a non-vanishing $\mathcal{C}\equiv\langle c_A^\dag c_B+c_A c_B^\dag \rangle - 2 {\rm Re} \{ \langle c_A^\dag \rangle\langle c_B\rangle\}$. In view of ${\bf (I)}$ this third requirement can be expressed as $${\bf (III)}\,\, \, \mathcal{C}= - 2 (\langle c_A^\dag c_A\rangle - |\langle c_A \rangle|^2)\neq 0\,,$$ and directly connects the correlations between A and B with the amount of radiation emitted from the first subsystem. Note that for a pure steady state, ${\mathcal{C}}\neq 0$ also implies that A and B are entangled. Finally, we remark that under stationary conditions, a non-vanishing $\mathcal{C}$ implies a constant flow of energy from A to B, while the total scattered light vanishes. In the examples discussed below this ‘coherent’ absorption of energy in B can be understood as a destructive interference of the signals scattered by the two subsystems.
The conditions ${\bf (I)}-{\bf (III)}$ will not be satisfied in general. However, given a system A described by a Hamiltonian $H_A$ and jump operator $c_A$ we can construct a perfect coherent absorber system B as follows. First, we point out that due to the unidirectional coupling the dynamics of A is unaffected by B, which can also be shown explicitly by tracing the ME over system B. In particular, the steady state $\rho_A^0$ of A is obtained by solving $\mathcal{L}_A\rho_A^0=0$, where ${\mathcal{L}}_A\rho_A \equiv -\rmi[H_A,\rho_A] + \gamma{\mathcal{D}}[c_A]\rho_A$, and assuming a unique solution we write its spectral decomposition as $\rho_A^0= \sum_k p_k |k\rangle\langle k|$. A pure state of the whole system is then given by $|\psi_0\rangle=\sum_k \sqrt{p_k} |k\rangle_A\otimes |\tilde k\rangle_B$, where we assumed A and B to have the same Hilbert space dimension and defined $|\tilde k\rangle= V|k\rangle$ in terms of an arbitrary unitary $V$ acting on ${| k \rangle}$. Now, we demonstrate in \[sec:app:absorber\] that conditions ${\bf (I)}$ and ${\bf (II)}$ can be satisfied by the choice $$\begin{aligned}
\label{eq:cB}
c_B&= - \sum_{n,m} \sqrt{\frac{p_n}{p_m}} \langle m| c_A| n\rangle |\tilde n\rangle\langle \tilde m|_B\,,\\
\label{eq:HB}
H_B&= - \frac{1}{2} \sum_{n,m} \left( \sqrt{\frac{p_n}{p_m}} \,A_{mn}+\sqrt{\frac{p_m}{p_n}} \,A_{nm}^* \right) |\tilde n\rangle\langle \tilde m|_B\,,\end{aligned}$$ where $A_{mn}= \langle m| H_{A,{\rm eff}}| n\rangle $ and $H_{A,{\rm eff}}=H_{A}-\rmi\frac{\gamma}{2}c_A^\dag c_A$ is the effective non-hermitian Hamiltonian associated with ${\mathcal{L}}_A$, and we assumed a positive and non-degenerate spectrum $\{p_k\}$. While equations and define a general absorber system B, we find that for many systems of interest the stationary state $\rho_A^0$ satisfies $\sqrt{p_n} \langle k | c_A|n\rangle =\sqrt{p_k} \langle n | c_A|k\rangle $ and $\sqrt{p_n} \langle k | H_{A,{\rm eff}}|n\rangle =\sqrt{p_k} \langle n | H_{A,{\rm eff}}|k\rangle $. In this case, the above relations simplify to $c_B=-Vc_AV^\dag $ and $H_B=-VH_AV^\dag$ [^2] such that up to a unitary basis transformation the absorber system is just the negative counterpart of A. In particular, this situation applies to the two examples of cascaded spin systems and cascaded non-linear cavities, which we describe in more detail in the following sections.
Cascaded spin networks {#sec:spin}
======================
Let us now be more specific and consider a set of $N$ driven spins coupled to a unidirectional bosonic bath as shown in . In [equation ]{}, the collective jump operator is now $c=\sum_i\sigma_-^i$ and the cascaded Hamiltonian in the frame rotating at the frequency ${\omega}_d$ of the external driving field reads $$\label{eq:Hcasc}
H_{\rm casc}= \sum_{i}\left(\frac{\delta_i}{2} \sigma_z^i+ \Omega_i \sigma_x^i\right)
- \rmi\frac{\gamma}{2}\sum_{j>i}\left(\sigma_+^j\sigma_-^i - \sigma_-^j\sigma_+^i\right).$$ Here the $\sigma_\mu^i$ are the usual Pauli operators on site $i$, the ${\Omega}_i$ are local Rabi frequencies, and the $\delta_i={\omega}_i-{\omega}_d$ are the detunings of the individual spin transition frequencies ${\omega}_i$ from the common classical driving frequency $\omega_d$. In the following, the basis states of the spins are denoted by ${| e \rangle}$, ${| g \rangle}$, such that $\sigma_-={|g\rangle\langlee|}$. The classical fields which are used to drive the spins can either be applied via additional local channels (assuming that the associated decay rate is much smaller than $\gamma$) or via a coherent field which is sent through the common waveguide. Note that by omitting the cascaded interaction in equation we recover the familiar Dicke model for multiple two level atoms decaying via a common field mode, where even for $\Omega=0$ a series of dark states can be identified from $c|\psi_0\rangle=0$. In contrast, in our cascaded setting these states are not stationary and non-trivial dark states can only emerge from an interplay between driving and cascaded coupling terms.
Construction of dark states
---------------------------
For $N=2$ the dark state condition ([**I**]{}) restricts $|\psi_0\rangle$ to the subspace spanned by $|gg\rangle$ and the singlet ${| S \rangle}=({| eg \rangle}-{| ge \rangle})/\sqrt{2}$. Condition ([**II**]{}) can then be satisfied for $\Omega_1=\Omega_2\equiv\Omega$ and any $\delta_1=-\delta_2\equiv\delta$, for which we obtain the unique and pure steady state $|\psi_0\rangle=|{\mathcal{S}}_\delta\rangle$, where $$\begin{aligned}
\label{eq:S2}
{| {\mathcal{S}}_\delta \rangle}=\frac{1}{\sqrt{1+{|\alpha|}^2}}\left({| gg \rangle}+\alpha {| S \rangle}\right),
\qquad
\alpha=\frac{2\sqrt{2}{\Omega}}{\rmi\gamma-2\delta}\,.\end{aligned}$$ The two spins thus realize a source and a matched absorber in the sense introduced in the previous section, and for the matrix representations of the various operators we can identify $c_A=\sigma_-$, $c_B=-V c_A V^\dag=\sigma_-$ and $H_B=-VH_AV^\dag$, using $V=\sigma_z$. For strong driving, ${\left|\alpha\right|}\gg1$, the state ${| {\mathcal{S}}_\delta \rangle}$ approaches the singlet ${| S \rangle}$, where the mutual correlation ${|{\mathcal{C}}|}\rightarrow 1$ is maximized.
While for larger $N$ a direct search for possible dark states is hindered by the exponential growth of the subspace defined by $c|\psi_0\rangle=0$, we can use the state as a starting point and solve the cascaded system iteratively “from left to right": Suppose that for $\Omega_i=\Omega$ and $\delta_1=-\delta_2$ the first two spins have evolved into the dark state ${| {\mathcal{S}}_{\delta_1} \rangle}$ such that no more photons are emitted into the waveguide. Then, the following two spins effectively see an empty waveguide and evolve into the dark state ${| {\mathcal{S}}_{\delta_3} \rangle}$, provided that $\delta_3=-\delta_4$. By iterating this argument we see that for any detuning profile with $\delta_{2i-1}=-\delta_{2i}$ ($i=1,2,\ldots$) the steady state of the ME is given by $$\begin{aligned}
\label{eq:psi0}
{| {\mathcal{S}}^0 \rangle}
={| {\mathcal{S}}_{\delta_1} \rangle}_{12} \otimes {| {\mathcal{S}}_{\delta_3} \rangle}_{34}\otimes \ldots \,,\end{aligned}$$ which is shown explicitly in \[sec:app:spin\]. In particular, for the homogenous case $\delta_i\simeq 0$, a strongly driven cascaded spin system relaxes into a chain of pairwise singlets.
![(a) Circuit model for constructing the state $|{\mathcal{S}}^\prime\rangle$ in [equation ]{} for the example $\Delta^\prime=(\delta_a,\delta_b,\ldots,-\delta_a,-\delta_b,\ldots)$. Each line represents a spin and is labeled by its detuning. A box connecting two lines denotes a unitary operation $U_i(\theta_i)$ and the corresponding exchange of detunings $\delta_i$ and $\delta_{i+1}$ (see text and inset). (b) Scaling of the von-Neumann entropy of the first $n$ spins in a network of $N=12$ nodes for $\Omega=2\gamma$. Upper curve: detuning profile as in (a) with ${|\delta_i|}=\gamma/3$. Lower curve: state corresponding to the detuning profile $\Delta^\prime=(0,\gamma,-\gamma,\gamma,\ldots,-\gamma,0)/3$.[]{data-label="fig:MultiQubitStates"}](Figure2_NJP_multiQubitStates){width="80.00000%"}
The dimer structure of the state ${| {\mathcal{S}}^0 \rangle}$ reflects the fact that radiation emitted from one node is immediately reabsorbed by the following one. We now consider situations where this reabsorption occurs by several of the following spins, leading to multi-partite entangled states. To this end, note that by starting from the state in [equation ]{} we can construct another dark state $|{\mathcal{S}}^\prime\rangle=U{| {\mathcal{S}}^0 \rangle}$ by any global unitary operation $U$ with $[U,c]=0$, while implementing the Hamiltonian $H'=U H_{\rm casc} U^\dag$ would ensure stationarity. However, for arbitrary unitary transformations $H^\prime$ will in general contain additional non-local terms, and we must hence restrict ourselves to unitaries $U$ under which $H_{\rm casc}$ is *form invariant*. As an example, we write $H_{\rm casc}\equiv H_{\rm casc}(\Delta)$, where $\Delta=(\delta_1,\delta_2,...)$ is the detuning profile, and introduce the nearest-neighbor operations $$\label{eq:U}
U_{i}(\theta_i)=\exp\left[\rmi\frac{\theta_i}{4}\left(\vec{\sigma}_i+\vec{\sigma}_{i+1}\right)^2\right]
\propto \exp\left[\rmi\frac{\theta_i}{2}\,\vec{\sigma}_i\cdot\vec{\sigma}_{i+1}\right]\,,$$ where $\vec{\sigma}_i=(\sigma_x^i,\sigma_y^i,\sigma_z^i)$. Then, by choosing $\tan(\theta_i)=(\delta_{i+1}-\delta_i)/\gamma$ we obtain $$\label{eq:formInvariance}
H'=U_i(\theta_i)\, H_{\rm casc} (\Delta) \, U_i^\dag(\theta_i) = H_{\rm casc} (\Delta^\prime)\,,$$ with a new detuning profile $\Delta^\prime=P_{i,i+1}\Delta$, where $P_{i,i+1}$ denotes the permutation of $\delta_i$ and $\delta_{i+1}$ (this is demonstrated in \[sec:app:spin\]). Thus, by starting from a set $\Delta^0$ of alternating detunings as defined before [equation ]{}, we can simply swap the detunings of nodes $i$ and $i+1$ to implement a new cascaded spin network with a unique stationary state $|{\mathcal{S}}^\prime\rangle=U_i(\theta_i){| {\mathcal{S}}^0 \rangle}$. By repeating this argument, we obtain a different pure steady state for each permutation $\Delta^\prime$ of $\Delta^0$. This class of states is given by $$\begin{aligned}
\label{eq:classOfDarkStates}
{| {\mathcal{S}}^\prime \rangle}=U(\Delta^0\rightarrow \Delta^\prime){| {\mathcal{S}}^0 \rangle}\,,\end{aligned}$$ where $U(\Delta^0\rightarrow \Delta^\prime)$ is a product of nearest-neighbor operations $U_i(\theta_i)$, specified by the sequence of nearest-neighbor transpositions required for transforming $\Delta^0$ into $\Delta^\prime$. A graphical representation of $U(\Delta^0\rightarrow \Delta^\prime)$ in terms of a circuit model is shown in (a) for a specific example.
![Tuning a six-spin network through bright and dark states by adjusting local detunings. The curves show purity and output intensity ${\langle c^\dag c \rangle}$, where the detunings $\delta_i$ are interpolated linearly between the profiles given on the horizontal axis. Red arrows indicate the entanglement structure of the dark states. The solid lines show the ideal case, while for the dashed lines we included a finite on-site decay $\kappa_0=0.0025\gamma$ (see ). Parameters are ${\Omega}=\gamma$, $\delta_0=\gamma/5$, $\delta_1=\gamma$, $\delta_2=\gamma/2$, $\delta_3=0$. []{data-label="fig:brightAndDarkStates"}](Figure3_NJP_darkAndBrightStates){width="80.00000%"}
Discussion
----------
For large detuning differences $|\delta_i-\delta_{i+1}|\gg\gamma$ the unitary transformations given in [equation ]{} are SWAP operations [@Nielsen2000] between neighboring sites. In this limit the states ${| {\mathcal{S}}^\prime \rangle}$ remain approximately two-partite entangled, but with singlets shared between arbitrary nodes in the network. In contrast, for $|\delta_i-\delta_{i+1}| \approx \gamma$ the $U_i$ correspond to highly entangling $\sqrt{{\rm SWAP}}$ operations. Then, the entanglement structure can be much richer and in general the states ${| {\mathcal{S}}^\prime \rangle}$ contain multi-partite entanglement between several or even all nodes. While in this case a full characterization is difficult, we point out that the $U_i$ conserve total angular momentum such that the ${| {\mathcal{S}}^\prime \rangle}$ approach multi-spin singlets in the strong driving limit. The amount of entanglement between subsystems now depends very much on the choice of the detuning profile, as can be seen from the two examples displayed in (b), showing oscillating and linearly growing block entropy, respectively. More generally, the cascaded network can be driven into different types of states by simply adjusting local detunings. This is illustrated in , where an adiabatic variation of the detunings in a six-node network is used to prepare pure steady states with 2-, 4- and 6-partite entanglement, separated by “bright" (mixed state) phases where the conditions in [equation ]{} are violated.
For assessing the relaxation time that characterizes the approach of the system towards the presented steady states it is sufficient to examine the simple detuning profile leading to ${| {\mathcal{S}}^0 \rangle}$, since the spectral properties of the Liouvillian are invariant under the transformations . Numerical calculations for small systems suggest that the preparation time for these states scales efficiently with the number of nodes $N$. In particular, the uniqueness of ${| {\mathcal{S}}^0 \rangle}$ implies that effects related to non-unique steady states found in related systems [@Gu2006] are absent in our case.
![Influence of imperfections on the entanglement properties of the steady state. (a-d) show the steady-state concurrence in a two-node network for various imperfections. (a) On-site decays (see text). (b) Dephasing of the spins modeled by additional Lindblad terms ${\mathcal{L}}^\prime\rho=\sum_i\frac{1}{2 T_2}{\mathcal{D}}[\sigma_z^i]\rho$. (c) Waveguide losses, where $\eta$ is the fraction of photons that gets lost between the nodes (modeled by adding a factor $\sqrt{1-\eta}$ to the last term in [equation ]{}). (d) Deviation from the asymmetric detuning condition modeled by a symmetric detuning offset $\delta_1=\delta_2=\epsilon$. In (a-c) we have chosen $\delta_1=\delta_2=0$. (e) Influence of on-site decays on concurrences $\mathcal{C}_{i,i+1}$ of reduced two-spin density matrices $\rho_{i,i+1}$ for $N=6$ spins with $\delta_i=0$. (f) Influence of on-site decays on the steady state four-partite entanglement in a four-spin network, quantified by the measure of Ref.[@Jungnitsch2011], which is bounded by $0.5$ in this case. The detuning profile is given by $\Delta^\prime=(0,-\gamma/2,\gamma/2,0)$. []{data-label="fig:imperfections"}](Figure4_NJP_imperfections){width="80.00000%"}
Imperfections {#sec:imperfections}
-------------
Under realistic conditions various imperfections like onsite decays or losses in the waveguide can violate the exact dark state condition and the system then evolves to a mixed (“bright") steady state. This is exemplified by the dashed lines in for the case of onsite decays, modelled by adding an additional term $\mathcal{L}'\rho=\kappa_0 \sum_i \mathcal{D}[\sigma_-^i]\rho$ to the ME . One clearly observes how the scattered intensity increases, while the purity drops as compared to the ideal case with $\kappa_0=0$. To study the entanglement properties in such non-ideal situations, we show in (a)-(d) the resulting steady state concurrence for various types of imperfections in a two-node network. In addition to onsite decays (panel (a)), we also consider intrinsic spin dephasing (panel (b)), waveguide losses (panel (c)), and small deviations from the ideal detuning profile (panel (d)). We see that different sources of imperfections lead to a qualitatively similar behavior and that in all cases the entanglement is quite robust and optimized for intermediate driving strengths $\Omega$.
For larger networks, the scattering of photons from the first nodes due to imperfections also affects successive spins, as shown in (e) for a six-spin dimer chain ${| {\mathcal{S}}^0 \rangle}$ in the presence of onsite losses. One clearly observes that the bi-partite entanglement in the dimers decreases with increasing number of previous nodes. To study the robustness of genuine multi-partite entanglement in the presence of imperfections, we employ the entanglement measure proposed in Ref. [@Jungnitsch2011]. It can be evaluated in a straight-forward way and (f) displays the results for a four-partite entangled steady state in the presence of onsite decays. The behavior of this measure qualitatively agrees with the results for the concurrence in the two-spin case (cf. (a)). That is, we observe a tradeoff between the maximal achievable entanglement and the robustness of the state. Finally, also shows that for a fixed $N$, different bi- and multi-partite entangled states are affected equally.
Cascaded non-linear cavities {#sec:kerr}
============================
![(a) Two driven cascaded cavities with Kerr-type non-linearities. For simplicity, we assume the cavities to be driven through their back mirrors (red arrows), whereas the dominant decay channel $\gamma$ is provided by the front mirrors. (b) Level scheme for the construction of the pure steady state in the occupation number basis ${| nm \rangle}\equiv{| n \rangle}_+{| m \rangle}_-$ of the $c_\pm$ modes ($N=n+m$ is the total photon number). Straight arrows denote Hamiltonian matrix elements, while wavy ones correspond to the decay of the symmetric mode. Blue dots indicate population in the dark state . []{data-label="fig:kerrSetup"}](Figure5_NJP_kerrSetup){width="80.00000%"}
The general concept of a coherent quantum absorber introduced in suggests that the formation of dark entangled states can exist also for systems other than spins. As a second non-trivial example where this can be shown explicitly, we now discuss a setting where the two cascaded systems A and B are represented by two Kerr non-linear cavities as depicted in (a). The resulting distribution scheme for continuous variable entanglement is an alternative to other cascaded settings considered in this context [@Peng2002]. We denote the two bosonic cavity modes by $a$ and $b$, and the dynamics of the system is governed by the ME with collective jump operator $c=a+b$ and cascaded Hamiltonian $$\begin{aligned}
H_{\rm casc}=H_A+H_B -\rmi\frac{\gamma}{2} (b^\dagger a - a^\dagger b)\,.\end{aligned}$$ Here, the Hamiltonian of the first cavity (system A, frequency ${\omega}_{c,A}$) is $$\begin{aligned}
\label{eq:kerr:HA}
H_A&=\Delta a^\dag a + K a^\dag a^\dag a a +\rmi \Omega (a^\dag - a) \,,\end{aligned}$$ where we have already moved to a frame rotating at the frequency ${\omega}_d$ of the external driving field, such that $\Delta={\omega}_{c,A}-{\omega}_d$ is the corresponding detuning of the cavity frequency and $\Omega$ the associated driving strength. Further, $K$ denotes the strength of the Kerr non-linearity. Motivated by our analysis of the cascaded spin system above, we assume that the Hamiltonian for the second cavity is given by $$\begin{aligned}
\label{eq:kerr:HB}
H_B&=-\Delta b^\dag b - K b^\dag b^\dag b b +\rmi \Omega(b^\dag - b)\,.\end{aligned}$$ Here we have chosen all constants to be identical to those used in $H_A$, such that the first two terms have opposite sign as compared to system A. As demonstrated below, this *educated guess* for $H_B$ ensures that the cascaded ME exhibits a dark state.
Steady state solution {#sec:kerr:solution}
---------------------
In order to show that the ME indeed exhibits a pure steady state for the local Hamiltonians chosen above we exploit once more the conditions given in [equation ]{}. From condition ([**I**]{}), i.e. $(a+b){| \psi_0 \rangle}=0$, it is clear that the steady state should contain zero quanta in the symmetric mode.[^3] Therefore, it is convenient to change to symmetric and anti-symmetric modes $c_\pm=(a\pm b)/\sqrt{2}$ and to write the dark state ansatz as ${| \psi_0 \rangle}={| 0 \rangle}_+ {\otimes}{| \chi_0 \rangle}_-$, with ${| \chi_0 \rangle}=\sum_n\alpha_n{| n \rangle}$, where ${| n \rangle}$ are Fock states in the occupation number basis. In order to exploit condition ([**II**]{}) we rewrite the cascaded Hamiltonian in terms of the symmetric and anti-symmetric modes, $$\begin{aligned}
H_{\rm casc}=\rmi\sqrt{2}\Omega c_+^\dag + \left(\Delta -\rmi\gamma/2 + K (\hat N-1) \right)\,c_+^\dag c_- + {\textrm{H.c.}}\,.\end{aligned}$$ Here $\hat N=c^\dag_+ c_+ + c^\dag_- c_-$ is the total number of quanta, which is conserved by all terms except for those $\propto\Omega$. Condition ([**II**]{}) is equivalent to $H_{\rm casc}{| \psi_0 \rangle}=\lambda{| \psi_0 \rangle}$, and by projecting this equation onto ${| \psi_0 \rangle}$ we see that it can only be fulfilled for $\lambda=0$ and hence $$\begin{aligned}
\label{eq:kerr:recursion}
\alpha_n=\sqrt{\frac{2}{n}} \frac{\epsilon}{x+n-1}\alpha_{n-1}\,,\quad
\epsilon=\frac{\Omega}{\rmi K}\,,\quad
x=\frac{i\Delta+\gamma/2}{\rmi K}\,.\end{aligned}$$ This recursion is readily solved, such that the unique solution of conditions ([**I**]{}) and ([**II**]{}) is given by $$\begin{aligned}
\label{eq:kerr:chiState}
{| \psi_0 \rangle}={| 0 \rangle}_+\otimes{| \chi_0 \rangle}_-\,,\quad
\textrm{with}\,\,\,
{| \chi_0 \rangle}=\frac{1}{{\mathcal{N}}}\sum_{n=0}^\infty
\frac{(\sqrt{2}\epsilon)^n}{\sqrt{n!}} \frac{\Gamma(x)}{\Gamma(x+n)}
{| n \rangle}\,.\end{aligned}$$ Here ${\mathcal{N}}=\left[{}_0F_2(x,x^*;2{|\epsilon|}^2)\right]^{1/2}$ and ${}_0F_2$ denotes the generalized hyper-geometric function [@Gradshteyn2007]. Due to the simple tensor-product structure of ${| \psi_0 \rangle}$ its only non-vanishing normally ordered moments are those of the $c_-$-mode, i.e. $$\begin{aligned}
{\langle (c_-^\dag)^n (c_-)^m \rangle}=
2^{\frac{n+m}{2}}
\frac{(\epsilon^*)^n\epsilon^m \Gamma(x^*)\Gamma(x)}{\Gamma(x^*+n)\Gamma(x+m)}
\frac{{}_0F_2(x^*+n,x+m;2{\left|\epsilon\right|}^2)}{{}_0F_2(x^*,x;2{\left|\epsilon\right|}^2)}\,,\end{aligned}$$ and the moments of the original modes thus read $$\begin{aligned}
\label{eq:kerr:moments}
{\langle (a^\dag)^n (b^\dag)^k (b)^l (a)^m \rangle}=
\frac{(-1)^{k+l}}{2^{(n+k+l+m)/2}}
{\langle (c_-^\dag)^{n+k} (c_-)^{l+m} \rangle}\,.\end{aligned}$$ Numerical studies suggest that there are no additional mixed steady states.
Entanglement properties of the steady state
-------------------------------------------
![Characterization of the steady state ${| \psi_0 \rangle}$ for $K/\gamma=0.01$ (first row) and $K/\gamma=0.5$ (second row). (a),(b) photon number in the first cavity normalized to resonant linear response. (c),(d) linear entropy $S_{\rm lin}$ of the first cavity as defined in [equation ]{}. In all panels the dashed lines indicate the critical driving strength $\Omega_c$, above which the semi-classical response becomes bi-stable (see, e.g., Ref.[@Yurke2006]).[]{data-label="fig:kerrFigure"}](Figure6_NJP_kerrResponse){width="80.00000%"}
Despite the simple structure of the steady state ${| \psi_0 \rangle}$ in the $c_\pm$ representation, it is generally entangled when written in terms of the original modes $a$ and $b$ according to $$\begin{aligned}
\label{eq:kerr:B}
{| \psi_0 \rangle}=\sum_{n,m=0}^\infty B_{nm} {| n \rangle}_A{| m \rangle}_B\,,
\,\,
B_{nm}=\alpha_{n+m}\, \frac{(-1)^m}{2^{(n+m)/2}}
\sqrt{\frac{(n+m)!}{n! m!}},\end{aligned}$$ where the $\alpha_n$ can be read off from [equation ]{}. We characterize the entanglement with respect to this bipartition by the mixedness of the reduced state $\rho_A^0=\tr_B\{{|\psi_0\rangle\langle\psi_0|}\}=BB^\dag$ of the first cavity, as measured by its linear entropy $$\begin{aligned}
\label{eq:kerr:Slin}
S_{\rm lin}=1-\tr\{(\rho^0_A)^2\}\,.\end{aligned}$$ The results for two different strengths of the non-linearity are displayed in . For a better orientation, we show in panels (a) and (b) the photon number of the first cavity, normalized to the resonant response in the linear case, i.e. $\Delta=K=0$. Here, one clearly observes the well-known behavior of a single Kerr-non-linear cavity, which is characterized by a deformation of the response curve for increasing $\Omega$. Above a certain critical driving strength ${\Omega}_c$ the classical response becomes bi-stable, while the quantum mechanical response curve exhibits a sharp step [@Drummond1980]. From panels (c) and (d) we see that the regions of pronounced non-linear response are those where the entropy is large, signaling a high degree of entanglement between the two cavities. In general, the features of the response are more washed out for larger non-linearities, where also the regions of high entropy are more extended. Note that the state ${| \chi_0 \rangle}$, and hence also ${| \psi_0 \rangle}$, is generally non-Gaussian, as can also be shown by evaluating appropriate measures [@Geroni2007]. Finally, we remark that for $K\rightarrow 0$ the steady state approaches a product of coherent states, i.e. ${| \psi_0 \rangle}\rightarrow{| \beta \rangle}_A\otimes{| -\beta \rangle}_B$, with $\beta=\epsilon/x$, where ${| \beta \rangle}$ denotes a coherent state of amplitude $\beta$. We are thus able to recover the classical limit in which no entanglement persists between the cavities.
The inverse coherent absorber problem
-------------------------------------
Finally, let us take a different view on the analysis presented so far and emphasize once more that cascaded systems may be solved in a successive fashion “from left to right". This means that tracing the full cascaded ME over the second subsystem B yields a closed equation for the first subsystem A, which in this case reads $$\begin{aligned}
\label{eq:kerr:MEA}
\dot\rho_A={\mathcal{L}}_A\rho_A \equiv -\rmi[\Delta a^\dag a + K a^\dag a^\dag a a +\rmi \Omega (a^\dag - a),\rho_A] + \gamma{\mathcal{D}}[a]\rho_A\,.\end{aligned}$$ In we have assumed that the stationary *mixed* state solution of this equation including its spectral decomposition is known, which allowed us to explicitly construct a corresponding perfect absorber system B, such that the whole system is driven into a pure steady state. However, in many cases solving for the steady state $\rho_A^0$ of [equation ]{} is a non-trivial problem by itself, which in the present case was first accomplished using phase space methods [@Drummond1980]. Note that by using an educated guess for the Hamiltonian $H_B$ of the absorber system and then solving for the *pure* steady state ${| \psi_0 \rangle}$ of the whole network, we have also indirectly solved the mixed state problem of a single cavity by computing $\rho_A^0=\tr_B\{{|\psi_0\rangle\langle\psi_0|}\}$. Its explicit matrix-elements in the Fock-state basis agree with the expressions found in the literature [@Kheruntsyan1999], as do its moments ${\langle (a^\dag)^n (a)^m \rangle}$ obtained as a special case of [equation ]{} [@Drummond1980]. The procedure presented above thus represents an elegant alternative way of obtaining the mixed steady state of the dispersive optical bi-stability problem.Ê
Avoiding to work with mixed states by using pure states on larger Hilbert spaces has found wide-spread use in the quantum information community, one of the reasons being that pure states can be manipulated more easily [@Nielsen2000]. From this point of view, the steady state ${| \psi_0 \rangle}$ possesses additional relevance as a *purification* of the density matrix $\rho_A^0$. This is an interesting result in itself, since obtaining such a purification generally requires knowledge of the spectral decomposition of $\rho_A^0$—a strong requirement even if $\rho_A^0$ is known explicitly. In the above example we were able to circumvent this difficulty by directly constructing the purification ${| \psi_0 \rangle}$ as a steady state of the cascaded ME. Although so far this procedure is limited to systems where the corresponding absorber Hamiltonian $H_B$ can be obtained by an *educated guess*, it is intriguing to think about cascaded quantum systems also as an analytic tool for calculating stationary states of non-trivial open quantum systems.
Implementations {#sec:implementations}
===============
The two key ingredients for realizing cascaded networks as discussed in the previous sections are (i) the implementation of low-loss non-reciprocal devices for directional routing of photons, and (ii) achieving a coupling of single quantum systems to a 1D waveguide that exceeds local decoherence channels. In the following, we discuss several implementations that fulfill these requirements, where a focus lies on optical and microwave setups. Requirement (i) is crucial for realizing the unidirectional coupling between the nodes and can in principle be fulfilled by standard circulators based on the Faraday effect (see, e.g., Ref.[@Gripp1995] for an optical implementation), but also by exploiting unidirectional edge modes in media with broken time-reversal symmetry [@Haldane2008; @Wang2008; @Wang2009]. However, in recent years there has been increasing interest in designing non-reciprocal on-chip devices which are integrable with, e.g., microwave circuitry [@Koch2010; @Kamal2011] or nano-fabricated photonic components [@Wang2005; @Feng2011]. Such elements would allow one to build up cascaded networks in a very controlled way in both the optical and the microwave domain, and thus constitute a promising way of implementing the scenarios discussed in this work.
![Possible realizations of a single node of a cascaded spin network as shown in . (a) generic setup where the unidirectional coupling is achieved by a cavity connected to a circulator, (b) non-reciprocal superconducting circuit based on the proposal of Ref. [@Koch2010], (c) optomechanical transducer based on toroidal cavities [@Stannigel2010; @Anetsberger2009].[]{data-label="fig:implementations"}](Figure7_NJP_implementations){width="80.00000%"}
Concerning the second requirement, we first discuss the case of the cascaded spin network presented in . Realizing a coupling of TLSs to a 1D waveguide could, e.g., be achieved along the lines of the experiments reported in Refs.[@Astafiev2010; @Akimov2007], where superconducting qubits or atoms are coupled to transmission lines or hollow core optical fibers, respectively. However, one can also do without a direct coupling of the TLS to the waveguide by using the generic and flexible approach depicted in (a). Here, the TLS is coupled to a cavity, whose output port is connected to a circulator or another non-reciprocal device. In the bad cavity limit $\kappa\gg g$, where $g$ is the TLS-cavity coupling, a series of these nodes results in the desired model , with an effective TLS-waveguide decay rate $\gamma=2g^2/\kappa$. A particular realization of this scheme in the context of circuit cavity quantum electrodynamics [@SchoelkopfNature2008; @Koch2010] is shown in (b). Finally note that other systems like optomechanical transducers ((c)) have been proposed to realize a similar unidirectional coupling [@Stannigel2010; @Anetsberger2009]. In the case of cascaded cavities discussed in , the most important requirement is the realization of strong Kerr-type non-linearities, which are of equal magnitude and opposite sign in the two cavities. Such a tunable non-linearity can be realized by a dispersive coupling of the cavity field to a suitable four-level system, which has been analyzed both for optical [@Imamoglu1997; @Hartmann2008] as well as superconducting microwave cavities [@Rebic2009].
In summary, we find that the current experimental capabilities for realizing strong TLS-photon interactions in various systems, combined with the development of novel non-reciprocal devices in the optical and microwave domain, enable the implementation and design of various cascaded quantum networks. Here, the dissipative state preparation schemes described in this work could serve as an interesting application exploring the unconventional physical properties of such devices.
Conclusions {#sec:conclusion}
===========
We have shown that photon emission and coherent reabsorption processes in cascaded quantum systems can lead to the formation of pure and highly entangled steady states. In the case of spin networks, this mechanism provides a tunable dissipative preparation scheme for a whole class of multi-partite entangled states, and instances of this scheme might serve as a basis for dissipative quantum communication protocols [@Vollbrecht2011]. For the case of two cascaded non-linear cavities we have identified a dissipative preparation scheme for non-Gaussian entangled states, and we have illustrated how cascaded systems can serve as a new analytic tool to evaluate the stationary states of driven-dissipative systems. More generally, our findings show that such driven cascaded networks realize a novel type of non-equilibrium quantum many-body system, which can be implemented with currently developed integrated optical systems or superconducting devices.
The authors thank B. Kraus for valuable discussions. This work was supported by the EU network AQUTE and the Austrian Science Fund (FWF) through SFB FOQUS and the START grant Y 591-N16.
General construction of coherent quantum absorbers {#sec:app:absorber}
==================================================
We present the arguments leading to the results quoted in . Given a system A in terms of its Hamiltonian $H_A$ and jump operator $c_A$, we seek a suitable system B described by $H_B$ and $c_B$ which perfectly reabsorbs the output field of A in steady state. In the following, we construct system B by requiring that the total cascaded system evolves into a pure steady state. The dynamics of the network is governed by the ME with Hamiltonian $$H_{\rm casc}= H_A+H_B- \rmi \frac{\gamma}{2}(c_A c_B^\dag - c^\dag_A c_B )\,$$ and collective jump operator $c=c_A+c_B$, and assuming that $c$ has no eigenvectors a pure state ${| \psi_0 \rangle}$ is a stationary state of [equation ]{} if and only if [@Kraus2008] $$\begin{aligned}
c{| \psi_0 \rangle}=0\,,\qquad
[H_{\rm casc},{|\psi_0\rangle\langle\psi_0|}]=0\,.\end{aligned}$$ As discussed in , the cascaded nature of the interaction allows us to solve for the reduced steady state $\rho_A^0$ of system A without knowing anything about B. Assuming uniqueness, we introduce its spectral decomposition $$\begin{aligned}
\rho_A^0=\sum_k p_k {|k\rangle\langlek|}\,,\end{aligned}$$ and assume that the eigenvalues $p_k$ are positive and non-degenerate. A potential pure steady state ${| \psi_0 \rangle}$ of the whole system can then be written as a purification of $\rho_A^0$, i.e., $|\psi_0\rangle=\sum_k \sqrt{p_k} |k\rangle_A\otimes |\tilde k\rangle_B$ [@Nielsen2000]. Here, $|\tilde k\rangle= V|k\rangle$, and $V$ denotes an arbitrary unitary and we assume subsystem B to have the same Hilbert space dimension as A.
In the following, we write out the tensor-products more explicitly, such that the dark-state condition $c{| \psi_0 \rangle}=0$ reads $$\begin{aligned}
\left(c_A{\otimes}{\mathbbm{1}}+ {\mathbbm{1}}{\otimes}c_B\right){| \psi_0 \rangle}=0\,,
\label{eq:app:darkStateCond}\end{aligned}$$ with $$\begin{aligned}
c_A&=\sum_{n,m} \langle n | c_A|m\rangle |n\rangle\langle m|\,, \quad
c_B=\sum_{n,m} \langle \tilde n | c_B|\tilde m\rangle |\tilde n\rangle\langle \tilde m|\,. \end{aligned}$$ To proceed, we plug the ansatz for $|\psi_0\rangle$ into [equation ]{}, yielding $$\sum_{k, n} \sqrt{p_k} \left( \langle n | c_A|k\rangle |n\rangle\otimes | \tilde k\rangle + \langle \tilde n | c_B|\tilde k\rangle |k\rangle\otimes | \tilde n\rangle\right)=0\,,
$$ and after relabeling the indices we obtain $$\sum_{k, n} \left( \sqrt{p_k} \langle n | c_A|k\rangle + \sqrt{p_n}\langle \tilde k | c_B|\tilde n\rangle \right) |k\rangle\otimes | \tilde n\rangle=0\,.
$$ Since the ${| k \rangle}$ form an orthonormal basis, this condition is equivalent to $$\label{eq:relationcAcB}
\sqrt{p_n} \langle \tilde k | c_B|\tilde n\rangle= - \sqrt{p_k} \langle n | c_A|k\rangle\,,$$ which fixes the operator $c_B$ to be $$\begin{aligned}
\label{eq:app:cB}
c_B&=& - \sum_{n,m} \sqrt{\frac{p_n}{p_m}} \langle m| c_A| n\rangle |\tilde n\rangle\langle \tilde m|\,.\end{aligned}$$
To construct the Hamiltonian $H_B$ for system B we exploit the second condition $[H_{\rm casc},{|\psi_0\rangle\langle\psi_0|}]=0$ needed for a pure steady state, which is equivalent to $H_{\rm casc}{| \psi_0 \rangle}=\lambda{| \psi_0 \rangle}$ for $\lambda\in\mathbbm{R}$. Note that $c|\psi_0\rangle=0$ implies $$\rmi \frac{\gamma}{2}(c_A{\otimes}c_B^\dag - c^\dag_A {\otimes}c_B ) |\psi_0\rangle
= \rmi \frac{\gamma}{2}(c^\dag_A c_A {\otimes}{\mathbbm{1}}- {\mathbbm{1}}{\otimes}c^\dag_B c_B ) |\psi_0\rangle\,,$$ such that this condition reads $$\label{eq:app:stationarityCond}
\left(H_{A,{{\rm eff}}}{\otimes}{\mathbbm{1}}+{\mathbbm{1}}{\otimes}H^\dag_{B,{{\rm eff}}}\right)|\psi_0\rangle = \lambda |\psi_0\rangle\,,$$ where we have introduced the effective non-hermitian Hamiltonians $H_{j,{\rm eff}}=H_j-\rmi\frac{\gamma}{2}c_j^\dag c_j$. Since a finite $\lambda$ would only lead to a global shift of $H_B$ below, we can assume $\lambda=0$ without loss of generality. We write the Hamiltonian as $$\begin{aligned}
H_B&=\sum_{n,m} \langle \tilde n | H_B|\tilde m\rangle |\tilde n\rangle\langle \tilde m|\,,\end{aligned}$$ and to determine the matrix elements in this expansion we start from [equation ]{} and proceed as for the dark state condition [equation ]{} with the replacements $c_A\rightarrow H_{A,{{\rm eff}}}$ and $c_B\rightarrow H_{B,{{\rm eff}}}^\dag$. As an intermediate result this yields $$\label{eq:app:HBintermediate}
\langle \tilde n | H_{B} |\tilde m\rangle= - \sqrt{\frac{p_n}{p_m}} \langle m | H_{A,{\rm eff}}|n\rangle-\rmi\frac{\gamma}{2} \langle \tilde n | c^\dag_{B}c_B |\tilde m\rangle \,.
$$ To express the right-hand side of this equation fully in terms of operators on A, we employ the identity $$\langle \tilde n | c^\dag_{B}c_B |\tilde m\rangle= \sqrt{\frac{1}{p_m p_n}} \langle m | c_A\rho_A^0 c_A^\dag| n\rangle\,,$$ which can be derived with the help of [equation ]{}, and then make use of the stationarity of system A, $$\gamma c_A\rho_A^0 c_A^\dag= \rmi (H_{A,{\rm eff}} \rho_A^0 - \rho_A^0 H_{A,{\rm eff}}^\dag )\,.$$ then becomes $$\begin{aligned}
\label{eq:app:HB}
\langle \tilde n | H_{B} |\tilde m\rangle
= - \frac{1}{2}\left( \sqrt{\frac{p_n}{p_m}} \langle m | H_{A,{\rm eff}}|n\rangle + \sqrt{\frac{p_m}{p_n}} \langle m | H_{A,{\rm eff}}^\dag |n\rangle \right) \,,\end{aligned}$$ which determines the Hamiltonian of system B. The expressions and are the results quoted in of the main text.
Dark states of cascaded spin networks {#sec:app:spin}
=====================================
We provide details regarding the cascaded spin network presented in . The system is described by the ME with collective jump operator $c=\sum_i\sigma^i_-$ and $H_{\rm casc}$ defined in [equation ]{}.
Uniqueness of the steady state ${| {\mathcal{S}}^0 \rangle}$
------------------------------------------------------------
We show by explicit construction that the state ${| {\mathcal{S}}^0 \rangle}$ given in [equation ]{} is the unique steady state of the network, provided that $\delta_{2i-1}=-\delta_{2i}$ and $\Omega_i=\Omega$ for $i=1,2,\ldots$. To do so, we exploit the fact that the cascaded interaction allows for a successive construction of the steady state and start with the case $N=2$. In this case, the steady state $\rho_2^0$ is obtained by solving ${\mathcal{L}}^{(2)}\rho_2^0=0$, where ${\mathcal{L}}^{(2)}={\mathcal{L}}_{12}$ is given by the block-wise Liouvillian $$\begin{aligned}
{\mathcal{L}}_{i,i+1}\rho=&-\rmi\left[\frac{\delta_i}{2}(\sigma^i_z-\sigma^{i+1}_z) + \Omega(\sigma^i_x + \sigma^{i+1}_x)\,,\,\rho\right] +\frac{\gamma}{2}{\mathcal{D}}[c_{i,i+1}]\rho\end{aligned}$$ with $c_{i,j}=\sigma^i_-+\ldots +\sigma^j_-$. We have already seen in the main text that a solution is given by $\rho_2^0={|{\mathcal{S}}_{\delta_1}\rangle\langle{\mathcal{S}}_{\delta_1}|}$, and its uniqueness can be shown by calculating the characteristic polynomial of ${\mathcal{L}}_{12}$ and realizing that there is only one zero eigenvalue for $\gamma>0$.
We continue with $N=4$ and write the ansatz for the steady state as $\rho_4^0={|{\mathcal{S}}_{\delta_1}\rangle\langle{\mathcal{S}}_{\delta_1}|}\otimes \mu$, where $\mu$ is a two-node density matrix. The four-node Liouvillian can be rewritten as $$\begin{aligned}
{\mathcal{L}}^{(4)}\rho= {\mathcal{L}}_{12}\rho + {\mathcal{L}}_{34}\rho
-\gamma \left( [c^\dag_{34},c_{12}\rho] + [\rho c^\dag_{12},c_{34}] \right)\,,\end{aligned}$$ and we note that ${\mathcal{L}}_{12}{|{\mathcal{S}}_{\delta_1}\rangle\langle{\mathcal{S}}_{\delta_1}|}=0$ as well as $c_{12}{| {\mathcal{S}}_{\delta_1} \rangle}=0$, such that the equation ${\mathcal{L}}^{(4)}\rho_4^0=0$ simplifies to ${\mathcal{L}}_{34}\mu=0$. However, this is just the two-node problem we have already solved and the unique solution for the four-node netwrok is thus given by $\rho_4^0={|{\mathcal{S}}_{\delta_1}\rangle\langle{\mathcal{S}}_{\delta_1}|}\otimes{|{\mathcal{S}}_{\delta_3}\rangle\langle{\mathcal{S}}_{\delta_3}|}$. By iterating this argument in blocks of two spins, we obtain the steady state ${| {\mathcal{S}}^0 \rangle}$ given in [equation ]{}.
Unitary form invariance of the master equation {#app:formInvariance}
----------------------------------------------
We briefly demonstrate that the cascaded Hamiltonian of the spin network is form-invariant under the unitary transformations as stated in [equation ]{}. In order to calculate $U_i H_{\rm casc} U_i^\dag$, we write $j\equiv i+1$ for brevity and rearrange $H_{\rm casc}$ as follows: $$\begin{aligned}
H_{\rm casc}&=\sum_{k\neq i,j}\left( \frac{\delta_k}{2}\sigma_z^k+ \Omega \sigma_x^k \right)
-\rmi\frac{\gamma}{2}\sum_{k>l}^{k,l\neq i,j}(\sigma_+^k\sigma_-^l - \sigma_-^k\sigma_+^l)\\
&-\rmi\frac{\gamma}{2}\left( c^\dagger_{i,j}c_{1,i-1} + c^\dagger_{j+1,N} c_{i,j} -{\textrm{H.c.}}\right)\\
&+\frac{\delta_i+\delta_j}{2} \frac{1}{2}(\sigma_z^i+\sigma_z^j)
+\Omega(\sigma_x^i+\sigma_x^j)\\
\label{eq:lastLineTransf}
&+\left[
\frac{\delta_i-\delta_j}{2} \frac{1}{2}(\sigma_z^i-\sigma_z^j)
-\rmi\frac{\gamma}{2}(\sigma_+^j \sigma_-^i - \sigma_-^j \sigma_+^i)
\right] \,,\end{aligned}$$ where we have again used the piecewise jump operator $c_{k,l}=\sigma^k_-+\ldots +\sigma^l_-$. Note that $U_i(\theta)$ commutes with the first three lines and we thus focus on the last one. We abbreviate the operators appearing there by $A=(\sigma_z^i-\sigma_z^j)/2$ and $B=\sigma^j_+\sigma^i_- - \sigma^j_- \sigma^i_+$ and observe that they transform into one another: $$\begin{aligned}
U_{i}(\theta) A U_{i}^\dag(\theta)= A \cos 2\theta + i B \sin 2\theta\\
U_{i}(\theta) B U_{i}^\dag(\theta)= B \cos 2\theta + i A \sin 2\theta\end{aligned}$$ Introducing the difference of detunings $\delta=(\delta_i-\delta_j)/2$, a non-trivial form-invariance of the Hamiltonian $H_{\rm casc}$ under $U_i(\theta)$ is realized if $$\begin{aligned}
& U_{i}(\theta)\left[ \delta A -\rmi \frac{\gamma}{2} B \right] U_{i}^\dag(\theta)
=-\delta A -\rmi \frac{\gamma}{2} B\end{aligned}$$ That is, we require the detunings to swap and the cascaded part to remain invariant. It is easy to check that this requirement results in two equations, which are both solved for the choice $\tan\theta=-2\delta/\gamma=(\delta_{i+1}-\delta_i)/\gamma$, which was to be demonstrated.
Bibliography {#bibliography .unnumbered}
============
[^1]: For $N=2$ this ME stands in contrast to Ref. [@Krauter2011; @Muschik2011], where the dynamics is purely dissipative with no coherent evolution.
[^2]: We use this as a short-hand notation for an equivalent matrix representation of the two operators, $\langle n| c_B|m\rangle =- \langle n| Vc_AV^\dag |m\rangle$, etc. Here, the same set of states is used on both systems, which is permissible due to their equal Hilbert-space dimension.
[^3]: As shown in Ref. [@Kraus2008] a pure stationary state could also exist if $|\psi_0\rangle$ is an eigenstate of the jump operator, i.e. $(a+b){| \psi_0 \rangle}=\lambda^\prime {| \psi_0 \rangle}$ with $\lambda^\prime\in\mathbb{C}$. However, in the present example this would imply that the symmetric mode is in a coherent state of amplitude $\lambda^\prime$. We do not expect this due to the non-linearity of the problem and it can indeed be shown that such an ansatz does not lead to a stationary solution unless $\lambda^\prime=0$.
|
---
abstract: 'The Einstein action for the gravitational field has some properties which make of it, after quantization, a rare prototype of systems with quantum configurations that do not have a classical analogue. Assuming spherical symmetry in order to reduce the effective dimensionality, we have performed a Monte Carlo simulation of the path integral with transition probability $e^{-\beta |S|}$. Although this choice does not allow to reproduce the full dynamics, it does lead us to find a large ensemble of metric configurations having action $|S|\ll \hbar$ by several magnitude orders. These vacuum fluctuations are strong deformations of the flat space metric (for which $S=0$ exactly). They exhibit a periodic polarization in the scalar curvature $R$. In the simulation we fix a length scale $L$ and divide it into $N$ sub-intervals. The continuum limit is investigated by increasing $N$ up to $\sim 10^6$; the average squared action $\langle S^2 \rangle$ is found to scale as $1/N^2$ and thermalization of the algorithm occurs at a very low temperature (classical limit). This is in qualitative agreement with analytical results previously obtained for theories with stabilized conformal factor in the asymptotic safety scenario.'
author:
- 'G. Modanese [^1]'
bibliography:
- 'QG2.bib'
title: 'Quantum-only metrics in spherically symmetric gravity'
---
Introduction
============
Efforts towards the unification of General Relativity and Quantum Mechanics into a coherent theory of Quantum Gravity were started long ago and have been intensifying in the last decades. In spite of big progress in loop Quantum Gravity [@rovelli2004quantum; @rovelli2014covariant], asymptotic safety [@reuter2018quantum] and discrete spacetime models [@hamber2008quantum; @hamber2019vacuum; @ambjorn2012nonperturbative; @loll2019quantum], “the revolution is still unfinished”, in the words of C. Rovelli. The spin-offs of this research work, however, are manifold and remarkable in their own right. In general, it is fair to say that the quest for unification has led to a better comprehension of both General Relativity and Quantum Mechanics.
We made some early contributions to Quantum Gravity in the covariant formulation by showing that the Wilson loop vanishes to leading order [@modanese1994wilson] and proposing an alternative expression for the static potential of two sources [@modanese1995potential]. This formula was used by Muzinich and Vokos [@muzinich1995long] and by Hamber and Liu [@hamber1995quantum], respectively in perturbation theory and in non-perturbative Regge calculus, to give an estimate of quantum corrections to the Newton potential. The vanishing of the Wilson loop (and of curvature correlations [@modanese1992vacuum]) to leading order is only one of the peculiar aspects of the quantum field theory of gravity, that sets it apart from other successful quantum field theories like QED and QCD.
One of the problems with defining quantum theories of gravity is that, unlike for other quantum mechanical systems, there is no action or Hamiltonian that would be bounded from below. As a result any formal definition of a path integral or partition function would be plagued by divergences and be dominated by configurations which have arbitrarily negative energy. This paper reports on a study in this context, where we only restrict to spherically symmetric configurations. The goal is to identify configurations with zero or almost zero action, which would all equally contribute to a quantum path integral (being unsuppressed by an action factor). This is done numerically and the paper reports on results as the number of steps in the discretisation is increased.
Although many possible extensions and generalizations of the Einstein action have been proposed [@nojiri2017modified], which could help in addressing open issues in cosmology, the Einstein action is the natural action at intermediate energies and arises directly from the quantization of massless spin 2 fields.
In our opinion, the indefinite sign of the Einstein action gives us a chance to explore a phenomenon that is otherwise unknown in quantum field theory and more generally in Quantum Mechanics, namely the existence of configurations for which the action is zero, like for the classical vacuum, but not a stationary point. We call them zero modes of the action and we have proven analytically their existence for the Einstein action as well as for a peculiar elementary quantum system (the massless harmonic oscillator [@modanese2016functional; @modanese2017ultra]).
The purpose of this work is to show that if the condition $S=0$ defining the zero modes is relaxed to $S/\hbar \ll 1$, then a large ensemble of these modes can be numerically constructed via a Metropolis – Monte Carlo algorithm. The condition $S/\hbar \ll 1$ implies that these modes can play an important role in the path integral, although they are very different from classical solutions. In this sense the Einstein action offers an example of a dynamical system with unique quantum properties and a possible prototype for similar systems in other branches of physics.
The algorithm for the generation of the zero modes ensemble has been presented in [@modanese2019metrics], but the application was limited to metric configurations at the Planck scale, and accordingly the discretization limited to $N=10^2$ space sub-intervals. Several authors have found, with various techniques, that the vacuum state of quantum gravity has a non-trivial structure at that scale. In particular, field configurations with spherical symmetry have been considered by [@preparata2000gas; @garattini2002spacetime]. One may wonder how this structure scales up to larger distances, and that is the main purpose of this work. After returning to more transparent physical units, we have performed several simulations looking for quantum zero modes with $S/\hbar \ll 1$ at a scale $L \gg L_P$ and in the continuum limit $N\to \infty$. It turns out that just the continuum limit allows to obtain such modes. The exact scaling dependence on $L$ and $N$ is reported in Sect. \[sec-scaling\].
Our results, though obtained in a different setting and with different methods, appear to be close to what is found in a paper by Bonanno and Reuter ([@bonanno2013modulated]; see also [@bonanno2019structure]). In this work, the authors have added an $R^2$ term to the action to make it bounded from below, and they find indications for a ground state which violates translational symmetry and displays a “rippled” structure. They argue that a “kinetic condensate” characterizes the vacuum state of asymptotically safe quadratic gravity theories, so that if this scenario is realized in the full theory, the vacuum state of gravity is the gravitational analogous to the Savvidy vacuum in Quantum Chromo-Dynamics. A more detailed comparison between our results and those of [@bonanno2013modulated] will be given in Sect. \[sec-limits\].
The outline of the paper is the following. In Sect. \[sec-dis\] we recall the form of the Einstein action reduced for spherically symmetric and time-independent metrics, first in the continuum version and then in the discretized version. We also recall our previous results from simulations at the Planck scale, made using a small number $N$ of sub-intervals. In Sect. \[sec-scaling\] we analyze the scaling properties of the discretized action with respect to $N$, up to $N\sim 10^6$, and also the scaling with respect to the inverse temperature $\beta$ of the Metropolis algorithm and the length scale $L$ of the vacuum fluctuations. In Sect. \[pol\] the observed polarization patterns of the metrics are reported and discussed. Sect. \[sec-2di\] offers for illustration purposes a mathematical example of zero modes in an oscillating 2D integral. Sect. \[sec-ext\] considers a possible extension to higher dimensions. Finally, Sect. \[sec-concl\] briefly summarizes our conclusions.
The discretized action {#sec-dis}
======================
Our physical model [@modanese2019metrics] is defined by the Einstein action of the gravitational field, computed for a metric with spherical symmetry and independent from time. The only field variable is the metric component $g_{rr}(r) = A(r)$, with $0 \le r < +\infty$. This kind of dimensional reduction of gravity has been already employed in several classical and quantum models.
The continuum action is $$S=\frac{\tau}{16\pi G} \int_0^\infty dr \sqrt{A}\left( \frac{rA'}{A^2}+1-\frac{1}{A}\right)
\label{S-cont}$$
This is derived from the usual expression of the Einstein action $\int \sqrt{g}Rd^4x$ in units such that $c=1$, and in which the integral over time has been replaced by a factor $\tau$, meaning that the metrics we are considering are stationary but have a limited duration $\tau$. This is clearly an approximation, and eventually we should introduce a function of time describing an adiabatic switch on/off; we expect the corresponding time derivatives to give a negligible contributions to the curvature for the values of $\tau$ and $L$ considered in this paper. (This can be checked for simplicity in the linearized approximation, where the scalar curvature is simply given by $R_{lin}=\partial_\mu \partial^\mu h_\nu^\nu-\partial_\mu \partial_\nu h^{\mu \nu}$; with the chosen form of the metric, the only time-dependent contribution is $\partial_0^2 h_{rr} \propto \tau^{-2} h_{rr} \ll \partial_r^2 h_{rr}$.)
In principle it is possible to improve the model by increasing the number of degrees of freedom, making the algorithms more complicated but still manageable, at least in two ways: (1) besides the component $g_{rr}$, consider as variable also the component $g_{00}(r)$, which at the moment is taken constant and equal to 1; (2) introduce a dependence on an angle $\theta$. The full expression of $R$ in this case is given in [@modanese2019metrics] and refs.
In the discretized version of the action the variable $r$ runs on an interval $(0,L)$ divided into $N$ parts. After defining $\delta=L/N$, we can say that $r$ takes the values 0, $\delta$, $2\delta$, ... , $N\delta$, or $\{h\delta, h=0,1,...,N\}$. The field takes values $A_h$, $h=0,1,...,N$, corresponding to $A(h\delta)$.
The boundary condition on the right end of the interval is $A(L)=A_N=1$, while on the left, for $A(0)$, we do not set any constraint. We suppose that $A(r)=1$ for $r \ge L$. As a consequence, we are not considering metric perturbations extended to infinity, but only fields different from flat space in $(0,L)$ (localized fluctuations). Clearly, $L$ must be regarded as one of the parameters for which we will need a scaling analysis.
Upon quantization the variables $A_0$, $A_1$, ... $A_N$ become the integration variables of a path integral, with measure given by the DeWitt super-metric (see [@hamber2008quantum], Sect. 2.4). Since the action is not positive definite, we suppose at the beginning that this path integral is of the Lorentzian kind, with weight $e^{iS/\hbar}$.
In the continuum action (\[S-cont\]) we replace the integral with a sum and so we obtain the discretized action $$S\simeq \frac{\tau L}{GN} \sum_{h=0}^N S_h
\label{S-disc}$$ with $$S_h=\sqrt{|\hat{A}_h|} \left( \frac{2h+1}{2\hat{A}_h^2} (A_{h+1}-A_h)+1-\frac{1}{\hat{A}_h} \right)$$ and $$\hat{A}_h=(A_{h+1}+A_h)/2$$
We are looking for anomalous fluctuations with respect to the trivial classical solution $A(r)=1$ everywhere, which gives $S=0$ (flat space). Our idea is to use the path integral as follows: if there is an ensemble of non-trivial metrics such that $S/\hbar \ll 1$, we suppose that they may describe important vacuum fluctuations. They do not need to be stationary points of the action like the classical configurations; they can also be exact zero modes of the action (for example, those we have found already with analytical techniques \[CQG\]) or modes with almost-zero action. An important requirement to make them relevant is that they must have a large volume in configuration space: the Montecarlo simulations will tell us if this is the case, and we may also expect (as confirmed in [@modanese2019metrics]) that according to the same simulations certain exact analytical zero modes will turn out to be too little probable to be physically relevant. A further discussion of this idea in relation to the stationary phase principle can be found in Sect. \[disc\].
Results at the Planck scale
---------------------------
In [@modanese2019metrics] we chose units such that $c=\hbar=G=1$, and we chose to explore a duration and length scale $\tau=1$, $L=1$ (Planck scale). We took $N=100$ in order to have a meaningful but “quick” discretization and we run a Metropolis algorithm [@newman1999monte] with a return probability $\exp(-\beta^2 S^2)$ or $\exp(-\beta|S|)$ in order to avoid the instability problems related to the indefinite sign of the action.
The result of the simulations is that for suitable values of the inverse temperature $\beta$ one finds an ensemble of equilibrium configurations in which $\langle S \rangle \sim \delta \cdot 10^{-7} \sim 10^{-9}$ and $\langle S^2 \rangle \sim \delta^2 \cdot 10^{-14}$ or less. The sum $\sum_{h=0}^N S_h$ is found to oscillate around zero with an amplitude $\sim 10^{-7}$.
In other words, after starting formally with a Lorentzian weight $e^{iS/\hbar}$ in order to skip the instability problems, and after realizing that the weight $e^{iS/\hbar}$ cannot be implemented numerically, the trick of using a weight $\exp(-\beta^2 S^2)$ or $\exp(-\beta|S|)$ in the algorithm is not meant as a solution of the instability or a way to study the full dynamics, but only as a way to obtain explicitly a set of fields with almost-zero action and a large volume in configuration space.
In this context, the choice of the inverse temperature $\beta$ is a matter of convenience. After some trials we find that if $\beta$ is too small (high temperature) the algorithm stabilizes quickly but the configurations obtained have values of the action fluctuating in a wide range. On the other hand, if $\beta$ is too large (low temperature) equilibrium cannot be obtained in a reasonable number of steps and the system keeps “drifting” in some direction. In general, a Metropolis algorithm is in equilibrium at a certain temperature when the ratio between the frequency of steps towards lower energy/action and the frequency of steps towards higher energy/action remains approximately constant. As seen from Tab. \[table2\], however, we can fulfil the condition above by choosing $\beta$ in a certain range. For instance, for $N=12800$ we easily obtain thermalization in the range $128\cdot 10^7 \le \beta \le 1024\cdot 10^7$.
The $\{A_h\}$ configurations of the equilibrium ensemble have a peculiar dependence on the coordinate $r$ ($r=\delta h$): a sort of polarization with a step in the middle of the interval and $A<1$ in the inner region, $A>1$ in the outer region. Intuitively this matches the expectation of a cancellation between contributions to the integral of $R$ over different regions, which is also a typical feature of some of the analytical zero modes [@modanese2007vacuum]. One can check that the inner region has always negative $R$, while the opposite holds for the outer region. We shall see below that when the number $N$ of sub-intervals in the discretized action grows (continuum limit), this simple pattern of “polarization into two regions” changes.
Scaling properties {#sec-scaling}
==================
Up to this point, what we can conclude is that at a length scale of the order of $L_P$ the discretized action permits the existence in the path integral of configurations which display strong deviations from flat space and polarization in $R$. In order to analyse the behavior at larger scale, we use natural units, in which $\hbar=c=1$. In these units the unit of length is 1 cm and the time $\tau$ is also expressed in cm$_t$, with the conversion 1 sec = 1 cm$_t \cdot (3\cdot 10^8)$. The Newton constant $G$ in natural units is equal to the square of the Planck length: $G \simeq L_P^2 \simeq 10^{-66}$ cm$^2$.
It follows that as magnitude order we can rewrite the discretized action (\[S-disc\]) as $$\frac{S}{\hbar} \simeq 10^{66}\tau \hat{S} \ \ \ \ ({\rm in \ natural \ units \ } \hbar=c=1)
\label{Ssuh}$$ $$\hat{S}=\frac{L}{N} \sum_{h=0}^N S_h
\label{Shat}$$
Due to the factor $10^{66}$, it seems very difficult to obtain $S/\hbar \ll 1$ in the discretized model, as soon as $L$ and $\tau$ are larger than the Planck scale $10^{-33}$ cm. If, however, we consider the continuum limit $N\to \infty$, it may be possible that the action $S/\hbar$ of the configurations built in our simulations becomes $\ll 1$, provided the scaling of $\hat{S}$ in $N$ is such that $\hat{S}\to 0$ quickly enough. For this reason we made several Metropolis simulations in order to compute $\hat{S}$ with increasing $N$.
Results at “macroscopic” $L$
----------------------------
In the first set of numerical trials that we performed in order to investigate the scaling for $N\to \infty$ we fixed the length $L$ to 10 units and the time $\tau$ to 1 unit. (Thinking of the field configurations in terms of vacuum fluctuations, $L$ represents their size and $\tau$ their duration.) The purpose of this choice is to compare and connect the present data, at the numerical level, with those obtained in our previous work. Here, however, the physical units employed are different and $L=10$ means $L=10$ cm. This is a macroscopic scale and certainly not our final target. We shall see that it is straightforward to pass from results at this scale to results at a length $L$ more appropriate to vacuum fluctuations (atomic and subatomic scale). That is because in the Metropolis algorithm employed the parameters $\beta$ and $L$ are multiplied by each other, so any reduction of $L$ can be compensated by an increase in the reciprocal temperature $\beta$ without affecting the thermal convergence of the algorithm.
In order to make contact with our previous data, let us start with a number of sub-intervals $N=100$ and proceed by repeatedly multiplying $N$ by 2. In this first series of trials we change $\beta$ in inverse proportion to $N$, in such a way that the factor $\beta L/N$ in the exponent of $\exp(-\beta |\hat{S}|)$ stays constant and optimal thermalization is achieved. The variation in $\hat{S}$ is due to the factor $1/N$ and to the fact that the sum $\sum_h S_h$ has more terms. The good news is that this finer subdivision leaves $\sum_h S_h$ almost unchanged and so $\hat{S}$ scales as $1/N$ (Tab. \[table1\]).
When $N$ is increased, we also need to increase the number of Montecarlo steps in order to obtain precise results, because at each step the algorithm changes at random one value of $A_h$ in the range $h=0,1,...,N$, by an amount $\pm \varepsilon$. The averages in the table are computed well after thermalization (after 50% of the steps for $N$ up to 25600, and then after 75% of the steps). The quantity $\langle e^{-\beta \hat{S} } \rangle$ is the average of the return probability for the steps in which $|\hat{S}|$ increases. This probability increases with $N$ (except for $N=100$ and $N=200$, where the polarization pattern is changing, see Sect. \[pol\]). $\langle \hat{S} \rangle$ and $\langle \hat{S}^2 \rangle$ are the averages of the action $\hat{S}$ and of its square. The most important quantity is $\langle \hat{S}^2 \rangle$, which tells us how much the action oscillates about its average. The dependence of $\langle \hat{S}^2 \rangle$ on $N$ reported in Tab. \[table1\] is also plotted in Fig. \[scaling\], from which a scaling very close to $1/N^2$ can be deduced. This means that in the continuum limit $N\to \infty$ the phase $S/\hbar$, eq. (\[Ssuh\]), can actually become $\ll 1$, in spite of the large dimensional factor $10^{66}$, especially if we start from microscopic values of $L$ and $\tau$ (see below). Moreover, there is a favourable scaling in the parameter $\beta$ (Tab. \[table2\]).
How should one interpret the increasing values of $\beta$ needed for equilibrium as $N$ grows? As explained at the beginning of Sect. \[sec-dis\], lower values of $\beta$ imply in general larger fluctuations of the action, while we need $\langle \hat{S}^2 \rangle$ to decrease at least as $1/N^2$ for the continuum limit to be effective. The interpretation is then that in the continuum limit the field configurations of the equilibrium ensemble have a very low temperature, i.e. they are very close to the minimum of the classical action. This is in qualitative agreement with the findings of [@bonanno2013modulated], namely that the true minimum of the stabilized action is obtained for a class of oscillating metrics, breaking translational invariance.
$N$ $\ \beta \ $ MC steps $\langle e^{-\beta |\hat{S}|} \rangle$ $\langle \hat{S} \rangle$ $\langle \hat{S}^2 \rangle$
-------- ------------------ ------------------- ------------------------------------------ --------------------------- -----------------------------
100 $10^7$ $2\cdot 10^9$ 0.17 $4.3\cdot 10^{-9}$ $2.0 \cdot 10^{-14}$
200 $2\cdot 10^7$ $2\cdot 10^9$ 0.11 $6.4\cdot 10^{-9}$ $5.1 \cdot 10^{-15}$
400 $4\cdot 10^7$ $4\cdot 10^9$ 0.022 $5.4\cdot 10^{-9}$ $1.3 \cdot 10^{-15}$
800 $8\cdot 10^7$ $8\cdot 10^9$ 0.024 $1.4\cdot 10^{-9}$ $3.2 \cdot 10^{-16}$
1600 $16\cdot 10^7$ $8\cdot 10^9$ 0.055 $3.0\cdot 10^{-10}$ $7.8 \cdot 10^{-17}$
3200 $32\cdot 10^7$ $8\cdot 10^9$ 0.14 $1.2\cdot 10^{-9}$ $2.3 \cdot 10^{-17}$
6400 $64\cdot 10^7$ $8\cdot 10^9$ 0.27 $1.5\cdot 10^{-9}$ $9.9 \cdot 10^{-18}$
12800 $128\cdot 10^7$ $16\cdot 10^9$ 0.25 $8.7\cdot 10^{-10}$ $2.8 \cdot 10^{-18}$
25600 $256\cdot 10^7$ $16\cdot 10^9$ 0.29 $3.4\cdot 10^{-10}$ $5.8 \cdot 10^{-19}$
204800 $2048\cdot 10^7$ $16\cdot 10^{10}$ $3.3\cdot 10^{-11}$ $5.3 \cdot 10^{-21}$
409600 $4096\cdot 10^7$ $16\cdot 10^{10}$ $1.5\cdot 10^{-11}$ $1.3 \cdot 10^{-21}$
: Average values of the return probability $\langle e^{-\beta \hat{S}} \rangle$, the action $\langle \hat{S} \rangle$ and the squared action $\langle \hat{S}^2 \rangle$ in dependence on $N$ (number of sub-intervals of $(0,L)$). Here $L=10$ cm. The inverse temperature $\beta$ changes in proportion to $N$, in order to maintain the factor $\beta/N$ constant. The discretized field components $A_h$ are randomly increased in the Montecarlo steps by $\pm \varepsilon$, with $\varepsilon=10^{-6}$. For the last two values of $N$ the calculation of $\langle e^{-\beta \hat{S}} \rangle$ (and of $\langle A_h \rangle$) was omitted in order to speed-up the algorithm and increase the precision in $\langle \hat{S}^2 \rangle$. See also plot of $\langle \hat{S}^2 \rangle$ in Fig. \[scaling\]. []{data-label="table1"}
$\ \beta \ $ $N$ MC steps $\langle e^{-\beta |\hat{S}|} \rangle$ $\langle \hat{S} \rangle$ $\langle \hat{S}^2 \rangle$
------------------ -------- ---------------- ------------------------------------------ --------------------------- -----------------------------
$128\cdot 10^7$ 12800 $16\cdot 10^9$ 0.25 $8.7\cdot 10^{-10}$ $2.8 \cdot 10^{-18}$
$256\cdot 10^7$ 12800 $16\cdot 10^9$ 0.16 $5.0\cdot 10^{-10}$ $8.2 \cdot 10^{-19}$
$512\cdot 10^7$ 12800 $16\cdot 10^9$ 0.095 $3.3\cdot 10^{-10}$ $2.8 \cdot 10^{-19}$
$1024\cdot 10^7$ 12800 $16\cdot 10^9$ 0.055 $2.2\cdot 10^{-10}$ $1.0 \cdot 10^{-19}$
: Scaling of $\langle e^{-\beta \hat{S}} \rangle$, $\langle \hat{S} \rangle$ and $\langle \hat{S}^2 \rangle$ in dependence on $\beta$ with $N$ fixed, $L=10$, $\varepsilon =10^{-6}$. Note the decrease of the average return probability $\langle e^{-\beta \hat{S}} \rangle$, coherent with the role of the inverse temperature $\beta$ in the thermalization process.[]{data-label="table2"}
![Scaling of the average squared action $\langle S^2 \rangle$ as a function of $N$. Log-log scale; dots represent values from Tab. \[table1\], while the dashed line represents a dependence $N^{-1}$ and the solid line a dependence $N^{-2}$. []{data-label="scaling"}](scaling-of-S2.pdf){width="7.0cm" height="5.1cm"}
Scaling in $\beta$ for microscopic $L$
--------------------------------------
Tab. \[table3\] shows the scaling of $\langle e^{-\beta \hat{S}} \rangle$, $\langle \hat{S} \rangle$ and $\langle \hat{S}^2 \rangle$ in dependence on $\beta$ for $L$ at a microscopic scale, namely $L =10^{-13}$ cm.
The first value of $\beta$ is chosen, to ensure thermalization, in such a way that the product $L\beta$ is the same as for data with $L=10$, $\beta=128\cdot 10^7$, $N=12800$; this implies that $\beta$ must now be equal to $128\cdot 10^{21}$.
The results for $\langle \hat{S} \rangle$ and $\langle \hat{S}^2 \rangle$ are seen in Tab. \[table3\] to scale in proportion to $L$, in comparison to the results in Tab. \[table1\]. This could have been predicted from the fact that the discretized action is proportional to $\delta=L/N$, and so are its variations $\Delta \hat{S}$ and $\Delta |\hat{S}|$ in the Montecarlo algorithm. We obtain here another confirmation that the algorithm scales as expected with respect to the parameters $L$ and $\beta$.
$\ \beta \ $ $N$ MC steps $\langle e^{-\beta |\hat{S}|} \rangle$ $\langle \hat{S} \rangle$ $\langle \hat{S}^2 \rangle$
--------------------- -------- ---------------- ------------------------------------------ --------------------------- -----------------------------
$128\cdot 10^{21}$ 12800 $16\cdot 10^9$ 0.25 $8.8\cdot 10^{-24}$ $2.9 \cdot 10^{-46}$
$256\cdot 10^{21}$ 12800 $16\cdot 10^9$ 0.15 $4.9\cdot 10^{-24}$ $8.0 \cdot 10^{-47}$
$512\cdot 10^{21}$ 12800 $16\cdot 10^9$ 0.093 $3.3\cdot 10^{-24}$ $2.8 \cdot 10^{-47}$
$1024\cdot 10^{21}$ 12800 $16\cdot 10^9$ 0.053 $2.2\cdot 10^{-24}$ $1.0 \cdot 10^{-47}$
: Scaling of $\langle e^{-\beta \hat{S}} \rangle$, $\langle \hat{S} \rangle$ and $\langle \hat{S}^2 \rangle$ in dependence on $\beta$ with $N$ fixed, $L =10^{-13}$ cm (“microscopic scale”), $\varepsilon =10^{-6}$.[]{data-label="table3"}
Polarization pattern {#pol}
--------------------
The simple bipolar polarization pattern observed in the averaged field values $\langle A_h \rangle$ for $N=100$ [@modanese2019metrics] changes when $N$ increases. Multiple oscillations begin to appear, with an envelope changing with $N$ (see an example in Fig. \[pol-1\], (a)), until for $N$ approximately greater than 3200 the situation stabilizes and all the oscillations have almost exactly the same amplitude (Fig. \[pol-1\], (b)). The number of oscillations does not depend on any of the physical parameters $N$, $L$, $\beta$ and $\varepsilon$. It appears to be a general “mathematical” feature of the minimum configuration of the discretized action $\sum_h S_h$. Fig. \[pol-2\] shows two details of Fig. \[pol-1\] (b), namely on sub-intervals with 1600 and 400 values of $h$. From these details we can see that the fixed total number of oscillations in the interval $(0,L)$ is approximately equal to $10^2$, even though, as mentioned, there appears to be no relation between this number and the physical parameters.
![(a) Polarization pattern for $N=800$ sub-intervals. (b) Polarization pattern for $N=6400$ sub-intervals. []{data-label="pol-1"}](20-3.pdf "fig:"){width="10.5cm" height="6.1cm"} ![(a) Polarization pattern for $N=800$ sub-intervals. (b) Polarization pattern for $N=6400$ sub-intervals. []{data-label="pol-1"}](20-7.pdf "fig:"){width="10.5cm" height="6.1cm"}
![(a) detail of the polarization pattern for $N=6400$ sub-intervals, showing $1/4$ of the total interval. (b) Showing $1/16$ of the total interval. []{data-label="pol-2"}](20-7-short.pdf "fig:"){width="10.5cm" height="6.1cm"} ![(a) detail of the polarization pattern for $N=6400$ sub-intervals, showing $1/4$ of the total interval. (b) Showing $1/16$ of the total interval. []{data-label="pol-2"}](20-7-short-2.pdf "fig:"){width="10.5cm" height="6.1cm"}
As discussed in [@modanese2019metrics], in polarized configurations of this kind there are contributions to the local curvature coming both from the plateaus and from the steps. When $N$ is large, the plateaus comprise hundreds of values of $h$ (the discretization index). For each $h$ we have a value of $\langle A_h \rangle$ at the end of the simulation, and on the plateaus these values are typically constant up to $10^{-3}$. (The values of $A_h$ displayed in Figs. \[pol-1\], \[pol-2\] are actually averages $\langle A_h \rangle$.) The steps comprise only a few values of $h$, with standard deviation of $\langle A_h \rangle$ of the order of $10^{-2}$. This shows that after the Monte Carlo algorithm has attained thermal equilibrium, this equilibrium is quite stable. As displayed in Tab. \[table1\], when $N$ increases thermalization requires a lower temperature.
![Contributions to the integral of $\cos[\hbar^{-1}(x^2-y^2)]$ in the square $\{0\le x\le 1, 0\le y \le 1\}$ obtained through an adaptive Monte Carlo integration with inverse temperature $\beta$. Parameters: (a) $\hbar=0.1$, $\beta=1$. (b) $\hbar=0.05$, $\beta=2$. (c) $\hbar=0.1$, $\beta=7.8\cdot 10^{-3}$. The contributions near the origin come from the stationary point of the phase, those along the diagonal from the zero mode $y=x$. In (b) the region near the stationary point is smaller because $\hbar$ is smaller, but the length of the zero mode is unaffected. In (c) the contributions of the disconnected zero modes $y=x\pm 2\pi \hbar$ also appear, because the temperature is much higher. []{data-label="staz-1"}](N50-beta-1.pdf "fig:"){width="7.0cm" height="5.1cm"} ![Contributions to the integral of $\cos[\hbar^{-1}(x^2-y^2)]$ in the square $\{0\le x\le 1, 0\le y \le 1\}$ obtained through an adaptive Monte Carlo integration with inverse temperature $\beta$. Parameters: (a) $\hbar=0.1$, $\beta=1$. (b) $\hbar=0.05$, $\beta=2$. (c) $\hbar=0.1$, $\beta=7.8\cdot 10^{-3}$. The contributions near the origin come from the stationary point of the phase, those along the diagonal from the zero mode $y=x$. In (b) the region near the stationary point is smaller because $\hbar$ is smaller, but the length of the zero mode is unaffected. In (c) the contributions of the disconnected zero modes $y=x\pm 2\pi \hbar$ also appear, because the temperature is much higher. []{data-label="staz-1"}](griglia-4-beta-2.pdf "fig:"){width="7.0cm" height="5.1cm"} ![Contributions to the integral of $\cos[\hbar^{-1}(x^2-y^2)]$ in the square $\{0\le x\le 1, 0\le y \le 1\}$ obtained through an adaptive Monte Carlo integration with inverse temperature $\beta$. Parameters: (a) $\hbar=0.1$, $\beta=1$. (b) $\hbar=0.05$, $\beta=2$. (c) $\hbar=0.1$, $\beta=7.8\cdot 10^{-3}$. The contributions near the origin come from the stationary point of the phase, those along the diagonal from the zero mode $y=x$. In (b) the region near the stationary point is smaller because $\hbar$ is smaller, but the length of the zero mode is unaffected. In (c) the contributions of the disconnected zero modes $y=x\pm 2\pi \hbar$ also appear, because the temperature is much higher. []{data-label="staz-1"}](N50-beta7-8125e-3.pdf "fig:"){width="7.0cm" height="5.1cm"}
Discussion, conclusions {#disc}
=======================
A 2D integral with stationary phase and zero modes {#sec-2di}
--------------------------------------------------
A key concept of this work, already discussed analytically in [@modanese2007vacuum; @modanese2016functional; @modanese2017ultra], is that of zero modes of the action. This relates to a peculiar property of the gravitational field, not easily found in other physical systems: the non-positivity of the action in the path integral $\int \, d[g_{\mu \nu}]e^{iS/\hbar}$.
A simple mathematical example can help to elucidate the idea of zero modes. Consider a 2D integral with oscillating integrand, of the form $$I=\int dx \int dy \cos[\phi(x,y)]f(x,y),\ \ \ {\rm with} \ \ \phi(x,y)=\hbar^{-1}(x^2-y^2)
\label{integrale}$$ where $f(x,y)$ is a smooth function, and suppose that $\hbar \ll 1$. We expect the main contribution to the integral to come from the region near the origin $x=0$, $y=0$, where the phase of the cosine is stationary. This would in fact be true if the phase was $\phi=[\hbar^{-1}(x^2+y^2)]$. However in this case the phase is zero, even if not stationary, along the lines $y=\pm x$. Could the infinite region along these lines give a contribution to the integral comparable to the region near the origin? This can be verified using an algorithm for numerical integration similar to an adaptive Monte Carlo. The algorithm samples the integrand at random starting from the origin and moving in small steps $(\delta x$, $\delta y)$. Each step is accepted unconditionally if it gives an increase $\delta |\cos \phi|$ positive, or else accepted with probability $e^{\beta \delta |\cos \phi|}$. In this way, the sampling points are more dense in the regions where there are larger contributions to the integral, and the effect can be tuned varying the inverse temperature $\beta$.
Let us reduce the integration region to the square $\{ 0\le x \le 1,\ 0\le y \le 1\}$ and divide it into, for example, $50 \times 50$ cells of side $a=0.02$ with indices $i$, $j$. If the number of sampling points falling in the cell $(i,j)$ is $c_{ij}$ and the sum of the values of the integrand at those points is $s_{ij}$, the integral is approximated by $$I \simeq a^2 \sum_{i,j} \frac{s_{ij}}{c_{ij}}$$
Fig. \[staz-1\] represents with a density plot the contributions of the individual cells in a case where the function is simply $f(x,y)=1$ (see caption for details). If we compute instead the average $\langle r \cos \phi \rangle$ over all sampling points, namely with $f(x,y)= \sqrt{x^2+y^2} $, we obtain at low temperature approximately 0.7 (half the diagonal), showing that the regions which contribute to the integral are in fact spread along the zero mode. However, when the temperature is increased (Fig. \[staz-1\], (c)) the strong destructive interference along the zero modes tend to cancel their contributions, leaving only the contribution near the origin. This can also be seen from the fact that the average $\langle r \cos \phi \rangle$ decreases.
In the simple case of the 2D integral in $x$, $y$ of eq. (\[integrale\]) all these properties can be easily predicted, because we can plot the integrand and we know that the main contributions arise in the regions where the integrand is large and are directly proportional just to the area of these regions. One can also predict that being zero modes 1-dimensional, in the limit of small $\hbar$ they do not contribute to the 2D integral.
Extension to higher dimension {#sec-ext}
-----------------------------
For a path integral in infinite dimensions, with a non-polynomial action, all this [*a priori*]{} information is not available. Even if we are able to solve the exact equation for the zero modes (analogue of $x^2-y^2=0$ in the 2D example) [@modanese2007vacuum; @modanese2019metrics], it is hard to assess the “volume” of the solutions in the functional space, and even harder to asses this volume for the weaker but crucial condition $S\ll \hbar$.
An higher-dimensional extension of the polynomial example above could be in principle the following: consider the integral $$\int d^mx \int d^ny \, \cos[\hbar^{-1}(x_1^2+...+x_m^2-y_1^2-...-y_n^2)]$$ and the zero modes of the phase, which satisfy the equation $$x_1^2+...+x_m^2-y_1^2-...-y_n^2=0$$ The dimension of these modes is $(m+n-1)$ (for example, for a phase proportional to $(x_1^2+x_2^2-y_1^2)$ the zero mode is a conical surface), so for $m,n\to \infty$ they might indeed contribute to the integral.
To complete the analogy, note that in the gravitational case the contribution to the adaptive Monte Carlo coming from the configurations which make the action stationary appears to be actually negligible.
Limitations of the present approach and comparison with other methods {#sec-limits}
---------------------------------------------------------------------
The use of the absolute value of the action in the Euclidean path integral allows to circumvent the stability issues. It is admittedly a strong assumption, whose validity should be further checked, and which does not hold for the dynamics of configurations with action $|S|\gg \hbar$, such that the phase factor in the path integral is rapidly oscillating.
For the practical purposes of a numerical simulation in the region $|S|\ll \hbar$, however, using the absolute value appears to be not very different from other stabilization techniques, like the introduction of an $R^2$ term [@bonanno2013modulated]. Being the flat space configuration, with $R=0$ everywhere, a stationary point, the absolute value does not produce any discontinuity in the derivatives of the action. In previous versions of the simulations we used the squared action instead of the absolute value, obtaining similar results. The algorithm could be further adapted to the insertion of an $R^2$ term.
One can safely state, in any case, that results of simulations with the absolute value are exact for a theory with action $|S_{E.H.}|$, which does not coincide in general with the Einstein-Hilbert theory, but has the same classical field equations, obtained minimizing $|S_{E.H.}|$. The logic here would be similar to that of theories with lagrangian $f(R)$ [@nojiri2017modified], even though at the level of perturbative quantum field theory only the Einstein-Hilbert action represents massless particles with spin 2.
On another front, we note that in the present approach it is impossible to address the diffeomorphism symmetry as clearly as done, for instance, in the Regge calculus with full simulation of the quantum dynamics ([@hamber2008quantum; @hamber2019vacuum] and refs.). Even in the spherically symmetric case presented here, an invariance under reparametrisations of the coordinate $r$ remains. When one generates a new metric configuration, in principle it is possible that the geometry is not actually being changed, but one is just doing such a reparametrization. In practice, however, this is extremely unlikely when randomly changing one of the discrete variables $A_h$ at a time, as it happens in our algorithm. Furthermore, [*a posteriori*]{} we can be sure that the configurations found at equilibrium are really different from the flat space we started from.
In these configurations, the coordinate distance cannot be interpreted as a physical distance, the latter being given instead by the usual expression $ds^2=g_{rr}dr^2$. This means, for instance, that the real length of the upper plateaus in Fig. \[pol-2\] is definitely larger than the length of the lower plateaus.
The analogies between our results and those of Ref. [@bonanno2013modulated] are stimulating, but several differences should be noticed, in addition to the different stabilization methods:
\(1) In [@bonanno2013modulated], the degree of freedom in the metric is a conformal factor, while here it is the component $g_{rr}$ in a stationary approximation.
\(2) The authors of [@bonanno2013modulated] search for the vacuum state by minimizing the action through a rigorous analytical approach, while we rely on numerical simulations. That is why they interpret the rippled spacetime obtained as one that becomes flat upon averaging over a periodicity volume, i.e. after a purely [*classical*]{} coarse graining. In our discretized model, we interpret the rippled spacetime obtained as an ensemble of purely quantum states with no classical counterpart. We also find, however, that a proper continuum limit is possible only in the low temperature limit (large $\beta$), which takes us back to the classical theory. In this sense there is a qualitative agreement between the two approaches. There might also be a connection between our concept of zero modes of the action and the restricted-space minimization of Sect. 2 in [@bonanno2013modulated].
Conclusion {#sec-concl}
----------
In this article, we have studied the gravitational path integral of spherically symmetric space-times independent in time using numerical Monte Carlo methods. The system is first reduced to the spherically symmetric and time-independent setting, before it is discretized in radial direction. The reduced Einstein-Hilbert action is Wick-rotated and only its absolute value is considered in the path integral, since it is not bounded from below. The goal is to explore the configurations with almost vanishing action since these might contribute in the full Lorentzian path integral. In the numerical studies oscillations are found that suggest large deviations from the classical vacuum solution.
Usually, one expects path integrals to be dominated by classical solutions given appropriate boundary data. However, this does not seem to be the case here as typical configurations appear to significantly deviate from the classical vacuum solution and show oscillatory behaviour. The interpretation of these configurations is not completely clear and open questions remain on whether it would be realistic to find them in the full Lorentzian path integral and, if yes, if this would result in observational consequences.
A non-perturbative Monte Carlo algorithm for the discretized action like that employed in this work (and in much more complete form by Hamber, Ambj[ø]{}rn and co-workers [@hamber2008quantum; @hamber2019vacuum; @ambjorn2012nonperturbative; @loll2019quantum]) seems to be at present one of the best tools available for exploring quantum metrics closely connected to the classical vacuum state like the polarized configurations we have found in this work. The astounding detailed structure of these configurations (Sect. \[pol\]) and their stability and reproducibility are intriguing, and possibly part of more general patterns valid beyond the approximations made here (spherical symmetry, $g_{00}=1$, modes almost stationary in time).
Some physical comparisons can be drawn, as in [@bonanno2013modulated; @bonanno2019structure], to kinetic condensates in other quantum field theories [@lauscher2000rotation] and to anti-ferromagnetic systems in statistical physics ([@branchina1999antiferromagnetic] and refs.).
We have shown that the average squared action $\langle \hat{S}^2 \rangle$ of the polarized configurations scales as $1/N^2$ up to a number $N$ of sub-intervals of the order of $10^6$, for any length scale $L$. If this behavior can be extrapolated to larger $N$, their adimensional action $S/\hbar\simeq 10^{66}\tau \hat{S}$ can be $\ll 1$ also at scales much larger than the Planck scale.
Future work should be devoted to an extension of the simulations to the case with angular and time dependence, and to a phenomenological comparison with observational constraints on gravitational vacuum fluctuations [@amelino2001phenomenological; @quach2015gravitational].
[^1]: Email address: giovanni.modanese@unibz.it
|
---
abstract: 'Tunneling two level systems (TLS), present in dielectrics at low temperatures, have been recently studied for fundamental understanding and superconducting device development. According to a recent theory by Burin *et al.*, the TLS bath of any amorphous dielectric experiences a distribution of Landau-Zener transitions if exposed to simultaneous fields. In this experiment we measure amorphous insulating films at millikelvin temperatures with a microwave field and a swept electric field bias using a superconducting resonator. We find that the maximum dielectric loss per microwave photon with the simultaneous fields is approximately the same as that in the equilibrium state, in agreement with the generic material theory. In addition, we find that the loss depends on the fields in a way which allows for the separate extraction of the TLS bath dipole moment and density of states. This method allows for the study of the TLS dipole moment in a diverse set of disordered films, and provides a technique for continuously inverting their population.'
author:
- 'M. S. Khalil'
- 'S. Gladchenko'
- 'M. J. A. Stoutimore'
- 'F. C. Wellstood'
- 'A. L. Burin'
- 'K. D. Osborn'
bibliography:
- 'References2.bib'
title: 'Landau-Zener population control and dipole measurement of a two level system bath'
---
In quantum computing, two-level systems (TLS) in dielectrics have been found to function as an environmental bath for superconducting quantum elements [@Martinis2005; @Simmonds2004; @Constantin2009; @Gao2008] and as quantum memory bits in a hybrid quantum computer [@Neeley2008]. The environmental impact of the deleterious bath has led to improved materials [@Oh2006; @Cicak2010; @Paik2010] for superconducting qubits. In recent qubit designs[@Geerlings2012; @Paik2011] the geometrical architecture allows only for a small amount of electrical energy storage in the deleterious amorphous metal oxides. Over four decades ago, a now standard model of TLS was introduced which describes charged nanoscale systems moving independently in a distribution of double well potentials, presumably created by undercoordinated bonds [@Phillips1972; @Anderson1972]. Recent measurements of individual TLSs under application of a strain field are in agreement with this model [@Grabovskij2012]. Although the TLS effects are generally known, the precise identity of the atomic defects or bonds that comprise the TLS and dipole moments from a given material are generally not known [@Pohl2002; @Southworth2009; @Queen2013]. Furthermore, it was found that the sudden application of strain or electric fields can result in an immediate change in the TLS density, followed by a slow glassy relaxation to the equilibrium state [@Salvino1994; @Rogge1996; @Natelson1998], possibly caused by weak TLS-TLS interactions [@Carruzzo1994; @Burin1995].
In the case of resonant microwave measurements, the loss tangent is proportional to the weighted TLS density–the TLS density times the dipole moment squared. Experiments on individual TLS provide important quantum properties [@Palomaki2010; @Neeley2008; @Shalibo2010; @Stoutimore2012], but have previously been restricted to an alumina tunneling barrier and must characterize many TLS, one at a time, in order to extract an average dipole moment of the film. The Landau-Zener effect has been used to study a wide variety of qubit systems, including superconducting circuits [@Oliver2005; @Berns2008; @Sillanpaa2006], silicon-dopants [@Dupont-Ferrier2013], and quantum dots [@Cao2013]. A recent theory using this effect predicts that TLS can be characterized using the quantum dynamics created by two simultaneous fields [@Burin2013]. Experimental realization of this theory, discussed below, reveals new information about the dynamics of the tunneling systems in amorphous films.
Here we describe measurements of the high-frequency $(\hbar\omega \gg k_BT)$ loss tangent of amorphous PECVD deposited $\mathrm{Si_3N_4}$ films [@Paik2010] in a non-equilibrium regime. This regime is reached by sweeping an electric field bias while probing the loss with a microwave (ac) field at millikelvin temperatures. As expected for TLS-laden films, the loss tangent decreases as the microwave power increases. However, we also found that with a sufficiently large bias sweep rate, the loss tangent in the non-equilibrium regime recovers the value of its linear-response steady-state measurement. We compared our loss tangent measurements at high sweep rates to the newly-proposed model, based on Landau-Zener dynamics of a conventional TLS distribution [@Burin2013]. By confirming this theory experimentally, we show that the standard TLS model is appropriate for studying a new non-equilibrium regime in amorphous solids, and also show a new method for extracting the TLS dipole moment. Agreement with the theory implies that the population of the TLS bath can be controlled.
Measurements were made with a thin-film superconducting aluminum 4.7 GHz resonator (see Fig. \[fig:TempDep\](a)) composed of a meandering inductor and four 250 nm thick amorphous $\mathrm{Si_3N_4}$-dielectric parallel-plate capacitors in an electrical bridge design (see Fig. \[fig:TempDep\](b)). Arms of the bridge are nominally identical, and a lead allows application of a voltage bias $V_{bias}$ creating a DC voltage difference of $V_{bias}/2$ across each capacitor. We apply resonant microwaves to the system via a coplanar waveguide transmission line that couples the microwave fields into the capacitors. The resonator coupled to the transmission line creates a notch filter, but it is nominally uncoupled from the bias line at resonance due to the balanced electrical design. From measurements of the microwave transmission through the coplanar waveguide [@Khalil2012], we extract the internal quality factor $Q_i=1/\tan\delta$, equal to the inverse loss tangent of the dielectric films, and the coupling quality factor of 6500. The device is mounted in a sealed copper box attached to the mixing chamber of a dilution refrigerator and measured at 33-200 mK. Filtered transmission lines and a cold low-noise HEMT amplifier allow resonator measurements with less than a single average photon excitation.
The standard TLS model assumes a broad number density distribution, $d^{2}n=d\Delta d\Delta_0 P_0/\Delta_0$, of double well TLS with energy $E_{TLS} = \sqrt{\Delta^2+\Delta_0^2}$, dependent on the tunneling energy $\Delta_0$ and asymmetry energy $\Delta$. The model yields a dielectric loss tangent $\tan\delta=\tan\delta_0 \tanh\left(\hbar\omega/2k_BT\right)/\sqrt{1+\left(E_{ac}/E_c\right)^2}$ [@VONSCHICKFUS1977], where $k_B$ is the Boltzmann constant, $T$ is the temperature, $\tan\delta_0=(\pi P_0p^2)/(3\epsilon)$, $P_0$ is the TLS spectral and spatial density, $p$ is the dipole moment of the TLS, $\epsilon$ is the dielectric permittivity, $E_{ac}$ is the ac field amplitude, and $E_c=\hbar/(p\tau)$. $\tau$ is a characteristic TLS lifetime that depends on the decoherence limit, such that $\tau=\sqrt{T_{1,min}T_2/3}$ for constant coherence time $T_2$ and $\tau=8\sqrt{T_{1,min}T_{2,min}}/(3\pi)$ for the spontaneous emission limit, $T_2=2T_1$; $T_1=(E_{TLS}/\Delta_0)^2T_{1,min}$ is the relaxation time. The loss tangent is mainly sensitive to TLS resonant dynamics near $\hbar\omega$. At low field amplitudes the linear equilibrium response $\tan\delta=\tan\delta_0$ is determined from Fermi’s Golden Rule, while at moderate fields the maximum Rabi frequency, $\Omega_{R0}=pE_{ac}/\hbar$, exceeds the TLS decoherence rate, $\Omega_{R0}>>1/\tau$, such that saturation occurs. Figure \[fig:TempDep\](c) shows $\tan\delta$ at zero field bias at two temperatures as a function of the RMS field $E_{ac, RMS}=E_{ac}/\sqrt{2}$. As expected, it decreases as the $E_{ac}$ increases and the low ac-field $\tan\delta$ decreases as $T$ increases. The $T$ dependence of the $\tan\delta$ is caused both by thermal saturation of the TLS and by changes in $\tau$ dependent on $T$.
According to the standard double-well model, the bias field $E_{bias}$ should adjust the asymmetry energy between the wells ($\Delta\rightarrow\Delta+2\vec{p}\cdot\vec{E}_{bias}$). Application of fixed voltage biases (data shown in Fig. \[fig:TempDep\](c) were taken with zero bias voltage) produce detectable but small variations of order 1% in the equilibrium loss tangent. These small changes are expected in the standard model distribution because the change in asymmetry results in the same TLS population near resonance within statistical variations.
In contrast, when we drive the resonator with sufficient microwave amplitude to saturate TLS in steady state we find that the loss tangent is sensitive to sufficiently rapid changes in the bias voltage. To observe how the sweep rate affects the loss, we measure the resonator response as a function of time while applying a square waveform, low-pass filtered with a single time constant of 8.5 ms, to $V_{bias}$. The bias voltage increases at 0.53 s such that $V_{bias}$ exponentially approaches 40 M V/m , while the voltage decreases at 1.78 seconds in an exponential approch towards 0 M V/m. The magnitude of the electric field sweep rate $\mid\dot{E}_{\textrm{bias}}\mid$ is shown in the Fig. \[fig:TempDep\](d) inset. Approximately 20 waveform cycles were averaged and a 5 ms time resolution was used to extract $\tan\delta$ at each time slice, which is shown in Fig. \[fig:TempDep\](d).
While the TLS density stays approximately constant as the bias is varied (according to the static measurements discussed above), Fig. \[fig:TempDep\](d) shows that $\tan\delta$ increases dramatically when the bias is swept, i.e. when the electric field has a significant sweep rate magnitude. Comparing Fig. \[fig:TempDep\](c) and \[fig:TempDep\](d) (with arrows shown) reveals that for both temperatures and during the fastest sweep rates, the (strong non-equilibrium) loss tangent approximately equals the linear-response equilibrium loss tangent, and it is smaller for lower sweep rates. Figure \[fig:TempDep\](e) shows this strong non-equilibrium $\tan\delta$ for several temperatures with the linear-response equilibrium $\tan\delta$; note that the former value is less than $5\%$ higher than the latter value. This correspondence between strong non-equilibrium $\tan\delta$ and the equilibrium linear-response $\tan\delta$ has been seen in multiple samples. The solid fit curve shows the expected thermal saturation for equilibrium TLS, indicating that the high sweep rate phenomena are related. We note that most of the return to the steady-state loss occurs while the bias voltage is changing at a rate $\mid\dot{E}_{\textrm{bias}}\mid> 10^{-4}$ V/m/s. For negligible sweep rates, $\mid\dot{E}_{\textrm{bias}}\mid < 10^{-4}$ V/m/s, a small amount of slow dynamics can be seen in the 33 mK data, which might possibly contain glassy relaxation phenomena, but regardless of the mechanism the 200 mK data returns to steady state equilibrium relatively fast indicating that coherence is involved in the slow 33 mK dynamics. Below we analyze the main nonequilibrium phenomenon, the loss tangent as a function of bias sweep rate, using the theory of Burin *et al.* [@Burin2013], which is predicted to apply (universally) to all amorphous dielectrics.
In our system the Rabi frequency, $\Omega_R$, of a TLS can be larger than the decoherence rate for a TLS, but it is always much smaller than the resonance frequency such that multiple photon processes can be ignored. A swept electric field bias changes the TLS energy at a rate of $\hbar v=p\dot{E}_{bias}\cos(\theta)(\Delta/E_{TLS})$ [@Burin2013], where $\theta$ is the angle between the field and the TLS dipole. Below we explore the non-equilibrium loss tangent in the fast sweep regime corresponding to $\Omega_{R0}^2/v_0<<1$, where $v_0=v(\Delta=\hbar\omega, \theta=0)$ is the maximum sweep rate, $\Omega_{R0}=\Omega_R(\Delta_0=\hbar\omega, \theta=0)$ is the maximum Rabi frequency on resonance, and as discussed earlier $\Delta_0$ is the TLS tunneling rate. When $v_0$ is sufficiently small, the steady-state equilibrium loss should be recovered due to TLS decoherence processes. For an individual TLS that is swept through the ac-field frequency, the probability of an adiabatic transition of the TLS-photon field from $|g,n\rangle$ to $|e,n-1\rangle$ is $P=1-e^{-\gamma}$, where $\gamma=(\pi\Omega^2_R)/(2v)$. This creates a non-equilibrium loss tangent [@Burin2013] of $$\tan\left(\delta\right)=\frac{16\pi P_0}{\epsilon E^2_{ac}}\int^{\hbar\omega}_0\frac{d\Delta_0}{\Delta_0}\frac{\hbar^2v(1-e^{-\gamma})}{\sqrt{1-\left(\frac{\Delta_0}{\hbar\omega}\right)^2}}.
\label{eq:tandNE}$$ In the fast sweep limit, $v_0\gg\Omega_{R0}^2$, the TLS pass through resonance rapidly such that they Landau-Zener tunnel to remain in their ground state $(|g,n\rangle)$ with a high probability (and excited state with a low probability). This low probability of individual excitation comes with a high rate of TLS crossings such that the loss tangent is actually higher than in steady-state equilibrium, and in agreement with theory, is approximately equal (see Fig. 1(e)) to the linear response loss tangent calculated from Fermi’s golden rule.
The data points in Fig. \[fig:SCurve\](a) show loss tangent measurements (normalized to maximum loss tangent), as a function of the bias rate. The thick blue symbols are produced by sweeping the bias with a 8.5 ms time-constant filter (similar to shown in Fig. \[fig:TempDep\](d) ) of varying amplitudes, and plotting the loss tangent for the maximum bias rate (the pulse amplitude divided by the rise time). The thin red symbols are produced by using a single wave form with a (slower) 50 ms time-constant filter and plotting the instantaneous loss tangent values against the instantaneous bias rate versus time (similar to the 33 mK data of Fig. \[fig:TempDep\](d) versus inset quantity, but with a longer exponential tail). The data shown are taken from upward voltage steps in bias for both data types; data from downward steps in bias (not shown) are nearly indistinguishable. Measurements were performed for three different applied microwave amplitudes (+, x, $\triangle$). While the amplitude of the input microwave field is constant for any given curve, the microwave field across the capacitors is also influenced by the loss tangent; the low sweep rate (steady-state) microwave field is given as a label for each curve in Fig. \[fig:SCurve\](a). The microwave field for the middle curves (x) vary from $E_{ac}=4.55$ V/m at the highest bias rate to $E_{ac}=15.52$ V/m at the lowest bias rates (the steady-state regime). Eq. (\[eq:tandNE\]) by itself is only applicable to our results at the fastest bias rates, $\xi>1$, where $\xi=(2v_0)/(\pi\Omega^2_{R0})$ is the dimensionless sweep rate, because it neglects TLS relaxation times, $T_1$ and $T_2$, which limit the loss in the steady-state regime ($\xi \ll 1$). At very slow bias rates, $\xi\ll1$, the response is adiabatic so the loss tangent approaches the steady-state loss tangent.
To model the loss for bias rates between the steady-state regime $(\xi\ll 1)$ and the strong non-equilibrium regime $(\xi \gg 1)$, we used a Monte-Carlo averaged solution of the TLS density matrix with $T_2=2T_1$. We fit the simulation to the data by varying the dipole moment (see solid curves in Fig. \[fig:SCurve\](a)) and use a loss tangent floor of $\tan\delta_{floor}=1.8\times 10^{-5}$ found from steady-state saturation measurements. We find excellent agreement using a single dipole moment, indicating that multiple dipole moments are not apparent in the dynamics. The fit revealed a TLS dipole moment of $p=7.9$ Debye and a value of $P_0=4.9 \times 10^{43} J^{-1}m^{-3}$ for the TLS spectral population density. The spontaneous emission limited relaxation time used in the fit is $T_{1,min}=3.0$ $\mu s$, and this can be alternatively extracted from the steady state loss tangent measurement using the extracted $p$. This moment also agrees with fits of only the fastest parts of the curves in Fig. \[fig:SCurve\](a) ($\xi>1$) to a numerical evaluation of Eq. (\[eq:tandNE\]). The error from the data analysis gives an imprecision of 3% for $p$, but the accuracy is limited to approximately 10% by a room-temperature calibration of the ac power. The same analysis applied to a second $\mathrm{Si_3N_4}$ film type, with a different stoichiometry and linear response loss tangent $(\tan\delta_0 \simeq 10^{-4})$, yielded the same dipole moment. This $p$ is comparable to that found for TLS in an amorphous $\mathrm{Al_2O_3}$ Josephson junction tunneling barrier [@Martinis2005]. However, we note that unlike previous dipole measurements in tunneling barriers our technique doesn’t require large statistical measurement of individual strongly coupled TLS but instead allows for ensemble TLS dipole measurement in a deposited film using a relatively fast measurement. Since this is an ensemble measurement it can be compared a previous measurements of a bulk (but not deposited) $\mathrm{SiO_2}$ insulator [@Golding1979].
In Fig. \[fig:SCurve\](b) we plotted one curve from each of the three ac drives in Fig. \[fig:SCurve\](a), scaling the bias rate on the x-axis by the parameter $\xi$ and scaling the y-axis by subtracting the constant bias steady-state loss tangent value $(\tan\delta_{ss})$ for each of the three sets of microwave amplitudes to account for the different steady-state losses. Here the dimensionless sweep rate $\xi$ is calculated for the same extracted dipole moment, $p=7.9$ D. The location of the step in loss tangent on the x-axis of Fig. \[fig:SCurve\](b) is expected from the theory. However, the collapse of the data to one curve shows that the scaled loss tangent (y-axis) quantity allows a check of the dipole moment without a Monte-Carlo simulation. In Fig. \[fig:SCurve\](b) the vertical lines represent critical bias rate (coded by symbols and color), $\xi_c=1/(\Omega_{R0}\sqrt{T_{1,min}T_2})$, above which the TLS dynamics is described by Landau-Zener theory and is not limited by TLS relaxation times. The model also predicts that when the bias rate is slowed such that the loss tangent is reduced to below $\tan\delta_0/2$, TLSs are continuously population inverted as they pass through the ac field frequency. Therefore data above $\xi_c$ and with a loss tangent below $\tan\delta_0/2$ represent a regime where TLS passing through resonance are inverted.
In conclusion, we have used a swept electric field to continuously spectrally-tune the broad distribution of TLS bath states in an insulating film within a microwave circuit. The swept bias field is applied in a complimentary way to the microwave field, and by utilizing the Landau-Zener effect a continuous population control near resonance is achieved, as monitored by the microwave loss tangent. Under application of a fast-swept bias field and a moderate microwave drive excitation, the population is primarily left in the ground state and the measured non-equilibrium loss tangent becomes approximately equal to the equilibrium (i.e. linear response) loss tangent. As the sweep rate is lowered or the ac power is increased, coherent population inversion occurs for TLS states in the bath passing through resonance. We find good agreement with a theory based on the Landau-Zener effect and apply this theory to extract the TLS dipole moment and density of states which characterizes the TLS bath dynamically. This technique presents the first TLS dipole moment measurement in a deposited film. The TLS bath can be manipulated for long periods of time, allowing one to create new environments on chip through population control. Although the TLS population is only controlled for TLS passing through an exciting field, a fast bias rate may also invert the TLS population over a useful spectral range of frequencies. Furthermore, using the TLS bath in this way could possibly allow for lasing from a disordered set of inverted states [@Burin2014SST].
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abstract: 'Both in the theory of orthogonal polynomials and special functions and in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $(P_n(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products, i.e., the product of any two $P_m(x),P_n(x)$ is a conical combination of the polynomials $P_{|m-n|}(x),\ldots,P_{m+n}(x)$. Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. In 1970, Gasper was able to determine the set $V$ of all pairs $(\alpha,\beta)\in(-1,\infty)^2$ for which the corresponding Jacobi polynomials $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, normalized by $R_n^{(\alpha,\beta)}(1)=1$, satisfy nonnegative linearization of products. In 2005, Szwarc asked to solve the analogous problem for the generalized Chebyshev polynomials $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. $(1-x^2)^{\alpha}|x|^{2\beta+1}\chi_{(-1,1)}(x)\,\mathrm{d}x$. In this paper, we give a solution and show that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products if and only if $(\alpha,\beta)\in V$, so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves and characterize strict positivity of the linearization coefficients, as well as we simplify Gasper’s original proof.'
address:
- 'Department of Mathematics, Chair for Mathematical Modelling, Chair for Mathematical Modeling of Biological Systems, Technical University of Munich, Boltzmannstr. 3, 85747 Garching b. München, Germany'
- 'Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany'
author:
- Stefan Kahler
bibliography:
- 'bibliographyjacobigencheb\_fin.bib'
title: 'Jacobi polynomials, their quadratic transformations and nonnegative linearization'
---
[^1]
Introduction {#sec:intro}
============
In the theory of orthogonal polynomials, it is of special interest under which—general or specific class-related—conditions a suitably normalized orthogonal polynomial sequence $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ satisfies the “nonnegative linearization of products” property, i.e., the product of any two polynomials $P_m(x),P_n(x)$ is contained in the conical hull of $\{P_k(x):k\in\mathbb{N}_0\}$; in other words, nonnegative linearization of products means that the coefficients appearing in the (Fourier) expansions of $P_m(x)P_n(x)$ w.r.t. the basis $\{P_k(x):k\in\mathbb{N}_0\}$ are always nonnegative. In this paper, we consider sequences $(P_n(x))_{n\in\mathbb{N}_0}$ which are orthogonal w.r.t. a probability (Borel) measure $\mu$ on the real line with $|\mathsf{supp}\;\mu|=\infty$ and $\mathsf{supp}\;\mu\subseteq(-\infty,1]$. Moreover, we assume $(P_n(x))_{n\in\mathbb{N}_0}$ to be normalized by $P_n(1)=1\;(n\in\mathbb{N}_0)$ (which, due to the conditions on $\mathsf{supp}\;\mu$ which yield that all zeros are located in $(-\infty,1)$, is always possible). Orthogonality is then given by $$\label{eq:orthogonalityrelation}
\int_{\mathbb{R}}\!P_m(x)P_n(x)\,\mathrm{d}\mu(x)=\frac{\delta_{m,n}}{h(n)}$$ with some function $h:\mathbb{N}_0\rightarrow(0,\infty)$. Under these circumstances, nonnegative linearization of products just corresponds to the property that the product of any two polynomials $P_m(x),P_n(x)$ is a convex combination of $P_{|m-n|}(x),\ldots,P_{m+n}(x)$, or to the nonnegativity of all linearization coefficients $g(m,n;k)$ defined by the expansions $$\label{eq:productlinear}
P_m(x)P_n(x)=\sum_{k=|m-n|}^{m+n}g(m,n;k)P_k(x),$$ where $\sum_{k=|m-n|}^{m+n}g(m,n;k)=1$. Observe that the summation in starts with $k=|m-n|$ due to orthogonality. Another obvious consequence of orthogonality is that $g(m,n;|m-n|),g(m,n;m+n)>0$. In particular, it is well-known that $(P_n(x))_{n\in\mathbb{N}_0}$ satisfies the three-term recurrence relation $P_0(x)=1$, $P_1(x)=(x-b_0)/a_0$, $$P_1(x)P_n(x)=a_n P_{n+1}(x)+b_n P_n(x)+c_n P_{n-1}(x)\;(n\in\mathbb{N}),$$ where $a_0>0$, $b_0=1-a_0$ and the sequences $(a_n)_{n\in\mathbb{N}},(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$ and $(b_n)_{n\in\mathbb{N}}\subseteq[0,1)$ satisfy $a_n+b_n+c_n=1\;(n\in\mathbb{N})$. Using and , one clearly has $$\label{eq:lincoeffh}
g(m,n;k)=h(k)\int_{\mathbb{R}}\!P_m(x)P_n(x)P_k(x)\,\mathrm{d}\mu(x)$$ and, in particular, $$h(n)=\frac{1}{g(n,n;0)}.$$ In accordance with the literature, we call the nonnegativity of all linearization coefficients $g(m,n;k)$ “property (P)”. It is well-known that property (P) enforces the uniqueness of $\mu$. Furthermore, property (P) gives rise to a certain “polynomial hypergroup” structure, including associated Banach algebras and the fruitful possibility to bring Gelfand theory into the theory of orthogonal polynomials [@La83]. Hence, nonnegative linearization of products is not only interesting with regard to a better understanding of general or specific orthogonal polynomials, but also has high relevance for harmonic analysis and, in particular, for the theory of Banach algebras. Within such polynomial hypergroups, the classes of Jacobi polynomials and generalized Chebyshev polynomials play a special role concerning product formulas and duality structures [@CMS91; @CS90; @Ga71; @Ga72; @La80; @La83].\
Given a specific sequence $(P_n(x))_{n\in\mathbb{N}_0}$, deciding whether property (P) is satisfied or not may be difficult: in many cases, the $g(m,n;k)$ are not explicitly known or explicit representations are of involved, cumbersome or inappropriate structure. In a series of papers starting with [@Sz92] and extending earlier work of Askey [@As70], Szwarc has provided some general criteria that can be helpful. However, to our knowledge there is no such general criterion which is strong enough to cover the full parameter range for which the Jacobi polynomials $$\begin{aligned}
R_n^{(\alpha,\beta)}(x)&={}_2F_1\left(\left.\begin{matrix}-n,n+\alpha+\beta+1 \\ \alpha+1\end{matrix}\right|\frac{1-x}{2}\right)=\\
&=\sum_{k=0}^n\frac{(-n)_k(n+\alpha+\beta+1)_k}{(\alpha+1)_k}\frac{(1-x)^k}{2^k k!}\end{aligned}$$ [@KLS10 (9.8.1)] satisfy property (P).[^2] Moreover, we are not aware of an explicit representation of the corresponding linearization coefficients which allows to easily identify *all* $(\alpha,\beta)$ such that property (P) is fulfilled.\
In some more details, the situation concerning the Jacobi polynomials is as follows: starting with the full (positive-definite case) parameter range $(\alpha,\beta)\in(-1,\infty)^2$ and defining $$a:=\alpha+\beta+1>-1,\;b:=\alpha-\beta\in(-1-a,1+a)$$ and a proper subset $V$ of $(-1,\infty)^2$ via $$V:=\left\{(\alpha,\beta)\in(-1,\infty)^2:a^2+2b^2+3a\geq3\frac{(a+1)(a+2)}{(a+3)(a+5)}b^2,b\geq0\right\},$$ Gasper has shown the following [@Ga70b Theorem 1] (or [@Ga75 Theorem 3]):
\[thm:gasper\] Let $\alpha,\beta>-1$. The following are equivalent:
1. $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies property (P), i.e., all $g_R(m,n;k)$ are nonnegative.
2. $(\alpha,\beta)\in V$.
Besides the original proof given in [@Ga70b], Gasper found a very different one in [@Ga83]. The second proof is based on the continuous $q$-Jacobi polynomials and an explicit corresponding linearization formula in terms of ${}_{10}\phi_9$ basic hypergeometric series due to Rahman [@Ra81b]. In the following, we shall always refer to Gasper’s first proof [@Ga70b].\
Recall that $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ is given by the measure $$\label{eq:jacobimeasure}
\mathrm{d}\mu_R(x)=\frac{\Gamma(\alpha+\beta+2)}{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}(1-x)^{\alpha}(1+x)^{\beta}\chi_{(-1,1)}(x)\,\mathrm{d}x$$ and the normalization $R_n^{(\alpha,\beta)}(1)=1\;(n\in\mathbb{N}_0)$ [@Ch78 Chapter V §2 (B)] [@Is09 (4.0.2)].\
Throughout the paper, like in Theorem \[thm:gasper\] or we use additional appropriate subscripts or superscripts when referring to the Jacobi polynomials $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ and generalized Chebyshev polynomials $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ (see below). Moreover, we use an addition superscript “$+$” when referring to the sequence $(R_n^{(\alpha,\beta+1)}(x))_{n\in\mathbb{N}_0}$. For instance, there will occur linearization coefficients $g_R(m,n;k)$, $g_R^+(m,n;k)$ and $g_T(m,n;k)$. Observe that a transition from $\beta$ to $\beta+1$, which will play a crucial role in this paper, corresponds to a transition from $(a,b)$ to $(a+1,b-1)$.\
Concerning Theorem \[thm:gasper\], it is not difficult to see that (ii) is necessary for (i); in fact, Gasper has shown that if $b<0$, then $g_R(1,1;1)<0$, whereas if $b\geq0$ and $(\alpha,\beta)\notin V$, then $g_R(2,2;2)<0$. The implication “(ii) $\Rightarrow$ (i)” is very nontrivial, however. The subcase $(\alpha,\beta)\in\Delta$, where $\Delta\subsetneq V$ is given by $$\Delta:=\{(\alpha,\beta)\in(-1,\infty)^2:a,b\geq0\},$$ is easier and was already solved in [@Ga70a], and concerning the special case $\alpha\geq\beta\geq-1/2$ Koornwinder gave a less computational proof via addition formulas [@Ko78]. Moreover, if $(\alpha,\beta)\in\Delta$, then the nonnegativity of the $g_R(m,n;k)$ can be seen via explicit representations in terms of ${}_9F_8$ hypergeometric series given by Rahman [@Ra81a (1.7) to (1.9)].[^3] Alternatively, the case $(\alpha,\beta)\in\Delta$ can also be obtained from one of the aforementioned general criteria of Szwarc [@Sz92]. The simplest subcase is given by $\alpha=\beta\geq-1/2$, for which the nonnegativity of the $g_R(m,n;k)$ follows from Dougall’s formula [@AAR99 Theorem 6.8.2] for ultraspherical polynomials.\
Despite the more involved arguments which are required to establish property (P) for the small region $V\backslash\Delta$, there is no reason for restricting to $\Delta$ when studying the associated hypergroups or Banach algebras, nor could there be identified a general advantage or benefit when restricting to $\Delta$. For instance, in [@Ka15 Theorem 3.1] (or [@Ka16b Theorem 3.1]) we have shown that the $\ell^1$-algebra (in the polynomial hypergroup sense, see [@La83; @La05]) associated with $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, $(\alpha,\beta)\in V$, is weakly amenable (i.e., there exist no nonzero bounded derivations into the dual module $\ell^{\infty}$, which acts on the $\ell^1$-algebra via convolution [@La07; @La09b]) if and only if $\alpha<0$, and the proof for $\{(\alpha,\beta)\in(-1,0)\times(-1,\infty):a=0\}\subseteq\Delta$ does not differ from the proof for $V\backslash\Delta$: both cases are traced back to the interior of $\Delta$ via the same argument in terms of inheritance via homomorphisms. Besides these relations between $V$ and $\Delta$, this example shows that the Banach algebraic properties may strongly vary even within the same class of orthogonal polynomials satisfying property (P). Hence, also in considering other example classes it is desirable to find various—or even *all*—sequences $(P_n(x))_{n\in\mathbb{N}_0}$ such that property (P) holds (e.g., by characterizing a corresponding parameter range).\
In this paper, we extend Gasper’s result Theorem \[thm:gasper\] to the class of generalized Chebyshev polynomials[^4], which are the quadratic transformations of the Jacobi polynomials. This solves a problem posted in [@Sz05] by Szwarc who asked to determine the parameter range for which these polynomials satisfy property (P). For any $\alpha,\beta>-1$, the corresponding sequence of generalized Chebyshev polynomials $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ is given by $$\begin{aligned}
\label{eq:genchebeven} T_{2n}^{(\alpha,\beta)}(x)&:=R_n^{(\alpha,\beta)}(2x^2-1),\\
\label{eq:genchebodd} T_{2n+1}^{(\alpha,\beta)}(x)&:=x R_n^{(\alpha,\beta+1)}(2x^2-1).\end{aligned}$$ $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ is orthogonal w.r.t. the measure $$\mathrm{d}\mu_T(x)=\frac{\Gamma(\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}(1-x^2)^{\alpha}|x|^{2\beta+1}\chi_{(-1,1)}(x)\,\mathrm{d}x$$ and satisfies the normalization $T_n^{(\alpha,\beta)}(1)=1\;(n\in\mathbb{N}_0)$ [@Ch78 Chapter V §2 (G)] [@Is09 (4.0.2)].\
It is an obvious consequence of Theorem \[thm:gasper\] and that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ can satisfy property (P) only if $(\alpha,\beta)\in V$; moreover, Szwarc has already shown that property (P) is fulfilled for all $(\alpha,\beta)\in\Delta$ [@Sz92], cf. also [@Sz05]. The special case $\alpha\geq\beta+1$ was already shown in [@La83]. In Theorem \[thm:mainfull\] below, we will obtain that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies property (P) if and only if $(\alpha,\beta)\in V$; hence, the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Having in mind , we will also precisely characterize the pairs $(\alpha,\beta)\in(-1,\infty)^2$ for which all $g_T(m,n;k)$ with at least one odd entry $m,n$ are nonnegative. Moreover, we will present a simpler proof for Gasper’s result Theorem \[thm:gasper\]. Our approach will also enable us to characterize (strict) positivity of the $g_R(m,n;k)$. Furthermore, we give characterizations concerning a certain oscillatory behavior of the $g_R(m,n;k)$. Before coming to these main results (Section \[sec:mainresults\]), in Section \[sec:preliminaries\] we recall some basic ingredients, including a recurrence relation for the $g_R(m,n;k)$ and a relation between the $g_T(m,n;k)$ and the $g_R(m,n;k)$, $g_R^+(m,n;k)$. The proofs will be given in Section \[sec:proofs\]. At several stages, our arguments are based on appropriate decompositions of multivariate polynomials. To find such decompositions (nested sums of suitable factorizations), we also used computer algebra systems (Maple). However, the final proofs can be understood without any computer usage.
Preliminaries {#sec:preliminaries}
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Let $\alpha,\beta>-1$, and let $a,b$ be defined as in Section \[sec:intro\]. We first recall that $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies the recurrence relation $R_0^{(\alpha,\beta)}(x)=1$, $R_1^{(\alpha,\beta)}(x)=(x-b_0^R)/a_0^R$, $$\label{eq:recJacobi}
R_1^{(\alpha,\beta)}(x)R_n^{(\alpha,\beta)}(x)=a_n^R R_{n+1}^{(\alpha,\beta)}(x)+b_n^R R_n^{(\alpha,\beta)}(x)+c_n^R R_{n-1}^{(\alpha,\beta)}(x)\;(n\in\mathbb{N})$$ with $$\label{eq:reccoeffJacobi}
\begin{split}
a_0^R=\frac{a+b+1}{a+1},\;a_n^R&=\frac{(a+1)(n+a)(2n+a+b+1)}{(a+b+1)(2n+a)(2n+a+1)}\;(n\in\mathbb{N}),\\
b_0^R=-\frac{b}{a+1},\;b_n^R&=\frac{4b n(n+a)}{(a+b+1)(2n+a-1)(2n+a+1)}\;(n\in\mathbb{N}),\\
c_n^R&=\frac{(a+1)n(2n+a-b-1)}{(a+b+1)(2n+a-1)(2n+a)}\\
\end{split}$$ [@Ga70a (4)]. It is well-known that $$\label{eq:alphabetachange}
R_n^{(\beta,\alpha)}(x)=(-1)^n\frac{(\alpha+1)_n}{(\beta+1)_n}R_n^{(\alpha,\beta)}(-x)$$ [@Is09 (4.1.4), (4.1.6)]. Moreover, $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies the recurrence relation $T_0^{(\alpha,\beta)}(x)=1$, $T_1^{(\alpha,\beta)}(x)=x$, $$\label{eq:recgencheb}
x T_n^{(\alpha,\beta)}(x)=a_n^T T_{n+1}^{(\alpha,\beta)}(x)+c_n^T T_{n-1}^{(\alpha,\beta)}(x)\;(n\in\mathbb{N})$$ with $$\label{eq:reccoeffgencheb}
\begin{split}
a_{2n-1}^T&=\frac{2n+a+b-1}{4n+2a-2},\;a_{2n}^T=\frac{n+a}{2n+a},\\
c_{2n-1}^T&=\frac{2n+a-b-1}{4n+2a-2},\;c_{2n}^T=\frac{n}{2n+a}
\end{split}$$ [@La83 3 (f)]. Using , , and , it is easy to relate the $g_T(m,n;k)$ to the $g_R(m,n;k)$ and $g_R^+(m,n;k)$: first observe that $g_T(m,n;k)=0$ if $m+n+k$ is odd (this is just a trivial consequence of symmetry). Next, write $$\begin{aligned}
T_{2m}^{(\alpha,\beta)}(x)T_{2n}^{(\alpha,\beta)}(x)&=R_m^{(\alpha,\beta)}(2x^2-1)R_n^{(\alpha,\beta)}(2x^2-1)=\\
&=\sum_{k=|m-n|}^{m+n}g_R(m,n;k)R_k^{(\alpha,\beta)}(2x^2-1)=\\
&=\sum_{k=|m-n|}^{m+n}g_R(m,n;k)T_{2k}^{(\alpha,\beta)}(x)\end{aligned}$$ to conclude that $$\label{eq:gTReven}
g_T(2m,2n;2k)=g_R(m,n;k).$$ Expanding $$\begin{aligned}
&T_{2m+1}^{(\alpha,\beta)}(x)T_{2n+1}^{(\alpha,\beta)}(x)=\\
&=x^2R_m^{(\alpha,\beta+1)}(2x^2-1)R_n^{(\alpha,\beta+1)}(2x^2-1)=\\
&=x^2\sum_{k=|m-n|}^{m+n}g_R^+(m,n;k)R_k^{(\alpha,\beta+1)}(2x^2-1)=\\
&=x\sum_{k=|m-n|}^{m+n}g_R^+(m,n;k)T_{2k+1}^{(\alpha,\beta)}(x)=\\
&=\sum_{k=|m-n|}^{m+n}g_R^+(m,n;k)[a_{2k+1}^T T_{2k+2}^{(\alpha,\beta)}(x)+c_{2k+1}^T T_{2k}^{(\alpha,\beta)}(x)]=\\
&=\sum_{k=|m-n|+1}^{m+n+1}a_{2k-1}^T g_R^+(m,n;k-1)T_{2k}^{(\alpha,\beta)}(x)+\sum_{k=|m-n|}^{m+n}c_{2k+1}^T g_R^+(m,n;k)T_{2k}^{(\alpha,\beta)}(x),\end{aligned}$$ one obtains $$\label{eq:gTodd}
\begin{split}
&g_T(2m+1,2n+1;2k)=\\
&=\begin{cases} c_{2|m-n|+1}^T g_R^+(m,n;|m-n|), & k=|m-n|, \\ a_{2m+2n+1}^T g_R^+(m,n;m+n), & k=m+n+1, \\ a_{2k-1}^T g_R^+(m,n;k-1)+c_{2k+1}^T g_R^+(m,n;k), & \mbox{else}. \end{cases}
\end{split}$$ Finally, $g_T(2m+1,2n;2k+1)$ and $g_T(2m,2n+1;2k+1)=g_T(2n+1,2m;2k+1)$ relate to via $$\label{eq:gTmixed}
g_T(2m+1,2n;2k+1)=\frac{h_T(2k+1)}{h_T(2n)}g_T(2m+1,2k+1;2n),$$ which is a consequence of . The equations to can already be found in [@La83]; due to their crucial role and for the sake of self-containedness, we have recalled their short standard proofs like above.\
One of our central tools will be a recurrence relation for the $g_R(m,n;k)$ which is taken from [@Ga70b] and relies on earlier work of Hylleraas [@Hy62]. Of course, it suffices to consider the case $n\geq m\geq1$; following [@Ga70b], we then use a more convenient notation and write $$\begin{aligned}
s:=n-m,\\
j:=k-s.\end{aligned}$$ [@Ga70b (2.1)] states that the linearization coefficients $g_R(m,n;k)=g_R(m,m+s;s+j)$ are linked to each other (w.r.t. $j$) via the following recursion: for $1\leq j\leq2m-1$, one has $$\label{eq:gRrecurrence}
\begin{split}
\theta(m,m+s;j)g_R(m,m+s;s+j+1)=&\iota(m,m+s;j)g_R(m,m+s;s+j)\\
&+\kappa(m,m+s;j)g_R(m,m+s;s+j-1),
\end{split}$$ where $\theta(m,m+s;.),\iota(m,m+s;.),\kappa(m,m+s;.):\{1,\ldots,2m-1\}\rightarrow\mathbb{R}$ read $$\begin{aligned}
\label{eq:gRlambda} \theta(m,m+s;j):=&(2m-j+a-1)(2m+2s+j+a+1)\\
\notag &\times\frac{(2s+j+1)(2s+2j+a-b+1)}{(2s+2j+a+1)(2s+2j+a+2)}(j+1),\\
\label{eq:gRmu} \iota(m,m+s;j):=&b\left[(2m-j)(2m+2s+j+2a)\frac{2s+j+1}{2s+2j+a+1}(j+1)\right.\\
\notag &\left.-(2m-j+1)(2m+2s+j+2a-1)\frac{2s+j}{2s+2j+a-1}j\right],\\
\label{eq:gRnu} \kappa(m,m+s;j):=&(2m-j+1)(2m+2s+j+2a-1)\\
\notag &\times\begin{cases} 0, & j-1=s=a=0, \\ \frac{(2s+j+a-1)(2s+2j+a+b-1)}{(2s+2j+a-2)(2s+2j+a-1)}(j+a-1), & \mbox{else}. \end{cases}\end{aligned}$$ Moreover, one has $$\begin{aligned}
\label{eq:gRs} g_R(m,m+s;s)&=\frac{\binom{m+s}{m}\binom{2m+a-1}{m}\binom{m+s+\frac{a-b-1}{2}}{m}}{\binom{2m}{m}\binom{2m+2s+a}{2m}\binom{m+\frac{a+b-1}{2}}{m}},\\
\label{eq:gR2ms} g_R(m,m+s;s+2m)&=\frac{\binom{2m+2s+a-1}{m+s}\binom{2m+a-1}{m}\binom{2m+s+\frac{a+b-1}{2}}{2m+s}}{\binom{4m+2s+a-1}{2m+s}\binom{m+s+\frac{a+b-1}{2}}{m+s}\binom{m+\frac{a+b-1}{2}}{m}}\end{aligned}$$ and $$\begin{aligned}
\label{eq:gRs1} &g_R(m,m+s;s+1)=\\
\notag &=\frac{4b m(m+s+a)(2s+a+2)}{(2m+2s+a+1)(2m+a-1)(2s+a-b+1)}g_R(m,m+s;s),\\
\label{eq:gR2ms1} &g_R(m,m+s;s+2m-1)=\\
\notag &=\frac{4b m(m+s)(4m+2s+a-2)}{(4m+2s+a+b-1)(2m+2s+a-1)(2m+a-1)}g_R(m,m+s;s+2m)\end{aligned}$$ [@Ga70b (2.2), (2.3), (2.4), (2.9)]. After these preliminaries, we can now motivate and state our main results.
Motivation and statement of the main results {#sec:mainresults}
============================================
Concerning the generalized Chebyshev polynomials, we deal with the following problems:
1. Szwarc’s problem, cf. above: find all pairs $(\alpha,\beta)\in(-1,\infty)^2$ such that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies property (P), i.e., such that all $g_T(m,n;k)$ are nonnegative.
2. Find all pairs $(\alpha,\beta)\in(-1,\infty)^2$ such that all $g_T(m,n;k)$ with at least one odd entry $m,n$ are nonnegative.
The pairs $(\alpha,\beta)\in(-1,\infty)^2$ such that all $g_T(m,n;k)$ with two even entries $m,n$ are nonnegative are exactly the $(\alpha,\beta)\in V$, which is an obvious consequence of and Theorem \[thm:gasper\]. Hence, it will be interesting to compare the resulting set of (B) to $V$.\
The solutions to (A) and (B) will be given in Theorem \[thm:mainfull\] and Theorem \[thm:mainodd\], respectively. We want to motivate these results by establishing two necessary conditions for the pairs $(\alpha,\beta)$ which are as in (B): given any $\alpha,\beta>-1$ and arbitrary $m,n\in\mathbb{N}$ with $n\geq m$, we use the notation of the previous sections and compute $$\label{eq:necfirst}
\begin{split}
&(2m+a)(2m+2s+a+2)\frac{2s+a+b+1}{2s+a+2}\left(\frac{c_{2s+3}^T}{a_{2s+1}^T}\frac{g_R^+(m,m+s;s+1)}{g_R^+(m,m+s;s)}+1\right)=\\
&=4bm^2+4b(s+a+1)m+a(2s+a+b+1)
\end{split}$$ via and . Making also use of , which yields $$\frac{g_R^+(m,m+s;s+2)}{\underbrace{g_R^+(m,m+s;s+1)}_{\neq0}}=\frac{\iota^+(m,m+s;1)}{\underbrace{\theta^+(m,m+s;1)}_{>0}}+\frac{\kappa^+(m,m+s;1)}{\theta^+(m,m+s;1)}\frac{g_R^+(m,m+s;s)}{g_R^+(m,m+s;s+1)}$$ for $b\neq1$, and combining this with and to , we furthermore obtain $$\label{eq:necsecond}
\begin{split}
&4(b-1)(2m+a-1)(2m+2s+a+3)\frac{(s+1)(2s+a+b+3)}{2s+a+4}\\
&\times\left(\frac{c_{2s+5}^T}{a_{2s+3}^T}\frac{g_R^+(m,m+s;s+2)}{g_R^+(m,m+s;s+1)}+1\right)=\\
&=(4m-4)(m+s+a+2)\left[(a^2+2b^2+3a)(s+1)-a(a+1)s\right]\\
&+(a+1)(2s+a+b+3)\left[(a+2b)(2s+2-b)+a^2+2b^2+3a\right]\;(b\neq1).
\end{split}$$ If $b<0$, then the right hand side of becomes negative for (all) sufficiently large $m\in\mathbb{N}$, whereas $(2m+a)(2m+2s+a+2)\frac{2s+a+b+1}{2s+a+2}$ is always positive. Hence, if $b<0$, then $\frac{c_{2s+3}^T}{a_{2s+1}^T}\frac{g_R^+(m,m+s;s+1)}{g_R^+(m,m+s;s)}+1$ is negative for sufficiently large $m\in\mathbb{N}$. Since $g_R^+(m,m+s;s)$ is always positive, the latter yields the negativity of $g_T(2m+1,2m+2s+1;2s+2)=a_{2s+1}^T g_R^+(m,m+s;s)+c_{2s+3}^T g_R^+(m,m+s;s+1)$ for sufficiently large $m\in\mathbb{N}$.\
Now assume that $a^2+2b^2+3a<0$. On the one hand, one necessarily has $b<1$ then (because $a^2+2+3a=(a+1)(a+2)>0$), so $4(b-1)(2m+a-1)(2m+2s+a+3)\frac{(s+1)(2s+a+b+3)}{2s+a+4}$ is always negative. On the other hand, if $s=0$, then the right hand side of becomes negative for (all) sufficiently large $m\in\mathbb{N}$. Hence, $\frac{c_5^T}{a_3^T}\frac{g_R^+(m,m;2)}{g_R^+(m,m;1)}+1$ is positive for sufficiently large $m\in\mathbb{N}$. Since, due to , $g_R^+(m,m;1)$ is negative, we obtain the negativity of $g_T(2m+1,2m+1;4)=a_3^T g_R^+(m,m;1)+c_5^T g_R^+(m,m;2)$ for sufficiently large $m\in\mathbb{N}$.\
Putting all together, we see that every pair $(\alpha,\beta)$ which fits into (B) has to satisfy both $b\geq0$ and $a^2+2b^2+3a\geq0$. Our following result deals with the converse and shows that these two conditions already *characterize* (B).
\[thm:mainodd\] Let $\alpha,\beta>-1$. The following are equivalent:
1. For all $m,n\in\mathbb{N}_0$ such that at least one of these numbers is odd, all linearization coefficients $g_T(m,n;k)$ are nonnegative.
2. $(\alpha,\beta)\in V^{\prime}$, where $$V^{\prime}:=\left\{(\alpha,\beta)\in(-1,\infty)^2:a^2+2b^2+3a\geq0,b\geq0\right\}\supsetneq V.$$
If $(\alpha,\beta)\in V^{\prime}\backslash\Delta$ and $m,n\in\mathbb{N}_0$ are such that at least one of these numbers is odd, and if $k\in\{|m-n|,\ldots,m+n\}$ is such that $m+n+k$ is even, then $g_T(m,n;k)$ is (strictly) positive.
As a consequence of Theorem \[thm:mainodd\], we will obtain our second main result and the answer to (A):
\[thm:mainfull\] Let $\alpha,\beta>-1$. The following are equivalent:
1. $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies property (P), i.e., all $g_T(m,n;k)$ are nonnegative.
2. $(\alpha,\beta)\in V$.
With respect to Theorem \[thm:mainfull\], we note that if $b\geq0$ and $(\alpha,\beta)\notin V$, then $g_T(4,4;4)<0$, which is a consequence of $g_R(2,2;2)<0$ (cf. Section \[sec:intro\]) and . Theorem \[thm:mainfull\] can be regarded as a sharpening of Gasper’s result Theorem \[thm:gasper\] because the nontrivial direction “(ii) $\Rightarrow$ (i) (Theorem \[thm:gasper\])” is trivially implied by “(ii) $\Rightarrow$ (i) (Theorem \[thm:mainfull\])” and . As a consequence of Theorem \[thm:mainfull\], we obtain that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ induces a polynomial hypergroup and an associated $\ell^1$-algebra [@La83] whenever $(\alpha,\beta)\in V$.\
Our argument via equation above shows a typical aspect of the strategy which will be used in the following proofs: once a decomposition like in is known, which allows to directly see the signs of the relevant parts, it may be easily (yet more or less tediously) verified by comparing the expansions, or by comparing common zeros and leading coefficients and so on. Hence, the actual task is not proving such decompositions but finding them.\
Besides the results on generalized Chebyshev polynomials, we will address the difficult direction of Gasper’s result Theorem \[thm:gasper\] and present a simpler proof than Gasper’s original one. Compared to the latter, our modified proof will be less computational and avoid Descartes’ rule of signs.\
We give some further characterizations. The first of them deals with the zeros of the canonical continuation of the coefficient function $\iota(m,m+s;.)$ to $[1,2m-1]$. The remaining ones deal with positivity and a certain oscillatory behavior of the $g_R(m,n;k)$.
\[prp:gRmu\] Let $\alpha,\beta>-1$, and let for $m\in\mathbb{N}$ and $s\in\mathbb{N}_0$ the function $\iota(m,m+s;.):[1,2m-1]\rightarrow\mathbb{R}$ be defined by $$\begin{split}
\iota(m,m+s;j):=&b\left[(2m-j)(2m+2s+j+2a)\frac{2s+j+1}{2s+2j+a+1}(j+1)\right.\\
\notag &\left.-(2m-j+1)(2m+2s+j+2a-1)\frac{2s+j}{2s+2j+a-1}j\right].
\end{split}$$ If $b\neq0$ (i.e., if $\alpha\neq\beta$), then the following are equivalent:
1. For all $m\in\mathbb{N}$ and $s\in\mathbb{N}_0$, $\iota(m,m+s;.)$ has at most one zero.
2. $a>-\frac{11}{8}+\frac{1}{8}\sqrt{73}\approx-0.30699953$.
Moreover, if $a\geq-11/8+\sqrt{73}/8$, then $\iota(m,m+s;1)\geq0\;(m\geq2,s\geq0)$ if and only if $b\geq0$. Furthermore, if $a\geq0$, then $\iota(m,m+s;1)\geq0\;(m\geq1,s\geq0)$ if and only if $b\geq0$.
\[thm:gasperpositivevariant\] Let $\alpha,\beta>-1$. The following are equivalent:
1. All $g_R(m,n;k)$ are (strictly) positive.
2. $(\alpha,\beta)$ is located in the interior of $V$.
\[cor:oscillatory\] Let $\alpha,\beta>-1$, and let $\widetilde{g_R}(m,n;k)$ denote the linearization coefficients belonging to the sequence $(R_n^{(\beta,\alpha)}(x))_{n\in\mathbb{N}_0}$. Then the following hold:
1. All numbers $(-1)^{m+n+k}\widetilde{g_R}(m,n;k)$ are nonnegative if and only if $(\alpha,\beta)\in V$.
2. All numbers $(-1)^{m+n+k}\widetilde{g_R}(m,n;k)$ are (strictly) positive if and only if $(\alpha,\beta)$ is located in the interior of $V$.
Note that Theorem \[thm:gasperpositivevariant\] can be regarded as a another sharpening of Theorem \[thm:gasper\] because the nontrivial direction “(ii) $\Rightarrow$ (i) (Theorem \[thm:gasper\])” is implied by “(ii) $\Rightarrow$ (i) (Theorem \[thm:gasperpositivevariant\])” via continuity.\
Comparing Theorem \[thm:gasper\], Theorem \[thm:mainfull\] and Theorem \[thm:gasperpositivevariant\], one may ask whether all $g_T(m,n;k)$ are (strictly) positive if $(\alpha,\beta)$ is located in the interior of $V$. However, this is not true: recall that for any choice of $(\alpha,\beta)\in(-1,\infty)^2$ one has $g_T(m,n;k)=0$ if $m+n+k$ is odd.\
The announced simpler proof of Gasper’s result Theorem \[thm:gasper\] and the proof of Theorem \[thm:gasperpositivevariant\] will essentially rely on the framework provided by Proposition \[prp:gRmu\]. Proposition \[prp:gRmu\] unifies and extends results which are already contained in [@Ga70a; @Ga70b]. Concerning some parts which are already contained in these references, we give a new proof which avoids Descartes’ rule of signs and is more elementary. Concerning the functions $\theta(m,m+s;.)$ and $\kappa(m,m+s;.)$, we will only need that $$\label{eq:thetapos}
\theta(m,m+s;.)|_{\{1,\ldots,2m-2\}}>0\;(m\geq2)$$ and $$\label{eq:kappapos}
\kappa(m,m+s;.)|_{\{2,\ldots,2m-1\}}>0\;(m\geq2),$$ which is an obvious consequence of and (and was also used in [@Ga70b]). Corollary \[cor:oscillatory\] will play an important role for the proof of Theorem \[thm:mainodd\].
Proofs {#sec:proofs}
======
We first establish Proposition \[prp:gRmu\].
The most interesting part is the direction “(ii) $\Rightarrow$ (i)”, which was obtained via Descartes’ rule of signs in [@Ga70b]. We present an alternative proof which just uses the mean value theorem.\
Let $b\neq0$, $m\in\mathbb{N}$ and $s\in\mathbb{N}_0$. We first assume that $s\neq0$ or $a\geq0$ and write $\iota(m,m+s;j)=b[f(j+1)-f(j)]$ with $f:[1,2m]\rightarrow(0,\infty)$, $$f(j):=(2m-j+1)(2m+2s+j+2a-1)\frac{2s+j}{2s+2j+a-1}j.$$ If the function $\iota(m,m+s;.)$ had two different zeros $j_1,j_2\in[1,2m-1]$, $j_1<j_2$, then the condition $b\neq0$ would imply that $0=f(j_1+1)-f(j_1)=f(j_2+1)-f(j_2)$ and therefore $f(j_2+1)-f(j_1+1)=f(j_2)-f(j_1)$. We now distinguish two cases.\
On the one hand, if $j_2\geq j_1+1$, then the mean value theorem yields the existence of $j_1^{\prime}\in(j_1,j_1+1)$ and $j_2^{\prime}\in(j_2,j_2+1)$ with $0=f^{\prime}(j_1^{\prime})=f^{\prime}(j_2^{\prime})$.\
On the other hand, if $j_2<j_1+1$, then the mean value theorem yields $j_1^{\prime\prime}\in(j_1,j_2)$ and $j_2^{\prime\prime}\in(j_1+1,j_2+1)$ such that $f^{\prime}(j_1^{\prime\prime})=f^{\prime}(j_2^{\prime\prime})=\frac{f(j_2)-f(j_1)}{j_2-j_1}$. Now if $f(j_2)=f(j_1)$, then we have $0=f^{\prime}(j_1^{\prime})=f^{\prime}(j_2^{\prime})$ for $j_1^{\prime}:=j_1^{\prime\prime}$ and $j_2^{\prime}:=j_2^{\prime\prime}$. If, however, $f(j_2)>f(j_1)$, then $f^{\prime}(j_1^{\prime\prime})=f^{\prime}(j_2^{\prime\prime})>0$, and the mean value theorem yields some $j_3^{\prime\prime}\in(j_2,j_1+1)$ with $f^{\prime}(j_3^{\prime\prime})=\frac{f(j_1+1)-f(j_2)}{j_1+1-j_2}=\frac{f(j_1)-f(j_2)}{j_1+1-j_2}<0$, so we can find $j_1^{\prime}\in(j_1^{\prime\prime},j_3^{\prime\prime})$, $j_2^{\prime}\in(j_3^{\prime\prime},j_2^{\prime\prime})$ such that $0=f^{\prime}(j_1^{\prime})=f^{\prime}(j_2^{\prime})$. Finally, if $f(j_2)<f(j_1)$, we can conclude in an analogous way.\
Hence, in any case there would be $j_1^{\prime},j_2^{\prime}\in(1,2m)$ with $j_1^{\prime}<j_2^{\prime}$ and $0=f^{\prime}(j_1^{\prime})=f^{\prime}(j_2^{\prime})$. We now decompose $f=u v$ with $u,v:[1,2m]\rightarrow(0,\infty)$, $$\begin{aligned}
u(j)&:=(2m+2s+j+2a-1)(2s+j)j,\\
v(j)&:=\frac{2m-j+1}{2s+2j+a-1}.\end{aligned}$$ For any $j\in[1,2m]$, one has $f^{\prime}(j)=0$ if and only if $u^{\prime}(j)/u(j)+v^{\prime}(j)/v(j)=0$, or equivalently $$\label{eq:gRplusmumonotony}
\begin{split}
\frac{1}{2m-j+1}&=\frac{1}{2m+2s+j+2a-1}+\frac{1}{2s+j}+\frac{1}{j}-\frac{2}{2s+2j+a-1}=\\
&=\frac{1}{2m+2s+j+2a-1}+\frac{1}{2s+j}+\frac{2s+a-1}{j(2s+2j+a-1)}.
\end{split}$$ At this stage, we distinguish two cases again.\
*Case 1:* $s\neq0$. Since $2s+a-1>0$, we see that the right hand side of equation is strictly decreasing w.r.t. $j\in[1,2m]$, whereas the left hand side is strictly increasing. We thus obtain that $f^{\prime}$ can have at last one zero, a contradiction.\
*Case 2:* $s=0$. Then, by the assumption, $a\geq0$. Moreover, reduces to $$\frac{1}{2m-j+1}=\frac{1}{2m+j+2a-1}+\frac{1}{j}+\frac{a-1}{j(2j+a-1)}$$ or, equivalently, $$\frac{2j+2a-2}{(2m-j+1)(2m+j+2a-1)}=\frac{2j+2a-2}{j(2j+a-1)}.$$ Since $a\geq0$, every zero of $f^{\prime}|_{(1,2m]}$ must therefore satisfy $$(2m-j+1)(2m+j+2a-1)=j(2j+a-1).$$ We now define $\eta:[1,2m]\rightarrow\mathbb{R}$, $$\eta(j):=j(2j+a-1)-(2m-j+1)(2m+j+2a-1).$$ Since $$\eta^{\prime}(j)=6j+3a-3>0$$ for all $j\in[1,2m]$, the function $\eta$ is strictly increasing and we obtain that $f^{\prime}|_{(1,2m]}$ can have at last one zero, which yields a contradiction again.\
Hence, if $s\neq0$ or $a\geq0$, then $\iota(m,m+s;.)$ has at most one zero. We now rewrite $$\iota(m,m;j)=-\frac{b}{(2j+a-1)(2j+a+1)}\chi_m(j)$$ with $\chi_m:[1,2m-1]\rightarrow\mathbb{R}$, $$\begin{aligned}
\chi_m(j):=&(2m-j+1)(2m+j+2a-1)j^2(2j+a+1)\\
&-(2m-j)(2m+j+2a)(j+1)^2(2j+a-1)=\\
=&(4m+a+1)(2m+j+2a-1)j^2\\
&-(2m-j)(2j+a-1)[(2m+j+2a)(2j+1)+j^2].\end{aligned}$$ Taking into account the condition $b\neq0$ once again, concerning the assertion “(i) $\Leftrightarrow$ (ii)” it is then left to establish the directions “(i’) $\Rightarrow$ (ii)” and “(ii’) $\Rightarrow$ (i’)” with
1. For all $m\in\mathbb{N}$, $\chi_m$ has at most one zero in $[1,2m-1]$.
2. $a\in\left(-\frac{11}{8}+\frac{1}{8}\sqrt{73},0\right)$.
We now consider $\chi_m$ on the whole real line (by canonical extension) and conclude as follows:\
“(i’) $\Rightarrow$ (ii)”: if $a\leq-11/8+\sqrt{73}/8$, then $$\begin{aligned}
\chi_2(1)&=-16a^2-44a-12=\\
&=-16\left[a-\left(-\frac{11}{8}+\frac{1}{8}\sqrt{73}\right)\right]\left[a-\left(-\frac{11}{8}-\frac{1}{8}\sqrt{73}\right)\right]\geq\\
&\geq0,\\
\chi_2(2)&=-12(a+1)(a+2)<0,\\
\chi_2(3)&=4a^2+88a+196>0.\end{aligned}$$ Consequently, $\chi_2$ has both a zero in $[1,2)$ and a zero in $(2,3)$, which violates (i’).\
“(ii’) $\Rightarrow$ (i’)”: the case $m=1$ is trivial, so let $m\geq2$. Then the estimation $a\in\left(-\frac{1}{3},0\right)$ implies $$\begin{aligned}
\chi_m(-1)&=(4a-4)(m+1)(m+a-1)<0,\\
\chi_m(0)&=(4-4a)m(m+a)>0,\\
\chi_m(1)&=-(4+12a)m(m+a)+4(a+1)(2a+1)\leq\\
&\leq\chi_2(1)=\\
&=-16\left[a-\left(-\frac{11}{8}+\frac{1}{8}\sqrt{73}\right)\right]\left[a-\left(-\frac{11}{8}-\frac{1}{8}\sqrt{73}\right)\right]<\\
&<0.\end{aligned}$$ Hence, together with $\lim_{j\to-\infty}\chi_m(j)=\infty$ we obtain that $\chi_m$ has a zero in $(-\infty,-1)$, that $\chi_m$ has a zero in $(-1,0)$ and that $\chi_m$ has also a zero in $(0,1)$. As a polynomial in $j$ of degree four, however, this implies that $\chi_m$ can have at most one zero in $[1,2m-1]$.\
The second part of the proposition is a consequence of the representations $$\begin{aligned}
&\iota(m,m+s;1)=\\
&=\frac{4b}{(2s+a+1)(2s+a+3)}\\
&\times\left[((2s+3)(2s+a+1)-(4s+2))(m-1)(m+s+a+1)+a(a+1)\right]=\\
&=\frac{4b}{(2s+a+1)(2s+a+3)}\\
&\times\left[((2s+3)(2s+a+1)-(4s+2))(m-2)(m+s+a+2)\right.\\
&\left.+(2(s+a+2)(2s+a+2)+4s+5a+5)s+4a^2+11a+3\right],\end{aligned}$$ the estimation $$(2s+3)(2s+a+1)-(4s+2)>(2s+3)\left(2s-\frac{1}{3}+1\right)-(4s+2)=\frac{2}{3}s(6s+5)\geq0$$ (because $a>-1/3$) and the factorization $$4a^2+11a+3=4\left[a-\left(-\frac{11}{8}+\frac{1}{8}\sqrt{73}\right)\right]\left[a-\left(-\frac{11}{8}-\frac{1}{8}\sqrt{73}\right)\right].$$
We will now give the announced simplified proof of Gasper’s result Theorem \[thm:gasper\].\
We establish the easy direction “(i) $\Rightarrow$ (ii)” as in [@Ga70b] and state the short proof just for the sake of completeness: if $b<0$, then and (or also and ) show that $$g_R(1,1;1)=b_1^R=\frac{4b}{(a+3)(a+b+1)}<0;$$ if $b\geq0$ but $(\alpha,\beta)\notin V$, then the equations to and yield $$g_R(2,2;2)=\frac{4[(a^2+2b^2+3a)(a+3)(a+5)-3(a+1)(a+2)b^2]}{(a+3)(a+5)(a+6)(a+b+1)(a+b+3)}<0.$$ We now come to our modified proof for the interesting direction “(ii) $\Rightarrow$ (i)”, so let $(\alpha,\beta)\in V$, let $m\in\mathbb{N}$ and let $s\in\mathbb{N}_0$. We have to show that $g_R(m,m+s;s+j)\geq0$ for all $j\in\{0,\ldots,2m\}$. As in [@Ga70b], we use two-sided induction and proceed as follows: to yield $g_R(m,m+s;s)>0$, $g_R(m,m+s;s+1)\geq0$, $g_R(m,m+s;s+2m)>0$ and $g_R(m,m+s;s+2m-1)\geq0$.[^5] If $m=1$, we are already done (cf. also and ). Hence, assume that $m\geq2$ from now on; it is then left to show that $g_R(m,m+s;s+j)\geq0$ for all $j\in\{2,\ldots,2m-2\}$.\
and yield $\theta(m,m+s;1)>0$ and $\kappa(m,m+s;2m-1)>0$, and via the equations to and , we compute $$\label{eq:simplifygasperI}
\begin{split}
&\frac{(2m+a-1)(2s+a-b+1)(2m+2s+a+1)(2s+a+3)}{4m(m+s+a)(2s+a+1)g_R(m,m+s;s)}\\
&\times\theta(m,m+s;1)g_R(m,m+s;s+2)=\\
&=\frac{(2m+a-1)(2s+a-b+1)(2m+2s+a+1)(2s+a+3)}{4m(m+s+a)(2s+a+1)g_R(m,m+s;s)}\\
&\times[\iota(m,m+s;1)g_R(m,m+s;s+1)+\kappa(m,m+s;1)g_R(m,m+s;s)]=\\
&=(b^2+a)(2m-4)(2m+2s+2a+4)2s\\
&+(a^2+2b^2+3a)[(2m-4)(2m+2s+2a+4)+2s(2s+2a+8)+(a+3)(a+5)]\\
&-3(a+1)(a+2)b^2
\end{split}$$ and $$\label{eq:simplifygasperII}
\begin{split}
&\frac{(2m+a-1)(2m+2s+a-1)(4m+2s+a-3)(4m+2s+a+b-1)}{4m(m+s)(4m+2s+a-1)g_R(m,m+s;s+2m)}\\
&\times\kappa(m,m+s;2m-1)g_R(m,m+s;s+2m-2)=\\
&=\frac{(2m+a-1)(2m+2s+a-1)(4m+2s+a-3)(4m+2s+a+b-1)}{4m(m+s)(4m+2s+a-1)g_R(m,m+s;s+2m)}\\
&\times[\theta(m,m+s;2m-1)g_R(m,m+s;s+2m)\\
&-\iota(m,m+s;2m-1)g_R(m,m+s;s+2m-1)]=\\
&=(b^2+a)(2m-4)(2m+2s-4)(4m+2s+2a)\\
&+(a^2+2b^2+3a)[(2m-4)(6m+6s+4a+4)+2s(2s+2a+8)+(a+3)(a+5)]\\
&-3(a+1)(a+2)b^2.
\end{split}$$ Since $b^2+a\geq0$ (if $a<0$, this is a consequence of the decomposition $2b^2+2a=a^2+2b^2+3a-a(a+1)$), the preceding calculations imply that $g_R(m,m+s;s+2)\geq0$ and $g_R(m,m+s;s+2m-2)\geq0$.[^6] In particular, we are done if $m=2$ and thus assume that $m\geq3$ from now on.\
Since $7/2-\sqrt{145}/2<-1$, since all $(\alpha,\beta)\in V$ with $a<7/2+\sqrt{145}/2$ satisfy the estimations $$\begin{aligned}
0&<\frac{1}{(a+3)(a+5)}\left[\left(\frac{7}{2}+\frac{1}{2}\sqrt{145}\right)-a\right]\left[a-\left(\frac{7}{2}-\frac{1}{2}\sqrt{145}\right)\right]=\\
&=2-3\frac{(a+1)(a+2)}{(a+3)(a+5)},\\
0&\leq a^2+2b^2+3a-3\frac{(a+1)(a+2)}{(a+3)(a+5)}b^2<\\
&<a^2+\left[2-3\frac{(a+1)(a+2)}{(a+3)(a+5)}\right](a+1)^2+3a=\\
&=4\frac{(a+2)}{(a+3)(a+5)}(4a^2+11a+3)=\\
&=16\frac{(a+2)}{(a+3)(a+5)}\left[a-\left(-\frac{11}{8}+\frac{1}{8}\sqrt{73}\right)\right]\left[a-\left(-\frac{11}{8}-\frac{1}{8}\sqrt{73}\right)\right],\end{aligned}$$ and since $-11/8-\sqrt{73}/8<-1$ and $7/2+\sqrt{145}/2>-11/8+\sqrt{73}/8$, we obtain that $a>-11/8+\sqrt{73}/8$ (which has already been observed in [@Ga70b]). Therefore, we can apply Proposition \[prp:gRmu\] and, like in [@Ga70b], obtain the existence of an $N\in\{1,\ldots,2m-1\}$ such that $\iota(m,m+s;j)\geq0$ for $1\leq j\leq N$ and $\iota(m,m+s;j)<0$ for those $1\leq j\leq2m-1$ which satisfy $j\geq N+1$. The remaining proof follows [@Ga70b]. We make use of and and distinguish two cases:\
*Case 1:* $N\geq3$. Then and induction yield $$\begin{aligned}
&g_R(m,m+s;s+j+1)=\\
&=\frac{\iota(m,m+s;j)}{\theta(m,m+s;j)}g_R(m,m+s;s+j)+\frac{\kappa(m,m+s;j)}{\theta(m,m+s;j)}g_R(m,m+s;s+j-1)\geq\\
&\geq0\;(2\leq j\leq N-1).\end{aligned}$$ This shows the nonnegativity of $g_R(m,m+s;s+3),\ldots,g_R(m,m+s;s+N)$.\
*Case 2:* $N\leq2m-3$. In this case, and induction yield $$\begin{aligned}
&g_R(m,m+s;s+j-1)=\\
&=\frac{\theta(m,m+s;j)}{\kappa(m,m+s;j)}g_R(m,m+s;s+j+1)-\frac{\iota(m,m+s;j)}{\kappa(m,m+s;j)}g_R(m,m+s;s+j)\geq\\
&\geq0\;(N+1\leq j\leq2m-2),\end{aligned}$$ which establishes the nonnegativity of $g_R(m,m+s;s+N),\ldots,g_R(m,m+s;s+2m-3)$.\
If $N\leq2$, then $N<2m-3$ and the nonnegativity of $$g_R(m,m+s;s+3),\ldots,g_R(m,m+s;s+2m-3)$$ is a consequence of Case 2. If $N\geq2m-2$, then $N>3$ and the nonnegativity of $$g_R(m,m+s;s+3),\ldots,g_R(m,m+s;s+2m-3)$$ is a consequence of Case 1. Finally, if $3\leq N\leq2m-3$, then the combination of both cases yields the nonnegativity of $$g_R(m,m+s;s+3),\ldots,g_R(m,m+s;s+2m-3).$$
As already observed in [@Ga70a; @Ga70b], the proof (direction “(ii) $\Rightarrow$ (i)”) considerably simplifies in the special case $a\geq0$, i.e., for $(\alpha,\beta)\in\Delta$. On the one hand, for $a\geq0$ the functions $\theta(m,m+s;.)$ and $\kappa(m,m+s;.)$ are nonnegative on their full domains, see and ; hence, one can avoid the computations of $g_R(m,m+s;s+2)$ and $g_R(m,m+s;s+2m-2)$. On the other hand, recall that our proof of (the important ingredient) direction “(ii) $\Rightarrow$ (i)” of Proposition \[prp:gRmu\] was simpler for $a\geq0$, too.
The equivalence follows from a straightforward modification of the preceding proof.
If $(\alpha,\beta)$ is located in the interior of $\Delta\subsetneq V$, then the positivity of all $g_R(m,n;k)$ can also be seen via Rahman’s formulas and .
As a consequence of , the linearization coefficients are connected to each other via $$(-1)^{m+n+k}\widetilde{g_R}(m,n;k)=\frac{(\alpha+1)_m(\alpha+1)_n}{(\alpha+1)_k}\frac{(\beta+1)_k}{(\beta+1)_m(\beta+1)_n}g_R(m,n;k).$$ Hence, the assertions are consequences of Theorem \[thm:gasper\] and Theorem \[thm:gasperpositivevariant\], cf. also the remarks at the end of [@Ga70b Section 1].
We now come to our results on the generalized Chebyshev polynomials.\
Since Theorem \[thm:mainodd\] implies that the set of all pairs $(\alpha,\beta)\in(-1,\infty)^2$ such that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies property (P) is given by $V\cap V^{\prime}$, cf. the remarks in the previous section, and since $V\subseteq V^{\prime}$, Theorem \[thm:mainfull\] follows from Theorem \[thm:mainodd\].\
The implication “(i) $\Rightarrow$ (ii)” of Theorem \[thm:mainodd\] was established in the previous section. In view of Szwarc’s earlier result, which already shows that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies property (P) at least for all $(\alpha,\beta)\in\Delta$ (cf. Section \[sec:intro\]), the converse “(ii) $\Rightarrow$ (i)” is a consequence of the assertion made in the second part of Theorem \[thm:mainodd\].\
In view of these observations, and in view of , Theorem \[thm:mainodd\] and Theorem \[thm:mainfull\] trace back to the following lemma:
\[lma:centrallemma\] Let $(\alpha,\beta)\in V^{\prime}\backslash\Delta$, and let $m\in\mathbb{N}$, $s\in\mathbb{N}_0$. Then $g_T(2m+1,2m+2s+1;2s+2j)>0$ for all $j\in\{0,\ldots,2m+1\}$.
Our task is to establish Lemma \[lma:centrallemma\], which will be done via Corollary \[cor:oscillatory\] and an auxiliary result.\
For the rest of the section, we always assume that $(\alpha,\beta)\in V^{\prime}\backslash\Delta$ and that $m\in\mathbb{N}$, $s\in\mathbb{N}_0$.\
Under these conditions, we have $$a\in\left(-\frac{1}{3},0\right)$$ and $$b\in(-a,1+a)\subseteq(0,1).$$ The inequality $a>-1/3$ follows from $0\leq a^2+2b^2+3a<a^2+2(1+a)^2+3a=(a+2)(3a+1)$ (and can also be found in [@Ga70b]), and the inequality $b>-a$ is due to $2b^2-2a^2=a^2+2b^2+3a-3a(a+1)>0$.\
We now define two auxiliary functions $p:\{1,\ldots,2m-1\}\rightarrow\mathbb{R}$, $q:\{1,\ldots,2m-1\}\rightarrow(0,\infty)$ by $$\begin{aligned}
p(j):=\frac{c_{2s+2j+3}^T}{a_{2s+2j+1}^T}\frac{\iota^+(m,m+s;j)}{\theta^+(m,m+s;j)},\\
q(j):=\frac{c_{2s+2j+1}^T c_{2s+2j+3}^T}{a_{2s+2j-1}^T a_{2s+2j+1}^T}\frac{\kappa^+(m,m+s;j)}{\theta^+(m,m+s;j)};\end{aligned}$$ concerning well-definedness, observe that $\theta^+(m,m+s;.)$ and $\kappa^+(m,m+s;.)$ are positive on their full domains due to and . Using and to , one can compute $$\label{eq:pqdecompose}
\begin{split}
p(j)&=p^{\infty}(j)+\frac{p^{\ast}(j)}{(2m-j+a)(2m+2s+j+a+2)},\\
q(j)&=q^{\infty}(j)+\frac{q^{\ast}(j)}{(2m-j+a)(2m+2s+j+a+2)},
\end{split}$$ where the four functions $p^{\infty}:\{1,\ldots,2m-1\}\rightarrow(-1,\infty)$, $p^{\ast},q^{\infty},q^{\ast}:\{1,\ldots,2m-1\}\rightarrow(0,\infty)$ are independent of $m$ and given by $$\label{eq:pqdecomposeexplicit}
\begin{split}
&p^{\infty}(j)=\\
&=\frac{2s+2j+a+2}{(2s+j+1)(2s+2j+a)(2s+2j+a+b+1)(j+1)}\\
&\times\left[b(2s+j+1)(2s+2j+a)(j+1)+(1-b)(2s+j)(2s+2j+a+1)j\right]-1,\\
&p^{\ast}(j)=(1-b)\frac{(2s+j+a)(2s+2j+a+1)(2s+2j+a+2)(j+a)(2s+2j+1)}{(2s+j+1)(2s+2j+a)(2s+2j+a+b+1)(j+1)},\\
&q^{\infty}(j)=\frac{(2s+2j+a+2)(2s+j+a)(2s+2j+a-b+1)(j+a)}{(2s+j+1)(2s+2j+a)(2s+2j+a+b+1)(j+1)},\\
&q^{\ast}(j)=\\
&=\frac{2s+2j+a+2}{(2s+j+1)(2s+2j+a)(2s+2j+a+b+1)(j+1)}\\
&\times(1-a)(2s+j+a)(2s+2j+a+1)(j+a)(2s+2j+a-b+1).
\end{split}$$ The superscript “$\infty$” is used because $p^{\infty}$ and $q^{\infty}$ are just the limits of $p$ and $q$ if $m$ tends to infinity.\
As a first consequence of and , we obtain that $p$ maps into $(-1,\infty)$.\
The following lemma provides an inequality in $p$ and $q$ which will be central in the proof of Lemma \[lma:centrallemma\].
\[lma:pqinequality\] Let $(\alpha,\beta)\in V^{\prime}\backslash\Delta$ and $m\geq2$, $s\in\mathbb{N}_0$. Then for every $j\in\{1,\ldots,2m-2\}$ the inequality $$\label{eq:pqinequality}
[1+p(j+1)][q(j)-p(j)]<q(j+1)$$ is valid.
The basic idea is to use and to isolate $m$ in an appropriate way. Let $j\in\{1,\ldots,2m-2\}$. We decompose $$\label{eq:factorm}
\begin{split}
&q(j+1)-[1+p(j+1)][q(j)-p(j)]=\\
&=q^{\infty}(j+1)-[1+p^{\infty}(j+1)][q^{\infty}(j)-p^{\infty}(j)]\\
&+\frac{q^{\ast}(j+1)-p^{\ast}(j+1)[q^{\infty}(j)-p^{\infty}(j)]}{(2m-j+a-1)(2m+2s+j+a+3)}\\
&-\frac{[1+p^{\infty}(j+1)][q^{\ast}(j)-p^{\ast}(j)]}{(2m-j+a)(2m+2s+j+a+2)}\\
&-\frac{p^{\ast}(j+1)[q^{\ast}(j)-p^{\ast}(j)]}{(2m-j+a-1)(2m-j+a)(2m+2s+j+a+2)(2m+2s+j+a+3)}
\end{split}$$ and compute $$\label{eq:omegaj}
\begin{split}
\omega_j:=&q^{\infty}(j+1)-[1+p^{\infty}(j+1)][q^{\infty}(j)-p^{\infty}(j)]=\\
=&\frac{(b-a)b[2s(2s+2j+a+2)+(j+a)(2j+4)+1-a]}{(2s+j+1)(2s+j+2)(j+1)(j+2)}\\
&\times\frac{(2s+2j+a+2)(2s+2j+a+4)}{(2s+2j+a+b+1)(2s+2j+a+b+3)}>\\
>&0.
\end{split}$$ Combining with , we obtain $$\label{eq:factormmod}
\begin{split}
&\frac{(2m-j+a-1)(2m-j+a)(2m+2s+j+a+2)(2m+2s+j+a+3)}{\omega_j}\\
&\times[q(j+1)-[1+p(j+1)][q(j)-p(j)]]=\\
&=[(2m-j+a-1)(2m+2s+j+a+3)+\alpha_j]\\
&\times[(2m-j+a)(2m+2s+j+a+2)+\beta_j]+\rho_j=\\
&=[(2m-j+a-1)((2m-j+a-1)+\sigma_j+1)+\alpha_j]\\
&\times[((2m-j+a-1)+1)((2m-j+a-1)+\sigma_j)+\beta_j]+\rho_j
\end{split}$$ with $$\begin{aligned}
\alpha_j&:=\frac{q^{\ast}(j+1)-p^{\ast}(j+1)[q^{\infty}(j)-p^{\infty}(j)]}{\omega_j},\\
\beta_j&:=-\frac{[1+p^{\infty}(j+1)][q^{\ast}(j)-p^{\ast}(j)]}{\omega_j},\\
\rho_j&:=-\frac{p^{\ast}(j+1)[q^{\ast}(j)-p^{\ast}(j)]}{\omega_j}-\alpha_j\beta_j,\\
\sigma_j&:=2s+2j+3.\end{aligned}$$ We now define $f:\left[\frac{j-a+1}{2},\infty\right)\rightarrow\mathbb{R}$, $$\begin{split}
f(x):=&[(2x-j+a-1)((2x-j+a-1)+\sigma_j+1)+\alpha_j]\\
&\times[((2x-j+a-1)+1)((2x-j+a-1)+\sigma_j)+\beta_j]+\rho_j
\end{split}$$ and *claim* that $f$ maps into $(0,\infty)$; once the claim is proven, inequality will follow via $m\in\left[\frac{j-a+1}{2},\infty\right)$ and . To establish the claim, we first compute $$\begin{split}
f^{\prime}(x)=&[4(2x-j+a-1)+2\sigma_j+2]\\
&\times\left[((2x-j+a-1)+1)((2x-j+a-1)+\sigma_j)+\beta_j\right.\\
&\left.+(2x-j+a-1)((2x-j+a-1)+\sigma_j+1)+\alpha_j\right].
\end{split}$$ Then, two further tedious calculations yield $$\begin{aligned}
f\left(\frac{j-a+1}{2}\right)=&\alpha_j(\sigma_j+\beta_j)+\rho_j=\\
=&\frac{(2s+j+a+1)(2s+2j+3)(2s+2j+a+3)(j+a+1)}{b[2s(2s+2j+a+2)+(j+a)(2j+4)+1-a]}\\
&\times[b(2s+j+1)(j+1)+(2-a)(1-b)(2s+2j+a+1)]>\\
>&0\end{aligned}$$ and, for any $x\geq(j-a+1)/2$, $$4(2x-j+a-1)+2\sigma_j+2\geq2\sigma_j+2>0$$ and $$\begin{aligned}
&((2x-j+a-1)+1)((2x-j+a-1)+\sigma_j)+\beta_j\\
&+(2x-j+a-1)((2x-j+a-1)+\sigma_j+1)+\alpha_j\geq\\
&\geq\sigma_j+\beta_j+\alpha_j=\\
&=\frac{1}{b[2s(2s+2j+a+2)+(j+a)(2j+4)+1-a]}\\
&\times\left[b\left((1-a)(2s+j+2)(2s+j+a)(2s+2j+a+3)\right.\right.\\
&\left.\left.+(1-a)(2s+2j+a+3)(j+1)(j+a+1)\right.\right.\\
&\left.\left.+(2s+j+2)(2s+j+a+1)(j+a)(2j+4)\right.\right.\\
&\left.\left.+2s(2s+2j+3)(2s+2j+a+2)+(j+1)(j+a)(2j+4)\right.\right.\\
&\left.\left.+(1-a)(2s+2j+3)\right)\right.\\
&\left.+(1-b)\left((2s+j)(2j+a+2)+(2+a)j+2+3a\right)\right.\\
&\left.\times(2s+2j+a+1)(2s+2j+a+3)\right]>\\
&>0.\end{aligned}$$ Hence, $f^{\prime}$ maps into $(0,\infty)$, which finishes the proof.
We now come to the proof of Lemma \[lma:centrallemma\].
As a consequence of Corollary \[cor:oscillatory\], all numbers $$(-1)^j g_R^+(m,m+s;s+j),\;j\in\{0,\ldots,2m\},$$ are positive (observe that $(\beta+1,\alpha)$ is located in the interior of $\Delta$).[^7] Hence, we may define $\phi:\{1,\ldots,2m\}\rightarrow(-\infty,0)$, $$\phi(j):=\frac{c_{2s+2j+1}^T}{a_{2s+2j-1}^T}\frac{g_R^+(m,m+s;s+j)}{g_R^+(m,m+s;s+j-1)}.$$ As a consequence of , we have $$\begin{aligned}
&p(j)+\frac{q(j)}{\phi(j)}=\\
&=\frac{c_{2s+2j+3}^T}{a_{2s+2j+1}^T}\frac{\iota^+(m,m+s;j)}{\theta^+(m,m+s;j)}\\
&+\frac{c_{2s+2j+1}^T c_{2s+2j+3}^T}{a_{2s+2j-1}^T a_{2s+2j+1}^T}\frac{\kappa^+(m,m+s;j)}{\theta^+(m,m+s;j)}\frac{a_{2s+2j-1}^T}{c_{2s+2j+1}^T}\frac{g_R^+(m,m+s;s+j-1)}{g_R^+(m,m+s;s+j)}=\\
&=\frac{c_{2s+2j+3}^T}{a_{2s+2j+1}^T}\frac{g_R^+(m,m+s;s+j+1)}{g_R^+(m,m+s;s+j)}\end{aligned}$$ and obtain the recurrence relation $$\phi(j+1)=p(j)+\frac{q(j)}{\phi(j)}\;(1\leq j\leq2m-1).$$ We now use this recurrence relation and induction to show that $$\label{eq:phieven}
\phi(2j)<-1$$ and $$\label{eq:phiodd}
\phi(2j-1)>-1$$ for all $j\in\{1,\ldots,m\}$. As a consequence of and $a+2b>-a>0$, we see that $\phi(2)<-1$. Moreover, making use of , which yields $$\begin{aligned}
&\frac{g_R^+(m,m+s;s+2m-2)}{\underbrace{g_R^+(m,m+s;s+2m-1)}_{\neq0}}=\\
&=-\frac{\iota^+(m,m+s;2m-1)}{\underbrace{\kappa^+(m,m+s;2m-1)}_{>0}}+\frac{\theta^+(m,m+s;2m-1)}{\kappa^+(m,m+s;2m-1)}\frac{g_R^+(m,m+s;s+2m)}{g_R^+(m,m+s;s+2m-1)},\end{aligned}$$ and combining this with , to and , we obtain that $$\begin{split}
&4(b-1)\frac{(2m+a-1)(2m+s+a)(2m+2s+a-1)(4m+2s+a-b-1)}{4m+2s+a-2}\\
&\times\left(\frac{a_{4m+2s-3}^T}{c_{4m+2s-1}^T}\frac{g_R^+(m,m+s;s+2m-2)}{g_R^+(m,m+s;s+2m-1)}+1\right)=\\
&=(2m+a-1)(2m+2s+a-1)\\
&\times\left[(a^2+2b^2+3a)(2m+s-1)-a(a+1)(2m+s-2)+(2+2a)b^2\right]\\
&+(a+1)b(2-b)(4m+2s+a)(4m+2s+2a-1).
\end{split}$$ Therefore, we obtain that $\phi(2m-1)>-1$.\
If $m=1$, then and are already verified to hold for all $j\in\{1,\ldots,m\}$ by the preceding calculations; hence, we assume that $m\geq2$ from now on. Let $j\in\{1,\ldots,m-1\}$ be arbitrary but fixed and assume that $\phi(2j)<-1$. Then $$\phi(2j+1)=p(2j)+\frac{q(2j)}{\phi(2j)}>p(2j)-q(2j).$$ Since $p$ maps into $(-1,\infty)$, we obtain $$(1+p(2j+1))\phi(2j+1)>(1+p(2j+1))(p(2j)-q(2j)),$$ and now Lemma \[lma:pqinequality\] implies that $$(1+p(2j+1))\phi(2j+1)>-q(2j+1).$$ Since $\phi(2j+1)<0$, the latter equation yields $$\phi(2j+2)=p(2j+1)+\frac{q(2j+1)}{\phi(2j+1)}<-1.$$ Finally, let $j\in\{2,\ldots,m\}$ be arbitrary but fixed and assume that $\phi(2j-1)>-1$. We have $$\frac{1}{\phi(2j-2)}=\frac{1}{q(2j-2)}(\phi(2j-1)-p(2j-2))>-\frac{1}{q(2j-2)}(1+p(2j-2)),$$ so $$1+p(2j-2)>-q(2j-2)\frac{1}{\phi(2j-2)}.$$ Since $$0>\phi(2j-2)=p(2j-3)+\frac{q(2j-3)}{\phi(2j-3)},$$ we can conclude that $$(1+p(2j-2))\left(p(2j-3)+\frac{q(2j-3)}{\phi(2j-3)}\right)<-q(2j-2).$$ We now apply Lemma \[lma:pqinequality\] again and obtain $$(1+p(2j-2))\left(p(2j-3)+\frac{q(2j-3)}{\phi(2j-3)}\right)<(1+p(2j-2))(p(2j-3)-q(2j-3)).$$ Since $p$ maps into $(-1,\infty)$, this shows that $$p(2j-3)+\frac{q(2j-3)}{\phi(2j-3)}<p(2j-3)-q(2j-3)$$ or, equivalently, $\phi(2j-3)>-1$, which finishes the induction. Hence, and are established to hold for all $j\in\{1,\ldots,m\}$ (for any $m\geq1$). Combining this with the positivity of all numbers $(-1)^j g_R^+(m,m+s;s+j)$ (see above) and , we can conclude that all $$\begin{aligned}
&g_T(2m+1,2m+2s+1;2s+2j)=\\
&=a_{2s+2j-1}^T\cdot\underbrace{(-1)^{j-1}g_R^+(m,m+s;s+j-1)}_{>0}\cdot\underbrace{(-1)^{j-1}(1+\phi(j))}_{>0},\end{aligned}$$ $j\in\{1,\ldots,2m\}$, are positive. Since the positivity of $g_T(2m+1,2m+2s+1;2s)$ and $g_T(2m+1,2m+2s+1;4m+2s+2)$ is clear, the proof is complete.
Correction of Rahman’s hypergeometric representations
=====================================================
In this short appendix, we correct small mistakes in Rahman’s hypergeometric representations [@Ra81a (1.7) to (1.9)] of the linearization coefficients $g_R(m,n;k)$ belonging to the Jacobi polynomials $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$. For every $m\in\mathbb{N}$ and $s\in\mathbb{N}_0$, one has $$\label{eq:Rahmaneven}
\begin{split}
&g_R(m,m+s;s+j)=\\
&=\frac{\alpha+\beta+1+2s+2j}{\alpha+\beta+1}(m+\alpha+\beta+1)_m\\
&\times\frac{(\alpha+1)_{s+j}(\beta+1)_{m+s}(\alpha+\beta+1)_{2s+j}(\alpha+\beta+1)_j(m+s)!}{(\alpha+1)_s(\alpha+1)_m(\beta+1)_{s+j}(\alpha+\beta+2)_{2m+2s+j}s!j!}\\
&\times\frac{(-m)_{\frac{j}{2}}(\alpha+\beta+m+s+1)_{\frac{j}{2}}}{\left(-m-\frac{\alpha+\beta}{2}\right)_{\frac{j}{2}}(\alpha+s+1)_{\frac{j}{2}}}\\
&\times\frac{(-m-\alpha)_{\frac{j}{2}}(\beta+m+s+1)_{\frac{j}{2}}\left(\frac{1}{2}\right)_{\frac{j}{2}}}{\left(\frac{1}{2}-m-\frac{\alpha+\beta}{2}\right)_{\frac{j}{2}}(s+1)_{\frac{j}{2}}(\alpha+1)_{\frac{j}{2}}}\\
&\times{}_9F_8\left(\begin{matrix}\alpha,1+\frac{\alpha}{2},\alpha+\frac{1}{2},\frac{\alpha-\beta}{2},\frac{\alpha-\beta+1}{2},\alpha+\beta+m+s+1+\frac{j}{2}, \\ \frac{\alpha}{2},\frac{1}{2},\frac{\alpha+\beta}{2}+1,\frac{\alpha+\beta+1}{2},-\beta-m-s-\frac{j}{2},\end{matrix}\right.\\
&\left.\left.\begin{matrix}-m+\frac{j}{2},-s-\frac{j}{2},-\frac{j}{2} \\ \alpha+m+1-\frac{j}{2},\alpha+s+1+\frac{j}{2},\alpha+1+\frac{j}{2}\end{matrix}\right|1\right)
\end{split}$$ for even $j\in\{0,\ldots,2m\}$ and $$\label{eq:Rahmanodd}
\begin{split}
&g_R(m,m+s;s+j)=\\
&=\frac{\alpha+\beta+1+2s+2j}{\alpha+\beta+1}(m+\alpha+\beta+1)_m\\
&\times\frac{(\alpha+1)_{s+j}(\beta+1)_{m+s}(\alpha+\beta+1)_{2s+j}(\alpha+\beta+1)_j(m+s)!}{(\alpha+1)_s(\alpha+1)_m(\beta+1)_{s+j}(\alpha+\beta+2)_{2m+2s+j}s!j!}\\
&\times\frac{(-m)_{\frac{j+1}{2}}(\alpha+\beta+m+s+1)_{\frac{j+1}{2}}}{\left(-m-\frac{\alpha+\beta}{2}\right)_{\frac{j+1}{2}}(\alpha+s+1)_{\frac{j+1}{2}}}\\
&\times\frac{(-m-\alpha)_{\frac{j-1}{2}}(\beta+m+s+1)_{\frac{j-1}{2}}\left(\frac{3}{2}\right)_{\frac{j-1}{2}}}{\left(\frac{1}{2}-m-\frac{\alpha+\beta}{2}\right)_{\frac{j-1}{2}}(s+1)_{\frac{j-1}{2}}(\alpha+2)_{\frac{j-1}{2}}}\\
&\times\frac{\alpha-\beta}{\alpha+\beta+1}{}_9F_8\left(\begin{matrix}\alpha+1,\frac{\alpha+3}{2},\alpha+\frac{1}{2},\frac{\alpha-\beta}{2}+1,\frac{\alpha-\beta+1}{2}, \\ \frac{\alpha+1}{2},\frac{3}{2},\frac{\alpha+\beta}{2}+1,\frac{\alpha+\beta+3}{2},\end{matrix}\right.\\
&\left.\left.\begin{matrix}\alpha+\beta+m+s+\frac{3}{2}+\frac{j}{2},-m+\frac{1}{2}+\frac{j}{2},\frac{1}{2}-s-\frac{j}{2},\frac{1-j}{2} \\ \frac{1-j}{2}-\beta-m-s,\alpha+m+\frac{3}{2}-\frac{j}{2},\alpha+s+\frac{3}{2}+\frac{j}{2},\alpha+\frac{3}{2}+\frac{j}{2}\end{matrix}\right|1\right)
\end{split}$$ for odd $j\in\{0,\ldots,2m\}$, which corrects [@Ra81a (1.7), (1.8)]. This shows the nonnegativity of the $g_R(m,n;k)$ for $(\alpha,\beta)\in\Delta$. For the subcase $\alpha\geq\beta\geq-1/2$, the nonnegativity of the $g_R(m,n;k)$ can also seen via the representation $$\label{eq:Rahmanspecial}
\begin{split}
g_R(m,m+s;s+j)=&\frac{\alpha+\beta+1+2s+2j}{\alpha+\beta+1}\cdot\frac{(m+s)!}{s!j!}\\
&\times\frac{(\beta+1)_{m+s}(\alpha+\beta+1)_{2m}}{(\alpha+1)_m(\beta+1)_s(\alpha+\beta+1)_m}\\
&\times\frac{(\alpha+\beta+1)_{2s+j}(-2m)_j(2\alpha+2\beta+2m+2s+2)_j}{(\alpha+\beta+2)_{2m+2s+j}(-2m-\alpha-\beta)_j}\\
&\times\frac{(\alpha-\beta)_j}{(2\beta+2s+2)_j}\\
\times{}_9F_8&\left(\begin{matrix}\beta+s+\frac{1}{2},1+\frac{\beta+s+\frac{1}{2}}{2},\beta+\frac{1}{2},\beta+m+s+1,-m-\alpha, \\ \frac{\beta+s+\frac{1}{2}}{2},s+1,-m+\frac{1}{2},\alpha+\beta+m+s+\frac{3}{2},\end{matrix}\right.\\
&\left.\left.\begin{matrix}\frac{\alpha+\beta+1}{2}+s+\frac{j}{2},\frac{\alpha+\beta+2}{2}+s+\frac{j}{2},\frac{1-j}{2},-\frac{j}{2} \\ \frac{\beta-\alpha}{2}+\frac{2-j}{2},\frac{\beta-\alpha}{2}+\frac{1-j}{2},\beta+s+1+\frac{j}{2},\beta+s+\frac{3}{2}+\frac{j}{2}\end{matrix}\right|1\right),
\end{split}$$ which is valid for all $m\in\mathbb{N}$, $s\in\mathbb{N}_0$ and $j\in\{0,\ldots,2m\}$ and which corrects a typo in [@Ra81a (1.9)]. Note that the expressions in to may not be well-defined if $(\alpha,\beta)$ is an element of the boundary of $\Delta$; in this case, the formulas have to be interpreted as limits.
[^1]: The research was begun when the author worked at Technical University of Munich, and the author gratefully acknowledges support from the graduate program TopMath of the ENB (Elite Network of Bavaria) and the TopMath Graduate Center of TUM Graduate School at Technical University of Munich. The main part of the research was done at RWTH Aachen University.
[^2]: Recall that $(a)_0=1$ and $(a)_n=\prod_{k=1}^n(a+k-1)\;(n\in\mathbb{N})$. Since $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ is normalized such that $R_n^{(\alpha,\beta)}(1)=1$, one has $R_n^{(\alpha,\beta)}(x)=n!P_n^{(\alpha,\beta)}(x)/(\alpha+1)_n$ if $(P_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ denotes the standard normalization of the Jacobi polynomials.
[^3]: The formulas [@Ra81a (1.7) to (1.9)] contain small mistakes. We will correct them in the appendix, see to .
[^4]: Some authors prefer to call these “generalized ultraspherical polynomials” or “generalized Gegenbauer polynomials”, and some authors use the expression “generalized Chebyshev polynomials” for different things.
[^5]: Of course, the positivity of $g_R(m,m+s;s)$ and $g_R(m,m+s;s+2m)$ is also clear from general results, cf. Section \[sec:intro\].
[^6]: The equations and allow us to obtain the nonnegativity of $g_R(m,m+s;s+2)$ and $g_R(m,m+s;s+2m-2)$ in a much faster way than Gasper estimated in [@Ga70b]. Besides the avoidance of Descartes’ rule of signs in the proof of Proposition \[prp:gRmu\] (which will be applied below), this is our essential simplification of Gasper’s argument given in [@Ga70b].
[^7]: Alternatively, the positivity of the numbers $(-1)^j g_R^+(m,m+s;s+j)$ can be obtained from and Rahman’s formula (take into account that $\beta+1>0>\alpha>-1/2$).
|
---
abstract: 'The Hamiltonian formulation of the teleparallel equivalent of general relativity is considered. Definitions of energy, momentum and angular momentum of the gravitational field arise from the integral form of the constraint equations of the theory. In particular, the gravitational energy-momentum is given by the integral of scalar densities over a three-dimensional spacelike hypersurface. The definition for the gravitational energy is investigated in the context of the Kerr black hole. In the evaluation of the energy contained within the external event horizon of the Kerr black hole we obtain a value strikingly close to the irreducible mass of the latter. The gravitational angular momentum is evaluated for the gravitational field of a thin, slowly rotating mass shell.'
author:
- |
J. W. Maluf$\,^{*}$, J. F. da Rocha-Neto$\,^{\S}$\
T. M. L. Toríbio, and K. H. Castello-Branco\
Instituto de Física,\
Universidade de Brasília\
C. P. 04385\
70.919-970 Brasília DF\
Brazil\
title: Energy and angular momentum of the gravitational field in the teleparallel geometry
---
PACS numbers: 04.20.Cv, 04.20.Fy, 04.90.+e
(\*) e-mail: wadih@fis.unb.br
(§) present address: Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, São Paulo, SP, Brazil
[**I. Introduction**]{}
Teleparallel theories of gravity have been considered long time ago in connection with attempts to define the energy of the gravitational field[@Mol]. By studying the properties of solutions of Einstein’s equations that describe the gravitational field of isolated material systems, it is concluded that a consistent expression for the energy [*density*]{} of the gravitational field would be given in terms of second order derivatives of the metric tensor. It is known that there exists no covariant, nontrivial expression constructed out of the metric tensor, both in three and in four dimensions, that contain such derivatives. However, covariant expressions that contain second order derivatives of tetrad fields are feasible. Thus it is legitimate to conjecture that the difficulties regarding the problem of defining the gravitational energy-momentum is related to the geometrical description of the gravitational field, rather than being an intrinsic drawback of the theory[@Grishchuk].
It is usually asserted in the literature that the principle of equivalence prevents the localizability of the gravitational energy. However, an expression for the gravitational field energy has been pursued since the early days of general relativity. A considerable amount of effort has been devoted to finding viable expressions other than pseudotensors (more recently the idea of quasi-local energy, i.e., energy associated to a closed spacelike two-surface, in the context of the Hilbert-Einstein action integral, has emerged as a tentative description of the gravitational energy[@Brown]). The search for a consistent expression for the gravitational energy is undoubtedly a longstanding problem in general relativity. The argument based on the principle of equivalence regarding the nonlocalizability of the gravitational energy is controversial and not generally accepted[@Grishchuk]. The principle of equivalence does not preclude the existence of scalar densities on the space-time manifold, constructed out of tetrad (or triad) fields, that may eventually yield the correct description of the energy properties of the gravitational field. Such densities may be given in terms of the torsion tensor, which cannot be made to vanish at a point by a coordinate transformation. Møller[@Mol] was probably the first one to notice that the tetrad description of the gravitational field allows a more satisfactory treatment of the gravitational energy-momentum.
The dynamics of the gravitational field can be described in the context of the teleparallel geometry, where the basic geometrical entity is the tetrad field $e^a\,_\mu$, ($a$ and $\mu$ are SO(3,1) and space-time indices, respectively). Teleparallel theories of gravity are defined on the Weitzenböck space-time[@Weit], endowed with the affine connection
$$\Gamma^\lambda_{\mu\nu}=e^{a\lambda} \partial_\mu
e_{a\nu}\;.\eqno(1.1)$$
The curvature tensor constructed out of Eq. (1.1) vanishes identically. This connection defines a space-time with teleparallelism, or absolute parallelism[@Schouten]. This geometrical framework was considered by Einstein[@Einstein] in his attempt at unifying gravity and electromagnetism.
Gravity theories in this geometrical framework are constructed out of the torsion tensor. An infinity of such theories defined by a Lagrangian density, quadratic in the torsion tensor, has been investigated by Hayashi and Shirafuji[@Hay] (who denote $e^a\,_\mu$ as [*parallel vector fields*]{}). Among such infinity of theories a particular one is distinguished, because the tetrad fields that are solutions of this particular theory yield a metric tensor that is a solution of Einstein’s equations. The teleparallel equivalent of general relativity (TEGR)[@Hehl; @Kop; @Muller; @Nester1; @Nester2; @Maluf1; @Per] constitutes an alternative geometrical description of Einstein’s equations.
A simple expression for the gravitational energy arises in the Hamiltonian formulation of the TEGR[@Maluf1] in the framework of Schwinger’s time gauge condition[@Schwinger]. The energy density is given by a scalar density in the form of a total divergence that appears in the Hamiltonian constraint of the theory[@Maluf2]. The investigations carried out so far confirm the consistency and relevance of this energy expression.
A recent approach to the localization of the gravitational energy has been considered in the Lagrangian framework of the TEGR by Andrade, Guillen and Pereira[@Per2]. It has been shown in the latter reference the existence of an expression for the gravitational energy density that is a true space-time tensor, and that reduces to Møller’s energy-momentum density of the gravitational field.
The Hamiltonian formulation of the TEGR, with no [*a priori*]{} restriction on the tetrad fields, has recently been established[@Maluf3]. Its canonical structure is different from that obtained in Ref. [@Maluf1], since it is not given in the standard ADM form[@ADM]. In this framework we again arrive at an expression for the gravitational energy, in strict similarity with the procedure adopted in Ref. [@Maluf2], namely, by interpreting the Hamiltonian constraint equation as an energy equation for the gravitational field. Likewise, the gravitational momentum can be defined. The constraint algebra of the theory suggests that certain momentum components are related to the gravitational angular momentum. It turns out to be possible to define, in this context, the angular momentum of the gravitational field.
In this article we investigate the definition of gravitational energy that arises in Ref. [@Maluf3], in the framework of the Kerr metric tensor[@Kerr]. The whole formulation developed in Ref. [@Maluf3] is carried out without enforcing the time gauge condition. It turns out, however, that consistent values for the gravitational energy are achieved by requiring the tetrad field to satisfy ([*a posteriori*]{}) the time gauge condition.
We investigate the irreducible mass $M_{irr}$ of the Kerr black hole. It is the total mass of the black hole at the final stage of Penrose’s process of energy extraction, considering that the maximum possible energy is extracted. It is also related to the energy contained within the external event horizon $E(r_+)$ of the black hole (the surface of constant radius $r=r_+$ defines the external event horizon). Every expression for local or quasi-local gravitational energy must necessarily yield the value of $E(r_+)$ in close agreement with $2M_{irr}$, since we know beforehand the value of the latter as a function of the initial angular momentum of the black hole[@Chris]. The evaluation of $2M_{irr}$ is a crucial test for any expression for the gravitational energy. $E(r_+)$ has been obtained by means of different energy expressions in Ref. [@Bergqvist]. Our expression for the gravitational energy is the only one that yields a satisfactory value for $E(r_+)$, strikingly close to $2M_{irr}$, and that arises in the framework of the Hamiltonian formulation of the gravitational field.
In the Hamiltonian formulation of the TEGR[@Maluf3] there arises a set of primary constraints $\Gamma^{ik}$ that satisty the angular momentum algebra. Following the prescription for defining the gravitational energy, the definition of the gravitational angular momentum arises by suitably interpreting the integral form of the constraint equation $\Gamma^{ik}=0$ as an angular momentum equation. We apply this definition to the gravitational field of a thin, slowly rotating mass shell. In the limit of slow rotation we obtain a realistic measure of the angular momentum of the field in terms of the moment of inertia of the source.
Notation: space-time indices $\mu, \nu, ...$ and SO(3,1) indices $a, b, ...$ run from 0 to 3. Time and space indices are indicated according to $\mu=0,i,\;\;a=(0),(i)$. The tetrad field $e^a\,_\mu$ yields the definition of the torsion tensor: $T^a\,_{\mu \nu}=\partial_\mu e^a\,_\nu-\partial_\nu e^a\,_\mu$. The flat, Minkowski space-time metric is fixed by $\eta_{ab}=e_{a\mu} e_{b\nu}g^{\mu\nu}= (-+++)$.\
[**II. The Hamiltonian constraint equation as an energy equation for the gravitational field**]{}
We summarize here the Hamiltonian formulation obtained in Ref. [@Maluf1], where Schwinger’s time gauge is assumed. The Hamiltonian density constructed out of triads $e_{(i)j}$ restricted to the three-dimensional spacelike hypersurface, and of the momenta canonically conjugated $\Pi^{(i)j}$, is given by
$$H=NC+N^iC_i+\Sigma_{mn}\Pi^{mn}+
{1\over {8\pi G}}\partial_k(NeT^k)+\partial_k(\Pi^{jk}N_j)
\;,\eqno(2.1)$$
where $N$ and $N^i$ are lapse and shift functions, $\Sigma_{mn}=-\Sigma_{nm}$ are Lagrange multipliers, $G$ is the gravitational constant and $\Pi^{ij}=e_{(k)}\,^i \Pi^{(k)j}$. The constraints are defined by
$$C=\partial_j(2keT^j)-ke\Sigma^{kij}T_{kij}
-{1\over{4ke}}\biggl(\Pi^{ij}\Pi_{ji}-{1\over 2}\Pi^2\biggr)\;,
\eqno(2.2)$$
$$C_k=-e_{(j)k}\partial_i\Pi^{(j)i}-\Pi^{(j)i}T_{(j)ik}
\;,\eqno(2.3)$$
where $e=det(e_{(i)j})$ and $k={1\over {16\pi G}}$. The tensor $\Sigma^{kij}$ reads
$$\Sigma^{kij}={1\over 4}(T^{kij}+T^{ikj}-T^{jki})+
{1\over 2}(g^{kj}T^i-g^{ki}T^j)\;.\eqno(2.4)$$
The trace of the torsion tensor is $T^i=g^{ik}T_k=g^{ik}e^{(m)j}T_{(m)jk}$. The definition of $\Sigma^{kij}$ yields
$$\Sigma^{kij}T_{kij}={1\over 4}T^{kij}T_{kij}+
{1\over 2}T^{kij}T_{ikj}-T^iT_i\;.$$
The first two terms on the right hand side of (2.2) are equivalent to the scalar curvature $R(e_{(i)j})$ on the three-dimensional spacelike hypersurface,
$$2\partial_j (eT^j)-e\Sigma^{kij}T_{kij}=
eR(e_{(i)j})\;.\eqno(2.5)$$
The integral form of the Hamiltonian constraint equation $C(x)=0$ can be interpreted as an energy equation[@Maluf2],
$$\int d^3x\, \partial_j(2keT^j)=
\int d^3x \biggl\{ke\Sigma^{kij}T_{kij}+
{1\over{4ke}}\biggl(\Pi^{ij}\Pi_{ji}-{1\over 2}\Pi^2\biggr)
\biggr\}\;.\eqno(2.6)$$
We identify Eq. (2.6) as an energy equation because the integral of the left hand side of this equation over the whole three-dimensional space yields the Arnowitt-Deser-Misner energy[@ADM],
$${1\over{8\pi G}}\int d^3 x \partial_j(eT^j)=
{1\over {16\pi G}}\int_S dS_k(\partial_i h_{ik}-
\partial_k h_{ii})=E_{ADM}\;.\eqno(2.7)$$
The right hand side of Eq. (2.7) is obtained by requiring the asymptotic behaviour
$$e_{(i)j}\simeq \eta_{ij}+{1\over 2}h_{ij}({1\over r})
\;,\eqno(2.8)$$
in the limit $r \rightarrow \infty$. $\eta_{ij}$ is the spatial sector of Minkowski’s metric tensor and $h_{ij}$ is the first term in the asymptotic expansion of $g_{ij}$. Therefore we define the gravitational energy enclosed by a volume $V$ of the three-dimensional space as[@Maluf2]
$$E_g={1\over{8\pi G}}\int_V d^3x\partial_j(eT^j)\;.\eqno(2.9)$$
The expression above has been applied to several configurations of the gravitational field. The most relevant application is the evaluation of the irreducible mass of the Kerr black hole[@Maluf4].
[**III. Gravitational energy expression in terms of tetrad fields**]{}
An expression for the gravitational energy density also arises in the framework of the Hamiltonian formulation of general relativity in the teleparallel geometry[@Maluf3], without posing any [*a priori*]{} restriction on the tetrad fields, again interpreting the integral form of the constraint equations as energy-momentum equations for the gravitational field.
The Hamiltonian formulation developed in Ref. [@Maluf3] is obtained from the Lagrangian density in empty space-time defined by
$$L(e)\;=\;-k\,e\,\biggl( {1\over 4} T^{abc}T_{abc} +
{1\over 2}T^{abc}T_{bac}-T^aT_a\biggr)\;,\eqno(3.1)$$
where $e=det(e^a\,_\mu)$, $T_{abc}=e_b\,^\mu e_c\,^\nu T_{a \mu \nu}$ and the trace of the torsion tensor is given by $T_b=T^a\,_{ab}\;.$ The Hamiltonian is obtained by just rewriting the Lagrangian density in the form $L=p\dot q -H$. It has not been made use of any kind of projection of metric variables to the three-dimensional spacelike hypersurface. Since there is no time derivative of $e_{a0}$ in (3.1), the corresponding momentum canonically conjugated $\Pi^{a0}$ vanishes identically. Dispensing with surface terms the total Hamiltonian density reads[@Maluf3]
$$H(e_{ai},\Pi^{ai})
=e_{a0}C^a+\alpha_{ik}\Gamma^{ik}+\beta_k\Gamma^k\;,\eqno(3.2)$$
where $\lbrace C^a, \Gamma^{ik}$ and $\Gamma^k\rbrace$ constitute a set of primary constraints, and $\alpha_{ik}$ and $\beta_k$ are Lagrange multipliers. Explicit details are given in Ref. [@Maluf3]. The first term of the constraint $C^a$ is given by a total divergence in the form $C^a=-\partial_k \Pi^{ak}+\cdot\cdot\cdot\; $. In similarity with Eq. (2.6) we identify this total divergence on the three-dimensional spacelike hypersurface as the energy-momentum density of the gravitational field. The total energy-momentum is defined by
$$P^a=-\int_V d^3x\,\partial_i \Pi^{ai}\;,\eqno(3.3)$$
where $V$ is an arbitrary space volume. It is invariant under coordinate transformations on the spacelike manifold, and transforms as a vector under the global SO(3,1) group (we will return to this point later on). The definition above generalizes expression (2.9) to tetrad fields that are not restricted by the time gauge condition. However, both expressions are equivalent, as we will see ahead, if the time gauge condition is imposed. After implementing the primary constraints $\Gamma^{ik}$ and $\Gamma^k$, the expression of the momenta $\Pi^{ak}$ reads
$$\Pi^{ak}\;=\;k\,e\biggl\{
g^{00}(-g^{kj}T^a\,_{0j}-
e^{aj}T^k\,_{0j}+2e^{ak}T^j\,_{0j})$$
$$+g^{0k}(g^{0j}T^a\,_{0j}+e^{aj}T^0\,_{0j})
\,+e^{a0}(g^{0j}T^k\,_{0j}+g^{kj}T^0\,_{0j})
-2(e^{a0}g^{0k}T^j\,_{0j}+e^{ak}g^{0j}T^0\,_{0j})$$
$$-g^{0i}g^{kj}T^a\,_{ij}+e^{ai}(g^{0j}T^k\,_{ij}-
g^{kj}T^0\,_{ij})-2(g^{0i}e^{ak}-g^{ik}e^{a0})
T^j\,_{ji} \biggr\}\;.\eqno(3.4)$$
With appropriate boundary conditions expression (3.3) yields the ADM energy. Let us consider asymptotically flat space-times and assume that in the limit $r \rightarrow \infty$ the tetrad fields have the asymptotic behaviour
$$e_{a\mu} \simeq \eta_{a\mu}
+ {1\over 2}h_{a\mu}({1\over r})\;,\eqno(3.5)$$
where $\eta_{a\mu}$ is Minkowski’s metric tensor and $h_{a\mu}$ is the first term in the asymptotic expansion of $g_{\mu \nu}$. Asymptotically flat space-times are defined by Eq. (3.5) together with $\partial_\mu g_{\lambda \nu}=O({1\over r^2})$, or $\partial_\mu e_{a\nu}=O({1\over r^2})$. Considering the $a=(0)$ component in Eq. (3.3) and integrating over the whole three-dimensional spacelike hypersurface we find, after a long but straightforward calculation, that
$$P^{(0)}=E = - \int_{V\rightarrow \infty} d^3x \partial_k
\Pi^{(0)k} =-2k\int_{V \rightarrow \infty} d^3x
\partial_k(eg^{ik}e^{(0)0}T^j\,_{ji})$$
$$={1\over {16\pi G}}\int_{S\rightarrow \infty}dS_k(\partial_i
h_{ik}-\partial_k h_{ii}) = E_{ADM}\;.\eqno(3.6)\;$$
We will prove that expressions (2.9) and (3.3) coincide if we require the time gauge condition. In order to prove it, let us rewrite $\Pi^{(0)k}$ as
$$\Pi^{(0)k}=e^{(0)}\,_i\Pi^{(ik)}
+e^{(0)}\,_i\Pi^{\lbrack ik \rbrack}
+e^{(0)}\,_0\Pi^{0k}\;,\eqno(3.7)$$
where $(..)$ and $\lbrack .. \rbrack$ denote symmetric and anti-symmetric components, respectively. In the time gauge condition we have $e_{(j)}\,^0=e^{(0)}\,_i=0$, and therefore Eq. (3.7) reduces to $\Pi^{(0)k}=e^{(0)}\,_0\Pi^{0k}$. An expression for $\Pi^{0k}$ can be obtained by requiring the vanishing of the constraint $\Gamma^k$[@Maluf3],
$$\Gamma^k \;=\;\Pi^{0k}\; +\;2k\,e\, (
g^{kj}g^{0i}T^0\,_{ij}-g^{0k}g^{0i}T^j\,_{ij}
+g^{00}g^{ik}T^j\,_{ij} )
\;.\eqno(3.8)$$
In the time gauge we have $T^0\,_{ij}=0$ and therefore from $\Gamma^k=0$ we arrive at
$$\Pi^{0k}=
2ke(g^{0k}g^{0i}-g^{00}g^{ik})T^j\,_{ij}\;.\eqno(3.9)$$
All quantities in Eq. (3.9) are four-dimensional field quantities. Let us now rewrite Eq. (3.9) in terms of field quantities restricted to the three-dimensional spacelike hypersurface by means of the lapse and shift functions, $N$ and $N^i$, respectively. In view of the relations $e=N\,^3e$,$\;\;$ $g^{0i}=N^i / N^2$ and $g^{ik}=\,^3g^{ik}-(N^iN^k) / N^2$, Eq. (3.9) can be written as
$$\Pi^{0k}={2\over N}k\;(\,^3e) (\,^3g^{ik}) T^j\,_{ij}\;.$$
The superscript 3 indicates that the quantity is projected on the spacelike hypersurface. Note that $T^j\,_{ij}$ is still given in terms of four-dimensional field quantities.
We make use of a 3+1 decomposition for the tetrad fields according to $e^{ai}=\,^3e^{ai}+(N^i / N)\eta^a$, $\;e^a\,_i=\,^3e^a\,_i$, $\;\eta^a=-Ne^{a0}$ and $e^a\,_0=\eta^a N+\,^3e^a\,_i N^i$. The tetrad fields $\,^3e_{ai}$ and $\,^3e^{ai}$ are related to each other by means of the metric tensor $g_{ij}$ and its inverse $\,^3g^{ij}$. With the help of these relations we can rewrite $T^j\,_{ij}$ in terms of quantities on the spacelike hypersurface in the time gauge condition, in which case we have $\eta^a=\delta^a_{(0)}$ and $e^{(0)}\,_0=N$. We eventually arrive at
$$\Pi^{(0)k}=2keg^{ik}g^{jm}e^{(l)}\,_m T_{(l)ij}\;,\eqno(3.10)$$
where we have eliminated the superscript 3. It is straightfoward to verify that $\Pi^{(0)k}=-T^k$, where $T^k$ and $T_{(l)ij}$ are precisely the same quantities that appear in section II, and in particular in expression (2.9). Therefore in the time gauge condition we have
$$P^{(0)}=-\int_V d^3x\,\partial_i \Pi^{(0)i}=
{1\over{8\pi G}}\int_V d^3x\partial_j(eT^j)\;.\eqno(3.11)$$
Differently from the quasi-local energy expressions[@Brown], Eq. (3.11) is an integral of a scalar density over finite space volumes, which can be transformed into a surface integral. Therefore our expression is not bound, in principle, to belong to any class of quasi-local energies. There is no need of subtraction terms in the present framework. And yet Eq. (3.11) does satisfy the usual requirements for a quasi-local energy expression. According to the latter requirements the quasi-local energy expression must (i) vanish for the Minkowski space-time; (ii) yield the ADM and Bondi mass in the appropriate limits; (iii) yield the appropriate value for weak and spherically symmetric gravitational fields and (iv) yield the irreducible mass of the Kerr black hole. The Bondi energy in the TEGR has been discussed in Ref. [@Maluf5], and the latter requirement is discussed in section V.
[**IV. The determination of tetrad fields**]{}
In the framework of the teleparallel geometry the gravitational field can be described by an anholonomic transformation between a reference space-time and the physical space-time. We will briefly recall the difference between holonomic and anholonomic transformations. Let us consider two sets of coordinates, $q^a=(t,x,y,z)$ and $x^\mu=(t,r,\theta,\phi)$, related by the coordinate transformation $dq^a=e^a\,_\mu dx^\mu$ such that
$$e^a\,_\mu={{\partial q^a}\over{\partial x^\mu}}=
\pmatrix{1&0&0&0\cr
0&\sin\theta\, \cos\phi & r\,\cos\theta\,\cos\phi
& -r\,\sin\theta\,\sin\phi\cr
0&\sin\theta\, \sin\phi & r\,\cos\theta\,\sin\phi
& r\,\sin\theta\,\cos\phi\cr
0&\cos\theta & -r\,\sin\theta & 0\cr}\;. \eqno(4.1)$$
The relation $dq^a=e^a\,_\mu dx^\mu$ can be integrated over the whole space-time, and therefore the transformation $q^a \rightarrow x^\mu$ corresponds to a single-valued global transformation. In this case the transformation is called holonomic and both coordinate sets describe the same space-time.
However, in the general case the relation $dq^a=e^a\,_\mu dx^\mu$ cannot be globally integrated, since $e^a\,_\mu$ may not be a gradient function of the type $\partial_\mu q^a $. If the quantities $e^a\,_\mu$ are such that $\partial_\mu e^a\,_\nu - \partial_\nu e^a\,_\mu \ne 0$, then the transformation is called anholonomic.
For the tetrads given by (4.1) the torsion tensor $T^a\,_{\mu\nu}=\partial_\mu e^a\,_\nu - \partial_\nu e^a\,_\mu$ vanishes. It is known that $T^a\,_{\mu\nu}$ vanish identically if and only if $e^a\,_\mu$ are gradient vectors[@Schouten2]. In the framework of the TEGR the gravitational field corresponds to a configuration such that $T^a\,_{\mu\nu}\ne 0.$ Thus every gravitational field is described by a space-time that is anholonomically related to the four-dimensional Minkowski space-time, which is taken as the reference space-time. Consequently the tetrad fields to be considered must necessarily yield a vanishing torsion tensor in the limit of vanishing physical parameters (such as mass, angular momentum and charge), in which case the tetrad field must reduce to expression (4.1), or, in the case of arbitrary coordinates, to the form $e^a\,_\mu = \partial_\mu q^a$.
The idea of describing the gravitational field as the gauge field of the Poincaré group is rather widespread. In view of the general acceptance of this idea, there is a unjustified prejudice against gravitational theories that do not exhibit local SO(3,1) symmetry. Rather than being a drawback of the present formulation, the requirement of a global set of tetrad fields for the description of the space-time is a natural feature of teleparallel theories[@Einstein] and of the teleparallel geometry.
Before addressing the problem of obtaining the appropriate set of tetrad fields out of a given metric tensor, it is instructive to analyze the construction of tetrads for the flat space-time, since a number of features that take place in this context carry over to the general case of an arbitrary space-time metric tensor. We will consider two sets of tetrads that describe the flat space-time, and that reveal the relationship between the reference space-time with coordinates $q^a$ and the physical space-time with coordinates $x^\mu$.
For a given arbitrary function $\omega(t)$ let us consider a transformation between two rotating cartesian coordinate systems, $q^0=t$, $q^1=x^1\cos\omega(t)-x^2\sin\omega(t)$, $q^2=x^1\sin\omega(t)+x^2\cos\omega(t)$, $q^3=x^3$. The tetrads are given by
$$e^a\,_\mu(t,x,y,z)=
\pmatrix{1&0&0&0\cr
-(x^1\sin\omega+x^2\cos\omega)\dot\omega
&\cos\omega&-\sin\omega &0\cr
(x^1\cos\omega-x^2\sin\omega)\dot\omega
&\sin\omega &\cos\omega&0\cr
0&0&0&1}\;.\eqno(4.2)$$
These tetrads describe a flat space-time with cartesian coordinates $x^\mu$ that is rotating with respect to the reference space-time with coordinates $q^a$. We notice the appearance of anti-symmetric components in the spatial sector of $e^a\,_\mu$. This is a general feature in cartesian coordinates: under an infinitesimal rotation a rotated vector $\tilde V$ is related to the vector $V$ by means of the relation $\tilde V=RV$; the rotation matrix is given by $R=1+\omega_iX^i$, where $\omega_i$ are arbitrary parameters and the generators $X^i$ are anti-symmetric matrices. Therefore the emergence of anti-symmetric components in the sector $e_{(i)j}(t,x,y,z)$ is expected if the two space-times are rotating with respect to each other.
Another transformation of general character is a Lorentz boost, $q^{(0)}=\gamma(t+(v/c^2)x^1)$, $q^{(1)}=\gamma(x^1+vt)$, $q^{(2)}=x^2$ and $q^{(3)}=x^3$, where $\gamma=1/\sqrt{1-v^2/c^2}$ (assuming the velocity of light $c\ne 1$). The two space-times have different time scales. The tetrads read
$$e^a\,_\mu(t,x,y,z)=
\pmatrix{ \gamma&(v/c^2)\gamma &0&0\cr
v\gamma &\gamma&0&0\cr
0&0 &1&0\cr
0&0&0&1}\;.\eqno(4.3)$$
The tetrads above do not satisfy the time gauge condition because of the emergence of the term $e^{(0)}\,_1=(v/c^2)\gamma$. Under an arbitrary boost transformation there will arise terms such that $e^{(0)}\,_k\ne 0$, which violate the time gauge condition $e_{(i)}\,^0=0$. The main feature of the time gauge condition is to lock the time axes of the reference space-time and of the physical space-time.
In the absence of the gravitational field, $e^a\,_\mu(t,x,y,z)=\delta^a_\mu$ is the unique set of tetrads that describes a reference space-time with coordinates $q^a$ that is neither related by a boost transformation nor rotating with respect to the physical space-time with coordinates $x^\mu$. The features above should also carry over to the case of an arbitrary gravitational field. As we will see, they are essential in the description of the energy properties of the gravitational field. Likewise, for a given space-time metric tensor the set of tetrad fields that in cartesian coordinates satisfy the properties
$$e_{(i)j}=e_{(j)i}\;,\eqno(4.4a)$$
$$e_{(i)}\,^0=0\,,\eqno(4.4b)$$
establish a [*unique*]{} reference space-time that is neither related by a boost transformation, nor rotating with respect to the physical space-time. Equations (4.4b) fix six degrees of freedom of the tetrad field. The importance of Eqs. (4.4a,b) to the definition of the gravitational energy will be discussed at the end of section V.
Let us consider now the Kerr space-time. In terms of Boyer-Lindquist[@BL] coordinates the Kerr metric tensor is given by
$$ds^2=-{\psi^2 \over\rho^2}dt^2-
{{2\chi \sin^2\theta}\over \rho^2}d\phi\,dt
+{\rho^2 \over \Delta}dr^2+
\rho^2 d\theta^2+
{{\Sigma^2 \sin^2\theta}\over \rho^2}d\phi^2 \;,\eqno(4.5)$$
where $\rho^2=r^2+a^2 cos^2\theta$, $\Delta= r^2 +a^2 -2mr$, $\chi=2amr$ and
$$\Sigma^2=(r^2+a^2)^2-\Delta a^2\,\sin^2\theta\;,$$
$$\psi^2=\Delta -a^2\,\sin^2\theta\;.$$
Each set of tetrad fields defines a teleparallel geometry. For a given space-time metric tensor $g_{\mu\nu}$, there exists an infinite set of tetrad fields that yield $g_{\mu\nu}$. From the point of view of the metrical properties of the space-time, any two set of tetrads out of this infinity corresponds to viable (but distinct) teleparallel configurations[@Nester2]. However, the description of the gravitational field energy requires at least boundary conditions. In the framework of the teleparallel geometry the correct description of the gravitational energy-momentum singles out a unique set of tetrad fields. In the following we will consider the most relevant tetrad configurations. The first one is based on the weak field approximation first suggested by Møller, given by expression (3.5),
$$e^M_{a\mu} \simeq \eta_{a\mu}
+ {1\over 2}h_{a\mu}\;,\eqno(4.6a)$$
together with the symmetry condition on $h_{a\mu}$,
$$h_{a\mu}=h_{\mu a}\;.\eqno(4.6b)$$
Note that Eq. (4.6a) is demanded not only in the asymptotic limit, but at every space-time point. Although the weak field limit fixes the expression of $e^M_{a\mu}$, the resulting expression is taken to hold in the strong field regime. The expression that satisfies Eq. (4.6) and that yields Eq. (4.5) is given by
$$e^M_{a\mu}=\pmatrix{
-{\psi \over \rho}\sqrt{1+M^2y^2} & 0 & 0 &
-{{\chi N y}\over {\psi \rho}}\sin^2\theta\cr
{{\chi y}\over {\Sigma \rho}}\sin\theta\,\sin\phi &
{\rho \over \sqrt{\Delta}}\sin\theta\,\cos\phi &
\rho\, \cos\theta\,\cos\phi &
-{\Sigma \over \rho}\sqrt{1+M^2N^2y^2}\sin\theta\,\sin\phi\cr
-{{\chi y}\over{\Sigma \rho}}\sin\theta\, \cos\phi &
{\rho \over \sqrt{\Delta}} \sin\theta\,\sin\phi &
\rho\, \cos\theta\,\sin\phi &
{\Sigma \over \rho}\sqrt{1+M^2N^2y^2}\sin\theta\,\cos\phi \cr
0 & {\rho \over \sqrt{\Delta}}\cos\theta &
-\rho\, \sin\theta & 0 \cr}\,,\eqno(4.7)$$
where
$$y^2={{ 2N\sqrt{1+M^2}-(1+N^2)}\over
{4M^2N^2-(1-N^2)^2}}\;,$$
$$M={\chi \over{\Sigma \psi}}\sin\theta\;,$$
$$N={{\psi r}\over \Sigma}\;.$$
The second set of tetrad fields to be considered satisfies the weak field approximation
$$e_{(i)j}\simeq \eta_{ij}+{1\over 2} h_{ij}\;,\eqno(4.8a)$$
$$h_{ij}=h_{ji}\;,\eqno(4.8b)$$
together with Schwinger’s time gauge condition, $e_{(k)}\,^0=e^{(0)}\,_j=0$ (Eq. (4.4b)). Note that Eqs. (4.8a,b) are essentially equivalent to Eq. (4.4a). Conditions (4.8) are assumed to fix the expression of $e^a\, _\mu$ also in the strong field regime. The set of tetrad fields that satisfies Eqs. (4.8), (4.4b) and that yields Eq. (4.5) reads
$$e^S_{a\mu}=\pmatrix{
-{1\over \rho}\sqrt{\psi^2+{\chi^2\over \Sigma^2}\sin^2\theta} &
0&0&0\cr
{\chi \over {\Sigma \rho}}\sin\theta\,\sin\phi &
{\rho \over \sqrt{\Delta}}\sin\theta\,\cos\phi &
\rho\,\cos\theta\,\cos\phi &
-{\Sigma \over \rho} \sin\theta\,\sin\phi\cr
-{\chi \over{\Sigma \rho}}\sin\theta\,\cos\phi &
{\rho \over \sqrt{\Delta}}\sin\theta\,\sin\phi &
\rho\,\cos\theta\,\sin\phi &
{\Sigma \over \rho}\sin\theta\,\cos\phi\cr
0&{\rho \over \sqrt{\Delta}}\cos\theta&-\rho\,\sin\theta&0\cr}
\,.\eqno(4.9)$$
We note finally that both Eqs. (4.7) and (4.9) reduce to Eq. (4.1) if we make $m=a=0$.
[**V. The irreducible mass of the Kerr black hole**]{}
In this section we will apply expression (3.3) to the evaluation of the irreducible mass $M_{irr}$ of the Kerr black hole. This is the most important test for any gravitational energy expression, local or quasi-local, since the geometrical setting corresponds to an intricate configuration of the gravitational field, and since the value of $M_{irr}$ is known from the work of Christodoulou[@Chris].
In order to obtain $M_{irr}$ we will calculate the $a=(0)$ component of Eq. (3.3) by fixing $V$ to be the volume within the $r= r_+$ surface, where $r_+=m+\sqrt{m^2-a^2}$ is the external horizon of the Kerr black hole. Therefore we will consider
$$P^{(0)}=E=-\int_S dS_i \,\Pi^{(0)i}=
-\int_S d\theta d\phi \,\Pi^{(0)1}(r,\theta,\phi) \;,\eqno(5.1)$$
where the surface $S$ is determined by the condition $r= r_+$. The expression of $\Pi^{(0)1}$ will be obtained by considering Eq. (4.7). In view of Eq. (3.11) there is no need to calculate Eq. (5.1) out of Eq. (4.9), since it has already been evaluated[@Maluf4].
In the Appendix we present the expressions of the components of the torsion tensor constructed out of the tetrad configuration Eq. (4.7). The component $\Pi^{(0)1}$ is then obtained from the definition (3.4) by means of simple (albeit long) algebraic manipulations. The expression of $\Pi^{(0)1}(r,\theta,\phi)$ for the tetrad expression (4.7) reads
$$\Pi^{(0)1}={{k\Sigma y}\over \rho}\sin\theta\,\biggl\{
-2\biggl(1+N\Omega +{\rho^2 \over{y\Sigma}}\biggr)
+{{2\sqrt{\Delta}}\over \Sigma}\partial_r \Sigma
+{{\sqrt{\Delta} N}\over \Omega}\biggl(
{M^2\over \chi}\partial_r \chi
+ {2\over \Sigma}\partial_r \Sigma\biggr) \biggr\}\;,\eqno(5.2)$$
where the definitions of $y, N$ and $M$ are given after expression (4.7) and
$$\Omega=\sqrt{1+M^2}\;.$$
On the surface $r=r_+$ we have $\Delta(r_+)=0$, $M^2(r_+)=-1$ and $\Omega(r_+)=0$. Therefore the last term in Eq. (5.2) is indefinite. It must be calculated by taking the limit $r\rightarrow r_+$. We find
$$\lim_{r\rightarrow r_+}
{{\sqrt{\Delta} N}\over \Omega}\biggl(
{M^2\over \chi}\partial_r \chi
+ {2\over \Sigma}\partial_r \Sigma\biggr)=
-{{a^2\sin^2\theta}\over m}\biggl(
{\sqrt{m^2-a^2}\over {2mr_+}}+
{r_+ \over{2mr_+ -a^2\sin^2\theta}}\biggr)\;.$$
The other terms in Eq. (5.2) do not pose any problem, and thus we can obtain the expression of the energy contained within the external event horizon of the Kerr black hole, that follows from the tetrad configuration (4.7). The final expression arises as a function of the angular momentum per unit mass $a$. It is given by (we are assuming $G=1$)
$$E\lbrack e^M_{a\mu}\rbrack = {m\over 4}
\int_0^\pi d\theta\,\sin\theta\,\bigg[
\sqrt{p^2+\lambda^2 \cos^2\theta}+
{{py}\over \sqrt{p^2+\lambda^2 \cos^2\theta}}$$
$$+{ {2p^3y} \over{(p^2+\lambda^2 \cos^2\theta)^{3\over 2}}}-
{ {y(p-1) \sqrt{p^2+\lambda^2 \cos^2\theta}}\over 2}
\biggr]\;,\eqno(5.3)$$
where
$$p=1+\sqrt{1-\lambda^2}\;\;\;\;,\;\;\;\;
a=\lambda m,\;\;\;\; 0\leq \lambda \leq 1\;.$$
For the tetrad configuration Eq. (4.9) we have[@Maluf4]
$$E\lbrack e^S_{a\mu}\rbrack =
m\biggl[ {\sqrt{2p}\over 4}+{{6p-\lambda^2}\over 4\lambda} \ln
\biggl( {{\sqrt{2p} +\lambda}\over p} \biggr) \biggr]
\;.\eqno(5.4)$$
Expressions (5.3) and (5.4) must be compared with $2M_{irr}$, where $M_{irr}$ is given by[@Chris] $M_{irr}={1\over 2}\sqrt{{r_+}^2+a^2}$. In our notation we have
$$2M_{irr}=m\sqrt{2p}\;.\eqno(5.5)$$
In the limit $a\rightarrow 0$ all energy expressions yield $2m$, which is the value obtained by several different approaches[@Bergqvist]. In figure 1 we have plotted $\varepsilon =E / m$ against $\lambda$, where $0\le \lambda \le 1$. Each value of $\lambda$ characterizes an angular momentum state of the black hole. The hope was that the tetrad field given by Eq. (4.7) would explain the tiny difference between the numerical values of Eqs. (5.4) and (5.5). However, the deviation of expression (5.3) from $2M_{irr}$ indicates that the tetrad configuration Eq. (4.7) is not appropriate to the description of gravitational energy. The latter is most correctly described by requiring the tetrad configuration Eq. (4.9), that satisfies Schwinger’s time gauge condition together with Eq. (4.4a).
The choice of the tetrad field given by Eq. (4.9) amounts to choosing the unique reference space-time that is neither related by a boost transformation nor rotating with respect to the physical space-time.
For an arbitrary space volume $V$ the gravitational energy is defined relationally, in the sense that it depends on the choice of the reference space-time. If the tetrad fields are required to satisfy conditions (4.4a,b) for a metric tensor that exhibits asymptotic boundary conditions similar to Eq. (3.5), then for asymptotically flat space-times the physical space-time coincides with the reference space-time in the limit $r \rightarrow \infty$. If, however, we choose a reference space-time that is, for instance, rotating (about the $z$ axis, say) with respect to the space-time defined by the Kerr solution, then the irreducible mass of the black hole, calculated with respect to this reference space-time, will be different from expression (5.4), the difference residing in rotational effects. Therefore in similarity to the ordinary concept of energy, the gravitational energy depends on the rotational state of the reference frame. Rotational and boost effects are eliminted by requiring conditions (4.4a,b) on the tetrad fields.
The agreement between Eqs. (5.4) and (5.5) is the most important result so far obtained from definitions (2.9) and (3.3). To our knowledge, the latter are the only energy definitions that yield a value satisfactorily close to $2M_{irr}$, and that arise from the structure of the Hamiltonian formulation of the theory.
Before closing this section we note that the time gauge condition (4.4b) breaks the SO(3,1) symmetry group into the global SO(3). Therefore in this case $P^a$ given by Eq. (3.3) is no longer a true SO(3,1) vector.
2.0cm [**VI. Angular momentum of the gravitational field**]{}
In the context of Einstein’s general relativity rotational phenomena is certainly not a completely understood issue. The prominent manifestation of a purely relativistic rotational effect is the dragging of inertial frames. If the angular momentum of the gravitational field of isolated systems has a meaningful notion, then it is reasonable to expect the latter to be somehow related to the rotational motion of the physical sources.
The angular momentum of the gravitational field has been addressed in the literature by means of different approaches. The oldest approach is based on pseudotensors[@LL; @BT], out of which angular momentum superpotentials are constructed. An alternative approach assumes the existence of certain Killing vector fields that allow the construction of conserved integral quantities[@Komar]. Finally, the gravitational angular momentum can also be considered in the context of Poincaré gauge theories of gravity[@PGT], either in the Lagrangian or in the Hamiltonian formulation. In the latter case it is required that the generators of spatial rotations at infinity have well defined functional derivatives. From this requirement a certain surface integral arises, whose value is interpreted as the gravitational angular momentum.
The main motivation for considering the angular momentum of the gravitational field in the present investigation resides in the fact that the constraints $\Gamma^{ik}$[@Maluf3],
$$\Gamma^{ik}=-\Gamma^{ki}=
2\Pi^{[ik]}-2\,k\,e \biggl( -g^{im}g^{kj}T^0\,_{mj}+
(g^{im}g^{0k}-g^{km}g^{0i})T^j\,_{mj} \biggr)
\;,\eqno(6.1)$$
satisfy the angular momentum algebra,
$$\lbrace \Gamma^{ij}(x),\Gamma^{kl}(y)\rbrace=\biggl(
g^{il}\Gamma^{jk}+g^{jk}\Gamma^{il}-
g^{ik}\Gamma^{jl}-g^{jl}\Gamma^{ik}\biggr)\delta(x-y)
\;,\eqno(6.2)$$
Following the prescription for defining the gravitational energy out of the Hamiltonian constraint of the TEGR, we interpret the integral form of the constraint equation $\Gamma^{ik}=0$ as an angular momentum equation, and therefore we define the angular momentum of the gravitational field $M^{ik}$ according to
$$M^{ik}=2\int_V d^3x\, \Pi^{[ik]}
= 2k\int_V d^3x\, e \biggl[ -g^{im}g^{kj}T^0\,_{mj}+
(g^{im}g^{0k}-g^{km}g^{0i})T^j\,_{mj} \biggr]
\;,\eqno(6.3)$$
for an arbitrary volume $V$ of the three-dimensional space.
In Einstein-Cartan type theories there also appear constraints that satisfy the Poisson bracket given by Eq. (6.2). However, such constraints arise in the form $\Pi^{[ik]}=0$, and so a definition similar to Eq. (6.3), i.e., interpreting the constraint equation as an equation for the angular momentum of the field, is not possible.
Since definition (6.3) is a three-dimensional integral we will consider a non-singular space-time metric that exhibits rotational motion. One exact solution that is everywhere regular in the exterior and interior regions of the rotating source is the metric associated to a thin, slowly rotating mass shell as described by Cohen[@Cohen]. In the limit of small angular momentum this metric corresponds to the asymptotic form of Kerr’s metric tensor. The main motivation for considering this metric is the construction of a realistic source for the exterior region of the Kerr space-time, and therefore to match the latter region to a singularity-free space-time. For a shell of radius $r_0$ and total mass $m=2\alpha$ as seen by an observer at infinity, the metric reads
$$ds^2=-V^2dt^2+\psi^4\lbrack dr^2+r^2d\theta^2+
r^2\sin^2\theta(d\phi-\Omega dt)^2\rbrack\;,\eqno(6.4)$$
where
$$V={{ r_0-\alpha}\over{r_0 + \alpha}}\;,$$
$$\psi = \psi_0 =1+ {\alpha \over r_0}\;\;\;\;,\;\;\;\;
\Omega=\Omega_0=const.\;,$$
for $r < r_0$, and
$$V={{r-\alpha}\over{r+\alpha}}\;,$$
$$\psi=1+{\alpha \over r}\;\;\;\;,\;\;\;\;
\Omega=\biggl({{r_0 \psi_0^2}\over{r \psi^2}}\biggr)^3
\Omega_0\;,$$
for $r> r_0$.
The set of tetrad fields that satisfy conditions (4.4a,b) is given by
$$e_{a\mu}=\pmatrix{ -V&0&0&0\cr
\Omega r\psi^2 \sin\theta\,\sin\phi &
\psi^2 \,\sin\theta\,\cos\phi &
r\psi^2\,\cos\theta\,\cos\phi &
-r\psi^2\,\sin\theta\,\sin\phi\cr
-\Omega r\psi^2 \sin\theta\,\cos\phi &
\psi^2 \,\sin\theta\,\sin\phi &
r\psi^2\,\cos\theta\,\sin\phi &
r\psi^2\,\sin\theta\,\cos\phi\cr
0 & \psi^2\,\cos\theta & -r\psi^2\,\sin\theta & 0 \cr}
\;.\eqno(6.5)$$
The determinant of $e_{a\mu}$ is $e=V r^2\psi^6\,\sin\theta$.
The nonvanishing components of the torsion tensor that are needed in the following read
$$T^{(1)}\,_{12}= r\partial_r \psi^2\,\cos\theta\,\cos\phi\;,$$
$$T^{(2)}\,_{12}= r\partial_r \psi^2\,\cos\theta\,\sin\phi\;,$$
$$T^{(3)}\,_{12}= -r\partial_r \psi^2\,\sin\theta\;,$$
$$T^{(1)}\,_{13}= -r\partial_r \psi^2\,\sin\theta\,\sin\phi\;,$$
$$T^{(2)}\,_{13}= r\partial_r \psi^2\,\sin\theta\,\cos\phi\;.$$
The anti-symmetric components $\Pi^{\lbrack ik \rbrack}$ can be easily evaluated. We obtain
$$\Pi^{\lbrack 13 \rbrack}(r,\theta,\phi)=
4k\alpha{\Omega \over V}\psi\,\sin\theta\;,$$
for $r>r_0$, $\Pi^{\lbrack 13 \rbrack}(r,\theta,\phi)=0$ for $r<r_0$, and $\Pi^{\lbrack 12 \rbrack}(r,\theta,\phi)=
\Pi^{\lbrack 23 \rbrack}(r,\theta,\phi)=0$ for any value of $r$. In cartesian coordinates the only nonvanishing component of the total angular momentum is given by
$$M^{12}=2\int d^3x\,\Pi^{\lbrack 12\rbrack}(x,y,z)=
4\pi\int^\pi_0 d\theta \int_0^\infty dr\,r \,\sin^2\theta\,
\Pi^{\lbrack 13 \rbrack}
(r,\theta,\phi)$$
$$=\alpha \, \int_0^\pi d\theta\,\sin^3\theta
\int_{r_0}^\infty dr\;r\psi {\Omega \over V}\;.\eqno(6.6)$$
The integral above is finite, well behaved and can be exactly computed. However, we are interested only in the limit $r_0 >>\alpha$, in which case Cohen identifies $J=1/2 (r_0\psi_0^2)^3\Omega_0$ as the Newtonian value for the angular momentum of a rotating mass shell[@Cohen]. In this limit the calculation is straightforward. We find
$$M^{12} \simeq {{8\alpha}\over {3r_0}}J
={{4m}\over{3r_0}}J\;.\eqno(6.7)$$
We identify $M^{12}$ as the angular momentum of the gravitational field. Substituting the expression of $J$ in Eq. (6.7) and considering that in the limit $r_0>>\alpha$ we have $\psi_0=1+\alpha/r_0 \simeq 1$, we arrive at
$$M^{12}=\biggl( {2\over 3} mr_0^2\biggr) \Omega_0\;.
\eqno(6.8)$$
$\Omega_0 =\Omega(r_0)$ is the induced angular velocity of inertial frames inside the shell[@Brill]. The term between the parentheses in the expression above corresponds to the moment of inertia of a rotating shell of radius $r_0$ and mass $m$. For small $\alpha$, $\Omega_0$ and the angular velocity of the shell $\omega_s$ are related via $\Omega_0=\omega_s(4m/3r_0)$[@Brill]. Therefore in the Newtonian limit $r_0>>\alpha$ we have $M^{12}=({\Omega_0/\omega_s})J$, where $J=(2/3)mr_0^2 \omega_s$.
The metric tensor (6.4) is likely to be the only exact solution of Einstein’s equations whose expression for the classical angular momentum of the source is precisely known.
In order to assess the significance of the above result, we will evaluate the angular momentum associated to the metric tensor (6.4) by means of Komar’s integral $Q_K$[@Komar],
$$Q_K={1\over {8\pi}}\oint_S
\sqrt{-g}\,\varepsilon_{\alpha \beta\mu\nu}
\nabla^{\lbrack \alpha} \xi^{\beta\rbrack}dx^\mu \wedge dx^\nu
\;,\eqno(6.9)$$
where $S$ is a spherical surface of radius $R\rightarrow \infty$, $\xi^\mu$ is the Killing vector field $\xi^\mu=\delta^\mu_3$ and $\nabla$ is the covariant derivative constructed out of the Christoffel symbols $\Gamma^\lambda_{\mu\nu}$. The integral $Q_K$ reduces to
$$Q_K={1\over {2\pi}}\oint_S\,\sqrt{-g}\,g^{0\mu}
\Gamma^1_{\mu 3}\,d\theta \,d\phi\;.\eqno(6.10)$$
By substituting Eq. (6.4) and taking the limit $S\rightarrow \infty$ we obtain
$$Q_K={4\over 3}(r_0\psi_0^2)^3\Omega_0
\simeq {4\over 3} r_0^3\Omega_0={{16}\over 9}mr_0^2\omega_s
={8\over 3}J\;.\eqno(6.11)$$
In the equation above we are considering $r_0>>\alpha$. We observe that definitions (6.3) and (6.9) yield distinct results. In order to make clear the distinction it is useful to rewrite both espressions, (6.7) and (6.11), in laboratory (CGS) units. Thus we make $m=(G/c^2)M$ and $\omega_s=\Omega_s/c$, where $M$ is given in grams, and $\Omega_s$ in radians per second. In addition, we make the replacement $1/(16\pi) \rightarrow c^3/(16\pi G)$ in the multiplicative factor of both expressions, in order to yield the correct dimension to the integrals. We arrive at
$$M^{12}=\biggl({G\over c^2}\biggr){{4M}\over{3r_0}} \biggl(
{2\over 3} M r_0^2\Omega_s\biggr)\;,\eqno(6.7')$$
$$Q_K={8\over 3}\biggl({2\over 3}Mr_0^2\Omega_s\biggr)
\;.\eqno(6.11')$$
We note that $G/c^2=0,74\times 10^{-28} g/cm$. Both expressions have angular momentum units.
One expects the gravitational angular momentum to be of the order of magnitude of the intensity of the gravitational field. We observe that Komar’s integral yields a value proportional to the angular momentum of the [*source*]{}, whereas $M^{12}$ is much smaller than $Q_K$. Indeed, the gravitational field of a mass shell of typical laboratory values is negligible, and consequently the gravitational angular momentum should be negligible as well. Therefore $M^{12}$ yields a realistic value for the angular momentum of the gravitational field, in contrast to $Q_K$.
The advantage of definition (6.4) is that it does not depend on the existence of Killing vector fields. The conclusion is that the angular momentum of the space-time of a rotating mass shell, according to the definition (6.3), is proportional to the induced angular velocity $\Omega_0$ of inertial frames.
The investigations carried out so far in the context of the Kerr solution are not yet conclusive. Although the calculations in the Boyer-Lindquist coordinates are extremely intricate, the indications are that $M^{12}$ diverges. Considering the metric tensor given by Eq. (4.5) and the related definitions, we calculate the anti-symmetric components of the momenta $\Pi^{\lbrack ik \rbrack}$ in the time gauge, i.e., out of tetrads (4.9). They are given by
$$\Pi^{\lbrack 12 \rbrack}(r,\theta,\phi)=0\;,\eqno(6.12a)$$
$$\Pi^{\lbrack 13 \rbrack}(r,\theta,\phi)=
{{k\chi\, \sin\theta}\over
\sqrt{\psi^2\Sigma^2+\chi^2\, \sin^2\theta}}
\biggl(1+{\rho^2\over \Sigma}-{\sqrt{\Delta}\over\Sigma}
\partial_r \Sigma\biggr)\;,\eqno(6.12b)$$
$$\Pi^{\lbrack 23 \rbrack}(r,\theta,\phi)={{k\chi}\over\sqrt{
\Delta (\psi^2\Sigma^2+\chi^2\, \sin^2\theta)}}
\biggl( \cos\theta\biggl({\rho^2\over \Sigma}-1\biggr)-
{{\sin\theta}\over \Sigma}\partial_\theta \Sigma\biggr)
\;.\eqno(6.12c)$$
Transforming to cartesian coordinates we obtain
$$M^{12}=\int d^3x \Pi^{\lbrack 12 \rbrack}(x,y,z)$$
$$=2\pi\int_0^\infty dr \int_0^\pi d\theta \biggl(
r\,\sin\theta\, \Pi^{\lbrack 13 \rbrack}(r,\theta,\phi)+
r^2\,\sin\theta\,\cos\theta\,
\Pi^{\lbrack 23 \rbrack}(r,\theta,\phi)
\biggr)\;,\eqno(6.13)$$
and $M^{13}=M^{23}=0$. The evaluation of Eq. (6.13) out of expressions (6.12) yields a divergent result. The latter is positively and negatively divergent in the external ($r_+$) and internal ($r_-$) horizons of the black hole, respectively. Moreover, in the region $r_-<r<r_+$, $M^{12}$ acquires an imaginary component. A possible interpretation is that the Boyer-Lindquist coordinates are not suitable to the present analysis. In any way, integration over the whole spacelike section of the Kerr space-time is a nontrivial operation.
It must be noted that the Kerr black hole has no classical analog. The interpretation of the angular momentum parameter $a$ of the Kerr solution is not straightforward, since in the Newtonian theory of gravitation the gravitational field of a body does not depend on its rotational motion. The parameter $a$ is identified with the angular momentum per unit mass of the source only after reducing the exterior region of the Kerr metric to a Lense-Thirring type metric by successive approximations[@Adler]. [**VII. Discussion**]{}
In this paper we have investigated the definitions of energy and angular momentum of the gravitational field that arise in the Hamiltonian formulation of the TEGR. We have compared the most important achievement, i.e., the calculation of the irreducible mass of the Kerr black hole, with the result previously obtained in the framework of the same theory, but with the Hamiltonian formulation established under the [*a priori*]{} imposition of the time gauge condition. The two results agreed. In fact, both energy expressions coincide by requiring the time gauge condition, if the latter is imposed [*a posteriori*]{} in the $a=(0)$ component of expression (3.3).
The relevance of Eq. (5.4) is further enhanced if we observe that the Brown-York method[@Brown] for the evaluation of quasi-local gravitational energy fails in obtaining a value close to the irreducible mass of the Kerr black hole. Although the calculations in the framework of this method are quite intricate, recently it has been carried out[@DM]. It has been shown that the gravitational energy within $r_+$ is close to $2M_{irr}$ only for $a/m <0.5$ (fig. 1 of Ref. [@DM]).
Definitions for the gravitational energy in the context of the teleparallel equivalent of general relativity have already been proposed in the literature. In Ref. [@Nester1] an expression for the gravitational energy arises from the surface term of the total Hamiltonian (Eqs. (3.18) and (3.19) of Ref. [@Nester1]). A similar quantity is suggested in Ref. [@Blagojevic], according to Eq. (3.8) of the latter reference. Both expressions are equivalent to the integral form of the total divergence of the Hamiltonian density developed in Ref. [@Maluf3] (Eq. (27) of the latter reference),
$$E=\int_{V\rightarrow \infty}d^3x\, \partial_k
(e_{a0}\Pi^{ak})=\oint_{S\rightarrow \infty}
dS_k\,(e_{a0}\Pi^{ak}).$$
The three expressions yield the same value for the [*total*]{} energy of the gravitational field. However, since these three expressions contain the lapse function in the integrand, none of them is suitable to the calculation of the irreducible mass of the Kerr black hole, in which case we consider a finite surface of integration, because the lapse funtion vanishes on the external event horizon of the black hole (recalling the 3+1 decomposition in section III, $e^a\,_0=\eta^a N+\,^3e^a\,_iN^i $. In the time gauge we have $\eta^a=\delta^a_{(0)}$ and $e^{(0)}\,_i=0$). The energy expressions of Refs. [@Nester1; @Blagojevic] are not to be applied to a finite surface of integration; rather, they yield the total energy of the space-time.
The energy expression (3.3) is defined with respect to a given reference space. Tetrad fields that satisfy conditions (4.4a,b) establish a unique reference space-time that is neither boost related nor rotating with respect to the physical space-time. These conditions uniquely associate a set of tetrad fields to an arbitrary metric tensor. Therefore in the present framework it does not suffice to assert that the reference space-time is Minkowski’s space-time. It is also necessary to enforce the soldering of the reference space-time to the physical space-time by means of Eqs. (4.4a,b). We conjecture that for a given space volume the latter conditions yield the [*minimum*]{} value for the energy expression (3.3).
[*Acknowledgements*]{}
T. M. L. T., J. F. R. N. and K. H. C. B. are grateful to the Brazilian agency CAPES for financial support.
**APPENDIX**
We present here the components of the torsion tensor obtained out of the tetrad configuration Eq. (4.7), that satisfies Møller’s weak field approximation:
$$\begin{aligned}
{T^{(0)}}_{01} & = & \sqrt{1+M^2y^2}\left(\fff{\psi}
{\rho^2}\partial_r\rho - \fff{1}{\rho} \partial_r\psi\right)
- \fff{\psi My}{\rho\sqrt{1+M^2y^2}}\left(y \partial_r M
+ M\partial_r y\right) ,
\nonumber
\\
{T^{(0)}}_{13} & =
& \fff{yN\chi}{\rho \psi}
\sen^2\theta \left(\fff{1}{y}\partial_r y
+ \fff{1}{\chi} \partial_r \chi + \fff{1}{N} \partial_r N -
\fff{1}{\rho} \partial_r \rho -
\fff{1}{\psi} \partial_r \psi\right) ,
\nonumber
\\
{T^{(1)}}_{01} & = & -
\fff{y\chi}{\rho\Sigma}\sen\theta\sen\phi
\left(\fff{1}{\chi}\partial_r\chi + \fff{1}{y} \partial_r y
- \fff{1}{\rho} \partial_r\rho -
\fff{1}{\Sigma} \partial_r\Sigma\right) ,
\nonumber
\\
{T^{(1)}}_{03} & = & -\fff{y\chi}{\rho \Sigma}
\sen \theta \cos \phi ,
\nonumber
\\
{T^{(1)}}_{12} & = & \cos\theta\cos\phi \left(\partial_r\rho
-\fff{\rho}{\sqrt{\Delta}}\right) -
\fff{1}{\sqrt{\Delta}}\sen\theta\cos\phi\,\partial_\theta\rho,
\nonumber
\\
{T^{(1)}}_{13} & = &
\sen\theta\sen\phi\left[\fff{\rho}{\sqrt{\Delta}}-
\fff{\Sigma}{\rho}
\sqrt{1+N^2M^2y^2}\left(\fff{1}{\Sigma}\partial_r \Sigma
- \fff{1}{\rho}\partial_r\rho\right) - \right.
\nonumber
\nonumber
\\
&&\left. -
\fff{\Sigma N^2M^2y^2}{\rho\sqrt{1 + N^2M^2y^2}}
\left(\fff{1}{N}\partial_r N + \fff{1}{M} \partial_r M +
\fff{1}{y}\partial_r y\right)\right] ,
\nonumber
\\
{T^{(2)}}_{01} & = &
\fff{y\chi}{\rho\Sigma}\sen\theta\cos\phi \left(\fff{1}{\chi}
\partial_r \chi + \fff{1}{y} \partial_r y -
\fff{1}{\rho} \partial_r\rho -
\fff{1}{\Sigma} \partial_r \Sigma\right) ,
\nonumber
\\
{T^{(2)}}_{03} & =
& -\fff{y\chi}{\rho\Sigma} \sen\theta \sen \phi ,
\nonumber
\\
{T^{(2)}}_{12} & = &
\cos \theta \sen\phi \left(\partial_r\rho
- \fff{\rho}{\sqrt{\Delta}}\right) -
\fff{1}{\sqrt{\Delta}} \sen \theta
\sen \phi\, \partial_\theta \rho ,
\nonumber
\\
{T^{(2)}}_{13} & = &
-\sen\theta \cos\phi \left[\fff{\rho}{\sqrt{\Delta}}
- \fff{\Sigma}{\rho}
\sqrt{1+N^2M^2y^2} \left(\fff{1}{\Sigma}\partial_r \Sigma
- \fff{1}{\rho}\partial_r\rho\right) -\right.
\nonumber
\nonumber
\\
&&\left.- \fff{\Sigma}{\rho\sqrt{1 + N^2M^2 y^2}}
\left(\fff{1}{N}\partial_r N +
\fff{1}{M} \partial_r M +
\fff{1}{y} \partial_r y\right)\right] ,
\nonumber
\\
{T^{(3)}}_{12} & = &
-\sen\theta (\partial_r \rho - \fff{\rho}{\sqrt{\Delta}})
- \fff{1}{\sqrt{\Delta}} \cos \theta\, \partial_\theta \rho.
\nonumber\end{aligned}$$
1.0cm
[99]{}
C. Møller, [*Tetrad Fields and Conservation Laws in General Relativity*]{}, Proceedings of the International School of Physics “Enrico Fermi", edited by C. Møller (Academic Press, London, 1962); [*Conservation Laws in the Tetrad Theory of Gravitation*]{}, Proceedings of the Conference on Theory of Gravitation, Warszawa and Jablonna 1962 (Gauthier-Villars, Paris, and PWN-Polish Scientific Publishers, Warszawa, 1964) (NORDITA Publications No. 136).
S. V. Babak and L. P. Grishchuk, Phys. Rev. D[**61**]{}, 024038 (1999).
J. D. Brown and J. W. York, Jr., Phys. Rev. D[**47**]{}, 1407 (1993); S. A. Hayward, Phys. Rev. D[**49**]{}, 831 (1994); Phys. Rev. D[**53**]{}, 1938 (1996); J. D. Brown, S. R. Lau and J. W. York, Jr., Phys. Rev. D[**59**]{}, 064028 (1999); C.-C. Chang, J. M. Nester and C. M. Chen, Phys. Rev. Lett. [**83**]{}, 1897 (1999).
R. Weitzenböck, [*Invarianten Theorie*]{} (Nordhoff, Groningen, 1923).
J. A. Schouten, [*Ricci Calculus*]{}, 2nd ed. (Springer-Verlag, London, 1954), p. 142.
A. Einstein, [*Riemannsche Geometrie unter Aufrechterhaltung des Begriffes des Fernparallelismus*]{} ([*Riemannian Geometry with Maintaining the Notion of Distant Parallelism*]{}), Sitzungsberichte der Preussischen Akademie der Wissenshcaften, phys.-math. Klasse (June 7th), 217-221 (1928); [*Auf die Riemann-Metrik und den Fernparallelismus gegründete einheitliche Feldtheorie*]{} ([*Unified Field Theory based on Riemann Metrics and Distant Parallelism*]{}), Mathematische Annalen [**102**]{}, 685-697 (1930) (English translations available under www.lrz.de/$^{\sim}$aunzicker/ae1930.html).
K. Hayashi, Phys. Lett. [**69**]{}B, 441 (1977); K. Hayashi and T. Shirafuji. Phys. Rev. D[**19**]{}, 3524 (1979); Phys. Rev. D[**24**]{}, 3312 (1981).
F. W. Hehl, in [*Proceedings of the 6th School of Cosmology and Gravitation on Spin, Torsion, Rotation and Supergravity*]{}, Erice, 1979, edited by P. G. Bergmann and V. de Sabbata (Plenum, New York, 1980); F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’eman, Phys. Rep. [**258**]{}, 1 (1995).
W. Kopczyński, J. Phys. A [**15**]{}, 493 (1982); Ann. Phys. (N.J.) [**203**]{}, 308 (1990).
F. Müller-Hoissen and J. Nitsch, Phys. Rev. D[**28**]{}, 718 (1983); Gen. Rel. Grav. [**17**]{}, 747 (1985).
J. M. Nester, Int. J. Mod. Phys. A [**4**]{}, 1755 (1989).
J. M. Nester, J. Math. Phys. [**33**]{}, 910 (1992).
J. W. Maluf, J. Math. Phys. [**35**]{}. 335 (1994).
V. C. de Andrade and J. G. Pereira, Phys. Rev. D[**56**]{}, 4689 (1997).
J. Schwinger, Phys. Rev. [**130**]{}, 1253 (1963).
J. W. Maluf, J. Math. Phys. [**36**]{}, 4242 (1995).
V. C. de Andrade, L. C. T. Guillen and J. G. Pereira, Phys. Rev. Lett. [**84**]{}, 4533 (2000).
J. W. Maluf and J. F. da Rocha-Neto, Phys. Rev. D[**64**]{}, 084014 (2001) (gr-qc/0002059).
R. Arnowitt, S. Deser and C. W. Misner, in [*Gravitation: an Introduction to Current Research*]{}, Edited by L. Witten (Wiley, New York, 1962).
R. P. Kerr, Phys. Rev. Lett. [**11**]{}, 237 (1963).
D. Christodoulou, Phys. Rev. Lett. [**25**]{}, 1596 (1970).
G. Bergqvist, Class. Quantum Grav. [**9**]{}, 1753 (1992).
J. W. Maluf, E. F. Martins and A. Kneip, J. Math. Phys. [**37**]{}, 6302 (1996).
J. W. Maluf and J. F. da Rocha-Neto, J. Math. Phys. [**40**]{}, 1490 (1999).
J. A. Schouten, [*Tensor Analysis for Physicists*]{}, 2nd ed. (Dover, New York, 1989), p. 82.
R. H. Boyer and R. W. Lindquist, J. Math. Phys. [**8**]{}, 265 (1967).
L. D. Landau and E. M. Lifshitz, [*The Classical Theory of Fields*]{} (Pergamon Press, Oxford, 1980). We note that the first Russian edition dates from 1939.
P. G. Bergmann and R. Thomson, Phys. Rev. [**89**]{}, 401 (1953).
A. Komar, Phys. Rev. [**113**]{}, 934 (1959); J. Winicour, “Angular Momentum in General Relativity” in [*General Relativity and Gravitation*]{}, edited by A. Held (Plenum, New York, 1980); A. Ashtekar, “Angular Momentum of Isolated Systems in General Relativity”, in [*Cosmology and Gravitation*]{}, edited by P. G. Bergmann and V. de Sabbata (Plenum, New York, 1980).
K. Hayashi and T. Shirafuji, Prog. Theor. Phys. [**73**]{}, 54 (1985); M Blagojević and M. Vasilić, Class. Quantum Grav. [**5**]{}, 1241 (1988); T. Kawai, Phys. Rev. D[**62**]{} (2000) 104014; T. Kawai, K. Shibata and I. Tanaka, Prog. Theor. Phys. [**104**]{}, 505 (2000) (gr-qc/0007025).
J. M. Cohen, J. Math. Phys. [**8**]{}, 1477 (1967).
D. R. Brill and J. M. Cohen, Phys. Rev. [**143**]{}, 1011 (1966).
R. Adler, M. Bazin and M. Schiffer, [*Introduction to General Relativity*]{} (McGraw-Hill, Tokyo, 1975), p. 257.
M. H. Dehghani and R. B. Mann, Phys. Rev. D[**64**]{}, 044003 (2001).
M. Blagojević and M. Vasilić, Phys. Rev. D[**64**]{}, 044010 (2001).
**FIGURE CAPTIONS**
[**Figure 1**]{} - Energy within the external event horizon of the Kerr black hole as a function of the angular momentum. The figure displays $\varepsilon = E/m$ against $\lambda$ for expressions (5.3) and (5.4). The lower curve represents $2M_{irr}$ given by Eq. (5.5). The one right above it, almost coinciding with the lower curve, corresponds to Eq. (5.4). The upper curve corresponds to Eq. (5.3).
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abstract: 'We present 40 quasar absorption line systems at intermediate redshifts ($z \sim 1$), with focus on one of the most kinematically complex known, as examples of how the unique capabilities of space–based and ground–based facilities can be combined to glean much broader insights into astrophysical systems.'
author:
- 'Chris Churchill$^{1}$, Rick Mellon$^{1}$, Jane Charlton$^{1}$, & Buell Januzzi$^{2}$'
title: Multiphase Gas in Intermediate Redshift Galaxies
---
Hubble and the More Complete Picture
====================================
Within the field of quasar absorption lines, one long–standing question is how the halos and ISM of earlier epoch galaxies compare or relate, in an evolutionary sense, to those of the present epoch. The look–back time to $z=1$ covers well more than half the age of the universe. Furthermore, spectral and morphological properties of absorbing galaxies are accessible with present day ground–based and spaced–based observatories (Steidel, Dickinson, & Persson 1994; Steidel 1998). Thus, absorption line studies at intermediate redshifts provide an opportunity to examine the gaseous evolution of galaxies.
The ISM and halos of local galaxies are comprised of many ionization phases, including diffused ionized gas, extended coronae, and denser low ionization regions often located in front of shock fronts (e.g.Dahlem 1998). In absorption, simultaneous study of both the low and high ionization phases in our Galaxy have been required to constrain the ionization mechanisms, chemical abundance variations, and the dust properties (e.g. Savage & Sembach 1996).
A significant obstacle in the face of rapid progress with studies employing absorption lines, however, is that the strongest transitions of the cosmologically most abundant elements lie in the far to near ultraviolet (UV) portion of the electromagnetic spectrum. Fortunately, at $z\sim1$, the near UV transitions, which are most often associated with neutral and low ionization ions[^1], are redshifted into the visible. Thus, they can be observed from the ground with large aperture telescopes. However, the far UV transitions, associated with moderate and high ionization ions[^2], are redshifted to the near UV; a study of the high ionization component requires a spaced–based telescope, i.e. [*HST*]{}. The [*HST*]{} archive is rich with $R=1300$ FOS spectra of quasars, the majority due to the QSO Absorption Line Key Project (Bahcall et al. 1993).
The C0.1em [**IV**]{}–Mg0.1em [**II**]{} Kinematics Connection: Multiphase Gas
==============================================================================
We used HIRES/Keck spectra ($R\sim 6$ ) and archival FOS/[*HST*]{} spectra ($R\sim 230$ ) to place constraints on the ionization and multiphase distribution of absorbing gas at $z=0.4$ to $z=1$. In Figure 1, we present $\lambda 2796$ and the [[[C]{}0.1em[iv]{} $\lambda\lambda 1548, 1550$]{}]{} doublet for each of 40 systems (note that the velocity scale for is 500 and for is 3000 ). Ticks above the HIRES spectra give the velocities of the Voigt profile sub–components and ticks above the FOS data give the expected location of these components for the doublet. The labels “D”, “L”, and “Bl” denote detection, limit, and blend, respectively. The systems are presented in order of increasing kinematic spread from the upper left to lower right.
Based upon a highly significant correlation between the equivalent widths and the kinematics, it is inferred that most intermediate redshift galaxies have multiphase gaseous structures (Churchill et al. 1999, 2000). The low ionization gas is in multiple, narrow components, $\left< b \right> \simeq 5$ , and the high ionization gas is kinematically spread out with $\left< b
\right> \simeq 70$ (using the doublet ratio method). This is an effective velocity dispersion, for the FOS spectra are of too low resolution to resolve velocity splittings below $\sim 500$ .
Case Study; The Complex Triple System at z=0.93
===============================================
The three systems at $z=0.9254$, $0.9276$, and $0.9343$ along the line of sight to PG $1206+459$ exhibit complex kinematics and exceptionally strong , , and absorption. We investigated the ionization and spatial distribution of these systems using detailed photoionization models (Cloudy; Ferland 1996).
In the top panels of Figure 2, the HIRES/Keck spectra of the $\lambda 2600$ transition and of the [[[Mg]{}0.1em[ii]{} $\lambda\lambda 2796, 2803$]{}]{} doublet are shown with a Voigt profile model spectrum superimposed; the ticks give the component centers. The systemic redshifts of the three systems, A, B, and C, are labeled. The lower two panels show the normalized FOS/[*HST*]{} spectrum (histogram) with tuned model predictions (not fits) superimposed (see Churchill & Charton 1999). The dotted–line is a single–phase model, assuming all absorption arises due to ionization balance in the clouds; a single phase of gas fails to account for the high ionization absorption strengths. The solid spectrum is a two–phase model, which allows the higher ionization gas to reside in a separate phase.
Based upon the photoionization modeling, a highly ionized phase, not seen in , is required to account for the observed , , and absorption. An “effective” Doppler width of $50
\leq b \leq 100$ is consistent with the complex, blended data. The physical size of the high ionization component is less than 30 kpc, with the best values between 10 and 20 kpc.
Based upon the sizes and effective Doppler widths, we infer that the highly ionized material is analogous to the Galactic coronae (Savage et al. 1997), material stirred up by energetic mechanical processes, such as galactic fountains. In this scenario, the gas is concentrated around the individual galaxies which presumably provide a source of support, heating, and chemical enrichment.
It seems promising that the answer to the posed question (§ 1) may be forthcoming when [*HST*]{} resolves the FOS profiles with STIS and COS.
Thanks are due to S. Kirhakos, C. Steidel, and D. Schneider for their contributions to the work presented here. I am especially grateful to all who work to make [*HST*]{} a unique platform for astronomy.
Bahcall, J. N. et al. 1993, ApJS, 87, 1
Churchill, C. W., & Charlton, J. C. 1999, AJ, 118, 59
Churchill, C. W., et al. 1999, ApJ, 519, L43
Churchill, C. W., et al. 2000, ApJ, 543, in press
Dahlem, M. 1998, PASP, 109, 1298
Ferland, G. 1996, [*Hazy*]{}, University of Kentucky Internal Report
Savage B. D. & Sembach K. M. 1996, ARA&A, 34, 279
Steidel, C. C., Dickinson, M. & Persson, E. 1994, ApJ, 437, L75
Steidel, C. C. 1998, in Galactic Halos: A UC Santa Cruz Workshop, ASP Conf. Series, V136, ed. D. Zaritsky (San Francisco : PASP), 167
[^1]: Meaning ions with ionization potentials in the range of a few to $\sim
30$ eV.
[^2]: Meaning those with ionization potentials ranging between $\sim 30$ and $\sim 50$ eV and between $\sim 50$ and $140$ eV, respectively.
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abstract: 'We use the leading singularity technique to determine the planar six-particle two-loop MHV amplitude in ${\cal N}=4$ super Yang-Mills in terms of a simple basis of integrals. Our result for the parity even part of the amplitude agrees with the one recently presented in [@Bern:2008ap]. The parity-odd part of the amplitude is a new result. The leading singularity technique reduces the determination of the amplitude to finding the solution to a system of linear equations. The system of equations is easily found by computing residues. We present the complete system of equations which determines the whole amplitude, and solve the two-by-two blocks analytically. Larger blocks are solved numerically in order to test the ABDK/BDS iterative structure.'
author:
- Freddy Cachazo
- Marcus Spradlin
- Anastasia Volovich
title: 'Leading Singularities of the Two-Loop Six-Particle MHV Amplitude'
---
Brown-HET-1551
1 cm
Introduction
============
Scattering amplitudes in massless gauge theories are remarkable objects with many properties hidden in the complexity of their Feynman diagram expansion. It has been known for decades that much information about an amplitude can be gleaned just from knowing the structure of its singularities (see [@SMatrix]). In Yang-Mills theories, extensive use of branch cut singularities was shown to tame much complexity in the computation of loop amplitudes via techniques that came to be known as the unitarity based method [@Bern:1994zx; @Bern:1994cg; @Bern:1995db; @Bern:1996je; @BDDKSelfDual; @Bern:1997sc; @Bern:2004cz].
One of the most surprising features of Yang-Mills theory and gravity is that their tree level amplitudes can be completely determined by exploiting only their behavior near a subset of their singularities [@BCFTree; @BCFW; @ArkaniHamed:2008yf]. Another surprise, this time in ${\cal N}=4$ SYM, is that the problem of computing one-loop amplitudes can be reduced to that of computing tree amplitudes [@BCFLoop]. The key to such striking simplification is that although the loop amplitude possesses poles and many branch cuts with a complicated structure of intersections, it is completely determined by the structure of the highest codimension singularities. These are known as the leading singularities [@SMatrix] and are computed by cutting all propagators in the diagram.
Applying the same technique at higher loops was first attempted in [@Buchbinder:2005wp] and refined in [@Bern:2007ct; @Cachazo:2008dx]. In [@Buchbinder:2005wp] and in [@Bern:2007ct] both the leading singularity as well as subleading ([*i.e.*]{}, lower codimension) singularities were used to constrain the form of higher loop amplitudes. In [@Cachazo:2008vp], the leading singularity was shown to be much more powerful than expected. It turns out that in massless theories, whenever all propagators are cut, one is actually studying many isolated singularities at the same time. The proposal of [@Cachazo:2008vp], building on observations made in [@Cachazo:2008dx], is to use each isolated singularity independently.
The new leading singularity technique outlined in [@Cachazo:2008vp] has three notable features. Firstly, for any amplitude under consideration it builds a natural set of integrals, which we call the geometric integrals, that can be used to construct a basis. Secondly, the coefficients of the integrals can be determined by solving [*linear*]{} equations. Finally, these linear equations are easily obtained by computing residues using Cauchy’s theorem. The utility of this new technique was demonstrated in [@Cachazo:2008vp], where it was shown that it easily reproduces the result for the two-loop five-particle amplitude in ${\cal N}=4$ SYM previously computed in [@TwoLoopFiveB] using the unitarity based method.
In this paper we apply the leading singularity technique to a much more challenging case, the planar two-loop six-particle MHV amplitude ${A}^{(2)}_{6,{\rm MHV}}$. The parity-even part of the normalized amplitude ${A}^{(2)}_{6,{\rm MHV}}/{A}^{\rm tree}_{6,{\rm MHV}}$ was computed recently in [@Bern:2008ap] using the unitarity based method. This was already an impressive display of computational power. In the five particle case studied in [@TwoLoopFiveA; @TwoLoopFiveB] the parity-odd part is noticeably of a higher degree of complexity than the parity-even part, and there is no reason to suspect that this would not be the case also for $n=6$. Here we find that the full coefficients (both the even and odd parts) emerge by solving the relatively simple linear equations presented explicitly below.
For six particles we find a new phenomenon which was not encountered in the $n=4,5$ cases studied in [@Cachazo:2008vp]: while there are 177 geometric integrals, the equations only fix 159 linear combinations of their coefficients, leaving 18 linear combinations undetermined. The reason is that the set of geometric integrals is overcomplete, so we cannot have expected to find a unique solution. One can show that by using well-known techniques [@Melrose:1965kb; @vanNeerven:1983vr] it is possible to build linear relations between seemingly independent integrals. In section IV we analyze the relevant reduction identities and identify 18 relations amongst the integrals in the geometric basis. Thus there is no ambiguity beyond that required by reduction identities, so we conclude that ${ A}_{6,{\rm MHV}}^{(2)}$ is in fact completely determined by its leading singularities.
It is important to stress that the leading singularity method turns loop integrals into contour integrals which are finite in four dimensions and knows nothing about how one might choose to regulate the infrared divergences that typically appear when carrying out the loop integrals. In dimensional regularization, amplitudes occasionally contain additional terms in the integrand which vanish in $D=4$. These so-called “$\mu$-terms” (see for example [@Bern:2002tk] for a thorough treatment) cannot be detected by the leading singularity in $D=4$.
One motivation for computing the MHV six-particle amplitude, beyond its serving as a testing ground for the leading singularity method, is to study the proposed iterative relation between MHV loop amplitudes known as the ABDK/BDS ansatz [@ABDK; @BDS]. The ansatz has been shown to hold for four particles up to three loops [@ABDK; @BDS] and for five particles up to two loops [@TwoLoopFiveA; @TwoLoopFiveB]. However, it was shown to break down for the parity-even part of the two-loop six-particle MHV amplitude in [@Bern:2008ap]. In this paper we find numerical evidence that the parity-odd part of the amplitude does satisfy the ABDK/BDS ansatz.
Outline of the Calculation
==========================
The object of interest is the planar six-particle two-loop MHV amplitude in ${\cal N} = 4$ super Yang-Mills. The goal is to find a compact expression for this amplitude as a linear combination of relatively simple integrals. The leading singularity method [@Cachazo:2008vp] provides both a natural set of integrals to work with, as well as a system of linear equations which determine the coefficients of those integrals.
In this section we provide a detailed outline of the steps involved in setting up the calculation. The first three subsections are relevant to NMHV as well as MHV amplitudes, since the homogeneous part of the system of linear equations is helicity independent. In subsection II.D we compute the inhomogeneous terms for the MHV helicity configuration. The final linear equations which determine the coefficients of the MHV amplitude are presented explicitly in section III.
Review of the Leading Singularity Method
----------------------------------------
Suppose we are interested in calculating some $L$-loop scattering amplitude ${ A}$. On the one hand, the amplitude may of course be represented as a sum over Feynman diagrams $F_j$, $$\label{eq:one}
{ A}(k) = \sum_j \int \prod_{a=1}^L d^d \ell_a\
F_j(k,\ell)\,,$$ where $k$ are external momenta and $\ell_a$ are the loop momenta. However it is frequently the case, especially in theories as rich as ${\cal N} = 4$ SYM, that directly calculating the sum over Feynman diagrams would be impractical. Rather the calculation proceeds by expressing ${ A}$ as a linear combination of relatively simple integrals in some appropriate basis $\{I_i\}$, $$\label{eq:two}
{ A}(k) = \sum_i c_i(k) \int \prod_{a=1}^L d^d \ell_a\
I_i(k,\ell)\,,$$ and then determining the coefficients $c_i$ by other means, such as the unitarity based method. If the set of integrals $\{I_i\}$ is overcomplete, then the coefficients $c_i(k)$ are not uniquely defined.
The basic idea underlying the leading singularity method is that the sum over Feynman diagrams in (\[eq:one\]) possesses singularities which must be properly reproduced by any representation (\[eq:two\]) of the amplitude in terms of simpler integrals. At the same time, any singularities in the set of integrals which are not present in the sum over Feynman diagrams must be spurious.
The most common kind of singularities in Feynman diagrams are poles, associated to collinear or multi-particle singularities, and branch cuts, associated to unitarity cuts. These branch cuts can themselves possess branch cuts leading to higher codimension singularities. The latter are computed by cutting propagators or equivalently [@Cachazo:2008dx] by promoting the loop integral to be a contour integral. (The observation that the Lorentz invariant phase space integral of a null vector can be recast as a contour integral was first discussed in [@CSW].) The contour is chosen to reproduce the behavior of the delta-functions in the cut calculations. In general, this gives rise to contours which compute the residue on several isolated singularities at the same time.
For example, consider the one-loop massless scalar box. Replacing each propagator by a delta-function leads us to consider the integral $$\int d^4 \ell\,
\delta(\ell^2)
\delta((\ell - k_1)^2)
\delta((\ell - k_1 - k_2)^2)
\delta((\ell + k_4)^2)\,.$$ For generic external momenta $k_i$ these delta-functions localize the $\ell$ integral onto two discrete points in complex $\ell$-space ($\mathbb{C}^4$). With the leading singularity method we do not replace propagators by delta-functions, but rather we consider two separate $T^4$ contours in $\mathbb{C}^4$, each of which computes the residue of the integrand on only one of the two isolated singularities. Then by equating (\[eq:one\]) and (\[eq:two\]) and performing the integral $$\label{eq:method}
\sum_i c_i(k) \int_\Gamma d^4 \ell\, I_i(k, \ell)
=
\int_\Gamma d^4 \ell\,
\sum_j F_j(k, \ell)$$ we obtain one linear equation on the coefficients $c_i$ for each contour $\Gamma$. At $L$ loops each contour is a $T^{4L}$ inside $\mathbb{C}^{4 L}$. Since the number of isolated singularities in a generic $L$-loop diagram can be as high as $2^L$ (simple diagrams can have fewer isolated singularities), the leading singularity method gives rise to an exponentially large (in $L$) number of linear equations for the coefficients $c_i$. We note that the homogeneous part of these linear equations (the left-hand side of (\[eq:method\])) depends only on the set of integrals $\{I_i\}$ and the choice of contours, while the details of which particular helicity configuration is under consideration enters only into the inhomogeneous terms on the right-hand side.
Choosing Useful Contours
------------------------
The formula (\[eq:method\]) gives a linear equation on the coefficients $c_i$ for any contour $\Gamma$ in ${\mathbb{C}}^{4 L}$. Of course if we choose some random contour $\Gamma$ then we will typically get the useless equation $0=0$. In order to get useful equations we should use $T^{4 L}$ contours which calculate residues at the known singularities of the right-hand side. It is clear that in the sum over Feynman diagrams, singularities occur when internal propagators go on-shell.
![The five indepedent 8-propagator topologies. Each diagram represents a sum of those Feynman diagrams in which all of the 8 indicated propagators are present. In each diagram the external momenta are labeled clockwise beginning with $k_1$ at the position of the arrow. Also in each diagram $p$ is the loop momentum in the left loop and $q$ is the momentum in the right loop. []{data-label="EightPropagatorTopologies"}](fig1.eps)
For six particles at two-loops there are two classes of useful $T^8$ contours. The most obvious $T^8$ contours are those which are chosen to calculate the residue at points in $\mathbb{C}^8$ where eight propagators go on-shell simultaneously. These contours are associated with the five different topologies shown in . Actually each topology in is a diagrammatic shorthand for four distinct $T^8$ contours. For example, the singularities of Feynman diagrams with topology $(D)$ are situated at the locus $$\begin{aligned}
S_{(D)} = \{ (p,q) \in \mathbb{C}^4 \times \mathbb{C}^4 &:&
p^2 = 0, ~ (p + k_6)^2 = 0, ~ (p + k_{456})^2 = 0, ~ (p - k_{12})^2 = 0,\cr
&&
q^2 = 0, ~ (q + k_1)^2 = 0, ~ (q + k_{12})^2 = 0, ~ (p + q)^2 = 0 \}\,.\end{aligned}$$ For generic external momenta $k_i$ $S_{(D)}$ consists of four distinct points in $\mathbb{C}^8$ of the form $(p^{(i)}, q^{(j)})$ for $i,j=1,2$. Correspondingly there are four different contours $\Gamma$ associated with topology $(D)$, one which computes the residue of the integrand at each of these four isolated singularities.
![The eight independent 7-propagator topologies. See for details. []{data-label="SevenPropagatorTopologies"}](fig2.eps)
The less obvious $T^8$ contours are those in which only seven propagators are apparent but an eighth singularity appears due to a Jacobian. These contours are associated with the eight different topologies shown in . For example, let us consider topology $(F)$. For fixed loop momentum $p$ the singularities in the $q$ integral occur at the locus $$S_{(F)q} = \{ q \in \mathbb{C}^4 : q^2 = 0, ~ (q + k_1)^2 = 0, ~
(q + k_{12})^2 = 0, ~ (q + p)^2 = 0 \}\,,$$ which consists of two points $\{q^{(1)}, q^{(2)}\}$ in $\mathbb{C}^4$. For each of these two singularities there is a contour $\Gamma_q$ such that integrating $q$ over $\Gamma_q$ computes the residue at that singularity. Integrating over either contour produces the same Jacobian factor $$\label{eq:jacobian}
\int_{\Gamma_q} d^4 q\ \frac{1}{q^2 (q + k_1)^2 (q+ k_{12})^2 (q + p)^2}
= {1 \over 2} {1 \over (k_1 + k_2)^2 (p - k_1)^2}\,.$$ The new singularity $1/(p - k_1)^2$ combines with the three remaining singularities manifest in topology $(F)$ so that the integral over $p$ can be localized by integrating over contours which compute the residue at the points $$S_{(F)p} = \{ p \in \mathbb{C}^4 : p^2 = 0, ~ (p + k_6)^2 = 0, ~
(p - k_{12})^2 = 0, ~ (p - k_1)^2 = 0\}\,.$$
We proceed analagously for each of the eight topologies shown in . In each case we first integrate the right-hand loop momentum $q$ and then use the additional singularity generated by the Jacobian to integrate the left-hand loop momentum $p$, thereby completely localizing the integral onto a set of discrete points.
For the MHV amplitude it turns out that the linear equations generated by the 13 types of contours described in are sufficient to uniquely determine all coefficients, although there certainly are additional contours that could be used to generate additional equations from (\[eq:method\]). We have checked that a class of additional equations are indeed satisfied by the MHV coefficients, thus providing a strong consistency check on the coefficients obtained from solving the equations in section III.
The Geometric Integrals
-----------------------
Although equation (\[eq:method\]) can be used to generate linear equations for coefficients in an arbitrary basis $\{I_i\}$, the leading singularity method suggests a natural set of integrals in which the left-hand side is easy to compute and the individual integrals have a geometric interpretation, in a sense we now explain.
![The geometric integrals for the two-loop six-particle amplitude in ${\cal N}=4$ SYM. In each diagram the external momenta are labeled clockwise beginning with $k_1$ at the position of the arrow and the number in parentheses denotes the number of independent permutations of the diagrams. As discussed in section IV, this is an overcomplete set: the 177 integrals here obey 18 linear relations. []{data-label="GeometricBasis"}](fig3.eps)
The procedure to determine the natural set of integrals in which to represent an amplitude starts by realizing that in ${\cal N}=4$ SYM tadpoles, bubbles, and triangles are unneccesary (see [@Bern:2006ew] for a thorough discussion). This means that we have to start by considering all topologies of sums over Feynman diagram with no triangles or bubbles. At one-loop, only sums of Feynman diagrams with the topology of a box are needed. For six particles at two-loops, we find five topologies with eight propagators, shown in , and eight topologies with seven propagators, shown in .
After each topology is identified, a first approximation for reproducing all of the leading singularities is to use just the scalar integrals with all of the appropriate topologies. In general, it turns out that such integrals are not enough to reproduce the singularities of Feynman diagrams. This will manifest itself in the failure of the linear equations (\[eq:method\]) to have any solutions, indicating that the set of integrals must be enlarged.
The next step is to introduce scalar integrals with additional propagators. At one-loop this step gives rise to pentagons in addition to boxes, which turns out to be sufficient for any $n$. At two loops we add the scalar pentagon-pentagon integrals shown in . At this stage some of the equations are solved ([*i.e.*]{} some of the leading singularities are correctly reproduced), while others are not.
It is then necessary to supplement additional integrals which must have non-zero residue on the missing singularities and zero residue on the ones which already work, in order to avoid spoiling them. The way to ensure that one has zero residue on a given pole is to include a zero in the numerator of the integrand which cancels the corresponding pole. In this form integrals with scalar numerators appear. Note that only numerators which cancel poles appear naturally.
The process of expanding the set of integrals by including additional numerator factors ends when one is able to solve all of the equations (\[eq:method\]). We call the integrals that are naturally constructed in this manner geometric integrals. The set of geometric integrals might be overcomplete if the equations do not determine a unique solution. For the six-particle MHV amplitude at two loops this process leads to the 177 geometric integrals shown in . For NMHV amplitudes it is possible that additional integrals, such as a pentagon-boxes with two numerators, might be required.
The Inhomogeneous Terms for the MHV Amplitude
---------------------------------------------
As mentioned above, it is obvious from (\[eq:method\]) that the homogeneous part of the system of linear equations is helicity independent, so the above discussion of the contours and geometric integrals applies to both the MHV and NMHV configurations. The inhomogeneous part on the right-hand side of (\[eq:method\]) is easily obtained for any contour $\Gamma$ by computing the product of tree amplitudes sitting at the ‘blobs’ in the corresponding topology in or . This product is evaluated at the value $(p^{(i)}, q^{(j)})$ of the loop momenta where the contour $\Gamma$ localizes the integral.
For MHV configurations there is an enormous simplification since it turns out that this product of tree amplitudes always comes out to be $0$ or $1$ times the corresponding tree-level amplitude, as can easily be shown by using the technique introduced in [@Cachazo:2008dx] where sums over the full ${\cal N}=4$ supermultiplet in the internal lines, which complicate the computation [@Buchbinder:2005wp; @Bern:2007ct], are automatically done by using simple identities.
For the contours associated with the 13 topologies shown in we find that the right-hand side of (\[eq:method\]) for an MHV configuration are $$\begin{aligned}
\label{eq:RHS}
(A): &&\qquad
\delta_{\langle p,1\rangle} \delta_{\langle q,4\rangle}
A^{\rm tree}_{6,{\rm MHV}}\,, \cr
(B): &&\qquad \delta_{\langle p,6\rangle} \delta_{\langle q,1\rangle}
A^{\rm tree}_{6,{\rm MHV}}\,, \cr
(C): &&\qquad
0\,,\cr
(D): &&\qquad
\delta_{\langle p,6\rangle} A^{\rm tree}_{6,{\rm MHV}}\,, \cr
(E): &&\qquad
\delta_{\langle p,6\rangle} \delta_{\langle q,2\rangle}
A^{\rm tree}_{6,{\rm MHV}}\,, \cr
(F): &&\qquad \delta_{\langle p,6\rangle} A^{\rm tree}_{6,{\rm MHV}}\,, \cr
(G): &&\qquad 0\,,\cr
(H): &&\qquad 0\,,\cr
(I): &&\qquad \delta_{\langle p,6\rangle} \delta_{\langle p,q\rangle}
A^{\rm tree}_{6,{\rm MHV}}\,, \cr
(J): &&\qquad 0\,,\cr
(K): &&\qquad \delta_{[p,6]} \delta_{[q,1]}A^{\rm tree}_{6,{\rm MHV}}\,, \cr
(L): &&\qquad 0\,,\cr
(M): &&\qquad 0\,.\end{aligned}$$ The Kronecker deltas such as $\delta_{\langle p,1\rangle}$ for topology $(A)$ arise because, in that example, the solution for $p$ is either of the form $\lambda_p \propto
\lambda_1$ or $\widetilde{\lambda}_p
\propto \widetilde{\lambda}_1$, and the product of tree amplitudes vanishes on latter solution.
The MHV Equations and Their Solutions
=====================================
We now assemble all of the ingredients prepared in section II for the MHV amplitude. By evaluating (\[eq:method\]) on all of the contours associated with the topologies shown in (together with all of their cyclic and mirror-image permutations), and using (\[eq:RHS\]) on the right-hand side, we find a system of 396 linear equations for the 177 coefficients of the geometric integrals in .
Generically, 396 linear equations in 177 variables have no solution, but in this case we find that the equations are in fact underdetermined: they only fix 159 linear combinations of the 177 coefficients, leaving 18 free parameters. In other words, we find that there are 18 linear combinations of the geometric integrals in which have vanishing leading singularity on all of the contours described by . One logically possible conclusion from such a result might have been that the leading singularity method is not enough to uniquely determine the two-loop six-particle MHV amplitude, which would have been disappointing.
Fortunately, as mentioned in the introduction, it turns out that integral reduction identities imply that the set of geometric integrals is overcomplete. In other words, there are linear combinations of the 177 geometric integrals which not only have vanishing leading singularity but actually vanish identically. In section IV we analyze these reduction identities and show that there are 18 linear combinations of the geometric integrals which vanish, precisely accounting for the abovementioned ambiguity in solving the leading singularity equations. The conclusion is therefore that the two-loop six-particle MHV amplitude is in fact uniquely determined by knowledge of its leading singularities.
Presentation of the Equations
-----------------------------
In order to demonstrate that the leading singularity method is not just black magic, we present here explicit expressions for the equations which determine all coefficients $A^{(2)}_{6,{\rm MHV}}$, including both the parity-even and odd terms. As just discussed, 18 coefficients out of 177 are actually redundant. In order to somewhat simplify the presentation of the equations we will choose a convenient ‘gauge’ which uniquely fixes all of the ambiguity. This amounts to choosing a basis of geometric integrals.
Several such choices are possible; the choice we make here is to spend the 18 gauge parameters by setting to zero the 6 coefficients $c_{12}$ and the 12 coefficients $c_{24}$. Once this is done all remaining coefficients are uniquely determined. A nice advantage of this choice is that several other coefficients turn out to also vanish identically: the 12 coefficients $c_{22}$, the 12 coefficients $c_{18}$, the 6 coefficients $c_{16}$ and the 3 coefficients $c_{23}$ are all zero.
Ultimately then there are only 126 nonzero coefficients, associated with just 15 out of the 21 integrals shown in . We label the $j$-th permutation of coefficient $c_i$ as $c_i^{(j)}$. Permutation $j$ maps the labeling $(1,2,3,4,5,6)$ of the external momenta to: $$\begin{aligned}
&& 1: (1,2,3,4,5,6)\,, \quad
~ 2: (2,3,4,5,6,1)\,, \quad
~ 3: (3,4,5,6,1,2)\,, \quad
~ ~ 4: ( 4,5,6,1,2,3)\,, \nonumber \\
&& 5: (5,6,1,2,3,4)\,, \quad
~ 6: (6,1,2,3,4,5)\,, \quad
%
~ 7: (6,5,4,3,2,1)\,, \quad
~~ 8: (5,4,3,2,1,6)\,, \nonumber \\
&& 9: (4,3,2,1,6,5)\,, \quad
10: (3,2,1,6,5,4)\,, \quad
11: (2,1,6,5,4,3)\,, \quad
12: (1,6,5,4,3,2)\,. \quad\end{aligned}$$
Since all of the coefficients for the MHV amplitude are proportional to the tree amplitude, we can go ahead and divide the right-hand side of all equations by $A^{\rm tree}_{6,{\rm MHV}}$. Equivalently we can say that solving the equations below yields the integral coefficients for the normalized amplitude $A^{(2)}_{6,{\rm MHV}}/A^{\rm tree}_{6,{\rm MHV}}$. A final comment is that we move all of the Jacobian factors (see for example eq. (\[eq:jacobian\])) to the right-hand side of the equations.
Finally we are ready to present the equations obtained by considering the contours associated with the topologies in . In each equation $p$ and $q$ are understood to be evaluated at the locations $(p^{(i)}, q^{(j)})$ of all the leading singularities.
Topology A: $$c_1^{(2)} +
\frac{
(p-k_{234})^2 (q-k_{34})^2 c_{13}^{(2)} +
(p + k_{61})^2 (q - k_{234})^2 c_{13}^{(5)}}
{(p+q-k_{234})^2}
= 4 s_{12} s_{23} s_{45} s_{56}
\delta_{\langle p,1\rangle} \delta_{\langle q,4\rangle}$$
Topology D: $$\label{eq:TopD}
c_{19}^{(2)} + (p-k_1)^2 c_8^{(6)} =
4 s_{12} (p - k_1)^2
(s_{123} s_{345} - s_{12} s_{45}) \delta_{\langle p,6\rangle}$$
Topology E: $$\begin{aligned}
&&
c_{21}^{(3)} + (p - k_{12})^2 c_{10}^{(3)} +
\frac{
(p - k_{12})^2 (q + k_{234})^2 c_{13}^{(2)}
+ (p - k_1)^2 (q - k_{61})^2 c_{13}^{(5)}}
{(q - k_1)^2}
\cr
&&
\qquad\qquad
= 4 s_{23} s_{45} s_{56} (p - k_{12})^2
\delta_{\langle p,6\rangle} \delta_{\langle q,2\rangle}\end{aligned}$$
Topology F: $$\label{eq:TopF}
c_2^{(6)} + \frac{c_{19}^{(2)}}{(p + k_{456})^2} = 4
s_{12}^2 s_{61} \delta_{\langle p,6\rangle}$$
Topology H: $$\begin{aligned}
&&
c_4^{(6)}
+ \frac{
(p - k_{12})^2 (q - k_{56})^2 c_{13}^{(2)}
+ (p - k_1)^2 (q - k_6)^2 c_{13}^{(5)}}
{(p + k_{56})^2 (q + k_{12})^2}
\cr
&&
+ \frac{c_{20}^{(6)} + (q - k_6)^{2} c_9^{(6)}}{(q + k_{12})^2}
+ \frac{c_{20}^{(12)} + (p - k_1)^2 c_9^{(12)}}{(p + k_{56})^2}
=0\end{aligned}$$
Topology I: $$\begin{aligned}
&&
c_3^{(6)} +
\frac{
(p - k_{12})^2 (q - k_{56})^2 c_{13}^{(2)}
+ (p - k_1)^2 (q - k_6)^2 c_{13}^{(5)}}
{(p + k_{56})^2 (q + k_1)^2}
\cr
&&
+\frac{c_{20}^{(6)} + (q - k_6)^2 c_9^{(6)}}
{(q + k_1)^2}
+ \frac{c_{20}^{(3)} + (p - k_{12})^2 c_9^{(3)}}
{(p + k_{56})^2}
= 4 s_{123} (s_{123} s_{345} - s_{12} s_{45})
\delta_{\langle p,6\rangle} \delta_{\langle p,q\rangle}\end{aligned}$$
Topology J: $$\begin{aligned}
\label{eq:topRMHV}
&&
c_7^{(5)}
+\frac{c_{21}^{(12)} + (p - k_{61})^2 c_{10}^{(12)}}{(p - k_6)^2}
+\frac{c_{21}^{(5)} + (q - k_{56})^2 c_{10}^{(5)}}{(q - k_6)^2}
\cr
&&
+\frac{
(p + k_{345})^2 (q - k_{56})^2 c_{13}^{(2)}
+ (p - k_{61})^2 (q - k_6)^2 c_{13}^{(5)}}
{(p - k_6)^2 (q + k_{123})^2}
\cr
&&
+\frac{c_{21}^{(2)} + (p - k_{61})^2 c_{10}^{(2)}}{(p + k_{345})^2}
+\frac{c_{21}^{(9)} + (q - k_{56})^2 c_{10}^{(9)}}{(q + k_{123})^2}
\cr
&&
+\frac{
(p - k_{61})^2 (q - k_{456})^2 c_{13}^{(1)}
+ (p - k_6)^2 (q - k_{56})^2 c_{13}^{(4)}}
{(p + k_{345})^2 (q - k_6)^2}
= 0\end{aligned}$$
Topology K: $$\begin{aligned}
&& c_{17}^{(6)} +
\frac{
+ (p - k_{12})^2 (q - k_{56})^2 c_{13}^{(2)}
+(p-k_1)^2 (q-k_6)^2 c_{13}^{(5)}}
{(p+k_{456})^2(q+k_{123})^2}
\cr
&& + \frac{c_{21}^{(6)} + (q-k_6)^2 c_{10}^{(6)}}{(q+k_{123})^2}
+ \frac{c_{21}^{(12)} + (p-k_1)^2 c_{10}^{(12)}}{(p+k_{456})^2}
= 4 s_{12} s_{56} (q - k_6)^2 \delta_{\langle p,q\rangle}\end{aligned}$$
Topology L: $$\begin{aligned}
&&c_5^{(3)} +
\frac{
(p - k_{12})^2 (q - k_{56})^2 c_{13}^{(2)}
+ (p - k_1)^2(q -k_6)^2 c_{13}^{(5)}}
{(p + k_{456})^2 (q + k_1)^2}
\cr
&&
+ \frac{c_{21}^{(6)} + (q - k_6)^2 c_{10}^{(6)}}{(q + k_1)^2}
+ \frac{c_{20}^{(3)} + (p - k_{12})^2 c_9^{(3)}}{(p + k_{456})^2}
= 0\end{aligned}$$
Topology M: $$\begin{aligned}
&&c_6^{(6)}
+ \frac{
(p - k_{12})^2 (q - k_{56})^2 c_{13}^{(2)}+
(p-k_1)^2 (q-k_6)^2 c_{13}^{(5)}}
{(p+k_{56})^2(q+k_{123})^2}
\cr
&&
+ \frac{c_{21}^{(12)} + (p-k_1)^2 c_{10}^{(12)}}{(p+k_{56})^2}
+ \frac{c_{20}^{(6)} + (q - k_6)^2 c_9^{(6)}}{(q+k_{123})^2}
+ \frac{c_{19}^{(2)} + (p - k_1)^2 c_8^{(2)}}{(p - k_{12})^2}
=0\end{aligned}$$
The equations for topologies $(B)$, $(C)$ and $(G)$ turn out to be redundant for the MHV amplitude with the choice of basis described above.
Analytic Results for a $2 \times 2$ Block
-----------------------------------------
The structure of the equations is sufficiently complicated that it is not clear whether it is possible to find useful analytic solutions, so in practice we resort to solving them numerically. However the equations from topologies ${(D)}$ and ${(F)}$ are exceptionally simple and only involve the coefficients $c_2$, $c_8$ and $c_{19}$, so they can easily be solved analytically as we now demonstrate.
### Topology $(F)$
The four contour integrals for topology $(F)$ localize the integral at the four points $(p^{(i)}, q^{(j)})_{i,j=1,2}$ given by $$\begin{aligned}
q^{(1)}=\left(-\lambda_1+\frac{\langle 1,6 \rangle}{\langle 2,6 \rangle}
\lambda_2\right) \widetilde{\lambda}_1\,, && \qquad
p^{(1)}=
\frac{\langle 2,1 \rangle}{\langle 2,6 \rangle} \lambda_6
\widetilde{\lambda}_1\,
\cr
q^{(2)}=\lambda_1
\left(-\widetilde{\lambda}_1+\frac{[1,6]}{[2,6]}
\widetilde{\lambda}_2\right)\,, && \qquad
p^{(2)}=
\frac{[2,1]}{[2,6]} \lambda_1 \tilde{\lambda}_6\,.\end{aligned}$$ Equation (\[eq:TopF\]) is then $$\label{eq:linF}
c_2^{(6)}+\frac{c_{19}^{(2)}}{(p^{(1)}+k_{456})^2}=4 s_{12}^2 s_{61}\,, \qquad
c_2^{(6)}+\frac{c_{19}^{(2)}}{(p^{(2)}+k_{456})^2}=0\,.$$ Note that even though there are four different contours we only obtain two independent equations since (\[eq:TopF\]) is independent of $q$. This is a generic feature whenever a contour is such that it chops off a massless box [@Cachazo:2008vp].
Solving (\[eq:linF\]) yields the coefficients $$\label{eq:sol19}
c_2^{(6)} = {4 s_{12}^2 s_{61} \over 1-a}, \qquad
c_{19}^{(2)} = {4 s_{12}^2 s_{61} \over 1- a} (p^{(1)} + k_{456})^2\,$$ where $$a=\frac{(p^{(1)}+k_{456})^2}{(p^{(2)}+k_{456})^2}\,.$$ It is frequently useful to separate coefficients into their parity-even and parity-odd parts. Since parity exchanges $\langle i,j\rangle
\leftrightarrow [i,j]$, it evidently takes $a \to 1/a$. If we denote the parity conjugate of a coefficient $c$ by $\bar{c}$ then we see that the even and odd parts of $c_2^{(6)}$ and $c_{19}^{(2)}$ are simply $$\begin{aligned}
{1 \over 2}(c_2^{(6)}+\bar{c}_2^{(6)}) =2 s_{16} s_{12}^2\,,&& \qquad
{1 \over 2}
(c_2^{(6)}-\bar{c}_2^{(6)})=2
s_{16} s_{12}^2 \left(\frac{1+a}{1-a} \right)\,,\cr
{1 \over 2}(
c_{19}^{(2)}+\bar{c}_{19}^{(2)}) =0\,, && \qquad
{1 \over 2} (c_{19}^{(2)}-\bar{c}_{19}^{(2)}) =
4 s_{12}^2 s_{61}\frac{(p^{(1)}+k_{456})^2}{1-a}\,.\end{aligned}$$ The parity-even parts of these coefficients agree precisely with those obtained in [@Bern:2008ap] via the unitarity based method. Here we see that these coefficients can be obtained simply by solving two equations in two variables, and moreover the parity-odd parts automatically come along for free.
### Topology ${(D)}$
For topology $(D)$ the contour integrals localize the integral at the points $$\begin{aligned}
p^{(1)}=
(\alpha \lambda_6+\beta \lambda_3)
\widetilde{\lambda}_6\, && \qquad
q^{(1)}= \frac{[1,2]}{[2,6]} \lambda_1 \widetilde{\lambda}_6\,,
\cr
p^{(2)}=\lambda_6 ({\alpha} \widetilde{\lambda}_6+\gamma
\widetilde{\lambda}_3)\, && \qquad
q^{(2)}=
\frac{\langle 1,2 \rangle}{\langle 2,6\rangle}
\lambda_6 \widetilde{\lambda}_1\,,\end{aligned}$$ where $$\alpha=\frac{s_{13}+s_{23}}{s_{36}},\quad
\beta=\frac{-s_{12}+\alpha (s_{12}+s_{26})}{\langle 3|1+2|6]},\quad
\gamma=\frac{-s_{12}+\alpha (s_{12}+s_{26})}{[3|1+2|6\rangle}\,.$$ Equation (\[eq:TopD\]) then gives $$\label{eq:linD}
c_{19}^{(2)}+\frac{c_8^{(6)}}{(p^{(1)}-k_1)^2}=
4 s_{12} (s_{345} s_{456}-s_{12} s_{45})\, \qquad
c_{19}^{(2)}+\frac{c_8^{(6)}}{(p^{(2)}-k_1)^2}=0\,.$$ We can eliminate $c_{19}^{(2)}$ to solve for $c_8^{(6)}$, finding $$\begin{aligned}
{1 \over 2} (c_{8}^{(6)} + \bar{c}_{8}^{(6)}) &=&
2 s_{12} ( s_{123} s_{345} - s_{12} s_{45})\,\cr
{1 \over 2} (c_{8}^{(6)} - \bar{c}_{8}^{(6)}) &=&
2 s_{12} (s_{123} s_{345} -s_{12} s_{45}) \left(\frac{c+1}{c-1} \right)\,,\end{aligned}$$ where $$c=\frac{(p^{(1)}-k_1)^2}{(p^{(2)}-k_1)^2}\,.$$ Again the even part of $c_8$ agrees precisely with the unitary based calculation of [@Bern:2008ap]. Of course the equation (\[eq:linD\]) provides a consistency condition on the coefficient $c_{19}^{(2)}$ that we already obtained in (\[eq:sol19\]).
The Parity-Even Part
--------------------
In the previous subsection we solved for the coefficients $c_2$, $c_8$ and $c_{19}$ analytically and demonstrated that their parity-even parts agree with the results of [@Bern:2008ap]. In order to check the validity of the leading singularity method it is important for us to compare the parity-even parts of all remaining coefficients as well. The first obstacle is the fact that [@Bern:2008ap] used a basis containing the integral shown in . According to our criteria we do not consider this a geometric integral since the propagator does not serve to cancel any pole. (The motivation for using $I_{11}$ in [@Bern:2008ap] was that the integral is manifestly dual conformally invariant [@Drummond:2006rz; @DrummondVanishing].)
We show in the next section that reduction identities can be used to express $I_{11}$ as a linear combination of geometric integrals, so secretly already contains $I_{11}$. However in order to facilicate comparison of our results with those of [@Bern:2008ap] it is convenient to explicitly add $I_{11}$ to the set of integrals. Then we have a set of $177 + 12 = 189$ integrals which is overcomplete by $18 + 12 = 30$ elements. Encouragingly, we find that it is possible to find a solution of the equations in which the non-dual conformally invariant integrals $I_{19}$–$I_{24}$ all have coefficients whose parity-even part vanishes. This is a necessary condition for agreement with [@Bern:2008ap] since the parity-even part of the amplitude was expressed there in terms of dual conformally invariant integrals. Moreover we find that after choosing this ‘gauge’ there is no further ambiguity in the basis; the linear equations furnish a unique solution.
![An extra 12 integrals we add to the set of geometric integrals in order to facilitate comparison with [@Bern:2008ap]. These integrals are dual conformally invariant but not geometric. Nevertheless they can be expressed as linear combinations of the geometric integrals in using first class identities (see section IV).[]{data-label="strangeintegral"}](fig4.eps)
As indicated above, the equations are sufficiently complicated that we found it necessary to solve them numerically. Let us note however that by ‘numerically’ we always mean that we choose all of the spinors $\lambda_i, \widetilde{\lambda}_i$ to be random rational numbers (subject to momentum conservation, of course). Then the coefficients obtained by solving the equations always come out to be rational numbers, so they can be compared to the results of [@Bern:2008ap] with absolute precision. By repeated successful comparison for many different random values of the spinors, we are able to report complete agreement with the parity-even parts of the coefficients obtained from the leading singularity method with those obtained in [@Bern:2008ap], recorded here in the $(1)$ permutation: $$\begin{aligned}
c_1 &=&
%%%%% begin : coeff1
s_{61} s_{34} s_{123} s_{345} + s_{12} s_{45} s_{234} s_{345} +
s_{345}^2 (s_{23} s_{56} - s_{123} s_{234})\,,\cr
%%%%% end : coeff1
c_2 &=&
%%%%% begin : coeff2
2 s_{12} s_{23}^2\,,\cr
%%%%% end : coeff2
c_3 &=&
%%%%% begin : coeff3
s_{234} (s_{123} s_{234} - s_{23} s_{56})\,,\cr
%%%%% end : coeff3
c_4 &=&
%%%%% begin : coeff4
s_{12} s_{234}^2\,,\cr
%%%%% end : coeff4
c_5 &=&
%%%%% begin : coeff5
s_{34} (s_{123} s_{234} - 2 s_{23} s_{56})\,,\cr
%%%%% end : coeff5
c_6 &=&
%%%%% begin : coeff6
- s_{12} s_{23} s_{234}\,,\cr
%%%%% end : coeff6
c_7 &=&
%%%%% begin : coeff7
2 s_{123} s_{234} s_{345} - 4 s_{61} s_{34} s_{123} - s_{12} s_{45}
s_{234} - s_{23} s_{56} s_{345}\,,\cr
%%%%% end : coeff7
c_8 &=&
%%%%% begin : coeff8
2 s_{61} (s_{234} s_{345} - s_{61} s_{34})\,,\cr
%%%%% end : coeff8
c_9 &=&
%%%%% begin : coeff9
s_{23} s_{34} s_{234}\,,\cr
%%%%% end : coeff9
c_{10} &=&
%%%%% begin : coeff10
s_{23} (2 s_{61} s_{34} - s_{234} s_{345})\,,\cr
%%%%% end : coeff10
c_{11} &=&
%%%%% begin : coeff11
s_{12} s_{23} s_{234}\,,\cr
%%%%% end : coeff11
c_{12} &=&
%%%%% begin : coeff12
s_{345} (s_{234} s_{345} - s_{61} s_{34})\,,\cr
%%%%% end : coeff12
c_{13} &=&
%%%%% begin : coeff13
- s_{345}^2 s_{56}\,.
%%%%% end : coeff13\end{aligned}$$ However, as mentioned in the introduction we note that the leading singularity method is completely blind to the integrals $I_{14}$ and $I_{15}$ in [@Bern:2008ap] since they have integrands that vanish in $D=4$.
Reductions
==========
Methods of Reduction
--------------------
Repeatedly throughout the paper we have mentioned and used the new feature that happens for six or more particles; loop integrals often satisfy linear relations which can be used to write one in terms of others (see for example [@vanNeerven:1983vr], as well as [@Bern:1993kr] for dimensionally regulated versions). Interestingly, there are relations even among what we call geometric integrals. Also important for us will be relations among integrals that appear in the manifestly dual conformal invariant expression of [@Bern:2008ap] and our geometric integrals.
In this section we discuss in detail how these relations arise since it is a crucial step in completing the proof that the leading singularity method does determine the amplitude uniquely. Recall that out of the $177$ coefficients the leading singularity fixes $159$ thus leaving $18$ free parameters. We will now account for these as a consequence of relations or what we call reduction identities. In other words, the set of geometric integrals which naturally appears in the process of matching leading singularities is overcomplete.
We distinguish between two different kind of identities; the ones that are valid on any contour of integration and the ones that are valid only on $T^8$ contours where loop momenta can be taken to be in four-dimensions. We call these first and second class identities, respectively.
### First Class
Identities of the first class are those which are valid on any contour of integration, in particular, on the real contours where integrals must be regulated, [*e.g.*]{} using dimensional regularization.
First consider for example the pentagon-box integral $$\includegraphics{figa.eps}$$ -.3cm with numerator factor $(p - k_1)^2$, which is a permutation of the one shown in . Clearly, this integral is not geometric since the numerator is not a zero which cancels a pole of the integral. However, this integral is dual conformal invariant and it appears in the representation of the even part of the amplitude obtained in [@Bern:2008ap]. The goal is to write this integral as a linear combination of the geometric integrals in .
Let us write the numerator as $p^2-2k_1\cdot p$. The first term cancels a propagator and gives rise to a double-box integral of type $I_{17}$ in . The second term can be decomposed by using that external momenta are kept in four dimensions. This means that only the four-dimensional component, $p_{[4]}$, of $p$ contributes, [*i.e.*]{} $k_1\cdot p = k_1\cdot p_{[4]}$. Given any four dimensional vector, one can write it as a linear combination of four axial vectors $$\vartheta_i^\mu =
\epsilon^{\mu\nu\rho\sigma}\ell^{i+1}_\nu\ell^{i+2}_\rho\ell^{i+3}_\sigma$$ where $\ell^i$, with $i=1,\ldots,4$, form a basis of four-vectors. In the case of interest, we choose to write $p_{[4]}$ as $$p_{[4]}^\mu =
\frac{1}{\epsilon^{\mu\nu\rho\sigma}
\ell_\mu^1\ell^{2}_\nu\ell^{3}_\rho\ell^{4}_\sigma}
\sum_{i=1}^4 (p_{[4]}\cdot \ell_i)\vartheta_i^\mu \label{eq:verma}$$ with the choice $$\ell_1 =k_6\,, \quad \ell_2=k_5+k_6\,, \quad \ell_3 =k_4+k_5+k_6\,, \quad
\ell_4 = -k_1-k_2\,.$$ This a standard construction in the scattering amplitude literature [@vanNeerven:1983vr; @Bern:1993kr].
Since all of the $\ell_i$ are written in terms of external particle momenta, they are completely four-dimensional so we are free to replace $p_{[4]}$ by $p$ in the coefficients $(p_{[4]} \cdot \ell_i)$. Writing each coefficient as $2p\cdot\ell_i =
(p+\ell_i)^2-p^2-\ell_i^2$, one finds that the term $k_1\cdot p$ in the original integral can be decomposed in terms of numerators which give rise to geometric integrals. Perhaps the only term which might require some explanation is the one corresponding to $\ell_4=-k_1-k_2$ since it is one which does not cancel a propagator. In this case the factor $(p-k_1-k_2)^2$ becomes a numerator which is easily seen to be a zero that cancels a pole which removes a leading singularity and hence gives rise to a geometric integral.
From this example it is clear what the necessary conditions are for the existence of first class relations among integrals. The first condition is that there be at least four propagators (including $1/p^2$) involving the same loop variable and only external momenta. If the number of such propagators is exactly four then there must be at least two different ways of putting a numerator which only involves the loop variable of interest and external momenta. This was the case considered above. If the number of propagators is at least five, then a relation can be obtained if at least one numerator is available. If the number of propagators is six or more, then no numerators are needed.
In our case, with six external particles, the maximum number of propagators in a single loop which only depend on a single loop variable and external momenta is five. This diagram is a hexagon-box. In fact, it is easy to show that a hexagon-box with a numerator can be reduced. We leave this as an exercise for the reader since such an integral did not have to be included in the original set of geometric integrals, at least in the MHV case.
Using identities of the first class we will be able to show that all six $I_{12}$ integrals can be written in terms of $I_{13}$ integrals plus other geometric integrals. This shows $6$ of the $18$ relations we have to account for. Also using first class relations we will show that $6$ of the $12$ $I_{24}$ integrals can be written in terms of the remaining six and other geometric integrals. Summarizing, after using all first class identities we end up with $6$ relations left to explain. These turn out to be second class identities.
### Second Class
The identities discussed above are valid in any contour because the loop momentum is always contracted with external momenta and hence it can be treated effectively as four dimensional. We now turn to identities which only hold on the $T^8$ contours where the loop momentum integrals can be taken completely in four dimensions. These identities will not hold in dimensional regularization. In fact, their failure to hold exactly is precisely proportional to integrals with numerators made out of the $-2\epsilon$-dimensional component of the loop momenta. We obtain identities by reducing integrals of this type to find linear combinations of other integrals which must sum to zero for four dimensional loop momenta.
Consider first an integral with numerator, $p_{[-2\epsilon]}\cdot
q_{[-2\epsilon]} = p\cdot q - p_{[4]}\cdot q_{[4]}$. The first term can be written as $2p\cdot q = (p+q)^2-p^2-q^2$. In order to treat the second term one has to find a convenient basis to expand $p_{[4]}$ and one for $q_{[4]}$. The relevant diagram must have at least four propagators that only depend on $p$ and at least four that only depend on $q$ and external momenta. With six external particles there is only one possible diagram (up to relabeling). This is the double pentagon integral $$\includegraphics{figb.eps}$$ -.3cm Let us choose to write $p_{[4]}^\mu$ using the reference vectors $$\label{eq:refa}
\ell_1 = k_6\,, \quad \ell_2 = k_5+k_6\,,\quad \ell_3 =
k_4+k_5+k_6\,,\quad \ell_4 = k_1 \label{eq:dosa}$$ while writing $q_{[4]}^\mu$ using $$\label{eq:refb}
\ell_1 = k_1\,, \quad \ell_2 = k_1+k_2\,, \quad \ell_3 = k_1+k_2+k_3\,,
\quad \ell_4 =k_6 . \label{eq:huno}$$ Plugging these into (\[eq:verma\]) and calculating $p_{[4]}\cdot q_{[4]}$, we find an expansion containing geometric integrals and some further integrals that can very easily be further decomposed into geometric ones. These identities give rise to six relations among the remaining six $I_{24}$ integrals and the rest of the geometric integrals. The reader might wonder how can one get six equations if the starting point, which is the $p_{[-2\epsilon]}\cdot q_{[-2\epsilon]}$ pentagon-pentagon integral, has a 4-fold symmetry implying that there are only three independent such integrals. However, the decomposition process breaks that symmetry since one needs to make a choice of basis vectors as in equations (\[eq:refa\]) and (\[eq:refb\]). There are two independent choices, leading to a total of six independent reduction identities.
One could also consider integrals with a factor of $p_{[-2\epsilon]}^2
= p^2 - p_{[4]}^2$ in the numerator. However the only way to use a reduction of such an integral in the case of six particles is if $p$ is the loop momentum in a hexagon inside the hexagon-box integral. The hexagon-box does not appear in since it is never needed in order to solve the equations, so we have no need for such reduction identities.
Summary of All Integral Reductions
----------------------------------
Here we summarize all of the relevant reduction identities that can be derived using the techniques explained above. First we have the identity schematically represented as $${\hbox{\lower 12.5pt\hbox{
\includegraphics{int11.eps}
}}} = {\hbox{\lower 12.5pt\hbox{
\includegraphics{int10.eps}
}}} + {\hbox{\lower 12.5pt\hbox{
\includegraphics{int21.eps}
}}} + {\rm boxes\,,}$$ which we use to indicate that the integral on the left can be expressed as a linear combination of the integrals on the right. It is straightforward to work out all of the precise coefficients, but they are not important for our analysis. The important point is the conclusion that the integral shown in can be reduced to integrals already present in .
It is also straightforward to derive the first class relation $${\hbox{\lower 12.5pt\hbox{
\includegraphics{int12.eps}
}}}
=
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int13.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int24.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int11.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int20.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int09.eps}
}}}
+ {\rm boxes}\,.$$ This is again a schematic relation: the integrals on the right-hand side can appear in various rotated or flipped incarnations.
The final first class relation is $${\hbox{\lower 12.5pt\hbox{
\includegraphics{int24.eps}
}}}
=
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int24flip.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int23.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int20.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int19.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int21.eps}
}}}$$ which implies that of the 12 apparently independent integrals of the type $I_{24}$ shown in , in fact only 6 are linearly independent.
Next we summarize the second class reduction formula, discussed in section IV.A.2, that is obtained by starting with a double pentagon with $p_{[-2 \epsilon]}
\cdot q_{[-2 \epsilon]}$ numerator. As explained above this analysis leads to 6 independent identities. Schematically these identities take the form $$\begin{aligned}
0 &=&
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int23.eps}
}}} \times p_{[-2\epsilon]}\cdot q_{[-2 \epsilon]}
\cr
&=&
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int01.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int21.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int10.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int09.eps}
}}}
+
{\hbox{\lower 12.5pt\hbox{
\includegraphics{int20.eps}
}}}
+
{\rm boxes}\,.
\nonumber\end{aligned}$$ Again the integrals on the right-hand side can appear in various different permutations.
Taking everything into account, we find that the set of geometric integrals in is overcomplete by $6 + 6 + 6 = 18$ elements. This precisely accounts for the 18 free parameters we found in solving the linear equations for the MHV coefficients. (This increases from 18 to $18 + 12 = 30$, as discussed in section III.C, when the integral $I_{11}$ is thrown into the mix.)
Numerical Check of the ABDK/BDS Ansatz
======================================
One of the most interesting properties of scattering amplitudes in ${\cal N}=4$ SYM is that the structure of infrared divergences in higher loop amplitudes is very simply related to those of lower loop amplitudes [@KnownIR]. In [@ABDK; @BDS], it was conjectured that this simplicity persists, at least for MHV amplitudes, to the finite terms as well. The precise form of the conjecture at two loops, in dimensional regularization to $D = 4 - 2 \epsilon$, is $$\label{eq:jaso}
M_{n,{\rm MHV}}^{(2)}(\epsilon) =
\frac{1}{2}M_{n,{\rm MHV}}^{(1)}(\epsilon)^2
+ f^{(2)}(\epsilon) M_{n,{\rm MHV}}^{(1)}(2\epsilon) - {\pi^4 \over 72} +
{\cal O}(\epsilon)\,,$$ where $M^{(L)}_{n,{\rm MHV}}
= A^{(L)}_{n,{\rm MHV}}/A^{\rm tree}_{n,{\rm MHV}}$ is the normalized $L$-loop amplitude and $f^{(2)}(\epsilon) = -(\zeta(2) + \zeta(3) \epsilon + \zeta(4) \epsilon^2 +
\cdots)$. The conventions implicit in equation (\[eq:jaso\]) require that every loop momentum integral $p$ be normalized with the factor $$-i \pi^{-D/2} e^{\epsilon \gamma} \int d^D p\,.$$
The simple structure (\[eq:jaso\]) holds perfectly for $n=4$ and $n=5$ [@ABDK; @TwoLoopFiveA; @TwoLoopFiveB]. However it apparently fails beginning at $n=6$ particles. This was found in [@Bern:2008ap] by computing the parity-even part of $M^{(2)}_{6,{\rm MHV}}$ numerically and finding disagreement with the right-hand side of (\[eq:jaso\]). Here we do not have anything to add to this issue except that the parity-even piece of our full answer agrees with the result of [@Bern:2008ap], thus providing independent confirmation.
Since the leading singularity method allows us to obtain the parity-odd parts of all coefficients with no more effort than the parity-even parts, we are in a position to test the parity-odd part of (\[eq:jaso\]) for $n=6$. Note that the loop momentum integrals must necessarily be evaluated numerically (except for $n=4$, where analytic results are known through three loops) with current state-of-the-art technology (in particular we use [@MB; @CUBA]).
Restricting to the parity-odd part of (\[eq:jaso\]) yields $$\label{eq:oddabdk}
M^{(2)}_{6, {\rm MHV~odd}}(\epsilon) = M^{(1)}_{6, {\rm MHV~even}}(\epsilon)
M^{(1)}_{6, {\rm MHV~odd}}(\epsilon) + {\cal O}(\epsilon).$$ The one-loop amplitude is [@BDDKSelfDual], $$\begin{aligned}
M^{(1)}_{6, {\rm MHV~even}}(\epsilon)
&=& - {1 \over 2 \epsilon^2} \sum_{i=1}^6 (-s_{i,i+1})^{-\epsilon}
+ {\cal O}(1)\,,\cr
M^{(1)}_{6, {\rm MHV~odd}}(\epsilon)
&=&
-{1 \over 4} \sum_{i=1}^6
\Big(
\langle i|i{+}1|i{+}2|i{+}3|i{+}4]
-
[i|i{+}1|i{+}2|i{+}3|i{+}4\rangle
\Big) P_{i{+}5,i{+}6}(\epsilon)\,,\end{aligned}$$ where $P_{i{+}5,i{+}6}(\epsilon)$ is the one-loop pentagon integral with external legs $i{+}5$ and $i{+}6$ joined and with a factor of $p_{[-2\epsilon]}^2$ in the numerator.
We have evaluated (\[eq:oddabdk\]) numerically at two independent kinematic points with randomly generated values of $\lambda_i$ and $\widetilde{\lambda}_i$ for the six external particles. Denoting the left- and right-hand sides of (\[eq:oddabdk\]) by $L$ and $R$ respectively, we find $$\begin{aligned}
L(\epsilon) &=& - \frac{4 \times 10^{-16}}{\epsilon^4}
+ {4 \times 10^{-15} \over \epsilon^3}
+ \frac{1(2) \times 10^{-11}}{\epsilon^2}
- \frac{0.430(7)}{\epsilon}
- 0.9(1) + {\cal O}(\epsilon)\,,\cr
R(\epsilon) &=&\qquad\qquad\qquad\qquad~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- \frac{0.428(2)}{\epsilon} -
0.92(1) + {\cal O}(\epsilon)
\label{eq:res1}\end{aligned}$$ at the first point and $$\begin{aligned}
L(\epsilon) &=& { < 10^{-16} \over \epsilon^4}
- {8 \times 10^{-15} \over \epsilon^3}
+ {7(5) \times 10^{-12} \over \epsilon^2}
- {15.902(5) \over \epsilon}
- 60.46(6) + {\cal{O}}(\epsilon)\,,
\cr
R(\epsilon) &=&\qquad\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- \frac{15.915(6) }{\epsilon}
- 60.38(4) + {\cal{O}}(\epsilon)
\label{eq:res2}\end{aligned}$$ at the second. In these expressions the value in parentheses denotes the estimated numerical error in the last digit as reported by [@CUBA].
We emphasize that the cancellation of the divergent terms in (\[eq:res1\]) is not a check of the ABDK/BDS conjecture, but rather a check on our application of the leading singularity method. This is so because it is a known fact [@KnownIR], not a conjecture, that (\[eq:jaso\]) must hold for the infrared divergent terms. Had we gotten a nonzero result, it would have signalled an error in our calculation of the integral coefficients. Note that the cancellation in $L(\epsilon)$ is highly nontrivial in the sense that the result shown is obtained after summing of order 100 contributions which are typically of order $1$. It is the fact that we see the cancellation persisting to order ${\cal{O}}(\epsilon^0)$ that strongly suggests that the parity-odd part of the two-loop six-particle MHV amplitude indeed satisfies the ABDK/BDS conjecture (\[eq:oddabdk\]).
However it is important to note once again that since the leading singularity method is not sensitive to any “$\mu$”-terms we cannot rule out the possibility that there may be additional such contributions to $M^{(2)}_{6,{\rm MHV~odd}}$. Given our apparently successful check of (\[eq:oddabdk\]) there are three possibilities: (1) the two-loop amplitude does not contain any parity-odd $\mu$-terms (this is indeed the case for $n=5$ particles [@TwoLoopFiveB]), (2) the amplitude does contain parity-odd $\mu$-terms but they contribute only at ${\cal O}(\epsilon)$, or (3) the amplitude contains parity-odd $\mu$-terms which spoil the ABDK/BDS relation (\[eq:jaso\]).
Conclusion
==========
In this paper we employed the leading singularity method to determine the integral coefficients of the planar two-loop six-particle MHV amplitude in ${\cal N} = 4$ YM, the parity-even parts of which were recently obtained in [@Bern:2008ap] using the unitarity based method. One advantage of the leading singularity method is that the full coefficients, including parity-odd parts, emerge from solving the relatively simple set of linear equations displayed explicitly in section III.A.
The leading singularity method has previously proven succesful [@Cachazo:2008vp] at reproducing the $n=4$ [@Bern:1997nh] and $n=5$ [@TwoLoopFiveB] particle amplitudes at two loops. However, there is currently no proof that a general amplitude in ${\cal N} = 4$ Yang-Mills is uniquely determined by its leading singularities only. It is a logical possibility that finding a representation of an amplitude in terms of simpler integrals which faithfully reproduce all of the leading singularities is not a sufficient condition to guarantee correctness of the representation, although clearly it is a necessary condition.
Here we find that the $n=6$ particle amplitude is in fact completely determined by its leading singularities, although establishing this fact required that considerable attention be given to the choice of basis for the integrals and reduction identities which relate various integrals to each other. This was necessitated by the fact that the full set of linear equations we found does not have a unique solution. Fortunately we found that all of the ambiguities could be accounted for by taking into account reduction identities.
One nice feature of the leading singularity method is that the procedure naturally provides a set of ‘geometric’ integrals for any amplitude under consideration. The set of geometric integrals does not coincide with the manifestly dual conformally invariant [@Drummond:2006rz; @DrummondVanishing] basis used to express the parity-even part of the $n=6$ amplitude in [@Bern:2008ap]. We expect this to be true in general. This is not surprising given the fact that by using the leading singularity technique both even and odd parts of the amplitude are computed simultaneously, whereas already for five particles the odd part of the amplitude [@TwoLoopFiveB] is not expressible in terms of integrals with manifest dual conformal properties. Our results indicate that also for $n=6$ the odd part of the amplitude cannot be expressed in terms of dual conformally invariant integrals alone.
Another motivation for computing the MHV six-particle amplitude, beyond its serving as a testing ground for the leading singularity method, is to study the so-called ABDK/BDS conjecture for MHV amplitudes [@ABDK; @BDS]. Although the conjecture was shown to be violated by the parity-even part of the $n=6$ particle amplitude [@Bern:2008ap], following earlier doubts that had been raised in [@AMTrouble; @BNST; @Lipatov], we provide numerical evidence that the parity-odd part of the amplitude does satisfy the ABDK/BDS relation.
Equivalently, one can say that the parity-odd part evidently cancels out when one takes the logarithm of the resummed amplitude (see [@BDS]). Although we do not know of any proof that this has to be the case, the result is consistent with the structure seen at strong coupling [@AM], which is manifestly parity-invariant. It is also consistent with the astounding but still mysterious equivalence between scattering amplitudes and lightlike Wilson loops in ${\cal{N}} = 4$ YM that has been observed at one-loop [@DrummondVanishing; @BrandhuberWilson] and at two-loops through $n=6$ particles [@DHKSTwoloopBoxWilson; @ConformalWard; @HexagonWilson; @Drummond:2008aq], since the ‘vanilla’ Wilson loop does not carry any helicity information. It is a very interesting open problem to determine if the amplitude/Wilson loop equivalence can be extended to other helicity configurations by appropriately dressing the Wilson loop.
Acknowledgments {#acknowledgments .unnumbered}
===============
M. S. and A. V. are grateful to Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, C. Tan, C. Vergu and C. Wen for many helpful comments and collaboration on related questions. We thank D. A. Kosower for sharing Mathematica files containing some results from the unitarity based method. The authors are also grateful to the Institute for Advanced Study for hospitality during the origination of this work. This work was supported in part by the NSERC of Canada and MEDT of Ontario (F. C.), the US Department of Energy under contract DE-FG02-91ER40688 (M. S. (OJI) and A. V.), the US National Science Foundation under grants PHY-0638520 (M. S.) and PHY-0643150 CAREER (A. V.).
[99]{}
R. J. Eden, P. V. Landshoff, D. I. Olive, J. C. Polkinghorne, “The Analytic S-Matrix,” Cambridge University Press, 1966.
Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, unitarity and collinear limits,” Nucl. Phys. B [**425**]{}, 217 (1994) \[arXiv:hep-ph/9403226\]. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B [**435**]{}, 59 (1995) \[arXiv:hep-ph/9409265\]. Z. Bern and A. G. Morgan, Nucl. Phys. B [**467**]{}, 479 (1996) \[arXiv:hep-ph/9511336\]. Z. Bern, L. J. Dixon and D. A. Kosower, Ann. Rev. Nucl. Part. Sci. [**46**]{}, 109 (1996) \[arXiv:hep-ph/9602280\]. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Phys. Lett. B [**394**]{}, 105 (1997) \[arXiv:hep-th/9611127\]. Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. B [**513**]{}, 3 (1998) \[arXiv:hep-ph/9708239\]. Z. Bern, L. J. Dixon and D. A. Kosower, JHEP [**0408**]{}, 012 (2004) \[arXiv:hep-ph/0404293\]. R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B [**715**]{}, 499 (2005) \[arXiv:hep-th/0412308\]. R. Britto, F. Cachazo, B. Feng and E. Witten, Phys. Rev. Lett. [**94**]{}, 181602 (2005) \[arXiv:hep-th/0501052\]. N. Arkani-Hamed and J. Kaplan, JHEP [**0804**]{}, 076 (2008) \[arXiv:0801.2385 \[hep-th\]\]. R. Britto, F. Cachazo and B. Feng, \[arXiv:hep-th/0412103\]. E. I. Buchbinder and F. Cachazo, JHEP [**0511**]{}, 036 (2005) \[arXiv:hep-th/0506126\]. Z. Bern, J. J. M. Carrasco, H. Johansson and D. A. Kosower, Phys. Rev. D [**76**]{}, 125020 (2007) \[arXiv:0705.1864 \[hep-th\]\]. F. Cachazo and D. Skinner, arXiv:0801.4574 \[hep-th\]. F. Cachazo, arXiv:0803.1988 \[hep-th\]. Z. Bern, M. Czakon, D. A. Kosower, R. Roiban and V. A. Smirnov, Phys. Rev. Lett. [**97**]{}, 181601 (2006) \[arXiv:hep-th/0604074\].
Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, M. Spradlin, C. Vergu and A. Volovich, arXiv:0803.1465 \[hep-th\]. D. B. Melrose, Nuovo Cim. [**40**]{}, 181 (1965). W. L. van Neerven and J. A. M. Vermaseren, Phys. Lett. B [**137**]{}, 241 (1984). Z. Bern, A. De Freitas and L. J. Dixon, JHEP [**0203**]{}, 018 (2002) \[arXiv:hep-ph/0201161\]. C. Anastasiou, Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Rev. Lett. [**91**]{}, 251602 (2003) \[arXiv:hep-th/0309040\]. Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D [**72**]{}, 085001 (2005) \[arXiv:hep-th/0505205\]. F. Cachazo, M. Spradlin and A. Volovich, Phys. Rev. D [**74**]{}, 045020 (2006) \[arXiv:hep-th/0602228\]. F. Cachazo, P. Svrcek and E. Witten, JHEP [**0409**]{}, 006 (2004) \[arXiv:hep-th/0403047\]. Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, Phys. Rev. D [**75**]{}, 085010 (2007) \[arXiv:hep-th/0610248\]. J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev, JHEP [**0701**]{}, 064 (2007) \[arXiv:hep-th/0607160\]. Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. B [**412**]{}, 751 (1994) \[arXiv:hep-ph/9306240\]. J. M. Drummond, G. P. Korchemsky and E. Sokatchev, Nucl. Phys. B [**795**]{}, 385 (2008) \[arXiv:0707.0243 \[hep-th\]\]. M. Czakon, Comput. Phys. Commun. [**175**]{}, 559 (2006) \[arXiv:hep-ph/0511200\]. T. Hahn, Comput. Phys. Commun. [**168**]{}, 78 (2005) \[arXiv:hep-ph/0404043\]. R. Akhoury, Phys. Rev. D [**19**]{}, 1250 (1979);\
J. C. Collins, Phys. Rev. D [**22**]{}, 1478 (1980);\
A. Sen, Phys. Rev. D [**24**]{}, 3281 (1981);\
G. P. Korchemsky, Phys. Lett. B [**220**]{}, 629 (1989);\
L. Magnea and G. Sterman, Phys. Rev. D [**42**]{}, 4222 (1990);\
G. P. Korchemsky and G. Marchesini, Phys. Lett. B [**313**]{} (1993) 433;\
S. Catani, Phys. Lett. B [**427**]{}, 161 (1998) \[arXiv:hep-ph/9802439\];\
G. Sterman and M. E. Tejeda-Yeomans, Phys. Lett. B [**552**]{}, 48 (2003) \[arXiv:hep-ph/0210130\]. Z. Bern, J. S. Rozowsky and B. Yan, Phys. Lett. B [**401**]{}, 273 (1997) hep-ph/9702424. L. F. Alday and J. Maldacena, JHEP [**0711**]{}, 068 (2007) \[arXiv:0710.1060 \[hep-th\]\]. R. C. Brower, H. Nastase, H. J. Schnitzer and C.-I. Tan, arXiv:0801.3891 \[hep-th\]. J. Bartels, L. N. Lipatov and A. Sabio Vera, arXiv:0802.2065 \[hep-th\]. L. F. Alday and J. Maldacena, JHEP [**0706**]{}, 064 (2007) \[arXiv:0705.0303 \[hep-th\]\]. A. Brandhuber, P. Heslop and G. Travaglini, Nucl. Phys. B [**794**]{}, 231 (2008) \[arXiv:0707.1153 \[hep-th\]\]. J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, Nucl. Phys. B [**795**]{}, 52 (2008) \[arXiv:0709.2368 \[hep-th\]\]. J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, arXiv:0712.1223 \[hep-th\]. J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, arXiv:0712.4138 \[hep-th\]. J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, arXiv:0803.1466 \[hep-th\].
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author:
- |
Wei Wang[^1]\
School of Mathematics and Statistics, Xi’an Jiaotong University,\
No. 28 Xianning West Rd., Xi’an, Shaanxi, P.R. China, 710049
title: '[A simple arithmetic criterion for graphs being determined by their generalized spectra]{}[^2]'
---
A graph $G$ is said to be determined by its generalized spectrum (DGS for short) if for any graph $H$, $H$ and $G$ are cospectral with cospectral complements implies that $H$ is isomorphic to $G$.
It turns out that whether a graph $G$ is DGS is closely related to the arithmetic properties of its walk-matrix. More precisely, let $A$ be the adjacency matrix of a graph $G$, and let $W =[e, Ae, A^2e,\cdots,A^{n-1}e]$ ($e$ is the all-one vector) be its *walk-matrix*. Denote by $\mathcal{G}_n$ the set of all graphs on $n$ vertices with $\det(W)\neq 0$. In \[Wang, Generalized spectral characterization of graphs revisited, The Electronic J. Combin., 20 (4),(2013), $\#P_4$\], the author defined a large family of graphs $$\mathcal{F}_n = \{G \in{\mathcal{G}_n}|~\frac{\det(W)}{2^{\lfloor\frac{n}{2}\rfloor}}~\mbox{is~square-free~and}~2^{n/2+1}\not|\det(W)\}$$ (which may have positive density among all graphs, as suggested by some numerical experiments) and conjectured every graph in $\mathcal{F}_n$ is DGS.
In this paper, we show that the conjecture is actually true, thereby giving a simple arithmetic condition for determining whether a graph is DGS.\
[**AMS classification: 05C50**]{}\
[**Keywords:**]{} [Spectra of graphs; Cospectral graphs; Determined by spectrum.]{}
Introduction
============
The spectra of graphs encodes a lot of combinatorial information about the given graphs, and thus has long been a useful tool in dealing with various problems in Graph Theory, even if they have nothing to do with graph spectra in the appearance.
A fundamental problem in the theory of graph spectra is: “What kinds of graphs are determined by the spectrum (DS for short)?" The problem dates back to more than 50 years ago and originates from Chemistry, which has received a lot of attention from researchers in recent years.
It was commonly believed that every graph is DS until the first counterexample was found by Collatz and Sinogowitz [@CS] in 1957. Since then, various constructions of cospectral graphs (i.e., graphs having the same spectrum) have been studied extensively and a lot of results are presented in literature. For example, Godsil and McKay [@GM] invented a powerful method call *GM-switching*, which can produce lots of pairs of cospectral graphs (with cospectral complements). An even more striking result was given by Schwenk [@Sch], stating that almost all trees are not DS.
However, less results are known about DS graphs, and it turns out that proving graphs to be DS is much more difficult than constructing cospectral graphs. Up to now, all the known DS graphs have very special properties, and the techniques (e.g., the eigenvalue interlacing technique) involved in proving them to be DS depend heavily on some special properties of the spectra of these graphs, and cannot be applied to general graphs. For the background and some known results about this problem, we refer the reader to [@DH; @DH1] and the references therein.
The above problem clearly depends on the spectrum concerned. In [@WX; @WX1], Wang and Xu gave a method for determining whether a graph $G$ is determined by its generalized spectrum (DGS for short, see Section 2 for details), which works for a large family of general graphs. The key observation is as follows:
Let $G$ and $H$ be two graphs that are cospectral with cospectral complements. Then there exists an orthogonal matrix $Q$ with $Qe=e$ ($e$ is the all-one matrix) such that $Q^TA(G)Q=A(H)$, where $A(G)$ and $A(H)$ are the adjacency matrices of $G$ and $H$, respectively. Moreover, the $Q$ can be chosen to be a rational matrix (under mild restrictions). Thus, if we can show that every rational orthogonal matrix $Q$ with $Qe=e$ such that $Q^TA(G)Q$ is a $(0,1)$-matrix with zero diagonal must be a permutation matrix, then $G$ is clearly DGS. This seems, at first glance, as difficult as the original problem. However, the authors managed to find some algorithmic methods to achieve this goal, by using some arithmetic properties of the walk-matrix associated with the given graph.
In Wang [@WWW], the author continued this line of research by showing that the DGS-property of a graph $G$ is actually closely related to whether the determinant of the walk-matrix $\det(W)$ is square-free (for odd primes). More precisely, the author defined a large family of graphs $\mathcal{F}_n$ (see Section 2 for details) that consists of graphs $G$ with $\frac{\det(W)}{2^{\lfloor\frac{n}{2}\rfloor}}$ (this is always an integer; see Section 3) being an odd square-free integer. Then he was able to show that for any graph $G\in{\mathcal{F}_n}$, if $Q$ is a rational orthogonal matrices $Q$ with $Qe=e$ such that $Q^TA(G)Q$ is a $(0,1)$-matrix with zero diagonal, then $2Q$ must be an integral matrix, and further proposed the following conjecture:
\[Wang [@Wang; @WWW]\]Every graph in $\mathcal{F}_n$ is DGS.
The main objective of this paper is to show that the above conjecture is actually true. Thus we have the following theorem.
\[Main\] Conjecture 1 is true.
The proof of above theorem is based on our previous work in [@WX; @WWW], and a new insight in dealing with the case $p=2$.
The paper is organized as follows: The next section, we review some previous results that will be needed in the sequel. In Section 3, we present the proof of Theorem \[Main\]. In Section 4, we give an extension of the Theorem 1.1. Conclusions and future work are given in Section 5.
Preliminaries
=============
For convenience of the reader, in this section, we will briefly review some known results from [@WX; @WWW].
Throughout, let $G=(V,E)$ be a simple graph with $(0,1)$-adjacency matrix $A=A(G)$. The *spectrum* of $G$ consists of all the eigenvalues (together with their multiplicities) of the matrix $A(G)$. The spectrum of $G$ together with that of its complement will be referred to as *the generalized spectrum* of $G$ in the paper (for some notions and terminologies in graph spectra, see [@CDS]).
For a given graph $G$, we say that $G$ is *determined by its spectrum* (DS for short), if any graph having the same spectrum as $G$ is necessarily isomorphic to $G$. (Of course, the spectrum concerned should be specified.)
The *walk-matrix* of a graph $G$, denoted by $W(G)$ or simply $W$, is defined as $[e, Ae, A^2e,\cdots,A^{n-1}e]$ ($e$ denotes the all-one vector henceforth). There is a well-known combinatorial interpretation of $W$, that is, the $(i,j)$-th entry of $W$ is the number of walks of $G$ starting from vertex $i$ with length $j-1$. It turns out that the arithmetic properties of $\det(W)$ is closely related to wether $G$ is DGS or not, as we shall see later.
A graph $G$ is called *controllable graph* if $W$ is non-singular (see also [@G]). Denote by $\mathcal{G}_n$ the set of all controllable graphs on $n$ vertices. The following theorem lies at the heart of our discussions.
Let $G\in{\mathcal{G}_n}$. Then there exists a graph $H$ that is cospectral with $G$ w.r.t. the generalized spectrum if and only if there exists a rational orthogonal matrix $Q$ such that $Q^TA(G)Q = A(H)$ and $Qe = e$.
Define $$\mathcal{Q}_G=\left\{ \begin{array}{rrr}Q~\mbox{is~a~rational~orthogonal}&\vline&Q^TAQ ~\mbox{is~a~ symmetric~(0,1)-matrix}\\
~\mbox{matrix with}~Qe=e~~~~&\vline&\mbox{with~zero~diagonal~~~~~~~~~~}
\end{array}\right\},$$ where $e$ is the all-one vector. We have the following theorem:
\[XXX\] Let $G\in\mathcal{G}_n$. Then $G$ is DS w.r.t. the generalized spectrum if and only if the set $\mathcal{Q}_G$ contains only permutation matrices.
By the theorem above, in order to determine whether a given graph $G\in\mathcal{G}_n$ is DGS or not w.r.t. the generalized spectrum, one needs to determine all $Q$’s in $\mathcal{Q}_G$ explicitly. At first glance, this seems to be as difficult as the original problem. However, we have managed to overcome this difficulty by introducing the following useful notion.
The *level* of a rational orthogonal matrix $Q$ with $Qe=e$ is the smallest positive integer $\ell$ such that $\ell Q$ is an integral matrix. Clearly, $\ell$ is the least common denominator of all the entries of the matrix $Q$. If $\ell = 1$, then clearly $Q$ is a permutation matrix.
Recall that an $n$ by $n$ matrix $U$ with integer entries is called *unimodular* if $\det(U) = \pm1$. The *Smith Normal Form* (SNF in short) of an integral matrix $M$ is of the form $diag(d_1,d_2,\cdots,d_n)$, where $d_i$ is the $i$-th elementary divisor of the matrix $M$ and $d_i|d_{i+1}~(i = 1,2,\cdots,n-1)$ hold. The following theorem is well known.
For every integral matrix $M$ with full rank, there exist unimodular matrices $U$ and $V$ such that $ M = USV = Udiag(d_1,d_2,\cdots,d_n)V$ , where $S$ is the SNF of the matrix $M$.
The following theorem shows that the level a rational orthogonal matrix $Q\in{\mathcal{Q}(G)}$ always divides the $n$-th elementary divisor of the walk-matrix.
\[L0\] Let $W$ be the walk-matrix of a graph $G\in{\mathcal{G}_n}$, and $Q\in{\mathcal{Q}(G)}$ with level $\ell$. Then we have $\ell|d_n$, where $d_n$ is the $n$-th elementary divisor of the walk-matrix $W$.
By the above theorem, $\ell$ is a divisor of $d_n$, and hence is a divisor of $\det(W)$. However, not all divisors of $\det(W)$ can be a divisor of $\ell$, as shown by the following theorem.
\[FF\] Let $G\in{\mathcal{G}_n}$. Let $Q \in{\mathcal{ Q}_G}$ with level $\ell$, and $p$ be an odd prime. If $p|\det(W)$ and $p^2\not|\det(W)$, then $p$ cannot be a divisor of $\ell$.
Motivated by above theorem, in [@WWW], the author introduced a large family of graphs (which might have density around 0.2, as suggested by some numerical experiments; see Section 4): $$\label{Defi}
\mathcal{F}_n = \{G \in{\mathcal{G}_n}|~\frac{\det(W)}{2^{\lfloor\frac{n}{2}\rfloor}}~\mbox{is~an odd~square-free~integer}\}.$$
As a simple consequence of Theorem \[FF\], we have
\[Old\] Let $G\in{\mathcal{F}_n}$. Let $Q\in{\mathcal{Q}_G}$ with level $\ell$. Then either $\ell=2^m$ for some integer $m\geq 0$.
Thus, if we can eliminate the possibility that $2\not|\ell$, then Theorem \[Main\] follows immediately. In the next section, we will show this is actually the true, which gives a proof of Theorem \[Main\].
Proof of Theorem 1.1
====================
In this section, we give the proof of Theorem \[Main\]. Before doing so, we need several lemmas below, the first few of which are taken from [@WWW]. In what follows, we will use the finite $\textbf{F}_p$ and ${\rm mod}~p$ (for a prime $p$) interchangeably.
\[NB\] Let $G\in{\mathcal{G}_n}$. If there is a rational orthogonal matrix $Q\in{\mathcal{Q}_G}$ with level $\ell$ such that $2|\ell$, then there exists a (0,1)-vector $u$ with $u\not\equiv 0~(\rm{mod}~2)$ such that $$\label{EE1}
u^TA^ku\equiv 0~({\rm mod}~4),~{\rm for}~ k=0,1,2,\cdots,n-1.$$ Moreover, $u$ satisfies $W^Tu\equiv~0~(\rm {mod}~ 2)$.
$Q\in{\mathcal{Q}_G}$ implies that $Q^TAQ = B$ for some $(0,1)$-matrix $B$ which is the adjacency matrix of a graph $H$. Let $\bar{u}$ be the $i$-th column of $\ell Q$ with $\bar{u}\not\equiv 0~(\rm{mod}~2)$ (such a $\bar{u}$ always exists by the definition of the level of $Q$). It follows from $Q^TA^kQ = B^k$ that $\bar{u}^TA^k\bar{u} = \ell^2(B^k)_{i,i}\equiv 0~(\rm mod~4)$. Let $\bar{u}= u+ 2v$, where $u$ is a $(0,1)$-vector and $v$ is an integral vector. Then $$\bar{u}^TA^k\bar{u} = u^TA^ku + 4u^TA^kv + 4v^TA^kv\equiv 0~ (\rm~mod~4).$$ Thus, Eq. (\[EE1\]) follows. To show the last assertion, notice that $Q^TA^kQ = B^k$ and $Qe = e$, it follows that $$Q^T[e,Ae,\cdots,A^{n-1}e]=[e,Be,\cdots,B^{n-1}e],$$ i.e., $W(G)^TQ=W(H)$ is an integral matrix. Thus $W(G)^T u \equiv 0~(\rm~mod~2)$ holds. This completes the proof.
\[Le1\]$e^TA^le$ is even for any integer $l\geq 1$.
We give a short proof for completeness. Let $A^l:=(b_{ij})$. Note that $$\begin{aligned}
e^TA^{l}e &=& {\rm Trace}(A^l) +\sum_{i\neq j}b_{ij}\\
& =& {\rm Trace}(A^l) + 2\sum_{1\leq i<j\leq n}b_{ij}\\
&\equiv& {\rm Trace}(A^l)~(\rm mod~2).\end{aligned}$$ Moreover, we have ${\rm Trace}(A^l) = {\rm Trace}(AA^{l-1}) =\sum_{i,j}a_{ij}\tilde{b}_{ij}=2\sum_{i<j}a_{ij}\tilde{b}_{ij}$, where$A^{l-1}:=(\tilde{b}_{ij})$. Thus the lemma follows.
\[H1\] ${\rm rank}_2(W)\leq \lceil\frac{n}{2}\rceil$, where ${\rm rank}_2(W)$ denotes the rank of $W$ over the finite field $\textbf{F}_2$.
Let $\det(W)=\pm 2^\alpha p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}$ be the standard prime decomposition of $\det(W)$. Then $\alpha\geq \lfloor\frac{n}{2}\rfloor$.
\[PO\] Let $G\in{\mathcal{F}_n}$. Then the SNF of $W$ is $$S=diag(\underbrace{1,1,\cdots,1}_{\lceil\frac{n}{2}\rceil},\underbrace{2,2,\cdots,2b}_{\lfloor\frac{n}{2}\rfloor}),$$ where the number of 2 in the diagonal of $S$ is $\lfloor\frac{n}{2}\rfloor$ and $b$ is an odd square-free integer. Moreover, we have $rank_2(W)=\lceil\frac{n}{2}\rceil$.
By the definition of $\mathcal{F}_n$, we have $\det(W)=\pm 2^{\lfloor\frac{n}{2}\rfloor}p_1p_2\cdots p_s$, where $p_i$ is an odd prime number for each $i$. Thus the SNF of $W$ can be written as $S=diag(1,1,\cdots,1,2^{l_1},2^{l_2},\cdots, 2^{l_t}b)$, where $b=p_1p_2\cdots p_s$ is an odd square-free integer. It follows from Lemma \[H1\] that $rank_2(W)\leq \lceil\frac{n}{2}\rceil$, i.e., $n-t\leq \lceil\frac{n}{2}\rceil$. Thus, we have $t\geq n-\lceil\frac{n}{2}\rceil=\lfloor\frac{n}{2}\rfloor$. Moreover, we have $l_1+l_2+\cdots+l_t=\lfloor\frac{n}{2}\rfloor$, since $\det(W)=\pm \det(S)$. It follows that $l_1=l_2=\cdots=l_t=1$ and $t=\lfloor\frac{n}{2}\rfloor$, and $rank_2(W)=n-t=\lceil\frac{n}{2}\rceil$.
\[MN\] Let $G\in{\mathcal{G}_n}$ and ${\rm rank}_2(W)=\lceil\frac{n}{2}\rceil$. Then any set of $\lfloor\frac{n}{2}\rfloor $ independent column vectors of $W$ (when $n$ is odd, the first column of $W$ is not included) forms a set of fundamental solutions to $W^Tx\equiv 0~(\rm mod~2)$.
We distinguish the following two cases.\
**Case 1**. $n$ is even. Let $W^TW=(w_{ij})_{n\times n}$, where $w_{ij}=e^TA^{i+j-2}e$. It follows from Lemma \[Le1\] and the fact $n$ is even that $W^TW\equiv ~0~(\rm mod~2)$. Notice that the dimension of the the solution space of $W^Tx\equiv 0~(\rm mod~2)$ is $n-{\rm rank}_2(W)=\frac{n}{2}$. Using the assumption ${\rm rank}_2(W)=\frac{n}{2}$ again, we know that any $\frac{n}{2}$ independent column vectors of $W$ forms a set of fundamental solutions to $W^Tx\equiv 0~(\rm mod~2)$.
**Case 2**. $n$ is odd. Let $\hat{W}$ be the matrix obtained from $W$ by deleting its first column. Similar to Case 1, we have $W^T\hat{W}\equiv 0~(\rm mod~2)$. Note the dimension of the solution space of $W^Tx\equiv 0~(\rm mod~2)$ is $n-{\rm rank}_2(W)=n-\frac{n+1}{2}=\frac{n-1}{2}$. Moreover, we have ${\rm rank}_2(\hat{W})\geq {\rm rank_2(W})-1=\frac{n-1}{2}$. Therefore, any $\frac{n-1}{2}$ columns from $\hat{W}$, or equivalently, from $W$ (except the first column), forms a set of fundamental solutions to $W^Tx\equiv 0~(\rm mod~2)$.
Combing Cases 1 and 2, the proof is complete.
\[SACH\] Let $P_G(x)=x^n+c_1x^{n-1}+\cdots+c_{n-1}x+c_n$ be the characteristic polynomial of graph $G$. Then $$c_i=\sum_{H\in{\mathcal{H}_i}}(-1)^{p(H)}2^{c(H)},$$ where $\mathcal{H}_i$ the set of elementary graphs with $i$ vertices in $G$; $p(H)$ is the number of components of $H$ and $c(H)$ is the number of cycles in $H$.
\[beauty\] Let $M$ be an integral symmetric matrix. If $M^2\equiv O~(\rm mod~2)$, then $Me\equiv 0~(\rm mod~2)$.
Let $M=(m_{ij})$. Then the $(i,i)$-th entry of $M^2$ is $\sum_{j=1}^{n}m_{ij}^2\equiv \sum_{j=1}^{n}m_{ij}\equiv 0~(\rm mod~2)$, which gives that $Me\equiv 0~(\rm mod~2)$.
Next, we fix some notations. Set $k=\lceil\frac{n}{2}\rceil$. Let $\tilde{W}$ be the matrix defined as follows: if $n$ is even, $\tilde{W}$ consists of the first $k$ columns of $W$, i.e., $\tilde{W}=[e,Ae,\cdots,A^{k-1}e]$ ; if $n$ is odd, $\tilde{W}$ consists of the first $k$ columns of $W$, except the first column, i.e., $\tilde{W}=[Ae,A^2e,\cdots,A^{k-1}e]$. Let $W_1=[e,A^2e,\cdots,A^{2n-2}e]$. Similarly, $\tilde{W}_1$ is defined as $\tilde{W}_1=[e,A^2e,\cdots,A^{2k-2}]$ if $n$ is even; and $\tilde{W}_1=[A^2e,A^4e,\cdots,A^{2k-2}]$ if $n$ is odd.
\[NBA\] Using notations above, we have
(i) ${\rm rank}_2(\tilde{W}_1)={\rm rank}_2(W_1)$; (ii) ${\rm rank}_2(\tilde{W})={\rm rank}_2(W)$.
We only prove the case that $n$ is even, the case that $n$ is odd can be proved in a similar way.
\(i) Let $P_G(x)=x^n+c_1x^{n-1}+\cdots+c_{n-1}x+c_n$ be the characteristic polynomial of graph $G$. By Sach’s Theorem \[SACH\], $c_i$ is even when $i$ is odd, since the number of cycles must be larger than or equal to one in an elementary subgraph of $G$ with odd number of vertices.
By Hamilton-Cayley’s Theorem, we have $$A^n+\sum_{i=1}^{n}c_iA^{n-i}\equiv A^n+\sum_{j=1}^{n/2}c_{2j}A^{n-2j}\equiv 0~(\rm mod~2).$$ It follows that $A^ne$ is the linear combinations of $e,A^2e,\cdots,A^{n-2}e$. Thus, $A^{n+m}e$ is the linear combinations of $e,A^2e,\cdots,A^{n-2}e$ for any $m\geq 1$. That is, the last $k$ columns of $W_1$ can be expressed as linear combinations of the first $k$ columns of $W_1$. So (i) follows.
\(ii) By (i), we have Let $A^{n}+c_2A^{n-2}+\cdots+c_{n-2}A^2+c_nI=0$. Let $M=A^{n/2}+c_2A^{(n-2)/2}+\cdots+c_{n-2}A+c_nI$. Then we have $$\begin{aligned}
M^2&\equiv&(A^{n/2}+c_2A^{(n-2)/2}+\cdots+c_{n-2}A+c_nI)^2\\
&\equiv &A^{n}+c_2^2A^{n-2}+\cdots+c_{n-2}^2A^2+c_n^2I\\
&\equiv &A^{n}+c_2A^{n-2}+\cdots+c_{n-2}A^2+c_nI\\
&\equiv& 0~(\rm mod~2).\end{aligned}$$ Then, by Lemma \[beauty\], we have $Me=A^{n/2}e+c_2A^{(n-2)/2}e+\cdots+c_{n-2}Ae+c_ne\equiv 0~(\rm mod~2)$. That is, $A^{n/2}e$ can be expressed as the linear combinations of the first $k$ columns of $W$, and the same is true for $A^{n/2+m}e$, for any $m\geq 0$. That is, any column of $W$ can be expressed as linear combinations of the first $k$ columns of $W$. So (ii) follows.
\[core\]Let $G\in{\mathcal{F}_n}$. Then we have ${\rm rank}_2(\frac{W^T\tilde{W}_1}{2})=k$ if $n$ is even; and ${\rm rank}_2(\frac{W^T\tilde{W}_1}{2})=k-1$ if $n$ is odd, where $k=\lceil n/2\rceil$.
We distinguish the following two cases:
\(i) $n$ is even. Write $W=[\tilde{W}_1,\tilde{W}_2]P$, where $P$ is a permutation matrix and $\tilde{W}_2=[Ae,A^3e,\cdots,A^{n-1}e]$. First we show that ${\rm rank}_2(\frac{W^TW}{2})=n$. Actually, notice that $G\in{\mathcal{F}_n}$, we have $\det(W)=\pm 2^{n/2}b$, where $b$ is an odd integer. It follows that $\det(W^TW)= 2^nb^2$, i.e., $\det(\frac{W^TW}{2})=b^2$. Note that $b$ is odd, the assertion follows immediately. Now we have $\frac{W^TW}{2}=[\frac{W^T\tilde{W}_1}{2},\frac{W^T\tilde{W}_2}{2}]P$. It follows that the column vectors of the matrix $\frac{W^T\tilde{W}_1}{2}$ are linearly independent (since $\frac{W^TW}{2}$ has full rank), over $\textbf{F}_2$.
\(ii) $n$ is odd. Construct a new matrix $\hat{W}=[2e,\tilde{W}_1,\tilde{W}_2]$. Notice that $\frac{W^T\hat{W}}{2}$ is now always an integral matrix. Since $\det(W)=2^{(n-1)/2}b$ $(b$ is odd), we have $\det(W^T\hat{W})=\det(W)\det(\hat{W})=\pm 2\det^2(W)=2^{n}b^2$ (since $\det(\hat{W})=\pm 2\det(W)$). It follows that $\frac{W^T\hat{W}}{2}=[W^Te,\frac{W^T\tilde{W}_1}{2},\frac{W^T\tilde{W}_2}{2}]$ has full rank $n$. Therefore, ${\rm rank}_2(\frac{W^T\tilde{W}_1}{2})$ equals the number of columns of $\tilde{W}_1$, which is $k-1$ when $n$ is odd.
Combining Cases (i) and (ii), the lemma is true. The proof is complete.
The following lemma lies at the heart of the proof of Theorem \[Main\].
\[L1\] Let $G\in{\mathcal{F}_n}$. Let $Q\in{\mathcal{Q}_G}$ be a rational orthogonal matrix with level $\ell$, then $2\not|\ell$.
We prove the lemma by contradiction. Suppose on the contrary, $2|\ell$. It follows from Lemma \[NB\] that there exists a vector $u$ such that Eq. (\[EE1\]) holds. Note that $u$ is a solution to the system of linear equations $W^Tx\equiv 0~(\rm mod~2)$. Since $G\in{\mathcal{F}_n}$, it follows from Lemma \[PO\] that ${\rm rank}_2(W)=\lceil\frac{n}{2}\rceil$. According to Lemmas \[MN\] and \[NBA\], we can assume that $\{A^{i_{1}}e,A^{i_{2}}e,\cdots,A^{i_{k}}\}$ is a set of fundamental solutions to $W^Tx\equiv 0~(\rm mod~2)$, where $k:=\lfloor n/2\rfloor$, and $i_1=0,i_2=1,\cdots,i_k=n/2-1$ if $n$ is even and $i_1=1,i_2=2,\cdots,i_k=(n-1)/2$ if $n$ is odd.
Write $\tilde{W}=[A^{i_1}e,A^{i_2}e,\cdots,A^{i_k}e]$. Then $u$ can be written as the linear combinations of the column vectors of $\tilde{W}$, i.e., there is a vector $v\not \equiv 0~(\rm mod~2)$ such that $u\equiv\tilde{W}v~(\rm mod~2)$. So we have $u=\tilde{W}v+2\beta$ for some integral vector $\beta$. It follows that $$\begin{aligned}
u^TA^lu&=&(\tilde{W}v+2\beta)^TA^l(\tilde{W}v+2\beta)\\
&=&v^T\tilde{W}^TA^l\tilde{W}v+2v^T\tilde{W}^TA^l\beta+2\beta^TA^l\tilde{W}v+4\beta^TA^l\beta\\
&=&v^T\tilde{W}^TA^l\tilde{W}v+4v^T\tilde{W}^TA^l\beta+4\beta^TA^l\beta\\
&\equiv&v^T\tilde{W}^TA^l\tilde{W}v~(\rm mod~4).\end{aligned}$$
By Eq. (\[EE1\]), we have $v^T\tilde{W}^TA^l\tilde{W}v\equiv 0~(\rm mod~4)$, for $l=0,1,2,\cdots,n-1$. Notice that $$\begin{aligned}
\tilde{W}^TA^l\tilde{W}=\left[\begin{array}{cccc}
e^TA^{2i_1+l}e & e^TA^{i_1+i_2+l}e &\cdots & e^TA^{i_1+i_k+l}e\\
e^TA^{i_1+i_2+l}e & e^TA^{2i_2+l}e &\cdots & e^TA^{i_2+i_k+l}e\\
\vdots&\vdots&\ddots&\vdots\\
e^TA^{i_1+i_k+l}e & e^TA^{i_2+i_k+l}e &\cdots & e^TA^{2i_k+l}e
\end{array}\right]\end{aligned}$$
Let $M:=\tilde{W}^TA^l\tilde{W}$. A key observation is that $M$ is always a symmetric matrix with every entry being a multiple of two. Actually, this follows from Lemma \[Le1\]. But we have to distinguish two cases: (i) when $n$ is even, Lemma \[Le1\] always can be applied except the case that $i_1=l=0$. While in this case, the $(1,1)$-entry of $M$ is $e^Te=n$ which is even; (ii) when $n$ is odd, we have $i_1=1$, thus applying Lemma \[Le1\] directly leads to the desired assertion.
Let $v=(v_1,v_2,\cdots,v_k)^T$. Then we have $$M_{ij}v_iv_j+M_{ji}v_jv_i=2M_{ij}v_iv_j\equiv 0~(\rm mod~4),$$ for $i\neq j$, since $M_{ij}$ is even by the above discussions. Therefore, we have $$\begin{aligned}
v^T\tilde{W}^TA^l\tilde{W}v&=&\sum_{i,j}M_{ij}v_iv_j\\
&\equiv &(e^TA^{2i_1+l}e)v_1^2+(e^TA^{2i_2+l}e)v_2^2+\cdots+(e^TA^{2i_k+l}e)v_k^2\\
&\equiv& (e^TA^{2i_1+l}e)v_1+(e^TA^{2i_2+l}e)v_2+\cdots+(e^TA^{2i_k+l}e)v_k\\
&\equiv& [e^TA^{2i_1+l}e,e^TA^{2i_2+l}e,\cdots,e^TA^{2i_k+l}e]v\\
&\equiv& 0~~(\rm mod~4),\end{aligned}$$ for $l=0,1,\cdots,n-1$. The second congruence equation follows since $(e^TA^{2i_j+l}e)v_j^2\equiv (e^TA^{2i_j+l}e)v_j~(\rm mod~4)$ for every $1\leq j\leq k$.
Let $\tilde{W}'$ be an $n$ by $k$ matrix defined as follows: $$\begin{aligned}
\tilde{W}'&:=&\left[\begin{array}{cccc}
e^TA^{2i_1}e & e^TA^{2i_2}e &\cdots & e^TA^{2i_k}e\\
e^TA^{2i_1+1}e & e^TA^{2i_2+1}e &\cdots & e^TA^{2i_k+1}e\\
\vdots&\vdots&\ddots&\vdots\\
e^TA^{2i_1+n-1}e & e^TA^{2i_2+n-1}e &\cdots & e^TA^{2i_k+n-1}e
\end{array}\right]\\
&=& \left[\begin{array}{c}
e^T\\
e^TA\\
\vdots\\
e^TA^{n-1}
\end{array}\right]
\left[\begin{array}{cccc}
A^{2i_1}e & A^{2i_2}e & \cdots &A^{2i_k}e
\end{array}\right]\\
&=& W^T\left[\begin{array}{cccc}
A^{2i_1}e & A^{2i_2}e & \cdots &A^{2i_k}e
\end{array}\right]\\
&=&W^T\tilde{W}_1,
\end{aligned}$$ where $\tilde{W}_1=[A^{2i_1}e ,A^{2i_2}e, \cdots,A^{2i_k}e]$.
Thus, we have $W^T\tilde{W}_1v\equiv 0~(\rm mod~4)$. Notice that $\frac{W^T\tilde{W}_1}{2}$ is always an integral matrix according to Lemma \[Le1\] and the definition of $\tilde{W}_1$. It follows that $$\frac{W^T\tilde{W}_1}{2}v\equiv 0~(\rm mod~2).$$
However, by Lemma \[core\], ${\rm rank}_2(\frac{W^T\tilde{W}_1}{2})=k$ and hence, $\frac{W^T\tilde{W}_1}{2}$ has full column rank. It follows that $v\equiv 0~(\rm mod~2)$; a contradiction. This completes the proof.
Now, we are ready to present the proof of Theorem \[Main\].
Let $G\in{\mathcal{F}_n}$. Let $Q\in{\mathcal{Q}_G}$ with level $\ell$. Then by Theorem \[Old\], we have $p\not|\ell$ for any odd prime $p$. By Lemma \[L1\], we have $2\not|\ell$. It follows that $\ell=1$ and hence, $Q$ is a permutation matrix. By Theorem \[XXX\], $G$ is DGS. The proof is complete.
An extension beyond Theorem 1.1
===============================
In the previous section, we have shown that graphs with $\frac{\det{W}}{2^{\lfloor\frac{n}{2}\rfloor}}$ being square-free is always DGS. Notice graphs with above property has the following SNF: $$S=diag(\underbrace{1,1,\cdots,1}_{\lceil\frac{n}{2}\rceil},\underbrace{2,2,\cdots,2b}_{\lfloor\frac{n}{2}\rfloor}),$$ where $b$ is an odd square-free integer. A natural question is: Can we enlarge the family of graph $\mathcal{F}_n$?
Generally, we cannot expect an affirmative answer to this question if we allow $b$ is not square-free. In [@WWW], the author have given an example of non-DGS graph of order 12 with $\det(W)=2^6\times 3^2\times 157\times1361\times 2237$, which shows Theorem 1.1 is best possible in the sense that we cannot guarantee that $G$ is DGS if $\frac{\det{W}}{2^{\lfloor\frac{n}{2}\rfloor}}$ has prime divisor with exponent larger than one.
However, based on the proof in Lemma \[L1\], we are able to give a method to determine DGS-property for graphs that are not in $\mathcal{F}_n$. Next, we try to give a method for determine the DGS-property for graphs whose walk-matrices have the following SNF: $$\label{SNF}
diag(\underbrace{1,1,\cdots,1}_{\lceil\frac{n}{2}\rceil},\underbrace{2^{l_1},2^{l_2},\cdots,2^{l_t}b}_{\lfloor\frac{n}{2}\rfloor})$$
\[LK1\] Let $G\in{\mathcal{G}_n}$. Suppose that ${\rm rank}_2(W)=\lceil\frac{n}{2}\rceil$ and the SNF of $W$ is given as in Eq. (\[SNF\]), where $b$ is a square-free integer. Let $Q\in{\mathcal{Q}_G}$ be a rational orthogonal matrix with level $\ell$. Let $W_1:=[e,A^2e,A^4e,\cdots,A^{2n-2}e]$. If $$\{x|\frac{W^TW_1}{2}x\equiv~0~({\rm mod}~2)\}\subset \{x|Wx\equiv 0~({\rm mod}~2)\},$$ then $2\not|\ell$.
The proof is similar to that of Lemma \[L1\]. A sketch.
Suppose on the contrary $2|\ell$. It follows from Lemma \[NB\] that there exists a vector $u$ such that Eq. (\[EE1\]) holds. Note that $u$ is a solution to the system of linear equations $W^Tx\equiv 0~(\rm mod~2)$. According to Lemma \[MN\], any solution of $W^Tx\equiv 0~(\rm mod~2)$, can be written as linear combinations of the column vectors of $W$ (when $n$ is odd, replace $W$ with $\hat{W}$). It follows that $u$ can be written as the linear combinations of the column vectors of $W$, i.e., there is a vector $v\not \equiv 0~(\rm mod~2)$ such that $u\equiv Wv~(\rm mod~2)$.
Using the similar arguments as in the remaining proof of Lemma \[L1\], we have $W^TW_1v\equiv 0~(\rm mod~4)$. Notice that $W^TW_1\equiv 0~(\rm mod~2)$. We have $\frac{W^TW_1}{2}v\equiv 0~(\rm mod~2)$, which implies that $u=Wv\equiv0~(\rm mod~2)$ by the assumption of the lemma; a contradiction. Therefore $2\not|\ell$. This completes the proof.
Combining the above lemma and Theorem 2.5, we have the following theorem.
Let $G\in{\mathcal{G}_n}$. Suppose that ${\rm rank}_2(W)=\lceil\frac{n}{2}\rceil$ and the SNF of $W$ is given as in Eq. (\[SNF\]), where $b$ is a square-free integer. Then $G$ is DGS.
We give an example as an illustration. Let the adjacency matrix of graph $G$ be given as follows:
[$$A=\left[\begin{array}{cccccccccccccccccccc}
{0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1}\\
{1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1}\\
{ 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0}\\
{1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1}\\
{ 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1}\\
{1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1}\\
{ 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1}\\
{1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1}\\
{0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0}\\
{0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0}\\
{1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0}\\
{1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0}\\
{1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0}\\
{1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0}\\
{ 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0}\\
{0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0}\\
{1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1}\\
{1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0}\\
{1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1}\\
{1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0}
\end{array}\right].$$]{} It can easily be computed by using Mathematica 5.0 that $\det(W)=-2^{13}b$, where $b=7\times 11\times 383\times 210857\times
231734663160530708115251000501057$. The SNF of $W$ is as follows: $$diag(\underbrace{1,1,\cdots,1}_{10}\underbrace{2,2,\cdots,2,2^2,2^2,2^2}_{10}b).$$ Moreover, it can be verified that Eq. (4) also holds. Thus, $G$ is DGS according to Theorem 4.2.
Conclusions and future work
===========================
In this paper, we have given a simple arithmetic criterion for determining whether a graph $G$ is DGS, in terms of whether the determinant of walk-matrix $\det(W)$ divided by $2^{\lfloor\frac{n}{2}\rfloor}$ is an odd square-free. It is noticed that the definition of $\mathcal{F}_n$ is so simple that the membership of a graph can easily be checked.
We have performed a series of numerical experiments to see how large the family of graphs $\mathcal{F}_n$ is. The graphs are generated randomly independently from the probability space $\mathcal{G}(n,\frac{1}{2})$ (see e.g. [@B]). At each time, we generated 1,000 graphs randomly, and counted the number of graphs that are in $\mathcal{F}_n$. Table 1 records one of such experiments (note the results may be varied slightly at each run of the algorithm). The first column is the order $n$ of the graphs generated varying from 10 to 50. The second column records the number of graphs that are belonged to $\mathcal{F}_n$ among the randomly generated 1,000 graphs, and the third column is the corresponding fractions.
[c|c|c]{}\
$n$ & \# Graphs in $\mathcal{F}_n$ & The Fractions\
10 & 211 & 0.211\
15 & 201 & 0.201\
20 & 213 &0.213\
25 & 216 &0.216\
30 & 233 &0.233\
35 & 229 &0.229\
40 & 198 &0.198\
45 & 202 &0.202\
50 & 204 &0.204\
We can see from Table 1 that graphs in $\mathcal{F}_n$ has a density around 0.2. It would be an interesting future work to show that this is actually the case.
[22]{}
B. Bollobás, Modern Graph Theory, Springer-Verlag, NewYork, 2002.
L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg, 21 (1957), pp. 63-77
D. M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, NewYork, 1982.
E. R. van Dam, W. H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl., 373 (2003) 241-272.
E. R. van Dam, W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Mathematics, 309 (2009) 576-586.
C.D. Godsil, B.D. McKay, Constructing cospectral graphs, Aequation Mathematicae, 25 (1982) 257-268.
C.D. Godsil, Controllable subsets in graphs, Annals of Combinatorics, 16 (2012) 733-744.
A.J. Hoffman, R. R. Singleton, Moore graphs with diameter 2 and 3, IBM Journal of Research and Development 5 (4) (1960) 497-504.
A.J. Schwenk, Almost all trees are cospectral, in: F. Harary (Ed.), New Directions in the Theory of Graphs, Academic Press, NewYork, 1973, pp. 275-307.
W. Wang, C. X. Xu, A sufficient condition for a family of graphs being determined by their generalized spectra, European J. Combin., 27 (2006) 826-840.
W. Wang, C.X. Xu, An excluding algorithm for testing whether a family of graphs are determined by their generalized spectra, Linear Algebra and its Appl., 418 (2006) 62-74.
W. Wang, On the Spectral Characterization of Graphs, Phd Thesis, Xi’an Jiaotong University, 2006. (In Chinese)
W. Wang, Generalized spectral characterization revisited, The Electronic J. Combin., 20 (4) (2013), $\#$ P4.
[^1]: The corresponding author. E-mail address:wang$\_$weiw@163.com.
[^2]: This work is supported by the National Natural Science Foundation of China (No. 11471005)
|
---
abstract: 'We study single hole dynamics in the bilayer Heisenberg and Kondo Necklace models. Those models exhibit a magnetic order-disorder quantum phase transition as a function of the interlayer coupling $J_{\bot}$. At strong coupling in the disordered phase, both models have a single-hole dispersion relation with band maximum at ${\pmb p} = (\pi,\pi)$ and an effective mass at this ${\pmb p}-$point which scales as the hopping matrix element $t$. In the Kondo Necklace model, we show that the effective mass at ${\pmb p} = (\pi,\pi)$ remains finite for all considered values of $J_{\bot}$ such that the strong coupling features of the dispersion relation are apparent down to weak coupling. In contrast, in the bilayer Heisenberg model, the effective mass diverges at a finite value of $J_{\bot}$. This divergence of the effective mass is unrelated to the magnetic quantum phase transition and at weak coupling the dispersion relation maps onto that of a single hole doped in a planar antiferromagnet with band maximum at ${\pmb p} = (\pi/2,\pi/2)$. We equally study the behavior of the quasiparticle residue in the vicinity of the magnetic quantum phase transition both for a mobile and static hole. In contrast to analytical approaches, our numerical results do not unambiguously support the fact that the quasiparticle residue of the static hole vanishes in the vicinity of the critical point. The above results are obtained with a generalized version of the loop algorithm to include single hole dynamics on lattice sizes up to $20 \times 20$.'
author:
- 'C. Brünger and F.F. Assaad'
title: 'Single hole dynamics in the Kondo Necklace and Bilayer Heisenberg models on a square lattice.'
---
Introduction
============
The modeling of heavy fermion systems is based on an array of localized spin degrees of freedom coupled antiferromagnetically to conduction electrons. Those models show competing interactions which lead to magnetic quantum phase transitions as a function of the antiferromagnetic exchange interaction $J$. Kondo screening of the localized spins, dominant at large $J$, favors a paramagnetic heavy fermion ground state, where the localized spins participate in the Luttinger volume. In contrast, the RKKY interaction favors magnetic ordering and is dominant at small values of $J$. There has recently been renewed interest concerning the understanding this quantum phase transition. In particular, recent Hall experiments [@Paschen04] suggest the interpretation that in the vicinity of the quantum phase transition the localized spins drop out of the Luttinger volume. Starting from the paramagnetic phase, this transition from a large to small Fermi surface should coincide with a effective mass divergence of the heavy fermion band.
Motivated by the above, we consider here a very simplified situation namely that of a doped hole in the Kondo insulating state as realized by the Kondo necklace and related models. Although this is not of direct relevance for the study of the Fermi surface, it does allow us to investigate the form of the quasiparticle dispersion relation from strong to weak coupling for a variety of models. Our aim here is two fold. On one hand we address the question of the divergence of the effective mass as a function of coupling for different models, and on the other hand the fate of the quasiparticle residue in the vicinity of the quantum phase transition.
The KLM emerges from the periodic Anderson model (PAM), where we have localized orbitals (LO) with on-site Hubbard interaction $U_{f}$ and extended orbitals (EO), which form a conduction band with dispersion $\varepsilon({\pmb p})=-2t\left(\cos p_{x} + \cos p_{y}\right)$. The overlap between the LOs and the EOs within each unit cell is described by the hybridization matrix element $V$. For large $U_{f}$ charge fluctuations on the localized orbitals becomes negligible and the PAM maps via the Schrieffer-Wolff transformation onto the KLM [@Schrieffer66; @Tsunetsugu97]: $$\begin{aligned}
\label{KLM}
\hat{H}_{\text{KLM}}=\sum_{{\pmb p},\sigma}\varepsilon ({\pmb p})
\hat{c}^{\dagger}_{{\pmb p}\sigma}\hat{c}_{{\pmb p}\sigma}
+\mathcal{J}\sum_{i}\hat{{\pmb S}}^{c}_{{\pmb i}}\hat{{\pmb S}}^{f}_{{\pmb i}}\,\, .\end{aligned}$$ Here $\hat{{\pmb S}}^{c}_{{\pmb i}}$ and $\hat{{\pmb S}}^{f}_{{\pmb i}}$ are spin $1/2$ operators for the extended orbitals and the localized orbitals respectively. In the first term, which represents the hopping processes, the fermionic operators $\hat{c}^{\dagger}_{{\pmb p}\sigma}$ ($\hat{c}_{{\pmb p}\sigma}$ ) create (annihilate) electrons in the conduction band with wave vector ${\pmb p}$ and $z$-component of spin $\sigma$. At half-filling – one conduction electron per localized spin – the two-dimensional KLM is an insulator and shows a magnetic order-disorder quantum phase transition at a critical value of $\mathcal{J}_{c}/t=1.45\pm 0.05$ [@Capponi00].
![\[lattice\][(a) Isotropic bilayer Heisenberg model. (b) Kondo Necklace model, that is related to the $U$KLM. In both cases the system dimerizes for large $J_{\bot}$, so that the AF ordering breaks down.]{}](Pic/BHM.eps "fig:"){height="3cm"} ![\[lattice\][(a) Isotropic bilayer Heisenberg model. (b) Kondo Necklace model, that is related to the $U$KLM. In both cases the system dimerizes for large $J_{\bot}$, so that the AF ordering breaks down.]{}](Pic/KNM.eps "fig:"){height="3cm"}
By taking into account an additional Coulomb repulsion $U$ between electrons within the conduction band, one obtains a modification of the KLM, the $U$KLM:
$$\begin{aligned}
\hat{H}_{U\text{KLM}}
&=&\sum_{{\pmb p},\sigma}\varepsilon ({\pmb p})
\hat{c}^{\dagger}_{{\pmb p}\sigma}\hat{c}_{{\pmb p}\sigma}
+\mathcal{J}\sum_{i}\hat{{\pmb S}}^{c}_{{\pmb i}}\hat{{\pmb S}}^{f}_{{\pmb i}}
\nonumber\\
& &+U\sum_{{\pmb i}}\left(\hat{n}_{{\pmb i}\uparrow}-\tfrac{1}{2}\right)
\left(\hat{n}_{{\pmb i}\downarrow}-\tfrac{1}{2}\right)\,\, .\end{aligned}$$
Here, $\hat{n}_{{\pmb i}\sigma}=\hat{c}^{\dagger}_{{\pmb i}\sigma}\hat{c}_{{\pmb i}\sigma}$ is the density operator for electrons with spin $\sigma$ in the conduction band. The additional Coulomb repulsion displaces the quantum critical point towards smaller value of $\mathcal{J}_{c}/t$. However the physics, in particular the single hole dynamics, remains unchange [@Feldbacher02]. This allows us to take the limit $U/t \rightarrow \infty $ to map the UKLM onto a Kondo necklace model (KNM) which we write as: $$\begin{aligned}
\label{BHM_model}
\hat{H}
& = & J_{\bot}\sum_{{\pmb i}}\hat{{\pmb S}}^{(1)}_{{\pmb i}}\hat{{\pmb S}}^{(2)}_{{\pmb i}} +
\sum_{\langle{\pmb i}{\pmb j}\rangle}\sum_{m}J^{(m)}_{\|}
\hat{{\pmb S}}^{(m)}_{{\pmb i}}\hat{{\pmb S}}^{(m)}_{{\pmb j}}.
\label{starth}\end{aligned}$$ Here $\hat{{\pmb S}}^{(m)}_{{\pmb i}}$ is a spin $1/2$ operator, which acts on a spin degree of freedom at site ${\pmb i}$. $J^{(m)}_{\|}$ stands for the intralayer exchange and the upper index $m=1,2$ labels the two different layers. The interlayer exchange, formerly the AF coupling $\mathcal{J}$ between LOs and EOs, is now characterized by $J_{\bot}$. Clearly, since we have motivated the KNM from a strong coupling limit of the UKLM, we have to set: $$J^{(1)}_{\|}\equiv J_{\|} \quad J^{(2)}_{\|}=0
\quad\text{for the KNM.}$$
The above models all have in common that the only interaction between the localized spins stems from the RKKY interaction. This in turn leads to the fact that at $\mathcal{J} =0$ for the KLM and UKLM or $J_{\bot} =0$ for the KNM the ground state is macroscopically degenerate. To lift the [*pathology*]{} we finally consider a Bilayer Heisenberg Model (BHM) in which an independent exchange between the localized spins is explicitly included in the Hamiltonian. Hence we will equally consider an Isotropic BHM which takes the form of Eq. (\[BHM\_model\]) with: $$J^{(1)}_{\|}=J^{(2)}_{\|}\equiv J_{\|}
\quad\text{for the isotropic BHM.}$$ Both the KNM and BHM systems are sketched in FIG. \[lattice\].
The main results and organization of the paper are the following. In section \[methods\] we give a short overview of the quantum Monte Carlo (QMC) method. We use a generalization of the loop algorithm which allows for the calculation of the imaginary time Green’s function of the doped hole [@Brunner00]. Dynamical information is obtained with a stochastic Maximum Entropy method [@Beach04; @Sandvik98]. In the first part of section \[dynamics\] we present our results for the spin dynamics. This includes the determination of the quantum critical point for the isotropic BHM as well as the Kondo Necklace model (KNM) by QMC methods. In the second part of that section we analyze the single particle spectral function. It turns out, that there are significant differences between the models. We can identify two classes of models: In the isotropic BHM the dispersion is continously deformed with decreasing interplanar coupling $J_{\bot}/J_{\|}$ resulting in a displacement of the maximum from ${\pmb p}=(\pi,\pi)$ to ${\pmb p}=(\tfrac{\pi}{2},\tfrac{\pi}{2})$. In other words, the effective mass – as defined by the inverse curvature of the quasiparticle dispersion relation – at ${\pmb p}=(\pi,\pi)$ diverges at a finite value of the interplanar coupling. This divergence of the effective mass is not related to the magnetic order-disorder transition. In contrast, in the KLM related models, UKLM and KNM, the maximum of the quasiparticle dispersion relation is pinned at ${\pmb p}=(\pi,\pi)$ irrespective of the value of the interplanar coupling. In those models the effective mass at ${\pmb p}=(\pi,\pi)$ grows as a function of decreasing interplanar coupling, but remains finite.
In section \[qpr\] we turn to the analysis of the quasi particle residue (QPR) across the quantum phase transition. To gain intuition, we first carry out an approximate calculation in the lines of Ref. [@Sushkov00]. The physics of the spin system may be solved in the framework of a bond mean-field calculation. Here, the disordered phase is described in terms of a condensate of singlets between the planes and gaped spin 1 excitations (magnons). At the critical point the magnons condense at the AF wave vector thus generating the static antiferromagnetic order. Within this framework one can compute the coupling of the mobile hole with the magnetic fluctuations and study the hole dynamics within a self-consistent Born approximation. The result of the calculation shows that the quasiparticle weight at wave vectors on the magnetic Brillouin zone \[$\epsilon({\pmb p}) = \epsilon({\pmb p} + {\pmb Q}) $ with $\pmb{Q} = (\pi,\pi)$ \] vanish as the square root of the spin gap. In contrast the QMC determination of the quasiparticle residue on lattices up to $20 \times 20$ for static and dynamical holes does not unambiguously support this point of view.
Numerical Methods {#methods}
=================
We use the world line QMC method with loop updates [@Evertz97] to investigate the physics of the BHM and KNM. To investigate the spin dynamics we compute both the spin stiffness as well as the dynamical spin structure factor. Our analysis of the single hole dynamics is based on the calculation of the imaginary time Green’s function. Analytical continuation with the use of the stochastic Maxent Method provides the spectral function and the quasiparticle residue is extracted from the asymptotic behavior of the imaginary time Green’s function. Below, we discuss in more details the calculation of each observables.
### Spin Stiffness {#spin-stiffness .unnumbered}
To probe for long-ranged magnetic order we introduce a continuous twist in spin space which, when cumulated along the length $L$ along (e.g.) the $x$-axis, amounts to a twist of angle $\phi$ around a certain spin axis ${\pmb e}$. This means thus the boundary conditions read: $\hat{{\pmb S}}_{{\pmb i}+L{\pmb e}_{x}}=R\left[{\pmb e},\phi\right]\hat{{\pmb S}}_{{\pmb i}}$, where $R\left[{\pmb e},\phi\right]$ is a matrix describing an $SO(3)$ rotation around the axis ${\pmb e}$ by the angle $\phi$. The spin stiffness is then defined as $$\begin{aligned}
\rho_{s} = \left. -\frac{1}{L^{d-2}}\frac{1}{\beta}\frac{\partial^{2}}{\partial
\phi^{2}}\ln Z(\phi) \right|_{\phi=0}\end{aligned}$$ with $\beta$ as inverse temperature, $L$ as the linear size of the system, $d$ the dimensionality and $ Z(\phi) $ the twist dependent partition function. In the presence of long-range order $\rho_{s}$ takes a finite value and in a disordered phase it vanishes.
Within the world-line algorithm, the spin stiffness is related to the winding number $\mathcal{W}_{x}$ of the world line configurations along the axis of cumulatively twisted spins (e.g. $x$-axis). In particular, in the limit $\Delta\tau\to 0$ it takes the simple form $$\begin{aligned}
\rho_{s}=\frac{1}{L^{d}}\frac{1}{\beta}\mathcal{W}^{2}_{x}.\end{aligned}$$
### Spin Correlations {#spin-correlations .unnumbered}
Within the QMC it is easy to obtain the spin correlations $\langle S^{z}_{{\pmb i}}(\tau)S^{z}_{{\pmb j}}(0)\rangle$ in real space and imaginary time $\tau$, where the imaginary time evolution of the spin operator reads $S^{z}_{{\pmb q}}(\tau)=e^{\tau\hat{H}}S^{z}_{{\pmb q}}e^{-\tau\hat{H}}$. Its representation in momentum space is related to the dynamical spin susceptibility $S({\pmb q},\omega)$ via: $$\begin{aligned}
\langle S^{z}_{{\pmb q}}(\tau)S^{z}_{-{\pmb q}}(0)\rangle = \frac{1}{\pi}
\int d\omega e^{-\tau\omega}S({\pmb q},\omega).\end{aligned}$$ By using the Stochastic Maximum Entropy (ME) method [@Beach04] we can extract the dynamical spin susceptibility. For large $\tau$ the spin correlation function is dominated by the lowest excitation: $$\begin{aligned}
\underset{\tau\rightarrow\infty}{\lim}
\langle S^{z}_{{\pmb q}}(\tau)S^{z}_{-{\pmb q}}(0)\rangle
\propto e^{-\Omega({\pmb q})\tau}\end{aligned}$$ where $\Omega({\pmb q})$ stands for momentum dependent gap to the first spin excitation. Thus, we obtain the gap energy $\Delta$ from the asymptotic behavior of the spin correlations: $\Delta \equiv \min \left[ \Omega({\pmb q}) \right]$.
### The Green’s Function {#the-greens-function .unnumbered}
To incorporate the dynamics of a single hole into the KNM and BHM, we consider the $tJ$-model $$\begin{aligned}
\hat{H}_{tJ}&=&
\mathcal{P}_{S}\Big{[}- \sum_{\langle {\pmb i}{\pmb j} \rangle,\sigma} t_{\pmb{i}\pmb{j}}
(\hat{c}^{\dagger}_{{\pmb i}\sigma}\hat{c}_{{\pmb j}\sigma}
+\hat{c}^{\dagger}_{{\pmb j}\sigma}\hat{c}_{{\pmb i}\sigma})
\nonumber\\
&&+\displaystyle\sum_{\langle {\pmb i}{\pmb j} \rangle}
J_{\pmb{i}\pmb{j}}\big{\{}\hat{{\pmb S}}_{{\pmb i}}\hat{{\pmb S}}_{{\pmb j}}
-\tfrac{1}{4}\hat{n}_{{\pmb i}}\hat{n}_{{\pmb j}}\big{\}}\Big{]}\mathcal{P}_{S}
\label{tj}\end{aligned}$$ which describes the more general case of arbitrary filling. Here, $ {\pmb i} $ and ${\pmb j}$ denote lattice sites of the bilayer BHM, $t_{{\pmb i}{\pmb j}}$ the hopping amplitude, $J_{{\pmb i}{\pmb j}} $ the exchange, $\hat{n}_{{\pmb j}}=\hat{c}^{\dagger}_{{\pmb i}\sigma}\hat{c}_{{\pmb i}\sigma}$, and the sums run over nearest inter- and intraplane neighbors. Finally $\mathcal{P}_{S}$ is a projection operator onto the subspace $S$ with no double occupation. We apply a mapping, introduced by Angelucci [@Angelucci95], which separates the spin degree of freedom and the occupation number. $$\begin{aligned}
\begin{array}{lcrclcl}
|\uparrow\rangle & \longrightarrow & |1,\Uparrow\rangle &\hphantom{xxx}&
\hat{c}_{{\pmb i}\uparrow} & \longrightarrow &
\hat{\sigma}^{z,+}_{{\pmb i}}\hat{f}^{\dagger}_{{\pmb i}}-\hat{\sigma}^{z,-}_{{\pmb i}}\hat{f}_{{\pmb i}}\vphantom{\frac{1}{1}}\\
|\downarrow\rangle & \longrightarrow & |1,\Downarrow\rangle &\hphantom{xxx}&
\hat{c}^{\dagger}_{{\pmb i}\uparrow} & \longrightarrow &
\hat{\sigma}^{z,+}_{{\pmb i}}\hat{f}_{{\pmb i}}-\hat{\sigma}^{z,-}_{{\pmb i}}\hat{f}^{\dagger}_{{\pmb i}}\vphantom{\frac{1}{1}}\\
|0\rangle & \longrightarrow & |0,\Uparrow\rangle &\hphantom{xxx}&
\hat{c}_{{\pmb i}\downarrow} & \longrightarrow & (\hat{f}_{{\pmb i}}+\hat{f}^{\dagger}_{{\pmb i}})\hat{\sigma}^{+}_{{\pmb i}}
\vphantom{\frac{1}{1}}\\
|\uparrow\downarrow\rangle & \longrightarrow & |0,\Downarrow\rangle &\hphantom{xxx}&
\hat{c}^{\dagger}_{{\pmb i}\downarrow} & \longrightarrow & \hat{\sigma}^{-}_{{\pmb i}}(\hat{f}^{\dagger}_{{\pmb i}}+\hat{f}_{{\pmb i}})
\vphantom{\frac{1}{1}}
\end{array}\end{aligned}$$ $\hat{f}^{\dagger}_{{\pmb i}}$ and $\hat{f}_{{\pmb i}}$ are spinless fermion operators which act on the charge degree of freedom and create (annihilate) a hole at site $i$: $\hat{f}^{\dagger}_{{\pmb i}}|1,\sigma\rangle = |0,\sigma\rangle$, $\hat{\sigma}^{\pm}_{{\pmb i}}$ are ladder operators for the spin degree of freedom and $\hat{\sigma}^{z,\pm}_{{\pmb i}}=\frac{1}{2}(1\pm\hat{\sigma}^{z}_{{\pmb i}})$ are projector operators acting on the spin degree of freedom. Within this base the Hamilton of the $tJ$-model (\[tj\]) writes: $$\begin{aligned}
\tilde{H}_{tJ}
& = & \tilde{\mathcal{P}}_{S}\Big{[} \sum_{\langle {\pmb i}{\pmb j} \rangle}
t_{{\pmb i}{\pmb j}}
\big{[}\hat{f}^{\dagger}_{{\pmb j}}\hat{f}_{{\pmb i}}\tilde{P}_{{\pmb i}{\pmb j}}+{\mathrm{h.c.}}\big{]}
\nonumber\\
& & +\frac{1}{2}\sum_{\langle {\pmb i}{\pmb j} \rangle} J_{{\pmb i}{\pmb j}} (\tilde{P}_{{\pmb i}{\pmb j}}-1)
\tilde{\Delta}_{{\pmb i}{\pmb j}}\Big{]}\tilde{\mathcal{P}}_{S}
\label{angeluccihamilton}\end{aligned}$$ where $\tilde{P}_{{\pmb i}{\pmb j}}=\tfrac{1}{2}(\hat{\vec{\sigma}}_{{\pmb i}}\hat{\vec{\sigma}}_{{\pmb j}}+1)$ and $\tilde{\Delta}_{{\pmb i}{\pmb j}} =1-\hat{f}^{\dagger}_{{\pmb i}}\hat{f}_{{\pmb i}}-\hat{f}^{\dagger}_{j}\hat{f}_{{\pmb j}}$ $\tilde{\mathcal{P}}_{S} = \prod_{{\pmb i}}\big{(}1-\hat{f}^{\dagger}_{{\pmb i}}\hat{f}_{{\pmb i}}\hat{\sigma}^{-}_{{\pmb i}}
\hat{\sigma}^{+}_{{\pmb i}}\big{)}$ is a projection operator in Angelucci representation which projects into the subspace $S$. This representation (\[angeluccihamilton\]) has two important advantages which facilitate numerical simulations: (i) Because the Hamiltonian commutes with the projection operator: $[\tilde{H}_{tJ} ,\tilde{\mathcal{P}}_{S}]=0$, the bare Hamiltonian ($\tilde{H}_{tJ}$ without projections) generates only states of subspace $S$ provided that the initial state is in the relevant subspace. (ii) The Hamiltonian is bilinear in the spinless fermion operators. Within the Angelucci representation the Green’s function reads: $$\begin{aligned}
G_{{\pmb j}{\pmb i}}(\tau) = \langle\hat{\sigma}^{z,+}_{{\pmb j}}(\tau) \hat{f}_{{\pmb j}}(\tau)
\hat{\sigma}^{z,+}_{{\pmb i}}(0) \hat{f}^{\dagger}_{{\pmb i}}(0) \rangle\,\, .\end{aligned}$$ The time evolution in imaginary time is given by: $\hat{\sigma}^{z,+}_{{\pmb j}}(\tau) \hat{f}_{{\pmb j}}(\tau)=e^{\tau\tilde{H}_{tJ}}
\hat{\sigma}^{z,+}_{{\pmb j}}\hat{f}_{{\pmb j}} e^{-\tau\tilde{H}_{tJ}}$. The authors of Ref. [@Brunner00] show in details how to implement the Green’s function into the world line algorithm of our QMC simulation. The spin dynamics is simulated with the loop algorithm. For each fixed spin configuration, one can readily compute the Green’s function since the Hamiltonian is bilinear in the spinless fermion operators $\hat{f}$.
From the Green’s function $G_{{\pmb p}}(\tau)$ we can extract the single particle spectral function $A({\pmb p},\omega)$ with the Stochastic Maximum Entropy: $$\begin{aligned}
G_{{\pmb p}}(\tau)
=\frac{1}{\pi}\int^{\infty}_{0}d\omega e^{-\tau\omega} A({\pmb p},-\omega)\,\, .\end{aligned}$$ In the $T=0$ limit the asymptotic form of the Green’s function reads: $$\begin{aligned}
G_{{\pmb p}}(\tau) =
|\langle\psi^{N-1}_{0}|\hat{c}_{{\pmb p}}|\psi^{N}_{0}\rangle|^{2}e^{-\mu\tau} \label{limesg}\end{aligned}$$ where $\mu$ is the chemical potential. As apparent, the prefactor, $$\mathcal{Z}_{\pmb p} = |\langle\psi^{N-1}_{0}|\hat{c}_{{\pmb p}}|\psi^{N}_{0}\rangle|^{2}\,\, ,$$ is nothing but the quasiparticle residue. Hence from the asymptotic form of the single particle Green’s function, we can read off the quasiparticle residue.
Spin and Hole Dynamics {#dynamics}
======================
In this section we present our results for the spin dynamics as well as for the spectral function of a doped mobile hole.
Spin Dynamics {#spindynamics}
-------------
All considered models, KLM, $U$KLM, KNM and BHM, show a quantum phase transition between an antiferromagnetic ordered phase and a disordered phase. It is believed, that all models belong to the same universality class. To demonstrate this generic property and to test our numerical method we determine the quantum critical point as well as critical exponents in the isotropic BHM and KNM. Fig. \[stiff\]a plots the spin stiffness for the KNM as a function of lattice size.
![\[stiff\][(a) Spin stiffness $\rho_{s}$ as a function of linear lattice size $L$ for different interplanar couplings $J_{\bot}/J_{\|}$ in the KN model. Extrapolation to the thermodynamic limit is carried out by fitting to the form $a + b/L$ (b) Extrapolated value of the spin stiffness as a function of $J_{\bot}/J_{\|}$. The dashes line corresponds to the fit according to the form of Eq. (\[Rhos\_fit\]).]{}](Pic/kondo_stiff_fit.eps "fig:"){height="4.5cm"}\
![\[stiff\][(a) Spin stiffness $\rho_{s}$ as a function of linear lattice size $L$ for different interplanar couplings $J_{\bot}/J_{\|}$ in the KN model. Extrapolation to the thermodynamic limit is carried out by fitting to the form $a + b/L$ (b) Extrapolated value of the spin stiffness as a function of $J_{\bot}/J_{\|}$. The dashes line corresponds to the fit according to the form of Eq. (\[Rhos\_fit\]).]{}](Pic/kondo_stiff_res.eps "fig:"){height="4.5cm"}
The extrapolated data is plotted in Fig. \[stiff\]b. We fit the data to the form: $$\begin{aligned}
\rho_s \propto\Big{[}\Big{(}\frac{J_{\bot}}{J_{\|}}\Big{)}_{c}-\Big{(}\frac{J_{\bot}}{J_{\|}}
\Big{)}\Big{]}^{\nu}\,\,
\label{Rhos_fit}\end{aligned}$$ to obtain $(J_{\bot}/J_{\|})_{c}=1.360\pm 0.017$ and a critical exponent of $\nu =0.582\pm 0.077$, which agrees (within the error bars) with the value of Ref. [@Troyer97]: $\nu=0.685\pm 0.035$. Similar data for the BHM localizes the quantum critical point at $(J_{\bot}/J_{\|})_{c}=2.5121\pm 0.0044$, which conforms roughly the literature value $(J_{\bot}/J_{\|})^{lit}_{c}=2.525\pm 0.002$ of Ref. [@Shevchenko00]. For the critical exponent we obtain $\nu=0.7357\pm 0.044$. Again this is in good agreement with the critical exponent specified in Refs. [@Troyer97]. In Ref. [@Kotov98] the BHM and the KNM are observed by dimer series expansions. Within this framework our numerical results are reflected quite well. FIG. \[magnondispersion\] plot the dynamical spin structure factor as a function of $J_{\bot}/J_{\|}$ for the BHM. In the deeply disordered phase the dispersion has a cosine-like shape. In the limit $J_{\bot}\to \infty$ the ground state wave function is a tensor product of singlets in each unit cell. Starting from this state, a magnon corresponds to breaking a singlet to form a triplet. In first order perturbation theory in $J_{\bot}/J_{\|}$, the magnon acquires a dispersion relation: $$\begin{aligned}
\Omega ({\pmb q})\approx J_{\bot}+\tfrac{1}{2}J_{\|}\gamma ({\pmb q})
\label{mdispstrong}\end{aligned}$$ with $\gamma({\pmb q})=2\left(\cos(q_{x})+\cos(q_{y})\right)$. This approximative approach is roughly consistent with the large-$J_{\bot}$ case in Fig. \[magnondispersion\]a. As as function of decreasing coupling $J_{\bot}$ the spin gap progressively closes (see Fig. \[spin\_gap\]) and at the critical coupling the magnons at ${\pmb q}=(\pi,\pi)$ condense to form the antiferromagnetic order. This physics is captured by the bond mean field approximation which we discuss below.\
![\[magnondispersion\][ Dynamical spin susceptibility, respectively magnon dispersion for different coupling ratios on a $12\times 12$ square lattice.]{}](Pic/spin3.5.eps "fig:"){height="4.0cm"}\
![\[magnondispersion\][ Dynamical spin susceptibility, respectively magnon dispersion for different coupling ratios on a $12\times 12$ square lattice.]{}](Pic/spin3.0.eps "fig:"){height="4.0cm"}\
![\[magnondispersion\][ Dynamical spin susceptibility, respectively magnon dispersion for different coupling ratios on a $12\times 12$ square lattice.]{}](Pic/spin2.5.eps "fig:"){height="4.0cm"}\
${\pmb q}$
### Bond Operator Mean Field Approach {#bond-operator-mean-field-approach .unnumbered}
The bond mean field approach [@Sachdev90] is a strong coupling approximation in $J_{\bot}$. The spins between layers dominantly form singlets and the density of triplets is “low”. This assumption allows one to neglect triplet-triplet interaction. The bond operator representation describes the system in a base of pairs of coupled spins, which can either be in a singlet or triplet state. $$\begin{aligned}
|s\rangle_{{{\pmb i}}}
&=& \hat{s}^{\dagger}_{{{\pmb i}}}|0\rangle_{{{\pmb i}}}=
\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle_{{{\pmb i}}}-|\downarrow\uparrow\rangle_{{{\pmb i}}}) \label{bonds}
\nonumber\\
|t_{x}\rangle_{{{\pmb i}}}
&=& \hat{t}^{\dagger}_{{{\pmb i}}x}|0\rangle_{{{\pmb i}}}=
\frac{-1}{\sqrt{2}}(|\uparrow\uparrow\rangle_{{{\pmb i}}}-|\downarrow\downarrow\rangle_{{{\pmb i}}})
\nonumber\\
|t_{y}\rangle_{{{\pmb i}}}
&=& \hat{t}^{\dagger}_{{{\pmb i}}y}|0\rangle_{{{\pmb i}}}=
\frac{{\mathrm{i}}}{\sqrt{2}}(|\uparrow\uparrow\rangle_{{{\pmb i}}}+|\downarrow\downarrow\rangle_{{{\pmb i}}})
\nonumber\\
|t_{z}\rangle_{{{\pmb i}}}
&=& \hat{t}^{\dagger}_{{{\pmb i}}z}|0\rangle_{{{\pmb i}}}=
\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle_{{{\pmb i}}}+|\downarrow\uparrow\rangle_{{{\pmb i}}}) \label{bondtz}\end{aligned}$$ The operators $\hat{t}^{\dagger} $ and $\hat{s}^{\dagger}$ satisfy Bose commutation rules provided that we impose the constraint $$\begin{aligned}
{\hat{s}^{\dagger}_{{\pmb i}}}{\hat{s}_{{\pmb i}}}+\sum_{\alpha}{\hat{t}^{\dagger}_{{\pmb i}\alpha}}{\hat{t}_{{\pmb i}\alpha}}= 1 \,\, .
\label{constraint}\end{aligned}$$ Since the original spin 1/2 degrees of freedom reads, $$\begin{aligned}
\hat{S}^{(1,2)}_{{{\pmb i}}\alpha}
& = & \tfrac{1}{2}(\pm{\hat{s}^{\dagger}_{{\pmb i}}}{\hat{t}_{{\pmb i}\alpha}}\pm{\hat{t}^{\dagger}_{{\pmb i}\alpha}}{\hat{s}_{{\pmb i}}}-{\mathrm{i}}\sum_{\beta\gamma}
\epsilon_{\alpha\beta\gamma}{\hat{t}^{\dagger}_{{\pmb i}\beta}}{\hat{t}_{{\pmb i}\gamma}})\,\, ,\end{aligned}$$ the Hamiltonian (\[starth\]) can be rewritten in the bond operator representation as: $$\begin{aligned}
\tilde{H}
& = & J_{\bot}\sum_{{\pmb i}}\big{(}
-\tfrac{3}{4}{\hat{s}^{\dagger}_{{\pmb i}}}{\hat{s}_{{\pmb i}}}+\tfrac{1}{4}\sum_{\alpha}{\hat{t}^{\dagger}_{{\pmb i}\alpha}}{\hat{t}_{{\pmb i}\alpha}}\big{)}
\nonumber\\
& & -\sum_{{\pmb i}}\mu_{{\pmb i}}\big{(}{\hat{s}^{\dagger}_{{\pmb i}}}{\hat{s}_{{\pmb i}}}+\sum_{\alpha}{\hat{t}^{\dagger}_{{\pmb i}\alpha}}{\hat{t}_{{\pmb i}\alpha}}-1\big{)}
\nonumber\\
& & +\frac{J_{\|}}{2}\sum_{\langle{\pmb i}{\pmb j}\rangle}\sum_{\alpha}
\big{(}{\hat{s}^{\dagger}_{{\pmb i}}}{\hat{s}^{\dagger}_{{\pmb j}}}{\hat{t}_{{\pmb i}\alpha}}{\hat{t}_{{\pmb j}\alpha}}+{\hat{s}^{\dagger}_{{\pmb i}}}{\hat{s}_{{\pmb j}}}{\hat{t}_{{\pmb i}\alpha}}{\hat{t}^{\dagger}_{{\pmb j}\alpha}}+{\mathrm{h.c.}}\big{)}
\nonumber\\
& & +\frac{J_{\|}}{2}\sum_{\alpha,\beta,\gamma}({\hat{t}^{\dagger}_{{\pmb i}\beta}}{\hat{t}_{{\pmb i}\gamma}}{\hat{t}^{\dagger}_{{\pmb j}\beta}}{\hat{t}_{{\pmb j}\gamma}}-
{\hat{t}^{\dagger}_{{\pmb i}\beta}}{\hat{t}_{{\pmb i}\gamma}}{\hat{t}^{\dagger}_{{\pmb j}\gamma}}{\hat{t}_{{\pmb j}\beta}})\,\, .
\nonumber\end{aligned}$$ $\mu_{{\pmb i}}$ is a Lagrange parameter which enforces locally the constraint (\[constraint\]). The interplanar part shows the characteristic Hamiltonian of two antiferromagnetically coupled spins whereas the intraplanar part includes the interaction between singlets and triplets of different bonds. We now follow the standard method of Sachdev and Bhatt [@Sachdev90]. In the disordered phase we expect a singlet condensate ($\bar{s}=\langle s\rangle\ne 0$) and impose the constraint only on average ($\mu_{\pmb i}=\mu$). As mentioned above we neglect triplet-triplet interactions. Apart from a constant we obtain the following mean field Hamiltonian in momentum space: $$\begin{aligned}
\hat{H}_{MFA}
& = & \sum_{\alpha}\sum_{{\pmb q}}A_{\pmb q}
\hat{t}^{\dagger}_{{\pmb q}\alpha}\hat{t}_{{\pmb q}\alpha}
\nonumber\\
& & +\sum_{\alpha}\sum_{{\pmb q}}\frac{B_{\pmb q}}{2}
(\hat{t}^{\dagger}_{{\pmb q}\alpha}\hat{t}^{\dagger}_{-{\pmb q}\alpha}+{\mathrm{h.c.}})
\label{mfh}\,\, ,\end{aligned}$$ where $$\begin{aligned}
A_{\pmb q}
& = & \frac{J_{\bot}}{4}-\mu+J_{\|}\bar{s}^{2}
\big{(}\cos(q_{x})+\cos(q_{y})\big{)}
\\
B_{\pmb q}
& = & J_{\|}\bar{s}^2
\big{(}\cos(q_{x})+\cos(q_{y})\big{)}\,\, .\end{aligned}$$
The parameter $\mu$ and $\bar{s}=\langle s\rangle$ are determined by the saddle-point equations: $\langle\partial\hat{H}_{MFA}/\partial\mu\rangle=0$ and $\langle\partial\hat{H}_{MFA}/\partial\bar{s}\rangle=0$. The Hamiltonian is diagonalized by a Bogoliubov transformation: $\hat{\alpha}^{\dagger}_{{\pmb q}\alpha}
= u_{{\pmb q}}\hat{t}^{\dagger}_{{\pmb q}\alpha}-
v_{{\pmb q}}\hat{t}_{-{\pmb q}\alpha}$. In terms of magnon creation and annihilation operators the Mean field Hamiltonian (\[mfh\]) writes: $$\begin{aligned}
\hat{H}_{MFA} = \sum_{\pmb q}\sum_{\alpha}\Omega({\pmb q})
\hat{\alpha}^{\dagger}_{{\pmb q}\alpha}\hat{\alpha}_{{\pmb
q}\alpha}\,\, .
\label{diagham}\end{aligned}$$ The Bogoliubov coefficients $u_{{\pmb q}}$ and $v_{{\pmb q}}$ satisfy the relation $u^{2}_{{\pmb q}}-v^{2}_{{\pmb q}}=1$, which follows from the bosonic nature of the magnons: $[\hat{\alpha}_{{\pmb q}},\hat{\alpha}^{\dagger}_{{\pmb q}'}]=\delta_{{\pmb q}{\pmb q}'}$. The coefficients are given by $$\begin{aligned}
u_{{\pmb q}},v_{{\pmb q}}
&=& \sqrt{\frac{A_{\pmb q}}{2\Omega({\pmb q})}\pm\frac{1}{2}}\,\, ,
\label{uv}\end{aligned}$$ where $\Omega({\pmb q})
= \sqrt{A^{2}_{\pmb q}-B^{2}_{\pmb q}}$ is the magnon dispersion. In the vicinity of the critical point it can be approximated by $$\begin{aligned}
\Omega({\pmb q})=\sqrt{\Delta^{2}+v^{2}_{s}({\pmb q}-{\pmb Q})^{2}}
\label{mdisp}\end{aligned}$$ with $\Delta$ the energy gap to magnon excitations, $v_{s}$ the magnon velocity and ${\pmb Q}=(\pi,\pi)$. Eq. (\[mdisp\]) gives an accurate description of the dispersion relation in the vicinity of the critical point (see FIG. \[magnondispersion\]c). At the critical point the gap $\Delta$ vanishes, so that the triplets can condense thus forming the AF static ordering.
![\[spin\_gap\][ (a) Spin correlation function ($J_{\bot}/J_{\|}=2.7$) at ${\pmb p}=(\pi,\pi)$ for a $12 \times 12$ lattice in the BHM. (Inverse temperature $\beta J_{\|}=30.0$, $\Delta\tau J_{\|}=0.02$) (b) Spin gap $\Delta$ at ${\pmb p}=(\pi,\pi)$ for different coupling ratios $J_{\bot}/J_{\|}$. The data for a $12\times 12$ lattice is fitted by $\Delta\propto\left(g-g_{c}\right)^{-z\nu}$ with $g=J_{\bot}/J_{\|}$ and literature values $g_{c}=2.525\pm0.002$ [@Shevchenko00] and $z=1$ [@Troyer97]]{}](Pic/spin_J2.7.eps "fig:"){height="4.5cm"} ![\[spin\_gap\][ (a) Spin correlation function ($J_{\bot}/J_{\|}=2.7$) at ${\pmb p}=(\pi,\pi)$ for a $12 \times 12$ lattice in the BHM. (Inverse temperature $\beta J_{\|}=30.0$, $\Delta\tau J_{\|}=0.02$) (b) Spin gap $\Delta$ at ${\pmb p}=(\pi,\pi)$ for different coupling ratios $J_{\bot}/J_{\|}$. The data for a $12\times 12$ lattice is fitted by $\Delta\propto\left(g-g_{c}\right)^{-z\nu}$ with $g=J_{\bot}/J_{\|}$ and literature values $g_{c}=2.525\pm0.002$ [@Shevchenko00] and $z=1$ [@Troyer97]]{}](Pic/spin_gap.eps "fig:"){height="4.5cm"}
Hole dynamics {#holedynamics}
-------------
We now dope our systems with a single mobile hole and restrict its motion to one layer thereby staying in the spirit of Kondo lattice models. To understand the coupling of the hole to magnetic fluctuations within the magnetic disordered phase we can extend the previously described bond mean-field approximation (See Eq. (\[diagham\])) to account for the hole motion. For this we introduce the operator $\hat{h}^{\dagger}_{{\pmb i}\sigma}$ ($\hat{h}_{{\pmb i}\sigma}$), that creates (anhilates) a hole with spin $\sigma$ in layer 1 at site ${\pmb i}$. $$\begin{aligned}
\hat{h}^{\dagger}_{{\pmb i}\sigma}|vac\rangle&=&|0\sigma\rangle_{{\pmb i}}\end{aligned}$$ $|\sigma_{1}\sigma_{2}\rangle_{\pmb i}$ denotes a dimer state at site ${\pmb i}$ with spin $\sigma_{1}$ in layer 1 and spin $\sigma_{2}$ in layer 2. The Hamiltonian now writes [@Vojta99]: $$\begin{aligned}
\hat{H}&=&\sum_{{\pmb q}}\Omega({\pmb q})
\hat{{\pmb \alpha}}^{\dagger}_{{\pmb q}}\hat{{\pmb \alpha}}_{{\pmb q}}
+\sum_{{\pmb p}}\varepsilon({\pmb p})
\hat{h}^{\dagger}_{{\pmb p}} \hat{h}_{{\pmb p}}
\label{holemagnonhamilton}\\
& & +\sum_{{\pmb p},{\pmb q}}g({\pmb p},{\pmb q})\,{\pmb \alpha}_{\pmb q}\cdot
\big{(}\hat{h}^{\dagger}_{{\pmb p}+{\pmb q}} {\pmb \sigma} \hat{h}_{{\pmb p}}\big{)} + \text{h.c.}
\nonumber\end{aligned}$$ with spinor $\hat{h}_{{\pmb p}}=(\hat{h}_{{\pmb p}\uparrow},\hat{h}_{{\pmb
p}\downarrow})$ and vector $\hat{{\pmb \alpha}}_{{\pmb q}}=(\hat{\alpha}_{{\pmb q}x},
\hat{\alpha}_{{\pmb q}y},\hat{\alpha}_{{\pmb q}z})$. ${\pmb \sigma}=(\sigma^{1},\sigma^{2},\sigma^{3})$ denotes the Pauli matrices. The coupling strength between the hole and magnons is given by $g({\pmb p},{\pmb q})$. We discuss $g({\pmb p},{\pmb q})$ in detail later in section \[qpr\]. For the bare hole dispersion the calculation yields $$\begin{aligned}
\varepsilon({\pmb p})=+t\bar{s}^{2}\big{(}\cos (p_{x})+\cos (p_{y})\big{)}\,\, .\end{aligned}$$
In the limit $J_{\bot}\to\infty$ the magnon excitation energy diverges (see Eq. (\[mdispstrong\])) and hence the coupling of the hole to magnetic excitations becomes negligible. In this limit the magnon excitations become quite rare, so that: $\bar{s}\equiv\langle s\rangle\approx 1$. Thus, in the strong coupling region we obtain from (\[holemagnonhamilton\]) a hole dispersion relation: $$\begin{aligned}
E({\pmb p})=t\big{(}\cos (p_{x})+\cos (p_{y})\big{)}\,\, .
\label{strongcoupling}\end{aligned}$$ This agrees with the result given by applying perturbation theory in $t/J_{\bot}$ [@Tsunetsugu97].
As apparent from Figs. \[dynamic\_spec\] and \[kondo\_spec\] this strong coupling behavior is reproduced by the Monte Carlo simulations where the dispersion exhibits a cosine form with maximum at ${\pmb p}=(\pi,\pi)$. The form of this dispersion relation directly reflects the singlet formation – in other words Kondo screening – between spin degrees of freedom on different layers. We note that this strong coupling behavior of the dispersion relation sets in at larger values of $J_{\bot}/J_{\|}$ for the BHM than for the KNM. This is quite reasonable since in the BHM the single bonds are coupled among each other within both layers.
\(a) $J_{\bot}/J_{\|}=10.0$\
![\[dynamic\_spec\][Spectrums of a mobile hole for a $12\times 12$ lattice in the BHM. The small dashed lines in (a) tag the dispersion of a free particle; in (d) they outline a dispersion of the form: $E ({\pmb p})=J_{\|}\left( \cos(p)_{x} + \cos(p_{y})\right)^{2}$.]{}](Pic/dynamic_spec_J10.0.eps "fig:"){height="4.5cm"}\
(b) $J_{\bot}/J_{\|}=2.4$\
![\[dynamic\_spec\][Spectrums of a mobile hole for a $12\times 12$ lattice in the BHM. The small dashed lines in (a) tag the dispersion of a free particle; in (d) they outline a dispersion of the form: $E ({\pmb p})=J_{\|}\left( \cos(p)_{x} + \cos(p_{y})\right)^{2}$.]{}](Pic/dynamic_spec_J2.4.eps "fig:"){height="4.5cm"}\
(c) $J_{\bot}/J_{\|}=2.0$\
![\[dynamic\_spec\][Spectrums of a mobile hole for a $12\times 12$ lattice in the BHM. The small dashed lines in (a) tag the dispersion of a free particle; in (d) they outline a dispersion of the form: $E ({\pmb p})=J_{\|}\left( \cos(p)_{x} + \cos(p_{y})\right)^{2}$.]{}](Pic/dynamic_spec_J2.0.eps "fig:"){height="4.5cm"}\
(d) $J_{\bot}/J_{\|}=1.0$\
![\[dynamic\_spec\][Spectrums of a mobile hole for a $12\times 12$ lattice in the BHM. The small dashed lines in (a) tag the dispersion of a free particle; in (d) they outline a dispersion of the form: $E ({\pmb p})=J_{\|}\left( \cos(p)_{x} + \cos(p_{y})\right)^{2}$.]{}](Pic/dynamic_spec_J1.0.eps "fig:"){height="4.5cm"} $\omega/J_{\|}$
With decreasing coupling ratio the bandwidth of the quasiparticle dispersion relation diminishes but the overall features of the strong coupling remain.
In the weak coupling limit we observe considerable differences between the single particle spectrum of the BHM and KNM. Let us start with the BHM. For this model the point $J_{\bot}/J_{\|} =0$ is well defined (i.e. the ground is non-degenerate on any finite lattice) and corresponds to two independent Heisenberg planes with mobile hole in the upper plane. The problem of the single hole in a two dimensional Heisenberg model has been addressed in the framework of the self-consistent Born approximation [@Martinez91], and yields a dispersion relation given by: $$\begin{aligned}
\label{SCB_Horsch}
E ({\pmb p})=J_{\|}\left( \cos(p)_{x} + \cos(p_{y})\right)^{2}\,\, .\end{aligned}$$ Since at $J_{\bot}/J_{\|} =0$ we have a well defined ground state we can expect that turning on a small value of $J_{\bot}/J_{\|}$ will not alter the single hole dispersion relation. This point of view is confirmed in Fig. \[dynamic\_spec\]. At $J_{\bot}/J_{\|} = 1$, the single hole dispersion relation follows of Eq. (\[SCB\_Horsch\]). Hence and as confirmed by Fig. \[dynamic\_spec\] the dispersion relation of a single hole in the BHM continuously deforms from the strong coupling form of Eq. (\[strongcoupling\]) to that of a doped hole in a planar antiferromagnet (see Eq. (\[SCB\_Horsch\])). Hence as a function of $J_{\bot}/J_{\|}$ there is a point where the effective mass (as defined by the inverse curvature of the dispersion relation) at $ {\pmb p} = (\pi,\pi) $ diverges. Upon inspection of the data (see Fig. \[dynamic\_spec\]), the point of divergence of the effective mass is not related to the magnetic quantum phase transition and since it occurs slightly below $\left( J_{\bot}/J_{\|}\right)_c$. This crossover between a dispersion with minimum at $ {\pmb p} = (\pi,\pi) $ and minimum at $ {\pmb p} = (\pi/2,\pi/2) $ with a crossover point lying inside the AF ordered phase is also documented in Ref. [@Vojta99].
The above argument can not be applied to the KNM, since the $J_{\bot}/J_{\|} =0$ point is macroscopically degenerate and hence is not a good starting point to understand the weak-coupling physics. Clearly the same holds for the KLM and UKLM. Inspection of the spectral data deep in the ordered phase of the KNM (see Fig. \[kondo\_spec\]c) shows that the maximum of the dispersion relation is still pinned at ${\pmb p}=(\pi,\pi) $ such that the strong coupling features stemming from Kondo screening is still present at weak couplings. For the KNM and up to the lowest couplings we have considered the effective mass at ${\pmb p} = (\pi,\pi) $ increases as a function of decreasing coupling strength but does not seem to diverge at finite values of $J_{\bot}/J_{\|} $. Precisely the same conclusion is reached in the framework of the KLM [@Capponi00] and UKLM [@Feldbacher02].
\(a) $J_{\bot}/J_{\|}=4.0$\
![\[kondo\_spec\][Spectrum of the KNM for a $12\times 12$ lattice. The dashed lines tag the dispersion of a free particle.]{}](Pic/kondo_spec_J4.0.eps "fig:"){height="4.5cm"}\
(b) $J_{\bot}/J_{\|}=2.0$\
![\[kondo\_spec\][Spectrum of the KNM for a $12\times 12$ lattice. The dashed lines tag the dispersion of a free particle.]{}](Pic/kondo_spec_J2.0.eps "fig:"){height="4.5cm"}\
(c) $J_{\bot}/J_{\|}=0.5$\
![\[kondo\_spec\][Spectrum of the KNM for a $12\times 12$ lattice. The dashed lines tag the dispersion of a free particle.]{}](Pic/kondo_spec_J0.5.eps "fig:"){height="4.5cm"} $\omega/J_{\|}$
Quasi Particle Residue {#qpr}
======================
In this section we turn out attention to the delicate issue of the quasiparticle residue in the vicinity of the magnetic quantum phase transition. We first address this question within the framework of the the mean-field model of Eq. (\[holemagnonhamilton\]) and compute the single particle Green’s function within the framework of the self-consistent Born approximation. In a second step, we attempt to determine the quasiparticle residue directly from the Monte Carlo data.
Analytical Approach {#Analytical.sec}
-------------------
Here we restrict our analysis to the BHM. and return to the Hamiltonian (\[holemagnonhamilton\]). The coupling between the hole and magnons $g({\pmb p},{\pmb q})$ reads: $$\begin{aligned}
\quad g({\pmb p},{\pmb q})=g_{a}({\pmb p},{\pmb q})+g_{b}({\pmb p},{\pmb q}).\end{aligned}$$ We identify the two coupling constants with the processes that are shown in Fig. \[coupling\]: $g_{a}({\pmb p},{\pmb q})$ is proportional to the hopping matrix element and hence describes the coupling of a mobile hole to magnetic background, whereas $g_{b}({\pmb p},{\pmb q})$ is proportional to $J^{(2)}_{\|}$ and describes the coupling of a hole at rest with the magnons. Our calculations give the following momentum dependent coupling strengths: $$\begin{aligned}
g_{a}({\pmb p},{\pmb q})&=&
-\frac{t\bar{s}}{\sqrt{N}}\big{(}\gamma({\pmb p}+{\pmb q})u({\pmb q})+\gamma({\pmb p})v({\pmb q})\big{)}\\
g_{b}({\pmb p},{\pmb q})&=&
-\frac{J^{(2)}_{\|} \bar{s}}{8\sqrt{N}}\gamma({\pmb q})\big{(}u({\pmb q})+v({\pmb q})\big{)}\end{aligned}$$ where $\gamma({\pmb q})=2\big{(}\cos (q_{x})+\cos (q_{y})\big{)}$. We concentrate on the coupling to critical magnetic fluctuations and hence set ${\pmb q}={\pmb Q}$ and place ourselves in the proximity of the quantum phase transition, on the disordered side. In this case $ \Omega( {\pmb Q} ) \rightarrow 0 $ and the coherence factors (see Eq. (\[uv\])) are both proportional to $ \Omega({\pmb q})^{-\frac{1}{2}} $. Since furthermore $\gamma({\pmb p}+{\pmb Q})=-\gamma({\pmb p})$ one arrives at the conclusion that $g_{a}({\pmb p},{\pmb Q})$ vanishes at the critical point.
![\[coupling\][Two possible processes where the hole can couple to magnons: (a) The hole moves to a next neighbor. (b) The hole is at rest.]{}](Pic/abb2.eps){width="40.00000%"}
There is hence no coupling via process (a) to critical fluctuations. In other words process (a) couples only to short range spin fluctuations. On the other hand in the vicinity of the critical point $g_{b}$ scales as $g_{b}({\pmb p},{\pmb q}) \propto \Omega({\pmb q})^{-\frac{1}{2}} $ so that we can only retain this term to understand the coupling to critical fluctuations. Summarizing we set: $$\begin{aligned}
g({\pmb p},{\pmb q}) \to g_{b} (\pmb q) \propto\frac{1}{\sqrt{\Omega(\pmb q)}}\,\, ,\end{aligned}$$ for the subsequent calculations. It is intriguing to note that in this simple approximation $ g_{b} (\pmb q) $ scales as $J^{(2)}_{\|} $, which is strictly speaking null in the KNM. However, such a coupling should be dynamically generated via an RKKY-type interaction.
With the above couple the first order self energy diagram for wave vectors satisfying $ \epsilon({\pmb p}) = \epsilon({\pmb p} + {\pmb Q} ) $ shows a logarithmic divergence as a function of the spin gap. Hence we have to sum up all diagrams. We do so in the non-crossing or self-consistent Born approximation which in the $T=0$ limit boils down to the following set of self-consistent equations. $$\begin{aligned}
\Sigma({\pmb p},\omega) & = & \frac{1}{N}\sum_{{\pmb q}}g^{2}({\pmb p},{\pmb q})
G({\pmb p}-{\pmb q},\omega-\Omega({\pmb q})) \nonumber \\
G({\pmb p},\omega) & = &
\frac{1}{\omega-\varepsilon({\pmb p})-\Sigma({\pmb p},\omega)}\end{aligned}$$ Here we use a magnon dispersion relation of the form $\Omega({\pmb q})=\sqrt{\Delta^{2}+v^{2}_{s}\left(1+\gamma({\pmb q})/4 \right)}$ with $\gamma({\pmb q})=2\big{(}\cos (q_{x})+\cos (q_{y})\big{)}$, which agrees in the limit ${\pmb q}\to{\pmb Q}=(\pi,\pi)$ with the form of Eq. (\[mdisp\]). Iterating the Green’s function up to the 15th order to ensure convergence, we calculate the spectrum, $\rho({\pmb p},\omega)=\frac{1}{\pi}Im[G({\pmb p},\omega)]$ via the imaginary part of the Green’s function and compute the quasi-particle residue (QPR) at the first pole of the spectrum. $$\begin{aligned}
\mathcal{Z}({\pmb p})=\Big{|}1-\frac{\partial}{\partial \omega}\Sigma'({\pmb
p},\omega)\Big{|}^{-1}_{\omega=\omega_{i}}\end{aligned}$$ Figure \[scb1\] shows the QPR for ${\pmb p}=(\pi,\pi)$ as a function of linear length $L$ of the square lattice for different values of the spin gap $\Delta$. The large-$L$ limit is indicated by a line.
![ \[scb1\] Self consistent Born approximation: QPR as a function of linear lattice size, $L$, for different spin gap energies, $\Delta$. ](Pic/scb_sizescaling.eps){height="4.5cm"}
Figure \[scb2\] plots the quasiparticle weight as a function of the spin gap for hole momenta ${\pmb p}=(\frac{\pi}{2},\frac{\pi}{2}),(0,\pi),(\pi,\pi)$.
![\[scb2\] Self consistent Born approximation: QPR in the vicinity of the quantum critical point for selected hole momenta (a) in a linear plot and (b) in a double logarithmic plot. $\Delta$ corresponds to the spin gap.](Pic/scb_qpr_vs_gap.eps "fig:"){height="4.5cm"}\
![\[scb2\] Self consistent Born approximation: QPR in the vicinity of the quantum critical point for selected hole momenta (a) in a linear plot and (b) in a double logarithmic plot. $\Delta$ corresponds to the spin gap.](Pic/scb_qpr_vs_gap_log.eps "fig:"){height="4.5cm"}
For hole momenta satisfying $ \epsilon({\pmb p }) = \epsilon({\pmb p } + {\pmb Q} ) $ ( ${\pmb p}=(\frac{\pi}{2},\frac{\pi}{2})$ and ${\pmb p}=(0,\pi)$ ) there is no energy denominator prohibiting the logarithmic divergence of the first order self-energy and the QPR shows an obvious decrease right up to a complete vanishing at the critical point. Furthermore, the data is consistent with $\mathcal{Z} \propto \sqrt{\Delta}$. The case ${\pmb p}=(\pi,\pi)$ is more complicated since $ \epsilon({\pmb p }) \neq \epsilon({\pmb p } + {\pmb Q} ) $. In first order, the self-energy remains bounded. The scattering of the hole of $ {\pmb Q}= (\pi,\pi) $ magnons leads to the progessive formation of shadow bands as the critical point is approached such that at the critical point, the relation $ E^{(1)}({\pmb p})= E^{(1)} ({\pmb p}+{\pmb Q})$ holds. This back folding of the band can lead to the vanishing of the QPR when higher order terms are included. Although the SCB results show a decrease of the QPR in the vicinity of the critical point, they are not accurate enough to answer the question of the vanishing of the QPR at this wave vector.
QMC approach
------------
![\[static\_green\][Green’s function in the vicinity of the phase transition ($J_{\bot}/J_{\|}=2.5$) for a static hole and various lattice sizes in the BHM (a) on a logarithmic plot and (b) on a plot where we adjusted the chemical potential in such a way that the Green’s function converges to a constant value. Within the error bars and for lattice sizes greater then $12\times 12$ there is no size scaling recognisable. (Inverse temperatures: $\beta J_{\|}=30$ ($L=12$), $\beta J_{\|}=50$ ($L=16$)), $\beta J_{\|}=70$ ($L=20$); $\Delta\tau J_{\|}=0.02$)]{} (c) QPR in the vicinity of the quantum critical point. ](Pic/static_green_logplot_J2.5.eps "fig:"){height="4.5cm"} ![\[static\_green\][Green’s function in the vicinity of the phase transition ($J_{\bot}/J_{\|}=2.5$) for a static hole and various lattice sizes in the BHM (a) on a logarithmic plot and (b) on a plot where we adjusted the chemical potential in such a way that the Green’s function converges to a constant value. Within the error bars and for lattice sizes greater then $12\times 12$ there is no size scaling recognisable. (Inverse temperatures: $\beta J_{\|}=30$ ($L=12$), $\beta J_{\|}=50$ ($L=16$)), $\beta J_{\|}=70$ ($L=20$); $\Delta\tau J_{\|}=0.02$)]{} (c) QPR in the vicinity of the quantum critical point. ](Pic/static_green_J2.5.eps "fig:"){height="4.5cm"} ![\[static\_green\][Green’s function in the vicinity of the phase transition ($J_{\bot}/J_{\|}=2.5$) for a static hole and various lattice sizes in the BHM (a) on a logarithmic plot and (b) on a plot where we adjusted the chemical potential in such a way that the Green’s function converges to a constant value. Within the error bars and for lattice sizes greater then $12\times 12$ there is no size scaling recognisable. (Inverse temperatures: $\beta J_{\|}=30$ ($L=12$), $\beta J_{\|}=50$ ($L=16$)), $\beta J_{\|}=70$ ($L=20$); $\Delta\tau J_{\|}=0.02$)]{} (c) QPR in the vicinity of the quantum critical point. ](Pic/static_qpr.eps "fig:"){height="4.5cm"}
As shown in section \[methods\] we can extract the QPR from the asymptotic behavior of the Green’s function. We first concentrate on the static hole in the BHM for which the QMC data is of higher quality that for the dynamic hole. Fig. \[static\_green\]a plots the Green’s funtion as a function of lattice size at $ J_{ \bot } / J_{ \| } =2.5 $. As apparent within the considered range of imaginary times no size and temperature effect is apparent. We fit the tail ( $5 < \tau J_{\|} < 6$) of the Green’s function to the form $\mathcal{Z} e^{-\tau \mu } $ and plot in Fig. \[static\_green\]b $G(\tau) e^{\tau \mu}$. In the large imgaginary time limit this quantitiy converges to the QPR $\mathcal{Z}$. The so obtained value of $\mathcal{Z}$ is plotted for values of $J_{\bot} / J_{\|} $ across the magnetic quantum phase transition. As apparent no sign of the vanishing of the QPR is apparent as we cross the quantum critcal point.
![ \[eta\_h\_static.fig\] [ $\eta_h$ (see Eq. (\[greenfit\])) as a function of $J_{\bot}/J_{\|} $ for a static hole in the BHM. ]{} ](Pic/etah.eps){height="4.5cm"}
Our QMC data allows a different interpretation. Following the work of Sachdev et al. [@Sachdev01] we fit the imaginary time Green function to the form: $$\begin{aligned}
G(\tau) \propto \tau^{-\eta_{h}}\exp \left( -\tau \mu \right)
\label{greenfit}\end{aligned}$$ in the the range $2.0< \tau J_{\|} <6.0$ as done in Ref. [@Sachdev01]. Clearly, if $\eta_h > 0 $ then the QPR vanishes. Our results for $\eta_h$ are plotted in Fig. \[eta\_h\_static.fig\]. At $J_{\bot}/J_{\|}=2.5$ our result, $ \eta_h =0.0875\pm 0.0085 $ compares very well to that quoted in Ref. [@Sachdev01], $\eta_{h}=0.087\pm 0.040$. The fact that the result of Ref. [@Sachdev01] is obtained on a $ 64 \times 64 $ lattice and ours on $20 \times 20$ confirms that for the considered imaginary time range, size effects are absent. Given the above interpretation of the data, Fig. \[eta\_h\_static.fig\] suggests that QPR of a static hole vanishes for all $J_{\bot}/J_{\|} \leq \left( J_{\bot}/J_{\|} \right)_c \simeq 2.5 $.
![\[dynamic\_green\][Green’s function of a dynamic hole (${\pmb p}=(\pi,\pi)$) in the BHM for a $12\times 12$ lattice at $J_{\bot}/J_{\|}=2.4$.]{} (b) QPR in the vicinity of the quantum critical point. ](Pic/dynamic_green_J2.4.eps "fig:"){height="4.5cm"} ![\[dynamic\_green\][Green’s function of a dynamic hole (${\pmb p}=(\pi,\pi)$) in the BHM for a $12\times 12$ lattice at $J_{\bot}/J_{\|}=2.4$.]{} (b) QPR in the vicinity of the quantum critical point. ](Pic/dynamic_qpr.eps "fig:"){height="4.5cm"}
The choice of the fitting function reflects different ordering of the limits $\tau \rightarrow \infty $ and $ N \rightarrow \infty$. On any finite size lattice the QPR is finite and hence it is appropriate to fit the tail of the Green’s function to the form $\mathcal{Z}(N) e^{-\tau \mu}$ to obtain a size dependend QPR, and subsequently take the thermodynamic limit. This strategy has been used successfully to show that the QPR of a doped mobile hole in a one dimensional Heisenberg chain vanishes [@Brunner00]. On the other hand, the choice of Eq. (\[greenfit\]) for fitting the data implies that we first take the thermodynamic limit. Only in this limit, can the assymptotic form of the Green’s function follow Eq. (\[greenfit\]) with $ \eta_h \neq 0$. The fact that both procedures yield different results sheds doubt on the small imaginary time range used to extract the quasiparticle residue. In particular, using data from $\tau J_{\|} = 2 $ onwards implies that we are looking at a frequency window around the lowest excitation of the order $\omega / J \simeq 0.5$. Given this, it is hard to resolve the difference between a dense spectrum and a well defined low-lying quasiparticle pole and a branch cut.
![\[knm\_qpr\][(a) Green’s function $G_{\pmb p}(\tau)$ in the Kondo Necklace model at ${\pmb p}=(\pi,\pi)$ for $J_{\bot}/J_{\|}=1.0,1.5,2.0$. (b) Extracted values for the QPR. The critical point is localised at $(J_{\bot}/J_{\|})_{c}=1.360\pm 0.017$.]{}](Pic/kondo_green.eps "fig:"){height="4.5cm"} ![\[knm\_qpr\][(a) Green’s function $G_{\pmb p}(\tau)$ in the Kondo Necklace model at ${\pmb p}=(\pi,\pi)$ for $J_{\bot}/J_{\|}=1.0,1.5,2.0$. (b) Extracted values for the QPR. The critical point is localised at $(J_{\bot}/J_{\|})_{c}=1.360\pm 0.017$.]{}](Pic/kondo_qpr_pipi_L12.eps "fig:"){height="4.5cm"}
We conclude this section by presenting data for a mobile hole in the BHM (see Fig. \[dynamic\_green\]) and KNM (see Fig. \[knm\_qpr\]). Recall that in our simulations we restrict the motion of the hole to a single plane. The data for the QPR in the above mentioned figures stem from fitting the tail of the Green’s function to the form $\mathcal{Z}e^{-\tau \mu}$. The fit to the form of Eq. (\[greenfit\]) yields values of $\eta_h$ which within the error bars are not distinguishable form zero.
Conclusion
==========
We have analyzed single hole dynamics across magnetic order-disorder quantum phase transitions as realized in the Kondo Necklace and bilayer Heisenberg models. The hole motion is restricted to the upper layer as appropriate for interpretation of the data in terms in Kondo physics. Both models have identical spin dynamics since the quantum phase transition is described by the $O(3)$ three-dimensional sigma model [@Troyer97]. On the other hand the single hole dynamics shows marked differences. In the strong coupling limit, deep in the disordered phase, the ground state of both models is well described by a direct product of singlets between the layers. This Kondo screening leads to a single hole dispersion relation with maximum at ${\pmb p}=(\pi,\pi)$. In the Kondo Necklace model, where the spin degrees of freedom on the lower layer interact indirectly through polarization of spin on the upper layer (RKKY type interaction), the single hole dispersion preserves it’s maximum at $\pmb{p} =(\pi,\pi) $ down to arbitrarily low interplanar couplings. This situation is very similar to the Kondo Lattice model of Eq. (\[KLM\]). In this case, down to $\mathcal{J}/t = 0.2$, substantially below the magnetic phase transition $\mathcal{J}_c/t = 1.45 \pm 0.05 $, the maximum of the the hole dispersion is pinned at $\pmb{p} = (\pi,\pi)$ and the effective mass at this ${\pmb p}$-point tracks the single ion Kondo temperature [@Assaad04a]. We note that this result is not supported by recent series expansions which show that there is a critical value of the coupling where the effective mass diverges [@Trebst06]. Hence the interpretation that in both the Kondo necklace and Kondo lattice models, the localized spins remained partially screen down to arbitrarily low values of the interlayer coupling. In other words, signatures of strong coupling physics in the single hole dispersion relation is present down to arbitrary low interplanar couplings.
In the bilayer Heisenberg model where there is an independent energy scale coupling the spins on the lower layer, the situation differs. At values of $ J_{\bot} < J_{\bot,c}$ the maximum of the single hole dispersion relation drifts towards $ {\pmb p} = (\pi/2,\pi/2) $ and the dispersion relation evolves continuously to that of a single hole doped in a planar antiferromagnetic [@Brunner00b]. Hence the interpretation that at weak couplings, Kondo screening in this model is completely suppressed. In other words, the small but finite $J_{\bot}$ results can be well understood starting form the $J_{\bot} =0 $ point.
We have equally, analyzed the quasiparticle residue across the magnetic order-disorder transition. In the disordered phase using a bond mean-field approximation, there are two processes in which the hole couples to magnetic fluctuations (see Fig. \[coupling\]): i) The hole propagates from one lattice site to another thereby rearranging the spin background. In the proximity of the critical point, and still within the bond-mean field approximation those processes do not couple to long range $\pmb{Q} = (\pi,\pi)$ magnetic fluctuations. A very similar result is obtained in the ordered phase [@Martinez91]. ii) In bilayer models the hole can remain immobile and the spin in the lower layer can flip. Those processes couple to critical magnetic fluctuations. Within a self-consistent Born approximation, this drives the quasiparticle residue to zero both for a static hole and mobile hole with momenta ${\pmb p} $ satisfying $ \epsilon({\pmb p} + {\pmb Q}) = \epsilon({\pmb p})$. We have attempted to confirm this point of view with Monte Carlo simulations. Within our quantum Monte Carlo approach, where the accuracy of the single particle Green’s function at large imaginary times is limited, we have found no convincing evidence of the vanishing of the quasiparticle residue both for a static and a mobile hole. Furhter work and algortihmic developments are required to clarify this delicate issue.
Acknowledgments. The calculations were carried out on the Hitachi SR8000 of the LRZ München. We thank this institution for generous allocation of CPU time. We have greatly profitted from discussions with M. Vojta and L. Martin. Financial support from the DFG under the grant number AS120/4-1 is acknowledged.
[10]{}
S. Paschen, T. Lühmann, S. Wirth, P. Gegenwart, O. Trovarelli, C. Geibel, F. Steglich, P. Coleman, and Q. Si, Nature [**432**]{}, 881 (2004).
J. R. Schrieffer and P. A. Wolff, Phys. Rev. [**149**]{}, 491 (1966).
H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys. [**69**]{}, 809 (1997).
S. Capponi and F. F. Assaad, Phs. Rev. B [**63**]{}, 155114 (2001).
M. Feldbacher, C. Jurecka, F. F. Assaad, and W. Brenig, Phys. Rev. B [**66**]{}, 045103 (2002).
M. Brunner, F. F. Assaad, and A. Muramatsu, Eur. Phys. J. B [**16**]{}, 209 (2000).
K. S. D. Beach, cond-mat/0403055 (2004).
A. Sandvik, Phys. Rev. B [**57**]{}, 10287 (1998).
O. P. Sushkov, Phys. Rev. B [**62**]{}, 12135 (2000).
H. G. Evertz, Adv. Phys. [**52**]{}, 1 (1997).
A. Angelucci, Phys. Rev. B [**51**]{}, 11580 (1995).
M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Jpn. [**66**]{}, 2957 (1997).
P. V. Shevchenko, A. W. Sandvik, and O. P. Sushkov, Phys. Rev. B [**61**]{}, 3475 (2000).
V. N. Kotov, O. Sushkov, Z. Weihong, and J. Oitmaa, Phys. Rev. Lett. [**80**]{}, 5790 (1998).
S. Sachdev and R. N. Bhatt, Phys. Rev. B [**41**]{}, 9323 (1990).
M. Vojta and K. W. Becker, Phys. Rev. B [**60**]{}, 15201 (1999).
G. Martínez and P. Horsch, Phys. Rev. B [**44**]{}, 317 (1991).
S. Sachdev, M. Troyer, and M. Vojta, Phys. Rev. Lett. [**86**]{}, 2617 (2001).
F. F. Assaad, Phys. Rev. B [**70**]{}, 020402 (2004).
S. Trebst, H. Monien, A. Grzesik, and M. Sigrist, Phys. Rev. B [**73**]{}, 165101 (2006).
M. Brunner, F. F. Assaad, and A. Muramatsu, Phys. Rev. B [**62**]{}, 12395 (2000).
|
---
abstract: 'Second harmonic generation and optical parametric amplification in negative-index metamaterials (NIMs) are studied. The opposite directions of the wave vector and the Poynting vector in NIMs results in a “backward” phase-matching condition, causing significant changes in the Manley-Rowe relations and spatial distributions of the coupled field intensities. It is shown that absorption in NIMs can be compensated by backward optical parametric amplification. The possibility of distributed-feedback parametric oscillation with no cavity has been demonstrated. The feasibility of the generation of entangled pairs of left- and right-handed counter-propagating photons is discussed.'
author:
- 'A. K. Popov'
- 'Vladimir M. Shalaev'
date: 'January 8, 2006'
title: 'Negative-Index Metamaterials: Second-Harmonic Generation, Manley-Rowe Relations and Parametric Amplification '
---
Introduction {#i}
============
Recent demonstration of a negative refractive index for metamaterials in the optical range [@NIMExp1; @NIMExp2] opens new avenues for optics and especially nonlinear optics. In parallel with progress for metal-dielectric metamaterials, experimental demonstrations of negative refraction in the near IR range have been made in a GaAs-based photonic crystals [@pc1] and in Si-Polyimide photonic crystals [@pc2]. Negative refractive-index metamaterials (NIMs) are also referred to as left-handed materials (LHMs). The sufficient (but not necessary) condition for a negative refractive index is simultaneously negative dielectric permittivity $\epsilon (\omega )$ and negative magnetic permeability $\mu
(\omega )$ [@Vesel]. Negative magnetic permeability in the optical range has been demonstrated in [@mu1; @mu2; @mu3]. NIMs exhibit highly unusual optical properties and promise a great variety of unprecedented applications. Optical magnetization, which is normally ignored in linear and nonlinear-optics of the ordinary, positive-index materials (PIMs) (i.e., right-handed materials, RHMs) plays a crucial role in NIMs.
The main emphasis in the studies of NIMs has been placed so far on linear optical effects. Recently it has been shown that NIMs including structural elements with non-symmetric current-voltage characteristics can possess a nonlinear magnetic response at optical frequencies [@Lap] and thus combine unprecedented linear and nonlinear electromagnetic properties. Important properties of second harmonic generation (SHG) in NIMs in the constant-pump approximation were discussed in [@Agr] for semi-infinite materials and in [@Lens] for a slab of a finite thickness. The propagation of microwave radiation in nonlinear transmission lines, which are the one-dimensional analog of NIMs, was investigated in [@Kozyr]. The possibility of exact phase-matching for waves with counter-propagating energy-flows has been shown in [@KivSHG] for the case when the fundamental wave falls in the negative-index frequency domain and the SH wave lies in the positive-index domain. The possibility of the existence of multistable nonlinear effects in SHG was also predicted in [@KivSHG].
As seen from our consideration below, the phase-matching of normal and backward waves is inherent for nonlinear optics of NIMs. We note here that the important advantages of interaction schemes involving counter-directed Poynting vectors in the process of optical parametric amplification in ordinary RHMs were discussed in early papers [@Har]. However, in RHMs such schemes impose severe limitations on the frequencies of the coupled waves because of the requirement that one of the waves has to be in the far-infrared range.
Absorption is one of the biggest problems to be addressed for the practical applications of NIMs. In [@Agr; @Lens], a transfer of the near-field image into SH frequency domain, where absorption is typically much less, was proposed as a means to overcome dissipative losses and thus enable the superlens.
In this paper, we demonstrate unusual characteristics in the spatial distribution of the energy exchange between the fundamental and second-harmonic waves. Both semi-infinite and finite-length NIMs are considered and compared with each other and with ordinary PIMs. Our analysis is based on the solution to equations for the coupled waves propagating in lossless NIMs beyond the constant-pump approximation. The Manley-Rowe relations for NIMs are analyzed and they are shown to be strikingly different from those in PIMs. We also propose a new means of compensating losses in NIMs by employing optical parametric amplification (OPA). This can be realized by using control electromagnetic waves (with frequencies outside the negative-index domain), which provide the loss-balancing OPA inside the negative-index frequency domain. We also predict laser oscillations without a cavity for frequencies in the negative-index domain and the possibility of the generation of entangled pairs of counter-propagating right- and left-handed photons.
The paper is organized as follows. Section \[shg\] discusses the unusual spatial distribution of the field intensities for SHG in finite and semi-infinite slabs of NIMs. The Manley-Rowe relations are derived and discussed here. The feasibility of compensating losses in NIMs by using the OPA is considered in Section \[opa\]. In this Section we also study cavity-less oscillations based on distributed feedback. Finally, a summary of the obtained results concludes the paper.
Second harmonic generation in NIMs {#shg}
==================================
Wave vectors and Poynting vectors in NIMs {#pv}
-----------------------------------------
We consider a loss-free material, which is left-handed at the fundamental frequency $\omega_1$ ($\epsilon_1<0$, $\mu_1<0$), whereas it is right-handed at the SH frequency $\omega_2=2\omega_1$ ($\epsilon_2>0$, $\mu_2>0$). The relations between the vectors of the electrical, $\mathbf{E}$, and magnetic, $\mathbf{H}$, field components and the wave-vector $\mathbf{k}$ for a traveling electromagnetic wave, $$\begin{aligned}
\mathbf{E}(\mathbf{r},t)&=&\mathbf{E}_0(\mathbf{r})\exp[-i(\omega t-\mathbf{k\cdot r})]+ c.c., \label{EM} \\
\mathbf{H}(\mathbf{r},t)&=&\mathbf{H}_0(\mathbf{r})\exp[-i(\omega t -\mathbf{k\cdot r})]+ c.c., \label{HM}\end{aligned}$$ are given by the following formulas $$\begin{aligned}
\mathbf{k}\times\mathbf{E}& = &({\omega}/{c})\mu\mathbf{H},\quad\mathbf{k}\times\mathbf{H} =- ({\omega}/{c}) \epsilon\mathbf{E}, \label{kh} \\
\sqrt{\epsilon}{E}(\mathbf{r},t)&=&-\sqrt{\mu}{H}(\mathbf{r},t), \label{eh}\end{aligned}$$ which follow from Maxwell’s equations. These expressions show that the vector triplet $\mathbf{E}$, $\mathbf{H}$ and $\mathbf{k}$ forms a right-handed system for the SH wave and a left-handed system for the fundamental beam. Simultaneously negative $\epsilon_i<0$ and $\mu_i<0$ result in a negative refractive index $n= -
\sqrt{\mu\epsilon}$. As seen from Eqs. (\[EM\]) and (\[HM\]), the phase velocity $\mathbf{v}_{ph}$ is co-directed with $\mathbf{k}$ and is given by $\mathbf{v}_{ph}=({\mathbf{k}}/{k})({\omega}/{k})=({\mathbf{k}}/{k})({c}/{|n|})$, where ${k}^{2}=n^{2}(\omega/{c})^{2}$. In contrast, the direction of the energy flow (Poynting vector) $\mathbf{S}$ with respect to $\mathbf{k}$ depends on the signs of $\epsilon $ and $\mu $: $$\begin{aligned}
\mathbf{S}(\mathbf{r},t) &=&\frac{c}{4\pi}[\mathbf{E}\times\mathbf{H}] =\frac{c^{2}}{4\pi\omega\epsilon}[\mathbf{H}\times\mathbf{k}\times\mathbf{H}]
= \notag \\
&=&\frac{c^{2}\mathbf{k}}{4\pi\omega\epsilon}H^{2} =\frac{c^{2}\mathbf{k}}{4\pi\omega\mu}E^{2}. \label{s}\end{aligned}$$ As mentioned, we assume here that all indices of $\epsilon$, $\mu$ and $n$ are real numbers. Thus, the energy flow $\mathbf{S}_1$ at $\omega_1$ is directed opposite to $\mathbf{k}_1$, whereas $\mathbf{S}_2$ is co-directed with $\mathbf{k}_2$.
SHG: Basic equations and the Manley-Rowe relations {#be}
--------------------------------------------------
We assume that an incident flow of fundamental radiation $\mathbf{ S}_{1}$ at $\omega _{1}$ propagates along the z-axis, which is normal to the surface of a metamaterial. According to (\[s\]), the phase of the wave at $\omega _{1}$ travels in the reverse direction inside the NIM (the upper part of Fig.\[f1\]). Because of the phase-matching requirement, the generated SH radiation also travels backward with energy flow in the same backward direction. This is in contrast with the standard coupling geometry in a PIM (the lower part of Fig.\[f1\]).
![SHG geometry and the difference between SHG in LHM and RHM slabs. []{data-label="f1"}](f1a.eps "fig:"){height=".3\textwidth"}![SHG geometry and the difference between SHG in LHM and RHM slabs. []{data-label="f1"}](f1b.eps "fig:"){height=".3\textwidth"}
Following the method of [@KivSHG], we assume that a nonlinear response is primarily associated with the magnetic component of the waves. Then the equations for the coupled fields inside a NIM in the approximation of slow-varying amplitudes acquire the form: $$\begin{aligned}
\frac{dA_{2}}{dz} &=&i\dfrac{\epsilon _{2}\omega
_{2}^{2}}{k_{2}c^{2}}4\pi
\chi _{eff}^{(2)}A_{1}^{2}\exp (-\Delta kz), \label{A2} \\
\frac{dA_{1}}{dz} &=&i\dfrac{\epsilon _{1}\omega
_{1}^{2}}{k_{1}c^{2}}8\pi \chi _{eff}^{(2)}A_{2}A_{1}^{\ast }\exp
(\Delta kz). \label{A1}\end{aligned}$$ Here, $\chi _{eff}^{(2)}$ is the effective nonlinear susceptibility, $\Delta k=k_{2}-2k_{1}$ is the phase mismatch, and $A_{2}$ and $A_{1}$ are the slowly varying amplitudes of the waves with the phases traveling against the z-axis: $${H}_{j}(z,t)={A}_{j}\exp [-i(k_{j}z+\omega _{j}t)]+c.c., \label{Az}$$ where, $\omega _{2}=2\omega _{1}$ and $k_{1,2}>0$ are the moduli of the wave-vectors directed against the z-axis. We note that according to Eq. (\[eh\]) the corresponding equations for the electric components can be written in a similar form, with $\epsilon _{j}$ substituted by $\mu _{j}$ and vice versa. The factors $\mu _{j}$ were usually assumed to be equal to one in similar equations for PIMs. However, this assumption does not hold for the case of NIMs, and this fact dramatically changes many conventional electromagnetic relations. The Manley-Rowe relations [@MR] for the field intensities and for the energy flows follow from Eqs. (\[s\]) - (\[A1\]): $$\frac{k_{1}}{\epsilon
_{1}}\dfrac{d|A_{1}|^{2}}{dz}+\frac{k_{2}}{2\epsilon
_{2}}\dfrac{d|A_{2}|^{2}}{dz}=0,\quad
\dfrac{d|S_{1}|^{2}}{dz}-\dfrac{d|S_{2}|^{2}}{dz}=0.
\label{MR1}$$ The latter equation accounts for the difference in the signs of $\epsilon _{1}$ and $\epsilon _{2}$, which brings radical changes to the spatial dependence of the field intensities discussed below.
We focus on the basic features of the process and ignore the dissipation of both waves inside the nonlinear medium; in addition, we assume that the phase matching condition $k_{2}=2k_{1}$ is fulfilled. The spatially-invariant form of the Manley-Rowe relations follows from equation (\[MR1\]): $$|A_{1}|^{2}/\epsilon _{1}+|A_{2}|^{2}/\epsilon _{2}=C, \label{I}$$ where $C$ is an integration constant. With $\epsilon _{1}=-\epsilon
_{2}$, which is required for the phase matching, equation (\[I\]) predicts that the *difference* between the squared amplitudes remains constant through the sample $$|A_{1}|^{2}-|A_{2}|^{2}=C, \label{D}$$ as schematically depicted in the upper part of Fig. \[f1\]. This is in striking difference with the requirement that the *sum* of the squared amplitudes is constant in the analogous case in a PIM, as schematically shown in the lower part of Fig. \[f1\]. We introduce now the real phases and amplitudes as $A_{1,2}=h_{1,2}\exp
(i\phi_{1,2})$. Then the equations for the phases, which follow from Eqs. (\[A2\]) and (\[A1\]), show that if any of the fields becomes zero at any point, the integral (\[I\]) corresponds to the solution with the constant phase difference $2\phi_{1}-\phi_{2}=\pi
/2$ over the entire sample.
The equations for the slowly-varying amplitudes corresponding to the ordinary coupling scheme in a PIM, shown in the lower part of Fig. \[f1\], are readily obtained from Eqs. (\[A2\]) - (\[Az\]) by changing the signs of $k_{1}$ and $k_{2}$. This does not change the integral (\[I\]); more importantly, the relation between $\epsilon_{1}$ and $\epsilon_{2}$ required by the phase matching now changes to $\epsilon_{1}=\epsilon_{2}$, where both constants are positive. The phase difference remains the same. Because of the boundary conditions $h_{1}(0)=h_{10}$ and $h_{2}(0)=h_{20}=0$, the integration constant becomes $C=h_{10}^{2}$. Thus, the equations for the real amplitudes in the case of a PIM acquire the form: $$\begin{aligned}
&&h_{1}(z)=\sqrt{h_{10}^{2}-h_{2}(z)^{2}}, \label{D3} \\
&&dh_{2}/{dz}=\kappa \lbrack h_{10}^{2}-h_{2}(z)^{2}], \label{h24}\end{aligned}$$ with the known solution $$\begin{aligned}
h_{2}(z) &=&h_{10}\tanh (z/z_{0}), \label{rhm2} \\
h_{1}(z) &=&h_{10}/\cosh (z/z_{0}),\,z_{0}=[\kappa h_{10}]^{-1}.
\label{rhm1}\end{aligned}$$ Here, $\kappa =({\epsilon_{2}\omega_{2}^{2}}/{k_{2}c^{2}})4\pi
\chi_{eff}^{(2)}$. *The solution has the same form for an arbitrary slab thickness*, as shown schematically in the lower part of Fig. \[f1\].
SHG in a NIM slab {#sl}
-----------------
Now consider phase-matched SHG in a lossless NIM slab of a finite length L. Equations (\[A2\]) and (\[D\]) take the form: $$\begin{aligned}
h_{1}(z)^{2} &=&C+h_{2}(z)^{2}, \label{D1} \\
dh_{2}/{dz} &=&-\kappa \lbrack C+h_{2}(z)^{2}]. \label{h12}\end{aligned}$$ Taking into account the *different boundary conditions in a NIM as compared to a PIM*, $h_{1}(0)=h_{10}$ and $h_{2}(L)=0$, the solution to these equations is as follows $$\begin{aligned}
h_{2} &=&\sqrt{C}\tan [\sqrt{C}\kappa (L-z)], \label{h22} \\
h_{1} &=&\sqrt{C}/\cos [\sqrt{C}\kappa (L-z)], \label{h11}\end{aligned}$$ where the integration parameter $C$ *depends on the slab thickness $L$* and on the amplitude of the incident fundamental radiation as $$\sqrt{C}\kappa L=\cos ^{-1}(\sqrt{C}/h_{10}). \label{C}$$ Thus, *the spatially invariant field intensity difference between the fundamental and SH waves in NIMs depends on the slab thickness, which is in strict contrast with the case in PIMs.* As seen from equation (\[D1\]), the integration parameter $C=h_{1}(z)^{2}-h_{2}(z)^{2}$ now represents the deviation of the conversion efficiency $\eta =h_{20}^{2}/h_{10}^{2}$ from unity: $(C/h_{10}^{2})=1-\eta $. Figure \[f2\] shows the dependence of this parameter on the conversion length $z_{0}=(\kappa
h_{10})^{-1}$.
![The normalized integration constant $C/h_{10}^{2}$ and the energy conversion efficiency $\protect\eta $ vs the normalized length of a NIM slab.[]{data-label="f2"}](f2.eps){width=".4\textwidth"}
The figure shows that for the conversion length of 2.5, the NIM slab, which acts as nonlinear mirror, provides about 80% conversion of the fundamental beam into a reflected SH wave. Figure \[f3\] depicts the field distribution along the slab. One can see from the figure that with an increase in slab length (or intensity of the fundamental wave), the gap between the two plots decreases while the conversion efficiency increases (comparing the main plot and the inset).
![The squared amplitudes for the fundamental wave (the dashed line) and SHG (the solid line) in a lossless NIM slab of a finite length. Inset: the slab has a length equal to one conversion length. Main plot: the slab has a length equal to five conversion lengths. The dash-dot lines show the energy-conversion for a semi-infinite NIM.[]{data-label="f3"}](f3.eps){height=".38\textwidth"}
SHG in a semi-infinite NIM {#si}
--------------------------
Now we consider the case of a semi-infinite NIM at $z>0$. Since both waves disappear at $z\rightarrow \infty $ due to the entire conversion of the fundamental beam into SH, $C=0$. Then equations (\[D1\]) and (\[h12\]) for the amplitudes take the simple form $$\begin{aligned}
&&h_{2}(z)=h_{1}(z), \label{D2} \\
&&dh_{2}/{dz}=-\kappa h_{2}^{2}. \label{h23}\end{aligned}$$ Equation (\[D2\]) indicates 100% conversion of the incident fundamental wave into the reflected second harmonic at $z=0$ in a lossless semi-infinite medium provided that the phase matching condition $\Delta k=0$ is fulfilled. The integration of (\[h23\]) with the boundary condition $h_{1}(0)=h_{10}$ yields $$h_{2}(z)=\dfrac{h_{10}}{(z/z_{0})+1},\,z_{0}=(\kappa h_{10})^{-1}.
\label{si}$$ Equation (\[si\]) describes a *concurrent decrease of both waves of equal amplitudes along the z-axis;* this is shown by the dash-dot plots in Fig. \[f3\]. For $z\gg z_{0}$, the dependence is inversely proportional to $z$. These *spatial dependencies, shown in Fig. \[f3\], are in strict contrast with those for the conventional process of SHG in a PIM*, which are known from various textbooks (compare, for example, with the lower part of Fig.\[f1\]).
Optical parametric amplification and difference-frequency generation in a NIM slab with absorption {#opa}
==================================================================================================
OPA: basic equations and Manley-Rowe relations {#mrr}
----------------------------------------------
As mentioned in Subsection \[pv\], $\mathbf{S}$ is counterdirected with respect to $\mathbf{k}$ in NIMs, because $\epsilon <0$ and $\mu <0$. We assume that a left-handed wave at $\omega _{1}$ travels with its wave-vector directed along the $z$-axis. Then its energy flow $\mathbf{S}_{1}$ is directed against the $z$-axis. We also assume that the sample is illuminated by a higher-frequency electromagnetic wave traveling along the axis $z$. The frequency of this radiation $\omega_{3}$ falls in a positive index range. The two coupled waves with co-directed wave-vectors $\mathbf{k}_{3}$ and $\mathbf{k}_{1}$ generate a difference-frequency idler at $\omega_{2}=\omega_{3}-\omega_{1}$, which has a positive refractive index. The idle wave contributes back into the wave at $\omega_{1}$ through three-wave coupling and thus enables optical parametric amplification (OPA) at $\omega_{1}$ by converting the energy of the pump field at $\omega_{3}$. Thus, the nonlinear-optical process under consideration involves three-wave mixing with wave-vectors co-directed along $z$. Note that the energy flow of the signal wave, $\mathbf{S}_{1}$, is directed against $z$, i.e., it is directed opposite to the energy flows of the two other waves, $\mathbf{S}_{2}$ and $\mathbf{S}_{3}$ (Fig. \[fig1\], the left part). Such a coupling scheme is in contrast with the ordinary phase-matching scheme for OPA, which is schematically shown in the right part of Fig. \[fig1\].
![The difference between OPA processes in LHM and PIM slabs.[]{data-label="fig1"}](fig1a.eps "fig:"){height=".3\textwidth"} ![The difference between OPA processes in LHM and PIM slabs.[]{data-label="fig1"}](fig1b.eps "fig:"){height=".3\textwidth"}
As above, we consider the magnetic type of the quadratic nonlinearity. For the magnetic field $${H}_{j}(z,t)={h}_{j}\exp [i(k_{j}z-\omega _{j}t)]+c.c., \label{hz1}$$ the nonlinear magnetization at the signal and idler frequencies is given by the equations $$\begin{aligned}
M_{1}^{NL} &=&2\chi_{eff}^{(2)}h_{3}h_{2}^{\ast }\exp
\{i[(k_{3}-k_{2})-\omega_{1}t]\}, \label{m1n1} \\
M_{2}^{NL} &=&2\chi_{eff}^{(2)}h_{3}h_{1}^{\ast }\exp
\{i[(k_{3}-k_{1})-\omega_{2}t]\}. \label{m2n1}\end{aligned}$$ Here, $j=1,2,3;\, \omega_{2}=\omega_{3}-\omega_{1};\,$ and $k_{j}=|n_{j}|\omega _{j}/c>0$. Then the equations for the slowly-varying amplitudes of the signal and the idler acquire the form $$\begin{aligned}
\frac{dh_{1}}{dz} &=&i\sigma_{1}h_{3}h_{2}^{\ast }\exp [i\Delta kz]+\frac{\alpha _{1}}{2}h_{1}, \label{h11} \\
\frac{dh_{2}}{dz} &=&i\sigma_{2}h_{3}h_{1}^{\ast }\exp [i\Delta kz]-\frac{\alpha_{2}}{2}h_{2}, \label{h21}\end{aligned}$$ where $\sigma_{j}=8\pi
\chi_{eff}^{(2)}{\epsilon_{j}\omega_{j}^{2}}/{k_{j}c^{2}}$, $\Delta
k=k_{3}-k_{2}-k_{1}$, and $\alpha _{j}$ are the absorption indices. The amplitude of the pump $h_{3}$ is assumed constant. We note the following *three fundamental differences* in equation (\[h11\]) as compared with the ordinary difference-frequency generation (DFG) through the three-wave mixing of co-propagating waves in a PIM. First, the sign of $\sigma_{1}$ is opposite to that of $\sigma_{2}$ because $ \epsilon_{1}<0$. Second, the opposite sign appears with $\alpha_{1}$ because the energy flow $\mathbf{S_{1}}$ is directed against the $z$-axis. Third, the boundary conditions for $h_{1}$ are defined at the opposite side of the sample as compared to $h_{2}$ and $h_{3}$ because their energy-flows $\mathbf{S_{1}}$ and $\mathbf{S_{2}}$ are counter-directed.
At $\alpha_{1}=\alpha_{2}=0$, one finds with the aid of Eqs. (\[h11\]), (\[h21\]) and (\[s\]): $$\begin{aligned}
&&\frac{d}{dz}\left[ \dfrac{S_{1z}}{\hbar {\omega_{1}}}-\dfrac{S_{2z}}{\hbar {\omega_{2}}}\right] =0, \label{MR1} \\
&&\dfrac{d}{dz}\left[\sqrt{\dfrac{\mu_{1}}{\epsilon_{1}}}\dfrac{
|h_{1}|^{2}}{\omega_{1}}+\sqrt{\dfrac{\mu_{2}}{\epsilon
_{2}}}\dfrac{|h_{2}|^{2}}{\omega_{2}}\right] =0. \label{MR21}\end{aligned}$$ These equations represent the Manley-Rowe relations [@MR], which describe the creation of pairs of *entangled counter-propagating photons* $\hbar{\omega_{1}}$ and $\hbar{\omega_{2}}$. The equations account for the opposite sign of the corresponding derivatives with respect to z. Equation (\[MR21\]) predicts that the *sum* of the terms proportional to the squared amplitudes of signal and idler remains constant through the sample, which is in contrast with the requirement that the *difference* of such terms is constant in the analogous case in a PIM. We note that according to Eqs. (\[eh\]) and (\[s\]) the corresponding equations for the electric components in the case of the quadratic electric nonlinearity can be written in a similar form with $\epsilon_{j}$ substituted by $\mu_{j}$. As seen from the equations below, this does not change either the results obtained or the main conclusions presented here; the same is true for the case of SHG. As mentioned in Section \[shg\], the factors $\mu_{j}$ were usually assumed equal to unity in equations for PIMs, which is not the case for NIMs.
OPA and DFG in NIMs
-------------------
We introduce the normalized amplitudes $a_{j}=\sqrt[4]{{\epsilon_{j}}/{\mu _{j}}}{h_{j}}/\sqrt{\omega _{j}};$ their squared values are proportional to the number of photons at the corresponding frequencies. The corresponding equations for such amplitudes acquire the form $$\begin{aligned}
\dfrac{da_{1}}{dz} &=&-i{g}a_{2}^{\ast }\exp [i\Delta kz]+\frac{\alpha_{1}}{2}a_{1}, \label{a11} \\
\dfrac{da_{2}}{dz} &=&i{g}a_{1}^{\ast }\exp [i\Delta kz]-\frac{\alpha_{2}}{2}a_{2}, \label{a21}\end{aligned}$$ where ${g}=(\sqrt{\omega_1\omega_2}/\sqrt[4]{\epsilon_1\epsilon_2/\mu_1\mu_2})
({8\pi}/{c}){\chi^{(2)}}h_{3}$. Accounting for the boundary conditions $a_{1}(z=L)=a_{1L}$, and $a_{2}(z=0)=a_{20}$ (where $L$ is the slab thickness), the solutions to equations (\[a11\]) and (\[a21\]) are as follows $$\begin{aligned}
a_{1}(z) &=&A_{1}\exp [(\beta_{1}+i\frac{\Delta k}{2})z]+ \notag \\
&+&A_{2}\exp [(\beta_{2}+i\frac{\Delta k}{2})z], \label{a1z1} \\
a_{2}^{\ast }(z) &=&\kappa_{1}A_{1}\exp [(\beta_{1}-i\frac{\Delta
k}{2})z]+
\notag \\
&+&\kappa_{2}A_{2}\exp [(\beta_{2}-i\frac{\Delta k}{2})z].
\label{a2z1}\end{aligned}$$ Here, $$\begin{aligned}
&&\beta_{1,2}={(\alpha_{1}-\alpha_{2})}/{4}\pm
iR,\,\kappa_{1,2}=[\pm {R}+is]/g, \label{bet1} \\
&&R=\sqrt{g^{2}-s^{2}},\,s=({\alpha_{1}+\alpha_{2}})/{4}-i{\Delta
k}/{2},\label{r1} \\
&&A_{1}=\{a_{1L}\kappa_{2}-a_{20}^{\ast}\exp
[(\beta_{2}+i\frac{\Delta k}{2})L]\}/D, \label{A11} \\
&&A_{2}=-\{a_{1L}\kappa_{1}-a_{20}^{\ast}\exp
[(\beta_{1}+i\frac{\Delta k}{2})L]\}/D, \label{A21} \\
&&D=\kappa_{2}\exp [(\beta_{1}+i\frac{\Delta k}{2})L]-\kappa_{1}\exp
[(\beta_{2}+i\frac{\Delta k}{2})L].\,\qquad \label{D1}\end{aligned}$$
At $a_{20}=0$, the *amplification factor for the left-handed wave* is given by $\eta_{a}(\omega_{1})=\left\vert
{a_{10}}/{a_{1L}}\right\vert ^{2}$, where $$\frac{a_{10}}{a_{1L}}=\dfrac{\exp \left[ -\left(
\dfrac{\alpha_{1}-\alpha_{2}}{4}+i\dfrac{\Delta k}{2}\right)
L\right]}{\cos
RL+\left[\dfrac{\alpha_{1}+\alpha_{2}}{4R}-i\dfrac{\Delta
k}{2R}\right] \sin RL}. \label{amp11}$$ Alternatively, at $a_{1L}^{\ast }=0$, the *conversion factor for the difference-frequency generation of the left-handed wave* is found as $\eta_{g}(\omega_{1})=\left\vert {a_{10}}/{a_{20}^{\ast
}}\right\vert ^{2}$, where $$\frac{a_{10}}{a_{20}^{\ast }}=\dfrac{-(g/R)\sin RL}{\cos RL+\left[
\dfrac{\alpha_{1}+\alpha_{2}}{4R}-i\dfrac{\Delta k}{2R}\right] \sin
RL}. \label{gen11}$$ Equation (\[amp11\]) shows that the amplification of the left-handed wave can be turned into a *cavity-less oscillation* when the denominator tends to zero. The conversion factor for DFG, $\eta_{g}$, experiences a similar increase. In the case of $\Delta
k=0$ and small optical losses $(\alpha_{1}+\alpha_{2})L\ll \pi$, equations (\[a1z1\]) and (\[a2z1\]) are reduced to $$\begin{aligned}
a_{1}^{\ast }(z) &\approx &\frac{a_{1L}^{\ast }}{\cos (gL)}\cos
({gz})+\frac{
ia_{20}}{\cos (gL)}\sin [{g(z-L)}],\quad \label{lla11} \\
a_{2}(z) &\approx &\frac{ia_{1L}^{\ast }}{\cos (gL)}\sin
({gz})+\frac{a_{20} }{\cos (gL)}\cos [{g(z-L)}].\quad
\label{lgen21}\end{aligned}$$ The output amplitudes are then given by $$\begin{aligned}
a_{10}^{\ast } &=&\frac{a_{1L}^{\ast }}{\cos (gL)}-ia_{20}\tan ({gL}),
\label{a101} \\
a_{2L} &=&ia_{1L}^{\ast }\tan ({gL})+\frac{a_{20}}{\cos ({gL})}.
\label{a2l1}\end{aligned}$$ Thus, the oscillation threshold value for the control field intensity in this case is given by $g_{t}=\pi /2L$. It increases with absorption and phase mismatch.
![The amplification factor $\protect\eta_{a}(\protect\omega_{1})$ (the solid line) and the efficiency of difference-frequency generation $\protect\eta_{g}(
\protect\omega_{1})$ (the dashed line) for the backward wave at z=0. $\protect\alpha_{1}L=1 $, $\protect\alpha_{2}L=1/2$. The upper plot: $\Delta k=0$.The lower plot $\Delta k=\protect\pi $.[]{data-label="fig2"}](fig2a.eps "fig:"){width=".35\textwidth"} ![The amplification factor $\protect\eta_{a}(\protect\omega_{1})$ (the solid line) and the efficiency of difference-frequency generation $\protect\eta_{g}(
\protect\omega_{1})$ (the dashed line) for the backward wave at z=0. $\protect\alpha_{1}L=1 $, $\protect\alpha_{2}L=1/2$. The upper plot: $\Delta k=0$.The lower plot $\Delta k=\protect\pi $.[]{data-label="fig2"}](fig2b.eps "fig:"){width=".35\textwidth"}
![Resonant changes in the distribution of the normalized intensity of the left-handed wave inside the slab of NIM, $\protect\eta_{a}(\protect\omega_{1})$ (the solid line) and $\protect\eta_{g}( \protect\omega_{1})$ (the dashed line), caused by the adjustment of the normalized intensity for the control field at $\omega_{3}$, $gL$. $\protect\alpha_{1}L=1$, $\protect\alpha_{2}L=1/2$, $\Delta k=0$.[]{data-label="fig3"}](fig3a.eps "fig:"){width=".36\textwidth"} ![Resonant changes in the distribution of the normalized intensity of the left-handed wave inside the slab of NIM, $\protect\eta_{a}(\protect\omega_{1})$ (the solid line) and $\protect\eta_{g}( \protect\omega_{1})$ (the dashed line), caused by the adjustment of the normalized intensity for the control field at $\omega_{3}$, $gL$. $\protect\alpha_{1}L=1$, $\protect\alpha_{2}L=1/2$, $\Delta k=0$.[]{data-label="fig3"}](fig3b.eps "fig:"){width=".36\textwidth"}
The dependence of the output intensity for the left-handed wave propagating in an absorptive NIM slab in the presence of the control field at $\omega_{3}$ and at $a_{20}=0$ is shown with the solid line in Fig. \[fig2\] for two representative cases of exact and partial phase-matching. The dash plots show the output in the case of DFG (at $a_{1L}=0$, $a_{2,0}\neq 0$). Amplification in the upper part of Fig. \[fig2\] reaches many orders in the first maximum and increases in the next maximums. It is seen that *the amplification can entirely compensate for absorption and even turn into oscillations* when the intensity of the control field reaches values given by a periodic set of increasing numbers. The larger the corresponding value, the greater is the amplification and the DFG output; the latter depends on the absorption for both waves and on the phase mismatch $\Delta k$. The conversion factor is larger in its maximums than the amplification factor because DFG is a one-step process, whereas OPA is a two-step process as discussed in Subsection \[mrr\]. As seen from Fig. \[fig2\], the output shows a resonance dependence on the intensity of the control field at $\omega_{3}$. Figure \[fig3\] depicts corresponding changes in the distribution of the negative-index field inside the slab in the vicinity of the first maximum at $\Delta k=0$.
Conclusion
==========
We have studied the unusual properties of second-harmonic generation (SHG) in metamaterials that have a negative refractive index for the fundamental wave and a positive index for its second harmonic (SH). The possibility of a left-handed nonlinear-optical mirror, which converts the incoming radiation into a reflected beam at the doubled frequency with efficiency that can approach 100% for lossless and phase-matched medium is considered. The most striking differences in the nonlinear propagation and the spatial dependence of the energy-conversion process for SHG in NIMs, as compared to PIMs, can be summarized as follows. In NIMs, the intensities of the fundamental and SH waves both decrease along the medium. Such unusual dependence and the apparent contradiction with the ordinary Manley-Rowe relations are explained by the fact that the energy flows for the fundamental and SH waves are counter-directed, whereas their wave-vectors are co-directed. Another interesting characteristic of SHG in NIMs is that the energy conversion at any point within a NIM slab depends on the total thickness of the slab. This is because SHG in a NIM is determined by the boundary condition for SH at the rear interface rather than the front interface of the slab.
We have shown the feasibility of compensating losses in NIMs by optical parametric amplification (OPA). In this process, the wave-vectors of all three coupled waves are co-directed, whereas the energy flow for the signal wave is counter-directed with respect to those for the pump and the idler waves. This process is characterized by properties that are in strict contrast with those known for conventional nonlinear-optical crystals. Such extraordinary features allow one to realize optical parametric oscillations (OPOs) without a cavity at frequencies where the refractive index is negative. We also showed that the OPA and OPO in NIMs enable the generation of pairs of entangled counter-propagating right- and left-handed photons inside the NIM slabs.
The backward energy flow for one of the coupled waves (whereas the wave-vectors of all the coupled waves are co-directed) is inherent for NIMs and it makes this process different from three-wave mixing in PIMs. This is also different from various processes in RHMs based on distributed gratings and feedback. The important advantage of the backward OPA and OPO in NIMs investigated here is the distributed feedback, which enables oscillations without a cavity. In NIMs, each spatial point serves as a source for the generated wave in the reflected direction, whereas the phase velocities of all the three coupled waves are co-directed. As mentioned, it is very hard to realize such a scheme in PIMs, while the OPA in NIMs proposed herein is free from the PIM limitations.
Acknowledgments
===============
The authors are grateful to V. V. Slabko for useful discussions and to S. A. Myslivets for help with numerical simulations. This work was supported in part by NSF-NIRT award ECS-0210445, by ARO grant W911NF-04-1-0350, and by DARPA under grant No. MDA 972-03-1-0020.
[99]{} V. M. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Optics Letters 30, 3356 (2005); first reported in arXiv: physics/0504091 (April. 13, 2005).
S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, Phys. Rev. Lett. 95, 137404 (2005); arXiv: physics/0504208 (2005)
A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, Phys. Rev. Lett. 93, 73902 (2004).
E. Schonbrun, M. Tinker, W. Park and J.-B. Lee, IEEE Photon. Technol. Lett. 17, 1196 (2005)
V.G. Veselago, Sov. Phys. Solid State **8**, 2854,(1967); V.G. Veselago, Usp. Fiz. Nauk **92**, 517 (1967) \[Sov. Phys. Usp. 10, 509,(1968)\].
S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, Science, vol. 306, pp. 1351-1353 (2004).
Z. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, Phys. Rev. Lett. 94, pp. 037402 (2005).
A. N. Grigorenko, A. K. Geim, N. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, Nature, **438**, 335 (2005).
M. Lapine, M. Gorkunov and K. H. Ringhofer, Phys. Rev. E **67**, 065601 (2003); A.A. Zharov, I.V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. **91**, 037401 (2003); M. Lapine and M. Gorkunov, Phys. Rev. E **70**, 66601 (2004); N. A. Zharova, I.V. Shadrivov, A.A. Zharov, and Yu. S. Kivshar, Optics Express **13**, 1291 (2005).
V.M. Agranovich, Y.R. Shen, R.H. Baughman and Zakhidov, Phys. Rev. B **69**, 165112(2004).
A.A. Zharov, N. A. Zharova, Il.V. Shadrivov and Yu. S. Kivshar, Appl. Phys. Lett. **87**, 091104-3 (2005).
A. B. Kozyrev, H. Kim, A. Karbassi and D. W. van der Weide, Appl. Phys. Lett. **87**, 121109 (2005).
I.V. Shadrivov, A.A. Zharov, and Yu. S. Kivshar, arXiv: physics/0506092 (2005).
S.E. Harris, Appl. Phys. Lett., **9**, 114, (1966); K. I. Volyak and A. S. Gorshkov, Radiotekhnika i Elektronika (Radiotechnics and Electronics) **18**, 2075 (1973) (Moscow, in Russian); A. Yariv, Quantum Electronics, 2d ed., New York: Wiley, 1975, Ch. 18.
J. M. Manley and H. E. Rowe, Proc. IRE **47**, 2115 (1959).
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abstract: 'The effect of a periodic pinning array on the vortex state in a 2D superconductor at low temperatures is studied within the framework of the Ginzburg-Landau approach. It is shown that attractive interaction of vortex cores to a commensurate pin lattice stabilizes vortex solid phases with long range positional order against violent shear fluctuations. Exploiting a simple analytical method, based on the Landau orbitals description, we derive a rather detailed picture of the low temperatures vortex state phase diagram. It is predicted that for sufficiently clean samples application of an artificial periodic pinning array would enable one to directly detect the intrinsic shear stiffness anisotropy characterizing the ideal vortex lattice.'
author:
- 'V.Zhuravlev$^{1}$ and T.Maniv$^{1,2}$'
title: 'Vortex states in 2D superconductor at high magnetic field in a periodic pinning potential. '
---
Introduction
============
The nature of the vortex lattice melting transition in 2D superconductors has been debated in the literature for many years [@mzvwrmp]. Early proposals [@doniahub]$^{,}$[@fisher80], based on the similarity to the Kosterlitz-Thouless-Halperin-Nelson-Young theory of melting in 2D solids[@kthny], have led to the conclusion that the melting transition is continuous. A weak first order melting transition was predicted more recently, however, by several Monte Carlo simulations [@tesanovic]$^{,}$[@humacdon] using the Ginzburg-Landau (GL) theory. It has been shown recently [@mzvwrmp]$^{,}$[@zm99]$^{,}$[@zm02] that shear motions of Bragg chains along the principal crystallographic axis of the vortex lattice cost a very small fraction of the SC condensation energy and are responsible for the low temperature vortex lattice melting.
This intrinsic anisotropy of the vortex lattice with respect to shear stress can not be easily detected experimentally since the orientation of the principal axis with respect to the laboratory frame depends on the local pinning potential, which in real superconductors is usually produced by random distribution of pinning centers. Indirect experimental detection of this hidden anisotropy may be achieved by means of the small angle neutron scattering (SANS) technique, due to the 1D nature of the effective thermal fluctuations in the vortex liquid state just above the melting point (see Ref.[@zm02]). A direct detection of this anisotropy (e.g. by means of SANS) could be possible if vortex solid phases with long range positional order were stabilized against the random influence of pinning impurities. This can be achieved by exposing the SC sample to an artificial periodic pinning array and tuning the magnetic flux density to an integer multiple of the pinning centers density. As will be shown in this paper, under certain conditions the artificial periodic pinning potential can stabilize weakly pinned vortex solid phases with long range positional order, which may exhibit the shear stiffness anisotropy characterizing the ideal vortex lattice.
Vortex matter interacting with periodic pinning arrays is currently a subject of intense experimental [@fiory78]$^{-}$[@terentiev00] and theoretical [@nelson79]$^{-}$[@pogosov02] investigations. Developments of nano-engineering techniques, such as e-beam lithography, make it possible to fabricate well defined periodic arrays of sub-micron antidotes, or magnetic dots, in SC films with low intrinsic pinning, enabling to study the effect of well controlled artificial pinning centers. These experiments have shown that under certain conditions the underlying artificial pinning centers can attract vortices very strongly, thus stabilizing vortex patterns with global translational symmetry against the randomize influence of the natural pinning centers.
From theoretical point of view the utilization of an external periodic pinning potential provides a convenient tool for testing different models of the vortex state by simplifying considerably the model calculations. At the same time, however, the interplay between the vortex-vortex interactions, which favor hexagonal vortex lattice symmetry, and the underlying periodic potential can lead to a variety of vortex configurations, depending on the pinning strength, in which vortices detach from pinning centers to form more closely packed vortex patterns.
As the interaction with a periodic substrate stabilizes the vortex system versus thermal fluctuations, it generally increases the melting temperature. However, as we shall see in this paper, deviation from the ideal hexagonal symmetry due to pinning reduces the phase dependent interaction between vortex chains [@zm02], making them less enduring under thermal fluctuations. In the weak pinning limit, where depinned floating state can occur, the corresponding phase diagram becomes rather complicated, due to the possibility of transitions between floating solid and pin solid phases [@reichhardt01].
In the present paper we study the influence of a periodic pinning substrate on the vortex state in 2D , extreme type II superconductors, at perpendicular high magnetic fields. Our approach is based on the previously developed theory of vortex lattice melting in pure superconductors ([@mzvwrmp]$^{,}$[@zm99]$^{,}$[@zm02]), carried out within the framework of the GL theory in the lowest Landau level (LLL) approximation. Specializing the calculation for a vortex system interacting with a square pinning array under the first matching magnetic field, we study in detail some key limiting regions of the vortex phase diagram, which enables us to determine its main qualitative features.
The model {#sec:1}
=========
We consider a 2D superconductor at high perpendicular magnetic field, interacting with a periodic substrate of pinning centers, located at $\left(
x_{i},y_{j}\right) $. A phenomenological Ginzburg-Landau functional, with an order parameter $\psi \left( x,y\right) $, is used to describe the superconducting (SC) part of the free energy, and a local periodic pinning potential [@tesanovic94]: $$V_{pin}=v_{0}\sum_{i,j}|\psi \left( x_{i},y_{j}\right) |^{2} \label{Vpin}$$ determines the interaction of the vortex state with pinning centers. We assume $%
v_{0}>0$, so that the pinning energy is minimal if the vortex core positions, determined by $\psi \left( x_{i},y_{j}\right) =0$, coincide with pinning centers.
Our main interest here is in the influence of the pinning potential on the vortex lattice melting process, so that the pinning energy $V_{pin}$ is restricted to the range of the vortex lattice melting energy, which is much smaller than the SC condensation energy. Since the latter is of the same magnitude as the cyclotron energy, it is justified to restrict the analysis to the LLL of the corresponding SC order parameter, which can be therefore written as a linear combination of ground Landau orbitals: $$\begin{aligned}
\psi \left( x,y\right) &=&\sum_{n}c_{n}\phi _{q_{n}}\left( x,y\right)
\label{LOrep} \\
c_{n}=|c_{n}|e^{i\varphi _{n}}; &\hspace{0.5cm}&\phi _{q}\left( x,y\right)
=e^{2iqx-\left( y+q\right) ^{2}} \nonumber\end{aligned}$$ where $q_{n}=qn$, $q=\pi /a_{x}$, and the amplitudes $c_{n}$ in the mean field approximation are related to the (spatial) mean square SC order parameter, $\Delta _{0}^{2}$ , through: $|c_{n}|^{2}=c_{0}^{2}=\sqrt{\frac{%
2q^{2}}{\pi }}\Delta _{0}^{2}$. In our notations all space variables are measured in units of magnetic length.
In this model, due to the Gaussian attenuation along the $y$-axis over a characteristic distance of the order of the magnetic length, the vortex cores (located at the zeros of $\psi \left( x,y\right) $ ) form a network of linear chains along the $x$-axis, each of which is determined mainly by a superposition of two neighboring Landau orbitals [@zm02]. The parameter $a_{x}$ is therefore equal to the inter-vortex distance within a chain, while $\pi /a_{x}$ is the inter-chain spacing in the $y$-direction (see Fig. \[fig:0\]). It should be noted that deviations of Landau orbital (LO) amplitudes from their mean field value, $c_{0}$, resulting in strong local distortions of the superfluid density and large increase of the corresponding free energy density, are neglected in comparison with variations of the phase variables, $\varphi _{n}$, provided the orbital direction is selected to be along the principal crystallographic axis [@zm99]$^{,}$[@zm02].
We select the pinning centers to form a rectangular lattice $$\left( x_{i},y_{j}\right) =\left( l_{x}i+x_{0},l_{y}j+y_{0}\right)
\label{eq:sqlat}$$ where $i,j=0,\pm 1,....$. The parameters $x_{0}$ and $y_{0}$ determine the relative position of the pin and vortex lattices. The nature of the vortex state in the presence of the pinning potential depends crucially on the ratio of the number of vortices, $N=\sqrt{N}\times \sqrt{N}$, to the number of pinning centers, $N_{p}=N_{p,x}\times N_{p,y}$ . Since the density of vortices depends on the external magnetic field strength $H$, one can tune this ratio by varying $H$. Of special interest are the matching fields $%
H=H_{\nu }$ , $\nu =1,2,...$, when the ratio $n_{\phi }\equiv N/N_{p}=\nu $ is an integer.
In matching fields one may distinguish between two different situations, when vortices are bound or unbound to pinning centers. If the pin lattice and the vortex lattice unit cells are commensurate along both $x$ and $y$ directions , i.e. $l_{x}=c_{x}a_{x}$ , $l_{y}=c_{y}\pi /a_{x}$, with $c_{x}$ and $c_{y}$ being integers, the pinning energy is equal to zero, since all the vortices coincide with pinning centers. In all other cases of matching fields , $c_{x}c_{y}=\pi \times {\mathtt{integer}}$ , non of the numbers $%
c_{x}\geq 1$ and $c_{y}\geq 1$ can be integer , and the lattice constants are incommensurate in both directions. It will be shown below that such a vortex configuration is in a floating state with respect to the pin lattice, similar to vortex states in mismatching magnetic fields.
![Schematic arrangement of a vortex lattice relative to the pin lattice.[]{data-label="fig:0"}](figure1.eps){width="6cm"}
Using the LO representation, Eq.(\[LOrep\]), of the SC order parameter in Eq.(\[Vpin\]) for the pinning energy, one may take advantage of the localized nature of the LOs and expand $V_{pin}$ in the small parameter $%
\lambda =e^{-q^{2}}$, which reflects the small overlap integral between adjacent orbitals contributing to the local superfluid density at the pinning centers. Retaining only dominant terms in $\lambda $ Eq. (\[Vpin\]) is reduced to the form: $$\begin{aligned}
V_{pin} &=&V_{0}\sqrt{\frac{2q^{2}}{\pi }}\sum_{k}\left[ u_{k}+2%
\sum_{m=1}e^{-q^{2}m^{2}/2}u_{k+m/2}\Phi _{k,m}\right] \nonumber \\
&\simeq &V_{0}\sqrt{\frac{2q^{2}}{\pi }}\sum_{k}\left[
u_{k}+2e^{-q^{2}/2}u_{k+1/2}\Phi _{k,1}\right] \nonumber \\
u_{k+m/2} &=&\frac{1}{N_{p,y}}\sum_{j}e^{-2\left( y_{j}+q\left( k+m/2\right)
\right) ^{2}} \nonumber \\
\Phi _{k,m} &=&\frac{1}{N_{p,x}}\sum_{i}\cos \left( \varphi _{k+m}-\varphi
_{k}+2mqx_{i}\right) \label{eq:3}\end{aligned}$$ where $V_{0}=v_{0}\Delta _{0}^{2}N_{p,x}N_{p,y}$. It should be stressed that this approximation is valid only for LOs along the principal axes since the minimal distance between them, $q=\pi /a_{x}$, is sufficiently large to ensure small and rapidly decreasing value of the overlap integrals between more distant orbitals.
If $c_{x}$ is not an integer, namely the rectangular pin lattice and vortex lattice are incommensurate in $x$-direction, then Eq.(\[eq:3\]) shows that $\Phi _{k,m}=0$. In this case the pinning energy does not depend on the phases (i.e. the relative horizontal positions ) of the Landau orbitals.
Expressing the functions $u_{k}$ and $u_{k+1/2}$ with the help of Poisson summation formula as $$\begin{aligned}
&&u_{k+m/2}=\frac{1}{N_{p,y}}\sum_{j}e^{-2\left( l_{y}j+y_{0}+q\left(
k+.5\right) \right) ^{2}}\approx \frac{1}{N_{p,y}}\sqrt{\frac{\pi }{%
2l_{y}^{2}}}\times \nonumber \\
&&\left[ 1+2\sum_{j}e^{-\frac{\pi ^{2}j^{2}}{2l_{y}^{2}}}\cos \left( \frac{%
2\pi j\left( q(k+\frac{m}{2})+y_{0}\right) }{l_{y}}\right) +...\right]
\label{eq:6}\end{aligned}$$ we note that when the lattices are incommensurate also along the $y$ axis ( i.e. when both $c_{x}$ and $c_{y}$ are not integer) the oscillating terms in $u_{k}$ are averaged to zero after summation over $k$. Thus, the pinning energy for incommensurate lattices is a constant $$V_{pin}=V_{0}\frac{q\sqrt{N}}{l_{y}N_{p,y}}=V_{0}$$ which does not depend on the mutual orientation of the vortex and the pin lattices. Note that the system size in $y$ direction is $L_{y}=q\sqrt{N}%
=l_{y}N_{p,y}$ , a relation connecting $\sqrt{N}$ to $N_{p,y}$. Obtained result is valid only for large system, $N\longrightarrow \infty $, where the boundary effect can be neglected.
For the sake of simplicity, we will consider in what follows a square pin lattice with $n_{\phi }=1$. In the commensurate situation the pinning energy is minimal ( i.e. equal to zero) when all vortices coincide with pinning centers. Deviations of vortices from this configuration in the form of shear distortions along the principal crystallographic axes are of special interest due to the relatively low SC energy involved. For the principal axis parallel to a side of the square unit cell, $c_{x}=c_{y}=1$ and $%
q^{2}=\pi $, and so, according to Eq. (\[eq:3\]), the pinning energy per single vortex is, up to small terms of the order $\sim e^{-2\pi }$, given by: $$\begin{aligned}
\frac{V_{pin}}{N} &=&v\frac{1}{\sqrt{N}}\sum_{k}\left[ a_{1}-a_{2}\cos
\left( \varphi _{k}-\varphi _{k-1}\right) \right] \nonumber \\
&\simeq &\kappa _{x}v\frac{1}{\sqrt{N}}\sum_{k}\left[ 1-\cos \left( \xi
_{k}\right) \right] \label{eq:15}\end{aligned}$$ where $v=V_{0}/N$ , $a_{1}=1-2e^{-\pi /2}\simeq .584$ and $a_{2}=2e^{-\pi
/2}\left( 1+2e^{-\pi /2}\right) \simeq .589$ . Note that in the above expressions we set $x_{0}=y_{0}=q/2$ so that the minimal pinning energy is obtained for $\xi _{k}=0$. Note also that for the undistorted square lattice in which $\varphi _{k}=\varphi _{k-1}$, the expression in the first line of Eq. (\[eq:15\]) is not strictly zero since $a_{1}\neq a_{2}$. The error, which is of order higher than the second in $e^{-\pi }$ , can be neglected in the approximation leading to Eq. (\[eq:15\]). The numbers $a_{1},a_{2}$ can be thus considered equal within this approximation, allowing us to introduce a single coefficient $\kappa _{x}\equiv a_{1}\simeq a_{2}\simeq
.59 $. The expression in the second line of Eq. (\[eq:15\]) yields the correct (i.e. zero) value for the undistorted lattice. It is written in terms of the variables, $\xi _{k}\equiv \varphi _{k}-\varphi _{k-1}$, describing the lateral positions of the vortex chains, which are generated mainly by interference between two neighboring LOs. This is consistent with the well known definition $u_{x}=\partial \varphi /\partial y$ of vortex displacement along the $x$ axis in the long wavelength limit [@Moore89].
To evaluate the excess pinning energy associated with shear distortion along the diagonal of the square unit cell the pin lattice may be conveniently described by two interpenetrating simple square sub-lattices with $%
c_{x}=1,c_{y}=2$ and $q^{2}=2\pi $ ( see Fig.\[fig:2n\] ). The corresponding interchain pinning energy for each of the sublattices can again be obtained from Eq. (\[eq:15\]), with $\kappa _{x^{\prime }}\simeq
.84$ and a phase shift of $\pi /2$ , which arises due to different shape of the unit cell.
![Primitive and none-primitive unit cell representations (solid and dotted lines respectively) of the square pin lattice used for description of shear distortion along the principal axes of the vortex lattice.[]{data-label="fig:2n"}](figure2.eps){width="6cm"}
The SC part of the free energy functional for the commensurate lattices described above ($a_{x}=\sqrt{\pi }$) is given by the following ($\xi _{k}$-dependent) expression [@zm99]: $$\frac{H_{sc}}{N}=-h_{\Box }-T_{\Box }\frac{1}{\sqrt{N}}\sum_{k}\left[ 1-\cos
(\xi _{k+1}-\xi _{k})\right] \label{eq:sqlaten}$$ where $h_{\Box }$ is the SC condensation energy (per unit flux) of the square vortex lattice, and $T_{\Box }=\frac{4\lambda _{sq}^{2}}{1+4\lambda
_{sq}}h_{\Box }$ is the shear distortion energy parameter, expressed through the dimensionless interchain coupling constant, $\lambda _{sq}=\exp (-\pi )$. Here $h_{\Box }=\varepsilon _{0}\frac{\beta _{A}}{\beta _{sq}}$ , where $%
\beta _{A}\simeq 1.159$ and $\beta _{sq}\simeq 1.18$ , are the values of the Abrikosov structure parameter for regular hexagonal and square lattices respectively, and $\varepsilon _{0}$ is the SC condensation energy of the former.
For the specific choice $\xi _{k}=\gamma k$, where $\gamma $ is a constant, the Bragg family of vortex chains along the principal axis,denoted $x$, is characterized by a lateral displacement, $\xi
_{k+1}-\xi _{k}=\gamma $, between neighboring chains. Evidently, the SC energy, $H_{sc}$ , for the undistorted square vortex lattice $\xi _{k}=0$ ( $\gamma =0$ ) ( see Eq. (\[eq:sqlaten\]) ), is equal to $Nh_{\Box }$. However, the minimum of the SC energy with respect to the collective tilt angle parameter $\gamma $ is reached for a triangular vortex lattice, determined by $\xi _{k}=\pi k$ ($%
\gamma =\pi $), whose unit cell is an isosceles triangle with a base ( along $x$-axis) and a height equal to $\sqrt{\pi }$ (see Fig. \[fig:3n\]). The corresponding SC energy is equal to $H_{sc}/N=-h_{\Box }-2T_{\Box }$. This value is lower than the SC energy of the square vortex lattice, and only slightly higher ( i.e. by $\sim .45\%$) than the SC energy of the equilateral triangular (Abrikosov) lattice, $H_{\triangle }/N=-\varepsilon
_{0}$.
![(a) The vortex lattice state with the lowest energy, which is commensurate with a square pinning lattice in the limit of zero pinning strength. (b) An alternative vortex lattice state, which may be favorable under square framework boundary conditions (see text).[]{data-label="fig:3n"}](figure3a.eps "fig:"){width="6cm"}\
*(a)*\
![(a) The vortex lattice state with the lowest energy, which is commensurate with a square pinning lattice in the limit of zero pinning strength. (b) An alternative vortex lattice state, which may be favorable under square framework boundary conditions (see text).[]{data-label="fig:3n"}](figure3b.eps "fig:"){width="6cm"}\
*(b)*
Vortex states for the lowest matching field
===========================================
Commensurate and incommensurate Ground states {#sec:3.1}
----------------------------------------------
The competition between the pinning energy, Eq. (\[eq:15\]), which favors vortices approaching the pinning points on a square lattice, and the SC energy, Eq. (\[eq:sqlaten\]), preferring triangular lattice configuration, leads to ’frustrated’ vortex structures, which depend on the relative pinning strength.
At zero temperature they can be obtained by minimizing the total energy, consisting of the SC and pinning parts. Since in the LO representation each orbital is $\sqrt{N}$-fold degenerate, the effective Hamiltonian is written as: $$\begin{aligned}
f_{\Box }&=&\frac{H_{sc}+V_{pin}}{N}=\frac{T_{\Box }}{\sqrt{N}}\sum_{k} \Big\{
-h_{\Box }/T_{\Box } \nonumber \\
&+&4p\left[ 1-\cos (\xi _{k})\right] -\left[ 1-\cos (\xi _{k+1}-\xi _{k})\right]
\Big\}\label{eq:low_ph}\end{aligned}$$ where the parameter $p\equiv \kappa _{x}v_{\Box }/4T_{\Box }$ determines the strength of the pinning potential relative to the inter vortex chain coupling. Under the constraints imposed by the requirement of commensurability between the vortex configuration and the pin lattice, the vortex chains are restricted to move laterally along the common $x$-axis of the underlying lattices (see Fig. \[fig:0\]). The corresponding displacements, $\xi _{k}$ , may be separated into two groups, corresponding to even and odd vortex chains, as follows: $$\xi _{k}=\left\{
\begin{array}{lll}
\theta _{l} & \textrm{for} & k=2l \\
\zeta _{l} & \textrm{for} & k=2l-1
\end{array}
\right.$$ so that $$\begin{aligned}
f_{\Box }&=&\frac{T_{\Box }}{\sqrt{N}}\sum_{l}\Big\{ -2h_{\Box }/T_{\Box }+
4p\left[ 2-\cos (\theta _{l})-\cos (\zeta _{l})\right] \nonumber \\
&-&\left[ 2-\cos (\theta _{l}-\zeta _{l})-\cos (\theta _{l-1}-\zeta _{l})\right]
\Big\} \label{eq:low_ph1}\end{aligned}$$
The calculation may be greatly simplified if we assume that the stationary point values within each group are all equal, that is: $\theta _{l}=\theta
_{c}$ and $\zeta _{l}=\zeta _{c}$. This restriction may be justified in the weak pinning regime $0<p<1$ , where the dominant SC energy part favors periodic triangular vortex structures, as shown in Fig. \[fig:3n\]a.
Substituting these values to Eq. (\[eq:low\_ph1\]) and minimizing the resulting functional one finds that the total energy has a minimum: $$\min \left( f_{\Box }\right) =-\frac{1}{\sqrt{N}}\sum_{k}\left[ 2T_{\Box
}(1-p)^{2}+h_{\Box }\right] \label{eq:low_ph2}$$ at $\theta _{c}=-\zeta _{c}=\theta _{0}$ where $$\cos (\theta _{0})=p. \label{eq:gr_st}$$ Thus, at zero pinning strength, $p=0$, the ground state energy per unit flux, $f_{\Box }=-2T_{\Box }-h_{\Box }$, corresponds to a triangular vortex lattice configuration, $\theta _{c}=-\zeta _{c}=\pi /2$ , whereas in the opposite extreme, when $p=1$ , the ground state energy, $f_{\Box }=h_{\Box }$, corresponds to a square vortex lattice, $\theta _{c}=-\zeta _{c}=0$ , which coincides with the underlying pin lattice. It should be stressed, however, that due to the constraints imposed by the requirement of commensurability with the pin square lattice, the triangular vortex structure obtained in the zero pinning strength limit, is not the equilateral (Abrikosov) lattice (see Fig.\[fig:3n\]a). This discontinuity indicates that the transition to the depinned (floating) vortex lattice should be of the first order (see below).
An illustration of the weak pinning ground state configuration is shown in Fig. \[fig:3n\]a. It is seen that odd and even vortex chains are shifted in opposite directions symmetrically with respect to the underlying substrate. The relative positions of the two lattices are determined by the strength of the pinning potential, $V_{0}$. In the zero pinning limit, $%
V_{0}\rightarrow 0$, the vortices in odd (even) chains approach lattice points which are shifted laterally by a quarter of a lattice constant, $%
\frac{1}{4}l_{x}$ ( $l_{x}=\sqrt{\pi }$ ), in the positive (negative) sense with respect to the square pin lattice, forming isosceles triangular lattice. Note that the asymmetric configuration, shown in Fig.\[fig:3n\]b, in which half of the vortex chains remain pinned to the underlying substrate, has energy $-2T_{\Box }(1-2p)-h_{\Box }$, which is only slightly (i.e. by a small, second order correction in $p$ ) higher than the energy given by Eq. (\[eq:low\_ph2\]). Such an asymmetric configuration may become energetically favorable (see Ref. [@reichhardt01]$^{,}$[@pogosov02]) due to, e.g. boundary conditions which are incompatible with the even-odd chain symmetry described above.
At sufficiently weak pinning, when the pinning energy becomes comparable to the difference between the SC energies of the commensurate isosceles triangular vortex lattice with $a_{x}^{2}=\pi $ , and the incommensurate equilateral triangular lattice with $a_{x}^{2}=\sqrt{3}\pi /2$ (Abrikosov lattice), the latter is preferable. To show this note that the energy, $%
-h_{\triangle }$ , of the equilateral triangular vortex lattice in the presence of incommensurate pin lattice is influenced only by the average pinning potential $v$, so that: $$-h_{\triangle }=-\varepsilon _{0}+v=-\varepsilon _{0}+\frac{4}{\kappa _{x}}%
pT_{\Box }$$ Comparing this value with that obtained in Eq. (\[eq:low\_ph2\]) for the commensurate, isosceles triangular lattice, $-2T_{\Box }(1-p)^{2}-h_{\Box }$, we find that for $p\leq $ $p_{c}\simeq .25$ the floating equilateral triangular lattice is the lowest energy state. This critical point can be thus identified as a transition point from pinned (commensurate) solid to a floating (incommensurate) solid state.
A second critical point exists in the strong pinning regime, i.e. at $p=1$, as indicated by Eq.(\[eq:gr\_st\]), which has no real solution at any $p>1$. At this critical point the vortex lattice coincides with the square pin lattice ( i.e. $\theta _{0}=0$ in Eq.(\[eq:gr\_st\])) and the pinning energy reaches its absolute minimum value (i.e. zero ). Since any further increase of the pinning strength above the critical value, $p=1$, can not be compensated by the SC energy terms in Eq.(\[eq:low\_ph1\]), the vortex configuration remains fixed at the square lattice structure for any $p\geq 1$. Thus, with increasing values of the parameter $p$ , the ground state vortex configuration changes continuously from a triangular lattice at $p=p_{c}$ , into a square lattice at $p=1$ , which does not changes with further increase of the pinning strength. This continuous transformation from a triangular lattice to a square lattice can be classified as a second order phase transition at $p=1$.
Commensurate equilibrium states at finite temperature {#sec:4}
-----------------------------------------------------
In the ideal vortex state at finite temperature thermal fluctuations associated with the low-lying shear excitations along the principal crystallographic axis destroy the long range phase coherence of the vortex state and lead to melting of the ideal vortex lattice at a temperature, $%
T_{m}$, well below the mean field $T_{c}$. This feature indicates an intrinsic anisotropy of the ideal vortex crystal [@zm02]: The characteristic excitation energy for sliding vortex chains along the principal axis ( denoted by $x$ ) parallel to a side of the unit cell is two orders of magnitude smaller than the SC condensation energy, and one order of magnitude smaller for fluctuations along the diagonal axis ( denoted by $%
x^{\prime }$). For all other crystallographic orientations the shear energy is of the order of the SC condensation energy.
{width="8cm"}
. \[fig:4\]
The nucleation of a SC crystallite can be established in such an ideal model by selecting boundary conditions which fix the position of a single vortex chain with respect to the laboratory frame. As shear fluctuations of parallel vortex chains diverge with the distance from the fixed chain [@zm99], a SC domain is restricted to nucleate only around a pinned chain, its transverse size shrinking to that of a single magnetic length as the temperature rises toward $T_{m}$. For the sake of simplicity, we avoid here the complication associated with the discontinuous nature of the vortex lattice melting process, which involves two principal families of easily sliding Bragg chains [@mzvwrmp], and restrict the analysis to a single family of vortex chains, i.e. that with the lowest crossover temperature $%
T_{cm}$ [@zm02]. A meaningful definition of $T_{cm}\left( a_{x}\right)
$ may invoke the phase correlation function, $$C_{n^{\prime },n}\equiv \langle e^{i\left( \varphi _{n^{\prime }}-\varphi
_{n}\right) }\rangle \label{Cn'n}$$ between Landau orbitals, $\ n^{\prime }$ and $n\ $,$\ $located near the fixed chain $n=0$ . $\ $Thus, melting of the entire vortex lattice occurs essentially when phase correlation between the nearest neighboring chains ( i.e. $\ n=1$, $n^{\prime }=2$ ) closest to the fixed chain is significantly suppressed (e.g. by a factor of 1/2). In the $p\rightarrow 0$ limit we use the expression derived in Ref.([@zm99]) to find: $$C_{n^{\prime }=2,n=1}\left( \widetilde{\tau }\right) \simeq \left( \frac{%
I_{1}(\widetilde{\tau })}{I_{0}(\widetilde{\tau })}\right) \left( \frac{%
I_{1/2}(\widetilde{\tau })}{I_{0}(\widetilde{\tau })}\right)$$ with $\widetilde{\tau }\equiv T_{\triangle }/T$ . Here the characteristic temperature, $T_{\triangle }\simeq \frac{4\lambda ^{2}}{1+4\lambda }%
\varepsilon _{0}$ with $\lambda =\exp (-\sqrt{3}\pi /2)$, corresponds to interaction between the principal LOs in the equilateral triangular lattice state. Note that the crossover between the vortex solid state at zero temperature, where $\widetilde{\tau }\rightarrow \infty $ , and $%
C_{n^{\prime }=2,n=1}\left( \widetilde{\tau }\right) \rightarrow 1$ , and the vortex liquid state at high temperature, where $\widetilde{\tau }%
\rightarrow 0$ , and $C_{n^{\prime }=2,n=1}\left( \widetilde{\tau }\right)
\rightarrow 0$ , occurs at about $\widetilde{\tau }\simeq 1.5$ , so that $%
T_{cm}\left( a_{x}\right) \approx .67T_{\triangle }$ (see Fig. \[fig:4\]). This crossover temperature is close to , though somewhat lower than the melting temperature, $T_{m}\simeq 1.2T_{\triangle }\simeq 2.8T_{\Box }$, predicted in Ref.( [@zm99]).
The presence of the periodic pinning potential stabilizes the vortex lattice against the violent phase fluctuations discussed above. This effect is nicely demonstrated by the phase correlation function $C_{n^{\prime },n}$ ( Eq.(\[Cn’n\]) ), which controls the mean superfluid density (see Eq.(\[LOrep\])) near the melting point. Assuming strong pinning, $p\gg 1$, and neglecting the small GL inter-vortex-chain coupling, the correlation function can be determined from the expression: $$C_{n^{\prime },n}\simeq \frac{\prod_{k}\int_{0}^{\pi }d\xi _{k}e^{i\left(
\varphi _{n^{\prime }}-\varphi _{n}\right) }e^{-4p\tau \cos (\xi _{k})}}{%
\prod_{k}\int_{0}^{\pi }d\xi _{k}e^{-4p\tau \cos (\xi _{k})}}
\label{Cn'nPin}$$
Using the identity $\varphi _{n}=\sum_{k=n_{0}}^{n}\xi _{k}$, where the value of $n_{0}$ can be found from boundary conditions which influence only the global phase of the SC order parameter, we find that $$\begin{aligned}
C_{n^{\prime },n} &\simeq &\left( \frac{I_{1}(4p\tau )}{I_{0}(4p\tau )}%
\right) ^{|\Delta n|} \nonumber \\
&\simeq &\exp \left( -|\Delta n|/8p\tau \right) \hspace{0.2cm}\textrm{for}%
\hspace{0.2cm}4p\tau \gg 1, \label{pin}\end{aligned}$$ where $\Delta n\equiv n^{\prime }-n$. This result contrasts with the correlation function obtained in the pure state [@zm99], which has the asymptotic form $$C_{n^{\prime },n}\propto \exp \left( -\frac{\overline{n}}{2\tau }|\Delta
n|^{2}\right) \quad\textrm{for} \quad\tau \gg 1 \label{depin}$$ where $\overline{n}=\frac{n^{\prime }}{3}+\frac{2n}{3}-\frac{1}{2}$ , with the $n=0$ chain being fixed. As discussed above (see also Refs.([@zm99]$^{,}$[@mzvwrmp]$^{,}$[@zm02])), fixing chain positions through boundary conditions is physically equivalent to introducing pinning potential into the GL functional, which is a crucial step for stabilizing the vortex lattice. The global stability of the vortex lattice in the presence of the periodic pinning potential is reflected in Eq.(\[pin\]), as compared with Eq.(\[depin\]), by the translational invariance of the former correlation function, as well as by its relatively weak (simple exponential) decay. To determine the crossover temperature from the square pin solid (SPS) to the vortex liquid we may follow the procedure described above and find the temperature $T_{cm}\left( a_{x},p\right) $ at which $C_{n^{\prime },n}$ in Eq.(\[Cn’nPin\]) for $|\Delta n|=1$ is reduced by a factor of 1/2 with respect to its zero temperature ($\tau \rightarrow \infty $ ) limit. This yields in the strong pinning limit, $p\gg 1$ (see Fig. \[fig:4\]), the linear dependence $$T_{cm}\left( a_{x},p\right) \approx .86\times 4pT_{\Box } \label{T_cm(p)}$$
Beside its influence on the vortex lattice melting transition, the pinning potential can change the vortex lattice structure, both continuously and discontinuously. The zero temperature limit was discussed in Sec.(\[sec:3.1\]). Above the critical value $p=1$ the lateral vortex positions coincide with the pin square lattice positions, $\xi _{l}=0$. For decreasing pinning strength below $p=1$, the configuration of the vortex lattice deviates continuously from the square structure to a lattice with vortices shifted along chains away from the pinning centers.
Similar second order SPS to TPS phase transition (as a function of $p$ ) is expected at finite temperatures. Indeed, as shown in Sec.(\[sec:3.1\]), the free energy functional $f_{\Box }$ in Eq. (\[eq:low\_ph\]) is minimized at the stationary points $\xi _{k}=\xi _{ck}\equiv \left( -1\right) ^{k}\theta _{0}$ , with $\cos \theta _{0}=p$ for $p\leq 1$ , and at $\xi _{ck}=0$ for $p\geq
1 $. Thus, expanding $f_{\Box }$ as a Taylor series in $\left( \xi
_{k}-\xi _{ck}\right) $ about its stationary points it is clear that for $%
p\geq 1$ (when $\xi _{ck}=0$ ) the expansion includes only *even* powers $\xi _{k}$ ( due to the symmetry of $f_{\Box }$ with respect to $\xi
_{k}\longrightarrow -\xi _{k}$ ). Thus, at any finite temperature, $T$ , the thermal mean values $\left\langle \xi _{k}\right\rangle $ are equal to zero for $p\geq 1$ , implying that for pinning strengths above the critical value $p=1$ the mean vortex positions coincide with the square pin lattice.
Phase diagram for the lowest matching field {#sec:5}
===========================================
The results of the previous sections enable us to draw a rather clear picture of the $V_{0}-T$ -phase diagram, as shown in Fig. \[fig:ph\_diag\]. In the strong pinning regime, $p\gg 1$, the pinning strength is so large that the gain in commensurate energy is larger than the vortex-vortex energy gain at any temperature, and so the floating solid phase is not favorable. Thus, the vortex lattice melting in this region should take place directly from the SPS to the liquid phase, as described by the asymptotic expression, Eq.(\[T\_cm(p)\]), which is equivalent to the straight line $p\approx
0.29T/T_{\Box }$ in the large $p$ regime of the phase diagram.
In the small $p$ regime the stable phase at low temperatures is the FS. Here the energy gain associated with creation of the closed packed equilateral triangular vortex lattice, exceeds the energy cost of the incommensurate state. This state remains stable up to a relatively high temperature $%
T\simeq 2.8T_{\Box }$ , above which it melts into a vortex liquid state. The phase boundary in this region is vertical (i.e. independent of $p$ ) since it is determined by the vortex-vortex coupling and not by the pinning energy (which is a constant in the floating state).
In the low temperatures region of the phase diagram our analysis shows the existence of two phase transitions: At small pinning, increasing $p$ above $p_{c}\approx .25$ transforms the FS discontinuously to a pin solid since the energy gain associated with the commensurate pin vortex solid exceeds the energy cost of distorting the closed packed equilateral triangular vortex lattice. The discontinuous nature of this transition is due to the fact that even infinitesimal deviation from a commesurate configuration rises the pinning energy by a finite amount (i.e. at least from $.6v$ to $v$).
It turns out that the pin vortex crystal just above the commensurate-incommensurate transition is not a square lattice, as found by Reichhardt *et al*. [@reichhardt01], but a triangular one, with a unit cell which depends on the pinning strength. At $T=0$ it is a parallelogram with equal base and height, which transforms continuously to a square at $p=1$. Similar continuous transition from a frustrated triangular pin lattice to the SPS takes place at the critical pinning strength $p=1$ at any temperature $T$. Interestingly, the corresponding horizontal transition line intersects the extrapolated SPS-L boundary line at $T\approx \allowbreak 3.\,\allowbreak 44T_{\Box }$, $p=1$ , that is in the close vicinity of the intersection between the vertical FS melting line, $T\simeq 2.8T_{\Box }$, and the SPS-TPS line.
![$V_{0}-T$ phase diagram: Solid lines - first order phase transitions from SPS to L phase (at large pinning strength), from TPS to FS (near zero temperature), and between FS and L phases. Dashed line - second order phase transition between partially pinned (TPS) and fully pinned (SPS) vortex crystals. The dashed-dotted line connects smoothly between the asymptotic SPS-L line and the low temperatures TPS-FS line (see text for explanation). []{data-label="fig:ph_diag"}](figure5.eps){width="10cm"}
It is not exactly known, however, how the FS-TPS boundary is extended beyond the zero temperature region. It is conceivable that its high temperature sector coincides with the low temperature sector of the SPS melting line. This is due to the fact that, at a fixed value of $p$, the driving force for both transitions are thermal fluctuations involving sliding vortex chains, which suppress the pinning energy gained in the commensurate phase (i.e. the term $-4p\cos (\xi _{k})$ in Eq.(\[eq:low\_ph\])). In the SPS-L transitions, where the vortex-vortex interaction is relatively small, this suppression leads to uncorrelated vortex chains, resulting in melting. In the TPS-FS transitions, where the the vortex-vortex coupling is relatively large, the suppression of the pinning energy results in mutually correlated vortex chains, which lose correlation with the underlying pinning lattice.
An intermediate pin solid phase of a triangular form has been also found in the London model calculation reported by Pogosov *et al*. [@pogosov02]. However, in contrast to the Ginzburg-Landau model, discussed here, they predicted the vortex configurations shown in Fig. \[fig:3n\]bas preferable below some critical value of the pinning potential strength. Above this value the symmetry of vortex lattice is changed discontinuously to the square symmetry of the pin lattice. Our proposed phase diagram, shown in Fig.\[fig:ph\_diag\], thus consists of 2 pin solid phases, a floating solid and a liquid phase, delimitted by 4 interphase boundary lines, which intersect at two nearby triple points. This result is similar to the phase diagram found by Reichhardt *et al*. [@reichhardt01], using molecular dynamics simulations. However, the intermediate TPS phase obtained in our calculation, is missing in Reichhardt *et al.* This seems to be due to the square boundary conditions imposed in the latter calculation. Another difference concerns the zero temperature limit of the PS-FS line, which seems to approach $p=0$ in Reichhardt *et al*..
Conclusions
===========
The influence of a periodic pinning potential on the vortex state of a 2D superconductor at temperatures well below the mean field $T_{c}$ has been studied within the framework of the GL functional integral approach. It is shown that attractive interaction of vortex cores to a commensurate pin lattice stabilizes vortex solid phases with long range positional order against violent shear fluctuations along the principal crystallographic axis. Exploiting a simple analytical approach we draw a rather detailed picture of the relevant vortex state $p-T$ (pinning strength-temperature) phase diagram. In agreement with previous numerical simulations [@reichhardt01], we have found a pinned, commensurate solid phase in the strong pinning-low temperature part of the phase diagram, which melts into a vortex liquid at high temperatures, and transforms into a floating (incommensurate) solid at low temperatures. We have shown that at low temperature, similar to Ref. [@pogosov02], there is an intermediate triangular phase, where vortices detaching from pinning centers remain strongly correlated with them. This pinned (frustrated) triangular solid transforms continuously into the fully pinned (square) solid phase at $p=1$, and discontinuously to a floating solid at small pinning strengths. The zero temperature limit of this commensurate-incommensurate transition line occurs at a finite pinning strength ($p=p_{c}\approx .25$).
It is predicted that for sufficiently clean samples, where random pinning is weak enough, application of an artificial periodic pinning array with an appropriate strength would stabilize a weakly pinned vortex solid phase with long range positional order. Exploiting the SANS method to the sample under these conditions one could therefore directly detect the shear stiffness anisotropy characterizing the ideal vortex lattice.
This research was supported in parts by a grant from the Israel Science Foundation founded by the Academy of Sciences and Humanities, and by the fund from the promotion of research at the Technion.
[99]{} T. Maniv, V. Zhuravlev, I.D. Vagner, and P. Wyder, Rev. Mod. Phys. **73**, 867 (2001).
S. Doniach and B.A. Huberman, Phys. Rev. Lett. 42, 1169 (1979).
Daniel S. Fisher, Phys. Rev. B 22, 1190 (1980).
J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973) ; B.I. Halperin and D. Nelson, Phys. Rev. Lett. 41, 121 1978 ; A.P. Young, Phys. Rev. B 19, 1855 (1979).
Zlatko Tesanovic and A.V. Andreev, Phys. Rev. B 49, 4064 (1994) .
J. Hu and A.H. MacDonald, Phys. Rev. B 56, 2788 (1997).
V. Zhuravlev and T. Maniv, Phys. Rev. B **60**, 4277 (1999).
V. Zhuravlev and T. Maniv, Phys. Rev. B **66**, 014529 (2002).
A.T. Fiory, A.F. Hebard, and S. Somekh, Appl. Phys. Lett. **32**, 73 (1978).
M. Baert, V.V. Metlushko, R. Jonckheere, V.V.Moshchalkov, and Y. Bruynseraede, Phys. Rev. Lett. **74**, 3269 (1995).
A. Castellanos, R. Wordenweber, G. Ockenfuss, A.v.d. Hart, and K. Keck, Appl. Phys. Lett. **71**, 962 (1997).
V.V. Moshchalkov, M. Baert, V.V. Metlushko, E. Rossel, M.J. Van Bael, K. Temst, Y. Bruynseraede, R. Jonckheere, Physica C **332**, 12 (2000).
J.I. Martin, M. Velez, J. Nogues, and I.K. Schuller, Phys. Rev. Lett. **79**, 1929 (1997).
D.J. Morgan and J.B. Ketterson, Phys. Rev. Lett. **80**, 3614 (1998).
M.J. Van Bael, K. Temst, V.V. Moshchalkov, and Y.Bruynseraede, Phys. Rev. B **59**, 14674 (1999).
A. Terentiev, D.B. Watkins, L.E. De Long, L.D. Cooley, D.J. Morgan, and J.B. Ketterson, Phys. Rev. B **61**, R9249 (2000).
D.R. Nelson and B.I. Halperin, Phys. Rev. B **19**, 2457 (1979).
L. Radzihovsky, Phys. Rev. Lett. **74**, 4923 (1995).
C. Reichhardt, C.J. Olson, and F. Nori, Phys. Rev. Lett. **78**, 2648 (1997); Phys. Rev. B **58**, 6534 (1998).
V. Marconi and D. Dominguez, Phys. Rev. Lett. **83**, 3061 (1999); C. Reichhardt and G.T. Zimanyi, Phys. Rev. B **61**, 14 354 (2000); G. Carneiro, *ibid.* **62**, R14 661 (2000).
C. Dasgupta and O.T. Valls, Phys. Rev. B **66**, 064518 (2002)
C. Reichhardt, C. J. Olson, R. T. Scalettar and G. T. Zimanyi, Phys. Rev. B **64**, 144509 (2001)
W. V. Pogosov, A. L. Rakhmanov, and V. V. Moshchalkov, Phys. Rev. B **67**, 014532 (2003); V.N. Rud’ko, O.N. Shevtsova, and S.V. Shiyanovsky, Low Temp. Phys. **22**, 999 (1996)
Z. Tesanovic and I.F. Herbut, Phys. Rev. B **50**, 10389 (1994)
M. A. Moore, Phys. Rev. B **39**, 136 (1989)
|
---
abstract: 'We propose different spectroscopic methods to explore the nature of the thermal excitations of a trapped Bose condensed gas: 1) a four photon process to probe the uniform region in the trap center: 2) a stimulated Raman process in order to analyze the influence of a momentum transfer in the resulting scattered atom momentum distribution. We apply these methods to address specifically the energy spectrum and the scattering amplitude of these excitations in a transition between two hyperfine levels of the gas atoms. In particular, we exemplify the potential offered by these proposed techniques by contrasting the spectrum expected, from the [*non conserving*]{} Bogoliubov approximation valid for weak depletion, to the spectrum of the finite temperature extensions like the [*conserving*]{} generalized random phase approximation (GRPA). Both predict the existence of the Bogoliubov collective excitations but the GRPA approximation distinguishes them from the single atom excitations with a gapped and parabolic dispersion relation and accounts for the dynamical screening of any external perturbation applied to the gas. We propose two feasible experiments, one concerns the observation of the gap associated to this second branch of excitations and the other deals with this screening effect.'
author:
- 'Patrick Navez$^1$, Kai Bongs$^2$'
title: Gap and screening in Raman scattering of a Bose condensed gas
---
Introduction
============
Nature of the elementary excitations
------------------------------------
The experimental discovery of the condensation of a Bose gas has confirmed the existence of the phonon-like nature of the collective excitations [@Ketterle; @Stringari]. The obtained measured energy spectrum not only is gapless as stated from the Hugenholtz-Pines theorem but is also in perfect agreement with the prediction of the Bogoliubov approach at zero temperature [@HP]. However, a second fundamental question arises as to whether these collective excitations are the elementary building constituents for the normal part of the fluid as assumed in the Bogoliubov approximation. Most standard textbooks rely on this quasiparticle hypothesis in order to determine the finite temperature gas properties [@Leggett; @books]. In contrast, in the theoretical description of a plasma, distinction is made between the elementary excitations (ions) and the collective ones (plasmons). As discussed in previous works [@gap; @gap2; @condenson; @Graham], there are no fundamental reasons to exclude this distinction also in a Bose gas.
Precisely, suppose a bulk gas of total and condensed densities $n$ and $n_0$ embedded in an volume $V$ where atoms of mass $m$ interact through the s wave channel with a scattering length $a$. The Bogoliubov approximation predicts that the elementary excitations of momentum ${\mathbf{k}}$ are phonon-like with a dispersion relation given by $\epsilon_{1,{\mathbf{k}}}^B=\sqrt{2gn_{\mathbf{0}}\epsilon^{}_{{\mathbf{k}}}+ \epsilon^{2}_{{\mathbf{k}}}}$ where $\epsilon_{\mathbf{k}}=\hbar^2{\mathbf{k}}^2/2m$ and $g=4\pi a \hbar^2/m$. Nevertheless, its [*non conserving*]{} property (violation of the mass conservation law) [@HM] restricts its validity for a weakly depleted Bose gas and thus limits its use to low temperature. As opposed to that, the so-called generalized random phase approximation (GRPA) or equivalently the time-dependent Hartree-Fock (TDHF) approximation is instead [*conserving*]{} and valid for the whole range of temperature. This alternative approach distinguishes explicitly these collective phonon-like excitations from the atom-like elementary excitations with the parabolic dispersion relation $\epsilon^{HF}_{1,{\mathbf{k}}}=\epsilon_{\mathbf{k}}
+g(2n- n_{\mathbf{0}}\delta_{{\mathbf{k}},{\mathbf{0}}})$ [@Reidl; @Levitov; @Zhang; @gap2]. The constant term corresponds to the Hartree and Fock (HF) mean field energy part and takes into account the absence of exchange interaction energy between condensed atoms. Therefore, an energy gap exists between the thermal and condensed atoms $\epsilon^{HF}_{1,{\mathbf{k}}}-\epsilon^{HF}_{1,{\mathbf{0}}}= gn_{\mathbf{0}}+\epsilon_{\mathbf{k}}$.
Superfluidity due to total screening
------------------------------------
Another important reason to discriminate among the various theoretical approaches is to have an improved understanding of the superfluidity phenomenon. More precisely, we would like to answer the following question: Why, from a kinetic point of view, a superfluid can remain in a metastable motion without converting its kinetic energy into heat? Many explanations have been provided but, according to [@Leggett], [*the situation is not entirely clear*]{} as far as kinetic theory is concerned.
Instead, the equilibrium aspects based on the ensemble approach of the superfluid phenomenon of a Bose condensed gas are well understood. Using the $\eta$ ensemble which breaks the $U(1)$ symmetry associated to the particle number conservation, one can describe the superfluid motion (condensed mode) relatively to the normal fluid (non condensed modes) [@Graham]. Such a relative motion should not be considered as an equilibrium state but as a metastable state possibly subject to relaxation of a state of lower energy. Unfortunately, the ensemble approach does not explain the physical reasons for such a metastability. It just tells that an artificial breaking of symmetry allows you such a description. Only a non-equilibrium treatment can provide these explanations and therefore confirm the validity of the assumptions used in the ensemble approach.
The kinetic theory so far developed in the Bogoliubov approximation allows for such a metastability in the weak depletion limit [@Kirkpatrick]. Particle exchange between the normal and superfluid are regulated through a balanced Beliaev process of transforming one collective excitations to two collective excitations. A complete different scenario appears in the GRPA as it accounts for the dynamical screening of any external time-dependent potential that perturbs the gas atoms [@condenson; @gap2]. The ability of the macroscopic condensed wave function to deform locally its profile allows for a screening of any external perturbation that affects the energy transition probability of any atom-like excitation. In particular, under some stability conditions [@condenson], a total screening forbids individual energy transitions involving a condensed atom. In this sense, the condensed atoms are [*gregarious*]{} since they respond only collectively to a perturbation via the creation of a phonon-like excitation. If the external potential originates from the presence of another thermal atom, this total screening prevents the binary collision between this thermal atom and any condensed one. Therefore, contrary to the Bogoliubov approach, the metastability of the relative motion between the normal and super fluids in GRPA is explained from the absence of this exchange collision process.
Nevertheless, atom exchanges between the normal and the super fluids should always exist in any kinetic description, in particular to guarantee the process of condensate formation. This is the case for the GRPA, but provided that instability conditions are satisfied [@gap2]. For example, when the relative velocity between the two fluids exceeds the critical velocity given by the Landau criterion, the total screening phenomenon disappears and the binary collisions become again possible.
Experimental difficulties
-------------------------
Both gap and total screening phenomena have been predicted to appear in a Raman transition process between two hyperfine levels of a $^{87}Rb$ gas, but only in the bulk case [@gap2]. In comparison to other methods like radio frequency (RF) or Bragg spectroscopy, the possibility of momentum transfer and the distinction between scattered and unscattered atoms enable these observations. However, an experimental realization is still not simple in the real case of a trap since the gas inhomogeneity, combined with the short duration of the applied coupling potential, leads to additional broadenings of the spectral lines that prevent the resolution of the gap and screening structure. In this context, a RF spectroscopy would have probed the whole gas which includes thermal atoms of the outer and inner condensate regions. Therefore, the distinction between various theoretical approaches is extremely difficult as long as the transition amplitude and the dispersion relation of thermal atoms have a strong spatial dependence.
Setup proposals
---------------
In this letter, we propose different methods to probe the atoms more efficiently than the RF spectroscopy: 1) the Raman scattering is a two-photon process that offers also the possibility to transfer the momentum ${\mathbf{q}}$ to the scattered atoms and observe their resulting momentum distribution after expanding the gas; 2) a four photon scattering process, where two sets of two beams cross in the trap center, addresses selectively the homogeneous region of the gas (see Fig.\[f4\]).
We apply these methods for the case of a finite temperature trapped Bose gas in the GRPA, in a bid to challenge the Bogoliubov approach. To this end, we propose two concrete experimental setups that overcome the difficulties associated with the trap: 1) The gap is observed from the four-photon process; 2) The total screening is determined in a Raman scattering. Previous theoretical works [@condenson; @gap2] argue in favor of the [*conserving*]{} GRPA. Nevertheless, a comparison with the [*non conserving*]{} Bogoliubov approximation is of relevance as long as the second branch of individual excitations has not been observed.
Raman scattering
================
The GRPA approach
-----------------
In a Raman transition, we start from atoms initially in the hyperfine level $|1\rangle =
|F=1,m_F=-1\rangle$. Each mode ${\mathbf{k}}$ is characterized by its initial population $N_{\mathbf{0}}$ and $N_{{\mathbf{k}}\not=0}=1/(\exp[\beta(\epsilon^{HF}_{1,{\mathbf{k}}}-\mu)]-1)$ and its initial plane wave function $\psi_{1,{\mathbf{k}}} =
{\exp[i({\mathbf{k}}.{\mathbf{r}}-\epsilon^{HF}_{1,{\mathbf{k}}} t)]}/{\sqrt{V}}$ with the inverse temperature $\beta=1/k_B T$ and the chemical potential $\mu=g(2n-n_{\mathbf{0}})$. The application of a perturbation coupling potential $V_{{\mathbf{q}}}({\mathbf{r}},t)=
V_R \exp[i({\mathbf{q}}.{\mathbf{r}}-\omega t)]$ at $t \geq 0$ transfers a small fraction of them into the second level $|2\rangle =|F=2,m_F=1\rangle$ of internal frequency $\omega_0$. The determination of the second spinor component of the associated wavefunction $\psi_{2,{\mathbf{k}}}({\mathbf{r}},t)$ of the mode ${\mathbf{k}}$ evolves according to the time-dependant Hartree-Fock equation [@gap2]: $$\begin{aligned}
\label{p21}
\left[i\hbar{\partial_t}
+\frac{\hbar \nabla^2_{\mathbf{r}}}{2m}-\hbar\omega_0 -
g_{12}\sum_{\mathbf{k'\not= {\mathbf{0}}}} N_{\mathbf{k'}}
{|\psi_{1,{\mathbf{k'}}}|^2}\right] \psi_{2,{\mathbf{k}}}
= \nonumber \\
\left[V_{{\mathbf{q}}} +
g_{12}\sum_{\mathbf{k'}} N_{\mathbf{k'}}
{\psi^{*}_{1,{\mathbf{k'}}}} \psi_{2,{\mathbf{k'}}}\right]
\psi_{1,{\mathbf{k}}}\end{aligned}$$ where we define the intercomponent coupling $g_{12}=4\pi \hbar^2 a_{12}/m$. The solution is [@gap2]: $$\begin{aligned}
\label{psiR}
\psi_{2,{\mathbf{k}}}({\mathbf{r}},t)=\!
\int_{-\infty}^\infty \!\!\!\!\!\!\!d\omega'
\frac{ \int_0^\infty \!dt' e^{i(\omega'+i0)(t'-t)}V_{{\mathbf{q}}}({\mathbf{r}},t')\psi_{1,{\mathbf{k}}}({\mathbf{r}},t)}
{2\pi i{\cal K}_{12}({\mathbf{q}},\omega')(\hbar\omega'+i0-\hbar\omega_{{\mathbf{k}},{\mathbf{q}}})}
$$ where $\hbar \omega_{{\mathbf{k}},{\mathbf{q}}}=\epsilon^{HF}_{2,{\mathbf{k}}+{\mathbf{q}}}-\epsilon^{HF}_{1,{\mathbf{k}}}$ and $\epsilon^{HF}_{2,{\mathbf{k}}+{\mathbf{q}}}=\hbar\omega_0+\epsilon_{\mathbf{k+q}}
+g_{12}(n- n_{\mathbf{0}}\delta_{{\mathbf{k}},{\mathbf{0}}})$ is the atom mean field energy in the second level without the exchange term. These formulae resemble the one obtained from the non interacting Bose gas except for the HF mean field terms and the screening factor: $$\begin{aligned}
\label{K12}
{{\cal K}}_{12}({\mathbf{q}},\omega)=1-\frac{g_{12}}{V}\sum_{\mathbf{k}}
\frac{N_{{\mathbf{k}}}}
{\hbar \omega +i0- \hbar \omega_{\mathbf{k,q}}} \end{aligned}$$ Eq.(\[psiR\]) is interpreted in Fig.\[f2\] in terms of propagators whose poles determine the resonance frequencies. One pole is associated to the individual transition between atoms: $\omega= \omega_{{\mathbf{k}},{\mathbf{q}}}$ and the other is the zero of the screening factor and corresponds to the collective excitations associated to the gas rotation in the spin space: $\delta \omega =\omega-\omega_0 \sim [\epsilon_{\mathbf{q}}-(g-g_{12})n]/\hbar$ for $g_{12} \sim g$.
\[vc\]
Total screening corresponds to the singularity ${{\cal K}}_{12}({\mathbf{q}},\omega_{{\mathbf{0}},{\mathbf{q}}})
\rightarrow \infty$ and prevents any single condensed atom scattering [@gap2]. In a bulk gas, the transferred atom density for each mode is obtained from $n_{2,{\mathbf{k}}+{\mathbf{q}}}(t)=
|\psi^{(1)}_{2,{\mathbf{k}}}({\mathbf{r}},t)|^2 N_{\mathbf{k}}$ so that we deduce the total atom density[@Stringari; @gap2]: $$\begin{aligned}
n_2=\! \sum_{\mathbf{k}} n_{2,{\mathbf{k}}}=\!\!
\int_{-\infty}^\infty
\!\!\!\!\!\! d\omega' \frac{4 \sin^2(\omega't/2)}{\hbar \pi \omega'^2}
|V_R|^2 \chi_{12}'' ({\mathbf{q}},\omega-\omega')\end{aligned}$$ expressed in terms of the imaginary part of the intercomponent susceptibility function $\chi_{12} ({\mathbf{q}},\omega)=1/(g_{12} {{\cal K}}_{12}({\mathbf{q}},\omega))$.
The Bogoliubov approach
-----------------------
These results can be compared to the one obtained from the Bogoliubov [*non conserving*]{} approximation developed in [@Fetter; @gap; @gap2] which is valid only for a weakly depleted condensate. This approach implicitly assumes that the elementary excitations are the collective ones forming a basis of quantum orthogonal states for the description of the normal fluid. Consequently, this formalism predicts no gap and no screening. The creation-annihilation operators $c^\dagger_{i,{\mathbf{k}}}(t), c_{i,{\mathbf{k}}}(t)$ describing the various components in the momentum space evolve according to $c_{1,{\mathbf{k}}}(t)=e^{-i\mu t}(\sqrt{N_{{\mathbf{0}}}}\delta_{{\mathbf{k}},0}
+u_{+,{\mathbf{k}}}
e^{-i\epsilon^B_{1,{\mathbf{k}}}t}
b_{1,{\mathbf{k}}}+
u_{-,{\mathbf{k}}} e^{i\epsilon^B_{1,{\mathbf{k}}}t}
b^\dagger_{1,-{\mathbf{k}}})$ and $c_{2,{\mathbf{k}}}(t)=
e^{-i(\mu +\epsilon^B_{2,{\mathbf{k}}}) t}
c_{2,{\mathbf{k}}}$. In this expression, besides the collective excitation modes of phonon of energy $\epsilon_{1,{\mathbf{k}}}^B$, a second collective mode of rotation appears with energy $\epsilon_{2,{\mathbf{k}}}^B =\epsilon_{\mathbf{k}}+(g_{12}-g)n_{\mathbf{0}}$. $\mu=gn_{\mathbf{0}}$ is the chemical potential, $b_{1,{\mathbf{k}}}$ is the annihilation operator associated to the quasi-particle such that $\langle b^\dagger_{1,{\mathbf{k}}} b_{1,{\mathbf{k}}}\rangle=
1/(\exp( \beta \epsilon^B_{1,{\mathbf{k}}})-1)$ and $u_{\pm,{\mathbf{k}}}=\pm
((\epsilon_{\mathbf{k}}+gn_{{\mathbf{0}}})/2\epsilon^B_{1,{\mathbf{k}}}\pm
1/2)^{1/2}$. Reexpressing the intercomponent susceptibility $$\begin{aligned}
\label{BP0}
\chi_{12}({\mathbf{q}},\omega)=
\frac{i V}{\hbar} \int_0^\infty \!\!dt\
e^{i(\omega+i0)t}\langle [{\rho^{12}_{{\mathbf{q}}}}^\dagger(0),
\rho^{12}_{{\mathbf{q}}}(t)]\rangle\end{aligned}$$ in terms of the autocorrelation function of the excitation operator $\rho^{\alpha \beta}_{{\mathbf{q}}}(t)=\sum_{\mathbf{k}}
c^\dagger_{\alpha,{\mathbf{k}}}(t)c^{}_{\beta,{\mathbf{k+q}}}(t)/V$, we calculate in the Bogoliubov approximation: $$\begin{aligned}
\label{BP}
\chi^B_{12}({\mathbf{q}},\omega) =\sum_{\pm,{\mathbf{k}}}
\frac{\delta_{{\mathbf{k}},{\mathbf{0}}} N_{\mathbf{0}}/2\pm u^2_{\pm,{\mathbf{k}}}/(\exp(\pm \beta \epsilon^B_{1,{\mathbf{k}}})-1)}
{V(\hbar \delta\omega
+i0 \pm\epsilon^B_{1,{\mathbf{k}}}-\epsilon^{B}_{2,{\mathbf{k}}\pm{\mathbf{q}}})}\end{aligned}$$ In contrast to the GRPA, Eq.(\[BP\]) describes a spin rotation transition of the condensed fraction, one transition involves the excitation transfer from a phonon mode into a rotation mode and another the excitation creation in the two modes simultaneously.
Extension to the trap
---------------------
These formulae can be easily extended to the case of a harmonic trap $V_H({\mathbf{r}})=\sum_i m\omega_i^2 r_i^2/2$ of frequency $\omega_i$ by considering the local density approximation (LDA) [@books]. For a weakly inhomogeneous gas, the population in each mode becomes a local quantity $N_{\mathbf{k}} \rightarrow N_{\mathbf{k}}({\mathbf{r}})$. By making this replacement, the thermal density $n_T({\mathbf{r}})= \sum_{{\mathbf{k}}\not=0} N_{\mathbf{k}}({\mathbf{r}})/V$, the energies $\epsilon^{HF}_{i,{\mathbf{k}}}({\mathbf{r}})$, $\epsilon^B_{i,{\mathbf{k}}}({\mathbf{r}})$, the screening factor ${\cal K}_{12}({\mathbf{r}},{\mathbf{q}},\omega)$, the potential amplitude $V_R({\mathbf{r}})$ and $n_{2,{\mathbf{k}}}({\mathbf{r}},t)$ become local quantities as well. The zero mode density $n_{\mathbf{0}}({\mathbf{r}})=|\Psi_{\mathbf{0}}({\mathbf{r}})|^2$ is determined from: $$\begin{aligned}
\label{super2}
-\frac{\hbar^2 \nabla_{\mathbf{r}}^2\Psi_{\mathbf{0}}({\mathbf{r}})}{2m\Psi_{\mathbf{0}}({\mathbf{r}})}
+V_{H}({\mathbf{r}})+g
(|\Psi_{\mathbf{0}}({\mathbf{r}})|^2 +
2n_{T}({\mathbf{r}}) )
=\mu\end{aligned}$$ while the non zero ones are determined from the semi-classical expression: $$\begin{aligned}
\label{neqin}
N_{{\mathbf{k}}}({\mathbf{r}})=
\frac{1}{\exp{[\beta(\epsilon^{HF}_{1,{\mathbf{k}}}({\mathbf{r}})+V_{H}({\mathbf{r}})-\mu)]}-1}\end{aligned}$$ The set of Eqs.(\[super2\],\[neqin\]) is reduced to a one dimensional problem if we assume the ansatz $n_{{\mathbf{0}}}(\overline{r})$ where $\overline{r}
=\sqrt{2m V({\mathbf{r}})}/\overline{\omega}$ and $\overline{\omega}=(\omega_x \omega_y \omega_z)^{1/3}$. This ansatz is exact for a spherical trap and is accurate in the Thomas-Fermi limit $\omega_i \ll gn({\mathbf{0}})$. It leads to the profiles in Fig.\[f3\] for the condensed and normal fluids and shows excellent agreements with both experiments [@Aspect] and exact Monte-Carlo calculations [@Holzmann] in the determination of the density profile of a trapped Bose condensed gas. These generalizations allow the determination of the transferred momentum distribution $N_{2,{\mathbf{k}}}(t)=
\int d^3\, {\mathbf{r}} \,n_{2,{\mathbf{k}}}({\mathbf{r}},t) $ from which we deduce the transferred thermal atom number $N_{2,T}(t)=\sum_{\mathbf{k\not={\mathbf{q}}}}N_{2,{\mathbf{k}}}(t)$.
Experimental proposals
======================
The gap experiment: four-photon process
---------------------------------------
For ${\mathbf{q}}=0$ and $g_{12} \sim g$, the Raman spectrum becomes discrete in a homogeneous gas. The resonance frequencies correspond to a gap $\hbar \omega_{{\mathbf{k}},{\mathbf{q}}}({\mathbf{r}})=-g n({\mathbf{r}})$ associated to the exchange interaction energy for the single mode transition and to $(g_{12}-g)n({\mathbf{r}})$ for the collective mode transition [@Levitov]. In comparison, if the condensed atom spectrum is quite similar, the thermal atom one displays differences in the Bogoliubov approximation. Since the energy difference $\epsilon_{2,{\mathbf{k}}}({\mathbf{r}})-\epsilon^B_{1,{\mathbf{k}}}({\mathbf{r}})$ is ${\mathbf{k}}$ dependant, no gap is observed and the oscillations are smoothed out leading to a continuous spectrum.
In order to distinguish clearly between the discrete and the continuous spectra, the coupling potential acts specifically in the trap center in order to reduce inhomogeneous broadening. This is realized by means of four beams (see Fig.\[f4\]) [@Becker]: two gaussian astigmatic beams $\sigma^+$ polarized along the z axis of quantization with the intensity profile $I_1({\mathbf{r}})=I_{01} \exp(-2 r_x^2/w_1^2(r_z)-2r_y^2/w_2^2(r_z))$ and two others $\pi$ polarized along the y axis with $I_2({\mathbf{r}})=I_{02} \exp(-2 r_x^2/w_3^2(r_y)-2r_z^2/w_4^2(r_y))$ where $w_i(s)=w_i (1+(s\lambda)^2/(\pi^2 w_i^4))^{1/2}$. The sum of their frequency differences corresponds to the transition frequency $\omega$. Provided that $\lambda \ll w_i$, we define an effective waist $\overline{w}$ such that: $$\begin{aligned}
\frac{1}{{\overline{\omega}}\, {\overline{w}}^2}=
\frac{1}{\omega_x}(\frac{1}{w_1^2}+\frac{1}{w_3^2})=
\frac{1}{\omega_y w_2^2}=
\frac{1}{\omega_z w_4^2}\end{aligned}$$ In these conditions, the resulting potential $V_R({\mathbf{r}})=V_{R0}\exp(-2\overline{r}^2/\overline{w}^2)$ is optimized for an atom transfer in the most homogeneous region with ${\mathbf{q}}=0$. To fix the idea, we choose $\lambda =843 nm$ and $\overline{w}= 7 \mu m$ which reduces to about $10^4$ the thermal atom effective number that can be specifically addressed. Transferring a small fraction of about 10% and for a detection resolution of about 100 atoms, we obtain a signal to noise ratio of about 10. A relative difference in the scattering lengths is also needed to observe the gap resonance and is obtained from the application of an external magnetic field [@Kai]. These consideration leads to the spectra of Fig.\[f5\]. Note the two orders of magnitude between the two peak intensities and the oscillatory behavior of period $1/t=100Hz$ associated to the finite time resolution. The finite size of the beam provides an additional negligible frequency uncertainty of about $\hbar/(\sqrt{m\beta}\overline{w})$ in the resolution.
The screening experiment: Raman scattering
------------------------------------------
The absence of Raman transition due to screening is observed in the scattered atom momentum distribution. For a long time, the transient effects in Eq.(\[psiR\]) can be neglected leading to a constant transfer rate and, except for the fact that the external potential is screened, we recover the Fermi golden rule: $$\begin{aligned}
\label{FG}
\frac{N_{2,{\mathbf{k+q}}}(t)}{t}\stackrel{t \rightarrow \infty}{=}
\int d^3{\mathbf{r}}
\frac{2\pi V^2_R({\mathbf{r}})n_{{\mathbf{k}}}({\mathbf{r}})
\delta (\omega -\omega_{{\mathbf{k}},{\mathbf{q}}}({\mathbf{r}}))}
{\hbar^2 |{\cal K}_{12}({\mathbf{r}},{\mathbf{q}},\omega)|^2}\end{aligned}$$ Considering $g_{12} \sim g$, the transition energy is position dependant causing inhomogeneous broadening: $\hbar \delta \omega=k_z q_z/m+\epsilon_{\mathbf{q}}-gn({\mathbf{r}})$. In the absence of screening, a resonance maximum appears for $k_z=0$. The screening factor strongly reduces the Raman scattering and forbids it at this maximum i.e. $N_{2,k_x,k_y,q_z}(t)/t
\stackrel{t \rightarrow \infty}{\rightarrow} 0$ thus avoiding the condensed atom transfer. For simplicity, let $\omega_x=\omega_y$. The atoms are transferred by means of a Raman transition resulting from two gaussian symmetric laser beams such that their wavevector difference ${\mathbf{q}}$ is along the z axis and their frequency difference is the transition frequency $\omega$. For small $q_z$, the angle between the beams is small and the Raman potential has the gaussian circular profile $V_R=V_{R0}\exp(-2(r_x^2+r_y^2)/w_5^2(r_z))$. Once the atoms are transferred, the trap is switched off and after a time of flight, the density profile provides their momentum distribution.
A negative detuning is chosen in order to scatter the thermal atoms with $k_z$ positive in the trap center region and negative otherwise.
The graphs in Fig.\[f6\] illustrate well the total screening effect around $k_z=0$ for which the macroscopic wave function deforms its shape in order to attenuate locally the Raman potential, thus preventing single atom scattering. The left part of the distribution ($k_z < -6\mu m^{-1}$) shows the thermal atoms coming from the outer condensate region. The choice of $q_z$ is such that the LDA validity condition $q_z w_5 \gg 1$ is fulfilled but also such that, during the flight, the mean field energy does not affect much the momentum distribution. The interaction time must be much lower than the relaxation time associated with collisions $t \ll \tau \sim \sqrt{\beta m/8\pi a^2 n_T({\mathbf{0}})}$ to avoid the equilibrium relaxation of the momentum distribution. Its finite value creates an energy uncertainty that alters the validity of Eq.(\[FG\]) by not suppressing totally atom scattering at $k_z=0$. Also, this time must be adequately chosen to suppress the condensed fraction due to the Rabi flopping associated to the collective mode: $N_{2,{\mathbf{q}}}(t)=2\sin^2[(\hbar \delta \omega -\epsilon_{\mathbf{q}})t/2]N_{20,{\mathbf{q}}}$ where $N_{20,{\mathbf{q}}}\simeq 2\int d^3{\mathbf{r}}\, n_{{\mathbf{0}}}({\mathbf{r}})
[V_R({\mathbf{r}})/(\hbar \delta \omega -\epsilon_{\mathbf{q}})]^2=0.19 N_{2,T}$ for the case of Fig.\[f6\].
Phenomenological approach
-------------------------
Although theoretical statements argue in favor of GRPA, we cannot exclude that none of the two approximations reproduces correctly the physical observation. In such a case, we can use a phenomenological approach assuming a transition process from an excitation of unknown energy $\epsilon_{1,{\mathbf{k}}}^X$ to an excitation of energy $\epsilon_{2,{\mathbf{k+q}}}^X$ and a process of creation of two excitations of energy $\epsilon_{1,{\mathbf{-k}}}^X$ and $\epsilon_{2,{\mathbf{k+q}}}^X$. Using a four photon process interacting in the uniform region of the gas, the fraction of scattered atoms is then written under the form analog to Eq.(\[FG\]): $$\begin{aligned}
\label{phen}
\frac{N_{2,{\mathbf{k+q_z}}}}{t}\stackrel{t \rightarrow \infty}{\simeq}
\sum_\pm A_\pm ({\mathbf{q}}_z, {\mathbf{k}})
\delta(\omega \pm \epsilon_{1,{\mathbf{\pm k}}}^X +\epsilon_{2,{\mathbf{k+q}}}^X)\end{aligned}$$ where $A_\pm ({\mathbf{q}}_z, {\mathbf{k}})$ represent the associated amplitude for such transition processes. Experimentally, the imaging in two dimensions allows only the determination of $F(k_x,k_z)=\int_\infty^\infty dk_y N_{2,{\mathbf{k+q_z}}}$. Thus the quantity (\[phen\]) is determined from the Abel’s transformation: $$\begin{aligned}
N_{2,{\mathbf{k+q_z}}}=-\frac{1}{\pi}\int_{\sqrt{k_x^2+k_y^2}}^\infty
\frac{d F(y,k_z)}{dy}\,\frac{dy}{\sqrt{y^2-(k_x^2+k_y^2)}}\end{aligned}$$ By varying the parameters $q_z$ and $\omega$, the resonance positions in the ${\mathbf{k}}$ space allow to reconstruct the dispersion relations $\epsilon_{1,{\mathbf{k}}}^X$ and $\epsilon_{2,{\mathbf{k}}}^X$ for the excitations.
Conclusions
===========
We explored the many body properties of a trapped Bose gas that can be extracted from a two-level hyperfine transition in the GRPA and Bogoliubov approximation. The calculated spectra not only show the existence of a second branch of excitation but also the total screening of the external potential which prevents single condensed atom transitions. If the external potential originates from the presence of a thermal atom, this total screening prevents the binary collision between that thermal atom and any condensed one. In this scenario, the metastability of the relative motion between the normal and super fluids is explained by the absence of this exchange collision process [@condenson]. The experimental observation of these phenomena will improve our understanding of the exact nature of the elementary excitations and of the origin of metastable motions in superfluids.
PN gratefully acknowledges support from the Belgian FWO project G.0115.06, from the Junior fellowship F/05/011 of the KUL research council, and from the German AvH foundation. KB thanks EPSRC for financial support in grant EP/E036473/1.
D.M. Stamper-Kurn, A.P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D.E. Pritchard, and W. Ketterle, Phys. Rev. Lett. [**83**]{}, 2876 (1999). F. Zambelli, L. Pitaevskii, D.M. Stamper-Kurn and S. Stringari, Phys. Rev. A [**61**]{}, 063608 (2000); R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson, Rev. Mod. Phys. [**77**]{}, 187 (2005). N.M. Hugenholtz and D. Pines, Phys. Rev. [**116**]{}, 489 (1959). A. J. Leggett, Rev. Mod. Phys. [**73**]{}, 307 (2001). L. Pitaevskii and S. Stringari, [*Bose-Einstein Condensation*]{}, (Clarendon Press, 2003); C. J. Pethick and H. Smith, [*Bose-Einstein Condensation in Dilute Gases*]{}, (Cambridge University Press, 2001). P. Navez, Eur. J. Phys. B (in press). P. Navez, Physica A [**387**]{}, 4070 (2008). P. Navez, J. Low. Temp. Phys. [**138**]{}, 705-710 (2005); P. Navez, Physica A [**356**]{}, 241-278 (2005) P. Navez and R. Graham, Phys. Rev. A [**73**]{}, 043612 (2006). P.C. Hohenberg and P.C. Martin, Ann. Phys. (N.Y.) [**34**]{}, 291 (1965). M.Ö. Oktel and L.S. Levitov, Phys. Rev. Lett. [**83**]{}, 6 (1999); C.J. Pethick and H.T.C. Stoof, Phys. Rev. A [**64**]{} 013618 (2001). M. Fliesser, J. Reidl, P. Szépfalusy, R. Graham, Phys. Rev. A [**64**]{}, 013609 (2001); J. Reidl, A. Csordas, R. Graham, P. Szépfalusy, Phys. Rev. A [**61**]{}, 043606 (2000). C-H Zhang and H.A. Fertig, Phys. Rev. A [**75**]{}, 013601 (2007); C-H Zhang and H.A. Fertig, Phys. Rev. A [**74**]{}, 023613 (2006). T.R. Kirkpatrick and J.R. Dorfman, J. Low Temp. Phys. [**58**]{}, 301 (1985); [**58**]{}, 399 (1985); M. Imamovic-Tomasovic, A. Griffin, J. Low Temp. Phys. [**122**]{}, 617 (2001). W. B. Colson and A. L. Fetter, J. Low Temp. Phys. [**33**]{}, 231 (1978); S.K. Yip, Phys. Rev. A [**65**]{} 033621 (2002). F. Gerbier, J. H. Thywissen, S. Richard, M. Hugbart, P. Bouyer, and A. Aspect, Phys. Rev. A [**70**]{}, 013607 (2004). M. Holzmann, W. Krauth, and M. Naraschewski, Phys. Rev. A [**59**]{}, 2956 (1999). Such a four-photon process has been already been experimentally achieved (C. Becker, phD, Hamburg, 2009). M. Erhard, H. Schmaljohann, J. Kronjäger, K. Bongs, and K. Sengstock, Phys. Rev. A [**69**]{}, 032705 (2004); A. Widera, S. Trotzky, P. Cheinet, S. Folling, F. Gerbier, I. Bloch, V. Gritsev, M.D. Lukin, and E. Demler, Phys. Rev. Lett. [**100**]{}, 140401 (2008).
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---
abstract: 'In this paper, we propose a novel framework of air traffic management (ATM). The framework is in particular characterized by the trajectory planning of *weakly supervised* aircraft; the air traffic control (ATC) does not completely determine the trajectory of each aircraft unlike conventional planning methods, but determines *allowable safe sets* of trajectories. ATC requests pilots to select their own trajectories from the sets, and the pilots determine ones by pursuing their own aims. For example, the selection can be based on pilot preferences and airline strategies. This *two stage* ATM system contributes to simultaneously achieve the both objectives of the ATC and pilots such as fuel cost minimization under safety management. The effectiveness of the proposed ATM system is demonstrated in a simulation using actual air traffic data.'
author:
- 'Sho Yoshimura and Masaki Inoue, [^1]'
bibliography:
- 'reference.bib'
title: Trajectory Planning of Weakly Supervised Aircraft
---
[Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{}
Trajectory planning, air traffic management, optimization
Introduction
============
the growth of air traffic all over the world, air traffic management (ATM) systems need to be further developed and to be operated more efficiently. As reported in [@ICAO], ATM is defined as dynamic and integrated management of air traffic and airspace through the provision of facilities and seamless services in collaboration with parties. In this paper, we focus only on control and management problems of aircraft trajectories from departure to arrival.
We review the current ATM system and find its drawback. A sketch of the current system is given in [Fig. \[ATCcurrent\]]{}. As illustrated in the figure, overall airspace is divided to some smaller spaces. The current ATM is operated in a decentralized manner; in other words, a control task corresponding to each divided space is assigned to each air traffic control (ATC). Then, each ATC is responsible for safety of aircraft in an assigned space and manually designs aircraft trajectories. The design is based on positions of aircraft that are obtained from radar information. The trajectories designed by ATC are provided to pilots. A main limitation of the current ATM system lies in manually designing trajectories by ATC. With the growth of air traffic, the number of aircraft contained in airspace increases. This increases burdens for the corresponding ATC, which inhibits efficient trajectory design. In addition, lack of communication between ATCs, caused by the traffic growth, can further deteriorate the efficiency. Therefore, a computer-aided and human-assist ATM system where overall airspace is managed in a centralized manner, is required.
![Illustration of the current ATM. Control tasks of aircraft from departure to arrival are divided between airspace. For each airspace, an air traffic controller designs aircraft trajectories manually and provides them to pilots.\[ATCcurrent\]](./fig/ATCcurrent.pdf)
In the past two decades, various computer-aided methods of generating aircraft trajectories have been proposed[@miyazawa2013dynamic; @pallottino2002conflict; @bonami2013multiphase; @menon1999optimal; @han2017traffic; @tang2016optimal; @vela2010near; @hu2002optimal; @bicchi2000optimal; @toratani2018effects]. One of the main topics is conflict avoidance. Please see, e.g., [@menon1999optimal; @han2017traffic; @tang2016optimal; @vela2010near; @hu2002optimal; @bicchi2000optimal]. The conflict avoidance is involved in optimal trajectory generation problems[@tang2016optimal; @vela2010near; @hu2002optimal; @bicchi2000optimal; @menon1999optimal]. In the current ATM system and previous works[@pallottino2002conflict; @menon1999optimal; @han2017traffic; @vela2010near; @hu2002optimal; @bicchi2000optimal; @tang2016optimal; @toratani2018effects], ATC determines aircraft trajectories completely and provides them to pilots. Pilots must obey the provided trajectories.
The priority to ATC in trajectory determination plays a role for safety management of aircraft. In addition to the safety aim, ATC can achieve to improve airport usage, e.g., by maximizing airport throughput. We can say that the current ATM system meets aims of ATC. On the other hand, the priority to ATC may not be positively acceptable for pilots and airlines. Trajectories determined and enforced by ATC may be fuel-consuming or delayed ones, which are contrary to the aims of pilots and airlines. This paper addresses a novel ATM system design where trajectories are determined based on pilots and airlines aims in addition to safety constraints.
To show the feasibility of the ATM system design, we illustrate the existence of degree of freedom (DOF) in actual aircraft trajectories determined by ATC. Actual trajectories are depicted in [Fig. \[motivation\]]{}, where trajectory data is extracted from CARATS (Collaborative Actions for Renovation of Air Traffic Systems) Open Data[^2] [@MLIT2010; @matsuda2017arrival; @fukuda2015air; @wickramasinghe2017feasibility] on May 11, 2015. In the figure, the three solid lines represent actual trajectories. The translucent lines represent all trajectories where conflict avoidance is achieved. The DOF in trajectories, illustrated by the translucent lines, implies that the aims of pilots and airlines can be pursued in addition to safety constraints. Motivated from this fact, this paper studies a novel framework of ATM systems by utilizing this DOF in trajectory design.
![Example of all trajectories that satisfy ATC requirements including conflict avoidance. The three solid lines represent actual trajectories. The translucent solid lines represent trajectories that avoid aircraft conflicts. Here, the conflict avoidance is defined that aircraft-aircraft distance is longer than 3 NM. \[motivation\]](./fig/motivation.pdf)
In this paper, we propose a framework of ATM systems where ATC *weakly supervises* aircraft for the trajectories planning. An illustration of the proposed framework is shown in [Fig. \[framework\]]{}. In the ATM system, ATC designs *allowable safe sets* of aircraft trajectories, where conflict avoidance and other aims of ATC are achieved, as depicted by red disks in the figure. Then, ATC requests pilots to select their trajectories from the sets. The selection can be based on pilot preferences or airline strategies. It is noted that for the trajectory selection, communication between pilots, such as cooperation, competition and negotiation, is not required. Each pilot individually determines his/her trajectory by pursuing his/her own aims, e.g., by minimizing fuel costs or by reducing uncertainty of arrival time in a decentralized manner.
![Illustration of the proposed ATM framework. In this figure, the black points represent the initial and terminal positions of the target aircraft. ATC designs an *allowable safe set* of trajectories which is represented by the red disks. A trajectory selected from the set by the pilot is represented by the blue cross marks. \[framework\]](./fig/illust.pdf)
Designing *trajectory-sets* for aircraft has been also studied as flow corridors, corridors-in-the-sky, or airspace tubes [@hoffman2008principles; @takeichi2016benefit; @xue2009design; @yousefi2013dynamic]. The concept of the corridor means an exclusive lane designed in airspace where autonomy is allowed for each aircraft; each pilot can design his/her trajectory within the lane without instructions by ATC. This implies that the pilot must be responsible for avoiding an air miss or conflict with other aircraft by communicating with other pilots. This drawback is overcome in the proposed ATM system; each pilot needs not to address the conflict avoidance problem, but simply pursues his/her aims based on a given trajectory-set.
The rest of this paper is organized as follows. In Section II, the proposed framework of ATM systems and problems of trajectory planning are briefly stated. In Section III, a trajectory design problem addressed by ATC is formulated as an optimization problem. By solving the problem, trajectory-sets that avoid aircraft conflicts are obtained. In Section IV, in a similar manner to Section III, a trajectory design problem addressed by pilots is formulated as an optimization problem. At the end of the section, the proposed ATM system is reviewed. In Section V, we show the effectiveness of the proposed system in a numerical simulation with CARATS Open Data.
Weak Control in ATM
===================
Outline of Proposed ATM
-----------------------
In this subsection, we briefly show the outline of the proposed ATM system. The operation flow of the ATM system is given in [Fig. \[concept\]]{}. Operation of the ATM system is to control aircraft from departure to arrival. Trajectory planning runs recursively at some time interval. For each trajectory planning, ATC designs *allowable safe sets* of trajectories, and then requests pilots to select their trajectories from the sets. The selection by pilots is performed independently of each other. This two-stage trajectory planning achieves aims of pilots and airlines such as minimization of fuel consumption, while satisfying requirements by ATC such as conflict avoidance.
Recall that the requests from ATC to pilots are given by *allowable safe sets* of trajectories. The sets are utilized for pilots to minimize fuel costs based on detailed aircraft models and weather conditions. We can say that in the proposed ATM system pilots are *weakly supervised* by ATC. The weakness contributes to *separate* ATC management and pilots optimization; in other words, safe management is performed independently of pilots choices.
![Operation flow of the proposed ATM system. Trajectory planning runs recursively at some time interval. ATC designs *allowable safe sets* of trajectories, and then requests them to the pilots. Pilots select trajectories from the sets independently of each other. The selections are based on their own aims. \[concept\]](./fig/concept.pdf)
Problem Formulation
-------------------
Notation utilized in this paper is listed on [Table \[tab:variables\]]{}. In this subsection, problems of trajectory planning are briefly stated. Let $N$ be the number of target aircraft considered in the problem. In this paper, trajectories refer to positions of aircraft at equal time intervals. Trajectories are restricted to two dimensions for simplicity, but they are easily extended to three dimensions. The position of aircraft $i\in \mathcal{N}:= \{1,\ldots,N\}$ at time $k\in\{0,1,\ldots \}$ is denoted by $C_i(k):=[\,x_i(k)~y_i(k)\,]^{\top}$.
Variables of aircraft are defined below. Let $v_i(k)$, $\theta_i(k)$, $u_i(k)$, and $\psi_i(k)$ be the speed, angle, speed difference, and angle difference of aircraft $i$ at time $k$, respectively. The state of aircraft $i$ at time $k$ is denoted by $X_i(k):=[\,C_i^{\top}(k)~v_i(k)~\theta_i(k)\,]^{\top}$.
As stated above, each trajectory planning is composed of two stages 1) trajectories design and request by ATC side and 2) trajectory design by pilots side. First, we focus on the design problem addressed by ATC as follows. We assume that the initial time $t_i$, the terminal time $T_i$, the initial state $X_i(t_i)$, and the terminal state $X_i(T_i)$ for all aircraft $i\in \mathcal{N}$ are given. Then, ATC designs reference trajectory-sets $\mathcal{W}_i(k),~k\in\{t_i+1,\ldots, T_i-1\},~i\in \mathcal{N}$. Let $\mathcal{W}_i(k)$ be a disk region. Then, the problem is reduced to a problem of finding the center $C_{i}(k)$ and radius $r_i(k)$ of $\mathcal{W}_i(k)$. In the remainder of this paper, $\mathcal{W}_i(k)$ is described as $\mathcal{W}_i(k)=\{ C_{i}(k),r_i(k)\}$. Through the trajectory design, ATC pursues the following aims;
- maximizing the DOF in the trajectory-set, defined by $\| r_i\|$,
- guaranteeing safety of aircraft by conflict avoidance.
For simplification of notation, the set of time of aircraft $i$ is denoted by $\mathcal{T}_i(t_i):=\{t_i+1,\ldots,T_i-1\}$.
Letting $r:=[\,r_1\, \cdots \, r_N\,]^{\top}$, the trajectory design problem addressed by ATC is summarized as follows
\[prob\_atc\] For given $X_i(t_i)$, $X_i(T_i)$, $i\in \mathcal{N}$, find $\mathcal{W}_i(k)$, $k\in \mathcal{T}_i(t_i)$, $i\in \mathcal{N}$ maximizing $\|r\|$ subject to some constraints.
**Problem 1** corresponds to the design problem 1), which is illustrated in the operation flow of [Fig. \[concept\]]{}.
Next, we briefly mention a trajectory design problem addressed by pilots, which is solved next to **Problem 1**. Each pilot solves the problem independently of each other. Assume that the initial state $X_i(t_i)$, the terminal state $
X_i(T_i)$, the requested trajectory-set $\mathcal{W}_i(k)$, $k\in \mathcal{T}_i(t_i)$, and disturbances $d_i(k)$, $k\in \mathcal{T}_i(t_i)$ are given. Then, each pilot selects a trajectory $\hat{C}_i(k)$, $k\in \mathcal{T}_i(t_i)$ by pursuing his/her aims, e.g., minimizing a fuel cost. The trajectory design problem addressed by a pilot labeled by $i$, is summarized as follows.
\[prob\_pilot\] For given $X_i(t_i)$, $X_i(T_i)$, $\mathcal{W}_i(k)$, $k\in \mathcal{T}_i(t_i)$, find $\hat{C}_i(k)$, $k\in \mathcal{T}_i(t_i)$ minimizing some costs.
**Problem 2** corresponds to the design problem 2), which is illustrated in the operation flow of [Fig. \[concept\]]{}.
It is assumed that model information of aircraft and a weather condition are available for each pilot, while they are not available for ATC. Precise model information and a weather condition play important role of reducing a fuel cost. Therefore, the pilot utilizes such information to solve **Problem 2** in an efficient manner and to reduce the achievable cost.
Variables Description
---------------------- ------------------------------------------------------------------
$N$ number of target aircraft
$t_i$ initial time
$T_i$ terminal time
$\mathcal{W}_i$ reference trajectory-set (region defined as disk)
$C_{i}$ center of $\mathcal{W}_i$, i.e., $C_i = [\,x_i,\, y_i\,]^{\top}$
$r_i$ radius of $\mathcal{W}_i$
$C_{\mathrm{sta},i}$ standard trajectory
$\Delta$ deviation between standard trajectory and center of region
$\hat{C}_i$ trajectory to be designed by pilot
$C_{\mathrm{pre},i}$ trajectory pilot select at the last trajectory planning
$d_i$ disturbance
$x_i$ x coordinate of position
$y_i$ y coordinate of position
$v_i$ speed
$\theta_i$ angle
$u_i$ speed difference
$\psi_i$ angle difference
$X_i$ state, i.e., $X_i = [\,C_i^{\top},v_i,\theta_i\,]^{\top}$
: Definition of variables of aircraft $i$ in this paper.\[tab:variables\]
Trajectory Design by ATC
========================
In this section, we mathematically formulate the trajectory design problem addressed by ATC. **Problem \[prob\_atc\]**, which is briefly stated in Section II, is re-formulated as an optimization problem.
Cost Function
-------------
In this subsection, cost functions of the trajectory design problem for ATC are defined. The decision variables are speed differences $u:=[\,u_1\cdots u_N\,]^{\top}$, angle differences $\psi:=[\,\psi_1\cdots \psi_N\,]^{\top}$, and radii $r$ of all aircraft. Trajectory-sets to be designed are evaluated by the size of $r$ and the deviations from a standard trajectory $C_{\mathrm{sta},i}(k)$, $k\in \mathcal{T}_i(t_i)$. The cost function to evaluate $r$ is given by $$\begin{aligned}
\label{eq:J1}
J_1
=
-\sum_{i=1}^{N}\sum_{k=t_i+1}^{T_i-1}
\ln{\left(r_i(k)+\varepsilon\right)},\end{aligned}$$ where $\varepsilon$ is a positive constant. This role of $\varepsilon$ is to prevent divergence of $J_1$. In addition, the cost function to evaluate the deviations is given by $$\begin{aligned}
J_2
=
\sum_{i=1}^{N}\sum_{k=t_i+1}^{T_i-1}
\Delta_i(k)^2
+
\sum_{i=1}^{N}\sum_{k=t_i+1}^{T_i-2}
\left(
\Delta_i(k+1)-\Delta_i(k)
\right)^2,\end{aligned}$$ where $\Delta_i(k)=C_i(k)-C_{\mathrm{sta},i}(k)$. The role of the second term is to prevent a large angle change in one time step. By this $J_2$, we aim at generating realistic trajectories, while avoiding impractical ones. Then, the cost function utilized in the problem here is defined by $$\begin{aligned}
\label{eq:atc_cost}
J_{\mathrm{ATC}}(u,\psi,r) :=
J_1 + \alpha J_2,\end{aligned}$$ where $\alpha$ is a non-negative constant.
There is another candidate of the cost function $J_1$ for evaluating the radii $r_i$. For example, $\ln{\left(r_i(k)+\varepsilon\right)}$ in can be replaced by $1/(r_i(k)+\varepsilon )^2$. In this paper, numerically tractable $J_1$, given by , is considered.
Constraints
-----------
We give constraints derived from physical properties of aircraft, from ATM system requirements, and for ATM system operation. First, the constraints derived from the physical properties are given as follows.
- Aircraft dynamics\
Assume that the dynamics of all the target aircraft are the same. The dynamics of each aircraft is described by $$\begin{aligned}
\label{eq:dynamics}
\begin{bmatrix}
x_i(k+1)\\
y_i(k+1)\\
v_i(k+1)\\
\theta _i(k+1)\\
\end{bmatrix}
=
f\left(
\begin{bmatrix}
x_i(k)\\
y_i(k)\\
v_i(k)\\
\theta _i(k)\\
\end{bmatrix}
\right)
+
\begin{bmatrix}
0\\
0\\
u_i(k)\\
\psi _i(k)\\
\end{bmatrix},\end{aligned}$$ where $f$ is given by $$\begin{aligned}
\label{eq:f}
f\left(
\begin{bmatrix}
x_i(k)\\
y_i(k)\\
v_i(k)\\
\theta _i(k)\\
\end{bmatrix}
\right)
:=
\begin{bmatrix}
x_i(k) + v_i(k)\cos \theta _i(k)\\
y_i(k) + v_i(k)\sin \theta _i(k)\\
v_i(k)\\
\theta_i(k)\\
\end{bmatrix}.\end{aligned}$$ Here, recall that $u_i(k)$ and $\psi _i(k)$ are speed difference and angle difference at time $k\in \mathcal{T}_i(t_i)$ of aircraft $i\in \mathcal{N}$. In the model , they are the control input to aircraft.
- Angular difference\
Angular difference of each aircraft trajectory, denoted by $\psi_i(k)$, is restricted as follows. $$\begin{aligned}
\label{const_angle}
-\Psi \le \psi_i(k) \le \Psi, \quad \forall k\in\mathcal{T}_i(t_i),~\forall i\in\mathcal{N},\end{aligned}$$ where $\Psi$ is the maximum angle difference of a trajectory.
- Speed difference\
Speed difference of each aircraft trajectory, denoted by $u_i(k)$, is restricted as follows. $$\begin{aligned}
-U \le u_i(k) \le U, \quad \forall k\in\mathcal{T}_i(t_i),~\forall i\in\mathcal{N},\end{aligned}$$ where $U$ is the maximum speed difference to be applied.
- Speed\
Speed of each aircraft, denoted by $v_i(k)$, is restricted as follows. $$\begin{aligned}
\label{eq:velocity}
V_{\mathrm{min}}\le v_i(k) \le V_{\mathrm{max}},
\quad \forall k\in\mathcal{T}_i(t_i),~\forall i\in\mathcal{N},\end{aligned}$$ where $V_{\mathrm{min}}$ and $V_{\mathrm{max}}$ are the lower and upper bounds of $v_i(k)$.
Next, the constraints related to ATM system requirements are given as follows.
- Terminal constraint\
Constraints on the speed and angle at the terminal time $T_i$ are given. The deviation between the speed at $T_i$ and the reference $V_{\mathrm{ter}}$ is less than or equal to $\delta_v$, and the deviation between angle at $T_i$ and the reference $\Theta_{\mathrm{ter}}$ is less than or equal to $\delta_{\theta}$. They are described by $$\begin{aligned}
\label{eq:term_velocity}
&-\delta_v \le v_i(T_i)-V_{\mathrm{ter}} \le \delta_v,\quad \forall i\in\mathcal{N},\\
\label{eq:term_angle}
&-\delta_{\theta} \le \theta_i(T_i)-\Theta_{\mathrm{ter}} \le \delta_{\theta},\quad \forall i\in\mathcal{N}.\end{aligned}$$
- Constraint for conflict avoidance\
A constraint for conflict avoidance is given by $$\begin{aligned}
\label{eq:conflict}
&\|
C_i(k)-C_j(k)
\|
-
\left(
r_i(k) + r_j(k)
\right)
\ge
D,\\
\nonumber
&\forall k\in \{ \mathcal{T}_i(t_i)\cap \mathcal{T}_j(t_j) \}
,~\forall i\neq j\in \mathcal{N},\end{aligned}$$ where $D$ is a positive constant. We illustrate the meaning of with [Fig. \[fig:margin\]]{}. The left side of represents the allowable minimum distance between two aircraft, which is illustrated in [Fig. \[fig:margin\]]{}. Therefore, $D$ plays a role of a safety margin.
- Constraint for feasibility\
In this paper, feasibility means that the trajectory design problem performed by pilots are feasible. In other words, every pilot can select a trajectory from a trajectory-set requested by ATC. This is described as $$\begin{aligned}
\nonumber
&\| C_i(k+1)-C_i(k)\| -
\left(r_i(k+1)+r_i(k)\right) \\
\label{eq:const_r_1}
&\qquad \qquad \ge V_{\mathrm{min}},~\forall k\in\{t_i \cup \mathcal{T}_i\},~\forall i\in\mathcal{N},\\
\nonumber
&\| C_i(k+1)-C_i(k)\| +
\left(r_i(k+1)+r_i(k)\right) \\
\label{eq:const_r_2}
&\qquad \qquad \le V_{\mathrm{max}},~\forall k\in\{t_i \cup \mathcal{T}_i\},~\forall i\in\mathcal{N},\end{aligned}$$ where $r_i(t_i)=0$ and $r_i(T_i)=0$, $i\in \mathcal{N}$. We illustrate the meaning of and with [Fig. \[fig:r\_const\]]{}. Equation means that the minimum distance of every pair of adjacent regions is longer than or equal to the *realizable* minimum distance, i.e., the trajectory is realized for some $u_i$, $\psi_i$, $v_i$, $\theta_i$, $x_i$, and $y_i$ satisfying -. In a similar manner, equation means the maximum distance of every pair of adjacent regions is shorter than or equal to the realizable maximum distance.
Finally, a constraint for ATM system operation is given.
- Operation Constraint\
We define a constraint that works at re-planning of trajectories. Consider that a trajectory planning is performed at a time $k = t_i$. Then, we suppose that pilots select $\hat{C}_i(k)$, $k\in \mathcal{T}_i(t_i)$. It should be noted that *all* the future trajectories $\hat{C}_i(t_i+1)$ to $\hat{C}_i(T_i-1)$ are selected by pilots $i\in \mathcal{N}$ based on their aims. At the re-planning, we incorporate the aims into trajectories to be redesigned by ATC. Let us consider that trajectories are re-planed at a time $t_i+\tau$, where $\tau \in \{1, 2,\ldots, T_i -2 -t_i \}$. Then, we need to impose a constraint on trajectories redesigned by ATC. We let $C_{\mathrm{pre},i}(k) = \hat{C}_i(k)$, $k\in \mathcal{T}_i(t_i)$. Then, trajectory-sets $\mathcal{W}_i(k)$, $k\in \{t_i+\tau +1,\ldots,T_i-1\}$ to be redesigned must include the last selection $C_{\mathrm{pre},i}(t_i+\tau +1)$ to $C_{\mathrm{pre},i}(T_i-1)$ such that pilots can select the same *best* trajectories. This constraint is described by a condition on $C_i(k)$ and $r_i(k)$ to be redesigned as $$\begin{aligned}
\label{eq:operation}
\| C_{\mathrm{pre},i}(k) - C_{i}(k) \|
\le
r_i(k),~\forall k\in \mathcal{T}_i(t_i+\tau _i),~\forall i\in\mathcal{N}.\end{aligned}$$ The meaning of the equation is illustrated in [Fig. \[fig:operation\]]{}.
![Constraint for conflict avoidance. The minimum distance between the aircraft $i$ and $j$ at time $k$, which is described by the left side of , is shown.\[fig:margin\]](./fig/margin.pdf)
![Constraints for feasibility. The minimum and maxmum distance of adjacent regions at time $k$ and $k+1$, which are described by the left side of and , respectively, are shown.\[fig:r\_const\]](./fig/region-const.pdf)
![Operation constraint. The trajectory $C_{\mathrm{pre},i}$, which is selected at the last planning, must be included in $\mathcal{W}_i$, which is redesigned by ATC at the current planning. The trajectory $C_{\mathrm{pre},i}$ is represented by the blue solid line, while the regions $\mathcal{W}_i$ is represented by the red disks. \[fig:operation\]](./fig/operation.pdf)
Optimization Problem
--------------------
**Problem 1**, which is stated in Section II, is mathematically formulated in this subsection. To this end, we let $X:=[\, X_1^{\top} X_2^{\top} \cdots X_N^{\top}\,]^{\top}$. Then, the optimization problem addressed by ATC is summarized as follows.
For given $X_i(t_i)$, $X_i(T_i)$, $C_{\mathrm{pre},i}(k)$, $k\in \mathcal{T}_i(t_i)$, $i\in \mathcal{N}$, $$\begin{aligned}
&{\mathop{\rm minimize}\limits}_{u,\psi,r,X}~J_{\mathrm{ATC}}(u,\psi,r)\\
\nonumber
&\mathrm{subject~to~Eqs.}~{\eqref{eq:dynamics}}-{\eqref{eq:operation}}.\end{aligned}$$
In the following, the optimizer to **Problem 3** is denoted by $(u^{\ast}$, $\psi^{\ast}$, $r^{\ast}$, $X^{\ast})$. A trajectory-set $\mathcal{W}^{\ast}_i(k)$, $k\in \mathcal{T}_i(t_i)$ is obtained based on the solution to **Problem 3**. Then, $\mathcal{W}^{\ast}_i(k)$, $k\in \mathcal{T}_i(t_i)$ is provided to each pilot.
Note again that **Problem 3** is formulated without utilizing information on detailed aircraft dynamics and weather conditions. This is because that such information is not necessary for system management including the conflict avoidance. On the other hand, the information can be efficiently utilized by pilots, e.g., for reducing fuel costs. The optimization problem addressed by pilots utilizes aircraft models based on BADA data[@nuic2010bada] or weather situation based on global forecasting systems, e.g., GDPFS (The Global Data-Processing and Forecasting System)[@GDPFS].
In this paper, a DOF in trajectories is expressed by spatial regions, denoted by $\mathcal{W}_i(k) = \{C_i(k),\,r_i(k)\}$. It should be emphasized that the spatial DOF is equivalently transformed into temporal one. As studied above, a spatial DOF in trajectories is expressed by a spatial ball region that aircraft must be included at a *specified time* $k$. On the other hand, a temporal DOF is expressed by a time range at *any of which* aircraft passes a specified vertically-placed disk. The expressions are illustrated in [Fig. \[illust\_DOF\]]{}. The two expressions of DOF are equivalent each other.
![Spatial DOF and temporal DOF. A spatial DOF is a spatial ball region where an aircraft must be included at a *specified time*. A temporal DOF is a time range at *any of which* an aircraft must pass a vertically-placed disk. \[illust\_DOF\]](./fig/illust_DOF.pdf)
Since **Problem 3** is in the class of high-dimensional nonlinear optimization problems, the choice of initial values of decision variables is essential to obtain a *better* local optimum. In this paper, the initial guess of $u$ and $\psi$ is determined based on standard trajectories. Otherwise, at re-planning case, the initial guess of $C_i(k)$, $k\in \mathcal{T}_i(t_i)$ is based on pilots selections $C_{\mathrm{pre}}(k)$, $k\in \mathcal{T}_i(t_i)$, which are designed at the last planning.
Trajectory Design by Pilots
===========================
In Section III, the trajectory design problem addressed by *ATC* is formulated. In Section IV, we focus on a trajectory design problem addressed by *pilots*. In the design problem of this section, we focus only on a pilot labeled by $i\in\mathcal{N}$. It is assumed that the reference trajectory-set, denoted by $W_i^{\ast}(k)$, $k\in \mathcal{T}_i(t_i)$, is given and available for the design problem. Then, in a similar manner to the ATC case, the design problem by the pilot is formulated as an optimization problem. In this section, all decision variables of the optimization problem are denoted as $\hat{\cdot}$ in order to distinguish them from those in **Problem 3**.
Cost Function
-------------
The cost function to evaluate the aims of the pilot is given. The decision variables are speed difference $\hat{u}_i$ and angular difference $\hat{\psi}_i$. The cost function is described by $$\begin{aligned}
\label{eq:cost-pilot}
J_{\mathrm{pilot}}(\hat{u}_i,\hat{\psi}_i)
:=
\sum_{k=t_i}^{T_i-2}
\left(
\left( \dfrac{\hat{u}_i(k)}{U}\right)^2
+\left( \dfrac{\hat{\psi}_i(k)}{\Psi}\right)^2
\right).\end{aligned}$$ This cost function $J_{\mathrm{pilot}}$ evaluates fuel costs of aircraft labeled by $i\in \mathcal{N}$. The fuel costs are defined as a sum of squared control inputs of aircraft based on the definition of the work[@toratani2018effects]. In , variables $\hat{u}_i$ and $\hat{\psi}_i$ are normalized such that the effects of them are fairly evaluated.
Constraints
-----------
Constraints that come from physical properties of aircraft and from ATM system requirements are listed. Constraints from aircraft specification are the same as -, while those from aircraft dynamics are more precise than . We assume that information of disturbance $d$, which represents, e.g., wind condition of airspace, is available for trajectory design by the pilot. Then, aircraft dynamics are modified from to $$\begin{aligned}
\label{eq:dynamics-p}
\begin{bmatrix}
\hat{x}_i(k+1)\\
\hat{y}_i(k+1)\\
\hat{v}_i(k+1)\\
\hat{\theta}_i(k+1)\\
\end{bmatrix}
=
f\left(
\begin{bmatrix}
\hat{x}_i(k)\\
\hat{y}_i(k)\\
\hat{v}_i(k)\\
\hat{\theta}_i(k)\\
\end{bmatrix}
\right)
+
\begin{bmatrix}
0\\
0\\
\hat{u}_i(k)\\
\hat{\psi}_i(k)\\
\end{bmatrix}
+
\begin{bmatrix}
d_{x,i}(k)\\
d_{y,i}(k)\\
0\\
0\\
\end{bmatrix}.\end{aligned}$$ In this problem setting, disturbances $d_i(k)$ are only applied to aircraft positions, in other words, $d_i(k)=[\,d_{x,i}(k)~d_{y,i}(k)~0~0\,]^{\top}$ holds.
For constraints from ATM system requirements, in addition to and each pilot selects his/her own trajectory from the trajectory set, which is designed and requested by ATC. Recall here that an aircraft trajectory to be designed by the pilot is denoted by $\hat{C}_i(k)$ and that the trajectory-set designed by ATC is denoted by $\mathcal{W}^{\ast}_i(k)=\{ C^{\ast}_{i}(k),r^{\ast}_i(k)\}$. Then, the constraint on the trajectory selection is described by $$\begin{aligned}
\label{const_pilot}
r^{\ast}_i(k) &\ge \| C^{\ast}_{i}(k)-\hat{C}_i(k)\|_2,~\forall k\in \mathcal{T}_i(t_i).\end{aligned}$$
Optimization Problem
--------------------
The optimization problem addressed by each pilot is summarized as follows.
For given $X^{\ast}_i(t_i)$, $X^{\ast}_i(T_i)$, $\mathcal{W}^{\ast}_i(k)$, $k\in \mathcal{T}_i(t_i)$, $$\begin{aligned}
\label{eq:opt-p}
&{\mathop{\rm minimize}\limits}_{\hat{u}_i,\hat{\psi}_i,\hat{X}_i}~J_{\mathrm{pilot}}(\hat{u}_i,\hat{\psi}_i)\\
\nonumber
&\mathrm{subject~to~Eqs.}~{\eqref{eq:f}}-{\eqref{eq:term_angle}},{\eqref{eq:dynamics-p}},{\eqref{const_pilot}}.\end{aligned}$$
In the following, the optimizer to **Problem 4** is denoted by $(\hat{u}^{\dagger}_i$, $\hat{\psi}^{\dagger}_i$, $\hat{X}^{\dagger}_i)$. Then, the optimal trajectory is denoted by $\hat{C}^\dagger_i(k)$, $k\in \mathcal{T}_i(t_i)$.
![Operation flow of the proposed ATM system. Trajectory planning runs recursively at some time interval. In one trajectory planning, trajectory design is performed sequentially by ATC and pilots. Trajectory-sets and trajectories are determined based on the solutions to **Problem 3** and **4**, respectively. \[concept\_detailed\]](./fig/concept_detailed.pdf)
In this paper, no precise model of aircraft dynamics is utilized even in **Problem 4**. In a practical setting, need to be replaced by a more precise model. Utilizing more detailed aircraft dynamics, the problem is formulated in the same manner.
The whole operation flow of the proposed ATM system is reviewed. The block diagram shown as [Fig. \[concept\_detailed\]]{} illustrates a more detailed operation flow than that of [Fig. \[concept\]]{}. Each trajectory planning is composed of two stages; 1) trajectory-set design by ATC and 2) trajectory design by pilots. 1) ATC designs trajectory-sets by solving **Problem 3**. In the problem setting of **Problem 3**, given initial states $X_i(t_i)$, terminal states $X_i(T_i)$, trajectories selected by pilots at the last trajectory planning, denoted by $C_{\mathrm{pre},i}(k)$, $k\in \mathcal{T}_i(t_i)$, $i\in \mathcal{N}$, we aim at finding reference trajectory-sets $\mathcal{W}_i(k)$, $k\in \mathcal{T}_i(t_i)$. ATC requests pilots to select their trajectories from the designed sets $\mathcal{W}_i^{\ast}(k)$, $k\in \mathcal{T}_i(t_i)$. 2) Then, each pilot design his/her trajectory by solving **Problem 4** independently of each other. In the problem setting of **Problem 4**, given $X_i^{\ast}(t_i)$, $X_i^{\ast}(T_i),~i\in \mathcal{N}$, we aim at finding trajectories $\hat{C}_i(k)$, $k\in \mathcal{T}_i(t_i)$. If re-planning is required due to some real-world uncertainties, we let $C_{\mathrm{pre},i}(k)=\hat{C}^{\dagger}_i(k)$, $k\in \mathcal{T}_i(t_i)$. Then, the two stage design is performed again.
Numerical Simulations
=====================
We verify the effectiveness of the proposed ATM system, including aircraft trajectory design, in a numerical simulation. To this end, the proposed design method is applied to actual data extracted from CARATS Open Data. The data includes three trajectories from or arrive at HANEDA airport on May 11, 2015. These trajectories are illustrated in [Fig. \[fig:actual\_path\]]{}. In the figure, HANEDA airport is located at the origin. The aircraft trajectories that depart from HANEDA are represented by red and blue solid lines, while that arrives at HANEDA is represented by the green solid line.
![Actual trajectories are extracted from CARATS Open Data. The aircraft trajectory that departs from HANEDA is represented by red and blue solid lines, while that arrives at HANEDA is represented by the green solid line.\[fig:actual\_path\]](./fig/actual_traj.pdf)
Simulation Conditions
---------------------
In the simulation, trajectory planning runs at the initial time and middle time of the operation. At the middle time, only trajectory design by ATC is demonstrated in this section. Conditions on the first and second trajectory planning are given below.
Conditions on the first trajectory planning are given. For each aircraft $i\in \mathcal{N}=\{1,2,3\}$, the initial operation time $t_i$, the terminal operation time $T_i$, the initial state $X_i(t_i)$, and the terminal state $X_i(T_i)$ are given as follows.
- Aircraft 1 $$\begin{aligned}
&t_1=1,~T_1=12,~X_1(t_1)=[\,4.71~-8.42~16.4~-1.58\,]^{\top}\\
&X_1(T_1)=[\,-413~-97.5~22.2~2.63\,]^{\top}\end{aligned}$$
- Aircraft 2 $$\begin{aligned}
&t_2=2,~T_2=13,~X_2(t_2)=[\,4.50~-9.01~17.7~-1.56\,]^{\top}\\
&X_2(T_2)=[\,-452~-123~31.1~2.84\,]^{\top}\end{aligned}$$
- Aircraft 3 $$\begin{aligned}
&t_3=2,~T_3=15,~X_3(t_3)=[\,-406~-217~31.8~0.471\,]^{\top}\\
&X_3(T_3)=[\,5.91~-1.96~31.2~0.833\,]^{\top}\end{aligned}$$
Here, one time step corresponds to six minutes.
In the cost function , the parameter is chosen by $\alpha = 0.01$. To solve **Problem 3**, the initial values of $u$ and $\psi$ are computed based on the standard trajectories in actual data, while that of $r$ is zero.
In **Problem 4**, for simplicity, we suppose that the wind disturbance $d$ is constant and that the magnitude and angle of $d$ are $1/3$ and $\pi/4$, respectively. That is, $d_{x,i}(k)=0.236,~d_{y,i}(k)=0.236,~k\in \mathcal{T}_i,~i\in \mathcal{N}$. Note that only pilots utilize this wind condition. To solve **Problem 4**, the initial values of $\hat{u}$ and $\hat{\psi}$ are determined such that the center of the trajectory-sets requested by ATC are tracked.
Conditions on the second trajectory planning are given. The second trajectory planning is carried out at the fifth time step, i.e., $t_i=5,~i\in\mathcal{N}$. The parameter $\alpha$ of is fixed as $\alpha = 0.01$, which is the same as the first planning. To solve the problem, the initial values of $\hat{u}$ and $\hat{\psi}$ are the same as ones selected by pilots at the first trajectory planning. In addition, the initial value of $\hat{r}$ is zero.
Results
-------
The resulting trajectories of the first trajectory planning are shown in [Figs. \[fig:atc\_traj\]]{} and \[fig:pilot\_traj\]. Figure \[fig:atc\_traj\] shows the result of the trajectory design by ATC. In the figure, the reference trajectory-sets are represented by the red, blue, and green disks, while the trajectories that track the center of the sets are represented by the solid lines. The mean and standard deviation of the resulting $r_i(k)$, which is the DOF, are listed on [Table \[tab:r\_res1\]]{}. The DOFs of Aircraft 1 and Aircraft 2, which are adjoining, are the minimum and maximum one. This is because that the cost tends to be small when one DOF in adjoining trajectories is big and the other is small.
Figure \[fig:pilot\_traj\] shows the result of the trajectory designed by pilots. In the figure, the trajectories selected by pilots are represented by the red, blue, and green solid lines. We see that all the trajectories are included in the trajectory-sets requested by ATC, which are represented by the disks. This implies that conflict avoidance is achieved. The trajectories are quantitatively evaluated by the cost function $J_\mathrm{pilot}$, which is defined in and simulates fuel costs in some sense. The cost values of actual trajectories (Trajectory A), trajectories that track the center of the sets (Trajectory B), and the trajectories selected by pilots (Trajectory C) are listed on [Table \[tab:cost\_value\]]{}. This shows that the fuel costs of Trajectory C is the minimum for all aircraft. This result implies that although only a simple aircraft model and a performance index of fuel costs are used in this simulation, it is expected that performances are improved even when more detailed models and performance indexes are used.
Aircraft 1 Aircraft 2 Aircraft 3
-------------------- ------------ ------------ ------------
Mean $13.5$ $21.6$ $15.5$
Standard deviation $2.30$ $5.65$ $3.85$
: Comparison at the values of as fuel costs.\[tab:cost\_value\]
Aircraft 1 Aircraft 2 Aircraft 3 total
-------------- ------------ ------------ ------------ --------
Trajectory A $2.91$ $1.95$ $0.44$ $5.31$
Trajectory B $2.99$ $2.07$ $0.47$ $5.44$
Trajectory C $2.13$ $1.62$ $0.25$ $4.00$
: Comparison at the values of as fuel costs.\[tab:cost\_value\]
Figure \[fig:atc\_ope\_traj\] shows the result of the second trajectory planning. In the figure, the reference trajectory-sets redesigned by ATC are drawn on [Fig. \[fig:pilot\_traj\]]{}. The sets are represented by the deep red, blue, and green disks. It is confirmed that the constraint for operation is satisfied; the trajectories selected by pilots at the first trajectory planning are included in the trajectory-sets redesigned by ATC. This implies that each pilot can choose the same trajectory as the last planning if there is no update for the weather condition and pilots and airlines cannot receive demerits by the re-planning.
![Result of ATC re-design at the second trajectory planning. The sets redesigned by ATC are represented by deep red, blue, and green disks. The trajectories selected by pilots at the first trajectory planning are represented by the solid lines. \[fig:atc\_ope\_traj\]](./fig/atc_traj.pdf)
![Result of ATC re-design at the second trajectory planning. The sets redesigned by ATC are represented by deep red, blue, and green disks. The trajectories selected by pilots at the first trajectory planning are represented by the solid lines. \[fig:atc\_ope\_traj\]](./fig/pilot_traj.pdf)
![Result of ATC re-design at the second trajectory planning. The sets redesigned by ATC are represented by deep red, blue, and green disks. The trajectories selected by pilots at the first trajectory planning are represented by the solid lines. \[fig:atc\_ope\_traj\]](./fig/atc_ope_traj.pdf)
Conclusion
==========
In this paper, we proposed a novel framework of ATM systems where ATC *weakly supervises* aircraft. Aircraft trajectories, which are completely determined by ATC conventionally, are designed by ATC as *trajectory-sets*, and the sets are provided to pilots. Then, the pilots individually select their trajectories from the sets. We showed that both the safety requirement, which is the aim of ATC, and reduction of fuel consumption, which is the aim of pilots and airlines, were achieved in the ATM system. The authors expect that the proposed ATM system may not be directly implemented to real-world ATM. However, the essence of the weak supervision, i.e., idea of explicitly providing DOFs to pilots, can be utilized to practical ATM systems in a modified manner.
The optimization problems formulated for trajectory planning are nonlinear to some decision variables, which are numerically intractable. In future work, we need to reformulate them to more tractable ones.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by Grant for Basic Science Research Projects from Sumitomo Foundation and by Grant-in-Aid for Scientific Research (A), No. 18H03774 from JSPS.
[Sho Yoshimura]{} Sho Yoshimura was born in Chiba, Japan in 1993. He received the Bachelor’s degree in Engineering from Keio University in 2017. He is currently a Master Course student in Keio University. His research interests include air traffic management design and system identification for large-scale systems.
[Masaki Inoue]{} Masaki Inoue was born in Aichi, Japan, in 1986. He received the M.E. and Ph.D. degrees in engineering from Osaka University in 2009 and 2012, respectively. He served as a Research Fellow of the Japan Society for the Promotion of Science from 2010 to 2012. From 2012 to 2014, He was a Project Researcher of FIRST, Aihara Innovative Mathematical Modelling Project, and also a Doctoral Researcher of the Graduate School of Information Science and Engineering, Tokyo Institute of Technology. Currently, he is an Assistant Professor of the Faculty of Science and Technology, Keio University. His research interests include stability theory of dynamical systems. He received several research awards including the Best Paper Awards from SICE in 2013, 2015, and 2018, from ISCIE in 2014, from IEEJ in 2017, and the Takeda Best Paper Award from SICE in 2018. He is a member of IEEE, SICE, ISCIE, and IEEJ.
[^1]: S. Yoshimura and M. Inoue are with the Department of Applied Physics and Physico-Informatics, Keio University, Yokohama, Japan (e-mail: minoue@appi.keio.jp).
[^2]: CARATS Open Data includes 3D-position and time data of all IFR (Instrument Flight Rules) commercial flights in Japanese airspace.
|
---
abstract: 'We seek to determine whether state-of-the-art, black box face recognition techniques can learn first-impression appearance bias from human annotations. With FaceNet, a popular face recognition architecture, we train a transfer learning model on human subjects’ first impressions of personality traits in other faces. We measure the extent to which this appearance bias is embedded and benchmark learning performance for six different perceived traits. In particular, we find that our model is better at judging a person’s dominance based on their face than other traits like trustworthiness or likeability, even for emotionally neutral faces. We also find that our model tends to predict emotions for deliberately manipulated faces with higher accuracy than for randomly generated faces, just like a human subject. Our results lend insight into the manner in which appearance biases may be propagated by standard face recognition models.'
author:
- |
Ryan Steed\
Department of Computer Science\
George Washington University\
`ryansteed@gwu.edu`\
Aylin Caliskan\
Department of Computer Science\
George Washington University\
`aylin@gwu.edu`
bibliography:
- 'references2.bib'
title: 'Machines Learn Appearance Bias in Face Recognition\'
---
Introduction {#sec:introduction}
============
Researchers have raised concerns about the use of face recognition for, *inter alia*, police surveillance and job candidate screening [@DeborahRaji2020SavingAuditing]. For example, HireVue’s automated recruiting technology uses candidate’s appearance and facial expression to judge their fitness for employment [@Harwell2019AJob]. If a surveillance or hiring algorithm learns harmful human biases from annotated training data, it may systematically discriminate against individuals with certain facial features. We investigate whether industry-standard face recognition algorithms can learn to trust or mistrust faces based on human annotators’ perception of personality traits from faces. If off-the-shelf machine learning face recognition models draw trait inferences about the faces they examine, then any application domain using face recognition to make judgments, from surveillance to hiring to self-driving cars, is at risk of propagating harmful prejudices. In human beings, quick trait inferences should not affect important, deliberate decisions [@Willis2006FirstImpressions], but unconscious appearance biases that occur during rapid data annotation may embed and amplify appearance discrimination in machines. We show that the embeddings from FaceNet model can be used to predict human annotators’ first-impression appearance biases for six different personality traits.
Because the predictions made by machine learning models depend on both the training data and the annotations used to label them, systematic biases in either source of data could result in biased predictions. For instance, a dataset on employment information designed to predict which job candidates will be successful in the future might contain data regarding mainly European American men. If such a dataset reflects historical injustices, it is likely to unfairly disadvantage African American job candidates. Moreover, annotators could introduce human bias to the dataset by labeling items according to their implicit biases. If annotators for a computer vision task are presented with a photo of two employees, they might label a woman as the employee and the man standing next to her as the employer or boss. Such embedded implicit or sociocultural bias leads to biased and potentially prejudiced outcomes in decision making systems. Prior research shows that in human-centered data, *a priori* bias often includes harmful stereotypes and introduces problems of bias or unfairness into subsequent decision-making. In computer vision, models used in face detection or self-driving cars have been proven biased against genders and races [@Buolamwini2018GenderClassification; @Wilson2019PredictiveDetection].
![Unilever using AI powered job candidate assessment tool HireVue [@Harwell2019AJob].[]{data-label="fig:hirevure"}](figures/unileverAI.jpg){width="50.00000%"}
Section \[sec:problem\] outlines our research question, while section \[sec:related\] reviews related work. Section \[sec:dataset\] details the data used in our investigation, and Section \[sec:approach\] describes our approach. Our results are presented in Section \[sec:results\]. Section \[sec:discussion\] concludes.
Problem Statement {#sec:problem}
=================
In this research, we hypothesize that computer vision models not only embed racial or gender biases but also embed other human-like biases, including appearance biases caused by first impressions. Some examples of these biases include gender choices made by automated captioning systems and contextual cues used incorrectly by visual question answering systems [@Hendricks2018WomenModels; @Zhao2017MenConstraints; @Manjunatha2019ExplicitModels]. As of this writing, these algorithms are actively used in video interview screening of job applicants [@Escalante2017DesignInterviews] (Figure-\[fig:hirevure\]), self-driving cars [@Geiger2012AreSuite], surveillance [@Ko2008AApplications], anomaly detection [@Mahadevan2010AnomalyScenes], military unmanned autonomous vehicles [@Nex2014UAVReview], and cancer detection [@Bejnordi2017DiagnosticCancer].
But while many biases affecting machine learning systems are explicit and easily detected with error analysis, some “implicit" biases are consciously disavowed and are much more difficult to measure and counteract. Often, these biases take effect a split second after perception in human judgment. These biases can often be quantified by implicit association tests [@Greenwald1998MeasuringTest] or other psychological studies [@Willis2006FirstImpressions]. We investigate whether biases formed during the first impression of a human face get embedded in face recognition models. First impressions are trait inferences drawn from the facial structure and expression of other people [@Willis2006FirstImpressions]. Here, traits are personality attributes including attractiveness, competence, extroversion, dominance, likeability, and trustworthiness [@Hassin2000FacingPhysiognomy]. Specifically, we propose that standard machine learning techniques, including pre-trained face recognition models, propagate first-impression trait inferences, or appearance biases, based on facial structures. To determine if machine learning models acquire first appearance bias, we quantify the correlation between our model’s predicted trust scores and human subjects’ actual trait inferences.
But while many biases affecting machine learning systems are explicit and easily detected with error analysis, some “implicit" biases are consciously disavowed and are much more difficult to measure and counteract. Often, these biases take effect a split second after perception. These biases can often be quantified by implicit association tests [@Greenwald1998MeasuringTest] or other psychological studies [@Willis2006FirstImpressions]. Specifically, we examine biases formed during the first impression of a human face. First impressions are trait inferences drawn from the facial structure and expression of other people [@Willis2006FirstImpressions]. Here, traits are personality attributes including as attractiveness, competence, extroversion, dominance, likeability, and trustworthiness [@Hassin2000FacingPhysiognomy]. Specifically, we propose that standard machine learning techniques, including pre-trained face recognition models, propagate first-impression trait inferences, or appearance biases, based on facial structures. To determine if machine learning models acquire first appearance bias, we test for a correlation between our model’s predicted trust scores and human subjects’ actual trait inferences.
Related Work {#sec:related}
============
There is a wealth of literature measuring the stereotypes perpetuated by image classifiers and other machine learning models, from search results to automated captioning [@Kay2015UnequalOccupations; @Hendricks2018WomenModels; @Kleinberg2017HumanPredictions]. Previous applications of unsupervised machine learning methods demonstrated the existence of social and cultural biases embedded in the statistical properties of language, but little research has been conducted with respect to the biases in transfer learning models for faces or people and even less attention has been paid to the intersection of machine learning and first appearance bias [@Caliskan2017; @Torralba2011UnbiasedBias]. Tangentially, Jacques Junior et. al review the use of computer vision to anticipate personality traits [@JacquesJunior2018FirstAnalysis]. Most notably, Yang and Glaser use a novel long-short term memory (LSTM) approach to predict first impressions of the Big Five personality traits after 15 seconds [@Yang2017PredictionLSTM]. But are these first impressions preserved in datasets and off-the-shelf models used in transfer learning, and can even more rapid judgments such as first-impression bias be replicated?
There are a few relevant psychology studies devoted to measuring cognitive bias associated with human face recognition. In particular, @Willis2006FirstImpressions measure the immediate judgments people make about others’ faces on first sight, recording a spectrum of trait inferences, from trustworthiness to aggressiveness, after less than a second of exposure to computer-generated faces [@Willis2006FirstImpressions; @Todorov2017FaceImpressions]. @Oosterhof2008TheEvaluation identified 50 principal components in the 2D space that represent face shape to linearly generate face variations that capture a broad range of facial attributes and trait judgments. They further analyze the facial cues used to make evaluations about trustworthiness and dominance, identifying “approach/avoidance" expressions that signal trustworthiness and features that signal physical strength, or dominance.
Dataset {#sec:dataset}
=======
![Face (center) manipulated to appear 3SD more (left) and 3SD less (right) trustworthy than the average face [@Todorov2013ValidationFacial].[]{data-label="fig:face_distinct"}](figures/trustworthy_face.jpg "fig:"){width="0.3\linewidth" height="0.3\linewidth"} ![Face (center) manipulated to appear 3SD more (left) and 3SD less (right) trustworthy than the average face [@Todorov2013ValidationFacial].[]{data-label="fig:face_distinct"}](figures/neutral_face.jpg "fig:"){width="0.3\linewidth" height="0.3\linewidth"} ![Face (center) manipulated to appear 3SD more (left) and 3SD less (right) trustworthy than the average face [@Todorov2013ValidationFacial].[]{data-label="fig:face_distinct"}](figures/untrustworthy_face.jpg "fig:"){width="0.3\linewidth" height="0.3\linewidth"}
To test whether first impression trait inferences can be learned from facial cues visualized in Figure \[fig:face\_distinct\], we aggregated datasets of computer-generated faces used to measure appearance bias in two psychological studies [@Oosterhof2008TheEvaluation; @Todorov2013ValidationFacial]. In each experiment, human participants are shown a face for less than a second and then asked to rate the degree to which it exhibits a given trait on a 9-point scale. Each face is hairless and centered on a black background. These face models were generated with FaceGen, which uses a database of laser-scanned male and female human faces to create new, unique faces [@FaceGenFaces]. Together, these two sets provide a benchmark for first impression, appearance-based evaluations of personality traits by human participants.
Randomly Generated Faces
------------------------
The first dataset (<span style="font-variant:small-caps;">300 Random Faces</span>) includes 300 computer-generated, emotionally neutral, Caucasian male faces (Figure \[fig:face\_distinct\]). Male faces were used due to lack of hair; participants tend to categorize bald faces as male [@Todorov2013ValidationFacial]. In this study, the authors asked 75 Princeton University undergraduates to judge each face from this dataset on attractiveness, competence, extroversion, dominance, likeability, and trustworthiness [@Oosterhof2008TheEvaluation; @Todorov2011Data-drivenPerception]. Here, the ground-truth labels are the trustworthy scores provided by the study participants. So that the ground-truth labels for both datasets are distributed normally, we standardize each score $x_i$ by calculating its distance from the mean $z_i = \frac{x_i-\bar{x}}{s_x}$, where $\bar{x}$ is the mean and $s_x$ is the standard deviation (SD).
Faces Manipulated Along Trait Dimensions
----------------------------------------
For the second dataset (<span style="font-variant:small-caps;">Maximally Distinct Faces</span>), @Todorov2011Data-drivenPerception select 25 “maximally distinct" faces from a random sample of 1,000 randomly generated faces. Maximally distant faces are those faces whose principal components are separated by the maximum Euclidean distance. Using the correlations between a principal component face representation and empirical trait inference ratings from the human participants, each maximally distinct face is manipulated along each of the six trait dimensions to produce a set of faces to elicit a trait inference -3, -2, -1, 0, 1, 2, and 3 SD from the mean. Manipulations are numerical perturbations in the face representation vector based entirely on the correlations between the vector and human participants’ judgments of the face it represents. Though the perturbations themselves are not psychologically meaningful, these manipulations tend to produce faces that vary noticeably along the trait dimensions (Figure \[fig:face\_distinct\]). After the faces were produced, the target trait scores were validated by 15 different Princeton University student participants on the same 9-point scale as in the first study [@Todorov2013ValidationFacial].
Approach {#sec:approach}
========
To train a regression model to predict trustworthy trait inferences, we construct a transfer learning pipeline to leverage face representations extracted from a pre-trained, state-of-the-art face recognition model (Figure \[fig:pipeline\]). From the final layer of FaceNet, a popular open-source Inception-ResNet-V1 deep learning architecture, we extract a 128 dimensional feature vector from the pixels of each image in the two sets of labeled faces described in Section \[sec:dataset\] [@Schroff2015FaceNet:Clustering]. For thousands of images, extraction takes minutes. Rather than train FaceNet from scratch, we utilize a model with weights pre-trained using softmax loss on the MS-Celeb-1M dataset, a common face recognition benchmark [@Guo2016MS-Celeb-1M:Recognition]. Pre-training the model for feature extraction allows us to replicate the feature processing used commonly in black box industry models. The FaceNet model (10k stars on Github), and similar architectures such as OpenFace (13.1k stars on Github), are used by software developers, researchers, and industry groups [@Schroff2015FaceNet:Clustering; @Amos2016OpenFace:Applications].
If widely used black box face recognition models tend to learn appearance biases embedded in datasets, face recognition applications may make biased decisions that inequitably impact users with certain facial features. After feature extraction, we train six random forest regression models to predict appearance bias for each of the six traits measured: attractiveness, competence, dominance, extroversion, likeability, and trustworthiness. The human participants’ trait scores, multiplied by 100, serve as the ground-truth labels. The random forest includes 100 weak learners with no maximum depth, a minimum split size of two, and mean-squared-error split criterion. These hyper-parameters were chosen using a holdout test set, consisting of one image from the twenty-five maximally distinct faces and twelve images from the 300 random faces. Data and code used to produce the figures, tables, and machine learning pipeline (Figure \[fig:pipeline\]) in this work are open-sourced at *https://github.com/anonymous/repo*. The following section details two methods for training and testing this regression model.
![A transfer learning pipeline for predicting appearance bias. FaceNet, pre-trained on the MS-Celeb-1M benchmark dataset, extracts embeddings for each face. In **A**, a random forest regression model is trained on feature embeddings from the set of maximally distinct faces and the set of randomly generated faces. In **B**, the regression is trained only on the latter.[]{data-label="fig:pipeline"}](figures/bias-pipeline.png){width="\linewidth"}
Results {#sec:results}
=======
Cross-Fold Validation {#sec:cv}
---------------------
**Experiment A:** To test how well the random forest regression model learns appearance bias from the labeled faces, we shuffle the image embeddings extracted with FaceNet such that the 300 random faces and maximally distinct faces are mixed. The target labels are the original appearance bias measurements provided by human participants. Splitting the training data into 10 equal folds, we do the following for each fold: 1) train the regressor on the other 9 partitions; 2) record and plot appearance bias predictions for the current partition. Once all 10 partitions are processed, each image has a corresponding vector of predicted appearance bias scores, one for each trait measured.
Table \[tab:coefs\] displays goodness-of-fit and correlation statistics from the cross-validations for regressions on all six traits measured. Notably, our approach learns appearance bias to a high degree of precision for the maximally distinct faces ($\rho=.99)$, but the accuracy drops on randomly generated faces.
Testing on Randomly Generated Faces {#sec:random}
-----------------------------------
**Experiment B:** To better assess our model’s performance and investigate the disparity in predictive performance on the maximally distinct faces and the randomly generated faces, we train the regression model on only the maximally distinct faces and test on only the randomly generated faces. Prediction on the randomly generated faces in this experiment has a smaller correlation coefficient ($\bar{\rho}=.32)$. This result may be due in part by the lower sample size for the randomly generated faces, but is more likely a result of the higher variance in participants’ responses to randomly generated faces [@Todorov2011Data-drivenPerception]. Like the human participants, our model tends to agree more about judgments of deliberately manipulated faces than about judgments of randomly generated faces. Our approach learns appearance bias more accurately with respect to judgments of dominance than judgments of other traits, perhaps because dominance has been shown to be less correlated with facial cues than other traits [@Willis2006FirstImpressions]. Also like the human participants, our model is much more accurate at predicting dominance judgments for the randomly generated faces than it is at predicting other trait judgments.
Pearson’s correlation coefficient $\rho$ and root mean square error (RMSE) for regression predictions. In Experiment **A**, a random forest regression is fitted on both sets of faces and predictions are produced by 10-fold cross validation; in **B**, the regression is fitted on maximally distinct faces and tested on randomly generated faces. P-values are from the correlation t-test of $H_0: \rho=0$. \[tab:coefs\]
Discussion {#sec:discussion}
==========
We show that state-of-the-art face recognition techniques learn appearance biases from human annotators without special tuning or design, suggesting that biases from annotators may be creeping into face recognition applications in the wild. Significantly, we also find that appearance biases are propagated more easily from images which have been artificially manipulated to appear more or less trustworthy to human beings, and that bias about the dominant trait may be easier to learn. The results of this research will be particularly useful to AI and machine learning practitioners wishing to detect and mitigate bias in their systems, psychologists studying prejudice in human perception of faces, and any public policy concerning fairness and bias in technology. Further work is necessary to determine the extent to which first appearance bias exists in specific machine learning domains. Models using traditional or transfer learning may not be the only methods capable of learning trait inferences; zero-shot, semi-supervised, and multi-modal learning approaches should also be investigated to determine whether another approach exacerbates or mitigates bias.
Additionally, it is not clear whether and how appearance biases translate to algorithmic decision-making; additional research is necessary to identify appearance biases in commonly used datasets and associate those biases with algorithmic outcomes. An unsupervised approach, like image embeddings, may be useful for formulating even more accurate measures of implicit biases in image classification in the style of the Word Embedding Association Test [@Caliskan2017].
Conclusions {#sec:conclusion}
===========
As artificial intelligence systems have a greater role in human interactions, a stronger understanding of the types of biases that can pervade these systems and the effect they have on machine learning systems is necessary to enforce fair and ethical human-AI interactions. We find that machine learning models might easily replicate human beings’ first impressions of personality traits in other faces using state-of-the-art models and data. We develop a method for measuring the extent to which trust impressions are learned in a standard face recognition scenario and establish benchmark learning rates for six perceived social traits. Our model learns to perceive traits using similar facial features as human participants, as demonstrated by its improved performance on the dominance trait for randomly generated faces. Because trait impressions are learned more easily for faces which have been artificially manipulated, our results lend insight into the manner in which biases may be extracted and interpreted by standard face recognition models.
|
---
abstract: |
We consider a particle which moves on the $x$ axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point $x_1$ of a particle which starts at $x_0$ with velocity $v_0$. The probability that the particle has not yet arrived at $x_1$ after a time $t$, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement $m={\rm
max}_t[x(t)]$.
author:
- 'Theodore W. Burkhardt'
title: |
First-passage and extreme-value statistics of a particle\
subject to a constant force plus a random force
---
email: tburk@temple.edu\
\
Key words: random acceleration, random force, first passage, extreme statistics,\
stochastic process, non-equilibrium statistics
Introduction {#intro}
============
\[sec:intro\] In this paper we consider a particle which moves on the $x$ axis and is subject to both a constant force, such as gravity, and a random force in the form of Gaussian white noise. The Newtonian equation of motion is given by $$\begin{aligned}
&&\displaystyle{d^2x\over
dt^2}=g+\eta(t)\;,\label{eqmo}\\
&&\langle\eta(t)\rangle=0\;,\quad\langle \eta(t)\eta(t')\rangle=
2\Lambda\delta(t-t')\;,\label{randomforce}\end{aligned}$$ where $g$ is a constant.
Simple stochastic processes such as this are of both mathematical and physical interest. The approximately random collision forces experienced by a particular particle in a many particle system are often modelled by Gaussian white noise. Langevin’s equation [@fpeq] for the motion of a Brownian particle in a constant force field corresponds to Eq. (\[eqmo\]) with an additional viscous damping term of the form $-\lambda\thinspace dx/dt$ on the right-hand side. On setting $g=0$ in Eq. (\[eqmo\]) and regarding $t$ as a Cartesian coordinate instead of time, one may interpret the path $x(t)$ of the particle as a configuration of a semi-flexible polymer [@twb93]. For several applications of the process (\[eqmo\]) related to semi-flexible polymers and driven granular matter, see [@twb93; @bk; @twb07; @ybg] and references therein.
In this paper we study first-passage properties [@redner] of the process (\[eqmo\]). More precisely, we analyze the statistics of the first arrival at point $x_1$ of a particle which starts at $x_0$ with velocity $v_0$. Due to translational invariance no generality is lost in choosing $x_1$ to be the origin, and since we consider both positive and negative $g$, no generality is lost in choosing $x_0$ to be positive. Thus, “first passage" corresponds to the first exit of the particle from the positive $x$ axis. If the initial velocity $v_0$ is positive, the particle must return to its initial position at least once before exiting from the positive $x$ axis. Thus, in the limit $x_0\searrow 0$ with $v_0>0$, the first-passage statistics reduces to the statistics of first return of the particle to its initial position. Clearly, first-passage statistics, as defined here, is the same as the statistics of absorption of a particle moving on the half line $x=0$ with an absorbing boundary at $x=0$.
First-passage properties of the random acceleration process, corresponding to Eq. (\[eqmo\]) but without the constant term $g$ on the right hand side, are derived or reviewed in Refs. [@mck; @mw; @sinai; @twb93; @twb07]. In the remainder of this section we show how these results can be generalized to include the constant force. In Section \[statquan\] some statistical quantities of interest in connection with first passage are defined, and in Section \[results\] our explicit results are presented. In Section \[extrstat\] we show that the extreme-value statistics [@twb07; @gum; @gal; @ggetal; @twbetal] and first-passage statistics of the process (\[eqmo\]) are closely related. This is then used in deriving the distribution of the maximum displacement $m={\rm max}_t[x(t)]$ of a particle which begins at the origin with velocity $v_0$.
With no loss of generality we replace the parameters $g$ and $\Lambda$, introduced in Eqs. (\[eqmo\]) and (\[randomforce\]), by $g\to\gamma=\pm 1$ and $\Lambda=1$ throughout this paper, since this can be achieved by rescaling [@rescale] the variables $x$ and $t$. For $\gamma=-1$ and $\gamma=1$, the constant force drives a particle on the positive $x$ axis toward and away from the origin, respectively.
Integrating the equation of motion (\[eqmo\]) yields $$x(t)=x_0+v_0t+\textstyle{1\over 2}\displaystyle\thinspace\gamma t^2+\int_0^t
(t-t')\eta(t')dt'\label{x(t)}\;,$$ which, together with properties (\[randomforce\]) of the random force, implies the moments $$\langle x\rangle=x_0+v_0t+\textstyle{1\over 2}\displaystyle\gamma
t^2\;,\qquad\langle\left(x-\langle x\rangle\right)^2\rangle=\textstyle{1\over
3}\displaystyle\thinspace t^3\;.\label{moments}$$ Thus, the contribution of the random force on the right side of Eq. (\[x(t)\]) has typical size $t^{3/2}$. For large $t$ the constant force is more important than the random force, but for small $t$ the opposite is true.
It is convenient to define $P_\gamma(x,v;x_0,v_0;t)\thinspace dxdv$ as the probability that the position and velocity of a particle, moving according to Eq. (\[eqmo\]) with $g\to\gamma=\pm 1$ and $\Lambda=1$, evolve from $x_0$, $v_0$ to values between $x$ and $x+dx$, $v$ and $v+dv$ in a time $t$ without ever reaching $x=0$. The probability distribution $P_\gamma$ satisfies the time-dependent Fokker-Planck equation [@fpeq] $$\left({\partial\over\partial t}+v{\partial\over\partial
x}+\gamma{\partial\over
\partial v}-{\partial^2\over
\partial v^2}\right)P_\gamma(x,v;x_0,v_0;t)=0\thinspace,\label{fp}$$ with the initial condition $$P_\gamma(x,v;x_0,v_0;0)=\delta(x-x_0)\delta(v-v_0)\;\label{ic}$$ and the boundary condition $$P_\gamma(0,v;x_0,v_0;t)=0\;,\quad v>0\;.\label{bc}$$ This boundary condition ensures that only trajectories which exit the positive $x$ axis for the first time at time $t$ contribute to $P_\gamma(0,v;x_0,v_0;t)$. Trajectories which leave the positive $x$ axis at an earlier time and return to the positive $x$ axis are excluded. Equation (\[bc\]) is also the appropriate boundary condition for motion on the half line $x>0$ with an absorbing boundary at $x=0$.
In the absence of the constant force, the corresponding probability distribution $P_0(x,v;x_0,v_0;t)$ satisfies the same Fokker-Planck equation (\[fp\]), initial condition (\[ic\]), and boundary condition (\[bc\]), except that the term $\gamma\partial P/\partial v$ in Eq. (\[fp\]) is absent. This implies the relation $$P_\gamma(x,v;x_0,v_0;t)=\exp\left[\textstyle{1\over
2}\gamma(v-v_0)-\textstyle{1\over 4}t\right]P_0(x,v;x_0,v_0;t)\label{Psigma}$$ between the distributions with and without the constant force, which is central to our work.
In a classic paper on the first-passage properties of a randomly accelerated particle, McKean [@mck] derived the exact propagator $P_0(0,-v;0,v_0;t)$ for $v>0$ and $v_0>0$, corresponding to a particle which leaves the origin with velocity $v_0$ and returns for the first time at time $t$ with velocity $-v$ and speed $v$. His result and Eq. (\[Psigma\]) imply $$\begin{aligned}
&&P_\gamma(0,-v;0,v_0;t)\nonumber\\
&&\quad\qquad={\sqrt{3}\over2\pi t^2}\thinspace \exp\left[-\textstyle{1\over
2}\thinspace\gamma(v+v_0)-\textstyle{1\over 4}\thinspace t-(v^2-vv_0+v_0^2)/t\right]
\thinspace{\rm erf}\left(\sqrt{3vv_0\over t}\right)\;,\label{propmck1}\end{aligned}$$ where ${\rm erf}(z)$ denotes the standard error function [@as; @gr].
The Laplace transform $$\tilde{P}_\gamma(x,v;x_0,v_0;s)=\int_0^\infty dt\thinspace
e^{-st}P_\gamma(x,v;x_0,v_0;t)\;.\label{laplacetransformdef}$$ plays a central role in our work. Substituting Eq. (\[propmck1\]) on the right-hand side, using the integral representation [@as; @gr] ${\ \rm
erf}(z)=2\pi^{-1/2}z\int_0^1dy\thinspace\exp(-z^2y^2)$, and integrating over $t$ with the help of Ref. [@gr], we obtain $$\begin{aligned}
&&\tilde{P}_\gamma(0,-v;0,v_0;s)\nonumber\\
&&\qquad={3\over 2\pi}\left(vv_0\right)^{1/2}e^{-\gamma(v+v_0)/2}\int_0^1
dy\thinspace
\exp\left[-(4s+1)^{1/2}(v^2-vv_0+v_0^2+3vv_0y^2)^{1/2}\right]\nonumber\\
&&\qquad\times\left[
(v^2-vv_0+v_0^2+3vv_0y^2)^{-3/2}+(4s+1)^{1/2}(v^2-vv_0+v_0^2+3vv_0y^2)^{-1}
\right].\label{laplacetransform1}\end{aligned}$$ We will also need McKean’s result [@mck] for the Laplace transform, $$\begin{aligned}
&&\tilde{P}_\gamma(0,-v;0,v_0;s)\nonumber\\
&&\qquad={e^{-\gamma(v+v_0)/2}\over \pi^2 vv_0}\int_0^\infty
d\mu\thinspace\mu\thinspace{\sinh(\pi\mu)\over \cosh(\textstyle{1\over
3}\displaystyle\pi\mu)}\thinspace
K_{i\mu}(\sqrt{(4s+1)}\;v)K_{i\mu}(\sqrt{(4s+1)}\;v_0)\;, \label{laplacetransform2}\end{aligned}$$ where $K_\mu(z)$ is a modified Bessel function [@as; @gr]. Expressions (\[laplacetransform1\]) and (\[laplacetransform2\]) are particularly convenient for numerical and analytical calculations, respectively, and both expressions are used below.
The exact solution $\tilde{P}_0(x,v;x_0,v_0;s)$ of the Fokker-Planck equation for random acceleration on the half line $x>0$ with boundary condition (\[bc\]), is given in Ref. [@twb93], where it is derived from more general results of Marshall and Watson [@mw]. All of our results for first passage from an arbitrary initial point $x_0$ are based on this solution. Substituting it in Eq. (\[Psigma\]) and setting $x=0$, we obtain $$\tilde{P}_\gamma(0,-v;x_0,v_0;s)=e^{-\gamma(v+v_0)/2}\int_0^\infty dF e^{-F
x_0}\phi_{s+1/4,F}(-v)\psi_{s+1/4,F}(v_0) \;,\label{propmw}$$ where $$\begin{aligned}
\psi_{s,F}(v)&=&F^{-1/6}{\rm Ai}\left(F^{1/3}v+F^{-2/3}s\right)\;,\label{psi}\\
\phi_{s,F}(v)&=&\psi_{s,F}(v)-{1\over 2\pi}\int_0^\infty
dG\thinspace{\exp\left[-{2\over 3}s^{3/2}\left(F^{-1}+G^{-1}\right)\right]\over
F+G}\;\psi_{s,G}(-v)\;,\label{phi}\end{aligned}$$ and ${\rm Ai}(z)$ is the Airy function [@as]. Some important properties of the two set of basis functions $\psi_{s,F}(v)$ and $\phi_{s,F}(v)$ are discussed in Ref. [@twb93]. For example, $\phi_{s,F}(v)$ vanishes identically for $v>0$, so that Eq. (\[propmw\]) satisfies the boundary condition (\[bc\]).
Statistical quantities of interest {#statquan}
==================================
The “survival probability” or probability that a particle with initial position and velocity $x_0$ and $v_0$ has not yet left the positive $x$ axis after a time $t$ is given by $$Q_\gamma(x_0,v_0;t)=\int_{-\infty}^\infty dv\int_0^\infty dx\thinspace
P_\gamma(x,v;x_0,v_0;t)\;.
\label{Qdef}$$ According to Eqs. (\[fp\]), (\[bc\]), and (\[Qdef\]) $${\partial\over\partial t}Q_\gamma(x_0,v_0;t)=
-\int_0^\infty dv\;vP_\gamma(0,-v;x_0,v_0;t)\;.\label{probcons}$$ Thus, we interpret $$vP_\gamma(0,-v;x_0,v_0;t)\thinspace dv\thinspace dt\;,\label{current}$$ for $x_0>0$, as the probability that the particle reaches the origin for the first time at a time between $t$ and $t+dt$ with speed between $v$ and $v+dv$.
Several useful relations follow from this interpretation of the quantity (\[current\]). The survival probability defined by Eq. (\[Qdef\]), its limiting value for $t\to\infty$, and its Laplace transform can be written in the form $$\begin{aligned}
&&Q_\gamma(x_0,v_0;t)=1-\int_0^t dt'\int_0^\infty dv\thinspace v
P_\gamma(0,-v;x_0,v_0;t')\;,\label{Q}\\
&&Q_\gamma(x_0,v_0;\infty)=1-\int_0^\infty dv\thinspace v
\tilde{P}_\gamma(0,-v;x_0,v_0;0)\;,\label{limQ}\\
&&\tilde{Q}_\gamma(x_0,v_0;s)={1\over s}\left[1-\int_0^\infty dv\thinspace v
\tilde{P}_\gamma(0,-v;x_0,v_0;s)\right]\;.\label{Qtilde}\end{aligned}$$ The mean time to exit the positive $x$ axis for the first time is given by $$\begin{aligned}
T_\gamma(x_0,v_0)&=&{\int_0^\infty dt\thinspace t\int_0^\infty dv\thinspace v
P_\gamma(0,-v;x_0,v_0;t)\over \int_0^\infty dt \int_0^\infty dv\thinspace v
P_\gamma(0,-v;x_0,v_0;t)}\label{T1}\\
&=&-\left[1-Q_\gamma(x_0,v_0;\infty)\right]^{-1}\thinspace\lim_{s\to
0}{\partial\over\partial s}\int_0^\infty dv\thinspace v
\tilde{P}_\gamma(0,-v;x_0,v_0;s)\label{T2}\;.\end{aligned}$$
Since a particle which begins at the origin with a negative velocity immediately moves onto the negative $x$ avis, $Q_\gamma$ and $T_\gamma$ satisfy the boundary conditions $$\begin{aligned}
&&Q_\gamma(0,v_0;t)=0\;,\quad v_0<0\;,\quad t>0\;,\label{Q0}\\
&&T_\gamma(0,v_0)=0\;,\quad v_0<0\;.\label{T0} \end{aligned}$$
Finally, the probability that the speed of the particle on exiting from the positive $x$ axis for the first time is between $v$ and $v+dv$ is given by $G_\gamma(v;x_0,v_0)\thinspace dv$, where $$G_\gamma(v;x_0,v_0)={v\int_0^\infty dt\thinspace
P_\gamma(0,-v;x_0,v_0;t)\over\int_0^\infty dv\thinspace v\int_0^\infty dt\thinspace
P_\gamma(0,-v;x_0,v_0;t)}= {v\tilde{P}_\gamma(0,-v;x_0,v_0;0)\over
1-Q_\gamma(x_0,v_0;\infty)}\;,\label{velocitydist}$$ and the normalization $\int_0^\infty dv\thinspace G_\gamma(v;x_0,v_0)=1$ has been imposed.
Results
=======
Limit $Q_\gamma(x_0,v_0;\infty)$ of the Survival Probability {#subsec1}
------------------------------------------------------------
The survival probability $Q_\gamma(x_0,v_0;t)$ introduced in the preceding section can, in principle, be evaluated for arbitrary $t$ from Eqs. (\[propmck1\]), (\[propmw\]), and (\[Q\]). However, this involves integrating over $t'$ and $v$ and, for $x_0>0$, inverting a Laplace transform to go from $s$ to $t$, most of which must be performed numerically. In this section we consider the limiting value $Q_\gamma(x_0,v_0;\infty)$ or probability that in an infinite time the particle never leaves the positive $x$ axis, which can be obtained analytically.
In the absence of a constant force, $Q_0(x_0,v_0,t)$ decays as $t^{-1/4}$ in the long-time limit [@mck; @sinai; @twb93]. Thus, $Q_0(x_0,v_0,\infty)=0$, which means that the particle exits from the positive $x$ axis in an infinite time with probability 1. As shown in Eq. (\[x(t)\]), the constant force adds an extra term ${1\over 2}\thinspace\gamma t^2$ to the displacement $x(t)$ of the randomly accelerated particle. In the case $\gamma=-1$ of a constant force toward the origin, the particle leaves the positive $x$ axis sooner than without the constant force, so $$Q_{-1}(x_0,v_0;\infty)=0\;.\label{Q1}$$
In the case $\gamma=1$ of a constant force pushing the particle away from the origin, the corresponding probability $Q_1(x_0,v_0,\infty)$ does not vanish and is expected to increase as $x_0$ and $v_0$ increase. Recalling Eq. (\[Q0\]), substituting Eqs. (\[laplacetransform2\]) and (\[propmw\]) into Eq. (\[limQ\]), and proceeding as described in the Appendix, we obtain
$$Q_1(0,v_0;\infty)=\left\{\begin{array}{l}0\;,\quad v_0<0\;,\nonumber\\{\rm
erf}\left(\sqrt{\textstyle{3\over 2}\displaystyle\thinspace v_0}\right)\;,\quad
v_0>0\;,\end{array}\right.\label{Q2}$$
$$\begin{aligned}
&&Q_1(x_0,v_0;\infty)=1-{e^{-v_0/2}\over\sqrt{2\pi}}\nonumber\\&&\qquad\qquad
\times\int_0^\infty dF\thinspace F^{-7/6}\exp\left(-\thinspace{1\over
12F}-Fx_0\right){\rm Ai}\left(F^{1/3}v_0+\textstyle{1\over 4}\displaystyle\thinspace
F^{-2/3}\right)\;.\label{Q3}\end{aligned}$$
As in Eqs. (\[propmck1\]) and (\[psi\]), ${\rm erf}(z)$ denotes the error function, and ${\rm Ai}(z)$ is the Airy function, both defined as in Ref. [@as]. We found it useful to make the change of variables $F=u^{-6}$ in evaluating the integrals in Eqs. (\[Q3\]), (\[T4\]), and (\[calP3\]) numerically with [*Mathematica*]{}.
Equations (\[Q2\]) and (\[Q3\]) imply the asymptotic behavior $$\begin{aligned}
&&Q_1(0,v_0;\infty)\approx\left\{
\begin{array}{l}\displaystyle\left({6v_0\over\pi}\right)^{1/2}\;,\quad v_0\searrow
0\;,\\\displaystyle 1-\left({2\over 3\pi v_0}\right)^{1/2}e^{-3v_0/2}\;,\quad
v_0\to\infty\;,\end{array}\right.
\label{Qsmalllargev0}\\
&&Q_1(x_0,0;\infty)\approx\left\{
\begin{array}{l}\displaystyle{2^{5/6}3^{1/3}\over\Gamma\left({1\over 3}\right)}\thinspace
x_0^{1/6}\;,\quad x_0\searrow 0\;,\\\displaystyle 1-\left({3\over 8\pi^2
x_0}\right)^{1/4}e^{-(2x_0/3)^{1/2}}\;,\quad
x_0\to\infty\;,\end{array}\right.\label{Qsmalllargex0}\\
&&Q_1(x_0,v_0;\infty)\approx\left\{
\begin{array}{l}{\rm
erf}\left(\sqrt{\textstyle{3\over 2}\displaystyle\thinspace v_0}\right)
+\left[\displaystyle{\partial\over\partial
x}Q_1(x,v_0;\infty)\right]_{x=0}x_0\;,\quad
v_0>0\;,\quad x_0\searrow 0\;,\\
\displaystyle{3^{3/2}\over\sqrt{2\pi}}{x_0\over|v_0|^{5/2}}\thinspace
e^{-v_0/2-|v_0|^3/9x_0}\;,\quad v_0<0\;,\quad x_0\searrow 0\;,\\\displaystyle
1-\left({3\over 8\pi^2 x_0}\right)^{1/4}e^{-v_0-(2x_0/3)^{1/2}}\;,\quad
x_0\to\infty\;,\end{array}\right.\end{aligned}$$
In Fig. 1, $Q_1(x_0,v_0;t)$, as given by Eqs. (\[Q2\]) and (\[Q3\]), is plotted as a function of $v_0$ for several values of $x_0$. Note that $Q_1(x_0,v_0;t)$ increases monotonically with increasing $x_0$ and $v_0$, as expected.
If the random force is switched off, the particle trajectory becomes $x(t)=x_0+v_0t+{1\over 2}\gamma t^2$, which never reaches the origin for $\gamma=1$ and $v_0>-\sqrt{2x_0}$. Thus, each of the smooth curves in Fig. 1 is replaced by the unit step function $Q_1(x_0,v_0;\infty)=\theta(v_0+\sqrt{2x_0})$. The short vertical lines in Fig. 1 indicate the value of $v_0$ at which $Q_1(x_0,v_0;\infty)$ is discontinuous.
Mean First-Passage Time $T_\gamma(x_0,v_0)$ {#subsec2}
-------------------------------------------
The $t^{-1/4}$ decay of the survival probability $Q_0(x_0,v_0,t)$ of a randomly accelerated particle [@mck; @sinai; @twb93] is so slow that the mean time of its first exit from the positive $x$ axis, given by $T_0(x_0,v_0)=\int_0^\infty
dt\thinspace t\left[-\partial Q_0(x_0,v_0,t)/\partial t\right]$, is infinite. However, in the case $\gamma=-1$ of a constant force toward the origin in addition to the random force, the corresponding mean time $T_{-1}(x_0,v_0)$ is finite. Recalling Eq. (\[T0\]), substituting Eqs. (\[laplacetransform2\]), (\[propmw\]), (\[Q2\]), and (\[Q3\]) into Eq. (\[T2\]), and proceeding as described in the Appendix, we obtain $$T_{-1}(0,v_0)=\left\{\begin{array}{l}0\;,\quad
v_0<0\;,\label{l}\\\displaystyle\sqrt{{2v_0\over\pi}}e^{-v_0/2}+\left({2\over
3}\displaystyle+v_0\right)\left[2-{\rm erfc}\left(\textstyle\sqrt{{1\over
2}\displaystyle v_0}\right)\right]\nonumber\\\qquad -\displaystyle{2\over
3}\displaystyle\;e^{3v_0/2}\thinspace{\rm erfc}\left(\sqrt{2v_0}\right)\;,\quad
v_0>0\;,\end{array}\right.\label{T3}$$ $$\begin{aligned}
&&T_{-1}(x_0,v_0)={1\over 8}\sqrt{{3\over\pi}}\int_0^\infty dt\thinspace
t^{-3/2}\left(6x_0+2v_0t+t^2\right)\exp\left[-{3\over 4}\left({x_0+v_0t\over
t^{3/2}}-{1\over 2}\thinspace t^{1/2}\right)^2\right]\nonumber\\
&&\qquad\qquad -\;{e^{v_0/2}\over 2\pi}\int_0^\infty dF\thinspace
F^{-7/6}\exp\left(-\thinspace{1\over 12F}-Fx_0\right){\rm
Ai}\left(F^{1/3}v_0+\textstyle{1\over
4}\displaystyle\thinspace F^{-2/3}\right)\nonumber\\
&&\qquad\qquad\times\left[\sqrt{6\pi}-{\pi\over\sqrt{F}}\exp\left({1\over
6F}\right){\rm erfc}\left({1\over\sqrt{6F}}\right)\right]\;,\label{T4}\end{aligned}$$ where ${\rm erfc}(z)=1-{\rm erf}(z)$ is the complementary error function [@as].
Equations (\[T3\]) and (\[T4\]) imply the asymptotic behavior $$\begin{aligned}
&&T_{-1}(0,v_0)\approx\left\{
\begin{array}{l}\displaystyle\left({18v_0\over\pi}\right)^{1/2}\;,\quad v_0\searrow
0\;,\\\displaystyle 2v_0+\textstyle{4\over 3}\displaystyle\;,\quad
v_0\to\infty\;,\end{array}\right.
\label{Tsmalllargev0}\\
&&T_{-1}(x_0,0)\approx\left\{
\begin{array}{l}\displaystyle\left({2\over\pi}\right)^{1/2}3^{5/6}\thinspace{\Gamma\left({5\over 6}\right)
\over\Gamma\left({2\over 3}\right)}\; x_0^{1/6}\;,\quad x_0\searrow
0\;,\\\displaystyle (2x_0)^{1/2}+\textstyle{2\over 3}\displaystyle\;,\quad
x_0\to\infty\;.\end{array}\right.\label{Tsmalllargex0}\end{aligned}$$
The leading terms $2v_0$ and $(2x_0)^{1/2}$ in Eqs. (\[Tsmalllargev0\]) and (\[Tsmalllargex0\]) for $x_0=0$, $v_0\to \infty$ and for $v_0=0$, $x_0\to\infty$ are easy to understand. According to the discussion following Eq. (\[moments\]), the constant force is more important than the random force for large $t$, implying $x(t)\approx x_0+v_0t-{1\over 2}\thinspace t^2$, which vanishes at $T_{-1}(x_0,v_0)\approx v_0+\sqrt{v_0^2+2x_0}$.
For small $x_0$ and $v_0$ the particle tends to reach the origin quickly, so $T_\gamma(x_0,v_0)$ is small. For short times the random force is more important than the constant force (see discussion following Eq. (\[moments\])) and primarily responsible for the asymptotic behavior $T_{\pm 1}(0,v_0)\sim v^{1/2}$ and $T_{-1}(x_0,0)\sim x^{1/6}$ in Eqs. (\[Tsmalllargev0\]), (\[Tsmalllargex0\]), and (\[Tplussmalllargev0\]). We note that these same power laws for small $x_0$ and $v_0$ appear in the mean first exit time [@fr; @mp] of a randomly accelerated particle with initial position and velocity $x_0$, $v_0$ from the finite interval $0<x<L$.
In Fig. 2, $T_{-1}(x_0,v_0)$, as given by Eqs. (\[T3\]) and (\[T4\]), is plotted as a function of $v_0$ for several values of $x_0$. As expected, $T_{-1}(x_0,v_0;t)$ increases monotonically as $x_0$ and $v_0$ increase.
As discussed just above Eq. (\[Q2\]), for $\gamma=1$ the probability $Q_1(x_0,v_0;\infty)$ that the particle never leaves the positive $x$ axis is nonzero, in general, and it increases, as in Eqs. (\[Qsmalllargev0\]) and (\[Qsmalllargex0\]), as $v_0$ and $x_0$ increase. However, the mean first exit time for those trajectories which do leave the positive $x$ axis, defined by Eq. (\[T2\]), is finite. From Eqs. (\[laplacetransform2\]), (\[T2\]), (\[T0\]), and (\[Q2\]), we obtain $$T_{1}(0,v_0)=\left\{\begin{array}{l}0\;,\quad
v_0<0\;,\nonumber\\\displaystyle{2\over 3}-v_0+\left(\sqrt{{6v_0\over\pi}}-{2\over
3}\right)\;{e^{-3v_0/2}\over{\rm erfc}\left(\sqrt{{3v_0\over 2}}\right)}\;,\quad
v_0>0\;,\end{array}\right.\label{T5}$$ which has the asymptotic behavior $$T_1(0,v_0;\infty)\approx\left\{
\begin{array}{l}\displaystyle\left({2v_0\over 3\pi}\right)^{1/2}\;,\quad v_0\searrow
0\;,\\\displaystyle 2v_0-\left({2\pi v_0\over 3}\right)^{1/2}+{5\over 3}\;,\quad
v_0\to\infty\;.\end{array}\right. \label{Tplussmalllargev0}$$
For arbitrary $x_0$ and $v_0$, $T_1(x_0,v_0)$ follows from substituting Eqs. (\[propmw\]) and (\[Q3\]) into Eq. (\[T2\]). This leads to a lengthy expression, containing multiple integrals, which we were unable to simplify and omit here.
The results (\[T3\]) and (\[T5\]) for $T_{-1}(0,v_0)$ and $T_1(0,v_0)$ are compared in Fig. 2.
Distribution $G_\gamma(v;x_0,v_0)$ of the Particle Speed at First Passage {#subsec3}
-------------------------------------------------------------------------
For arbitrary initial position $x_0$ and initial velocity $v_0$, the distribution $G_\gamma(v;x_0,v_0)$ of the particle speed on exiting from the positive $x$ axis for the first time is determined by Eqs. (\[propmw\]), (\[velocitydist\]), and (\[Q3\]). Here we restrict our attention to the case of a particle which begins at the origin with $v_0>0$. The distribution $G_\gamma(v;0,v_0)$ of its speed on returning to the origin for the first time and leaving the positive $x$ axis can be readily evaluated by substituting Eqs. (\[laplacetransform1\]) and (\[Q2\]) into Eq. (\[velocitydist\]) and integrating over the variable $y$ numerically. The results, for several values of $v_0$, are shown in Fig. 3.
Each of the curves in Fig. 3 has a single peak. As $v_0$ increases, the peak shifts to larger values of $v$, as expected, and becomes broader. The peak position and width correspond roughly to the mean speed $\langle v\rangle_{_\gamma}$ at first return and the root-mean-square deviation $\sigma_{_\gamma}=\left\langle\left(v-\langle
v\rangle_{_\gamma}\right)^2\right\rangle_{_\gamma}^{1/2}$, where $$\langle v^n\rangle_{_\gamma}= \int_0^\infty dv\thinspace v^n
G_\gamma(v;0,v_0)\;.\label{momv}$$
Using Eqs. (\[laplacetransform2\]), (\[velocitydist\]), and (\[Q2\]) and the approach of the Appendix, we have calculated the first two moments of the speed at first return analytically, obtaining $$\begin{aligned}
&&\langle v\rangle_{_{-1}}=v_0+{4\over 3}\displaystyle +e^{-v_0/2}\nonumber\\
&&\quad\times\left[\left({2v_0\over\pi}\right)^{1/2}-\left(v_0+{2\over
3}\displaystyle\right)e^{v_0/2} {\rm erfc}\left(\sqrt{v_0\over 2}\right)-{2\over
3}\displaystyle\thinspace e^{2v_0}
{\rm erfc}\left(\sqrt{2v_0}\right)\right]\;,\label{momminus1}\\
&&\langle v^2\rangle_{_{-1}}=v_0^2+2v_0+2\langle v\rangle_{_{-1}}\;,\label{momminus2}\\
&&\langle v\rangle_{_{1}}={2\over 3}\left({e^{-3v_0/2}\over{\rm erfc}
\left(\sqrt{{3v_0\over 2}}\right)}-1\right)\;,\label{momplus1}\\
&&\langle v^2\rangle_{_{1}}=v_0^2-2v_0+{4\over 3}-\left[\left({2v_0\over
3\pi}\right)^{1/2}(v_0-5)+{4\over 3}\right]\left({3\over 2}\langle
v\rangle_{_{1}}+1\right)\;.\label{momplus2}\end{aligned}$$ For large $v_0$ the average speed at first return and the root-mean-square deviation from the average have the asymptotic forms $$\begin{aligned}
&&\langle v\rangle_{_{-1}}\approx v_0+{4\over 3}\;,\label{momminus1large}\\
&&\sigma_{_{-1}} \approx \left({4\over 3}\thinspace
v_0\right)^{1/2}\;,\label{rmsminus}\end{aligned}$$ and $$\begin{aligned}
&&\langle v\rangle_{_1}\approx\left({2\pi v_0\over
3}\right)^{1/2}\;,\label{momplus1large}\\
&&\sigma_{_1}\approx \left[{1\over
3}\thinspace(8-2\pi)v_0\right]^{1/2}\;,\label{rmsplus}\end{aligned}$$ which are qualitatively consistent with the evolution of the curves in Fig. 3 as $v_0$ increases.
For large $v_0$ the constant force is more important than the random force, and $x(t)\approx x_0+v_0t+\textstyle{1\over 2}\displaystyle\gamma t^2$. Thus, for $\gamma=-1$, the particle returns to its starting point with approximately the same speed it had initially. This is consistent with the asymptotic behavior in Eq. (\[momminus1large\]).
Extreme-Value Statistics {#extrstat}
========================
Consider a particle which begins at $x_0=0$ with velocity $v_0$ and moves according to Eq.(\[eqmo\]) with $g\to\gamma=\pm 1$. At some time in the interval $0<t<\infty$ the particle attains a maximum displacement $m={\rm max}_t[x(t)]$. For large $t$, $x(t)\approx{1\over 2}\gamma t^2$, as follows from the discussion below Eq. (\[moments\]). Thus, in the case $\gamma=1$ of a constant force in the positive direction, $m=\infty$. In this section we consider the less trivial question of the maximum displacement $m$ for $\gamma=-1$, and we derive the corresponding distribution ${\cal P}_{-1}(m,v_0)$. Distributions such as this play a central role in the field of extreme-value statistics [@twb07; @gum; @gal]. The extreme-value statistics of a generalized Gaussian process that includes random acceleration as a special case is studied in Refs. [@ggetal; @twbetal].
To derive the distribution ${\cal P}_{-1}(m,v_0)$, we begin by writing $${\cal P}_{-1}(m,v_0)={\partial\over\partial m}{\cal F}_{-1}1(m,v_0)\;,\label {calP1}$$ where ${\cal F}_{-1}(m,v_0)$ is the probability that, for a constant force in the [*negative*]{} direction, the displacement $x(t)$ of a particle which begins at the origin with velocity $v_0$ never exceeds $m$ in the time interval $0<t<\infty$. For $m<0$, ${\cal F}_{-1}(m,v_0)$=0, since the initial displacement $x_0=0$ already exceeds $m$. For $m>0$, ${\cal F}_{-1}(m,v_0)$ is the same as the probability that, for a constant force in the [*positive*]{} direction, a particle with initial position $m$ and initial velocity $-v_0$ never reaches the origin. This follows from the invariance of the probability under the coordinate transformation $x\to m-x$. Since this latter probability is precisely the survival probability $Q_1(m,-v_0;\infty)$ considered in Sections \[statquan\] and \[results\], $${\cal F}_{-1}(m,v_0)=\theta(m)\thinspace Q_1(m,-v_0;\infty)\;,\label{calF}$$ where $\theta(m)$ is the standard step function.
Making use of Eqs. (\[calP1\]) and (\[calF\]) and the expressions for $Q_1(0,v_0;\infty)$ and $Q_1(x_0,v_0;\infty)$ in Eqs. (\[Q2\]) and (\[Q3\]), we obtain $$\begin{aligned}
&&{\cal P}_{-1}(m,v_0)=\theta(-v_0)\thinspace{\rm erf}\left(\sqrt{\textstyle{3\over
2}\displaystyle\thinspace|v_0|}\right)\delta(m)\nonumber\\&&\quad
+\theta(m)\thinspace{e^{v_0/2}\over\sqrt{2\pi}}\int_0^\infty dF\thinspace
F^{-1/6}\exp\left(-\thinspace{1\over 12F}-Fm\right){\rm
Ai}\left(-F^{1/3}v_0+\textstyle{1\over 4}\displaystyle\thinspace
F^{-2/3}\right)\;\label{calP3}\end{aligned}$$ for the extreme-value distribution. The distribution vanishes for $m<0$ and is normalized so that $\int_{-\infty}^\infty dm\thinspace{\cal P}_{-1}(m,v_0)=1$, as follows from Eqs. (\[calP1\]) and (\[calF\]) and the boundary condition $Q_1(\infty,-v_0;\infty)=1$. The first term on the right side of Eq. (\[calP3\]) has its origin in the non-zero probability ${\rm erf}\left(\sqrt{\textstyle{3\over
2}\displaystyle\thinspace|v_0|}\right)$ (see Section \[subsec1\]) that a particle which begins at the origin with $v_0<0$ never returns to the origin, in which case the maximum displacement $m$ equals the initial value $x_0=0$.
The extreme-value distribution ${\cal P}_{-1}(m,v_0)$ is plotted as a function of $m$ for several positive and negative values of $v_0$ in Figs. 4a and 4b, respectively. In the absence of the random force, $x=v_0t-{1\over 2}t^2$, which implies ${\cal P}_{-1}(m,v_0)=\theta(-v_0)\delta(m)+\theta(v_0)\delta(m-{1\over
2}v_0^2)$. The random force broadens the delta functions, as seen in the figure.
For positive $v_0$, the peak in Fig. 4a shifts to larger values of $m$ and becomes broader as $v_0$ increases, as expected. The mean value and the root-mean-square deviation vary as $\langle m\rangle\approx {1\over 2}v_0^2+v_0$ and $\sigma\approx\left({2\over 3}v_0^3\right)^{1/2}$ for large positive $v_0$. For large $v_0$ the constant force is more important than the random force, and the leading term in $\langle m\rangle$ equals the maximum displacement ${1\over
2}v_0^2$ of a particle subject only to the constant force.
In the results for $v_0<0$ in Fig. 4b, the vertical line at $m=0$ represents the term proportional to $\delta(m)$ in Eq. (\[calP3\]). The most probable value of $m$, which maximizes ${\cal P}_{-1}(m,v_0)$, is zero for all negative $v_0$. The mean value of $m$ is positive and, for $v_0$ negative and large in magnitude, $\langle m\rangle\approx{4\over 3}({2|v_0|/3\pi })^{1/2}e^{-3|v_0|/2}$, and $\langle
m^2\rangle\approx{32\over 9}({2|v_0|^3/3\pi })^{1/2}e^{-3|v_0|/2}$.
Concluding Remarks {#conclusion}
==================
This completes our study of the first-passage and extreme-value statistics of the process (\[eqmo\]). In closing we note that in a mathematical tour de force, Marshall and Watson [@mw] derived the Laplace transform $\tilde{P}_{g,\lambda}(x,v;x_0,v_0;s)$ of the solution to the Klein-Kramers equation with the absorbing boundary condition (\[bc\]). The Klein-Kramers equation is the Fokker-Planck equation for the process [@fpeq] $$\displaystyle{d^2x\over dt^2}+\lambda{dx\over dt}=g+\eta(t)\;,\label{eqmo2}$$ which, unlike Eq. (\[eqmo\]), includes viscous damping and plays a central role in the theory of Brownian motion. In principle, all of the first-passage and extreme-value properties we have considered follow, for this more general process, from the expression of Marshall and Watson for $\tilde{P}_{g,\lambda}(x,v;x_0,v_0;s)$, which, however, involves an infinite double sum over special functions and is difficult to work with.
I am grateful to Zoltan Rácz for asking about the extreme statistics of the process (\[eqmo\]), which led to its inclusion in this paper. It is a pleasure to thank him and Dieter Forster for useful discussions and Robert Intemann for help with [*Mathematica*]{} and LaTex.
Calculational Details
=====================
The quantities $Q_\gamma(0,v_0;\infty)$ and $T_\gamma(0,v_0)$, defined in Eqs. (\[limQ\]) and (\[T2\]), can both be expressed in terms of the integral $$\begin{aligned}
&&\int_0^\infty dv\thinspace v
\tilde{P}_\gamma(0,-v;0,v_0;s)\nonumber\\
\qquad\qquad&&={e^{-\gamma v_0/2}\over \pi^2 v_0}\int_0^\infty
d\mu\thinspace\mu\thinspace{\sinh(\pi\mu)\over \cosh({1\over 3}\pi\mu)}\thinspace
F_\gamma(\mu,s)K_{i\mu}(\sqrt{(4s+1)}\;v_0)\;,\label{basiceq1}\\
&&\qquad\qquad F_\gamma(\mu,s)=\int_0^\infty dv\thinspace e^{-\gamma
v/2}K_{i\mu}(\sqrt{(4s+1)}\;v)\;,\label{Fgammamu}\end{aligned}$$ where we have used the expression for $\tilde{P}_\gamma(0,-v;0,v_0;s)$ in Eq. (\[laplacetransform2\]). Evaluating $F_\gamma(\mu,s)$ with the help of the integral representation [@as; @gr] $$K_{i\mu}(v)=\int_0^\infty dt\thinspace\cos(\mu t)\thinspace e^{-v\cosh
t}\label{intrep}$$ and substituting the result in Eq. (\[basiceq1\]), we obtain $$\begin{aligned}
&&\int_0^\infty dv\thinspace v \tilde{P}_\gamma(0,-v;x_0,v_0;s)={e^{-\gamma
v_0/2}\over\pi v_0}\textstyle{\left(4s+1-{1\over
4}\gamma^2\right)^{-1/2}}\nonumber\\
&&\qquad\displaystyle\times\int_0^\infty
d\mu\thinspace\mu\thinspace{\sinh\left[\mu\arccos\left({1\over
2}\gamma(4s+1)^{-1/2}\right)\right]\over\cosh\left({1\over
3}\pi\mu\right)}\thinspace K_{i\mu}(\sqrt{(4s+1)}\;v_0)\;.\label{basiceq2}\end{aligned}$$
First we consider the quantity $Q_\gamma(0,v_0;\infty)$, defined in Eq. (\[limQ\]). For $\gamma=-1$ and $s=0$, the $\arccos$ in Eq. (\[basiceq2\]) equals ${2\over 3}\pi$. From Eqs. (\[limQ\]) and (\[basiceq2\]) and the relation [@gr] $$\int_0^\infty d\mu\thinspace\mu\thinspace\sinh(b\mu)K_{i\mu}(v_0)={1\over
2}\thinspace\pi v_0\sin b\thinspace e^{-v_0\cos b}\;,\label{integ1}$$ we obtain $Q_{-1}(0,v_0;\infty)=0$ for $v_0>0$, in agreement with Eq. (\[Q1\]).
For $\gamma=1$ and $s=0$, the $\arccos$ in Eq. (\[basiceq2\]) equals ${1\over
3}\pi$, and expression (\[Q2\]) for $Q_1(0,v_0;\infty)$, with $v_0>0$, follows from Eqs. (\[limQ\]) and (\[basiceq2\]) and the relation $$\int_0^\infty d\mu\thinspace\mu\thinspace\tanh\left(\textstyle{1\over
3}\displaystyle\pi\mu\right)K_{i\mu}(v_0)={\sqrt{3}\over 2}\thinspace\pi v_0
e^{v_0/2}{\rm erfc}\left(\sqrt{{3v_0\over 2}}\right)\;,\label{integ2}$$ which we derived with the help of the integral representation (\[intrep\]).
We now turn to $Q_\gamma(x_0,v_0;\infty)$ for $x_0\neq 0$. The result $Q_{-1}(x_0,v_0;\infty)=0$ was established in the paragraph containing Eq. (\[Q1\]). Expression (\[Q3\]) for $Q_1(x_0,v_0;\infty)$ follows from Eqs. (\[propmw\])-(\[phi\]) and (\[limQ\]). The lengthy derivation will not be given here, but it is easy to see, with the help of Ref. [@twb93], that the result (\[Q3\]) satisfies the appropriate Fokker-Planck equation $(v_0\partial_{x_0}+\gamma\partial_{v_0}+
\partial_{v_0}^2)Q_\gamma(x_0,v_0;\infty)=0$ and to check, by numerical integration, that Eqs. (\[Q2\]) and (\[Q3\]) agree for $x_0=0$.
The results (\[T3\]) and (\[T5\]) for $T_\gamma(0,v_0)$ may be derived by substituting Eqs. (\[Q2\]) and (\[basiceq2\]) into Eq. (\[T2\]) and using Eqs. (\[integ1\]), (\[integ2\]) and some analogous integrals over $\mu$ which can be evaluated with the help of the integral representation (\[intrep\]). Expression (\[T4\]) for $T_{-1}(x_0,v_0)$ follows from substituting Eqs. (\[propmw\])-(\[phi\]) and (\[Q1\]) into Eq. (\[T2\]), but the derivation is long and will not be given here. Making use of Ref. [@twb93], we have confirmed that the result (\[T4\]) satisfies the appropriate Fokker-Planck equation $(v_0\partial_{x_0}+\gamma\partial_{v_0}+\partial_{v_0}^2)T_\gamma(x_0,v_0;\infty)=-1$ and checked by numerical integration that Eqs. (\[T3\]) and (\[T4\]) agree for $x_0=0$.
The moments $\langle v^n\rangle_{_\gamma}\thinspace,$ defined by Eqs. (\[velocitydist\]) and (\[momv\]), can be expressed in terms in terms of the integral $$\begin{aligned}
&&\int_0^\infty dv\thinspace v^{n+1} \tilde{P}_\gamma(0,-v;x_0,v_0;s)={e^{-\gamma
v_0/2}\over\pi
v_0}\left(-2{\partial\over\partial\gamma}\right)^n\left\{\textstyle\left(4s+1-{1\over
4}\gamma^2\right)^{-1/2}\right.\nonumber\\&&\qquad\left.\times\displaystyle\int_0^\infty
d\mu\thinspace\mu\thinspace{\sinh\left[\mu\arccos\left({1\over
2}\gamma(4s+1)^{-1/2}\right)\right]\over\cosh\left({1\over 3
}\pi\mu\right)}\thinspace K_{i\mu}(\sqrt{(4s+1)}\;v_0)\right\}\;.\label{basiceq3}\end{aligned}$$ This relation is the same as Eq. (\[basiceq2\]) except for the extra factor $v^n$ introduced by applying $(-2\partial/\partial\gamma)^n$ to the quantity $F_\gamma(\mu,s)$ in Eq. (\[Fgammamu\]). The steps leading from Eq. (\[basiceq3\]) to the results for $\langle v\rangle_{_\gamma}$ and $\langle
v^2\rangle_{_\gamma}$ in Eqs. (\[momminus1\])-(\[momplus2\]) are very similar to the steps, described above, from Eq. (\[basiceq2\]) to the final expressions for $Q_\gamma(0,v_0;\infty)$ and $T_\gamma(0,v_0)$.
[03]{} See, e.g., S. Chandrasekhar, Rev. Mod. Phys. [**15**]{}, 1 (1943); H. Risken,[*The Fokker-Planck Equation: Methods of Solution and Applications*]{}, 2nd edition (Springer, Berlin, 1989). T. W. Burkhardt, J. Phys. A [**26**]{}, L1157 (1993). T. W. Burkhardt and S. N. Kotsev, Phys. Rev. E [**73**]{}, 046121 (2006). T. W. Burkhardt, J. Stat. Mech., P07004 (2007). Y. Yang, T. W. Burkhardt and G. Gompper, Phys. Rev. E [**76**]{}, 011804 (2007). S. Redner, [*A Guide to First-Passage Processes*]{} (Cambridge, Cambridge U.K., 2001). H. P. McKean, J. Math. Kyoto Univ. [**2**]{}, 227 (1963). T. W. Marshall and E. J. Watson, J. Phys A [**18**]{}, 3531 (1985). Y. G. Sinai, [*Teor. Mat. Fiz.*]{} [**90**]{}, 323 (1992) \[English translation [*Theor. Math. Phys.*]{} [**90**]{}, 219 (1992)\]. E. Gumbel [*Statistics of Extremes*]{}, (Dover, New York, 1958). J. Galambos [*The Asymptotic Theory of Extreme Order Statistics*]{}, (Wiley, New York, 1978). G. Györgyi, N. R. Maloney, K. Ozogány, and Z. Rácz, Phys. Rev. E [**75**]{}, 021123 (2007). T. W. Burkhardt, G. Györgyi, N. R. Maloney, and Z. Rácz, Phys. Rev. E [**76**]{}, 041119 (2007). In terms of the dimensionless variables $\xi=|g|^3\Lambda^{-2}x$, $\tau=|g|^2\Lambda^{-1}t$, Eqs. (\[eqmo\]) and (\[randomforce\]) take the form $d^2\xi/d\tau^2=\pm 1+\zeta(\tau)\;,\ \langle\zeta(\tau)\rangle=0\;,\ \langle
\zeta(\tau)\zeta (\tau')\rangle= 2\delta(\tau-\tau')$. , edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1965). I. S. Gradshteyn and I. M. Ryzhik, [*Tables of Integrals, Series, and Products*]{} (Academic, New York, 1980). J. N. Franklin and E. R. Rodemich, [*SIAM J. Numer. Anal.*]{} [**5**]{}, 680 (1968). J. Masoliver and J. M. Porrà, [*Phys. Rev. Lett.*]{} [**75**]{}, 189 (1995).
![Probability $Q_1(x_0,v_0;\infty)$, given by Eqs. (\[Q2\]) and (\[Q3\]), that a particle subject to a random force plus a constant force pushing it away from the origin never leaves the positive $x$ axis. If the random force is switched off, $Q_1(x_0,v_0;\infty)$ becomes a unit step function, as discussed in the second paragraph below Eq. (\[Qsmalllargex0\]). The short vertical lines in the figure indicate the values of $v_0$ at which the step function jumps from 0 to 1 for $x_0=0$, 0.1, 1.[]{data-label="fig1"}](afirstpassfig1.eps){width="12cm"}
![(a) Mean time $T_{-1}(x_0,v_0)$ to exit the positive $x$ axis for the first time for a particle subject to a random force plus a constant force toward the origin, given by Eqs. (\[T3\]) and (\[T4\]). (b) Mean first exit times $T_{-1}(0,v_0)$ and $T_1(0,v_0)$, given in Eqs. (\[T3\]) and (\[T5\]) for constant forces toward and away from the origin, respectively. []{data-label="fig2"}](afirstpassfig2a.eps "fig:"){width="12cm"}1.5cm
\
![(a) Mean time $T_{-1}(x_0,v_0)$ to exit the positive $x$ axis for the first time for a particle subject to a random force plus a constant force toward the origin, given by Eqs. (\[T3\]) and (\[T4\]). (b) Mean first exit times $T_{-1}(0,v_0)$ and $T_1(0,v_0)$, given in Eqs. (\[T3\]) and (\[T5\]) for constant forces toward and away from the origin, respectively. []{data-label="fig2"}](afirstpassfig2b.eps){width="12cm"}
\
![Distribution $G_\gamma(v;0,v_0)$, given by Eqs. (\[laplacetransform1\]), (\[velocitydist\]), and (\[Q2\]), of the particle speed $v$ at first return, for $v_0=$ 0.2, 0.4, 0.8, and 1.6. The results in (a) and (b) are for constant forces directed toward and away from the origin, respectively. As $v_0$ increases, the peak becomes lower and broader and moves to the right.[]{data-label="fig3"}](afirstpassfig3a.eps "fig:"){width="12cm"}1.5cm
\
![Distribution $G_\gamma(v;0,v_0)$, given by Eqs. (\[laplacetransform1\]), (\[velocitydist\]), and (\[Q2\]), of the particle speed $v$ at first return, for $v_0=$ 0.2, 0.4, 0.8, and 1.6. The results in (a) and (b) are for constant forces directed toward and away from the origin, respectively. As $v_0$ increases, the peak becomes lower and broader and moves to the right.[]{data-label="fig3"}](afirstpassfig3b.eps){width="12cm"}
\
![Distribution ${\cal P}_{-1}(m,v_0)$, given by Eq. (\[calP3\]), of the maximum displacement $m$ attained by a particle which begins at the origin with velocity $v_0$ and moves according to Eq. (\[eqmo\]) with $g=-1$. The results in (a) are for $v_0=$ 0.4, 0.6, 0.8, and 1.0. As $v_0$ increases, the peak becomes lower and broader and moves to the right. The curves in (b) correspond, from top to bottom, to $v_0=$ -0.4, -0.6, -0.8, and -1. The vertical line at $m=0$ represents the term in $P_{-1}(m,v_0)$ proportional to $\delta(m)$.[]{data-label="fig3"}](afirstpassfig4a.eps "fig:"){width="12cm"}1.5cm
\
![Distribution ${\cal P}_{-1}(m,v_0)$, given by Eq. (\[calP3\]), of the maximum displacement $m$ attained by a particle which begins at the origin with velocity $v_0$ and moves according to Eq. (\[eqmo\]) with $g=-1$. The results in (a) are for $v_0=$ 0.4, 0.6, 0.8, and 1.0. As $v_0$ increases, the peak becomes lower and broader and moves to the right. The curves in (b) correspond, from top to bottom, to $v_0=$ -0.4, -0.6, -0.8, and -1. The vertical line at $m=0$ represents the term in $P_{-1}(m,v_0)$ proportional to $\delta(m)$.[]{data-label="fig3"}](afirstpassfig4b.eps){width="12cm"}
\
|
---
author:
- 'Osamu <span style="font-variant:small-caps;">Sakai</span>[^1]'
title: 'Band Calculations for Ce Compounds with AuCu$_{3}$-type Crystal Structure on the basis of Dynamical Mean Field Theory I. - CePd$_{3}$ and CeRh$_{3}$ - '
---
Introduction
============
Nonempirical band calculations for strongly correlated electron systems have been extensively developed on the basis of dynamical mean field theory (DMFT) [@A1; @A2]. The $4f$ electrons in Ce compounds are typical strongly correlated electrons [@A3; @A4; @A5]. Recently, a DMFT band calculation scheme for Ce compounds was developed in refs. and . In the present work, it is applied to Ce compounds with the AuCu$_{3}$-type crystal structure, which show a wide variety of $4f$ electronic states from the most itinerant limit to the localized limit.
The $4f$ state splits into the $j=5/2$ ground multiplet and the $j=7/2$ excited multiplet with a separation of about 0.3 eV owing to the spin-orbit interaction (SOI). The multiplet shows crystal-field splitting (CFS) of the order of 100 K. In cubic crystals, the $j=5/2$ multiplet splits into the ($j=5/2$)$\Gamma_{7}$ doublet and the ($j=5/2$)$\Gamma_{8}$ quartet. Hereafter, we call them the $\Gamma_{7}$ and $\Gamma_{8}$ states, respectively. It is important to take account of the SOI and CFS effects in $4f$ systems [@A3]. In DMFT, the correlated band electron problem is mapped onto the calculation of the single-particle excitation spectrum of the auxiliary impurity Anderson model in an effective medium. Reliable methods of solving the impurity Anderson model with CFS and the SOI effect are needed in the DMFT band calculation for $4f$ compounds. A theory named NCA$f^{2}$vc (noncrossing approximation including the $f^{2}$ state as a vertex correction) has been developed [@A8; @A9; @A10; @B1] and combined with the linear muffin-tin orbital (LMTO) method [@A11; @A12] to carry out the DMFT band calculation [@A7]. NCA$f^{2}$vc can include CFS and the SOI effect, and also the correct exchange process of the $f^{1} \rightarrow f^{0}, f^{2}$ virtual excitation. The calculation gives an accurate order of the Kondo temperature ($T_{\rm K}$).
The DMFT band calculation will be applied in a series of studies to Ce compounds with the AuCu$_{3}$-type structure: CePd$_{3}$, CeRh$_{3}$, CeIn$_{3}$, and CeSn$_{3}$. Each of these materials is classified as a typical example of strongly correlated $4f$ systems [@A5; @A13; @A14; @A15; @E1; @A16; @A17].
CePd$_{3}$ is a typical heavy Fermion system with $T_{\rm K}$ of about 250 K, and has been studied extensively by various methods [@A5; @A16; @A17; @A18; @A19; @A20; @A21; @A22; @A23; @A24; @A25; @A26; @A27; @A28; @A29]. It has a nonmagnetic Fermi liquid (FL) ground state at low temperatures. A controversy existed in the experimental works on inelastic magnetic excitation, but it has recently been resolved by a detailed study of the wave number vector (wave vector) dependence of spectra [@A17; @A19; @A20; @A21]. The wave-vector-integrated-spectrum of the magnetic excitation has a broad peak at approximately 55 meV. In the single-particle excitation, a broad peak with the $4f$ character was observed on the inverse photoemission spectrum (IPES) side (i.e., in the energy region above the Fermi energy ($E_{\rm F}$)). This seems to be consistent with the high $T_{\rm K}$ of this compound [@A22; @A23; @A24; @A25]. However, a strong peak structure has never been observed on the photoemission spectrum (PES) side (i.e., in the energy region below $E_{\rm F}$), contradicting other physical properties [@A26; @A27]. Recent careful study using the $3d$-$4f$ high-resolution resonant photoemission spectrum (RPES) revealed that the bulk component of the PES of this compound shows strong intensity at $E_{\rm F}$ consistently with high $T_{\rm K}$ [@A28]. The angle-resolved PES (ARPES) has also been studied recently [@A29]. It may be worthwhile whether to confirm these recent results of studies are reproduced or not by the first-principles DMFT band calculation. CeRh$_{3}$ is known as one of the compounds having the most itinerant 4f states [@A23; @A24; @A30; @A31; @A32; @A33; @A34].
CeSn$_{3}$ is also known to show the nonmagnetic FL ground state with a high characteristic temperature [@A35; @A36]. CeIn$_{3}$ has an antiferromagnetic ground state with a Neel temperature of $T_{\rm N}=11$ K. Recently, it was found that $T_{\rm N}$ decreases to zero under the pressure of 2.5 GPa, in addition, the transition to superconductivity occurs at $T_{\rm SC}=0.2$ K [@A37; @A38; @A39]. It will be interesting to study the change of the band structure under pressures by the DMFT calculation.
These compounds commonly have the AuCu$_{3}$-type crystal structure, which is classified into the simple cubic ([*sc*]{}) lattice. The DMFT band structure of these compounds will be reported in two papers. In CePd$_{3}$ and CeRh$_{3}$, the hybridization of $4f$ states with $4d$ states of the transition metal is very strong. The $4d$ states almost sink to below $E_{\rm F}$ in CePd$_{3}$ whereas they are located near $E_{\rm F}$ in CeRh$_{3}$ [@A40]. In this paper, calculations for these $4d$ compounds will be reported. Calculations for CeSn$_{3}$ and CeIn$_{3}$ will be given in a subsequent paper. Their $4f$ states hybridize with broad $5p$ states of ligand ions.
Calculated results of CePd$_{3}$ and CeRh$_{3}$ generally show reasonable agreement with experimental results of the PES, IPES, ARPES, and inelastic magnetic excitation by neutrons. However, the calculation gives a higher $T_{\rm K}$ than that expected from experiments when it is examined in detail. The DMFT band calculation for CePd$_{3}$ gives a Fermi surface (FS) structure similar to that obtained by the local density approximation (LDA) calculation at very low temperatures. When the temperature increases, $4f$ bands shift to the high-energy side and their lifetime broadening increases. This leads to the change of the FS structure into one that is similar to the FS of LaPd$_{3}$. At $T=150$ K, the FS has a different form from both FSs of LaPd$_{3}$ and the LDA band of CePd$_{3}$ as an intermediate stage of the change. The lifetime broadening overcomes the fine wave vector dependence of the $4f$ spectrum at $T=300$ K. In CeRh$_{3}$, the $4f$ band is located at about 0.9 eV above $E_{\rm F}$, and the dispersion of the DMFT band is almost identical to that of the LDA band in the energy region near $E_{\rm F}$. The density of states (DOS) has an appreciable value slightly below $E_{\rm F}$ in the ARPES. This low-binding energy part shows a weak wave vector dependence, though no flat bands do not exist in the vicinity of the Fermi energy.
In §2, we briefly give the formulation on the basis of the LMTO method. Results of the application to CePd$_{3}$ are shown in §3, and results for CeRh$_{3}$ are given in §4. A summary is given in §5. In the appendices, notes on the calculation of the total electron number are given. An efficient method of calculating the Cauchy integral using the spline interpolation scheme is also presented. This integral is frequently used in the DMFT calculation.
Formulation
===========
The method of calculation is described briefly because its details have been given in previous papers [@A6; @A7]. We consider the excitation spectrum of the following Hamiltonian: =[H]{}\_[LDA]{} + \_ (\_[,]{} c\^[+]{}\_[\^[a]{}]{} c\_[\^[a]{}]{} -n\^[LDA\*]{}\_[f]{})\^[2]{}. \[eq.Hamiltonian\] Here, $ c_{\phi^{\rm a}{\mbox{\boldmath $i \:$}}\Gamma\gamma}$ is the annihilation operator for the atomic localized state $
\phi^{\rm a}_{{\mbox{\boldmath $i \:$}}\Gamma\gamma}({\mbox{\boldmath $r \:$}})
$ at site ${\mbox{\boldmath $i \:$}}$ with the $\gamma$ orbital of the $\Gamma$-irreducible representation. The quantity $n^{\rm LDA*}_{{\mbox{\boldmath $i \:$}}f}$ is determined using the occupation number of the atomic $4f$ electron per Ce ion in the LDA calculation. We assume that the local Coulomb interaction acts only on the orbital $\phi^{\rm a}_{{\mbox{\boldmath $i \:$}}\Gamma\gamma}$.
The excitation spectrum is expressed by introducing the self-energy terms [@A6], \_[DMFT]{}=[H]{}\_[LDA]{}\
+ \_[,(,)]{} (\_(+[i]{}) +\^[a]{}\_-\_\^[LDA]{}) |\^[a]{}\_ > < \^[a]{}\_ |, \[eq.DMFT-Hamiltonian\] where $\VEP^{\rm a}_{\Gamma}$ is the single-electron energy level of the $4f$ state, and $\VEP_{\Gamma}^{\rm LDA}$ is the energy level in the LDA calculation. The self-energy $\Sigma_{\Gamma}(\VEP+{\rm i}\delta)$ is calculated by solving the auxiliary impurity problem with the use of NCA$f^{2}$vc; its outline is described in the Appendix of ref. .
In later calculations we will approximate the localized $4f$ state $\phi^{\rm a}$ by the band center orbital $\phi(-)$ because its localization is good for the $4f$ state. $\phi(-)$ has the logarithmic derivative $-\ell-1$ on the muffin-tin surface [@A11; @A12].
In the LMTO method, the Hamiltonian ${\cal H}_{\rm LDA}$ is diagonalized using the LMTO bases, \^[j]{}() =\_[q L]{}a\^[j]{}\_[q L]{} \^\_[q L]{}(). \[eq.psi\] Here, $a^{j{\mbox{\boldmath $k \:$}}}_{q L}$ is the expansion coefficient of the $j$th eigenvector on the LMTO base of the Bloch type, $\chi^{{\mbox{\boldmath $k \:$}}}_{q L}({\mbox{\boldmath $r \:$}})$, with the wave number vector ${\mbox{\boldmath $k \:$}}$, the angular momentum ($\ell,m$), and the spin ($\alpha$) at site $q$ in the unit cell, where $L \equiv (\ell,m,\alpha)$ [@A12].
The explicit expression of $\chi^{{\mbox{\boldmath $k \:$}}}_{q L}({\mbox{\boldmath $r \:$}})$ has been given in a previous paper [@A6]. Note that they are not orthogonal to each other, but the eigenvectors $\psi^{j{\mbox{\boldmath $k \:$}}}$ are orthonormal.
The DMFT band structure is calculated in the following way: (A) first the LDA part of the Hamiltonian ${{\cal H}}_{\rm LDA}$ is diagonalized for a given ${\mbox{\boldmath $k \:$}}$, (B) then the matrix equation of the Greenian is prepared in the manifold of the eigenvectors $\psi^{j{\mbox{\boldmath $k \:$}}}$. The Greenian equation for the given ${\mbox{\boldmath $k \:$}}$ is written as G(z;) = I, \[eq.G-eq\] where $I$ is the unit matrix and $D_{\rm LDA}({\mbox{\boldmath $k \:$}})$ is the diagonal matrix of the eigenenergies of ${\cal H}_{\rm LDA}$ with ${\mbox{\boldmath $k \:$}}$. The matrix elements of $\Sigma(z)$ are given by calculating the self-energy operator term of eq. (\[eq.DMFT-Hamiltonian\]) based on eq. (\[eq.psi\]).
The DOS on the atomic $4f$ state is given by \^[([band]{})]{}\_(;)=- \[\_G(+[i]{};)\], \[eq.rho-band\] where the projection operator is defined as \_=\_ |\^[a]{}\_[()]{}><\^[a]{}\_[()]{}|. \[eq.density-op\] The local DOS in the DMFT band calculation is obtained by summing $\rho^{({\rm band})}_{\Gamma}(\VEP;{\mbox{\boldmath $k \:$}})$ over ${\mbox{\boldmath $k \:$}}$ in the Brillouin zone: $\rho^{({\rm band})}_{\Gamma}(\VEP)=\frac{1}{N}\sum_{{\mbox{\boldmath $k \:$}}}\rho^{({\rm band})}_{\Gamma}(\VEP;{\mbox{\boldmath $k \:$}})$ [@A41]. Here, $N$ is the total number of unit cells.
The auxiliary impurity problem is solved by the NCA$f^{2}$vc method. The splitting of the self-energy due to the SOI and CFS effects is considered. As shown in ref. , this method gives an accurate order of the Kondo temperature when the result is compared with that of the more correct numerical renormalization group (NRG) calculation [@A7; @A43] in a simple model case.
Since the method of the self-consistent calculation in the DMFT has been described previously [@A6; @A7], we exclude the detailed explanation from this paper. First of all, we calculate the self-consistent LDA band by the LMTO method, and potential parameters, except for the $f$ levels, are fixed to those in the LDA calculation. (I) We calculate the atomic $4f$ density of states $\rho^{({\rm imp.})}_{\Gamma}(\VEP)$ ($4f$ DOS) for the auxiliary impurity Anderson model by the NCA$f^{2}$vc method with a trial energy dependence of the hybridization intensity (HI) and $4f$ levels[@D2], then calculate the local self-energy. (II) The DMFT band calculation is carried out using the self-energy term, and the local $4f$ DOS in the DMFT band is calculated. The calculation is iterated so that the $4f$ DOS of the local auxiliary impurity model and the DMFT band satisfy the self-consistent conditions [@D1; @A44].
The $4f$ level is adjusted in the DMFT self-consistent iterations under the condition that the $4f$ occupation number has a given target value, $n_{f}({\rm rsl.target})$, which is estimated from the LDA band calculation. The temperature dependence of the Fermi energy, $E_{\rm F}$, is neglected by fixing it at a value determined at a low temperature. It is estimated using the occupation number of the renormalized band (RNB) calculation, in which the self-energy is approximated by an expansion form up to the linear term in the energy variable at $E_{\rm F}$ (see Appendix A for the calculation of the total occupation number). The target $4f$ electron number $n_{f}({\rm rsl.target})$ is imposed on the occupation number calculated directly using the resolvents to stabilize the self-consistency iterations. The occupation number obtained by the integration of the $4f$ DOS, $n_{f}({\rm intg.})$, has a deviation within 1.0% from $n_{f}({\rm rsl.target})$ because many intermediate calculation process are included.
CePd$_{3}$
===========
Density of states
-----------------
![ $4f$ DOS of CePd$_{3}$ at $T=37.5$ K. The solid line shows the total $4f$ PES. The dashed line is the DOS of the ($j=5/2$)$\Gamma_{7}$ component, the dot-dash line is the DOS of the ($j=5/2$)$\Gamma_{8}$ component, and the two-dots-dash line is the DOS of the $j=7/2$ component. The Fermi energy $E_{\rm F}=0.6143$ Ry is indicated by the vertical dot-dash line. The inset shows spectra in the vicinity of $E_{\rm F}$. []{data-label="fig:CePd3-38-flsp"}](Fig/64339F1-CePd3-874-37.5-flspo.eps){width="8cm"}
![ Hybridization intensity (HI) [@D3] of CePd$_{3}$ at $T=37.5$ K. The dashed line is the HI of the ($j=5/2$)$\Gamma_{7}$ component, the dot-dash line is the HI of the ($j=5/2$)$\Gamma_{8}$ component, and the two-dots-dash line is the HI of the $j=7/2$ component. $E_{\rm F}$ is indicated by the vertical dot-dash line. The inset shows HI in the vicinity of $E_{\rm F}$. []{data-label="fig:CePd3-38-mix"}](Fig/64339F2-CePd3-874-37.5-mix.eps){width="8cm"}
![ Total DOS (dashed line) and $4f$ DOS (solid line) of CePd$_{3}$ at $T=37.5$ K. The spectra are broadened by an imaginary factor, $\gamma=0.01$ Ry, in the energy variable. $E_{\rm F}$ is indicated by the vertical dot-dash line. []{data-label="fig:CePd3-38-bdos"}](Fig/64339F3-CePd3-874-37.5-bdos.eps){width="8cm"}
[@cccc]{} &$\Gamma_{7}$ & $\Gamma_{8}$ & $j=7/2$\
$n_{\Gamma}^{({\rm imp.})}$ & 0.084 & 0.719 & 0.185\
$\VEP_{\Gamma}$(Ry) & -0.19027 &-0.19182 &-0.16493\
$\rho_{\Gamma}(E_{\rm F})(Ry^{-1})$ & 3.8 & 59.0 & 0.5\
$\bar{Z}^{-1}_{\Gamma}$ & 6.3 & 6.7 & 3.6\
$\bar{\VEP}_{\Gamma}$(Ry) & 0.0153 &0.0057 & 0.1184\
$\bar{\Gamma}_{\Gamma}$(Ry) & 2.7 $\times 10^{-4}$ & 16.0 $\times 10^{-4}$ & 3.0 $\times 10^{-4}$\
&\
In Fig. \[fig:CePd3-38-flsp\], we show the DOS of the $4f$ component ($4f$ DOS) for CePd$_{3}$ at $T=37.5$ K. The solid line shows the total $4f$ component of the PES ($4f$ PES). The dashed line is the DOS of the $\Gamma_{7}$ component and the dot-dash line is the DOS of the $\Gamma_{8}$ component. The two-dots-dash line is the DOS of the $j=7/2$ component. The CFS of the self-energy in the excited $j=7/2$ multiplet is neglected. The vertical dot-dash line indicates the Fermi energy $E_{\rm F}=0.6143$ Ry. The inset shows spectra in the energy region near $E_{\rm F}$. The $4f$ DOS has a large peak at 0.642 Ry, about 0.028 Ry (0.38 eV) above $E_{\rm F}$. This has mainly the $j=7/2$ character. The spin-orbit splitting on the IPES side is usually enhanced in Ce systems. The spectral intensity on the PES side consists mostly of the $\Gamma_{8}$ component. The PES has a sharp peak at $E_{\rm F}$ with a steep tail up to 0.094 Ry (1.3 eV) below $E_{\rm F}$. It also has a long tail with small structures reflecting the DOS of $4d$ states of Pd on the high-binding-energy side. In the steep tail region, shoulders appear at binding energies of 0.004 Ry (0.05 eV) and 0.024 Ry (0.33 eV). These may correspond, respectively, to the CFS and the SOI side band. The sharp peak at $E_{\rm F}$ with the steep and the long tail has been observed in high-resolution experiments by Kasai [*et al.*]{} [@A28]. The shoulder due to the SOI seems to be identified.
The Kondo temperature $T_{\rm K}$ and the CFS excitation energy are, respectively, estimated to be about 27 meV (310 K) and 41 meV (480 K) from magnetic excitation spectra, as will be shown in Fig. \[fig:CePd3-38-mag\] in the next subsection. We note that spectra do not show appreciable change even when the temperature is decreased to 18.75 K in the calculation.
The parameters and the calculated values are given in Table \[tab:CePd3\]. The LMTO band parameters [@A12] for states except for the $f$ component are fixed to those of the LDA calculation. $E_{\rm F}$ is fixed to the value determined by the occupied state in the RNB, as is discussed in a later section. The relative occupation number of the $\Gamma_{7}$ component to the $\Gamma_{8}$, 0.084/0.719 (0.12) is small compared with the 0.5 expected from the ratio of the degeneracy, but is very large compared with the value expected from the simple model of the CFS for an isolated ion with $\VEP_{\Gamma 7}-\VEP_{\Gamma 8}=0.00155$ Ry (240 K). The ratio does not change so greatly even when the temperature is raised: 0.089/0.723 (0.12) at 150 K and 0.108/0.707 (0.15) at 300 K. Moreover, the occupation of the $j=7/2$ component, 0.185, is not small, and is almost independent of $T$. A simple picture of the CFS for an isolated ion cannot be applied.
Usually, the electrostatic potential causes cubic CFS in $4f$ electron systems with a higher energy level of about 150 K for the $\Gamma_{7}$ state [@A47; @A48]. This is included in the present calculation. Even when it is neglected, the $4f$ DOS and the magnetic excitation spectrum are not greatly changed because the hybridization effect causes large CFS, of greater than 300 K [@A49].
The target value of the occupation number on the atomic $4f$ states, $n_{f}({\rm rsl.target})$, has been tentatively chosen to be 0.98 in the present calculation. This value is small compared with the $4f$ occupation number, 1.045, of the LDA band calculation for CePd$_{3}$. When we carry out the LDA calculation for compounds in which Ce ions are replaced by La ions, the occupation number on the $4f$(La) state usually amounts to 0.1 [@A50]. We use the occupation number as reduced to 94% of the LDA value as the atomic occupation number on the Ce $4f$ state. Kanai [*et al.*]{} concluded that $4f$ occupancy in CePd$_{3}$ is expected to be 0.92 on the basis of the results of resonant inverse photoemission (RIPES) experiments [@A25; @A45; @A46]. When we perform the DMFT band calculation by setting $n_{f}({\rm rsl. target})$ to be 0.92, $T_{\rm K}$ determined from the magnetic excitation spectrum is expected to be 500 K. On the other hand, $T_{\rm K}$ is greatly reduced to about 10 K when we choose 1.05.
In Fig. \[fig:CePd3-38-mix\], we show the trial HI [@D3] obtained in the DMFT band calculation at 37.5 K. It has very large peaks corresponding to the $4d$ band of Pd, but these peaks are located in the energy region deep below $E_{\rm F}$. The HI is not high in the energy region near $E_{\rm F}$. In particular, the HI of the $\Gamma_{7}$ component is low though it is high in the deeper energy region. The overall features of the HI in the DMFT band are similar to those calculated directly using the LDA band, but the HI in DMFT is increased in the vicinity of $E_{\rm F}$ to about twice the LDA value for the $\Gamma_{7}$ and $\Gamma_{8}$ components, and is decreased in the deep energy region. Moreover, the DMFT calculation causes fine structures of the HI in the vicinity of $E_{\rm F}$, which are shown in the inset of the figure. This contrasts with the HI in the LDA, which has a weak energy dependence in this region. The HI of the $\Gamma_{8}$ component in DMFT has a small peak at $E_{\rm F}$, while those of the $\Gamma_{7}$ and $j=7/2$ components have small peaks above or on both sides of $E_{\rm F}$. The reason for this different behavior is not clear at present, but we note that the peaks of the $4f$ DOS for the latter two cases are located above $E_{\rm F}$. When we do a calculation in the fictitious case that $\Gamma_{7}$ is mainly occupied by assigning a low energy level to it, the HI of $\Gamma_{7}$ has a small peak at $E_{\rm F}$. However, this result should not be used as a general rule, because the modification of HI in DMFT is delicately dependent on details of the band structures. The Kondo temperature is increased in the DMFT calculation in the CePd$_{3}$ case. We have obtained the Kondo temperature of 10 K in the single impurity calculation using the HI of the LDA band.
The total DOS at $T=37.5$ K is shown by the dashed line in Fig. \[fig:CePd3-38-bdos\]. The large peaks at about 0.3 and 0.45 Ry have the $4d$ character of Pd, and that at 0.8 Ry has the $5d$ character of Ce. These peaks are also obtained by the LDA calculation [@A51]. The sharp $4f$ peaks slightly above $E_{\rm F}$, which are called the $f^{1}$ peak in IPES, are also obtained in the LDA calculation[@A40]. Their intensity in DMFT is reduced compared with that in the LDA because a part of it is transferred to the intensity of the peak at about 1.0 Ry, which is called the $f^{2}$ peak. In the present calculation, the width of the $f^{2}$ peak is not large because the multiplet splitting of the $f^{2}$ final state is neglected.
In the analysis of RIPES experiments, the ratio of the $f^{1}$ peak to the total RIPES intensity has been given as 0.22 [@A25], whereas it is estimated to be about 0.2 in the present calculation. The present DMFT calculation seems to give results that emphasizes the hybridization effects strongly (i. e. the higher $T_{\rm K}$).
Magnetic excitation
-------------------
![ Magnetic excitation spectrum of CePd$_{3}$. The solid line shows the spectrum at $T=37.5$ K, the dot-dash line is the spectrum at $T=150$ K, and the two-dots-dash line is the spectrum at $T=300$ K. The dashed line is the spectrum in a fictitious case where matrix elements of the magnetic moment are restricted within the intra-$\Gamma_{8}$ manifold of space. []{data-label="fig:CePd3-38-mag"}](Fig/64339F4-CePd3-mag-corr.eps){width="8cm"}
In Fig. \[fig:CePd3-38-mag\] we show the magnetic excitation spectrum. The total magnetic excitation spectrum at $T=37.5$ K is shown by the solid line. It has a peak at about $E=0.003$ Ry (41 meV). The dashed line depicts the spectrum for a fictitious case in which the matrix elements of the magnetic moment are nonzero only in the manifold of $\Gamma_{8}$, and thus it may correspond to the excitation spectrum within the $\Gamma_{8}$ manifold. It has a peak at about $E=0.002$ Ry (27 meV), and leads to a faint shoulder in the solid line. The CFS excitation energy seems to be slightly larger than $T_{\rm K}$ in the present calculation. The calculated magnetic susceptibility is 1.9 $\times 10^{-3}$ emu/mol, whereas the experimental value is 1.5 $\times 10^{-3}$ emu/mol at $T \sim 0$ K [@A52; @A56]. We show magnetic excitation spectra at $T=150$ K and at $T=300$ K by the dot-dash line and the two-dots-dash line, respectively. The spectrum at $T=300$ K has a broad peak centered at about 0.003 Ry (41 meV). The shoulder shifts to $E \sim 0$ and becomes a peak. the overall features do not change so greatly when we neglect the electrostatic CFS of 150 K. The magnetic excitation spectrum observed in the wave-vector-integrated case has a peak at about 55 meV at $T= 10$ K, and the peak shifts to the low-energy side as T increases [@A19]. The calculated results seem to be generally consistent with those of the experiment. However, the peak at $E \sim 0$ is higher at $T=300$ K in the experiments [@A19]. The present DMFT calculation seems to give stronger HI compared with the value in the experiments. The detailed calculation of physical quantities using finely tuned parameters will be given in the future.
In ref. , excitation peaks with an energy of 15 meV and less than 3 meV are indicated. Low-energy peaks are not expected in the present calculation of the wave-vector-integrated spectra because $T_{\rm K}$ is not low. One possibility of the origin of the low-energy peaks may be the wave vector dependence of the magnetic excitation spectra, as noted in ref. .
RNB, and wave number-vector-dependent DOS
-----------------------------------------
![ Band dispersion of the renormalized band (RNB) picture for CePd$_{3}$ at $T=37.5$ K. The symbols under the horizontal axis denote the symmetry points and axes of the Brillouin zone of the $sc$ lattice. $E_{\rm F}$ is indicated by the horizontal dashed line. []{data-label="fig:CePd3-rnb"}](Fig/64339F5-CePd3-874-38K-rnb-greek-641.eps){width="10cm"}
![ Wave number vector (${\mbox{\boldmath $k \:$}}$) dependence of the DOS (k-DOS) for CePd$_{3}$ at $T=37.5$ K. ${\mbox{\boldmath $k \:$}}$ is moved from the $\Gamma$ point (bottom) to the R point (top) along the $\Lambda$ axis. The total DOS is shown, but the $4f$ component is dominant in this energy region of CePd$_{3}$. []{data-label="fig:CePd3-38-kspc"}](Fig/64339F6-CePd3-874-37.5-kspct.eps){width="8cm"}
![ k-DOS of CePd$_{3}$ at $T=150$ K. ${\mbox{\boldmath $k \:$}}$ is changed from the $\Gamma$ point (bottom) to the R point (top) along the $\Lambda$ axis. []{data-label="fig:CePd3-300-kspc"}](Fig/64339F7-CePd3-875-150-kspct.eps){width="8cm"}
![ Angle-resolved PES of CePd$_{3}$ at $T=37.5$ K. The surface is assumed to be (111). The representative ${\mbox{\boldmath $k \:$}}$ is swept from the $\Gamma$ (bottom) point to the M (top) point along the $\Sigma$ axis, and the intensities are averaged over the wave number vector normal to the (111) surface. This means that ${\mbox{\boldmath $k \:$}}$ also runs from the R (bottom) point to the X (top) point along the S axis. The Fermi energy $E_{\rm F}$ is indicated by the vertical dot-dash line. The total DOS is shown, but the intensity above the energy of 0.612 Ry is dominated by the $4f$ character. The intensity below 0.611 Ry has mainly the non-$4f$ character. []{data-label="fig:CePd3-38-PES"}](Fig/64339F8-CePd3-874-37.5K-arpes.eps){width="8cm"}
![ Angle-resolved PES of CePd$_{3}$ at $T=150$ K. For the definition of lines, see the caption of Fig. \[fig:CePd3-38-PES\]. []{data-label="fig:CePd3-300-PES"}](Fig/64339F9-CePd3-875-150K-arpes.eps){width="8cm"}
In Fig. \[fig:CePd3-rnb\], we show the RNB dispersion at $T=37.5$ K. The energy shift (the real part of the self-energy at $E_{\rm F}$: $\Re\Sigma_{\Gamma}(E_{\rm F}$)) and the mass renormalization factor ($1-\partial\Re\Sigma_{\Gamma}(\varepsilon)/\partial\varepsilon|_{E_{\rm F}}$), which are given in Table \[tab:CePd3\], are taken into account in this calculation. Narrow bands with the $j=5/2$ character appear slightly above $E_{\rm F}$, and those with the character of $j=7/2$ appear around the energy 0.645 Ry which is near the energy of the $j=7/2$ peak in the $4f$ DOS shown in Fig. \[fig:CePd3-38-flsp\].
The lowest $4f$ band sinks to below $E_{\rm F}$ near the $\Gamma$ and R points, and is located above $E_{\rm F}$ in the other regions. The dispersion of RNB corresponds well with the behavior of the ${\mbox{\boldmath $k \:$}}$-dependent density of states (k-DOS). For example, we show the k-DOS when ${\mbox{\boldmath $k \:$}}$ moves from the $\Gamma$ (bottom) to the R (top) point along the $\Lambda$ line in Fig. \[fig:CePd3-38-kspc\]. A peak of the DOS with mainly the $4f$ character is located below $E_{\rm F}$ at the $\Gamma$ and R points. Starting from the peak below $E_{\rm F}$ at the $\Gamma$ point, one of ridge lines runs above the Fermi energy across $E_{\rm F}$, and then connects to the peak below $E_{\rm F}$ at the R point. Another runs to the low-energy side of $E_{\rm F}$ up to 0.611 Ry at the halfway, and then turns back to connect to the peak below $E_{\rm F}$ at the R point. This “hanging” branch does not cross the Fermi energy.
Note that we have depicted the total spectral intensity, not the $f$-component, in Fig. \[fig:CePd3-38-kspc\]. In the energy region shown in the figure, the spectral intensity has mainly the $4f$ character. On the other hand, very sharp spectral peaks of a non-$4f$ character appears in the energy region outside of the figure. The hanging band on the $\Lambda$ axis has a stronger non-$4f$ character. This branch is a hybridization band between the $4f$ of Ce and a conduction band that has the character of the $sp$-free electron band and the $5d$ of Ce.
The dispersion of the RNB is qualitatively similar to the result of the band structure determined by Hasegawa and Yanase by the LDA calculation [@A53], although the width of the $4f$ band with $j=5/2$ is about 0.005 Ry in the RNB, whereas that of the LDA band is about 0.02 Ry. Both calculations give electron pockets at the $\Gamma$ and R points, and hole pockets centered on the T axis [@D4].
The Main features of the band dispersion near the Fermi energy are formed by the hybridization of the narrow $4f$ bands and the wide $sp$ bands. The $4f$ bands have dispersion characterized by the LMTO (linear combination of atomic orbitals) tight-binding bands of the $sc$ lattice. Although the $4f$ band width is reduced in DMFT, the qualitative features of the dispersion of the $4f$-$sp$ hybridized bands are not changed because the number of participating $sp$ bands is small and their dispersion is very rapid compared with that of $4f$ bands.
Here, we must note a weak point of the NCA$f^{2}$vc method, that is it does not automatically ensure the Fermi liquid sum rule, i.e., the integral of the total DOS below $E_{\rm F}$, $N({\rm total}; \rho)=\int{\rm d}\VEP\frac{1}{N}\sum_{{\mbox{\boldmath $k \:$}}}(-\frac{1}{\pi}\Im G(\VEP+{\rm i}\delta;{\mbox{\boldmath $k \:$}}))f(\VEP)$, is not equal to the occupation number of electrons calculated by the volume of the occupied states in the ${\mbox{\boldmath $k \:$}}$ space. The quantity $N({\rm total}; \rho)$ is a smaller value in CePd$_{3}$, and thus we obtain a higher Fermi energy. If we use it in the RNB calculation, the volume of the occupied states in the ${\mbox{\boldmath $k \:$}}$ space becomes large. In the case of CePd$_{3}$, which has an even total electron number, the balance between the electron and hole states is lost. In this study, we tentatively use the Fermi energy determined using the quantity $N({\rm total};{\rm RNB})=\frac{1}{N}\sum_{\lambda{\mbox{\boldmath $k \:$}}}f(E_{\rm RNB}(\lambda{\mbox{\boldmath $k \:$}}))$ to ensure the electron-hole balance in the RNB band, where $E_{\rm RNB}(\lambda{\mbox{\boldmath $k \:$}})$ is the energy obtained by the RNB calculation. (For more details, see Appendix B.)
When we calculate the RNB dispersion at $T=150$ K, the $4f$ band shifts slightly to the high-energy side. The $4f$ state at the $\Gamma$ point nears $E_{\rm F}$, and the $4f$ state at the R point is located on $E_{\rm F}$. Therefore, a part of the hanging band on the $\Lambda$ line rises above $E_{\rm F}$. Hole pockets on the T axis grow into larger hole regions around the R point. As the temperature increases further, the $4f$ state at the R point shifts up to above $E_{\rm F}$, and a large hole surface enclosing the R point appears. In other words, we have a connected electron Fermi surface (FS) that contains the X and M points inside it.
The primary structure of the FS of the RNB at high temperatures is similar to that of the FS of the LDA band of LaPd$_{3}$, but their fine topologies will differ. At $T=150$ K, the main part of the hanging band on the $\Lambda$ line is still located below $E_{\rm F}$. The hole sheets around the $\Gamma$ and R points are separated by an electron region on the $\Lambda$ axis. On the other hand, this entire hanging branch is located above $E_{\rm F}$ in LaPd$_{3}$, and thus the separation by the electron region does not occur.
We should, of course, note that the RNB picture has only limited meaning at high temperatures because the imaginary part is large [@D5]. The k-DOS at $T=150$ K for ${\mbox{\boldmath $k \:$}}$ along the $\Lambda$ line is shown in Fig. \[fig:CePd3-300-kspc\]. The widths of peaks become large, but we can see that the peak at the R point is located almost at $E_{\rm F}$, and the peak at $\Gamma$ nears $E_{\rm F}$. The trace of broad peaks shows a shift corresponding to that of the RNB dispersion. We can recognize that the ridge of the peaks of the hanging band crosses the Fermi energy, as noted in the previous paragraph. At $T=300$ K, the width of peaks becomes so large that it surpasses the fine ${\mbox{\boldmath $k \:$}}$ dependence of the spectra. However, the $4f$ peak near the Fermi energy still exists in the $4f$ DOS calculation, similar to that shown in Figs. \[fig:CePd3-38-flsp\] and \[fig:CePd3-38-bdos\]. It moves slightly to the high-energy side with increasing width at 300 K.
The present calculation has been performed with fixed $E_{\rm F}$ and $n_{f}({\rm rsl.target})$. The calculated total occupation number in the DMFT band increases to $N({\rm total} ; {\rm RNB})=34.304$ at $T=300$ K, although the energies of the $4f$ bands shift upward [@D6]. We point out that even when we move $E_{\rm F}$, the energy of $4f$ bands relative to the Fermi energy will not change greatly because the Kondo resonance peak usually shifts following the change of the Fermi energy.
Recently, the wave vector dependence of the PES has been observed under the $4d \rightarrow 4f$ resonant condition [@A29]. The experiment was carried out for the (111) surface of a thin film by sweeping the ${\mbox{\boldmath $k \:$}}$ vector along $\bar{\Gamma}$-K-M-K-$\bar{\Gamma}$ in the surface Brillouin zone (BZ). This may correspond to the following sweeping of ${\mbox{\boldmath $k \:$}}$ in the 3-dimensional [*sc*]{} BZ: the component parallel to the surface runs as $\Gamma$-($\Sigma$)-M-($\Sigma$)-$\Gamma$ with averaging over its components normal to the surface. This means that the sweeping of the parallel component R-(S)-X-(S)-R is also included.
In Fig. \[fig:CePd3-38-PES\], we show the PES when the representative wave vector moves from the $\Gamma$ (bottom) point to the M (top) point with average intensities for ${\mbox{\boldmath $k \:$}}$ normal to the surface. The total intensity is plotted in the figure, but the intensity above the energy of 0.612 Ry has mainly the $4f$ character. The spectra have a peak for ${\mbox{\boldmath $k \:$}}$ near $\Gamma$, and this corresponds well to the experimental results. The intensity below the energy of 0.611 Ry has a non$4f$ character. As noted previously, the contribution from the R point is also included at the representative $\Gamma$ point. At these points, the $4f$ component is located below $E_{\rm F}$ at $T=37.5$ K. In Fig. \[fig:CePd3-300-PES\], we show spectra at $T=150$ K. The peak of the intensity at $\Gamma$ nears $E_{\rm F}$ because of the shifting up of the $4f$ band. The thermal distribution effect and the very high intensity of the $4f$ DOS above $E_{\rm F}$ also have affect this spectral shape. The change of the peak can be checked in experiments.
CeRh$_{3}$
===========
![ $4f$ DOS of CeRh$_{3}$ at $T=10^{3}$ K. For the definition of lines, see the caption of Fig. \[fig:CePd3-38-flsp\]. $E_{\rm F}=0.6940$ Ry is indicated by the vertical dot-dash line. []{data-label="fig:CeRh3-flsp"}](Fig/64339F10-CeRh3-238-1-flsp.eps){width="8cm"}
![ Total DOS (dashed line) and $4f$ DOS (solid line) of CeRh$_{3}$ at $T=10^{3}$ K. See the caption of Fig. \[fig:CePd3-38-bdos\]. []{data-label="fig:CeRh3-bdos"}](Fig/64339F11-CeRh3-238-1-bdos.eps){width="8cm"}
![ Angle-resolved PES of CeRh$_{3}$ at $T=300$ K. For the definition of lines, see the caption of Fig. \[fig:CePd3-38-PES\]. The ${\mbox{\boldmath $k \:$}}$ point corresponding to the second solid line from the top is named ${\mbox{\boldmath $k \:$}}^{*}$ in the text. []{data-label="fig:CeRh3-kpes"}](Fig/64339F12-CeRh3-238-1-arpes-k.eps){width="8cm"}
![ Magnetic excitation spectra of CeRh$_{3}$ at $T=10^{3}$ K. The solid line is the total intensity and the dashed line is the result calculated using a fictitious model in which the matrix elements of the magnetic moment are restricted within the $j=5/2$ manifold of states. []{data-label="fig:CeRh3-mag"}](Fig/64339F13-CeRh3-238-1-mag.eps){width="8cm"}
[@cccc]{} &$\Gamma_{7}$ & $\Gamma_{8}$ & $j=7/2$\
$n_{\Gamma}^{({\rm imp.})}$ & 0.178 & 0.319 & 0.446\
$\VEP_{\Gamma}$(Ry) & -0.26923 &-0.26921 &-0.24426\
$\rho_{\Gamma}(E_{\rm F})$ (Ry$^{-1}$) & 0.27 & 1.40 & 0.99\
$\bar{Z}^{-1}_{\Gamma}$ & 2.2 & 1.9 & 1.9\
$\bar{\VEP}_{\Gamma}$(Ry) & 0.1218 &0.1216 & 0.1646\
$\bar{\Gamma}_{\Gamma}$(Ry) & $4.6 \times 10^{-3}$ & $12.6 \times 10^{-3}$ & $5.8 \times 10^{-3}$\
&\
In Fig. \[fig:CeRh3-flsp\], we show the $4f$ DOS of CeRh$_{3}$ at $10^{3}$ K. The $4f$ spectrum has a sharp and large peak at an energy about of 0.76 Ry, which is 0.066 Ry (0.9 eV) above the Fermi energy. This separation of energy from $E_{\rm F}$ is slightly smaller than that of the $4f$ band in the LDA, about 0.009 Ry (1.2 eV). The result seems to be consistent with those of RIPES experiments and their detailed analysis [@A24; @A30]. The PES has a relatively large peak at about 0.1 Ry (1.4 eV) below $E_{\rm F}$ and a peak at $E_{\rm F}$. The qualitative behavior of the present PES is similar to the result obtained by Harima in the LDA [@A54]. However, the binding energy of the peak at 0.1 Ry below $E_{\rm F}$ is slightly lower and the intensity is higher than that of the LDA. The peak at $E_{\rm F}$ is also slightly sharper. In the experiment, the sharp peak at $E_{\rm F}$ was observed, but the peak at 0.1 Ry below $E_{\rm F}$ has not been identified at present [@A31; @A32; @A33; @A34].
In Fig. \[fig:CeRh3-bdos\], we show the total and $4f$ DOS of CeRh$_{3}$ at $T=10^{3}$ K. The $4d$ band of Rh exhibits strong peaks of the DOS at about 0.4 and 0.6 Ry. The $4f$ component also has a small peak at about 0.6 Ry, as noted previously. The Fermi energy is located in the top region of the $4d$ band, and the hybridization intensity in this region is high. Dispersions of the RNB are almost identical to those of the LDA in the energy region very near $E_{\rm F}$, but the width of the $4f$ band, which is located at 0.76 R, is about 2/3 that of the LDA. A $f^{2}$ satellite peak appears in DMFT at about 1.2 Ry on the high-energy side. The ratio of the intensity of peak at 0.76 Ry ($f^{1}$ peak) to the total IPES intensity is estimated to be about 0.4 in the present calculation, while a slightly larger value, 0.6, was obtained in the experiment. Uozumi [*et al.*]{} predicted the $4f$ occupation number to be 0.86 [@A30], but we tentatively used $n_{f}({\rm rsl.target})=0.94$, which is 94% of the LDA value. The difference between these values is not small, but will not cause extreme differences in the calculation of the very strong hybridization limit. The mass enhancement factor is expected to be about 2, as given in Table \[tab:CeRh3\].
In Fig. \[fig:CeRh3-kpes\], we show the PES for the (111) surface when the representative wave vector moves from the $\Gamma$ point (bottom) to the M point (top). In the energy region between $E_{\rm F}$ and 0.6 Ry, the relative intensity of the $4f$ component is about 10% of the total. The spectra have fine peaks as if some flat bands exist slightly below $E_{\rm F}$. However, we note that no flat bands exist very near $E_{\rm F}$ in the RNB dispersion. Let us denote as ${\mbox{\boldmath $k \:$}}^{*}$ the representative ${\mbox{\boldmath $k \:$}}$ corresponding to the second solid line from the top (i.e., the ${\mbox{\boldmath $k \:$}}$ point on the $\Sigma$ axis at a distance of about 0.3 $\times$ length of the $\Sigma$ axis from the M point). The spectral intensity slightly below $E_{\rm F}$ is relatively large for ${\mbox{\boldmath $k \:$}}$ near ${\mbox{\boldmath $k \:$}}^{*}$. For the wave vectors near ${\mbox{\boldmath $k \:$}}^{*}$, several bands stay slightly below $E_{\rm F}$ around the M and X points when the normal components are varied. Calculated results show quantitatively similar behaviors to experimental results, but the careful separation of the surface and bulk components is necessary to enable a detailed comparison [@A55].
The magnetic excitation spectra are shown in Fig. \[fig:CeRh3-mag\]. They have a steep increase at the excitation energy of about 0.03 Ry. This energy may correspond to the energy from $E_{\rm F}$ to the low-energy edge of the peak at 0.76 Ry of the $4f$ DOS. The-low energy end of the spectrum mainly originates from the excitation within the $j=5/2$ components, but contribution of the $j=7/2$ components is not small even in the low-excitation-energy region. The $j=7/2$ components also join the Kondo effect. We may expect the Kondo temperature of this system to be about 0.03 Ry (4700 K). The calculated value of the magnetic susceptibility is $0.7 \times 10^{-4}$ emu/mol. The experimental magnetic susceptibility ($\chi \sim 4 \times 10^{-4}$ emu/mol) [@A56] may indicate a high $T_{\rm K}$ of several thousand K, but the calculated value of 4700 K seems to be too high. DOSs, both of the total and of the $4f$ component, do not have any gap, as seen in Fig. \[fig:CeRh3-flsp\], but the magnetic excitation spectrum has a shape indicating the existence of a pseudogap. We note that, in Fig. \[fig:CeRh3-mag\], the excitation spectra of only intra-$4f$ components are shown. A broad continuous $5d$ component will be superposed on these spectra.
The HI of CeRh$_{3}$ in the LDA calculation has a spectrum shape similar to that of the partial DOS on the $5d$ state of Rh, i.e., the spectral shape given by subtracting the $4f$ and $5d$ parts of the Ce ion from the total DOS in Fig. \[fig:CeRh3-bdos\]. The HI in DMFT is almost equal to that of the LDA in the high-energy region, but it has a steep dip at an energy slightly above $E_{\rm F}$. Similar behavior has been seen in the HI of the $j=7/2$ component of CePd$_{3}$ which has shown in Fig. \[fig:CePd3-38-mix\].
Summary and Discussion
======================
We have studied the electronic structures of CePd$_{3}$ and CeRh$_{3}$ on the basis of the DMFT calculation. The auxiliary impurity problem was solved by a method named NCA$f^{2}$vc, which includes the correct exchange process of the $f^{1} \rightarrow f^{0}$ and $f^{1} \rightarrow f^{2}$ virtual excitation. The splitting of the self-energy owing to the SOI and CFS effects was also considered.
The DMFT band calculation gives Fermi surface structures similar to those obtained by the LDA calculation in CePd$_{3}$ [@A53] at very low temperatures. Electron pockets appear at the bottom of the $4f$ band at the $\Gamma$ and R points. Hole pockets appear that are centered on the T symmetry axis.
The $4f$ band shifts to the high-energy side relative to the Fermi energy as the temperature increases. At the same time, the lifetime width of the $4f$ states increases. The band structures produced by the band overlap between the $4f$ and non-$4f$ components shift up to the high energy side of the Fermi energy. Therefore the primary structures of the band in the vicinity of the Fermi energy approaches to those of the $sp$ free-electron-like band of LaPd$_{3}$. However, some characteristic features of the LDA band of CePd$_{3}$ remain at higher temperatures. For example, a region that electrons occupy will appear on the $\Lambda$ axis in CePd$_{3}$ at $T \sim 150$ K, which is not expected in LaPd$_{3}$. In CePd$_{3}$, the lifetime broadening overcomes the ${\mbox{\boldmath $k \:$}}$ dependence of the $4f$ spectrum at room temperature, thus, the $4f$ state becomes a broad dispersionless state located above $E_{\rm F}$.
The ARPES of CePd$_{3}$ shows strong intensity near the representative $\Gamma$ point at low temperatures because the $4f$ band is located below the Fermi energy at the $\Gamma$ and R points. This result seems to be consistent with the recent experimental results [@A29]. The observed $4f$ component will be greatly reduced at room temperature because of the shift of the $4f$ band to the high-energy side.
The k-integrated magnetic excitation spectrum has a peak at 41 meV and a faint shoulder structure at about 27 meV at low temperatures. The temperature dependence of the excitation spectrum generally seems to be consistent with the results of experiments [@A19]. The magnetic CFS excitation energy is estimated to be about 41 meV, while the Kondo temperature is slightly smaller, 320 K (27 meV).
The calculated PES shows good correspondence with the bulk component obtained in recent high-resolution experiments [@A28]. The intensity ratio of the $f^{1}$ peak of the IPES to the total IPES is estimated to be about 0.2, while it was predicted to be 0.22 by the recent experiment analysis [@A25]. The HI is enhanced near the Fermi energy in the DMFT band compared with that of the LDA calculation. The present DMFT calculation seems to give electronic structures with slightly stronger HI than that expected from experiments for CePd$_{3}$.
The DMFT calculation for CeRh$_{3}$ gives an almost identical band structure to that obtained by the LDA calculation. However, the energy of the $f^{1}$ peak of the IPES in the former is slightly lower than that in the latter. In addition, the $f^{2}$ satellite peak appears in the DMFT band calculation. The calculated intensity ratio of the $f^{1}$ peak to the total IPES, 0.4, is comparable to, but smaller than the experimental value of 0.6. [@A30]. The calculated PES has a sharp peak at the Fermi energy. A peak reflecting the $4d$(Rh) DOS also appears, similarly to the result of the LDA calculation. The former has been observed, but the latter has not been identified in experiments [@A33; @A34].
The Fermi energy is located in the energy region of the $4d$ band of Rh in CeRh$_{3}$, therefore the HI near the Fermi energy is strong, about three times greater than that of CePd$_{3}$. The $4f$ band width of CeRh$_{3}$ is about twice that of CePd$_{3}$ in the LDA calculation. In the DMFT calculation, the characteristic energy scales are drastically different from each other, 300 and 5000 K.
In CeRh$_{3}$, the ARPES shows an appreciable DOS slightly density below the Fermi energy and is weakly dependent on the wave vector, although no flat $4f$ bands exist near the Fermi energy in the RNB dispersion. The spectra have stronger intensity halfway along the $\Sigma$ axis from the $\Gamma$ to M points of the Brillouin zone. This seems to be similar to the results of experiments, but further studies to separate the surface effects are necessary [@A55].
General features of the experimental results for CePd$_{3}$ and CeRh$_{3}$ are reproduced by the DMFT band calculation with the LMTO+NCA$f^{2}$vc scheme. However, the present DMFT calculation gives a higher Kondo temperature than that in the results of the detailed analysis of experiments. Moreover, there is some arbitrariness in the choice of the target value of the $4f$ occupation number, $n_{f}({\rm rsl.target})$. According to calculations for AuCu$_{3}$-type Ce compounds, accurate calculated results seem to be obtained when a $4f$ occupation number between 90% and 95% of the LDA value is used. We have used 94% in the present calculation, and we obtained 310 K (27 meV) for $T_{\rm K}$ of CePd$_{3}$ ($n_{f}({\rm rsl.target})=0.98$) and $ 4700$ K (0.03 Ry) for $T_{\rm K}$ of CeRh$_{3}$ ($n_{f}({\rm rsl.target})=0.94$) from the magnetic excitation spectrum. When we perform calculations using 90%, the $T_{\rm K}$ are $400$ K for CePd$_{3}$ ($n_{f}({\rm rsl.target})=0.94$) and $ 6300$ K for CeRh$_{3}$ ($n_{f}({\rm rsl.target})=0.90$). These values are not greatly different from the previous values because these compounds belong to a group of materials having high $T_ {\rm K} $. For materials with lower $T_{\rm K}$, $T_{\rm K}$ drastically depends on the choice of $n_{f}({\rm rsl.target})$. Careful treatment of the target value is necessary in such cases.
The effectiveness and some of the weaknesses (for example, the correct calculation of the occupied electron number) of the present DMFT scheme are recognized. Calculations of the $4f$ band state in CeIn$_{3}$ and CeSn$_{3}$, and also various Ce compounds will be carried out in the near future.
At the end of this paper, we refer the very early and recent application of methods similar to NCA$f^{2}$vc to the DMFT band calculation. Lægsgaard and Svane calculated the band structure of Ce pnictides in 1998 [@A57]. Recently, Haule [*et al.*]{} studied the $\alpha \rightarrow \gamma$ transition of Ce metal [@A58], and Shim [*et al.*]{} studied the electronic band structure of CeIrIn$_{5}$ [@A59]. The CFS of the self-energy was not considered in those studies.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank H. Shiba for encouragements, and Y. Kuramoto and J. Otsuki for important comments on the resolvent method, H. Harima for valuable comments on the band calculation method, and Y. Shimizu for valuable collaboration in the early stage of developing the LMTO+NCA$f^{2}$vc code. This work was partly supported by Grants-in-Aid for Scientific Research C (No. 21540372) from JSPS, and on Innovative Areas “Heavy Electrons”(No. 21102523) and for Specially Promoted Research (No. 18002008) from MEXT.
Occupation number
==================
The calculation based on the NCA$f^{2}$vc method usually does not satisfy the FL sum rule, i.e., the integral of the total DOS below the Fermi energy, $N({\rm total}; \rho)=\int{\rm d}\VEP\frac{1}{N}\sum_{{\mbox{\boldmath $k \:$}}}(-\frac{1}{\pi}\Im G(\VEP+{\rm i}\delta;{\mbox{\boldmath $k \:$}}))f(\VEP)$, is not equal to the value $N({\rm total};\ln G)=\int{\rm d}\VEP\frac{1}{N}\sum_{{\mbox{\boldmath $k \:$}}}(-\frac{1}{\pi}\Im\ln\det G(\VEP+{\rm i}\delta;{\mbox{\boldmath $k \:$}}))
(-\frac{\P f(\VEP)}{\P \VEP})$, where $f(\VEP)$ is the Fermi distribution function at $T=0$. In the RNB calculation, in which the self-energy term is approximated by $\Sigma_{\Gamma}(z)\approx\Re\Sigma_{\Gamma}(E_{\rm F})+\frac{\P\Re \Sigma_{\Gamma}(z)}{\P z}|_{E_{\rm F}}(z-E_{\rm F})$, we obtain real eigen energies $E_{\rm RNB}(\lambda{\mbox{\boldmath $k \:$}})$ where $\lambda$ is the band suffix. The number of occupied states in the RNB band, $N({\rm total};{\rm RNB})=\frac{1}{N}\sum_{{\mbox{\boldmath $k \:$}}}f(E_{\rm RNB}(\lambda{\mbox{\boldmath $k \:$}}))$, is expected to agree with $N({\rm total};\ln G)$ at $T=0$ if the imaginary part can be neglected near the Fermi energy. However, the imaginary component has a considerable magnitude in NCA$f^{2}$vc even at very low temperatures; this is partly because $T$ must maintain the condition $T \gtrsim 0.1T_{\rm K}$. For example, $N({\rm total};\rho)$, $N({\rm total};\ln G)$, and $N({\rm total};{\rm RNB})$ are 33.59, 34.16, and 34.01, respectively, at $T=18.75$ K in CePd$_{3}$. The difference between these values becomes serious when the detailed structure of the Fermi surface is discussed. In this study, we use $N({\rm total};{\rm RNB})$ to determine the Fermi energy $E_{\rm F}$ at $T=0$ because this quantity is directly related to the occupation number of electrons calculated from the volume of the occupied states in the ${\mbox{\boldmath $k \:$}}$ space.
Cauchy integral of using spline interpolation
==============================================
The Cauchy integral appears in various places in the DMFT calculation. Therefore, an efficient and accurate numerical calculation of the Cauchy integral is needed. We briefly explain a method of using the spline interpolation for the DOS. Let us assume that numerical data of DOS $\{y_{i}\}$ at energy points $\{x_{i}\}$ ($i=1, N_{\rm data number}$) are given. In the cubic spline interpolation [@A60], the DOS in the interval $[x_{i},x_{i+1}]$ is expressed as \_[3,i]{}(x) ={(x\_[i+1]{}-x)\^[3]{}M\_[i]{}+(x-x\_[i]{})\^[3]{}M\_[i+1]{}}\
+(y\_[i]{} -) +(y\_[i+1]{}-), where $h_{i}=x_{i+1}-x_{i}$. The quantity $M_{i}$ is the second derivative of the DOS at $x=x_{i}$, and is given by the usual procedure of the spline interpolation.
The integral of the interval $[x_{i},x_{i+1}]$ is calculated as \_[x\_[i]{}]{}\^[x\_[i+1]{}]{}x = (x\_[i+1]{}-z)\^[2]{} -(z-x\_[i]{})\^[2]{}\
+(x\_[i+1]{}-z) -(z-x\_[i]{})\
+(y\_[i]{} -) -(y\_[i+1]{}-)\
-\_[3,i]{}(z). \[B2\] The total integral is given by summing the contribution from each interval. Equation (\[B2\]) is expressed as a combination of powers of quantities $(x_{i+1}-z)$ and $(z-x_{i})$; therefore it is applicable for $z$ when $|z-\frac{x_{i}+x_{i+1}}{2}|$ is not much lager than $h_{i}$. When $z$ approaches the edge of the integration, i.e., $z \rightarrow x_{i}$ or $x_{i+1}$, the singularity of the logarithm term is removed by the counter contribution of the neighboring $[x_{i-1}, x_{i}]$ or $[x_{i+1},x_{i+2}]$ region. The round-off error due to the subtraction of logarithm terms of neighboring regions is not serious even when $z$ is extremely near the edge point, because the divergence of the logarithm is very weak. The integral (\[B2\]) is expected to be $O(z-\frac{x_{i}+x_{i+1}}{2})^{-1}$ when $|z-\frac{x_{i}+x_{i+1}}{2}| \gg h_{i}$. Therefore, mutual cancellation occurs among terms in (\[B2\]) in this limit. We find that the integral is re-expressed by a compact form in this case \_[x\_[i]{}]{}\^[x\_[i+1]{}]{}x\
= -\_[=1]{}\^\[()\^(-)\
-()\^(-)\]. This equation gives a highly accurate estimation of the integral even when terms are truncated up to $\nu \sim 10$.
In most cases, it is convenient to use the linear interpolation scheme with a very fine mesh for the DOS, \_[1,i]{}(x) =m\_[i]{}(x-)+, where $m_{i}=\frac{y_{i+1}-y_{i}}{h_{i}}$. The Cauchy integral is expressed by the following forms \_[x\_[i]{}]{}\^[x\_[i+1]{}]{}x = -m\_[i]{}h\_[i]{}-\_[1,i]{}(z), and for $|z-\frac{x_{i}+x_{i+1}}{2}| \gg h_{i}$, \_[x\_[i]{}]{}\^[x\_[i+1]{}]{}x = &-\_[=1]{}\^\[ ()\^(-)\
&- ()\^(+) \].
[9]{}
K. Held, I. A. Nekrasov, G. Keller, V. Eyert, N. Blümer, A. K. McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt: phys. Status Solidi B [**243**]{} (2006) 2599.
L. V. Pourovskii, B. Amadon, S. Biermann, and A. Georges: Phys. Rev. B [**76**]{} (2007) 235101.
A. C. Hewson: [*The Kondo Problems to Heavy Fermions*]{} (Cambridge University Press, Cambridge, 1993)
J. W. Allen: J. Phys. Soc. Jpn. [**74**]{} (2005) 34.
J. C. Parlebas and A. Kotani: J. Electron Spectrosc. Relat. Phenom. [**136**]{} (2004) 3.
O. Sakai, Y. Shimizu, and Y. Kaneta: J. Phys. Soc. Jpn. [**74**]{} (2005) 2517.
O. Sakai and Y. Shimizu: J. Phys. Soc. Jpn. [**76**]{} (2007) 044707.
O. Sakai, M. Motizuki, and T. Kasuya: [*Core-Level Spectroscopy in Condensed Systems Theory*]{}, ed. J. Kanamori and A. Kotani (Springer-Verlag, Berlin, 1988) p. 45.
J. Otsuki and Y. Kuramoto: J. Phys. Soc. Jpn. [**74**]{} (2006) 064707.
J. Kroha and P. Wölfle: J. Phys. Soc. Jpn. [**74**]{} (2005) 16.
For the resolvent method, see, for example, N. E. Bickers: Rev. Mod. Phys. [**59**]{} (1987) 845.
O. K. Andersen: Phys. Rev. B [**12**]{} (1975) 3060.
H. L. Skriver: [*The LMTO Method*]{} (Springer-Verlag, Berlin, 1984).
D. W. Lynch and J. H. Weaver: in [*Handbook on Physics and Chemistry of Rare Earths*]{}, ed. K. A. Gschneidner, Jr., L. Eyring, and S. H. Hüfner (Elsevier Science, Amsterdam, 1987) Vol. 10, p. 231.
F. U. Hillebrecht and M. Campagna: in [*Handbook on Physics and Chemistry of Rare Earths*]{}, ed. K. A. Gschneidner, Jr., L. Eyring, and S. H. Hüfner (Elsevier Science, Amsterdam, 1987) Vol. 10, p. 425.
Y. Ōnuki and A. Hasegawa: in [*Handbook on Physics and Chemistry of Rare Earths*]{}, ed. K. A. Gschneidner, Jr., and L. Eyring (Elsevier Science, Amsterdam, 1995) Vol. 20, p. 1.
M. R. Norman and D. D. Koelling: in [*Handbook on Physics and Chemistry of Rare Earths*]{}, ed. K. A. Gschneidner, Jr., L. Eyring, G. H. Lander, and G. R. Choppin (Elsevier Science, Amsterdam, 1993) Vol. 17, p. 1.
M. Loewenhaupt and K. H. Fisher: in [*Handbook on Physics and Chemistry of Rare Earths*]{}, ed. K. A. Gschneidner, Jr., and L. Eyring (Elsevier Science, Amsterdam, 1993) Vol. 16, p. 1.
E. Holland-Moritz and G. H. Lander: in [*Handbook on Physics and Chemistry of Rare Earths*]{}, ed. K. A. Gschneidner, Jr., L. Eyring, G. H. Lander, and G. R. Choppin (Elsevier Science, Amsterdam, 1994) Vol. 19, p. 1.
J. M. Lawrence, J. D. Thompson, and Y. Y. Chen: Phys. Rev. Lett. [**54**]{} (1985) 2537.
A. P. Murani, A. Servering, and W. G. Marshall: Phys. Rev. B [**53**]{} (1996) 2641.
S. M. Shapiro, C. Stassis, and G. Aeppli: Phys. Rev. Lett. [**62**]{} (1989) 94.
J. M. Lawrence, V. R. Fanelli, E. A. Goremychkin, R. Osborn, E. D. Bauer, K. J. McClellan, and A. D. Christianson: Physica B [**403**]{} (2008) 783.
M. Grioni, D. Malterre, P. Weibel, B. Dardel, and Y. Baer: Physica B [**186-188**]{} (1993) 38.
M. Grioni, P. Weibel, D. Malterre, Y. Baer, and L. Duò: Phys. Rev. B [**55**]{} (1997) 2056.
K. Kanai, Y. Tezuka, T. Terashima, Y. Muro, M. Ishikawa, T. Uozumi, A. Kotani, G. Schmerber, J. P. Kappler, J. C. Parlebas, and S. Shin: Phys. Rev. B [**60**]{} (1999) 5244.
K. Kanai, T. Terashima, A. Kotani, T. Uozumi, G. Schmerber, J. P. Kappler, J. C. Parlebas, and S. Shin: Phys. Rev. B [**63**]{} (2001) 033106.
R. D. Parks, S. Raaen, M. L. denBoer, and Y.-S. Chang: Phys. Rev. Lett. [**52**]{} (1984) 2176.
S. Ogawa, S. Suga, A. E. Bocquet, F. Iga, M. Kasaya, T. Kasuya, and A. Fujimori: J. Phys. Soc. Jpn. [**62**]{} (1993) 3575.
S. Kasai, S. Imada, A. Yamasaki, A. Sekiyama, F. Iga, M. Kasaya, and S. Suga: J. Electron Spectrosc. Relat. Phenom. [**156-158**]{} (2007) 441.
S. Danzenbächer, Yu. Kucherenko, M. Heber, D. V. Vyalikh, S. L. Molodtsov, V. D. Servedio, and C. Laubschat: Phys. Rev. B [**72**]{} (2005) 033104.
T. Uozumi, K. Kanai, S. Shin, A. Kotani, G. Schmerber, J. P. Kappler, and J. C. Parlebas: Phys. Rev. B [**65**]{} (2002) 045105.
E. Weschke, C. Laubschat, R. Ecker, A. Höhr, M. Domeke, and G. Kaindl: Phys. Rev. Lett. [**69**]{} (1992) 1792.
D. Malterre, M. Grioni, Y. Baer, L. Braicovich, L. Duò, P. Vavassori, and G. L. Olcese: Phys. Rev. Lett. [**73**]{} (1994) 2005.
R.-J. Jung, B.-H. Choi, S.-J. Oh, H.-D. Kim, E.-J. Cho, T. Iwasaki, A. Sekiyama, S. Imada, S. Suga, and J.-G. Park: Phys. Rev. Lett. [**91**]{} (2003) 157601.
Yu. Kucherenko, S. L. Molodtsov, and C. Laubschat: Phys. Rev. Lett. [**94**]{} (2005) 039709.
A. P. Murani: J. Phys C: Solid State Phys. [**33**]{} (1983) 6359.
I. Umehara, Y. Kurosawa, N. Nagai, M. Kikuchi, K. Satoh, and Y. Ōnuki: J. Phys. Soc. Jpn. [**59**]{} (1990) 2848.
N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. Haselwimmer, and G. G. Lonzarich: Nature [**394**]{} (1998) 39.
F. M. Grosche, I. R. Walker, S. R. Julian, N. D. Mathur, D. M. Freye, M. J. Steiner, and G. G. Lonzarich: J. Phys.: Condens. Matter [**13**]{} (2001) 2845.
R. Settai, H. Shishido, T. Kubo, A. Araki, T. C. Kobayashi, H. Harima, and Y. Ōnuki: J. Magn. Magn. Mater. [**310**]{} (2007) 541.
L. Severin and B. Johansson: Phys. Rev. B [**50**]{} (1994) 17886.
The summation over ${\mbox{\boldmath $k \:$}}$ is carried out on a mesh dividing the $\Gamma-X$ axis into 8 parts. Usually this gives a nearly smooth spectrum for $4f$ components, but gives fictitious fine structures for non-$4f$ components. The total DOS is calculated by smoothing with the introduction of the imaginary part $\gamma=0.01$ Ry in the energy variable.
O. Sakai, Y. Shimizu, and T. Kasuya: J. Phys. Soc, Jpn. [**58**]{} (1989) 3666.
In NCA$f^{2}$vc, the atomic energy levels are treated as independent input parameters. The energy level of the $f^{0}$ state is $E(f^{0}) =0$ and the energy level of the $f^{1}$ state is $E(f^{1},\Gamma)=\VEP_{\Gamma}$ when levels are measured from $E_{\rm F}$. The energy level of the $f^{2}$ state is assumed to be $E(f^{2}) = 2\VEP(f)+U$, where $\VEP(f)=\frac{1}{6}(2\VEP_{\Gamma_{7}}+4\VEP_{\Gamma_{8}})$ is the average of $f$ levels of the $j=5/2$ multiplet, and $U$ is the Coulomb constant.
The impurity $4f$ Green function is given using the $4f$ DOS $\rho^{\rm (imp.)}_{\Gamma}(x)$ as the Cauchy integral: $G_{\Gamma}^{\rm (imp.)}(z)=\int{\rm d}x\frac{\rho^{\rm (imp.)}_{\Gamma}(x)}{z-x}$. It is also expressed as $G_{\Gamma}^{\rm (imp.)}(z)=1/[z-\VEP^{\rm a}_{\Gamma}-\Sigma_{\Gamma}(z)-\Sigma_{\Gamma}^{\rm (h)}(z)]$, where $\VEP^{\rm a}_{\Gamma}$ is the impurity level, and $\Sigma_{\Gamma}(z)$ and $\Sigma^{\rm (h)}_{\Gamma}(z)$ are, respectively, the self-energies due to the Coulomb interaction and the hybridization [@A6]. The self-energy $\Sigma_{\Gamma}(z)$ is calculated as $\Sigma_{\Gamma}(z)=z-\VEP^{\rm a}_{\Gamma}-G^{\rm (imp.)}_{\Gamma}(z)^{-1}-\Sigma^{\rm (h)}_{\Gamma}(z)$ by solving the impurity model with the HI $-\Im\Sigma_{\Gamma}^{\rm (h)}(x+\I\delta)$. The self-consistent condition on the $4f$ DOS in DMFT is $\rho^{\rm (imp.)}_{\Gamma}(x)=\rho^{(\rm band)}_{\Gamma}(x)$. Under the self-consistent condition, the self-energy due to the hybridization is given by $\Sigma^{\rm (h)}_{\Gamma}(z)=z-\VEP^{\rm a}_{\Gamma}-G^{\rm (imp.)}_{\Gamma}(z)^{-1}-\Sigma_{\Gamma}(z)$.
We find that the maximum relative difference of the $4f$ DOS hardly decreases below 0.5, even when we advance the iteration process. The average of the relative differences of the Greenian on the imaginary axis is monitored as the convergence parameter, but this also hardly decreases below $0.3 \times 10^{-2}$. We usually stop the iteration process when this parameter reaches approximately $0.3 \times 10^{-2}$ and shows stationary behavior.
A. Furrer and H-G. Purwins: J. Phys. C [**9**]{} (1976) 1491.
U. Walter and E. Holland-Moritz: Z. Phys. B: Condens. Matter [**45**]{} (1981) 107.
When the CFS of the electrostatic term is not included, a large number of iteration loops are needed to reach a solution for large CFS because the calculation starts with a condition of equal energy levels: $\VEP^{\rm a}_{\Gamma 7}=\VEP^{\rm a}_{\Gamma 8}$.
When we perform the LDA calculation for a fictitious LaPd$_{3}$ with the lattice constant of CePd$_{3}$, the occupation number on the $4f$ state is estimated to be 0.188, which is larger than that of usual compounds, about 0.1. This large occupation is induced through the strong hybridization between the $4f$ state and the $4d$(Pd) state located in the energy region deep below $E_{\rm F}$. Some parts of the $4f$ occupation in LaPd$_{3}$ may be ascribed to the tail of the $4d$(Pd) extending to the Ce region.
Kasai [*et al.*]{} predicted the $4f$ occupation number to be 0.79 from their bulk PES analysis [@A28]. However, they analyzed the PES using the NCA calculation and obtained a small occupation number because the $f^{2}$ configuration was neglected. For NCA, see, for example, refs. and .
Y. Kuramoto: Z. Phys. B [**53**]{} (1983) 37.
Under the self-consistent condition, the self-energy due to the hybridization is given by $\Sigma^{\rm (h)}_{\Gamma}(z)=z-\VEP^{\rm a}_{\Gamma}-G^{\rm (imp.)}_{\Gamma}(z)^{-1}-\Sigma_{\Gamma}(z)$, where $\VEP^{\rm a}_{\Gamma}$ is the impurity level, $G^{\rm (imp.)}_{\Gamma}(z)$ is the impurity Green’s function and $\Sigma_{\Gamma}(z)$ is the self-energy due to the Coulomb interaction. The HI shown in Fig. \[fig:CePd3-38-mix\] is defined as $-\Im\Sigma_{\Gamma}^{\rm (h)}(x+\I\delta)$. This is the quantity corresponding to $\sum_{\nu{\mbox{\boldmath $k \:$}}}\pi|v_{\Gamma}(\nu{\mbox{\boldmath $k \:$}})|^{2}\delta(x-E_{\nu{\mbox{\boldmath $k \:$}}})$ in the usual impurity problem, where $v_{\Gamma}(\nu{\mbox{\boldmath $k \:$}})$ is the $c-f$ hybridization matrix and $E_{\nu{\mbox{\boldmath $k \:$}}}$ is the energy of the band state.
C. Koenig: Z. Phys. B: Condens. Matter [**50**]{} (1983) 33.
R. M. Galera, A. P. Murani, J. Pierre, and K. R. A. Ziebeck: J. Magn. Magn. Mater. [**63-64**]{} (1987) 594.
T. Gambke, B. Elchner, J. Schaafhausen, and H. Schaeffer: [*Valence Fluctuations in Solids*]{}, ed. L. M. Falicov, W. Hanke, and M. B. Maple (North-Holland, Amsterdam, 1981) p. 447.
A. Hasegawa and A. Yanase: J. Phys. Soc. Jpn. [**56**]{} (1987) 3990.
The $4f$ electron occupation number calculated using the RNB is 0.84 at $T=37.5$ K for CePd$_{3}$. This value is large compared with the value supposed from the impression of Fig. \[fig:CePd3-rnb\] in which only the bottoms of $4f$ bands are occupied. The $4f$ occupation number through the hybridization with conduction bands is considerable in the RNB picture of CePd$_{3}$. The LDA calculation also gives $4f$ bands where only their bottoms are occupied in the same way as in the RNB calculation, and leads to the $4f$ occupation number of 1.045.
The $4f$ DOS calculated by DMFT is shown in Fig. \[fig:CePd3-38-flsp\]. The spectrum intensity in the deep region from $E_{\rm F}$ (“the incoherent component” in the terminology of band theory and “the $f^{0}$ excitation part” in the terminology of the local Kondo problem) has a considerably high intensity. However, this deep energy component is disregarded in RNB, but its intensity is recovered as the full value of the intensity of the “coherent part” in the calculation of the occupied state. Only the band dispersion in the neighborhood of $E_{\rm F}$ and the count of the occupied ${\mbox{\boldmath $k \:$}}$ states at very low temperatures compared with the Kondo temperature have meaning in the RNB calculation.
Note that the temperature dependence of $N({\rm total};\rho)$ is not strong: it is 33.59 at $T=18.75$ K, and is 33.60 at $T=300$ K.
H. Harima: private communication. See Fig. 3 of ref. .
W. Schneider, S. L. Molodtsov, M. Richter, Th. Gantz, P. Englemann, and C. Laubschat: Phys. Rev. B [**57**]{} (1998) 14930.
J. Lægsgaad and A. Svane: Phys. Rev. B [**58**]{} (1998) 12817.
K. Haule, V. Oudovenko, S. Y. Savrasov, and G. Kotliar: Phys. Rev. Lett. [**94**]{} (2005) 036401.
J. H. Shim, K. Haule, and G. Kotliar: Science [**318**]{} (2007) 1618.
See for example, A. Ralston, and P. Rabinowitz: [*A First Course in Numerical Analysis*]{} (McGraw-Hill, New York, 1978) 2nd ed.
[^1]: E-mail address: sakai\_ym@star.ocn.ne.jp
|
---
abstract: |
We study the kinematics and cross section of the scattering process $e p \rightarrow e \Sigma^+$. The cross section is expressed in terms of complex form factors characterizing the hadron vertices. We estimate the cross section for small momentum transfer using known experimental information. To first order in the momentum transfer, we obtain a model independent result for the photon-exchange part of the cross section, which is completely determined by the decay width $\Gamma(\Sigma^+ \rightarrow p\gamma)$. For the kinematics of the parity violation experiment at MAMI, this first order result gives rise to a ratio of $(d\sigma/d\Omega)_{ep\rightarrow e\Sigma^+}$ $/(d\sigma/d\Omega)_{ep\rightarrow ep}
\simeq 4.0\times 10^{-15}$. The $Z^0$-exchange and interference parts give much smaller contributions due to the suppression of the flavor changing weak neutral current in the standard model. Feasibility of the experimental measurement is briefly discussed.
address: |
Center for Theoretical Physics\
Laboratory for Nuclear Science\
and Department of Physics\
Massachusetts Institute of Technology\
Cambridge, Massachusetts 02139\
author:
- 'Xuemin Jin [^1] and R. L. Jaffe [^2]'
date: 'MIT-CTP-2594, hep-th/9612316. [ ]{} December 1996'
title: 'WEAK HYPERON PRODUCTION IN $ep$ SCATTERING[^3]'
---
=10000 =10000 =10000 =5000 =10000 =100 =100
Submitted to: [*Physical Review D*]{}
Introduction
============
Currently, experimentalists at MAMI in Mainz are considering the possibility of measuring the scattering processes $e p\rightarrow e \Sigma^+$ or $e d \rightarrow e p \Lambda$ [@maas]. To our knowledge, there have been no theoretical investigations of such processes. It is thus important to explore the physics and the feasibility of the experimental measurement. Here we discuss the kinematics and cross section for $e p \rightarrow e \Sigma^+$. With minor alterations our analysis applies as well to $e n\rightarrow e \Lambda$.
In lowest order the scattering process $e p \rightarrow e \Sigma^+$ proceeds via the exchange of one photon or one $Z^0$ boson (see Fig. \[fig-1\]). Thus, the cross section consists of a pure $\gamma$, pure $Z^0$ and interference parts. The last two depend on the physics at the $pZ^0\Sigma^+$ vertex, a classic flavor changing weak neutral current, which is severely suppressed in the standard model. On the other hand, the photon-exchange part, $p \gamma \Sigma^+$, includes all gauge interactions of standard model: strong, weak, and electromagnetic. The same vertex appears in the weak radiative decay $\Sigma^+\rightarrow p \gamma$, which has been well measured experimentally [@pdt]. Despite substantial theoretical effort [@comment], hyperon weak radiative decays remain poorly understood. Measurement of $e p \rightarrow e \Sigma^+$ might provide more information about the vertex $p \gamma \Sigma^+$ and hence constrain theoretical models.
We express the cross section in terms of various form factors, which reflect the structure of the hadron vertices. We then estimate the cross section at small four-momentum transfer using known experimental information on the weak radiative decay, $\Sigma^+ \rightarrow p \gamma$, and the flavor changing weak neutral current. To first order in the momentum transfer, we obtain a model independent result for the photon-exchange part, which is completely determined by the weak radiative decay width $\Gamma(\Sigma^+\rightarrow p\gamma)$. For the kinematics of the parity violation experiment at MAMI ($\theta \simeq 35^0$ and $q^2\simeq -0.237$ GeV$^2$ for the elastic scattering $e p \rightarrow e p$, with $\theta$ the scattering angle and $q^2$ the squared four-momentum transfer) [@maas96], this first order result leads to a suppression factor of $4\times 10^{-15}$ relative to the $ep$ elastic scattering. The physical reasons for the suppression are the factor of $G_F^2$ from the weak hamiltonian [*and*]{} a factor of $q^2$ from electromagnetic gauge invariance that suppresses the cross section at small momentum transfer. Using the experimental result for the branching ratio of $K^0_L\rightarrow \mu^+\mu^-$, we estimate that the $Z^0$-exchange and interference parts give negligible contributions. We shall discuss briefly the feasibility of experimental measurements at available facilities.
This paper is organized as follows: In Sec. \[II\], we discuss the kinematics and derive the cross section. Sec. \[III\] gives an estimate of the cross section. Sec. \[IV\] is devoted to summary and conclusion.
Kinematics and cross section {#II}
============================
The kinematics for the process $e p\rightarrow e \Sigma^+$ is illustrated in Fig. \[fig-1\]. The four momentum of the initial and final states are denoted by $k = \{E,{\bf k}\}$ for the initial electron, $P = \{E_p,{\bf p}\}$ for the target (proton) ($P = \{M_p,{\bf 0}\}$ in the target rest frame), $k^\prime = \{E^\prime, {\bf k^\prime}\}$ for the outgoing electron, and $P^\prime = \{E_p^\prime, {\bf p^\prime}\}$ for the outgoing hyperon ($\Sigma^+$). One can then define the two invariants $$\begin{aligned}
q^2& \equiv & (k - k^\prime)^2 = (P^\prime - P)^2
= -4 E E^\prime \sin^2(\theta/2) = - Q^2 < 0\ ,
\label{invar-q}
\\*[7.2pt]
\nu& \equiv & P\cdot q = M_p (E - E^\prime)\ ,
\label{invar-pq}\end{aligned}$$ where the electron mass has been neglected (and will be henceforth). The scattering angle $\theta$ has been indicated in Fig. 1. Unless otherwise noted, the target rest frame will be adopted.
=7.0truecm
Since the final $\Sigma^+$ state is on-shell, one finds $$\begin{aligned}
& & 2 \nu + q^2 = M_{\Sigma}^2 - M_p^2\ ,
\label{nu-q}
\\*[7.2pt]
& & E^\prime = {M_p E - {1\over 2} (M_\Sigma^2 - M_p^2)
\over M_p + 2 E \sin^2(\theta/2)}\ .
\label{eprime}\end{aligned}$$ Note that there are only two independent variables, $E$ and $\theta$, and all the other quantities can be expressed in terms of them. The kinematic domain is thus given by the two conditions $q^2 < 0$ $(Q^2 > 0)$ and $2\nu = M_\Sigma^2 - M_p^2 - q^2$. Since $E^\prime > 0$, one has $$E > {1\over 2 M_p} \left(M_\Sigma^2 - M_p^2\right)\ ,$$ which implies a minimum initial electron energy of $E_{\rm min} \simeq 285$ MeV.
In the discussions to follow, we shall consider the scattering process with unpolarized beam and target and with final spins unobserved. The extension to other situations is straightforward. The differential cross section in this case can be written as: $$d \sigma = {e^4\over 4 E Q^4}
\int {d^3 k^\prime\over (2\pi)^3 E^\prime}
{d^3 p^\prime \over (2\pi)^3}{M_\Sigma\over E_p^\prime}
(2\pi)^4 \delta^4 (P^\prime + k^\prime - P - k)
\sum_{i=\gamma,\gamma Z,Z} \eta^i l^{\mu\nu}_i W^i_{\mu\nu}\ ,
\label{dsigma}$$ where $$\eta^\gamma = 1\ ,
\hspace*{1cm}
\eta^{\gamma Z} = {1\over \sin^2 2\theta_{\rm w}}\,
{Q^2\over Q^2+M_Z^2}\ ,
\hspace*{1cm}
\eta^Z = \left( \eta^{\gamma Z}\right)^2\, ,
\label{coupling-f}$$ with $\theta_{\rm w}$ the weak angle and $M_Z$ the $Z^0$ mass. Here we use the normalization conventions of Itzykson and Zuber [@book]. The leptonic tensor $l^{\mu\nu}_i$ is simply given by $$\begin{aligned}
l^{\mu\nu}_\gamma &=& k^\mu k^{\prime\nu} + k^\nu k^{\prime\mu} -
k\cdot k^\prime g^{\mu\nu}\ ,
\\*[7.2pt]
l^{\mu\nu}_{\gamma Z} &=& 2 g^e_{\rm v}
\left (k^\mu k^{\prime\nu} + k^\nu k^{\prime\mu} -
k\cdot k^\prime g^{\mu\nu} \right)
-2 g^e_{\rm a} i \epsilon^{\mu\nu\rho\sigma}k^\prime_\rho k_\sigma\ ,
\\*[7.2pt]
l^{\mu\nu}_Z &=& \left[ (g^e_{\rm v})^2 +(g^e_{\rm a})^2\right]
\left (k^\mu k^{\prime\nu} + k^\nu k^{\prime\mu} -
k\cdot k^\prime g^{\mu\nu} \right)
-2 g^e_{\rm v} g^e_{\rm a} i \epsilon^{\mu\nu\rho\sigma}k^\prime_\rho k_\sigma\ ,
\label{lmn}\end{aligned}$$ where $$g^e_{\rm v} = -{1\over 2} + 2 \sin^2\theta_{\rm w}\ , \hspace*{1cm}
g^e_{\rm a} = -{1\over 2}\ .
\label{lepton-coupling}$$ On the other hand, the hadronic tensor $W_{\mu\nu}^i$ describes the complicated structure of the vertices $p \gamma \Sigma^+$ and $p Z^0\Sigma^+$, $$\begin{aligned}
W_{\mu\nu}^\gamma &=&\sum_{\rm spins}
\langle \Sigma^+| J_\mu^\gamma |p\rangle
\langle \Sigma^+| J_\nu^\gamma |p\rangle^*\ ,
\\*[7.2pt]
W_{\mu\nu}^{\gamma Z} &=& \sum_{\rm spins}\left[
\langle \Sigma^+| J_\mu^\gamma |p\rangle
\langle \Sigma^+| J_\nu^Z |p\rangle^*
+\langle \Sigma^+| J_\mu^Z |p\rangle
\langle \Sigma^+| J_\nu^\gamma |p\rangle^*\right]\ ,
\\*[7.2pt]
W_{\mu\nu}^Z &=& \sum_{\rm spins}
\langle \Sigma^+| J_\mu^Z |p\rangle
\langle \Sigma^+| J_\nu^Z |p\rangle^*\ ,
\label{w-def}\end{aligned}$$ where $J_\mu^\gamma$ and $J_\mu^Z$ denote the electromagnetic and weak neutral currents, respectively. The matrix element $\langle \Sigma^+| J_\mu^j |p\rangle$ $\{j = \gamma, Z\}$, upon the use of Lorentz covariance, takes the following form: $$\begin{aligned}
\langle \Sigma^+ |J_\mu^j |p\rangle &\equiv&
\overline{U}(P^\prime)
\Bigl\{\gamma_\mu F_1^j(q^2)
+ q_\mu B^j(q^2)+{i \sigma_{\mu\nu} q^\nu
\over M_p + M_\Sigma} F_2^j (q^2)
\nonumber
\\*[7.2pt]
& & + \gamma_5\gamma_\mu F_3^j(q^2) + \gamma_5 q_\mu B_5^j(q^2)
+ {i\gamma_5\sigma_{\mu\nu}q^\nu\over M_p - M_\Sigma} F_4^j(q^2)
\Bigr\} U(P)\ ,
\label{vertex-gen}\end{aligned}$$ where $U(P^\prime)$ and $U(P)$ are the Dirac spinors of $\Sigma^+$ and proton, respectively. The form factors are in general complex. While $F_1^j, F_2^j$, and $B^j$ are parity-conserving, $F_3^j, F_4^j$, and $B^j_5$ are parity-violating.
From electromagnetic current conservation, one obtains the following relations: $$(M_p - M_\Sigma)\, F_1^\gamma = q^2 B^\gamma\ ,
\hspace*{1.5cm} (M_p + M_\Sigma)\, F_3^\gamma = q^2 B_5^\gamma\ .
\label{curr-cons}$$ Since there is no zero-mass particle involved except the photon and the photon propagator has been included explicitly, Eq. (\[curr-cons\]) implies that $$F_1^\gamma(0) = 0, \hspace*{2cm}F_3^\gamma(0) = 0\ .
\label{cons-fs}$$ Note that the condition $F_1^\gamma(0) = 0$ is different from the usual $F_1^\gamma(0) = 1$ (= electric charge of the proton) seen in the proton electromagnetic form factors. This difference arises from gauge invariance. While the proton couples minimally to the photon field $A_\mu$, the $p\gamma \Sigma^+$ vertex must be proportional to $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. Thus, there are only four independent form factors at the vertex $p \gamma \Sigma^+$, which we choose as $F_1^\gamma, F_2^\gamma, F_3^\gamma$, and $F_4^\gamma$. Since $J^Z_\mu$ is not conserved, the six form factors describing the vertex $p Z^0\Sigma^+$ are in general independent.
Lorentz covariance and electromagnetic current conservation allows one to separate $W_{\mu\nu}^\gamma$ into three distinct structures: $$\begin{aligned}
W_{\mu\nu}^\gamma
& \equiv & \left( -g_{\mu\nu}
+ {q_\mu q_\nu\over q^2}\right) {M_p\over M_\Sigma} W_1^\gamma(q^2)
+\left[\left( P_\mu - {P\cdot q\over q^2} q_\mu\right)
\left(P_\nu - {P\cdot q\over q^2}q_\nu\right)\right]
{W_2^\gamma(q^2)\over M_p M_\Sigma}
\nonumber
\\*[7.2pt]
& & -i \epsilon_{\mu\nu\rho\sigma} P^\rho q^\sigma
{W_3^\gamma(q^2)\over 2 M_p M_\Sigma}\ .
\label{w-g-decop}\end{aligned}$$ Here the introduction of the factors $M_p/ M_\Sigma$, $1/M_p M_\Sigma$, and $1/2 M_p M_\Sigma$ makes the $W^\gamma$’s dimensionless. On the other hand, $W_{\mu\nu}^i$ for $\{i=\gamma Z, Z\}$ can be decomposed into six structures: $$\begin{aligned}
W_{\mu\nu}^i
& \equiv & -g_{\mu\nu} {M_p\over M_\Sigma} W^i_1(q^2)
+P_\mu P_\nu {W_2^i(q^2)\over M_p M_\Sigma}
-i \epsilon_{\mu\nu\rho\sigma} P^\rho q^\sigma
{W_3^i(q^2)\over 2 M_p M_\Sigma}
+{\cal O}(q_\mu \, \mbox{or} \, q_\nu)\ .
\label{w-z-decop}\end{aligned}$$ Here terms proportional to $q_\mu$ or $q_\nu$ vanish upon contraction with $l_i^{\mu\nu}$. Carrying out the integral in Eq. (\[dsigma\]), we obtain the following expressions for the three parts of the cross section: $$\begin{aligned}
\left({d\sigma \over d\Omega}\right)_\gamma
&=&
\left({d\sigma \over d\Omega}\right)_0 \eta^\gamma
\left[ W_2^\gamma (q^2) + 2 \tan^2\left(\theta/ 2\right) W_1^\gamma (q^2)
\right]\ ,
\label{sig-gamma}
\\*[7.2pt]
\left({d\sigma \over d\Omega}\right)_{\gamma Z}
&=&
\left({d\sigma \over d\Omega}\right)_0 \eta^{\gamma Z}\,
\Bigl\{ 2g^e_{\rm v}\left[ W_2^{\gamma Z} (q^2)
+ 2 \tan^2\left(\theta/ 2\right) W_1^{\gamma Z} (q^2)
\right]
\nonumber
\\*[7.2pt]
& & \hspace*{3.5cm}
-2 g^e_{\rm a} {E+E^\prime\over M_p}\tan^2 \left(\theta/ 2\right)
W_3^{\gamma Z} (q^2)\Bigr\}\ ,
\label{sig-gammaz}
\\*[7.2pt]
\left({d\sigma \over d\Omega}\right)_Z
&=&
\left({d\sigma \over d\Omega}\right)_0 \eta^Z\,
\Bigl\{ \left[(g^e_{\rm v})^2+(g^e_{\rm a})^2\right] \left[ W_2^Z (q^2)
+ 2 \tan^2\left(\theta/ 2\right) W_1^Z (q^2)
\right]
\nonumber
\\*[7.2pt]
& & \hspace*{5cm}
-2 g^e_{\rm v} g^e_{\rm a}
{E+E^\prime\over M_p}\tan^2 \left(\theta/2\right)
W_3^Z (q^2)\Bigr\}\ ,
\label{sig-z}\end{aligned}$$ where $$\left({d\sigma \over d\Omega}\right)_0
= {\alpha_E^2
\over 4 E^2 \sin^4(\theta/2)}
\left\{{\cos^2\left(\theta/ 2\right)\over
2 [1+2 (E/M_p)\sin^2(\theta/2)]}\right\}\ ,
\label{sig-0}$$ with $\alpha_E$ the electromagnetic fine structure constant. Therefore, the cross section is determined by various $W^j_k$, which, in terms of the form factors, can be expressed as: $$\begin{aligned}
W_1^j & = & {1\over 2M_p^2}\left\{ |F_1^j+F_2^j|^2
\left[Q^2 + \left(M_p - M_\Sigma\right)^2\right]
+|F_3^j+F_4^j|^2\left[Q^2 + \left(M_p + M_\Sigma\right)^2\right]
\right\}\ ,
\label{w1-gamma}
\\*[7.2pt]
W_2^j & = & 2 \left\{ |F_1^j|^2 +|F_3^j|^2 +Q^2
\left[{|F_2^j|^2\over (M_p + M_\Sigma)^2}
+ {|F_4^j|^2 \over (M_p - M_\Sigma)^2}\right]
\right\}\ ,
\label{w2-gamma}
\\*[14.4pt]
W_3^j & = & -4\, {\rm Re}\left[(F_1^j+F_2^j) (F_3^j+F_4^j)^*\right]\ ,
\label{w3-gamma}\end{aligned}$$ for $\{j = \gamma, Z\}$, where ${\rm Re}$ denotes the real part. The expressions for $W_k^{\gamma Z}$, which contain the product of $F_k^\gamma$ and $F_k^Z$, have not been listed.
Estimate of the cross section {#III}
=============================
The hadron vertices $p \gamma \Sigma^+$ and $pZ^0\Sigma^+$ are very complicated, containing the interplay among the strong, weak, and (for $p\gamma\Sigma^+$) electromagnetic interactions. Obviously, it is difficult to calculate the form factors and cross section directly from the standard model. At this stage, the best one could do is to calculate the form factors in models. For the experiment conditions of interest we find it possible to calculate the cross section without invoking any explicit effective model, by making use of experimental information on the weak radiative decay $\Sigma^+ \rightarrow p \gamma$ and the flavor changing weak neutral current.
Let us first consider the photon-exchange part. It is easy to show that in our notation, the weak radiative decay width of the $\Sigma^+$ can be written as $$\Gamma (\Sigma^+ \rightarrow p \gamma)
= {e^2\over \pi} \left({M_\Sigma^2 - M_p^2\over 2 M_\Sigma}\right)^3
\left[ {|F_2^\gamma(0)|^2\over (M_p + M_\Sigma)^2} +
{|F_4^\gamma(0)|^2\over (M_p - M_\Sigma)^2 }\right]\ ,
\label{w-decay}$$ and the asymmetry parameter as $$\alpha_\gamma = 2 \left( {M_p + M_\Sigma \over M_p - M_\Sigma} \right)
\left\{ {{\rm Re} \left[F_2^\gamma(0) F_4^{^\gamma *} (0)\right]\over
|F_2^\gamma(0)|^2 +
\left[(M_p + M_\Sigma)/(M_p - M_\Sigma)\right]^2 |F_4^\gamma(0)|^2}\right\}\ .
\label{asy-par}$$ Experimentally, the branching ratio of $\Sigma^+ \rightarrow p \gamma$ is $(1.25\pm 0.07) \times 10^{-3}$, and $\alpha_\gamma = -0.76\pm 0.08$ [@pdt]. Note that $\tan^2(\theta/2)\sim q^2$ \[Eq. (\[invar-q\])\], and $F_1^\gamma(q^2)\sim q^2$, $F_3^\gamma(q^2)\sim q^2$ \[Eq. (\[cons-fs\])\] (at most) for small $q^2$. To first order in $q^2$, we have $$\begin{aligned}
W_2^\gamma(q^2) &+& 2\tan^2\left({\theta\over 2}\right) W_1^\gamma(q^2)
= -2 q^2 \left[ {|F_2^\gamma(0)|^2\over (M_p + M_\Sigma)^2}
+ {|F_4^\gamma(0)|^2\over (M_p - M_\Sigma)^2} \right]
\nonumber
\\*[7.2pt]
& +& 2\tan^2\left({\theta\over 2}\right)
\left[{(M_p^2 - M_\Sigma^2)^2\over 2M_p^2}\right]
\left[{|F_2^\gamma(0)|^2\over (M_p + M_\Sigma)^2}
+ {|F_4^\gamma(0)|^2\over (M_p - M_\Sigma)^2}\right]
+ {\cal O}[(q^2)^2]
\nonumber
\\*[7.2pt]
& =& -(2 q^2)\times 3.11\times 10^{-14} ({\rm GeV})^{-2}
+ 2\tan^2\left({\theta\over 2}\right)\times 5.05\times 10^{-15}
+ {\cal O}[(q^2)^2]\ .
\label{cross-expan}\end{aligned}$$ Here in the last step, we have used the experimental values for $\Gamma(\Sigma^+ \rightarrow p\gamma)$ and the lifetime of $\Sigma^+$. Therefore, the above first order result is model independent. To be concrete, we consider the kinematics of the parity violation experiment at MAMI [@maas96], $\theta = 35^0$ and $q^2 \simeq -0.237$ GeV$^2$ for the elastic scattering, which implies $q^2 \simeq -0.16$ GeV$^2$ for the process under consideration. Using the above estimate, we arrive at the following result: $${\left({\textstyle d\sigma\over \textstyle d\Omega}\right)_\gamma
\left(e p\rightarrow e \Sigma^+\right)
\over {\textstyle d\sigma\over \textstyle d\Omega}
\left(e p \rightarrow e p\right)}
\simeq 4\times 10^{-15}\ .
\label{result}$$ This indicates a severe suppression relative to elastic scattering. This suppression arises from the weak interaction, gauge invariance, and kinematics.
We observe from Eqs. (\[w1-gamma\]) and (\[w2-gamma\]) that $W_1^\gamma$ and $W_2^\gamma$ may increase as $Q^2$ gets larger, implying, perhaps, a larger value for the ratio of $(d\sigma/d\Omega)_{ep\rightarrow
e\Sigma^+}/(d\sigma/d\Omega)_{ep\rightarrow ep}$. On the other hand, as $Q^2$ goes to infinity, one expects all the form factors go to zero. Thus, there may be a chance that the ratio has a maximum at a non-zero momentum transfer, which, however, is unlikely to alter the result of Eq. (\[result\]) qualitatively.
The $Z^0$-exchange part and the interference part involve the flavor changing weak neutral current, which is suppressed (at the tree level) in the standard model. At the quark level, the relevant vertex is $s Z^0 d$. For the purposes at hand, we can parameterize this vertex in terms of a vector coupling $g^Z_{\rm V}$ and an axial vector coupling $g^Z_{\rm A}$. Note that the same vertex is also responsible for the decay $K^0_L\rightarrow \mu^+\mu^-$. Although the typical momentum scales in the two processes ($ep\rightarrow e\Sigma^+$ and $K_L\rightarrow \mu^+\mu^-$) differ by multiples of some typical hadronic scale, we will not make a significant error by treating $g^Z_{\rm V}$ and $g^Z_{\rm A}$ as constants. Since for electron and muon, $g^{e,\mu}_{\rm v}\sim 0$, only $g^Z_{\rm A}$ enters.
Thus we can make use of the experimental information on the decay $K^0_L\rightarrow \mu^+\mu^-$ to estimate $g^Z_{\rm A}$. Since the decay width of $K^0_L\rightarrow \mu^+\mu^-$ agrees with the standard model estimate (via $K^0_L\rightarrow \gamma\gamma \rightarrow \mu^+\mu^-$), we can safely assume $g^Z_{\rm A}$ contribution to $K^0_L$ decay to be below the limit of the experimental errors on the $K^0_L\rightarrow \mu^+\mu^-$ branching ratio. This gives: $$|g^Z_{\rm A}| \leq {1\over 2}\sin\theta_{\rm c} \cos\theta_{\rm w}
\left[{\Gamma (K^0_L\rightarrow \mu^+\mu^-) \over
\Gamma (K^+\rightarrow \mu^+\nu_\mu)}\right]^{1/2}
\simeq 2\times 10^{-6}\ ,
\label{fnc3-est}$$ where $\theta_{\rm c}$ is the Cabibbo angle. Here we have neglected phase space difference, which is expected to be small.
With the neglect of QCD binding effects, we can apply the above estimate directly to the vertex $p Z^0\Sigma^+$, where only the axial vector coupling contributes at $Q^2\sim 0$. We then obtain $${\left({\textstyle d\sigma\over \textstyle d\Omega}\right)_Z
\left(e p\rightarrow e \Sigma^+\right)
\over {\textstyle d\sigma\over \textstyle d\Omega}
\left(e p \rightarrow e p\right)}
\leq 7\times 10^{-22}\ .
\label{est-z}$$ Therefore, the cross section for the process $e p \rightarrow e\Sigma^+$ is dominated by the photon exchange part. The $Z^0$-exchange and the interference give negligible contributions because of the suppression of the flavor changing weak neutral current.
Summary and conclusion {#IV}
======================
In this paper, we have studied the kinematics and derived the cross section of the scattering process $e p \rightarrow e \Sigma^+$. The cross section can be expressed in terms of various form factors which are introduced to parameterize the physics at the vertices $p \gamma \Sigma^+$ and $pZ^0\Sigma^+$. We have estimated the cross section at small momentum transfer invoking known experimental information. In particular, we obtain a model independent first order (in momentum transfer) result for the photon-exchange part of the cross section, which is completely determined by the weak radiative decay width $\Gamma(\Sigma^+\rightarrow p\gamma)$. With this result and the kinematics of the parity violation experiment at MAMI, we found a factor of $4\times 10^{-15}$ suppression relative to the elastic process $e p\rightarrow e p$. This suppression largely results from the weak interaction involved and the gauge invariance at the vertex $p\gamma\Sigma^+$. The $Z^0$-exchange part and the interference part both depend on the flavor changing weak neutral current. Using the experimental result for the branching ratio of $K^0_L\rightarrow \mu^+\mu^-$, we estimate that these two parts are much smaller than the photon-exchange contribution.
In principle, the weak hyperon production process $e p\rightarrow e \Sigma^+$ discussed here can be measured experimentally in facilities with low energy electron beams. For low momentum transfer, MAMI, MIT-Bates, and TJNAF are the best candidates. However, our estimate shows that the cross section is severely suppressed relative to the $ep$ elastic scattering. To be specific, consider the parity violation experiment proposed at MAMI. There, about $10^{14}$ elastic events can be accumulated (in 700 hours) [@maas96], making a meaningful measurement unlikely at this time.
We are indebted to Frank Maas for suggesting the scattering process addressed here to us and providing useful information on the parity violation experiment at MAMI. We would like to thank Lisa Randall for useful conversations.
F.E. Maas, private communication.
L. Montanet [*et al.*]{}, Phys. Rev. D [**50**]{}, 1173 (1994).
There have been many theoretical works discussing hyperon radiative decays. For a recent review and references, see B. Bassalleck in [*Proceedings of the International Conference On Hypernuclear and Strange Particle Physics*]{}, \[Nucl. Phys. [**A585**]{}, 255c (1995)\].
F.E. Maas [*et al.*]{}, “The New Parity Violation Experiment at MAMI”, Proceedings of the Erice Summer school on the spin structure of the nucleon, Erice, Sicili, August 10-15, 1996.
C. Itzykson and J. Zuber, “[*Quantum Field Theory*]{}” (McGraw-Hill, New York, 1980).
[^1]: Email address: [jin@ctp02.mit.edu]{}
[^2]: Email address: [jaffe@mitlns.mit.edu]{}
[^3]: This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement \#DF-FC02-94ER40818.
|
---
abstract: '$T$- and $S$-duality rules among the gauge potentials in type II supergravities are studied. In particular, by following the approach of arXiv:1909.01335, we determine the $T$- and $S$-duality rules for certain mixed-symmetry potentials, which couple to supersymmetric branes with tension $T\propto g_s^{-n}$ ($n\leq 4$). Although the $T$-duality rules are rather intricate, we find a certain redefinition of potentials which considerably simplifies the duality rules. After the redefinition, potentials are identified with components of the $T$-duality-covariant potentials, which have been predicted by the $E_{11}$ conjecture. We also discuss the field strengths of the mixed-symmetry potentials.'
---
**Duality rules for more mixed-symmetry potentials**
Yuho Sakatani
[*Department of Physics, Kyoto Prefectural University of Medicine,*]{}\
[*Kyoto 606-0823, Japan*]{}\
[`yuho@koto.kpu-m.ac.jp`]{}
Introduction
============
Toroidally compactified 11D supergravity or type II supergravity has the $U$-duality symmetry but this is not manifest in the standard formulation. In order to exhibit the symmetry, the standard metric, scalar fields, and $p$-form gauge potentials are not enough [@hep-th/0104081]. In fact, we additionally need to introduce certain mixed-symmetry potentials, which are related to the standard potentials through a non-local relation, similar to the electric-magnetic duality. According to the $E_{11}$ conjecture [@hep-th/0104081; @hep-th/0307098], there are infinitely many mixed-symmetry potentials in each theory. By introducing an integer-valued parameter $\ell$, known as the level, the number of mixed-symmetry potentials with a fixed level $\ell$ is finite, and we can determine the full list of the mixed-symmetry potentials for each level $\ell$ (see [@0705.0752; @1907.07177] and references therein).
Although a list of mixed-symmetry potentials which constitutes the $U$-duality multiplets has been algebraically determined, their physical definitions are still obscure. In the case of the standard supergravity fields, their definitions can be fixed by the supergravity action, but the mixed-symmetry potentials do not appear in the standard supergravity action and it is not straightforward to specify their definitions. A possible way to specify their definitions is to construct the worldvolume actions for supersymmetric branes. As is well known, the Ramond–Ramond (R–R) fields couple to D-branes, and one can identify the definition of the R–R fields by looking at the Wess–Zumino (WZ) term. Similarly, mixed-symmetry potentials generally couple to certain exotic branes [@hep-th/9707217; @hep-th/9712047; @hep-th/9712075; @hep-th/9712084; @hep-th/9809039; @hep-th/9908094; @hep-th/0012051; @0805.4451; @1004.2521; @1209.6056] and their definitions can be fixed by constructing the WZ term for exotic branes. For example, the WZ term of the Kaluza–Klein (KK) monopole has been constructed in [@hep-th/9802199; @hep-th/9806169; @hep-th/9812188] and a precise definition of the dual graviton has been given. However, at present, worldvolume actions have been constructed only for a few exotic branes.
To make precise definitions of mixed-symmetry potentials, it is more straightforward to determine the $T$- and $S$-duality transformation rule. The mixed-symmetry potentials which couple to supersymmetric branes are related to the standard $p$-form potentials under $T$-/$S$-duality transformations. Then, by determining the duality rule, we can fix the convention for the mixed-symmetry potentials. Recently, a systematic approach to determine the duality rules has been proposed in [@1909.01335], and the $T$-/$S$-duality rules for the dual graviton have been determined. In this paper, we continue the analysis and obtain the $T$-/$S$-duality rules for more mixed-symmetry potentials. Concretely, we consider the duality web described in Figure \[fig:duality-web\]. There, each line (with a circled alphabet appended) corresponds to a $T$-duality that connects a type IIA brane and a type IIB brane. For example, the $T$-duality connects the $5^2_2$-brane in type IIA theory and the $5^1_2$-brane in type IIB theory. Since the $5^2_2$-brane and the $5^1_2$-brane minimally couple to the potentials ${{\mathscr{A}}}_{8,2}$ and ${{\mathsf{A}}}_{7,1}$, respectively, $T$-duality corresponds to a $T$-duality rule for ${{\mathscr{A}}}_{a_1\cdots a_7 {{y}}, a{{y}}} \leftrightarrow {{\mathsf{A}}}_{a_1\cdots a_7, a}$, where $x^{{{y}}}$ is the $T$-duality direction. We determine the $T$-duality rule for the 27 lines, –, including non-linear terms in the duality rules.
In Figure \[fig:duality-web\], by following the notation of [@hep-th/9809039], a $p$-brane in type II theory with tension $$\begin{aligned}
T_p = \frac{{g_s}^{-n}}{{l_s}\,(2\pi l_s)^p} \Bigl(\frac{R_{n_1}\cdots R_{n_{c_2}}}{{l_s}^{c_2}}\Bigr)^2\cdots \Bigl(\frac{R_{q_1}\cdots R_{q_{c_s}}}{{l_s}^{c_s}}\Bigr)^s\qquad (R_n:\text{ toroidal radii})\,,\end{aligned}$$ is denoted as a $p^{(c_s,\dotsc,c_2)}_{n}$-brane. In particular, the NS5-brane is denoted as $5_2\equiv 5^{(0,\dotsc,0)}_{2}$ and the fundamental string (F1) and the D$p$-brane may be denoted as $1_0\equiv 1^{(0,\dotsc,0)}_{0}$ and $p_1\equiv p^{(0,\dotsc,0)}_{1}$, respectively. The $T$-dualities – and – correspond to the standard $T$-dualities for the NS–NS fields and the R–R fields. The $T$-dualities – have been obtained in [@hep-th/9806169; @0712.3235; @0907.3614; @1701.07819; @1909.01335] and is obtained in [@hep-th/9908094]. To the author’s knowledge, other $T$-dualities are new.[^1] In our approach, type IIB fields are defined as ${\text{SL}}(2)$ $S$-duality tensors, and the $S$-duality rules are simple.
The structure of this paper is as follows. In section \[sec:sugra\], we fix our convention for the (bosonic) supergravity fields. The standard fields are defined through the action, and several higher $p$-form potentials are introduced through the electric-magnetic duality. Additional mixed-symmetry potentials are defined in section \[sec:1-form\] by finding a consistent parameterization of the $U$-duality-covariant 1-form field ${\mathcal A}_\mu^I$. In section \[sec:duality\], we determine the $T$- and $S$-duality rules by following the approach of [@1701.07819; @1909.01335]. In particular, in sections \[sec:D-potential\] and \[sec:EF-potential\], by considering certain field redefinitions, we show that our mixed-symmetry potentials can be packaged into the ${\text{O}}(10,10)$-covariant potentials ${{D}}_{M_1\cdots M_4}$, ${{E}}_{MN\dot{a}}$, and ${{F}}^+_{M_1\cdots M_{10}}$. In section \[sec:gauge\], we discuss the gauge symmetries and field strengths in each theory. Section \[sec:conclusions\] is devoted to conclusions and discussions. In Appendix \[app:notation\], we explain our conventions.
![Duality web studied in this paper. Type IIB branes are paired into $S$-duality multiplets, though only supersymmetric branes are shown explicitly. The M-theory uplifts of type IIA branes are displayed at the bottom. Several different names for branes are as follows: \[fig:duality-web\]](duality-web.pdf){width="\linewidth"}
Supergravity fields {#sec:sugra}
===================
In this section, we fix our conventions for the bosonic supergravity fields.
11D supergravity {#sec:11D-sugra}
----------------
In 11D supergravity, the bosonic fields are ${{\hat{g}}}_{ij}$ and ${{\hat{A}}}_{\hat{3}}$, for which the Lagrangian is $$\begin{aligned}
{\mathcal L}_{11} = \hat{*}\hat{R} -\tfrac{1}{2}\,{{\hat{F}}}_{\hat{4}}\wedge \hat{*}{{\hat{F}}}_{\hat{4}} - \tfrac{1}{3!}\,{{\hat{A}}}_{\hat{3}}\wedge {{\hat{F}}}_{\hat{4}}\wedge {{\hat{F}}}_{\hat{4}} \,,\end{aligned}$$ where we have defined ${{\hat{F}}}_{\hat{4}}\equiv {{\mathrm{d}}}{{\hat{A}}}_{\hat{3}}$. By introducing the dual field strength as $$\begin{aligned}
{{\hat{F}}}_{\hat{7}} \equiv -*_{11} {{\hat{F}}}_{\hat{4}} \,,
\label{eq:F4-F7}\end{aligned}$$ the equation of motion for ${{\hat{A}}}_{\hat{3}}$ is expressed as the Bianchi identity $$\begin{aligned}
{{\mathrm{d}}}{{\hat{F}}}_{\hat{7}} - \tfrac{1}{2}\,{{\hat{F}}}_{\hat{4}}\wedge {{\hat{F}}}_{\hat{4}} = 0\,. \end{aligned}$$ This suggests us to introduce the 6-form potential ${{\hat{A}}}_{\hat{6}}$ as $$\begin{aligned}
{{\hat{F}}}_{\hat{7}} \equiv {{\mathrm{d}}}{{\hat{A}}}_{\hat{6}} + \tfrac{1}{2}\,{{\hat{A}}}_{\hat{3}}\wedge {{\hat{F}}}_{\hat{4}}\,.\end{aligned}$$
Although the potential ${{\hat{A}}}_{\hat{6}}$ is not contained in the standard Lagrangian, it is necessary for manifesting the $U$-duality symmetry. For the manifest $U$-duality symmetry, in general, we need to introduce additional gauge potentials, which are generally mixed-symmetry tensors. Among these, we consider ${{\hat{A}}}_{{\hat{8}},{\hat{1}}}$, ${{\hat{A}}}_{{\hat{9}},{\hat{3}}}$, and ${{\hat{A}}}_{{\hat{10}},{\hat{1}},{\hat{1}}}$ in this paper. The dual graviton ${{\hat{A}}}_{{\hat{8}},{\hat{1}}}$ and the potential ${{\hat{A}}}_{{\hat{10}},{\hat{1}},{\hat{1}}}$ respectively appear in the worldvolume action of the KK monopole [@hep-th/9802199] and the M9-brane [@hep-th/9806069; @hep-th/9812225; @hep-th/9912030; @hep-th/0003240], and their definitions are rather established. However, for the $5^3$-brane (which couples to ${{\hat{A}}}_{{\hat{9}},{\hat{3}}}$), only the kinetic term has been constructed in [@1601.05589] and the WZ term including the potential ${{\hat{A}}}_{{\hat{9}},{\hat{3}}}$ has not been known. Thus the definition of ${{\hat{A}}}_{{\hat{9}},{\hat{3}}}$ is still unclear.
Here, instead of considering brane actions, we define the mixed-symmetry potentials by using the approach of [@1909.01335]. Namely, we consider the $E_{n(n)}$ $U$-duality-covariant 1-form ${\mathcal A}_1^I$, which appears when the eleven-dimensional spacetime is compactified on an $n$-torus. It is uniquely defined (as the generalized graviphoton [@1909.01335]), and by parameterizing ${\mathcal A}_1^I$ in terms of the mixed-symmetry potentials, we can fix the convention for the mixed-symmetry potentials. The concrete parameterizations are given in section \[sec:1-form\]. After providing the parameterizations, we can straightforwardly obtain the $T$- and $S$-duality transformation rules as explained in [@1909.01335].
For convenience, below we summarize the correspondence between each gauge potential and the supersymmetric brane, which electrically couples to the potential: $$\begin{aligned}
\begin{array}{|c|c|c|c|c|}\hline
{{\hat{A}}}_{\hat{3}} & {{\hat{A}}}_{\hat{6}} & {{\hat{A}}}_{{\hat{8}},{\hat{1}}} & {{\hat{A}}}_{{\hat{9}},{\hat{3}}} & {{\hat{A}}}_{{\hat{10}},{\hat{1}},{\hat{1}}} \\\hline
\text{M2} & \text{M5} & \underset{\text{\tiny(MKK)}\vphantom{|}}{6^1} & 5^3 & \underset{\text{\tiny(M9)}\vphantom{|}}{8^{(1,0)}\vphantom{{}^{\big|}}} \\\hline
\end{array} \ .\end{aligned}$$
Type IIA supergravity
---------------------
In order to obtain type IIA supergravity, we consider the standard 11D–10D map, $$\begin{aligned}
\begin{split}
{{\hat{g}}}_{ij}\,{{\mathrm{d}}}x^i\,{{\mathrm{d}}}x^j &= {\operatorname{e}^{-\frac{2}{3}\,{{\varphi}}}} {{\mathscr{g}}}_{mn}\,{{\mathrm{d}}}x^m\,{{\mathrm{d}}}x^n +{\operatorname{e}^{\frac{4}{3}\,{{\varphi}}}}\,({{\mathrm{d}}}x^z + {{\mathscr{C}}}_1)^2 \,,
\\
{{\hat{A}}}_{\hat{3}} &= {{\mathscr{C}}}_3 + {{\mathscr{B}}}_2\wedge {{\mathrm{d}}}x^{{{z}}} \qquad (m,n=0,\dotsc,9)\,,
\end{split}
\label{eq:11D-10D}\end{aligned}$$ where $x^{{{z}}}$ is the coordinate along the M-theory circle. Then, we obtain the type IIA Lagrangian $$\begin{aligned}
\begin{split}
{\mathcal L}_{\text{IIA}} &= {\operatorname{e}^{-2{{\varphi}}}}\bigl(* R + 4\,{{\mathrm{d}}}{{\varphi}}\wedge*{{\mathrm{d}}}{{\varphi}}-\tfrac{1}{2}\,{{\mathscr{H}}}_3\wedge *{{\mathscr{H}}}_3\bigr)
\\
&\quad - \tfrac{1}{2}\,\bigl({\mathcal G}_2\wedge *{\mathcal G}_2 + {\mathcal G}_4\wedge *{\mathcal G}_4 + {{\mathscr{B}}}_2\wedge {{\mathrm{d}}}{{\mathscr{C}}}_3 \wedge {{\mathrm{d}}}{{\mathscr{C}}}_3\bigr) \,,
\end{split}\end{aligned}$$ where we have defined $$\begin{aligned}
{{\mathscr{H}}}_3 \equiv {{\mathrm{d}}}{{\mathscr{B}}}_2 \,, \qquad
{\mathcal G}_2 \equiv {{\mathrm{d}}}{{\mathscr{C}}}_1 \,, \qquad
{\mathcal G}_4 \equiv {{\mathrm{d}}}{{\mathscr{C}}}_3 -{{\mathscr{H}}}_3\wedge {{\mathscr{C}}}_1\,,\end{aligned}$$ and used the identity $$\begin{aligned}
{{\hat{F}}}_{\hat{4}} = {\mathcal G}_4 + {{\mathscr{H}}}_3\wedge ({{\mathrm{d}}}x^{{{z}}}+{{\mathscr{C}}}_1)\,.\end{aligned}$$ Again, the equations of motion for ${{\mathscr{B}}}_2$ and ${{\mathscr{C}}}_3$ are expressed as Bianchi identities $$\begin{aligned}
{{\mathrm{d}}}{{\mathscr{H}}}_7 - {\mathcal G}_2\wedge * {\mathcal G}_4 - \tfrac{1}{2}\,{\mathcal G}_4\wedge {\mathcal G}_4 = 0 \,,\qquad
{{\mathrm{d}}}{\mathcal G}_6 - {{\mathscr{H}}}_3\wedge {\mathcal G}_4 =0 \,,\end{aligned}$$ where the dual field strengths are defined by $$\begin{aligned}
{{\mathscr{H}}}_7\equiv {\operatorname{e}^{-2{{\varphi}}}}* {{\mathscr{H}}}_3\,,\qquad {\mathcal G}_6 \equiv - * {\mathcal G}_4 \,.\end{aligned}$$ Then, we can introduce the dual potentials ${{\mathscr{C}}}_5$ and ${{\mathscr{B}}}_6$ as follows: $$\begin{aligned}
{\mathcal G}_6 \equiv {{\mathrm{d}}}{{\mathscr{C}}}_5 -{{\mathscr{H}}}_3\wedge {{\mathscr{C}}}_3\,, \qquad
{{\mathscr{H}}}_7 \equiv {{\mathrm{d}}}{{\mathscr{B}}}_6 - {\mathcal G}_6 \wedge {{\mathscr{C}}}_1 + \tfrac{1}{2}\,{\mathcal G}_4\wedge {{\mathscr{C}}}_3 - \tfrac{1}{2}\,{{\mathscr{H}}}_3 \wedge {{\mathscr{C}}}_3 \wedge {{\mathscr{C}}}_1 \,.\end{aligned}$$ Through the electric-magnetic duality, we obtain the following 11D–10D map: $$\begin{aligned}
{{\hat{A}}}_{\hat{6}} = {{\mathscr{B}}}_6 + \bigl({{\mathscr{C}}}_5 - \tfrac{1}{2!}\, {{\mathscr{C}}}_3\wedge {{\mathscr{B}}}_2\bigr)\wedge {{\mathrm{d}}}x^{{{z}}} \,,\qquad
{{\hat{F}}}_{\hat{7}} = {{\mathscr{H}}}_7 + {\mathcal G}_6 \wedge ({{\mathrm{d}}}x^{{{z}}} + {{\mathscr{C}}}_1) \,.\end{aligned}$$
On the other hand, the equation of motion for ${{\mathscr{C}}}_1$ is expressed as $$\begin{aligned}
{{\mathrm{d}}}{\mathcal G}_8 - {{\mathscr{H}}}_3\wedge {\mathcal G}_6 =0 \,,\qquad {\mathcal G}_8 \equiv * {\mathcal G}_2 \,,\end{aligned}$$ and we can introduce the 7-form potential ${{\mathscr{C}}}_7$ as $$\begin{aligned}
{\mathcal G}_8 \equiv {{\mathrm{d}}}{{\mathscr{C}}}_7 -{{\mathscr{H}}}_3\wedge {{\mathscr{C}}}_3\,.
\label{eq:G-8-def}\end{aligned}$$ The 11D uplift of this 8-form field strength is discussed in section \[sec:gauge\] \[see Eq. \].
In 11D supergravity, we have introduced non-standard potentials ${{\hat{A}}}_{{\hat{8}},{\hat{1}}}$, ${{\hat{A}}}_{{\hat{10}},{\hat{1}},{\hat{1}}}$, and ${{\hat{A}}}_{{\hat{9}},{\hat{3}}}$. Here, we consider the following simple 11D–10D map for these potentials:[^2] $$\begin{aligned}
\begin{split}
\begin{alignedat}{2}
{{\hat{A}}}_{\hat{8}, 1} &= {{\mathscr{A}}}_{8, 1} + {{\mathscr{A}}}_{7, 1}\wedge {{\mathrm{d}}}x^{{{z}}} \,, \qquad&
{{\hat{A}}}_{\hat{8}, {{z}}} &= \overline{{{\mathscr{A}}}}_{8} + {{\mathscr{A}}}_7 \wedge{{\mathrm{d}}}x^{{{z}}} \,,
\\
{{\hat{A}}}_{\hat{9}, 3} &= {{\mathscr{A}}}_{9, 3} + {{\mathscr{A}}}_{8, 3} \wedge {{\mathrm{d}}}x^{{{z}}} \,,\qquad&
{{\hat{A}}}_{\hat{9}, 2{{z}}} &= \overline{{{\mathscr{A}}}}_{9, 2} + {{\mathscr{A}}}_{8, 2} \wedge {{\mathrm{d}}}x^{{{z}}} \,,
\\
{{\hat{A}}}_{\hat{10},1,1} &= {{\mathscr{A}}}_{10,1,1} + {{\mathscr{A}}}_{9,1,1}\wedge {{\mathrm{d}}}x^{{{z}}} \,, \qquad&
{{\hat{A}}}_{\hat{10},{{z}},{{z}}} &= \overline{{{\mathscr{A}}}}_{10} + {{\mathscr{A}}}_9\wedge {{\mathrm{d}}}x^{{{z}}} \,.
\end{alignedat}
\end{split}\end{aligned}$$ The potential ${{\mathscr{A}}}_7$ is related to the R–R 7-form ${{\mathscr{C}}}_7$ introduced in and the 9-form ${{\mathscr{A}}}_9$ is related to the standard R–R 9-form ${{\mathscr{C}}}_9$ (see section \[sec:standard-T\]).
In the type IIA case, the correspondence between the potentials and the supersymmetric branes are summarized as follows:[^3] $$\begin{aligned}
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
{{\mathscr{B}}}_2 & {{\mathscr{C}}}_1 & {{\mathscr{C}}}_3 & {{\mathscr{C}}}_5 & {{\mathscr{B}}}_6 & {{\mathscr{C}}}_7 & {{\mathscr{A}}}_{7, 1} & {{\mathscr{A}}}_{8, 1} & {{\mathscr{C}}}_9 & {{\mathscr{A}}}_{8, 2} & {{\mathscr{A}}}_{8, 3} & {{\mathscr{A}}}_{9,1,1} & {{\mathscr{A}}}_{10,1,1} \\\hline
\text{F1} & \text{D0} & \text{D2} & \text{D4} & \underset{\text{\tiny(NS5)}\vphantom{|}}{5_2} & \text{D6} & \underset{\text{\tiny(KK5A)}\vphantom{|}}{5_2^1} & \underset{\text{\tiny(KK6A)}\vphantom{|}}{6_3^{1}\vphantom{{}^{\big|}}} & \text{D8} & 5^2_2 & 4^3_3 & \underset{\text{\tiny(KK8A)}\vphantom{|}}{7^{(1,0)}_3} & \underset{\text{\tiny(NS9A)}\vphantom{|}}{8^{(1,0)}_4} \\\hline
\end{array}\,. \end{aligned}$$
Type IIB supergravity
---------------------
The standard ${\text{SL}}(2)$ $S$-duality-invariant (pseudo) Lagrangian for type IIB supergravity is $$\begin{aligned}
{\mathcal L}_{\text{IIB}} &= *_{{\mathrm{E}}} {\mathsf{R}}+ \tfrac{1}{4}\,{{\mathsf{F}}}_{1 {{\alpha}}{{\beta}}} \wedge *_{{\mathrm{E}}}{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_1 - \tfrac{1}{2}\,{{\mathsf{m}}}_{{{\alpha}}{{\beta}}}\,{{\mathsf{F}}}^{{{\alpha}}}_3\wedge *_{{\mathrm{E}}}{{\mathsf{F}}}^{{{\beta}}}_3
- \tfrac{1}{4}\, {{\mathsf{F}}}_5\wedge *_{{\mathrm{E}}}{{\mathsf{F}}}_5 + \tfrac{1}{4}\,\epsilon_{{{\alpha}}{{\beta}}}\,{{\mathsf{A}}}_4\wedge {{\mathsf{F}}}^{{{\alpha}}}_3\wedge {{\mathsf{F}}}^{{{\beta}}}_3\,,\end{aligned}$$ where ${{\alpha}}={\bm{1}},{\bm{2}}$ are indices of ${\text{SL}}(2)$ doublets and $(\epsilon_{{{\alpha}}{{\beta}}})=(\epsilon^{{{\alpha}}{{\beta}}})=\bigl(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\bigr)$. The fundamental fields are $\{{{\mathsf{g}}}_{mn},\,{{\mathsf{m}}}_{{{\alpha}}{{\beta}}},\,{{\mathsf{A}}}^{{{\alpha}}}_2,\,{{\mathsf{A}}}_4\}$, and ${{\mathsf{g}}}_{mn}$ is the Einstein-frame metric, for which the Hodge star operator is denoted by $*_{{\mathrm{E}}}$. The scalar field ${{\mathsf{m}}}_{{{\alpha}}{{\beta}}}$ is symmetric ${{\mathsf{m}}}_{{{\alpha}}{{\beta}}}={{\mathsf{m}}}_{({{\alpha}}{{\beta}})}$ and satisfies $$\begin{aligned}
{{\mathsf{m}}}^{{{\alpha}}}{}_{{{\gamma}}}\,{{\mathsf{m}}}^{{{\gamma}}}{}_{{{\beta}}} = - {{\mathsf{m}}}^{{{\alpha}}{{\gamma}}}\,{{\mathsf{m}}}_{{{\gamma}}{{\beta}}} = - \delta^{{{\alpha}}}_{{{\beta}}} \,,\end{aligned}$$ where we have raised or lowered the ${\text{SL}}(2)$ indices as $v^{{{\alpha}}}=\epsilon^{{{\alpha}}{{\beta}}}\,v_{{{\beta}}}$ and $v_{{{\alpha}}}= v^{{{\beta}}}\,\epsilon_{{{\beta}}{{\alpha}}}$. The field strengths are defined by $$\begin{aligned}
{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_1 \equiv {{\mathsf{m}}}^{{{\alpha}}{{\gamma}}}\, {{\mathrm{d}}}{{\mathsf{m}}}_{{{\gamma}}}{}^{{{\beta}}} = {{\mathsf{F}}}^{({{\alpha}}{{\beta}})}_1 \,,\qquad
{{\mathsf{F}}}^{{{\alpha}}}_3 \equiv {{\mathrm{d}}}{{\mathsf{A}}}^{{{\alpha}}}_2 \,, \qquad
{{\mathsf{F}}}_5 \equiv {{\mathrm{d}}}{{\mathsf{A}}}_4 + \tfrac{1}{2}\, \epsilon_{{{\alpha}}{{\beta}}}\, {{\mathsf{F}}}^{{{\alpha}}}_3\wedge {{\mathsf{A}}}^{{{\beta}}}_2\,,\end{aligned}$$ which satisfy the Bianchi identities $$\begin{aligned}
{{\mathrm{d}}}{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_1 + \epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{F}}}^{{{\alpha}}{{\gamma}}}_1 \wedge {{\mathsf{F}}}^{{{\delta}}{{\beta}}}_1 = 0 \,,\qquad
{{\mathrm{d}}}{{\mathsf{F}}}^{{{\alpha}}}_3 = 0\,,\qquad
{{\mathrm{d}}}{{\mathsf{F}}}_5+ \tfrac{1}{2}\,\epsilon_{{{\alpha}}{{\beta}}}\, {{\mathsf{F}}}^{{{\alpha}}}_3\wedge {{\mathsf{F}}}^{{{\beta}}}_3 = 0 \,.\end{aligned}$$ The self-duality relation for the 5-form field strength, $$\begin{aligned}
{{\mathsf{F}}}_5 = *_{{\mathrm{E}}}{{\mathsf{F}}}_5\,,
\label{eq:F5-self}\end{aligned}$$ should be imposed at the level of equations of motion.
As is well-known, under the self-duality relation the equation of motion for ${{\mathsf{A}}}_4$ is equivalent to the last Bianchi identity. If we additionally define the dual field strengths as[^4] $$\begin{aligned}
{{\mathsf{F}}}^{{{\alpha}}}_7 \equiv {{\mathsf{m}}}^{{{\alpha}}}{}_{{{\beta}}}*_{{\mathrm{E}}} {{\mathsf{F}}}^{{{\beta}}}_3 \,, \qquad
{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_9 \equiv *_{{\mathrm{E}}} {{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_1 \,,
\label{eq:hodge-7-3-9-1}\end{aligned}$$ the equations of motion for ${{\mathsf{m}}}_{{{\alpha}}{{\beta}}}$ and ${{\mathsf{A}}}^{{{\alpha}}}_2$ also can be expressed as the Bianchi identities $$\begin{aligned}
{{\mathrm{d}}}{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_9 - {{\mathsf{F}}}^{({{\alpha}}}_3\wedge {{\mathsf{F}}}^{{{\beta}})}_7 = 0\,,\qquad
{{\mathrm{d}}}{{\mathsf{F}}}^{{{\alpha}}}_7 - {{\mathsf{F}}}^{{{\alpha}}}_3\wedge {{\mathsf{F}}}_5 = 0\,.\end{aligned}$$ They suggest us to introduce the higher potentials, ${{\mathsf{A}}}^{{{\alpha}}}_6$ and ${{\mathsf{A}}}^{{{\alpha}}{{\beta}}}_8$ as $$\begin{aligned}
{{\mathsf{F}}}^{{{\alpha}}}_7 &\equiv {{\mathrm{d}}}{{\mathsf{A}}}^{{{\alpha}}}_6 - {{\mathsf{F}}}^{{{\alpha}}}_3\wedge {{\mathsf{A}}}_4 + \tfrac{1}{3!}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{F}}}^{{{\gamma}}}_3 \wedge{{\mathsf{A}}}^{{{\delta}}}_2\wedge {{\mathsf{A}}}^{{{\alpha}}}_2\,,
\\
{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_9 &\equiv {{\mathrm{d}}}{{\mathsf{A}}}^{{{\alpha}}{{\beta}}}_8 - {{\mathsf{F}}}_3^{({{\alpha}}} \wedge {{\mathsf{A}}}_6^{{{\beta}})} + \tfrac{1}{4!}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{F}}}^{{{\gamma}}}_3\wedge {{\mathsf{A}}}^{{{\delta}}}_2\wedge {{\mathsf{A}}}^{{{\alpha}}}_2\wedge {{\mathsf{A}}}^{{{\beta}}}_2\,.\end{aligned}$$ In [@hep-th/0506013; @hep-th/0602280; @hep-th/0611036; @1004.1348], a 10-form potential was also introduced by considering the supersymmetry algebra (which is also predicted by $E_{11}$ [@hep-th/0511153]),[^5] and the field strength, in our convention, is defined as $$\begin{aligned}
&{{\mathsf{F}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{11} \equiv {{\mathrm{d}}}{{\mathsf{A}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10} - {{\mathsf{F}}}^{({{\alpha}}}_3\wedge {{\mathsf{A}}}^{{{\beta}}{{\gamma}})}_8 + \tfrac{1}{5!}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\mathsf{F}}}^{{{\zeta}}}_3\wedge {{\mathsf{A}}}^{{{\eta}}}_2\wedge {{\mathsf{A}}}^{{{\alpha}}}_2\wedge {{\mathsf{A}}}^{{{\beta}}}_2 \wedge {{\mathsf{A}}}^{{{\gamma}}}_2 \ (= 0)\,.\end{aligned}$$ This satisfies the Bianchi identity (without considering the dimensionality) $$\begin{aligned}
{{\mathrm{d}}}{{\mathsf{F}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{11} - {{\mathsf{F}}}^{({{\alpha}}}_3\wedge {{\mathsf{F}}}^{{{\beta}}{{\gamma}})}_9 =0\,.\end{aligned}$$
In this paper, we consider the following set of type IIB fields: $$\begin{aligned}
\{{{\mathsf{g}}}_{mn},\,{{\mathsf{m}}}_{{{\alpha}}{{\beta}}},\,{{\mathsf{A}}}^{{{\alpha}}}_2,\,{{\mathsf{A}}}_4,\,{{\mathsf{A}}}^{{{\alpha}}}_6,\,{{\mathsf{A}}}_{7,1},\,{{\mathsf{A}}}^{{{\alpha}}{{\beta}}}_8,\,{{\mathsf{A}}}^{{{\alpha}}}_{8,2},\,{{\mathsf{A}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10},\,{{\mathsf{A}}}_{9,2,1}\}\,,\end{aligned}$$ which transform covariantly under ${\text{SL}}(2)$ $S$-duality transformations. At the present stage, definitions of ${{\mathsf{A}}}_{7,1}$, ${{\mathsf{A}}}^{{{\alpha}}}_{8,2}$, ${{\mathsf{A}}}_{9,2,1}$, and ${{\mathsf{A}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10}$ are not specified. They are defined in section \[sec:1-form\].
In the following discussion, a redefinition of the 6-form potential $$\begin{aligned}
{{\bm{\mathsf{A}}}}^{{{\alpha}}}_6 \equiv {{\mathsf{A}}}^{{{\alpha}}}_6 - {{\mathsf{A}}}_4\wedge {{\mathsf{A}}}^{{{\alpha}}}_2\,, \end{aligned}$$ makes the $T$-duality rules slightly shorter. Thus, the 6-form ${{\bm{\mathsf{A}}}}^{{{\alpha}}}_6$ rather than ${{\mathsf{A}}}^{{{\alpha}}}_6$ is mainly used in this paper. For notational consistency, other fields also may be denoted by bold typeface, $$\begin{aligned}
{{\bm{\mathsf{A}}}}^{{{\alpha}}}_2\equiv {{\mathsf{A}}}^{{{\alpha}}}_2,\
{{\bm{\mathsf{A}}}}_4\equiv {{\mathsf{A}}}_4,\
{{\bm{\mathsf{A}}}}_{7,1}\equiv {{\mathsf{A}}}_{7,1},\
{{\bm{\mathsf{A}}}}^{{{\alpha}}{{\beta}}}_8\equiv {{\mathsf{A}}}^{{{\alpha}}{{\beta}}}_8,\
{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{8,2}\equiv {{\mathsf{A}}}^{{{\alpha}}}_{8,2},\
{{\bm{\mathsf{A}}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10}\equiv {{\mathsf{A}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10},\
{{\bm{\mathsf{A}}}}_{9,2,1}\equiv {{\mathsf{A}}}_{9,2,1}\,.\end{aligned}$$ The relation between the potentials and the supersymmetric branes are as follows: $$\begin{aligned}
\begin{array}{|c|c|c|c|c|c|c|c|}\hline
{{\bm{\mathsf{A}}}}^{{{\alpha}}}_2 & {{\bm{\mathsf{A}}}}_4 & {{\bm{\mathsf{A}}}}^{{{\alpha}}}_6 & {{\bm{\mathsf{A}}}}_{7,1} & {{\bm{\mathsf{A}}}}^{{{\alpha}}{{\beta}}}_{8} & {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{8,2} & {{\bm{\mathsf{A}}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10}& {{\bm{\mathsf{A}}}}_{9,2,1} \\\hline
\text{F1/D1} & \text{D3} & \text{D5/}\!\!\underset{\text{\tiny(NS5)}\vphantom{|}}{5_2\vphantom{{}^{\big|}}} & \underset{\text{\tiny(KK5B)}\vphantom{|}}{5_2^1} & \text{D7}/\!\!\underset{\text{\tiny(NS7B)}\vphantom{|}}{7_3} & 5^2_2/5^2_3 & \text{D9}/\!\!\!\underset{\text{\tiny(NS9B)}\vphantom{|}}{9_4} & \underset{\text{\tiny(KK7B)}\vphantom{|}}{6_3^{(1,1)}} \\\hline
\end{array}\,.\end{aligned}$$ It is noted that an ${\text{SL}}(2)$ $n$-plet ${{\bm{\mathsf{A}}}}_{\cdots}^{{{\alpha}}_1\cdots {{\alpha}}_{n-1}}$ $(n\geq 2)$ always couples to only two supersymmetric branes. The components which couple to supersymmetric branes are ${{\bm{\mathsf{A}}}}_{\cdots}^{{\bm{1}}\cdots {\bm{1}}}$ and ${{\bm{\mathsf{A}}}}_{\cdots}^{{\bm{2}}\cdots {\bm{2}}}$ as discussed in [@hep-th/0611036; @1009.4657] (see also [@1907.07177]).
Parameterization of the $U$-duality-covariant 1-form {#sec:1-form}
----------------------------------------------------
When 11D supergravity or type II supergravity is compactified to $d$-dimensions, the bosonic fields with one external index $\mu\ (=0,\dotsc,d-1)$ are packaged into the 1-form field ${\mathcal A}_\mu^I$, where $I$ is the index for the so-called the vector representation of the $U$-duality group $E_{n(n)}$ ($n=11-d$). Under the compactification, we decompose the indices in M-theory/type IIB theory as $$\begin{aligned}
\text{M-theory:}\quad &\{i\}=\{\mu,\underline{i}\} \qquad (\underline{i}=d,\dotsc,8,{{y}},{{z}})\,,\
\\
\text{Type IIB theory:}\quad &\{m\}=\{\mu,\underline{m}\} \qquad (\underline{m}=d,\dotsc,8,{{\mathsf{y}}})\,.\end{aligned}$$ Then the vector index $I$ in M-theory is decomposed into indices of ${\text{SL}}(n)$ tensors as [@hep-th/0104081] $$\begin{aligned}
({\mathcal A}_\mu^I) = \bigl({\mathcal A}_\mu^{\underline{i}},\ \tfrac{{\mathcal A}_{\mu;\underline{i}_1\underline{i}_2}}{\sqrt{2!}},\ \tfrac{{\mathcal A}_{\mu;\underline{i}_1\cdots \underline{i}_5}}{\sqrt{5!}},\ \tfrac{{\mathcal A}_{\mu;\underline{i}_1\cdots \underline{i}_7, \underline{k}}}{\sqrt{7!}},\ \tfrac{{\mathcal A}_{\mu;\underline{i}_1\cdots \underline{i}_8,\underline{k}_1\underline{k}_2\underline{k}_3}}{\sqrt{8!\,3!}},\ \tfrac{{\mathcal A}_{\mu;\underline{i}_1\cdots \underline{i}_9, \underline{k}, \underline{l}}}{\sqrt{9!}},\cdots \bigr) \,,\end{aligned}$$ where only the relevant components are shown. In type IIB theory, the vector index is denoted by ${\mathsf{I}}$ and it is decomposed into indices of ${\text{SL}}(n-1)\times {\text{SL}}(2)$ tensors as [@hep-th/0107181] $$\begin{aligned}
\begin{split}
(\bm{{\mathcal A}}_\mu^{\mathsf{I}}) = \bigl(&\bm{{\mathcal A}}_\mu^{\underline{m}},\ \bm{{\mathcal A}}^{{{\alpha}}}_{\mu;\underline{m}},\ \tfrac{\bm{{\mathcal A}}_{\mu;\underline{m}_1\underline{m}_2\underline{m}_3}}{\sqrt{3!}},\ \tfrac{\bm{{\mathcal A}}^{{{\alpha}}}_{\mu;\underline{m}_1\cdots \underline{m}_5}}{\sqrt{5!}},\ \tfrac{\bm{{\mathcal A}}_{\mu;\underline{m}_1\cdots \underline{m}_6,\underline{p}}}{\sqrt{6!}},
\\
&\tfrac{\bm{{\mathcal A}}^{{{\alpha}}{{\beta}}}_{\mu;\underline{m}_1\cdots \underline{m}_7}}{\sqrt{7!}},\ \tfrac{\bm{{\mathcal A}}^{{{\alpha}}}_{\mu; \underline{m}_1\cdots \underline{m}_7, \underline{p}_1\underline{p}_2}}{\sqrt{7!\,2!}},\ \tfrac{\bm{{\mathcal A}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{\mu; \underline{m}_1\cdots \underline{m}_9}}{\sqrt{9!}},\ \tfrac{\bm{{\mathcal A}}_{\mu; \underline{m}_1\cdots \underline{m}_8,\underline{p}_1\underline{p}_2,\underline{q}}}{\sqrt{8!\,2!}} ,\cdots \bigr)\,.
\end{split}\end{aligned}$$
Now, we parameterize each component of the 1-form field in terms of the bosonic fields introduced in the last subsections. In fact, the 1-form has the universal form [@1909.01335] $$\begin{aligned}
{\mathcal A}_\mu^I = {{\hat{N}}}_{\mu}{}^I + {{\hat{A}}}_\mu^{\underline{j}}\, {{\hat{N}}}_{\underline{j}}{}^I\quad (\text{M-theory})\,,\qquad
\bm{{\mathcal A}}_\mu^{{\mathsf{I}}} = {{\mathsf{N}}}_{\mu}{}^{{\mathsf{I}}} + {{\mathsf{A}}}_\mu^{\underline{n}}\, {{\mathsf{N}}}_{\underline{n}}{}^{{\mathsf{I}}}\quad (\text{type IIB theory})\,,\end{aligned}$$ where ${{\hat{N}}}$/${{\mathsf{N}}}$ are the 11D/10D fields and ${{\hat{A}}}_\mu^{\underline{i}}$/${{\mathsf{A}}}_\mu^{\underline{m}}$ are the graviphoton, which are defined by $$\begin{aligned}
{{\hat{A}}}_\mu^{\underline{i}}\equiv \bm{g}_{\mu\nu}\,{{\hat{g}}}^{\nu\underline{i}} \qquad \bigl[ (\bm{g}_{\mu\nu}) \equiv ({{\hat{g}}}^{\mu\nu})^{-1} \bigr]\,,\end{aligned}$$ and similar for ${{\mathsf{A}}}_\mu^{\underline{m}}$. Therefore, the parameterization of the 1-form field is equivalent to the parameterization of the 11D- or 10D-covariant field ${{\hat{N}}}$ or ${{\mathsf{N}}}$.
In this paper, we parameterize the 11D tensors $\{{{\hat{N}}}\}=\{{{\hat{N}}}_{j}{}^i,\ \tfrac{{{\hat{N}}}_{j;i_1i_2}}{\sqrt{2!}},\cdots \}$ as follows: $$\begin{aligned}
{{\hat{N}}}_{j}{}^i &=\delta_j^i \,,
\\
{{\hat{N}}}_{j;i_1i_2} &={{\hat{A}}}_{j i_1i_2} \,,
\\
{{\hat{N}}}_{j;i_1\cdots i_5} &= {{\hat{A}}}_{j i_1\cdots i_5} - 5\,{{\hat{A}}}_{j[i_1i_2}\, {{\hat{A}}}_{i_3i_4i_5]}\,,
\\
{{\hat{N}}}_{j;i_1\cdots i_7, k}
&\simeq {{\hat{A}}}_{j i_1\cdots i_7, k} - 21\, {{\hat{A}}}_{j [i_1\cdots i_5}\,{{\hat{A}}}_{i_6i_7] k}
+35\,{{\hat{A}}}_{j [i_1i_2}\,{{\hat{A}}}_{i_3i_4i_5}\,{{\hat{A}}}_{i_6i_7]k} \,,
\\
\begin{split}
{{\hat{N}}}_{j;i_1\cdots i_8, k_1k_2k_3} &\simeq
{{\hat{A}}}_{j i_1\cdots i_8, k_1k_2k_3}
+ 3\,{{\hat{A}}}_{j \bar{k}_1\bar{k}_2}\, {{\hat{A}}}_{i_1\cdots i_8, \bar{k}_3}
+ 14\,{{\hat{A}}}_{j [i_1i_2}\,{{\hat{A}}}_{i_3\cdots i_8]} \,{{\hat{A}}}_{k_1k_2k_3}
\\
&\quad - 84\,{{\hat{A}}}_{j [i_1|\bar{k}_1|}\,{{\hat{A}}}_{i_2 |\bar{k}_2\bar{k}_3|}\,{{\hat{A}}}_{i_3\cdots i_8]}
- 42\,{{\hat{A}}}_{j \bar{k}_1\bar{k}_2}\,{{\hat{A}}}_{[i_1i_2|\bar{k}_3|}\,{{\hat{A}}}_{i_3\cdots i_8]}
\\
&\quad
+ 28\,{{\hat{A}}}_{j [i_1\cdots i_5}\,{{\hat{A}}}_{i_6i_7i_8] k_1k_2k_3}
-210\,{{\hat{A}}}_{j [i_1i_2}\,{{\hat{A}}}_{i_3i_4i_5}\,{{\hat{A}}}_{i_6i_7|\bar{k}_1|}\,{{\hat{A}}}_{i_8] \bar{k}_2\bar{k}_3} \,,
\end{split}
\label{eq:N-M-9-3}
\\
\begin{split}
{{\hat{N}}}_{j;i_1\cdots i_9, k, l}
&\simeq {{\hat{A}}}_{j i_1\cdots i_9, k, l}
- 84\,{{\hat{A}}}_{j [i_1\cdots i_6 |k,l|}\, {{\hat{A}}}_{i_7i_8i_9]}
+378\,{{\hat{A}}}_{j [i_1\cdots i_5}\,{{\hat{A}}}_{i_6i_7 |k|}\, {{\hat{A}}}_{i_8i_9] l}
\\
&\quad -315\,{{\hat{A}}}_{j [i_1i_2}\,{{\hat{A}}}_{i_3i_4i_5}\,{{\hat{A}}}_{i_6i_7|k|}\,{{\hat{A}}}_{i_8i_9] l}\,.
\end{split}\end{aligned}$$ Here, the overlined indices are totally antisymmetrized; e.g. ${{\hat{A}}}_{j \bar{k}_1\bar{k}_2}\, {{\hat{A}}}_{i_1\cdots i_8, \bar{k}_3} = {{\hat{A}}}_{j [\bar{k}_1\bar{k}_2|}\, {{\hat{A}}}_{i_1\cdots i_8, |\bar{k}_3]}$. In addition, the equality $$\begin{aligned}
{{\hat{N}}}_{j; i_1\cdots i_p, j_1\cdots j_q, k_1\cdots k_r,\cdots}
\simeq (\cdots )_{j i_1\cdots i_p j_1\cdots j_q k_1\cdots k_r \cdots} \,,\end{aligned}$$ denotes that it is valid only for the indices satisfying the restriction rule, $$\begin{aligned}
\{i_1,\dotsc, i_p\} \supset \{j_1,\dotsc, j_q\} \supset \{k_1,\dotsc, k_r\} \supset \cdots\,.
\label{eq:M-restriction}\end{aligned}$$ Since the 1-form ${\mathcal A}_\mu^I$ is uniquely defined, the above parameterizations uniquely define our bosonic fields, in particular, the mixed-symmetry potentials, ${{\hat{A}}}_{\hat{8},\hat{1}}$, ${{\hat{A}}}_{\hat{9},\hat{3}}$, and ${{\hat{A}}}_{\hat{10},\hat{1},\hat{1}}$.
The detailed procedure, how to determine the above parameterization of ${{\hat{N}}}$ is explained in section 2 of [@1909.01335]. By considering the consistency between the M-theory and the type IIB parameterizations, the above parameterizations are uniquely determined (up to redefinitions of mixed-symmetry potentials). The same parameterization can be obtained also by constructing a matrix representation of the $E_{11}$ generators as discussed in section 3 of [@1909.01335]. In the second approach based on $E_{11}$, the parameterizations will be completely determined without requiring the restriction rule (see [@1909.01335] for more details).
Now, let us turn to the type IIB parameterization. In type IIB theory, we parameterize the ${\text{SL}}(2)$-covariant 10D tensors $\{{{\mathsf{N}}}\}$ as follows: $$\begin{aligned}
{{\mathsf{N}}}_{n}{}^m &= \delta_n^m \,,
\\
{{\mathsf{N}}}_{n;m}^{{{\alpha}}} &= {{\bm{\mathsf{A}}}}_{nm}^{{{\alpha}}} \,,
\\
{{\mathsf{N}}}_{n;m_1m_2m_3} &= {{\bm{\mathsf{A}}}}_{nm_1m_2m_3} - \tfrac{3}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_2m_3]} \,,
\\
{{\mathsf{N}}}^{{{\alpha}}}_{n;m_1 \cdots m_5} &= {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{nm_1 \cdots m_5} +5\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}_{m_2\cdots m_6]}
+ 5\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n[m_1}\, {{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_2m_3}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_4m_5]} \,,
\\
\begin{split}
{{\mathsf{N}}}_{n;m_1\cdots m_6, p} &\simeq {{\bm{\mathsf{A}}}}_{n m_1\cdots m_6, p} + \epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{np}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_1\cdots m_6}+10\,{{\bm{\mathsf{A}}}}_{n[m_1m_2m_3}\,{{\bm{\mathsf{A}}}}_{m_4m_5m_6]p}
\\
&\quad -30\,\epsilon_{{{\alpha}}{{\beta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}_{m_4m_5m_6]p}
\\
&\quad +\tfrac{15}{2}\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_4m_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_6]p} \,,
\end{split}
\\
\begin{split}
{{\mathsf{N}}}^{{{\alpha}}{{\beta}}}_{n;m_1\cdots m_7} &\simeq {{\bm{\mathsf{A}}}}^{{{\alpha}}{{\beta}}}_{n m_1\cdots m_7}
- 21\,{{\bm{\mathsf{A}}}}^{({{\alpha}}}_{n[m_2\cdots m_5}\, {{\bm{\mathsf{A}}}}^{{{\beta}})}_{m_6m_7]}
-105\, {{\bm{\mathsf{A}}}}_{n[m_2m_2m_3}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_4m_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_6m_7]}
\\
&\quad
-105\, {{\bm{\mathsf{A}}}}^{({{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\beta}})}_{m_2m_3}\,{{\bm{\mathsf{A}}}}_{m_4\cdots m_7]}
-\tfrac{105}{4}\, \epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_4m_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_6m_7]} \,,
\end{split}
\\
\begin{split}
{{\mathsf{N}}}^{{{\alpha}}}_{n; m_1\cdots m_7, p_1p_2}
&\simeq {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n m_1\cdots m_7, p_1p_2}
-2\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n \bar{p}_1}\, {{\bm{\mathsf{A}}}}_{m_1\cdots m_7 , \bar{p}_2}
\rlap{$\displaystyle{} -35\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1m_2m_3| p_1p_2|}\,{{\bm{\mathsf{A}}}}_{m_4\cdots m_7]}$}
\\
&\quad +\tfrac{21}{2}\, \epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1\cdots m_5}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_6m_7]}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{p_1p_2}
+ 7\, \epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n\bar{p}_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{[m_1|\bar{p}_2|} \,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_2\cdots m_7]}
\\
&\quad +210\,{{\bm{\mathsf{A}}}}_{n[m_1m_2|\bar{p}_1|}\,{{\bm{\mathsf{A}}}}_{m_3m_4m_5|\bar{p}_2|}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_6m_7]}
\\
&\quad
-105\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}_{m_4m_5|p_1p_2|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_6m_7]}
\\
&\quad
+\tfrac{1575}{8}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{m_4m_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|p_1p_2|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_6m_7]}
\\
&\quad +\tfrac{1365}{8} \, \epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n\bar{p}_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{[m_1m_2}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{m_3m_4}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{m_5|\bar{p}_2|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_6m_7]}
\\
&\quad -1470\, \epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_2|\bar{p}_1|}\, {{\bm{\mathsf{A}}}}^{{{\zeta}}}_{m_3m_4}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{m_5|\bar{p}_2|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_6m_7]}\,,
\end{split}
\\
\begin{split}
{{\mathsf{N}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{n;m_1\cdots m_9}
&\simeq {{\bm{\mathsf{A}}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{n m_1\cdots m_9}
-36\, {{\bm{\mathsf{A}}}}^{({{\alpha}}{{\beta}}}_{n[m_1\cdots m_7}\, {{\bm{\mathsf{A}}}}^{{{\gamma}})}_{m_8m_9]}
+378\,{{\bm{\mathsf{A}}}}^{({{\alpha}}}_{n[m_1\cdots m_5}\, {{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\gamma}})}_{m_8m_9]}
\\
&\quad
+2520\, {{\bm{\mathsf{A}}}}_{n[m_1m_2m_3}\, {{\bm{\mathsf{A}}}}^{({{\alpha}}}_{m_4m_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\gamma}})}_{m_8m_9]}
\\
&\quad
+1890\, {{\bm{\mathsf{A}}}}^{({{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}^{{{\gamma}})}_{m_4m_5}\, {{\bm{\mathsf{A}}}}_{m_6\cdots m_9]}
\\
&\quad
+189\, \epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}^{({{\alpha}}}_{m_4m_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\gamma}})}_{m_8m_9]}\,,
\end{split}
\\
\begin{split}
{{\mathsf{N}}}_{n;m_1\cdots m_8,p_1p_2,q}
&\simeq
{{\bm{\mathsf{A}}}}_{nm_1\cdots m_8 ,p_1p_2 ,q}
+\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{nq}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_1\cdots m_8 , p_1p_2}
+56\,{{\bm{\mathsf{A}}}}_{n[m_1\cdots m_6| , \bar{p}_1|}\, {{\bm{\mathsf{A}}}}_{m_7m_8] \bar{p}_2 q}
\\
&\quad
+168\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{n[m_1\cdots m_5| \bar{p}_1 , \bar{p}_2|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_8]q}
\rlap{$\displaystyle{}
-70\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{nq}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{[m_1\cdots m_4|p_1p_2|}\, {{\bm{\mathsf{A}}}}_{m_5\cdots m_8]}$}
\\
&\quad
+84\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{nq}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{[m_1\cdots m_5|\bar{p}_1|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_8]\bar{p}_2}
\\
&\quad
-\tfrac{560}{3} \,{{\bm{\mathsf{A}}}}_{n[m_1m_2|\bar{p}_1|}\, {{\bm{\mathsf{A}}}}_{m_3m_4m_5|\bar{p}_2|}\,{{\bm{\mathsf{A}}}}_{m_6m_7m_8]q}
\\
&\quad
+840\, \epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{n[m_1m_2|\bar{p}_1|}\,{{\bm{\mathsf{A}}}}_{m_3m_4m_5|q|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_8]\bar{p}_2}
\\
&\quad
-840\, \epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{n[m_1m_2|q|}\,{{\bm{\mathsf{A}}}}_{m_3m_4m_5|\bar{p}_1|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_8]\bar{p}_2}
\\
&\quad
-210\, \epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_2m_3}\, {{\bm{\mathsf{A}}}}_{m_4m_5 |p_1 p_2|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_8]q}
\\
&\quad
+1470\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_2m_3}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_4m_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|\bar{p}_1\bar{p}_2|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{m_8]q}
\\
&\quad
-23100\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{n[m_1}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{m_2|\bar{p}_1|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{m_3m_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{m_5|\bar{p}_2|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{m_6m_7}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{m_8]q} \,.
\end{split}\end{aligned}$$ Here, the equality $$\begin{aligned}
{{\mathsf{N}}}^{{{\alpha}}_1\cdots{{\alpha}}_n}_{n; m_1\cdots m_p, n_1\cdots n_q, p_1\cdots p_r,\cdots}
\simeq (\cdots)^{{{\alpha}}_1\cdots{{\alpha}}_n}_{n m_1\cdots m_p n_1\cdots n_q p_1\cdots p_r \cdots} \,,\end{aligned}$$ denotes that it is valid only for the indices satisfying the restriction rules $$\begin{aligned}
\{m_1,\dotsc, m_p\} \supset \{n_1,\dotsc, n_q\} \supset \{p_1,\dotsc, p_r\} \supset \cdots\,,\qquad
{{\alpha}}_1=\cdots={{\alpha}}_n\,.
\label{eq:B-restriction}\end{aligned}$$
Now, let us comment more on the restriction rules given in and . As already mentioned, our parameterizations of ${{\hat{N}}}$ and ${{\mathsf{N}}}$ are valid only for the restricted components. One of the motivations of this paper is to provide a firm ground to study the worldvolume dynamics of exotic branes. For that purpose, it will be enough to consider the restricted components, because the other components, which break the restriction rules, do not couple to any supersymmetric branes [@hep-th/0611036; @0708.2287; @1108.5067; @1109.2025; @1009.4657]. Components satisfying and breaking the rule are contained in different duality orbits, and they can be separated. Indeed, in our parameterizations, the restricted components of ${{\hat{N}}}$ and ${{\mathsf{N}}}$ are always parameterized by the restricted components of mixed-symmetry potentials. For example, in the parameterization of ${{\hat{N}}}_{j;i_1\cdots i_8, k_1k_2k_3}$ given in , as long as the rules $\{i_1,\dotsc, i_8\} \supset\{k_1,k_2,k_3\}$ is satisfied, the dual graviton ${{\hat{A}}}_{i_1\cdots i_8, \bar{k}_3}$ appearing on the right-hand side also satisfies the rule $\{i_1,\dotsc, i_8\} \supset\{k_3\}$.[^6] In this sense, our parameterizations respect the restriction rule, and the $T$-duality rules obtained in the next section connect only the restricted components in type IIA/IIB theories. In other words, components breaking the restriction rule do not appear in our $T$-duality rules.
Duality rules {#sec:duality}
=============
As it has been discussed in [@1909.01335], the two 1-forms ${\mathcal A}_\mu^I$ and $\bm{{\mathcal A}}_\mu^{\mathsf{I}}$ are the same object expressed in different bases. Indeed, they are related through a constant matrix $S$ as $$\begin{aligned}
{\mathcal A}_\mu^I = S^I{}_{\mathsf{I}}\,\bm{{\mathcal A}}_\mu^{\mathsf{I}}\,,\end{aligned}$$ which is called the linear map [@1701.07819] (see also [@hep-th/0402140]). In order to explain the linear map in more detail, let us further decompose the internal indices in M-theory and type IIB theory as $$\begin{aligned}
\text{M-theory:}\quad \{\underline{i}\}=\{{\mathsf{a}},{{\alpha}}\}\,,\qquad
\text{Type IIB theory:}\quad \{\underline{m}\}=\{{\mathsf{a}},{{\mathsf{y}}}\} \,,
\label{eq:internal-decomposition}\end{aligned}$$ where ${\mathsf{a}}=d,\dotsc,8$ and ${{\alpha}}={{y}},{{z}}$. Under the decomposition, components of the 1-forms $\{{\mathcal A}_\mu^I\}$ in M-theory and $\{\bm{{\mathcal A}}_\mu^{\mathsf{I}}\}$ in type IIB theory are decomposed into ${\text{SL}}(n-2)\times {\text{SL}}(2)$ tensors. Then, the linear maps connect the ${\text{SL}}(n-2)\times {\text{SL}}(2)$ tensors in the two theories. In this paper, we consider the following linear maps (which are extensions of [@1701.07819; @1909.01335]): $$\begin{aligned}
\begin{split}
&{\mathcal A}_\mu^{{\mathsf{a}}} \overset{\tiny\textcircled{a}}{=} \bm{{\mathcal A}}_\mu^{{\mathsf{a}}}\,, \quad
{\mathcal A}_\mu^{{{\alpha}}} \!\overset{\tiny\textcircled{c}\textcircled{e}}{=}\! \bm{{\mathcal A}}_{\mu;{{\mathsf{y}}}}^{{{\alpha}}} \,, \quad
{\mathcal A}_{\mu;{\mathsf{a}}_1{\mathsf{a}}_2} \overset{\tiny\textcircled{\raisebox{0.3pt}{g}}}{=} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1{\mathsf{a}}_2{{\mathsf{y}}}}\,, \quad
{\mathcal A}_{\mu;{\mathsf{a}}{{\alpha}}} \!\overset{\tiny\textcircled{\raisebox{-0.9pt}{d}}\textcircled{\raisebox{-0.9pt}{f}}}{=}\! \bm{{\mathcal A}}_{\mu;{\mathsf{a}}}^{{{\beta}}}\,\epsilon_{{{\beta}}{{\alpha}}} \,, \quad
{\mathcal A}_{\mu;{{y}}{{z}}} \overset{\tiny\textcircled{\raisebox{-0.9pt}{b}}}{=} \bm{{\mathcal A}}_{\mu}^{{{\mathsf{y}}}} \,,
\\
&{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5} \overset{\tiny\textcircled{o}}{=} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5{{\mathsf{y}}},{{\mathsf{y}}}} \,, \quad
{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_4{{\alpha}}} \!\overset{\tiny\textcircled{\raisebox{-0.7pt}{i}}\textcircled{n}}{=}\! \bm{{\mathcal A}}^{{{\beta}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_4{{\mathsf{y}}}}\,\epsilon_{{{\beta}}{{\alpha}}}\,,\quad
{\mathcal A}_{\mu;{\mathsf{a}}_1{\mathsf{a}}_2{\mathsf{a}}_3 {{y}}{{z}}} \overset{\tiny\textcircled{\raisebox{-0.9pt}{h}}}{=} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1{\mathsf{a}}_2{\mathsf{a}}_3} \,,
\\
&{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\alpha}},{\mathsf{a}}} \!\overset{\tiny\textcircled{r}\textcircled{t}}{\simeq}\! \bm{{\mathcal A}}^{{{\beta}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}},{\mathsf{a}}{{\mathsf{y}}}}\,\epsilon_{{{\beta}}{{\alpha}}} \,,\qquad
{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6({{\alpha}}_1,{{\alpha}}_2)} \!\overset{\tiny\textcircled{\raisebox{-0.9pt}{k}}\textcircled{x}}{\simeq}\! \bm{{\mathcal A}}^{{{\beta}}_1{{\beta}}_2}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}}}\,\epsilon_{{{\beta}}_1{{\alpha}}_1}\,\epsilon_{{{\beta}}_2{{\alpha}}_2} \,,
\\
&{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5{{y}}{{z}},{\mathsf{a}}} \overset{\tiny\textcircled{p}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5{{\mathsf{y}}},{\mathsf{a}}}\,,\qquad
{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5 {{y}}{{z}},{{\alpha}}} \!\overset{\tiny\textcircled{j}\textcircled{\raisebox{-0.9pt}{\~n}}}{=}\! \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5}^{{{\beta}}}\,\epsilon_{{{\beta}}{{\alpha}}} \,,
\\
&{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{y}}{{z}},{\mathsf{b}}_1{\mathsf{b}}_2{{\alpha}}} \!\overset{\tiny\textcircled{s}\textcircled{u}}{\simeq}\! \bm{{\mathcal A}}^{{{\beta}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}},{\mathsf{b}}_1{\mathsf{b}}_2}\,\epsilon_{{{\beta}}{{\alpha}}}\,,\qquad
{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{y}}{{z}},{\mathsf{a}}{{y}}{{z}}} \overset{\tiny\textcircled{q}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6,{\mathsf{a}}} \,,
\\
&{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7,{\mathsf{a}}} \overset{\tiny\textcircled{v}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{\mathsf{y}}},{\mathsf{a}}{{\mathsf{y}}},{{\mathsf{y}}}} \,,\qquad
{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{y}}{{z}},{\mathsf{a}},{\mathsf{b}}} \overset{\tiny\textcircled{w}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{\mathsf{y}}},{\mathsf{a}}{{\mathsf{y}}},{\mathsf{b}}} \,,
\\
&{\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{y}}{{z}}, ({{\alpha}}_1, {{\alpha}}_2)} \!\overset{\tiny\textcircled{l}\textcircled{y}}{\simeq}\! \bm{{\mathcal A}}^{{{\beta}}_1{{\beta}}_2}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7} \,\epsilon_{{{\beta}}_1{{\alpha}}_1}\,\epsilon_{{{\beta}}_2{{\alpha}}_2} \,,\quad
\rlap{$\displaystyle {\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8({{\alpha}}_1,{{\alpha}}_2,{{\alpha}}_3)} \!\overset{\tiny\textcircled{m}\textcircled{z}}{\simeq}\! \bm{{\mathcal A}}^{{{\beta}}_1{{\beta}}_2{{\beta}}_3}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8{{\mathsf{y}}}} \,\epsilon_{{{\beta}}_1{{\alpha}}_1}\,\epsilon_{{{\beta}}_2{{\alpha}}_2}\,\epsilon_{{{\beta}}_3{{\alpha}}_3}\,.$}
\end{split}
\label{eq:linear-map}\end{aligned}$$ These relate the M-theory fields (left-hand side) and the type IIB fields (right-hand side), and by rewriting the M-theory fields in terms of type IIA fields, we obtain the $T$-duality rules between type IIA/IIB theories. By decomposing the indices ${{\alpha}},{{\beta}}$ into ${{y}}\equiv{\bm{1}}$ and ${{z}}\equiv{\bm{2}}$, and taking into account of the restriction rule , we find that there are 27 linear maps.[^7] They correspond to the 27 $T$-duality lines – depicted in Figure \[fig:duality-web\]. In the following subsections, we obtain the 27 $T$-duality rules from the above linear maps.
Before proceeding, let us comment on the second restriction rule in type IIB theory . Apparently, it looks different from the first restriction rule, but in fact both of them can be understood as a consequence of the M-theory rule [@1907.07177]. As an example, let us consider the last linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8({{\alpha}}_1,{{\alpha}}_2,{{\alpha}}_3)} \simeq \bm{{\mathcal A}}^{{{\beta}}_1{{\beta}}_2{{\beta}}_3}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8{{\mathsf{y}}}} \,\epsilon_{{{\beta}}_1{{\alpha}}_1}\,\epsilon_{{{\beta}}_2{{\alpha}}_2}\,\epsilon_{{{\beta}}_3{{\alpha}}_3}$. The M-theory rule requires $$\begin{aligned}
\{{\mathsf{a}}_1,\cdots, {\mathsf{a}}_8,{{\alpha}}_1\}\supset \{{{\alpha}}_2\} \supset \{{{\alpha}}_3\}\,,\end{aligned}$$ and this leads to ${{\alpha}}_1={{\alpha}}_2={{\alpha}}_3$. Then, the corresponding type IIB field $\bm{{\mathcal A}}^{{{\beta}}_1{{\beta}}_2{{\beta}}_3}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8{{\mathsf{y}}}}$ need to satisfy ${{\beta}}_1={{\beta}}_2={{\beta}}_3$ and the second restriction rule in type IIB theory is derived.
Standard potentials {#sec:standard-T}
-------------------
Let us begin with a comparison of the linear maps – with the standard $T$-duality rules. For this purpose, we parameterize the type IIA fields and the $S$-duality-covariant type IIB fields by using familiar fields. We parameterize the 7-form and 9-form in type IIA theory as $$\begin{aligned}
{{\mathscr{A}}}_7 = {{\mathscr{C}}}_7 - \tfrac{1}{3!}\,{{\mathscr{C}}}_3\wedge{{\mathscr{B}}}_2\wedge{{\mathscr{B}}}_2\,,\qquad
{{\mathscr{A}}}_9 = {{\mathscr{C}}}_9 - \tfrac{1}{4!}\,{{\mathscr{C}}}_3\wedge{{\mathscr{B}}}_2\wedge{{\mathscr{B}}}_2\wedge{{\mathscr{B}}}_2 \,,
\label{eq:IIA-7-9}\end{aligned}$$ and parameterize the type IIB fields as follows: $$\begin{aligned}
({{\mathsf{m}}}_{{{\alpha}}{{\beta}}}) &\equiv {\operatorname{e}^{{{\Phi}}}} \begin{pmatrix} {\operatorname{e}^{-2\,{{\Phi}}}} +({{\mathsf{C}}}_0)^2 & {{\mathsf{C}}}_0 \\ {{\mathsf{C}}}_0 & 1 \end{pmatrix},\quad
({{\mathsf{A}}}^{{{\alpha}}}_2) \equiv \begin{pmatrix} {{\mathsf{B}}}_2 \\\ -{{\mathsf{C}}}_2\end{pmatrix} ,\quad
{{\mathsf{A}}}_4 \equiv {{\mathsf{C}}}_4 - \tfrac{1}{2}\,{{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2\,,
\label{eq:IIB-param}
\\
({{\mathsf{A}}}^{{{\alpha}}}_6) &\equiv \begin{pmatrix} {{\mathsf{C}}}_6 - \tfrac{1}{3!}\, {{\mathsf{C}}}_2 \wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2 \\ -\bigl({{\mathsf{B}}}_6 - \tfrac{2}{3!}\,{{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2\bigr) \end{pmatrix} ,\quad
({{\bm{\mathsf{A}}}}^{{{\alpha}}}_6) \equiv \begin{pmatrix} {{\bm{\mathsf{C}}}}_6 \\ - {{\bm{\mathsf{D}}}}_6 \end{pmatrix},
\\
({{\mathsf{A}}}^{{{\alpha}}{{\beta}}}_8) &\simeq \begin{pmatrix} {{\mathsf{A}}}^{{\bm{11}}}_8 \\ {{\mathsf{A}}}^{{\bm{22}}}_8 \end{pmatrix}
\equiv \begin{pmatrix}
{{\mathsf{C}}}_8 - \tfrac{1}{4!}\, {{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2 \wedge {{\mathsf{B}}}_2 \wedge {{\mathsf{B}}}_2 \\
{{\mathsf{E}}}_8 - \tfrac{3}{4!}\, {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2 \end{pmatrix}, \quad
({{\mathsf{A}}}^{{{\alpha}}}_{8,2}) \equiv \begin{pmatrix} {{\bm{\mathsf{D}}}}_{8,2} \\\ -{{\bm{\mathsf{E}}}}_{8,2} \end{pmatrix} ,
\\
({{\mathsf{A}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10}) &\simeq \begin{pmatrix} {{\mathsf{A}}}^{{\bm{111}}}_{10} \\ {{\mathsf{A}}}^{{\bm{222}}}_{10} \end{pmatrix}
\equiv \begin{pmatrix}
{{\mathsf{C}}}_{10} - \tfrac{1}{5!}\,{{\mathsf{C}}}_2 \wedge {{\mathsf{B}}}_2 \wedge {{\mathsf{B}}}_2 \wedge {{\mathsf{B}}}_2 \wedge {{\mathsf{B}}}_2 \\
-\bigl({{\mathsf{F}}}_{10} - \tfrac{4}{5!}\,{{\mathsf{B}}}_2 \wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2 \bigr) \end{pmatrix}.
\label{eq:10-form-param}\end{aligned}$$ These parameterizations are given such that the linear maps – in reproduce the standard $T$-duality rules for NS–NS fields, $$\begin{aligned}
\begin{split}
{{\mathscr{A}}}_\mu^a &\overset{\text{A--B}}{=} {{\mathsf{A}}}_\mu^a\,,\qquad
{{\mathscr{A}}}_\mu^{{{y}}} \overset{\text{A--B}}{=} {{\mathsf{B}}}_{\mu{{\mathsf{y}}}} + {{\mathsf{A}}}_\mu^{p}\,{{\mathsf{B}}}_{p{{\mathsf{y}}}} \,, \qquad
{{\mathsf{A}}}_\mu^{{{\mathsf{y}}}} \overset{\text{B--A}}{=} {{\mathscr{B}}}_{\mu {{y}}} + {{\mathscr{A}}}_\mu^{p}\,{{\mathscr{B}}}_{p{{y}}} \,,
\\
{{\mathscr{B}}}_{ab} &\overset{\text{A--B}}{=} {{\mathsf{B}}}_{ab} - \tfrac{{{\mathsf{B}}}_{a{{\mathsf{y}}}}\,{{\mathsf{g}}}_{b{{\mathsf{y}}}}-{{\mathsf{g}}}_{a{{\mathsf{y}}}}\,{{\mathsf{B}}}_{b{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,,\qquad
{{\mathscr{B}}}_{a{{y}}} \overset{\text{A--B}}{=} -\tfrac{{{\mathsf{g}}}_{a{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \,,
\\
{{\mathsf{B}}}_{ab} &\overset{\text{B--A}}{=} {{\mathscr{B}}}_{ab} - \tfrac{{{\mathscr{B}}}_{a{{y}}}\,{{\mathscr{g}}}_{b{{y}}}-{{\mathscr{g}}}_{a{{y}}}\,{{\mathscr{B}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\,,\qquad
{{\mathsf{B}}}_{ay} \overset{\text{B--A}}{=} -\tfrac{{{\mathscr{g}}}_{a{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \,,
\end{split}\end{aligned}$$ and R–R fields, $$\begin{aligned}
\begin{split}
{{\mathscr{C}}}_{a_1\cdots a_{n-1}{{y}}}&\overset{\text{A--B}}{=} {{\mathsf{C}}}_{a_1\cdots a_{n-1}} - \tfrac{(n-1)\,{{\mathsf{C}}}_{[a_1\cdots a_{n-2}|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_{n-1}]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,,
\\
{{\mathscr{C}}}_{a_1\cdots a_n} &\overset{\text{A--B}}{=} {{\mathsf{C}}}_{a_1\cdots a_n{{\mathsf{y}}}} - n\, {{\mathsf{C}}}_{[a_1\cdots a_{n-1}}\, {{\mathsf{B}}}_{a_n]{{\mathsf{y}}}} - \tfrac{n\,(n-1)\,{{\mathsf{C}}}_{[a_1\cdots a_{n-2}|{{\mathsf{y}}}|}\, {{\mathsf{B}}}_{a_{n-1}|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_n]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,,
\\
{{\mathsf{C}}}_{a_1\cdots a_{n-1}{{\mathsf{y}}}}&\overset{\text{B--A}}{=} {{\mathscr{C}}}_{a_1\cdots a_{n-1}} - \tfrac{(n-1)\,{{\mathscr{C}}}_{[a_1\cdots a_{n-2}|{{y}}|}\,{{\mathscr{g}}}_{a_{n-1}]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\,,
\\
{{\mathsf{C}}}_{a_1\cdots a_n} &\overset{\text{B--A}}{=} {{\mathscr{C}}}_{a_1\cdots a_n{{y}}} - n\, {{\mathscr{C}}}_{[a_1\cdots a_{n-1}}\, {{\mathscr{B}}}_{a_n]{{y}}} - \tfrac{n\,(n-1)\,{{\mathscr{C}}}_{[a_1\cdots a_{n-2}|{{y}}|}\, {{\mathscr{B}}}_{a_{n-1}|{{y}}|}\,{{\mathscr{g}}}_{a_n]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\,.
\end{split}\end{aligned}$$ Here, the indices $a_1,a_2,\cdots$ are 9D indices, which are orthogonal to the $T$-duality direction ${{y}}$ or ${{\mathsf{y}}}$. On the other hand, in the linear map , the indices are restricted to ${\mathsf{a}}_1,{\mathsf{a}}_2,\cdots$ which run over the internal $(n-2)$ dimensions \[recall \]. By assuming that the $T$-duality rules have the 9D covariance, we have extended the linear map by replacing ${\mathsf{a}}$ with $a$. This is always assumed in the following discussion.
Under the above parameterizations, the field strengths in type IIB supergravity become $$\begin{aligned}
\begin{split}
&\bigl({{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_1\bigr) = \begin{pmatrix} 1 & 0 \\ -{{\mathsf{C}}}_0 & 1 \end{pmatrix} \begin{pmatrix} {\operatorname{e}^{2{{\Phi}}}}{\mathcal G}_1 & {{\mathrm{d}}}{{\Phi}}\\ {{\mathrm{d}}}{{\Phi}}& - {\mathcal G}_1 \end{pmatrix} \begin{pmatrix} 1 & -{{\mathsf{C}}}_0 \\ 0 & 1 \end{pmatrix} , \quad
\bigl({{\mathsf{F}}}^{{{\alpha}}}_3\bigr) = \begin{pmatrix} 1 & 0 \\ -{{\mathsf{C}}}_0 & 1 \end{pmatrix} \begin{pmatrix} {{\mathsf{H}}}_3 \\ -{\mathcal G}_3 \end{pmatrix},
\\
&{{\mathsf{F}}}_5 = {\mathcal G}_5 \,, \qquad
\bigl({{\mathsf{F}}}^{{{\alpha}}}_7\bigr) = \begin{pmatrix} 1 & 0 \\ -{{\mathsf{C}}}_0 & 1 \end{pmatrix} \begin{pmatrix} {\mathcal G}_7 \\ -{{\mathsf{H}}}_7 \end{pmatrix} , \qquad
\bigl({{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_9\bigr) \simeq \begin{pmatrix} {{\mathsf{F}}}^{{\bm{11}}}_9 \\ {{\mathsf{F}}}^{{\bm{22}}}_9 \end{pmatrix} = \begin{pmatrix} {\mathcal G}_9 \\ {{\mathsf{H}}}_9 \end{pmatrix}\,,
\end{split}\end{aligned}$$ where we have defined $$\begin{aligned}
{{\mathsf{H}}}_3 &\equiv {{\mathrm{d}}}{{\mathsf{B}}}_2\,,\qquad {\mathcal G}_{2p+1} \equiv {{\mathrm{d}}}{{\mathsf{C}}}_{2p} - {{\mathsf{H}}}_3\wedge {{\mathsf{C}}}_{2p-2}\quad ({{\mathsf{C}}}_{-2}\equiv 0)\,,
\\
{{\mathsf{H}}}_7 &\equiv {{\mathrm{d}}}{{\mathsf{B}}}_6 - {{\mathsf{C}}}_4\wedge{{\mathrm{d}}}{{\mathsf{C}}}_2 -\tfrac{1}{2}\,{{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{H}}}_3 - {{\mathsf{C}}}_0\,{\mathcal G}_7\,,
\\
{{\mathsf{H}}}_9 &\equiv {{\mathrm{d}}}{{\mathsf{E}}}_8 - {{\mathsf{B}}}_6 \wedge {{\mathrm{d}}}{{\mathsf{C}}}_2 - \tfrac{1}{3!}\,{{\mathsf{H}}}_3\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\,.\end{aligned}$$ By using the Hodge star operator $*$ in the string frame ($*_{\mathrm{E}}\alpha_p={\operatorname{e}^{\frac{p-5}{2}{{\Phi}}}}*\alpha_p$), we obtain $$\begin{aligned}
&{\mathcal G}_p = (-1)^{\frac{p(p-1)}{2}} *{\mathcal G}_{10-p}\,,\qquad {{\mathsf{H}}}_7 = {\operatorname{e}^{-2{{\Phi}}}} * {{\mathsf{H}}}_3 \,,
\\
&{{\mathsf{H}}}_9 = -2{\operatorname{e}^{-2{{\Phi}}}}{{\mathsf{C}}}_0 *{{\mathrm{d}}}{{\Phi}}+ \bigl({{\mathsf{C}}}_0^2 - {\operatorname{e}^{-2{{\Phi}}}}\bigr)\,* {\mathcal G}_1\,.\end{aligned}$$ The electric-magnetic duality for ${{\mathsf{H}}}_9$ is considered further in section \[sec:S-duality\].
For later convenience, we also define the following 6-form, which has been used in [@0712.3235] $$\begin{aligned}
{\mathcal B}_6 \equiv {{\mathsf{B}}}_6 - {{\mathsf{C}}}_4\wedge {{\mathsf{C}}}_2 \,.\end{aligned}$$ We also introduce several redefinitions of the dual graviton. The type IIB dual graviton ${\mathcal N}_{7,1}$[^8] introduced in [@0712.3235] is related to our ${{\mathsf{A}}}_{7, 1}$ as $$\begin{aligned}
{\mathcal N}_{m_1 \cdots m_7, n} &\simeq {{\mathsf{A}}}_{m_1 \cdots m_7, n} -7\,{{\mathsf{B}}}_{[m_1\cdots m_6}\,{{\mathsf{B}}}_{m_7]n}
-\tfrac{105}{4}\, {{\mathsf{C}}}_{[m_1\cdots m_4}\,({{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7]n}-3\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{B}}}_{m_7]n})
{\nonumber}\\
&\quad -\tfrac{315}{4}\, {{\mathsf{C}}}_{[m_1m_2}\,{{\mathsf{C}}}_{m_3m_4}\,{{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{B}}}_{m_7]n}\,. \end{aligned}$$ As we see below, this is useful to simplify the $T$-duality rules (although the $S$-duality covariance is not manifest). In M-theory, we introduce $$\begin{aligned}
\bm{{{\hat{A}}}}_{i_1\cdots i_8 , j} \equiv {{\hat{A}}}_{i_1\cdots i_8 , j} - \tfrac{56}{3}\, {{\hat{A}}}_{[i_1\cdots i_6}\,{{\hat{A}}}_{i_7i_8] j}\,,\end{aligned}$$ and define $\pmb{{{\mathscr{A}}}}_{7,1}$ as $\pmb{{{\mathscr{A}}}}_{m_1\cdots m_7,n} \equiv \bm{{{\hat{A}}}}_{m_1\cdots m_7{{z}},n}$. More explicitly, we have $$\begin{aligned}
\pmb{{{\mathscr{A}}}}_{m_1\cdots m_7,n} &\simeq {{\mathscr{A}}}_{m_1 \cdots m_7, n}
+\tfrac{14}{3}\, {{\mathscr{B}}}_{[m_1\cdots m_6}\,{{\mathscr{B}}}_{m_7] n}
{\nonumber}\\
&\quad - 14\, {{\mathscr{C}}}_{[m_1\cdots m_5}\,{{\mathscr{C}}}_{m_6m_7]n}
+ 70\, {{\mathscr{C}}}_{[m_1m_2m_3}\,{{\mathscr{C}}}_{m_4m_5|n|}\,{{\mathscr{B}}}_{m_6m_7]} \,.\end{aligned}$$ The type IIA dual graviton $\pmb{{{\mathscr{A}}}}_{7,n}$ associated with a Killing direction (i.e. isometry direction) $n$ corresponds to ${\mathcal N}^{(7)}$ of [@0712.3235], and it is also useful to simplify the $T$-duality rules.
Potentials for solitonic five branes
------------------------------------
In this subsection, we consider the potentials that couple to the solitonic 5-branes (i.e. 5-branes with tensions $T\propto g_s^{-2}$). We begin by reproducing the known $T$-duality rules –. There, we demonstrate that redefinitions of potentials can make the $T$-duality rules simpler. After reproducing the known results, we obtain new $T$-duality rules –. In section \[sec:D-potential\], we find the field redefinitions, which make the $T$-duality rules considerably simple, and discuss the relation to the potentials studied in the context of the double field theory (DFT) [@hep-th/9302036; @hep-th/9305073; @hep-th/9308133; @0904.4664; @1006.4823].
#### \
By substituting our parameterizations into the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_4{{y}}} \!\overset{\tiny\textcircled{n}}{=}\! -\bm{{\mathcal A}}^{{\bm{2}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_4{{\mathsf{y}}}}$, we find $$\begin{aligned}
{{\mathscr{B}}}_{a_1\cdots a_5{{y}}}&\overset{\text{A--B}}{=} {{\mathsf{B}}}_{a_1\cdots a_5 {{\mathsf{y}}}}
-5\,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4a_5]}
-5\,\bigl({{\mathsf{C}}}_{[a_1\cdots a_4}-\tfrac{2\,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_4|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\,{{\mathsf{C}}}_{a_5]{{\mathsf{y}}}} \,,
\\
{{\mathsf{B}}}_{a_1\cdots a_5{{\mathsf{y}}}}
&\overset{\text{B--A}}{=}{{\mathscr{B}}}_{a_1\cdots a_5{{y}}}
+5\,{{\mathscr{C}}}_{[a_1\cdots a_4|{{y}}|}\,\bigl({{\mathscr{C}}}_{a_5]}-\tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_5]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)
{\nonumber}\\
&\quad\ +5\,\bigl({{\mathscr{C}}}_{[a_1a_2a_3}-\tfrac{3\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{C}}}_{a_4a_5]{{y}}} \,.\end{aligned}$$ They have been obtained in [@hep-th/9806169] (see Appendix \[app:sugra\] for their conventions).
They can be simplified by using the 6-form ${\mathcal B}_6$ [@0712.3235] $$\begin{aligned}
{{\mathscr{B}}}_{a_1\cdots a_5{{y}}}&\overset{\text{A--B}}{=} {\mathcal B}_{a_1\cdots a_5 {{\mathsf{y}}}}
+5\,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, \bigl({{\mathsf{C}}}_{a_4a_5]}- \tfrac{2\, {{\mathsf{C}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr) \,,
\\
{\mathcal B}_{a_1\cdots a_5{{\mathsf{y}}}}
&\overset{\text{B--A}}{=}{{\mathscr{B}}}_{a_1\cdots a_5{{y}}}
- 5\,\bigl({{\mathscr{C}}}_{[a_1a_2a_3}-\tfrac{3\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{C}}}_{a_4a_5]{{y}}} \,. \end{aligned}$$
On the other hand, if we employ the $S$-duality-covariant fields, we find $$\begin{aligned}
{{\mathscr{B}}}_{a_1\cdots a_5{{y}}}&\overset{\text{A--B}}{=} {{\bm{\mathsf{D}}}}_{a_1\cdots a_5 {{\mathsf{y}}}}
+5\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{C}}}_{a_4a_5]} - \tfrac{2\,{{\mathsf{C}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)
{\nonumber}\\
&\quad\ +\tfrac{5}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{C}}}_{a_4a_5]} - \tfrac{6\,{{\mathsf{C}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\,,
\label{eq:B6-B6-S}
\\
{{\bm{\mathsf{D}}}}_{a_1\cdots a_5 {{\mathsf{y}}}}
&\overset{\text{B--A}}{=} {{\mathscr{B}}}_{a_1\cdots a_5{{y}}}
-5\,\bigl({{\mathscr{C}}}_{[a_1a_2a_3}-\tfrac{2\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{C}}}_{a_4a_5]{{y}}}
{\nonumber}\\
&\quad\ +10\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,\bigl({{\mathscr{C}}}_{a_3}-\tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{B}}}_{a_4a_5]}\,.\end{aligned}$$ Since our linear maps have the $S$-duality covariance, the $T$-duality rules are covariant under $S$-duality. In the present example, we can uplift the $T$-duality rule into $$\begin{aligned}
{{\hat{A}}}_{a_1\cdots a_5{{\alpha}}}&\overset{\text{M--B}}{=} \Bigl[{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_1\cdots a_5 {{\mathsf{y}}}}
+5\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{A}}}^{{{\beta}}}_{a_4a_5]} - \tfrac{2\,{{\mathsf{A}}}^{{{\beta}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)
{\nonumber}\\
&\qquad +\tfrac{5}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{A}}}^{{{\beta}}}_{a_4a_5]} - \tfrac{6\,{{\mathsf{A}}}^{{{\beta}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\Bigr]\,\epsilon_{{{\beta}}{{\alpha}}}\,,\end{aligned}$$ which indeed reproduce by choosing ${{\alpha}}={{y}}$. On the other hand, ${{\alpha}}={{z}}$ gives $$\begin{aligned}
{{\mathscr{C}}}_{a_1\cdots a_5} - 5\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{B}}}_{a_4a_5]} &\overset{\text{A--B}}{=} {{\bm{\mathsf{C}}}}_{a_1\cdots a_5 {{\mathsf{y}}}}
+5\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{B}}}_{a_4a_5]} - \tfrac{2\,{{\mathsf{B}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)
{\nonumber}\\
&\qquad +\tfrac{5}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{B}}}_{a_4a_5]} - \tfrac{6\,{{\mathsf{B}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr) \,,\end{aligned}$$ which corresponds to the $T$-duality rule for R–R potentials obtained before. Each of the $T$-duality rules obtained in this paper has this kind of $S$-dual partner.
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5 {{y}}{{z}},{{y}}} \!\overset{\tiny\textcircled{\raisebox{-0.9pt}{\~n}}}{=}\! -\bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5}^{{\bm{2}}}$, we obtain [@hep-th/9806169] $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_6{{y}}, {{y}}}&\overset{\text{A--B}}{=} {{\mathsf{B}}}_{a_1\cdots a_6}
- \tfrac{6\,{{\mathsf{B}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
- 30\, {{\mathsf{B}}}_{[a_1a_2}\,{{\mathsf{C}}}_{a_3a_4}\, \bigl({{\mathsf{C}}}_{a_5a_6]} -\tfrac{4\,{{\mathsf{C}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr)
{\nonumber}\\
&\quad\ + \tfrac{20\, {{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\mathsf{C}}}_{a_4a_5}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \,,
\\
{{\mathsf{B}}}_{a_1\cdots a_6}
&\overset{\text{B--A}}{=}
{{\mathscr{A}}}_{a_1\cdots a_6{{y}},{{y}}}
-6\,{{\mathscr{B}}}_{[a_1\cdots a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]{{y}}}
-30\,{{\mathscr{C}}}_{[a_1\cdots a_4|{{y}}|}\,\bigl({{\mathscr{C}}}_{a_5}-\tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_5]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \bigr)\,{{\mathscr{B}}}_{a_6]{{y}}}
{\nonumber}\\
&\quad -10\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]{{y}}}
+30\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{C}}}_{a_3a_4|{{y}}|}\,\bigl({{\mathscr{B}}}_{a_5a_6]} -\tfrac{3\,{{\mathscr{B}}}_{a_5|{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr) \,.\end{aligned}$$ By using the potentials $\pmb{{{\mathscr{A}}}}_{7,1}$ and ${\mathcal B}_6$, we can simplify the duality rules as [@0712.3235] $$\begin{aligned}
\pmb{{{\mathscr{A}}}}_{a_1\cdots a_6{{y}}, {{y}}}&\overset{\text{A--B}}{=} {\mathcal B}_{a_1\cdots a_6}
- \tfrac{2\,{\mathcal B}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
+ 5\,{{\mathsf{C}}}_{[a_1\cdots a_4}\, \bigl({{\mathsf{C}}}_{a_5a_6} - \tfrac{2\,{{\mathsf{C}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr) \,,
\\
{\mathcal B}_{a_1\cdots a_6}
&\overset{\text{B--A}}{=}
\pmb{{{\mathscr{A}}}}_{a_1\cdots a_6{{y}},{{y}}}
-2\,{{\mathscr{B}}}_{[a_1\cdots a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]{{y}}}
-5\,{{\mathscr{C}}}_{[a_1\cdots a_4|{{y}}|}\, {{\mathscr{C}}}_{a_5a_6]{{y}}}
{\nonumber}\\
&\quad\
+30\,\bigl({{\mathscr{C}}}_{[a_1a_2a_3}-\tfrac{3\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]{{y}}} \,.\end{aligned}$$ Instead, if we use the $S$-duality covariant fields, we find $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_6{{y}}, {{y}}}&\overset{\text{A--B}}{=} {{\bm{\mathsf{D}}}}_{a_1\cdots a_6}
- \tfrac{6\,{{\bm{\mathsf{D}}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
+15\, {{\bm{\mathsf{A}}}}_{[a_1\cdots a_4}\, \bigl({{\mathsf{C}}}_{a_5a_6]} -\tfrac{2\,{{\mathsf{C}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr)
{\nonumber}\\
&\quad\
- \tfrac{40\, {{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4a_5}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
- \tfrac{30\, \epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{C}}}_{[a_1a_2} \, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,,
\\
{{\bm{\mathsf{D}}}}_{a_1\cdots a_6}
&\overset{\text{B--A}}{=}
{{\mathscr{A}}}_{a_1\cdots a_6{{y}},{{y}}}
-6 \,{{\mathscr{B}}}_{[a_1\cdots a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]{{y}}}
-15\,{{\mathscr{C}}}_{[a_1\cdots a_4|{{y}}|}\, {{\mathscr{C}}}_{a_5a_6]{{y}}}
{\nonumber}\\
&\quad\
+45\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{C}}}_{a_3a_4|{{y}}|}\,\bigl({{\mathscr{B}}}_{a_5a_6}-\tfrac{2\,{{\mathscr{B}}}_{a_5|{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \bigr)
{\nonumber}\\
&\quad\
+50\,\bigl({{\mathscr{C}}}_{[a_1a_2a_3}-\tfrac{3\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \bigr)\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]{{y}}}
{\nonumber}\\
&\quad\
-60\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,\bigl({{\mathscr{C}}}_{a_3}-\tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\, {{\mathscr{B}}}_{a_4a_5}\,{{\mathscr{B}}}_{a_6]{{y}}}\,.\end{aligned}$$ The $S$-duality counterpart is the $T$-duality rule , relating the R–R 7-form and 6-form.
In the following, we adopt $\{{{\mathscr{B}}}_6,\,\pmb{{{\mathscr{A}}}}_{7,1},\,{\mathcal B}_6,\,{\mathcal N}_{7,1}\}$ for the $S$-duality non-covariant expressions while $\{{{\mathscr{B}}}_6,\,{{\mathscr{A}}}_{7,1},\,{{\bm{\mathsf{D}}}}_6,\,{{\bm{\mathsf{A}}}}_{7,1}\}$ for the $S$-duality covariant expressions.
#### \
From ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5} \overset{\tiny\textcircled{o}}{=} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5{{\mathsf{y}}},{{\mathsf{y}}}}$, we obtain [@0712.3235] $$\begin{aligned}
{{\mathscr{B}}}_{a_1\cdots a_6}&\overset{\text{A--B}}{=}
{\mathcal N}_{a_1 \cdots a_6 {{\mathsf{y}}}, {{\mathsf{y}}}}
-6\, {\mathcal B}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\, {{\mathsf{B}}}_{a_6] {{\mathsf{y}}}}
-30\, {{\mathsf{C}}}_{[a_1 a_2 a_3|{{\mathsf{y}}}|}\, \bigl({{\mathsf{C}}}_{a_4a_5}-\tfrac{2\, {{\mathsf{C}}}_{a_4|{{\mathsf{y}}}|}\, {{\mathsf{g}}}_{a_5|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\, {{\mathsf{B}}}_{a_6] {{\mathsf{y}}}} \,,
\\
{\mathcal N}_{a_1\cdots a_6{{\mathsf{y}}},{{\mathsf{y}}}}
&\overset{\text{B--A}}{=} {{\mathscr{B}}}_{a_1\cdots a_6}
-\tfrac{6\,{{\mathscr{B}}}_{[a_1\cdots a_5|{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \,.\end{aligned}$$ For the $S$-duality covariant fields, we find [@1909.01335] $$\begin{aligned}
{{\mathscr{B}}}_{a_1\cdots a_6}&\overset{\text{A--B}}{=}
{{\bm{\mathsf{A}}}}_{a_1 \cdots a_6 {{\mathsf{y}}}, {{\mathsf{y}}}}
-15\, \epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, \bigl({{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\, {{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6] {{\mathsf{y}}}}
+\tfrac{2\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\, {{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\,,
\\
{{\bm{\mathsf{A}}}}_{a_1\cdots a_6{{\mathsf{y}}},{{\mathsf{y}}}}
&\overset{\text{B--A}}{=} {{\mathscr{B}}}_{a_1\cdots a_6}
-15\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{B}}}_{a_4a_5}\,\bigl({{\mathscr{C}}}_{a_6}-\tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)
-\tfrac{15\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \,,\end{aligned}$$ which are self-dual under $S$-duality.
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5{{y}}{{z}},{\mathsf{a}}} \overset{\tiny\textcircled{p}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_5{{\mathsf{y}}},{\mathsf{a}}}$, we obtain $$\begin{aligned}
\pmb{{{\mathscr{A}}}}_{a_1\cdots a_6{{y}}, b}&\overset{\text{A--B}}{\simeq}
{\mathcal N}_{a_1\cdots a_6{{\mathsf{y}}}, b} - \tfrac{1}{3}\,\tfrac{{\mathcal N}_{a_1\cdots a_6{{\mathsf{y}}}, {{\mathsf{y}}}}\,{{\mathscr{g}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}
-4\,{\mathcal B}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6] b}
{\nonumber}\\
&\quad\
-\bigl({\mathcal B}_{a_1\cdots a_6}-\tfrac{2\,{\mathcal B}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathsf{B}}}_{b{{\mathsf{y}}}}
+5\,{{\mathsf{C}}}_{[a_1\cdots a_4|b{{\mathsf{y}}}|}\,\bigl({{\mathsf{C}}}_{a_5a_6]} -\tfrac{2\,{{\mathsf{C}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)
{\nonumber}\\
&\quad\
-\tfrac{5}{2}\,\bigl({{\mathsf{C}}}_{[a_1\cdots a_4} - \tfrac{4\,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_4|{{\mathsf{y}}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr) \, {{\mathsf{C}}}_{a_5a_6]b{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
- 5\,{{\mathsf{C}}}_{[a_1\cdots a_4}\,\bigl({{\mathsf{C}}}_{a_5a_6]} -\tfrac{2\,{{\mathsf{C}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathsf{B}}}_{b{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-20\,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{C}}}_{a_4a_5} -\tfrac{2\,{{\mathsf{C}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_5|{{\mathsf{y}}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathsf{B}}}_{a_6]b} \,,
\\
{\mathcal N}_{a_1\cdots a_6{{\mathsf{y}}}, b}
&\overset{\text{B--A}}{\simeq}
\pmb{{{\mathscr{A}}}}_{a_1\cdots a_6{{y}}, b}
- \tfrac{\pmb{{{\mathscr{A}}}}_{a_1\cdots a_6{{y}}, {{y}}}\,{{\mathscr{g}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}
+4\,{{\mathscr{B}}}_{[a_1\cdots a_5|{{y}}|} \, \bigl({{\mathscr{B}}}_{a_6]b} - \tfrac{{{\mathscr{B}}}_{a_6]{{y}}}\,{{\mathscr{g}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)
{\nonumber}\\
&\quad\
-\tfrac{1}{3}\, {{\mathscr{B}}}_{a_1\cdots a_6}\,{{\mathscr{B}}}_{b{{y}}}
+\tfrac{6\,{{\mathscr{B}}}_{[a_1\cdots a_5|{{y}}|}\, {{\mathscr{g}}}_{a_6]{{y}}}\,{{\mathscr{B}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}
+2\,{{\mathscr{C}}}_{[a_1\cdots a_5}\,{{\mathscr{C}}}_{a_6]b{{y}}}
{\nonumber}\\
&\quad\
+\tfrac{15\,{{\mathscr{C}}}_{[a_1\cdots a_4|{{y}}|}\,{{\mathscr{C}}}_{a_5|b{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}
+\tfrac{5}{2}\, {{\mathscr{C}}}_{[a_1\cdots a_4|{{y}}|}\, \bigl({{\mathscr{C}}}_{a_5a_6]b} - \tfrac{{{\mathscr{C}}}_{a_5a_6]{{y}}}\,{{\mathscr{g}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr) \,.\end{aligned}$$ They are partially obtained in Eq. (5.13) of [@hep-th/9908094] under the truncation ${{\mathsf{B}}}_2 = 0={{\mathsf{C}}}_2$. The full result without the truncation is obtained in [@1909.01335]. The same $T$-duality map seems to be obtained in Eqs. (3.10) and (3.11) of [@0907.3614], although the relation to our potentials is not clear.
On the other hand, by using the $S$-duality covariant fields, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_6{{y}}, b}&\overset{\text{A--B}}{\simeq}
{{\bm{\mathsf{A}}}}_{a_1\cdots a_6{{\mathsf{y}}}, b} - \tfrac{{{\bm{\mathsf{A}}}}_{a_1 \cdots a_6 {{\mathsf{y}}}, {{\mathsf{y}}}}\,{{\mathsf{g}}}_{b{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
+ 6\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1 \cdots a_5 |{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6]b}
+ \tfrac{30\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1 \cdots a_4 |b{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+10\,\bigl({{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|} + \tfrac{{{\bm{\mathsf{A}}}}_{[a_1a_2|b{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_3|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr)\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6]{{\mathsf{y}}}}
+20\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6]b}
{\nonumber}\\
&\quad\
-30\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6]{{\mathsf{y}}}}
-\tfrac{10\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6]{{\mathsf{y}}}}\,{{\mathsf{g}}}_{b{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{30\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|b{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
+\tfrac{20\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4|b|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{60\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{15}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_6]{{\mathsf{y}}}} {\nonumber}\\
&\quad\
+\tfrac{45}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|b{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{15}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,,
\\
{{\bm{\mathsf{A}}}}_{a_1\cdots a_6{{\mathsf{y}}}, b}
&\overset{\text{B--A}}{\simeq}
{{\mathscr{A}}}_{a_1\cdots a_6{{y}}, b}
-{{\mathscr{B}}}_{a_1\cdots a_6}\,{{\mathscr{B}}}_{b{{y}}}
-6\,{{\mathscr{B}}}_{[a_1\cdots a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]b}
+6\,{{\mathscr{C}}}_{[a_1\cdots a_5}\,{{\mathscr{C}}}_{a_6]b{{y}}}
{\nonumber}\\
&\quad\
-10\,{{\mathscr{C}}}_{[a_1a_2a_3|b{{y}}|}\,\bigl({{\mathscr{C}}}_{a_4a_5a_6]}+\tfrac{3}{2}\,\tfrac{{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)
{\nonumber}\\
&\quad\
-15\,{{\mathscr{C}}}_{[a_1a_2a_3|b{{y}}|}\,\bigl({{\mathscr{C}}}_{a_4}- \tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_4|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{B}}}_{a_5a_6]}
{\nonumber}\\
&\quad\
+20\,{{\mathscr{C}}}_{[a_1a_2a_3}\,\bigl({{\mathscr{C}}}_{a_4a_5|b}- \tfrac{{{\mathscr{C}}}_{a_4a_5|{{y}}}\,{{\mathscr{g}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{B}}}_{|a_6]{{y}}}
-50\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{C}}}_{a_4|b{{y}}|}\,{{\mathscr{B}}}_{a_5a_6]}
{\nonumber}\\
&\quad\
+\tfrac{15}{2}\,{{\mathscr{C}}}_{[a_1a_2|b|}\,{{\mathscr{C}}}_{a_3a_4|{{y}}|}\,{{\mathscr{B}}}_{a_5a_6]}
+\tfrac{20\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{B}}}_{a_6]{{y}}}\,{{\mathscr{g}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{15\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{g}}}_{a_6]{{y}}}\,{{\mathscr{B}}}_{b{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}
+\tfrac{45\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{C}}}_{a_3a_4|{{y}}|}\,{{\mathscr{C}}}_{a_5|b|}\,{{\mathscr{g}}}_{a_6]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}
{\nonumber}\\
&\quad\
+15\,{{\mathscr{C}}}_{[a_1a_2a_3}\,\bigl({{\mathscr{C}}}_{a_4}-\tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_4|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{B}}}_{a_5a_6]}\,{{\mathscr{B}}}_{b{{y}}}
{\nonumber}\\
&\quad\
-45\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,\bigl({{\mathscr{C}}}_{a_3}-\tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,{{\mathscr{B}}}_{a_4a_5}\,{{\mathscr{B}}}_{a_6]b} \,.\end{aligned}$$ They are again self-dual under $S$-duality.
#### A short comment\
In the following, we present new results. The $T$-duality rules obtained below are rather lengthy, and we determine the maps only from type IIA fields to type IIB fields. However, in section \[sec:D-potential\], we find a redefinition of mixed-symmetry potentials, which transforms our potentials into the potentials ${{D}}_{6}$, ${{D}}_{7,1}$, and ${{D}}_{8,2}$. The $T$-duality rules for the new fields are very simple, and one can easily find the inverse map, if necessary.
#### \
The linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{y}}{{z}},{\mathsf{a}}{{y}}{{z}}} \overset{\tiny\textcircled{q}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6,{\mathsf{a}}}$ gives $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_7 {{\mathsf{y}}}, b{{\mathsf{y}}}}
&\overset{\text{A--B}}{\simeq} {{\bm{\mathsf{A}}}}_{a_1\cdots a_7, b}
-\tfrac{35 \,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\ -\tfrac{105}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{105}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|b|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{315}{4}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,,\end{aligned}$$ which is self-dual under $S$-duality.
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{z}},{\mathsf{a}}} \!\overset{\tiny\textcircled{r}}{\simeq}\! \bm{{\mathcal A}}^{{\bm{1}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}},{\mathsf{a}}{{\mathsf{y}}}}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_7, b} &\overset{\text{A--B}}{\simeq} {{\bm{\mathsf{D}}}}_{a_1\cdots a_7{{\mathsf{y}}}, b{{\mathsf{y}}}}
-7\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_6 |{{\mathsf{y}}},{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_7]b}
-\tfrac{42\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_5 |b{{\mathsf{y}}}, {{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}{\nonumber}\\
&\quad\
-\tfrac{21}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{C}}}}_{[a_1\cdots a_5|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
+ 21\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{C}}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|b|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-\tfrac{105\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{C}}}}_{[a_1\cdots a_4 |b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
+140\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4a_5 |b{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
- 70\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6 |b|}\,{{\mathsf{B}}}_{a_7]{{\mathsf{y}}}}
- \tfrac{70\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4a_5 |b{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
- 70\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7]}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{b{{\mathsf{y}}}}
+140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|b|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+105\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2 |b{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5a_6}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
+\tfrac{140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_4|{{\mathsf{y}}}|}{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5a_6}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|b{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_4|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{105\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2 |b{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{1575}{8}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\mathsf{B}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_6a_7]}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{b{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{2835}{4}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\mathsf{B}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|b|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_6|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+ \tfrac{315}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\mathsf{B}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_6|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{b{{\mathsf{y}}}}\,{{\mathsf{g}}}_{|a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \,.
\label{eq:A71-D82}\end{aligned}$$ Under the simplifying assumption ${{\mathsf{B}}}_2={{\mathsf{C}}}_2=0$, this map has been obtained in [@hep-th/9908094] \[the last line of Eq. (5.12)\], where $N^{(8)}$ corresponds to our ${{\bm{\mathsf{D}}}}_{8, b{{\mathsf{y}}}}$ (up to ${{\mathsf{B}}}_2={{\mathsf{C}}}_2=0$). The $S$-dual counterpart of this $T$-duality rule is obtained later in .
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{y}}{{z}},{\mathsf{b}}_1{\mathsf{b}}_2{{z}}} \!\overset{\tiny\textcircled{s}}{\simeq}\! \bm{{\mathcal A}}^{{\bm{1}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}},{\mathsf{b}}_1{\mathsf{b}}_2}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_7{{y}}, b_1b_2}
&\overset{\text{A--B}}{=} {{\bm{\mathsf{D}}}}_{a_1\cdots a_7{{\mathsf{y}}}, b_1b_2}
-7\,{{\bm{\mathsf{C}}}}_{[a_1\cdots a_6}\,{{\bm{\mathsf{A}}}}_{a_7] b_1b_2{{\mathsf{y}}}}
-\tfrac{7}{2} \,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{C}}}}_{[a_1\cdots a_6}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{|b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-\tfrac{35}{2} \,{{\bm{\mathsf{C}}}}_{[a_1a_2a_3 |b_1b_2{{\mathsf{y}}}|}\, \bigl({{\bm{\mathsf{A}}}}_{a_4\cdots a_7]} + \tfrac{2\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)
{\nonumber}\\
&\quad\
+\tfrac{105}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{C}}}}_{[a_1\cdots a_4 |\overline{b}_1{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5a_6}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]\overline{b}_2}
-\tfrac{105}{2} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{C}}}}_{[a_1a_2a_3 |b_1b_2{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+35\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_7]\overline{b}_2}
-\tfrac{315}{2} \,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}_{a_4a_5|\overline{b}_2{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+ 70\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6|\overline{b}_2|}\,{{\mathsf{B}}}_{a_7]{{\mathsf{y}}}}
+ \tfrac{105\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4 |b_1b_2{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+ \tfrac{105\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b_1b_2|}\,{{\bm{\mathsf{A}}}}_{a_3a_4a_5|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{105}{4} \,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+35\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|\overline{b}_1|}\,{{\mathsf{B}}}_{a_7]\overline{b}_2}
+\tfrac{35}{4} \,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|b_1b_2|}\,{{\mathsf{B}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
-\tfrac{315}{2} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2 |\overline{b}_1{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]}\,{{\mathsf{g}}}_{\overline{b}_2{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{70\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}}\,{{\mathsf{B}}}_{b_1b_2} \,{{\mathsf{g}}}_{|a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{35\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|\overline{b}_1{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6|\overline{b}_2|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{280\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\, {{\mathsf{B}}}_{a_6|\overline{b}_2|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4|\overline{b}_1}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{\overline{b}_2{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{35}{2} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{315}{4} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-315\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_6|\overline{b}_2|}\,{{\mathsf{B}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{2835}{8}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|\overline{b}_2{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+\tfrac{1575}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|\overline{b}_2|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+\tfrac{315}{4}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{\overline{b}_2{{\mathsf{y}}}}\,{{\mathsf{B}}}_{|a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{105}{8}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|\overline{b}_1{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]} \,{{\mathsf{g}}}_{\overline{b}_2{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{105}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6a_7]} \,{{\mathsf{g}}}_{\overline{b}_2{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{525}{4}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|\overline{b}_1{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6|\overline{b}_2|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{210\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{B}}}_{a_6|\overline{b}_2|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,.
\label{eq:D82A-D82B}\end{aligned}$$ The $S$-dual counterpart of this $T$-duality rule is found later in . It may be possible to make the expression simpler by finding some $S$-duality-covariant field redefinitions, but here do not attempt to find such redefinitions.
### $T$-dual-manifest redefinitions {#sec:D-potential}
Now, let us consider the field redefinition that makes the $T$-duality rules very simple. In the case of the R–R fields in type IIA/IIB theory, the R–R polyform in the $C$-basis is defined as $$\begin{aligned}
{{\mathscr{C}}}\equiv {{\mathscr{C}}}_1 + {{\mathscr{C}}}_3 + {{\mathscr{C}}}_5 + {{\mathscr{C}}}_7 + {{\mathscr{C}}}_9 \,, \qquad
{{\mathsf{C}}}\equiv {{\mathsf{C}}}_0 + {{\mathsf{C}}}_2 + {{\mathsf{C}}}_4 + {{\mathsf{C}}}_6 + {{\mathsf{C}}}_8 + {{\mathsf{C}}}_{10} \,.\end{aligned}$$ By considering a redefinition into the $A$-basis [@hep-th/0103233], $$\begin{aligned}
{{A}}\equiv {\operatorname{e}^{-{{\mathscr{B}}}_2\wedge}}{{\mathscr{C}}}\,,\qquad
{{A}}\equiv {\operatorname{e}^{-{{\mathsf{B}}}_2\wedge}}{{\mathsf{C}}}\,,
\label{eq:A-def}\end{aligned}$$ we find that the $T$-duality rules for the new fields are simple [@hep-th/0103149] $$\begin{aligned}
{{A}}_{a_1\cdots a_p} \overset{\text{A--B}}{=} {{A}}_{a_1\cdots a_p{{\mathsf{y}}}}\,,\qquad
{{A}}_{a_1\cdots a_p{{y}}} \overset{\text{A--B}}{=} {{A}}_{a_1\cdots a_p}\,.
\label{eq:A-T-dual}\end{aligned}$$ This is according to the fact that the $A$-basis transforms as an ${\text{O}}(10,10)$ spinor. As studied in [@hep-th/9907132; @1106.5452; @1107.0008], if we define the (real) gamma matrices $\{\Gamma^M\} = \{\Gamma^m,\,\Gamma_m\}$ that satisfy $$\begin{aligned}
\bigl\{\Gamma^M,\, \Gamma^N \bigr\} = \eta^{MN}\,,\qquad
(\eta^{MN})\equiv \begin{pmatrix} 0 & {{\delta}}^m_n \\ {{\delta}}_m^n & 0 \end{pmatrix},\qquad
(\Gamma_m)^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}= \Gamma^m \,,\qquad
(\Gamma^m)^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}= \Gamma_m \,,\end{aligned}$$ and also define the Clifford vacuum ${\lvert {0} \rangle}$ as $$\begin{aligned}
\begin{split}
\Gamma_m{\lvert {0} \rangle} =0\,,\qquad \langle 0\vert 0\rangle = 1 \,, \qquad \Gamma^{11}{\lvert {0} \rangle} = {\lvert {0} \rangle}\,,
\end{split}\end{aligned}$$ where ${\langle {0} \rvert}\equiv {\lvert {0} \rangle}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}$ and $\Gamma^{11}\equiv (-1)^{N_F}$ ($N_F\equiv \Gamma^m\,\Gamma_m$), we find that $$\begin{aligned}
{\lvert {{{A}}} \rangle}\equiv \sum_p \tfrac{1}{p!}\,{{A}}_{m_1\dots m_p}\,\Gamma^{m_1\cdots m_p}{\lvert {0} \rangle}\,,\qquad
\Gamma^{M_1\cdots M_p}\equiv \Gamma^{[M_1}\cdots \Gamma^{M_p]}\,,\end{aligned}$$ transforms as an ${\text{O}}(10,10)$ spinor. Here, the R–R field ${\lvert {{{A}}} \rangle}$ is defined to have a definite chirality $$\begin{aligned}
\Gamma^{11}{\lvert {{{A}}} \rangle} = \mp {\lvert {{{A}}} \rangle} \quad (\text{IIA/IIB}) \,. \end{aligned}$$ Under the factorized $T$-duality along the $x^y$-direction, it transforms as $$\begin{aligned}
{\lvert {{{A}}} \rangle} \to {\lvert {{{A}}'} \rangle} = \bigl(\Gamma^y -\Gamma_y\bigr)\,\Gamma^{11}\,{\lvert {{{A}}} \rangle} \,,\end{aligned}$$ and in terms of the components, this transformation rule gives the rules .
Similarly, the potentials which couple to the solitonic 5-branes also constitute an ${\text{O}}(10,10)$-covariant potential denoted by ${{D}}_{M_1\cdots M_4}$ [@1102.0934], where the 20D indices are totally antisymmetric. This tensor can be generally decomposed into ${\text{SL}}(10)$ tensors: $$\begin{aligned}
\begin{split}
{{D}}^{m_1m_2m_3m_4} &= \tfrac{1}{6!}\,\epsilon^{m_1\cdots m_4n_1\cdots n_6}\,{{D}}_{n_1\cdots n_6} \,,
\\
{{D}}^{m_1m_2m_3}{}_{m_4} &= \tfrac{1}{7!}\,\epsilon^{m_1m_2m_3n_1\cdots n_7}\, {{D}}_{n_1\cdots n_7,m_4}+\cdots \,,
\\
{{D}}^{m_1m_2}{}_{m_3m_4} &= \tfrac{1}{8!}\,\epsilon^{m_1m_2n_1\cdots n_8}\, {{D}}_{n_1\cdots n_8,m_3m_4}+\cdots \,,
\\
{{D}}^{m_1}{}_{m_2m_3m_4} &= \tfrac{1}{9!}\,\epsilon^{m_1n_1\cdots n_9}\, {{D}}_{n_1\cdots n_9, m_2m_3m_4}+\cdots \,,
\\
{{D}}_{m_1m_2m_3m_4} &= \tfrac{1}{10!}\,\epsilon^{n_1\cdots n_{10}}\,{{D}}_{n_1\cdots n_{10},m_1\cdots m_4}\,,
\end{split}\end{aligned}$$ where the ellipsis denote the irrelevant contribution from the potentials that do not couple to supersymmetric branes. Under the $T$-duality along the $x^y$-direction, this transforms as $$\begin{aligned}
{{D}}'_{M_1\cdots M_4} = \Lambda_{M_1}{}^{N_1}\cdots \Lambda_{M_4}{}^{N_4}\,{{D}}_{N_1\cdots N_4} \,, \qquad
(\Lambda_M{}^N) \equiv
\begin{pmatrix}
\bm{1}- e_y & e_y \\
e_y & \bm{1}- e_y
\end{pmatrix} ,\end{aligned}$$ where $e_y$ is a $10\times 10$ matrix, $e_y={\operatorname{diag}}(0,\dotsc,0,\underbrace{1}_y,0,\dotsc,0)$. By rewriting the transformation rule in terms of the component fields ${{D}}_6$, ${{D}}_{7,1}$, ${{D}}_{8,2}$, ${{D}}_{9,3}$, and ${{D}}_{10,4}$, we obtain $$\begin{aligned}
\begin{split}
{{D}}_{a_1\cdots a_6b_1\cdots b_n,b_1\cdots b_n} &\overset{\text{A--B}}{=} {{D}}_{a_1\cdots a_6b_1\cdots b_n{{\mathsf{y}}},b_1\cdots b_n{{\mathsf{y}}}}\quad (n=0,\dotsc,3)\,,
\\
{{D}}_{a_1\cdots a_5b_1\cdots b_n{{y}},b_1\cdots b_n} &\overset{\text{A--B}}{=} {{D}}_{a_1\cdots a_5b_1\cdots b_n{{\mathsf{y}}},b_1\cdots b_n}\quad (n=0,\dotsc ,4)\,,
\\
{{D}}_{a_1\cdots a_6b_1\cdots b_n{{y}},b_1\cdots b_n{{y}}} &\overset{\text{A--B}}{=} {{D}}_{a_1\cdots a_6b_1\cdots b_n,b_1\cdots b_n}\quad (n=0,\dotsc ,3)\,.
\end{split}
\label{eq:D-T-duality}\end{aligned}$$
Similar to the case of the R–R potential ${{A}}_{p}$, which is related to our potentials as , it is natural to expect that ${{D}}_{6+n,n}$ are also obtained by considering a redefinition of our mixed-symmetry potentials. Indeed, if we redefine the type IIA fields as $$\begin{aligned}
{{D}}_{m_1\cdots m_6} &\equiv {{\mathscr{B}}}_{m_1\cdots m_6} -3\,{{\mathscr{C}}}_{[m_1\cdots m_5}\,{{\mathscr{C}}}_{m_6]}\,,
\\
{{D}}_{m_1\cdots m_7, n}
&\simeq {{\mathscr{A}}}_{m_1\cdots m_7, n}
+ 7\, {{\mathscr{B}}}_{[m_1\cdots m_6} \,{{\mathscr{B}}}_{m_7]n}
- \tfrac{1}{2}\, {{\mathscr{C}}}_{m_1\cdots m_7} \,{{\mathscr{C}}}_{n}
- \tfrac{21}{2}\, {{\mathscr{C}}}_{[m_1\cdots m_5}\,{{\mathscr{C}}}_{m_6m_7]n}
{\nonumber}\\
&\quad + 70\, {{\mathscr{C}}}_{[m_1m_2m_3}\, {{\mathscr{C}}}_{m_4m_5 |n|}\, {{\mathscr{B}}}_{m_6m_7]}
+ 21\, {{\mathscr{C}}}_{[m_1\cdots m_5}\, {{\mathscr{B}}}_{m_6|n|}\, {{\mathscr{C}}}_{m_7]} \,,
\label{eq:D71A}
\\
{{D}}_{m_1\cdots m_8,\,n_1n_2}
&\simeq {{\mathscr{A}}}_{m_1\cdots m_8,\,n_1n_2}
- 4\,{{\mathscr{C}}}_{[m_1\cdots m_7}\,{{\mathscr{C}}}_{m_8]\bar{n}_1\bar{n}_2}
- 4\,{{\mathscr{C}}}_{[m_1\cdots m_7}\,{{\mathscr{C}}}_{m_8]}\,{{\mathscr{B}}}_{n_1n_2}
{\nonumber}\\
&\quad
- 56\,{{\mathscr{C}}}_{[m_1\cdots m_6|\bar{n}_1|}\,{{\mathscr{C}}}_{m_7}\,{{\mathscr{B}}}_{m_8]\bar{n}_2}
-168\,{{\mathscr{C}}}_{[m_1\cdots m_5}\,{{\mathscr{C}}}_{m_6}\,{{\mathscr{B}}}_{m_7|\bar{n}_1|}\,{{\mathscr{B}}}_{m_8]\bar{n}_2}
{\nonumber}\\
&\quad
+ 84\,{{\mathscr{C}}}_{[m_1\cdots m_5}\,{{\mathscr{B}}}_{m_6m_7}\,{{\mathscr{C}}}_{m_8]\bar{n}_1\bar{n}_2}
+140\,{{\mathscr{C}}}_{[m_1m_2m_3}\,{{\mathscr{B}}}_{m_4m_5}\,{{\mathscr{C}}}_{m_6m_7m_8]\bar{n}_1\bar{n}_2}
{\nonumber}\\
&\quad
-210\,{{\mathscr{C}}}_{[m_1m_2m_3}\,{{\mathscr{B}}}_{m_4m_5}\,{{\mathscr{B}}}_{m_6m_7}\,{{\mathscr{C}}}_{m_8]n_1n_2}\,,\end{aligned}$$ and type IIB fields as $$\begin{aligned}
{{D}}_{m_1\cdots m_6} &\equiv {{\mathsf{B}}}_{m_1\cdots m_6}-\tfrac{15}{2}\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6]}-\tfrac{1}{2}\,{{\mathsf{C}}}_0\, {{\mathsf{C}}}_{m_1\cdots m_6}\,,
\label{eq:D-B-6}
\\
{{D}}_{m_1\cdots m_7, n}
&\simeq {{\mathsf{A}}}_{m_1\cdots m_7, n}
+ \tfrac{7}{2}\, {{\mathsf{C}}}_{[m_1\cdots m_6}\, {{\mathsf{C}}}_{m_7] n}
- \tfrac{7}{2}\, {{\mathsf{C}}}_0\, {{\mathsf{C}}}_{[m_1\cdots m_6}\, {{\mathsf{B}}}_{m_7] n}
- \tfrac{105}{4}\, {{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7]n}
{\nonumber}\\
&\quad
+ \tfrac{105}{4}\, {{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{B}}}_{m_7]n}
+ \tfrac{315}{4}\, {{\mathsf{B}}}_{[m_1m_2}\,{{\mathsf{B}}}_{m_3m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7]n}\,,
\label{eq:D-B-71}
\\
{{D}}_{m_1\cdots m_8, n_1n_2}
&\simeq {{\bm{\mathsf{D}}}}_{m_1\cdots m_8, n_1n_2}
- 4\,{{\mathsf{C}}}_{[m_1\cdots m_7, |\bar{n}_1|}\,{{\mathsf{C}}}_{m_8]\bar{n}_2}
+ 4\,{{\mathsf{C}}}_0\,{{\mathsf{C}}}_{[m_1\cdots m_7, |\bar{n}_1|}\,{{\mathsf{B}}}_{m_8]\bar{n}_2}
{\nonumber}\\
&\quad
+ 28\,{{\mathsf{C}}}_{[m_1\cdots m_6}\,{{\mathsf{B}}}_{m_7|\bar{n}_1|}\,{{\mathsf{C}}}_{m_8]\bar{n}_2}
- 28\,{{\mathsf{C}}}_0\,{{\mathsf{C}}}_{[m_1\cdots m_6}\,{{\mathsf{B}}}_{m_7|\bar{n}_1|}\,{{\mathsf{B}}}_{m_8]\bar{n}_2}
{\nonumber}\\
&\quad
- 28\,{{\mathsf{C}}}_{[m_1\cdots m_5|\bar{n}_1|}\,{{\mathsf{C}}}_{m_6m_7m_8]\bar{n}_2}
+ 84\,{{\mathsf{C}}}_{[m_1\cdots m_5 |\bar{n}_1|}\,{{\mathsf{C}}}_{m_6m_7}\,{{\mathsf{B}}}_{m_8]\bar{n}_2}
{\nonumber}\\
&\quad
+ 420\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6 |n_1n_2|}\,{{\mathsf{B}}}_{m_7m_8]}
- \tfrac{35}{2}\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5\cdots m_8]}\,{{\mathsf{B}}}_{n_1n_2}
{\nonumber}\\
&\quad
-2520\,{{\mathsf{C}}}_{[m_1m_2m_3|\bar{n}_1|}\,{{\mathsf{B}}}_{m_4m_5}\,{{\mathsf{B}}}_{m_6|\bar{n}_2|} \,{{\mathsf{C}}}_{m_7m_8]}
{\nonumber}\\
&\quad
- 840\,{{\mathsf{C}}}_{[m_1m_2m_3|\bar{n}_1|}\,{{\mathsf{B}}}_{m_4m_5}\,{{\mathsf{B}}}_{m_6m_7} \,{{\mathsf{C}}}_{m_8]\bar{n}_2}
{\nonumber}\\
&\quad
- 420\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{B}}}_{m_5|\bar{n}_1|}\,{{\mathsf{B}}}_{m_6|\bar{n}_2|}\,{{\mathsf{C}}}_{m_7m_8]}
{\nonumber}\\
&\quad
+1365\,{{\mathsf{B}}}_{[m_1m_2}\,{{\mathsf{B}}}_{m_3|\bar{n}_1|}\,{{\mathsf{B}}}_{m_4|\bar{n}_2|}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8]}
{\nonumber}\\
&\quad
-2765\,{{\mathsf{B}}}_{[m_1m_2}\,{{\mathsf{B}}}_{m_3m_4}\,{{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7|\bar{n}_1|}\,{{\mathsf{C}}}_{m_8]\bar{n}_2}
{\nonumber}\\
&\quad
+ 805\,{{\mathsf{B}}}_{[m_1m_2}\,{{\mathsf{B}}}_{m_3m_4}\,{{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8]}\,{{\mathsf{C}}}_{n_1n_2} \,,
\label{eq:D-B-82}\end{aligned}$$ the complicated $T$-duality rules obtained in this subsection are surprisingly simplified as .
As a consistency check, let us express the 7-form field strength in type IIA/IIB theory by using the new 6-form ${{D}}_6$. Then, we obtain $$\begin{aligned}
\begin{split}
{{\mathscr{H}}}_7 &= {{\mathrm{d}}}{{D}}_6 - \tfrac{1}{2}\,\bigl({\mathcal G}_6 \wedge {{\mathscr{C}}}_1 - {\mathcal G}_4\wedge {{\mathscr{C}}}_3 + {\mathcal G}_2 \wedge {{\mathscr{C}}}_5\bigr) \,,
\\
{{\mathsf{H}}}_7 &= {{\mathrm{d}}}{{D}}_6 - \tfrac{1}{2}\,\bigl({\mathcal G}_7\,{{\mathsf{C}}}_0 - {\mathcal G}_5\wedge {{\mathsf{C}}}_2 + {\mathcal G}_3\wedge {{\mathsf{C}}}_4 - {\mathcal G}_1\wedge {{\mathsf{C}}}_6\bigr) \,,
\end{split}\end{aligned}$$ and they precisely coincide with the expression given in [@1102.0934] up to conventions. This shows that our ${{D}}_{6+n,n}$ $(n=0,1,2)$ are precisely the same as ${{D}}_{6+n,n}$ studied there, and they can be straightforwardly extended also to $n=3,4$. As shown in [@1903.05601], the field strength, ${{\mathscr{H}}}_7$ or ${{\mathsf{H}}}_7$, can be regarded as a particular component of[^9] $$\begin{aligned}
H_{MNP} \equiv \partial^Q {{D}}_{MNP Q} - \tfrac{1}{2}\,\overline{{\langle {{{A}}} \rvert}} \Gamma_{MNP} {\lvert {F} \rangle}\,, \qquad
{\lvert {F} \rangle} \equiv \Gamma^M\,\partial_M {\lvert {{{A}}} \rangle} \,,
\label{eq:HMNP}\end{aligned}$$ where $\overline{{\langle {A} \rvert}}\equiv {\langle {A} \rvert}\,C^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\equiv ({\lvert {A} \rangle})^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,C^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}$ with $C \equiv (\Gamma^0+ \Gamma_0) \cdots (\Gamma^9 + \Gamma_9)$. The indices $M,N,\cdots$ are raised/lowered by using $\eta_{MN}$ and the derivative $\partial_M$ can be understood as $(\partial_M)=(\partial_m,0)$. Then, we can show that ${{\mathscr{H}}}_7$ or ${{\mathsf{H}}}_7$ in type IIA or IIB theory is reproduced from $$\begin{aligned}
H_7 \equiv \tfrac{1}{7!\,3!}\,\epsilon_{m_1\cdots m_7n_1n_2n_3}\, H^{n_1n_2n_3} \,{{\mathrm{d}}}x^{m_1}\wedge\cdots\wedge {{\mathrm{d}}}x^{m_7}\,.\end{aligned}$$ Other components are also easily computed. For example, the component $H^{a_1a_2}{}_n$ associated with a Killing direction $n$ satisfying $n\not\in \{a_1,\,a_2\}$ gives the field strength of the dual graviton, $$\begin{aligned}
\iota_n H_{8, n} \equiv \tfrac{1}{7!\,2!}\,\epsilon_{m_1\cdots m_7 a_1a_2 n}\, H^{a_1a_2}{}_n \,{{\mathrm{d}}}x^{m_1}\wedge\cdots\wedge{{\mathrm{d}}}x^{m_7}\,.\end{aligned}$$ In type IIA/IIB theory, this reproduces $$\begin{aligned}
\begin{split}
\iota_n {{\mathscr{H}}}_{8, n} &= {{\mathrm{d}}}\iota_n {{D}}_{7, n} + \tfrac{1}{2}\,\bigl(\iota_n F_8 \, \iota_n{{A}}_1 - \iota_n F_6\wedge \iota_n{{A}}_3 + \iota_n F_4 \wedge \iota_n {{A}}_5 - \iota_n F_2 \wedge \iota_n{{A}}_7\bigr) \,,
\label{eq:def-AH7}
\\
\iota_n {{\mathsf{H}}}_{8, n} &= {{\mathrm{d}}}\iota_n {{D}}_{7, n} + \tfrac{1}{2}\,\bigl(\iota_n F_7\wedge \iota_n{{A}}_2 - \iota_n F_5\wedge \iota_n {{A}}_4 + \iota_n F_3\wedge \iota_n{{A}}_6 - \iota_n F_1\,\iota_n {{A}}_8\bigr)\,,
\end{split}\end{aligned}$$ where $F_{p+1}\equiv {{\mathrm{d}}}{{A}}_p$ (note that $\iota_n F_1={\pounds}_n {{A}}_0=0$). The 11D uplift or the $S$-duality-invariant expression is given respectively in section \[sec:gauge-IIA\] or \[sec:gauge-IIB\]. We can compute the other components as well, yielding the field strengths for mixed-symmetry potentials ${{D}}_{8,2}$ and ${{D}}_{9,3}$.
Here, it will be useful to comment on the notion of the level $n$. If we look at, for example, the right-hand side of , terms like ${{\mathsf{C}}}_{....}\,{{\mathsf{C}}}_{..}\,{{\mathsf{B}}}_{..}\,{{\mathsf{B}}}_{..}$ appear, but ${{\mathsf{C}}}_{....}\,{{\mathsf{B}}}_{..}\,{{\mathsf{B}}}_{..}\,{{\mathsf{B}}}_{..}$ never appears. This can be understood by considering the level, which has been introduced in the study of the $E_{11}$ conjecture [@hep-th/0207267; @hep-th/0212291]. In type II theories, a potential which couples to a brane with the tension $T\propto g_s^{-n}$ has the level $n$ [@0805.4451]. For example, ${{\mathsf{B}}}_2$ has level $0$ while the R–R potentials have level $1$. Since the potentials ${{D}}_{6+n,n}$ have level $2$, the level on the right-hand side of must be summed up to $2$, and it is the reason why ${{\mathsf{C}}}_{....}\,{{\mathsf{B}}}_{..}\,{{\mathsf{B}}}_{..}\,{{\mathsf{B}}}_{..}$ does not appear. The level is always respected in various equations, such as the parameterization of ${{\mathsf{N}}}$ given in section \[sec:1-form\], the $T$-duality rules, the field strengths such as , and the field redefinitions, and it helps when we find the complicated redefinitions such as .
Potentials for exotic branes
----------------------------
Here, we find the $T$-duality rules for mixed-symmetry potentials that couple to exotic branes with tensions $T\propto g_s^{-3}$ and $g_s^{-4}$. Since the potentials have many indices, the $T$-duality rules are generally more complicated than before. Thus, we again find only the $T$-duality rules, each of which maps a type IIA potential to type IIB potentials, and by using those, we identify the relation between our potentials and the manifestly $T$-duality-covariant potentials.
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{y}},{\mathsf{a}}} \!\overset{\tiny\textcircled{t}}{\simeq}\! -\bm{{\mathcal A}}^{{\bm{2}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}},{\mathsf{a}}{{\mathsf{y}}}}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_7{{y}}, b} &\overset{\text{A--B}}{\simeq} {{\bm{\mathsf{E}}}}_{a_1\cdots a_7{{\mathsf{y}}}, b{{\mathsf{y}}}}
-7\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_6 |{{\mathsf{y}}},{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_7]b}
-\tfrac{42\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_5 |b{{\mathsf{y}}}, {{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}{\nonumber}\\
&\quad\
-\tfrac{21}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{D}}}}_{[a_1\cdots a_5|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
+ 21\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{D}}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|b|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-\tfrac{105\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{D}}}}_{[a_1\cdots a_4 |b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
+140\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4a_5 |b{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
- 70\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6 |b|}\,{{\mathsf{C}}}_{a_7]{{\mathsf{y}}}}
- \tfrac{70\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4a_5 |b{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
- 70\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7]}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{b{{\mathsf{y}}}}
+140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|b|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+105\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2 |b{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5a_6}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
+\tfrac{140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4|{{\mathsf{y}}}|}{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5a_6}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|b{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3 |{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{105\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2 |b{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{1575}{8}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\mathsf{C}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_6a_7]}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{b{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{2835}{4}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\mathsf{C}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|b|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_6|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+ \tfrac{315}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\mathsf{C}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_6|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{b{{\mathsf{y}}}}\,{{\mathsf{g}}}_{|a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \,.
\label{eq:A81-E82}\end{aligned}$$ This is $S$-dual to the $T$-dual rule . Under ${{\mathsf{B}}}_2={{\mathsf{C}}}_2=0$, this map has been obtained in [@hep-th/9908094] \[the middle line of Eq. (5.12)\], where ${\mathcal N}^{(8)}$ corresponds to ${{\bm{\mathsf{E}}}}_{8, b{{\mathsf{y}}}}$ (under ${{\mathsf{B}}}_2={{\mathsf{C}}}_2=0$).
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{y}}{{z}},{\mathsf{b}}_1{\mathsf{b}}_2 {{y}}} \!\overset{\tiny\textcircled{u}}{\simeq}\! -\bm{{\mathcal A}}^{{\bm{2}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}},{\mathsf{b}}_1{\mathsf{b}}_2}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_7{{y}}, b_1b_2{{y}}}
&\overset{\text{A--B}}{\simeq} {{\bm{\mathsf{E}}}}_{a_1\cdots a_7{{\mathsf{y}}}, b_1b_2}
-7\,{{\bm{\mathsf{D}}}}_{[a_1\cdots a_6}\,{{\bm{\mathsf{A}}}}_{a_7] b_1b_2{{\mathsf{y}}}}
-\tfrac{7}{2} \,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{D}}}}_{[a_1\cdots a_6}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{|b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-\tfrac{35}{2} \,{{\bm{\mathsf{D}}}}_{[a_1a_2a_3 |b_1b_2{{\mathsf{y}}}|}\, \bigl({{\bm{\mathsf{A}}}}_{a_4\cdots a_7]} + \tfrac{2\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)
{\nonumber}\\
&\quad\
+\tfrac{105}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{D}}}}_{[a_1\cdots a_4 |\overline{b}_1{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_5a_6}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7]\overline{b}_2}
-\tfrac{105}{2} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{D}}}}_{[a_1a_2a_3 |b_1b_2{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+35\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_7]\overline{b}_2}
-\tfrac{315}{2} \,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}_{a_4a_5|\overline{b}_2{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+ 70\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}_{a_4a_5a_6|\overline{b}_2|}\,{{\mathsf{C}}}_{a_7]{{\mathsf{y}}}}
+ \tfrac{105\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4 |b_1b_2{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+ \tfrac{105\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b_1b_2|}\,{{\bm{\mathsf{A}}}}_{a_3a_4a_5|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{105}{4} \,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+35\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|\overline{b}_1|}\,{{\mathsf{C}}}_{a_7]\overline{b}_2}
+\tfrac{35}{4} \,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|b_1b_2|}\,{{\mathsf{C}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
-\tfrac{315}{2} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2 |\overline{b}_1{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]}\,{{\mathsf{g}}}_{\overline{b}_2{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{70\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}}\,{{\mathsf{C}}}_{b_1b_2} \,{{\mathsf{g}}}_{|a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{35\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|\overline{b}_1{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6|\overline{b}_2|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{280\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_5|{{\mathsf{y}}}|}\, {{\mathsf{C}}}_{a_6|\overline{b}_2|} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{140\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|\overline{b}_2{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{35}{2} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{315}{4} \,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b_1b_2|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_4|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-315\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_6|\overline{b}_2|}\,{{\mathsf{C}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{2835}{8}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|\overline{b}_2{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+\tfrac{1575}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|\overline{b}_2|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]}
{\nonumber}\\
&\quad\
+\tfrac{315}{4}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{\overline{b}_2{{\mathsf{y}}}}\,{{\mathsf{C}}}_{|a_5a_6} \,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{105}{8}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|\overline{b}_1{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]} \,{{\mathsf{g}}}_{\overline{b}_2{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{105}{2}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6a_7]} \,{{\mathsf{g}}}_{\overline{b}_2{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{525}{4}\,\tfrac{\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|\overline{b}_1{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6|\overline{b}_2|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{210\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_4|\overline{b}_1|}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{a_5|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_6|\overline{b}_2|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\,.
\label{eq:D83A-E82B}\end{aligned}$$ This is $S$-dual to the $T$-dual rule .
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7,{\mathsf{a}}} \overset{\tiny\textcircled{v}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{\mathsf{y}}},{\mathsf{a}}{{\mathsf{y}}},{{\mathsf{y}}}}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_8, b} &\overset{\text{A--B}}{\simeq} {{\bm{\mathsf{A}}}}_{a_1\cdots a_8{{\mathsf{y}}}, b{{\mathsf{y}}}, {{\mathsf{y}}}}
-84\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_5| b{{\mathsf{y}}},{{\mathsf{y}}}|}\,\bigl({{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7} -\tfrac{2\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\ +280\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,\bigl({{\bm{\mathsf{A}}}}_{a_4a_5a_6|b|} - \tfrac{{{\bm{\mathsf{A}}}}_{a_4a_5|b{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\ -560\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_4a_5|b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\ +140\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|{{\mathsf{y}}}|}\,\bigl({{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7a_8]} - \tfrac{8\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}}\,{{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr)\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{b{{\mathsf{y}}}}
{\nonumber}\\
&\quad\ +140\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_4a_5}\, {{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}} \,,\end{aligned}$$ which is self-dual under $S$-duality. Under ${{\mathsf{B}}}_2={{\mathsf{C}}}_2=0$, this map has been obtained in [@hep-th/9908094] \[the first line of Eq. (5.12)\], where $N^{(9)}$ may be related to ${{\bm{\mathsf{A}}}}_{9,n{{\mathsf{y}}},{{\mathsf{y}}}}$.
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{y}}{{z}},{\mathsf{a}},{\mathsf{b}}} \overset{\tiny\textcircled{w}}{\simeq} \bm{{\mathcal A}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{\mathsf{y}}},{\mathsf{a}}{{\mathsf{y}}},{\mathsf{b}}}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_8{{y}}, b, c} &\overset{\text{A--B}}{\simeq}
{{\bm{\mathsf{A}}}}_{a_1\cdots a_8{{\mathsf{y}}}, b{{\mathsf{y}}}, c}
-\tfrac{8\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_7|b{{\mathsf{y}}}, c{{\mathsf{y}}}, {{\mathsf{y}}}} \,{{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+8\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1\cdots a_7|{{\mathsf{y}}}, b{{\mathsf{y}}}|} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]c}
+\tfrac{56\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{[a_1\cdots a_6|b{{\mathsf{y}}}, c{{\mathsf{y}}}|} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7|{{\mathsf{y}}}|}\, {{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-28\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1\cdots a_6|{{\mathsf{y}}}, {{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|b} \,\bigl({{\bm{\mathsf{A}}}}^{{{\delta}}}_{|a_8]c}-\tfrac{2\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{{{\mathsf{y}}}c}\, {{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)
{\nonumber}\\
&\quad\
-84\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1\cdots a_5|b{{\mathsf{y}}}, c|}\, \bigl({{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7} - \tfrac{2\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|{{\mathsf{y}}}|} \, {{\mathsf{g}}}_{a_7|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr) \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{28\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_6|{{\mathsf{y}}},{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7a_8]} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{b{{\mathsf{y}}}} \,{{\mathsf{g}}}_{c{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
-\tfrac{84\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1\cdots a_5|b,{{\mathsf{y}}},{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}} \,{{\mathsf{g}}}_{c{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
- 28 \,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\, \bigl({{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1\cdots a_6} - \tfrac{12\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr) \,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_7|b|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_8]{{\mathsf{y}}}} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{c{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+168 \,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|} \,\bigl({{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|b|} - \tfrac{{{\bm{\mathsf{A}}}}^{{{\beta}}}_{|{{\mathsf{y}}}b|}\,{{\mathsf{g}}}_{a_6|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr)\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|c|} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{168\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1\cdots a_5|b|}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{c{{\mathsf{y}}}} \,{{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{ 84\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|b|} \,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}} \,{{\mathsf{g}}}_{c{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+70\,\bigl({{\bm{\mathsf{A}}}}_{[a_1\cdots a_4} -\tfrac{{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\mathsf{g}}}_{a_4|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr)\,{{\bm{\mathsf{A}}}}_{a_5a_6|b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}_{a_7a_8]c{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+630\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}_{a_4a_5|b{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]c}
{\nonumber}\\
&\quad\
+560\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\, {{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|c|} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-280\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|} \, {{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7a_8]} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{c{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\tfrac{1155\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}_{a_4a_5|b{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|c{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{ 525\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}_{a_4a_5|b{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}} \,{{\mathsf{g}}}_{c{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{ 210\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}_{a_4a_5|b{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|c|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7|{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{ 560\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\, {{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}} \,{{\mathsf{g}}}_{c{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{1120\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\, {{\bm{\mathsf{A}}}}_{a_4a_5a_6|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{c{{\mathsf{y}}}} \,{{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+105\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1\cdots a_4}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_5a_6} \,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_7|b|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_8]{{\mathsf{y}}}} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{c{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-105\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1\cdots a_4}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_5|b|} \,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|c|}\, {{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-210\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_4a_5} \,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|b|}\, \bigl({{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7a_8]}-\tfrac{3\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr) \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{c{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+210\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\bm{\mathsf{A}}}}_{[a_1a_2a_3|b|}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_4a_5} \,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|{{\mathsf{y}}}|}\, \bigl({{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7a_8]}-\tfrac{8\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr) \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{c{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-\tfrac{ 315\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{|b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|c{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{315}{2} \, \tfrac{\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7a_8]}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{b{{\mathsf{y}}}}\,{{\mathsf{g}}}_{c{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{315}{2} \, \tfrac{\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_6|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_7|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_8]{{\mathsf{y}}}}\,{{\mathsf{g}}}_{c{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{315}{2} \, \tfrac{\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_5|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6a_7}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|c{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{315}{2} \, \tfrac{\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\bm{\mathsf{A}}}}_{[a_1a_2|b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{a_3a_4}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_5|c|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_6|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_7|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
+\tfrac{3045}{2}\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1a_2} \,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5} \,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\, \,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_7a_8]} \,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{c{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-\tfrac{11865\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_3|b|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{a_6|{{\mathsf{y}}}|}\, {{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_7|{{\mathsf{y}}}}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{c{{\mathsf{y}}}}\, {{\mathsf{g}}}_{|a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\
-\tfrac{6405}{4} \,\tfrac{\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,\epsilon_{{{\zeta}}{{\eta}}}\, {{\bm{\mathsf{A}}}}^{{{\alpha}}}_{[a_1a_2}\,{{\bm{\mathsf{A}}}}^{{{\beta}}}_{a_3|{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\gamma}}}_{a_4a_5}\,{{\bm{\mathsf{A}}}}^{{{\delta}}}_{|b{{\mathsf{y}}}|}\,{{\bm{\mathsf{A}}}}^{{{\zeta}}}_{a_6a_7}\,{{\bm{\mathsf{A}}}}^{{{\eta}}}_{|c{{\mathsf{y}}}|} \,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \,.\end{aligned}$$
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{y}},{{y}}} \!\overset{\tiny\textcircled{x}}{=}\! \bm{{\mathcal A}}^{{\bm{22}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_6{{\mathsf{y}}}}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_7{{y}}, {{y}}}
&\overset{\text{A--B}}{=} {{\mathsf{E}}}_{a_1\cdots a_7{{\mathsf{y}}}}
-7\,\bigl({{\mathsf{B}}}_{[a_1\cdots a_6}-\tfrac{6\,{{\mathsf{B}}}_{[a_1\cdots a_5|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_6|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\,{{\mathsf{C}}}_{a_7]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\
-35 \,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,\bigl({{\mathsf{C}}}_{a_4a_5}\,{{\mathsf{C}}}_{a_6a_7]} - \tfrac{4\,{{\mathsf{C}}}_{a_4a_5}\,{{\mathsf{C}}}_{a_6|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_7]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr) \,.
\label{eq:A81-BNS7}\end{aligned}$$ The inverse map has been obtained in Eq. (4.7) of [@hep-th/9908094],[^10] and in our convention, we have $$\begin{aligned}
{{\mathsf{E}}}_{a_1\cdots a_7{{\mathsf{y}}}}
&\overset{\text{B--A}}{=}
{{\mathscr{A}}}_{a_1\cdots a_7{{y}}, {{y}}}
+7\, {{\mathscr{A}}}_{[a_1\cdots a_6|{{y}}, {{y}}}\, \bigl({{\mathscr{C}}}_{|a_7]} - \tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{|a_7]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)
{\nonumber}\\
&\quad\
+35\, \bigl({{\mathscr{C}}}_{[a_1a_2a_3}-\tfrac{3\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\, {{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \bigr)\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{C}}}_{a_6a_7]{{y}}}
{\nonumber}\\
&\quad\
+70\,\bigl(2\,{{\mathscr{C}}}_{[a_1a_2a_3}\,{{\mathscr{B}}}_{a_4|{{y}}|} + 3\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{B}}}_{a_3a_4}\bigr)\,{{\mathscr{C}}}_{a_5a_6|{{y}}|}\, \bigl({{\mathscr{C}}}_{a_7]} - \tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_7]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\,. \end{aligned}$$ The $S$-dual counterpart of this $T$-duality is the map connecting ${{\mathscr{C}}}_9$ and ${{\mathsf{C}}}_8$.
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7{{y}}{{z}},{{y}},{{y}}} \!\overset{\tiny\textcircled{y}}{=}\! \bm{{\mathcal A}}^{{\bm{22}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_7}$, we obtain $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_8{{y}}, {{y}}, {{y}}}
&\overset{\text{A--B}}{=} {{\mathsf{E}}}_{a_1\cdots a_8} -\tfrac{8\,{{\mathsf{E}}}_{[a_1\cdots a_7|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
+\tfrac{210\,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4a_5}\,{{\mathsf{C}}}_{a_6a_7}\,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}
{\nonumber}\\
&\quad\ - 315 \,{{\mathsf{C}}}_{[a_1a_2}\,{{\mathsf{C}}}_{a_3a_4}\,\bigl({{\mathsf{B}}}_{a_5a_6}\,{{\mathsf{C}}}_{a_7a_8]} - \tfrac{6\,{{\mathsf{B}}}_{a_5a_6}\,{{\mathsf{C}}}_{a_7|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_8]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr) \,.\end{aligned}$$ This is $S$-dual to the $T$-duality is the map connecting ${{\mathscr{C}}}_9$ and ${{\mathsf{C}}}_{10}$. We can also find the inverse map as $$\begin{aligned}
{{\mathsf{E}}}_{a_1\cdots a_8}
&\overset{\text{B--A}}{=} {{\mathscr{A}}}_{a_1\cdots a_8{{y}}, {{y}}, {{y}}}
-8\, {{\mathscr{A}}}_{[a_1\cdots a_7|{{y}}, {{y}}|}\, {{\mathscr{B}}}_{a_8]{{y}}}
+ 56\, {{\mathscr{A}}}_{[a_1\cdots a_6|{{y}}, {{y}}|}\,{{\mathscr{B}}}_{a_7|{{y}}|} \, \bigl({{\mathscr{C}}}_{a_8]}- \tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_8]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}} \bigr)
{\nonumber}\\
&\quad\ -70\, \bigl({{\mathscr{C}}}_{[a_1a_2a_3}-\tfrac{3\, {{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{g}}}_{a_3|{{y}}|}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)\, {{\mathscr{C}}}_{a_4a_5|{{y}}|}\, {{\mathscr{C}}}_{a_6a_7|{{y}}|}\,{{\mathscr{B}}}_{a_8]{{y}}}
{\nonumber}\\
&\quad\ +315\,{{\mathscr{C}}}_{[a_1a_2|{{y}}|}\,{{\mathscr{C}}}_{a_3a_4|{{y}}|}\,{{\mathscr{C}}}_{a_5a_6|{{y}}|}\,\bigl({{\mathscr{B}}}_{a_7a_8}-\tfrac{2\,{{\mathscr{B}}}_{a_7|{{y}}|}\,{{\mathscr{g}}}_{a_8]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr)
{\nonumber}\\
&\quad\ +1680\, {{\mathscr{B}}}_{[a_1a_2}\,{{\mathscr{B}}}_{a_3|{{y}}|}\,{{\mathscr{C}}}_{a_4a_5|{{y}}|}\,{{\mathscr{C}}}_{a_6a_7|{{y}}|}\,\bigl({{\mathscr{C}}}_{a_8]}- \tfrac{{{\mathscr{C}}}_{|{{y}}|}\,{{\mathscr{g}}}_{a_8]{{y}}}}{{{\mathscr{g}}}_{{{y}}{{y}}}}\bigr) \,.\end{aligned}$$
So far, we have considered the potentials which couple to exotic $(7-p+n)^{(n,p-n)}_3$-branes. Finally, let us consider the only map, which is associated with branes with tension $T\propto g_s^{-4}$.
#### \
From the linear map ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8{{y}},{{y}},{{y}}} \!\overset{\tiny\textcircled{z}}{=}\! -\bm{{\mathcal A}}^{{\bm{222}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8{{\mathsf{y}}}}$, we find $$\begin{aligned}
{{\mathscr{A}}}_{a_1\cdots a_9{{y}}, {{y}}, {{y}}}
&\overset{\text{A--B}}{=} {{\mathsf{F}}}_{a_1\cdots a_9{{\mathsf{y}}}}
-9\,\bigl({{\mathsf{E}}}_{[a_1\cdots a_8} - \tfrac{8\,{{\mathsf{E}}}_{[a_1\cdots a_7|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_8|{{\mathsf{y}}}|}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}} \bigr)\,{{\mathsf{C}}}_{a_9]{{\mathsf{y}}}}
{\nonumber}\\
&\quad\ -315\,{{\mathsf{C}}}_{[a_1a_2a_3|{{\mathsf{y}}}|}\,{{\mathsf{C}}}_{a_4a_5}\,{{\mathsf{C}}}_{a_6a_7}\,\bigl({{\mathsf{C}}}_{a_8a_9]} - \tfrac{6\,{{\mathsf{C}}}_{a_8|{{\mathsf{y}}}|}\,{{\mathsf{g}}}_{a_9]{{\mathsf{y}}}}}{{{\mathsf{g}}}_{{{\mathsf{y}}}{{\mathsf{y}}}}}\bigr)\,.
\label{eq:F1011A-F10B}\end{aligned}$$ The potentials ${{\mathscr{A}}}_{10,1,1}$ and ${{\mathsf{F}}}_{10}$ have level 4 while ${{\mathsf{E}}}_8$ has level 3, and again we can see that the levels indeed match on both sides.
### $T$-dual-manifest redefinitions {#sec:EF-potential}
Having obtained the $T$-duality rules, let us find the field redefinitions that map our mixed-symmetry potentials to the $T$-duality-covariant potentials. As studied in [@1108.5067], potentials ${{E}}_{8+n,p,n}$ ($n=0,1,2$ and $p=\text{odd/even}$ in type IIA/IIB theory) that couple to the exotic $(7-p+n)^{(n,p-n)}_3$-branes constitute the $T$-duality-covariant potential ${{E}}_{MN\dot{a}}$ $(\dot{a}=1,\dotsc,512)$. This transforms in the 87040-dimensional tensor-spinor representation of the ${\text{O}}(10,10)$ group. By using the notation of the ${\text{O}}(10,10)$ spinor given in section \[sec:D-potential\], we denote it as ${\lvert {{{E}}_{MN}} \rangle}$ that satisfies the following conditions: $$\begin{aligned}
{\lvert {{{E}}_{MN}} \rangle} = - {\lvert {{{E}}_{NM}} \rangle}\,,\qquad \Gamma^N\,{\lvert {{{E}}_{NM}} \rangle} = 0\,,\qquad \Gamma^{11}{\lvert {{{E}}_{MN}} \rangle} = \mp {\lvert {{{E}}_{MN}} \rangle} \quad (\text{IIA/IIB})\,.\end{aligned}$$ As discussed in [@1903.05601], if we truncate the components which do not couple to supersymmetric branes, ${\lvert {E_{MN}} \rangle}$ can be parameterized as $$\begin{aligned}
{\lvert {{{E}}^{mn}} \rangle} &= \sum_p \tfrac{1}{8!\,p!}\,\epsilon^{mn q_1\cdots q_8}\,{{E}}_{q_1\cdots q_8,r_1\cdots r_p}\,\Gamma^{r_1\cdots r_p}\,{\lvert {0} \rangle}\,,
\\
{\lvert {{{E}}^m{}_n} \rangle} &= \sum_p \tfrac{1}{9!\,p!}\,\epsilon^{m q_1\cdots q_9}\,{{E}}_{q_1\cdots q_9,r_1\cdots r_p,n}\,\Gamma^{r_1\cdots r_p}\,{\lvert {0} \rangle}\,,
\\
{\lvert {{{E}}_{mn}} \rangle} &= \sum_p \tfrac{1}{10!\,p!}\,\epsilon^{q_1\cdots q_{10}}\,{{E}}_{q_1\cdots q_{10},r_1\cdots r_p,mn}\,\Gamma^{r_1\cdots r_p}\,{\lvert {0} \rangle}\,.\end{aligned}$$ The constraint $\Gamma^N\,{\lvert {{{E}}_{NM}} \rangle} = 0$ is automatically satisfied under the restriction rule for the indices. Under the factorized $T$-duality along the $x^y$-direction, it transforms as $$\begin{aligned}
{\lvert {{{E}}'_{M_1M_2}} \rangle} = \Lambda_{M_1}{}^{N_1}\,\Lambda_{M_2}{}^{N_2}\, \bigl(\Gamma^y -\Gamma_y\bigr)\,\Gamma^{11}\,{\lvert {{{E}}_{N_1N_2}} \rangle} \,,\end{aligned}$$ and in terms of the components, we have $$\begin{aligned}
\begin{split}
{{E}}_{a_1\cdots a_{8+n}, b_1\cdots b_p, c_1\cdots c_n} &\overset{\text{A--B}}{\simeq} {{E}}_{a_1\cdots a_{8+n} {{\mathsf{y}}}, b_1\cdots b_p {{\mathsf{y}}}, c_1\cdots c_n {{\mathsf{y}}}} \,,
\\
{{E}}_{a_1\cdots a_{7+n}{{y}}, b_1\cdots b_p, c_1\cdots c_n} &\overset{\text{A--B}}{\simeq} {{E}}_{a_1\cdots a_{7+n} {{\mathsf{y}}}, b_1\cdots b_p {{\mathsf{y}}}, c_1\cdots c_n} \,,
\\
{{E}}_{a_1\cdots a_{7+n}{{y}}, b_1\cdots b_{p-1} {{y}}, c_1\cdots c_n} &\overset{\text{A--B}}{\simeq} {{E}}_{a_1\cdots a_{7+n} {{\mathsf{y}}}, b_1\cdots b_{p-1} , c_1\cdots c_n} \,,
\\
{{E}}_{a_1\cdots a_{8+n}{{y}}, b_1\cdots b_{p-1} {{y}}, c_1\cdots c_n{{y}}} &\overset{\text{A--B}}{\simeq} {{E}}_{a_1\cdots a_{8+n} , b_1\cdots b_{p-1} , c_1\cdots c_n} \,,
\end{split}
\label{eq:E-T-duality}\end{aligned}$$ where $n=0,1,2$ and $p=1,3,5,7$. The $T$-duality web for the family of potentials ${{E}}_{8+n,p,n}$, which contains our $T$-dualities –, can be summarized as follows: $$\begin{aligned}
\vcenter{\xymatrix@C=15pt@R=18pt{
{{E}}_{8,7} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{8,6} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{8,5} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{8,4} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{8,3} \ar@{<.>}[d]\ar@{<->}[r]^{\tiny\textcircled{u}} & {{E}}_{8,2} \ar@{<.>}[d]\ar@{<->}[r]^{\tiny\textcircled{t}} & {{E}}_{8,1} \ar@{<->}[d]^{\tiny\textcircled{v}}\ar@{<->}[r]^{\tiny\textcircled{x}} & {{E}}_{8} \ar@{<->}[d]^{\tiny\textcircled{y}} \\
{{E}}_{9,8,1} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{9,7,1} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{9,6,1} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{9,5,1} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{9,4,1} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{9,3,1} \ar@{<.>}[d]\ar@{<.>}[r] & {{E}}_{9,2,1} \ar@{<.>}[d]\ar@{<->}[r]^{\tiny\textcircled{w}} & {{E}}_{9,1,1} \ar@{<.>}[d] \\
{{E}}_{10,9,2} \ar@{<.>}[r] & {{E}}_{10,8,2} \ar@{<.>}[r] & {{E}}_{10,7,2} \ar@{<.>}[r] & {{E}}_{10,6,2} \ar@{<.>}[r] & {{E}}_{10,5,2} \ar@{<.>}[r] & {{E}}_{10,4,2} \ar@{<.>}[r] & {{E}}_{10,3,2} \ar@{<.>}[r] & {{E}}_{10,2,2}
}}.\end{aligned}$$
Through trial and error, we have found that the following redefinitions indeed map our mixed-symmetry potentials to the $T$-duality-covariant potentials ${{E}}_{8+n,p,n}$: $$\begin{aligned}
\text{\underline{Type IIA:}}
{\nonumber}\\
{{E}}_{m_1\cdots m_8 , n}
&\simeq {{\mathscr{A}}}_{m_1\cdots m_8, n}
- 56 \,\bigl({{\mathscr{B}}}_{[m_1\cdots m_5|n|} + \tfrac{5}{3} \,{{\mathscr{C}}}_{[m_1\cdots m_4|n|}\,{{\mathscr{C}}}_{m_5}\bigr)\, {{\mathscr{C}}}_{m_6m_7m_8]} \,,
\\
{{E}}_{m_1\cdots m_8 , n_1n_2n_3}
&\simeq {{\mathscr{A}}}_{m_1\cdots m_8 , n_1n_2n_3}
+\tfrac{280}{3} \,{{\mathscr{C}}}_{[m_1\cdots m_4|n_1n_2n_3|}\,{{\mathscr{C}}}_{m_5m_6m_7}\,{{\mathscr{C}}}_{m_8]}
{\nonumber}\\
&\quad -28\,{{\mathscr{B}}}_{[m_1m_2m_3 |n_1n_2n_3|}\,\bigl({{\mathscr{C}}}_{m_4\cdots m_8]} -5\, {{\mathscr{C}}}_{m_4m_5m_6}\,{{\mathscr{B}}}_{m_7m_8]}\bigr)
{\nonumber}\\
&\quad -280\,{{\mathscr{C}}}_{[m_1m_2m_3|\overline{n}_1\overline{n}_2|}\,{{\mathscr{C}}}_{m_4m_5m_6}\,{{\mathscr{C}}}_{m_7m_8]\overline{n}_3}
+56\,{{\mathscr{C}}}_{[m_1\cdots m_5}\,{{\mathscr{C}}}_{m_6m_7|n_1n_2n_3|}\,{{\mathscr{C}}}_{m_8]}
{\nonumber}\\
&\quad +56\,{{\mathscr{C}}}_{[m_1\cdots m_5}\,\bigl({{\mathscr{C}}}_{m_6m_7m_8]}\, {{\mathscr{C}}}_{\overline{n}_1} -3\, {{\mathscr{C}}}_{m_6m_7|\overline{n}_1|}\,{{\mathscr{C}}}_{m_8]} \bigr)\, {{\mathscr{B}}}_{\overline{n}_2\overline{n}_3}
{\nonumber}\\
&\quad -210\,{{\mathscr{C}}}_{[m_1m_2m_3}\,{{\mathscr{C}}}_{m_4m_5|\overline{n}_1|}\, {{\mathscr{C}}}_{m_6|\overline{n}_2\overline{n}_3|}\,{{\mathscr{B}}}_{m_7m_8]} \,,
\\
{{E}}_{m_1\cdots m_9 , n , p}
&\simeq {{\mathscr{A}}}_{m_1\cdots m_9 , n , p}
-36\,{{\mathscr{A}}}_{[m_1\cdots m_7| n, p|} \,{{\mathscr{B}}}_{m_8m_9]}
-84\,{{\mathscr{A}}}_{[m_1\cdots m_6| n , p|}\, {{\mathscr{C}}}_{m_7m_8m_9]}
{\nonumber}\\
&\quad -252\,{{\mathscr{B}}}_{[m_1\cdots m_6}\, {{\mathscr{C}}}_{m_7m_8|n|}\, {{\mathscr{B}}}_{m_9]p}
+21\, {{\mathscr{C}}}_{[m_1\cdots m_5}\, {{\mathscr{C}}}_{m_6\cdots m_9] n}\, {{\mathscr{C}}}_{p}
{\nonumber}\\
&\quad
+420\,{{\mathscr{C}}}_{[m_1\cdots m_4|n|}\,{{\mathscr{C}}}_{m_5m_6m_7}\, \bigl({{\mathscr{C}}}_{m_8m_9]p} -2\,{{\mathscr{B}}}_{m_8|p|}\,{{\mathscr{C}}}_{m_9]} \bigr)
{\nonumber}\\
&\quad -1575\,{{\mathscr{C}}}_{[m_1m_2m_3}\, {{\mathscr{C}}}_{m_4m_5|n|}\,{{\mathscr{C}}}_{m_6m_7|p|}\,{{\mathscr{B}}}_{m_8m_9]} \,,
\\
\text{\underline{Type IIB:}}
{\nonumber}\\
{{E}}_{m_1\cdots m_8}
&= {{\mathsf{E}}}_{m_1 \cdots m_8}
-28\,{{\mathsf{B}}}_{[m_1\cdots m_6}\,{{\mathsf{C}}}_{m_7m_8]}
+140\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8]}
{\nonumber}\\
&\quad +\tfrac{35}{3}\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5\cdots m_8]}\,{{\mathsf{C}}}_0 \,,
\\
{{E}}_{m_1\cdots m_8 , n_1n_2}
&\simeq
{{\bm{\mathsf{E}}}}_{m_1\cdots m_8, n_1n_2}
+420\,{{\mathsf{B}}}_{[m_1\cdots m_4 |n_1n_2|}\,{{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8]}
+ 56\,{{\mathsf{B}}}_{[m_1\cdots m_6}\,{{\mathsf{B}}}_{m_7|\overline{n}_1|}\,{{\mathsf{C}}}_{m_8]\overline{n}_2}
{\nonumber}\\
&\quad
-28\,{{\mathsf{B}}}_{[m_1\cdots m_6}\,\bigl({{\mathsf{C}}}_{m_7m_8] n_1n_2} + {{\mathsf{B}}}_{m_7m_8]}\,{{\mathsf{C}}}_{n_1n_2}\bigr)
{\nonumber}\\
&\quad
+\tfrac{70}{3} \,{{\mathsf{C}}}_{[m_1\cdots m_4|n_1n_2|}\,{{\mathsf{C}}}_{m_5\cdots m_8]}\,{{\mathsf{C}}}_0
-70\,{{\mathsf{C}}}_{[m_1\cdots m_4|n_1n_2|}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8]}
{\nonumber}\\
&\quad
-\tfrac{35}{6} \,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5\cdots m_8]}\,\bigl({{\mathsf{C}}}_{n_1n_2}+2\,{{\mathsf{B}}}_{n_1n_2}\,{{\mathsf{C}}}_0\bigr)
{\nonumber}\\
&\quad
+490\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6 |n_1n_2|}\,{{\mathsf{C}}}_{m_7m_8]}
+1260\, {{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7 |\overline{n}_1|}\,{{\mathsf{B}}}_{m_8]\overline{n}_2}
{\nonumber}\\
&\quad
- 630\,{{\mathsf{C}}}_{[m_1m_2|n_1n_2|}\,{{\mathsf{C}}}_{m_3m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{B}}}_{m_7m_8]}
- 455\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8]}\,{{\mathsf{B}}}_{n_1n_2}
{\nonumber}\\
&\quad
+\tfrac{1995}{2}\,{{\mathsf{C}}}_{[m_1m_2}\,{{\mathsf{C}}}_{m_3m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{B}}}_{m_7m_8]}\,{{\mathsf{B}}}_{n_1n_2}
{\nonumber}\\
&\quad
-\tfrac{1155}{4}\,{{\mathsf{B}}}_{[m_1m_2}\,{{\mathsf{B}}}_{m_3m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8]}\,{{\mathsf{C}}}_{n_1n_2}
{\nonumber}\\
&\quad
-2310\,{{\mathsf{C}}}_{[m_1m_2}\,{{\mathsf{C}}}_{m_3m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{B}}}_{m_7|\overline{n}_1|}\,{{\mathsf{B}}}_{m_8]\overline{n}_2}\,,
\\
{{E}}_{m_1\cdots m_9 , n_1n_2, p}
&\simeq {{\mathsf{A}}}_{m_1 \cdots m_9 , n_1n_2 , p}
-126\,{{\mathsf{A}}}_{[m_1\cdots m_5 |n_1n_2 , p|}\, \bigl({{\mathsf{C}}}_{m_6\cdots m_9]} -3\,{{\mathsf{B}}}_{m_6m_7}\, {{\mathsf{C}}}_{m_8m_9]} \bigr)
{\nonumber}\\
&\quad
+1260\,{{\mathsf{C}}}_{[m_1\cdots m_4|n_1n_2|}\, {{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8}\,{{\mathsf{C}}}_{m_9]p}
+ 420\,{{\mathsf{C}}}_{[m_1\cdots m_4|n_1n_2|}\, {{\mathsf{C}}}_{m_5m_6m_7|p|}\,{{\mathsf{C}}}_{m_8m_9]}
{\nonumber}\\
&\quad
-420\,{{\mathsf{C}}}_{[m_1\cdots m_4 |n_1n_2|}\, {{\mathsf{C}}}_{m_5m_6m_7|p|}\,{{\mathsf{B}}}_{m_8m_9]}\,{{\mathsf{C}}}_0
{\nonumber}\\
&\quad
+210\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6m_7|p|}\,\bigl({{\mathsf{B}}}_{m_8m_9]}\,{{\mathsf{C}}}_{n_1n_2} -3\, {{\mathsf{C}}}_{m_8m_9]}\,{{\mathsf{B}}}_{n_1n_2}\bigr)
{\nonumber}\\
&\quad
+315\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6|n_1n_2|}\,\bigl({{\mathsf{C}}}_{m_7m_8}\,{{\mathsf{B}}}_{m_9]p} - {{\mathsf{B}}}_{m_7m_8}\,{{\mathsf{C}}}_{m_9]p} \bigr)
{\nonumber}\\
&\quad
- 105\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5\cdots m_8}\,{{\mathsf{B}}}_{m_9]p}\,{{\mathsf{B}}}_{n_1n_2}\,{{\mathsf{C}}}_0
{\nonumber}\\
&\quad
+630\,{{\mathsf{C}}}_{[m_1\cdots m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8}\,{{\mathsf{B}}}_{m_9]p}\,{{\mathsf{B}}}_{n_1n_2}
{\nonumber}\\
&\quad
+2520\,{{\mathsf{C}}}_{[m_1m_2m_3|p|}\,{{\mathsf{C}}}_{m_4m_5}\,{{\mathsf{C}}}_{m_6m_7}\,{{\mathsf{B}}}_{m_8|\overline{n}_1|}\,{{\mathsf{B}}}_{m_9]\overline{n}_2}
{\nonumber}\\
&\quad
-1260\,{{\mathsf{C}}}_{[m_1m_2m_3|p|}\,{{\mathsf{B}}}_{m_4m_5}\,{{\mathsf{B}}}_{m_6m_7}\,{{\mathsf{C}}}_{m_8|\overline{n}_1|}\,{{\mathsf{C}}}_{m_9]\overline{n}_2}
{\nonumber}\\
&\quad
-2520\,{{\mathsf{C}}}_{[m_1m_2m_3|\overline{n}_1|}\,{{\mathsf{C}}}_{m_4m_5}\,{{\mathsf{C}}}_{m_6|\overline{n}_2|}\,{{\mathsf{B}}}_{m_7m_8}{{\mathsf{B}}}_{m_9]p}
{\nonumber}\\
&\quad
- 630\,{{\mathsf{B}}}_{[m_1m_2}\,{{\mathsf{B}}}_{m_3m_4}\,{{\mathsf{B}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8}{{\mathsf{C}}}_{m_9]p}\, {{\mathsf{C}}}_{n_1n_2} \,.\end{aligned}$$
Similarly, even for the potentials ${{\mathscr{A}}}_{10,1,1}$ and ${{\mathsf{F}}}_{10}$, if we consider the redefinitions, $$\begin{aligned}
{{F}}_{m_1\cdots m_{10}, b , c}
&\simeq {{\mathscr{A}}}_{m_1\cdots m_{10}, b , c}
-120\,{{\mathscr{A}}}_{[m_1\cdots m_7|b , c|}\, {{\mathscr{C}}}_{m_8m_9m_{10}]}
{\nonumber}\\
&\quad +1260\,{{\mathscr{B}}}_{[m_1\cdots m_5 |b|}\,{{\mathscr{C}}}_{m_6m_7m_8}\,{{\mathscr{C}}}_{m_9m_{10}]c}
+42\,{{\mathscr{C}}}_{[m_1\cdots m_6|b|}\,{{\mathscr{C}}}_{m_7m_8m_9}\,{{\mathscr{C}}}_{m_{10}]}\,{{\mathscr{C}}}_c
{\nonumber}\\
&\quad
+63\,{{\mathscr{C}}}_{[m_1\cdots m_5}\,\bigl({{\mathscr{C}}}_{m_6\cdots m_9|b|}\,{{\mathscr{C}}}_{m_{10}]} - {{\mathscr{C}}}_{m_6m_7m_8}\,{{\mathscr{C}}}_{m_9m_{10}]b}\bigr)\,{{\mathscr{C}}}_c
{\nonumber}\\
&\quad -1575\,{{\mathscr{C}}}_{[m_1\cdots m_4 |b|}\,{{\mathscr{C}}}_{m_5m_6m_7}\,{{\mathscr{C}}}_{m_8m_9|c|}\,{{\mathscr{C}}}_{m_{10}]} \,,
\\
{{F}}_{m_1\cdots m_{10}}
&= {{\mathsf{F}}}_{m_1\cdots m_{10}}
-45\,{{\mathsf{E}}}_{[m_1\cdots m_8}\,{{\mathsf{C}}}_{m_9m_{10}]}
+630\,{{\mathsf{B}}}_{[m_1\cdots m_6}\,{{\mathsf{C}}}_{m_7m_8}\,{{\mathsf{C}}}_{m_9m_{10}]}
{\nonumber}\\
&\quad + \tfrac{21}{2}\, {{\mathsf{C}}}_{[m_1\cdots m_6}\,\bigl({{\mathsf{C}}}_{m_7\cdots m_{10}]}\,{{\mathsf{C}}}_0 -3\,{{\mathsf{C}}}_{m_7m_8}\,{{\mathsf{C}}}_{m_9m_{10}]}\bigr)\,{{\mathsf{C}}}_0
{\nonumber}\\
&\quad - \tfrac{315}{2}\, {{\mathsf{C}}}_{[m_1\cdots m_4}\,\bigl({{\mathsf{C}}}_{m_5\cdots m_8}\,{{\mathsf{C}}}_0 +15\, {{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8}\bigr)\,{{\mathsf{C}}}_{m_9m_{10}]}
{\nonumber}\\
&\quad + 3780\,{{\mathsf{B}}}_{[m_1m_2}\,{{\mathsf{C}}}_{m_3m_4}\,{{\mathsf{C}}}_{m_5m_6}\,{{\mathsf{C}}}_{m_7m_8}\,{{\mathsf{C}}}_{m_9m_{10}]} \,,\end{aligned}$$ the $T$-duality rule is simplified as $$\begin{aligned}
{{F}}_{a_1\cdots a_9{{y}}, {{y}}, {{y}}} \overset{\text{A--B}}{=} {{F}}_{a_1\cdots a_9{{\mathsf{y}}}}\,.\end{aligned}$$ As is studied in [@1201.5819; @1210.1422], potentials that couple to the $(9-p)^{(p,0)}_4$-branes ($p=1,3,5,7,9/0,2,4,6,8$ in type IIA/IIB theory) are packaged into the self-dual ${\text{O}}(10,10)$ tensor ${{F}}^+_{M_1\cdots M_{10}}$. Then, the above potentials, ${{F}}_{10,1,1}$ and ${{F}}_{10}$, will be identified as the particular components of ${{F}}^+_{M_1\cdots M_{10}}$. The potential ${{F}}^+_{M_1\cdots M_{10}}$ contains the following family of potentials: $$\begin{aligned}
\vcenter{\xymatrix@C=0pt@R=18pt{
\text{\underline{IIA}}\quad & {{F}}_{10,9,9} \ar@{<.>}[rd] & & {{F}}_{10,7,7} \ar@{<.>}[ld] \ar@{<.>}[rd] & & {{F}}_{10,5,5} \ar@{<.>}[ld] \ar@{<.>}[rd] & & {{F}}_{10,3,3} \ar@{<.>}[ld] \ar@{<.>}[rd] & & {{F}}_{10,1,1} \ar@{<.>}[ld] \ar@{<->}[rd]^-{\tiny\textcircled{z}} & \\
\text{\underline{IIB}}\quad & & {{F}}_{10,8,8} & & {{F}}_{10,6,6} & & {{F}}_{10,4,4} & & {{F}}_{10,2,2} & & {{F}}_{10}
}}\,.\end{aligned}$$ What we have explicitly confirmed is only the rightmost arrow , but the existence of the ${\text{O}}(10,10)$-covariant potential ${{F}}^+_{M_1\cdots M_{10}}$ suggests the validity of other maps. Namely, we can define the family of potentials ${{F}}_{10,p,p}$ through the simple $T$-duality rules, $$\begin{aligned}
\begin{split}
{{F}}_{a_1\cdots a_9{{y}},b_1\cdots b_{p-1} {{y}}, b_1\cdots b_{p-1}{{y}}} &\overset{\text{A--B}}{\simeq} {{F}}_{a_1\cdots a_9{{\mathsf{y}}}, b_1\cdots b_{p-1}, b_1\cdots b_{p-1}} \,,
\\
{{F}}_{a_1\cdots a_9{{y}},b_1\cdots b_p, b_1\cdots b_p} &\overset{\text{A--B}}{\simeq} {{F}}_{a_1\cdots a_9{{\mathsf{y}}}, b_1\cdots b_p{{\mathsf{y}}}, b_1\cdots b_p{{\mathsf{y}}}}\,,
\end{split}\end{aligned}$$ for $p=1,3,5,7,9$, without any non-linear correction. Of course, in order to discuss the M-theory uplifts or the $S$-duality rules for ${{F}}_{10,p,p}$ ($p\geq 2$), we need to determine how they enter into the 1-form field, ${\mathcal A}_\mu^I$ or $\bm{{\mathcal A}}_\mu^{\mathsf{I}}$.
$S$-duality rule {#sec:S-duality}
----------------
In this paper, the type IIB fields are defined to be $S$-duality covariant, and under an ${\text{SL}}(2)$ transformation $\Lambda^{{{\alpha}}}{}_{{{\beta}}}$, the bosonic fields transform as $$\begin{aligned}
\begin{split}
&{{\mathsf{g}}}'_{mn} ={{\mathsf{g}}}_{mn}\,,\quad
{{\mathsf{m}}}'^{{{\alpha}}{{\beta}}} = \Lambda^{{{\alpha}}}{}_{{{\gamma}}}\,\Lambda^{{{\beta}}}{}_{{{\delta}}}\,{{\mathsf{m}}}^{{{\gamma}}{{\delta}}}\,, \quad
{{\mathsf{A}}}'^{{{\alpha}}}_2 = \Lambda^{{{\alpha}}}{}_{{{\beta}}}\,{{\mathsf{A}}}^{{{\beta}}}_2\,, \quad
{{\mathsf{A}}}'_4 = {{\mathsf{A}}}_4\,,
\\
&{{\mathsf{A}}}'^{{{\alpha}}}_6 = \Lambda^{{{\alpha}}}{}_{{{\beta}}}\,{{\mathsf{A}}}^{{{\beta}}}_6\,,\quad
{{\mathsf{A}}}'_{7,1} = {{\mathsf{A}}}_{7,1}\,,\quad
{{\mathsf{A}}}'^{{{\alpha}}{{\beta}}}_8 = \Lambda^{{{\alpha}}}{}_{{{\gamma}}}\,\Lambda^{{{\beta}}}{}_{{{\delta}}}\,{{\mathsf{A}}}^{{{\gamma}}{{\delta}}}_8 \,,\quad
{{\mathsf{A}}}'^{{{\alpha}}}_{8,2} = \Lambda^{{{\alpha}}}{}_{{{\beta}}}\,{{\mathsf{A}}}^{{{\beta}}}_{8,2} \,,
\\
&{{\mathsf{A}}}'^{{{\alpha}}_1{{\alpha}}_2{{\alpha}}_3}_{10} = \Lambda^{{{\alpha}}_1}{}_{{{\beta}}_1}\,\Lambda^{{{\alpha}}_2}{}_{{{\beta}}_2}\,\Lambda^{{{\alpha}}_3}{}_{{{\beta}}_3}\,{{\mathsf{A}}}^{{{\beta}}_1{{\beta}}_2{{\beta}}_3}_{10}\,,\quad
{{\mathsf{A}}}'_{9,2,1}={{\mathsf{A}}}_{9,2,1}\,.
\end{split}\end{aligned}$$ In particular, under the $S$-duality, $\Lambda=\bigl(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\bigr)$, the component fields are transformed as $$\begin{aligned}
\begin{split}
&{{\mathsf{g}}}'_{mn} ={{\mathsf{g}}}_{mn}\,,\quad
{{\mathsf{C}}}'_0 = - \tfrac{{{\mathsf{C}}}_0}{({{\mathsf{C}}}_0)^2+{\operatorname{e}^{-2{{\Phi}}}}}\,,\quad {\operatorname{e}^{-{{\Phi}}'}} = \tfrac{{\operatorname{e}^{-{{\Phi}}'}}}{({{\mathsf{C}}}_0)^2+{\operatorname{e}^{-2{{\Phi}}}}}\,,
\\
&{{\mathsf{B}}}'_2 =- {{\mathsf{C}}}_2\,,\quad {{\mathsf{C}}}'_2 = {{\mathsf{B}}}_2\,,\quad {{\mathsf{C}}}'_4 = {{\mathsf{C}}}_4 - {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2 \,,
\\
&{{\mathsf{C}}}'_6 = - \bigl({{\mathsf{B}}}_6 - \tfrac{1}{2!}\, {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\bigr) \,, \quad
{{\mathsf{B}}}'_6 = {{\mathsf{C}}}_6 - \tfrac{1}{2!}\, {{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\,,
\\
&{{\mathsf{C}}}'_8 = {{\mathsf{E}}}_8 - \tfrac{1}{3!}\, {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2 \,, \quad
{{\mathsf{E}}}'_8 = {{\mathsf{C}}}_8 - \tfrac{1}{3!}\, {{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\,,
\\
&{{\mathsf{C}}}'_{10} = - \bigl({{\mathsf{E}}}_{10} - \tfrac{1}{4!}\, {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\bigr) \,, \quad
{{\mathsf{F}}}'_{10} = {{\mathsf{C}}}_{10} - \tfrac{1}{4!}\, {{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\,,
\\
&{{\mathsf{A}}}'_{7,1} = {{\mathsf{A}}}_{7,1}\,,\quad {{\bm{\mathsf{D}}}}'_{8,2} =- {{\bm{\mathsf{E}}}}_{8,2}\,,\quad {{\bm{\mathsf{E}}}}'_{8,2} = {{\bm{\mathsf{D}}}}_{8,2}\,,\quad {{\mathsf{A}}}'_{9,2,1} = {{\mathsf{A}}}_{9,2,1} \,.
\end{split}
\label{eq:S-duality}\end{aligned}$$
It is sometimes useful to introduce the dual parameterization of ${{\mathsf{m}}}_{{{\alpha}}{{\beta}}}$, $$\begin{aligned}
({{\mathsf{m}}}_{{{\alpha}}{{\beta}}}) = {\operatorname{e}^{\Phi}} \begin{pmatrix}
{\operatorname{e}^{-2{{\Phi}}}} + ({{\mathsf{C}}}_0)^2 & {{\mathsf{C}}}_0 \\
{{\mathsf{C}}}_0 & 1
\end{pmatrix} \equiv {\operatorname{e}^{-\tilde{\phi}}} \begin{pmatrix}
1 & -\tilde{\gamma} \\
-\tilde{\gamma} & {\operatorname{e}^{2\tilde{\phi}}} + \tilde{\gamma}^2
\end{pmatrix} ,\end{aligned}$$ which is equivalent to $$\begin{aligned}
\tilde{\gamma} \equiv -\frac{{{\mathsf{C}}}_0}{({{\mathsf{C}}}_0)^2+{\operatorname{e}^{-2{{\Phi}}}}}\,,\qquad
{\operatorname{e}^{\tilde{\phi}}}\equiv \frac{{\operatorname{e}^{-{{\Phi}}'}}}{({{\mathsf{C}}}_0)^2+{\operatorname{e}^{-2{{\Phi}}}}}\,.\end{aligned}$$ Then, the $S$-duality rule becomes $$\begin{aligned}
{{\mathsf{C}}}'_0 = \tilde{\gamma}\,,\qquad {\operatorname{e}^{{{\Phi}}'}} = {\operatorname{e}^{-\tilde{\phi}}} \,. \end{aligned}$$ The electric-magnetic duality for ${{\mathsf{H}}}_9$ is also simplified as $$\begin{aligned}
{{\mathsf{H}}}_9 = {\operatorname{e}^{-2\tilde{\phi}}} *_{{\mathrm{E}}}\, {{\mathrm{d}}}\tilde{\gamma} \,.
\label{eq:EM-7_3}\end{aligned}$$
For the $T$-duality-covariant potentials, the $S$-duality transformation rules are complicated. For example, we find $$\begin{aligned}
{{D}}'_6 &= {{\mathsf{C}}}_6 -\tfrac{1}{2}\, {{\mathsf{C}}}_4\wedge {{\mathsf{B}}}_2 + \tfrac{\tilde{\gamma}}{2}\,\bigl({{D}}_6 + \tfrac{1}{2}\, {{\mathsf{C}}}_6\,{{\mathsf{C}}}_0 + \tfrac{1}{2}\, {{\mathsf{C}}}_4\wedge {{\mathsf{C}}}_2 -\tfrac{1}{2}\,{{\mathsf{B}}}_2\wedge{{\mathsf{C}}}_2^2 \bigr)\,,
\\
{{E}}'_8 &= {{\mathsf{C}}}_8 - {{\mathsf{C}}}_6\wedge{{\mathsf{B}}}_2 + \tfrac{1}{3}\, {{\mathsf{C}}}_4\wedge {{\mathsf{B}}}_2^2 + \tfrac{\tilde{\gamma}}{3!}\,\bigl({{\mathsf{C}}}_4^2 -2\,{{\mathsf{C}}}_4\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2 + {{\mathsf{C}}}_2^2 \wedge {{\mathsf{B}}}_2^2 \bigr)\,,
\label{eq:E8-S-dual}
\\
{{F}}'_{10} &= {{\mathsf{C}}}_{10} - {{\mathsf{C}}}_8\wedge {{\mathsf{B}}}_2 + \tfrac{1}{2}\,{{\mathsf{C}}}_6\wedge {{\mathsf{B}}}_2^2 - \tfrac{1}{8}\,{{\mathsf{C}}}_4\wedge {{\mathsf{B}}}_2^3 - \tfrac{1}{30}\,{{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2^4
{\nonumber}\\
&\quad + \tfrac{\tilde{\gamma}}{40}\,\bigl({{\mathsf{D}}}_6\wedge {{\mathsf{B}}}_2 + \tfrac{1}{2}\,{{\mathsf{C}}}_6\wedge {{\mathsf{B}}}_2\, {{\mathsf{C}}}_0 - 2\,{{\mathsf{C}}}_4^2 + \tfrac{9}{2}\,{{\mathsf{C}}}_4\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2 - \tfrac{5}{2}\,{{\mathsf{C}}}^2_2\wedge{{\mathsf{B}}}_2^2 \bigr) \wedge{{\mathsf{B}}}_2
{\nonumber}\\
&\quad - \tfrac{\tilde{\gamma}^2}{40}\,\bigl[2\,{{\mathsf{D}}}_6\wedge ({{\mathsf{C}}}_4 - {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2) + {{\mathsf{C}}}_6\wedge ({{\mathsf{C}}}_4 - {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2)\,{{\mathsf{C}}}_0
{\nonumber}\\
&\qquad\qquad + ({{\mathsf{C}}}_4^2 - 2\,{{\mathsf{C}}}_4\wedge{{\mathsf{C}}}_2 \wedge {{\mathsf{B}}}_2 + {{\mathsf{C}}}_2^2\wedge {{\mathsf{B}}}_2^2)\wedge{{\mathsf{C}}}_2 \bigr]\,.\end{aligned}$$ The $S$-duality rules for other $T$-duality-covariant potentials also can be obtained from .
Field strengths and gauge transformations {#sec:gauge}
=========================================
In this section, we summarize the field strengths and gauge transformations studied in the literature in terms of our mixed-symmetry potentials, and make a small progress.
11D/Type IIA supergravity {#sec:gauge-IIA}
-------------------------
In 11D supergravity, the field strengths ${{\hat{F}}}_{\hat{4}}$ and ${{\hat{F}}}_{\hat{7}}$ defined in section \[sec:11D-sugra\] are invariant under $$\begin{aligned}
\delta {{\hat{A}}}_{\hat{3}} = {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{2}}\,, \qquad
\delta {{\hat{A}}}_{\hat{6}} = {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{5}} - \tfrac{1}{2}\,{{\hat{A}}}_{\hat{3}}\wedge {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{2}} \,.\end{aligned}$$ Here, we discuss the field strengths for the mixed-symmetry potentials ${{\hat{A}}}_{\hat{8},\hat{1}}$ and ${{\hat{A}}}_{\hat{10},\hat{1},\hat{1}}$.
Since ${{\hat{A}}}_{\hat{8},\hat{1}}$ and ${{\hat{A}}}_{\hat{10},\hat{1},\hat{1}}$ are 11D uplifts of the R–R 7-form and 9-form, let us consider the 11D uplifts of the known R–R field strengths ${\mathcal G}_8$ and ${\mathcal G}_{10}$. The 10-form ${\mathcal G}_{10}$ is the electric-magnetic dual to the Romans mass [@Romans:1985tz], and in order to discuss the field strength of ${{\hat{A}}}_{\hat{10},\hat{1},\hat{1}}$, we need to introduce the mass deformation. Thus, let us begin by summarizing the gauge transformations and field strengths in massive type IIA supergravity. In massive type IIA supergravity, the field strength in the $A$-basis has the 0-form field strength $F_0\equiv m$, $$\begin{aligned}
F \equiv F_0 + F_2 + \cdots + F_8 + F_{10}= {{\mathrm{d}}}{{A}}+ m \,,\qquad {{A}}\equiv {{A}}_1 + {{A}}_3 + {{A}}_5 + {{A}}_7 + {{A}}_9\,.\end{aligned}$$ Accordingly, in the $C$-basis, the field strength is given by $$\begin{aligned}
{\mathcal G}\equiv {\mathcal G}_0 + {\mathcal G}_2 + \cdots + {\mathcal G}_8 + {\mathcal G}_{10} = {\operatorname{e}^{{{\mathscr{B}}}_2\wedge}} \bigl[{{\mathrm{d}}}({\operatorname{e}^{-{{\mathscr{B}}}_2\wedge}}{{\mathscr{C}}}) + m\bigr] = {{\mathrm{d}}}{{\mathscr{C}}}-{{\mathscr{H}}}_3\wedge {{\mathscr{C}}}+ {\operatorname{e}^{{{\mathscr{B}}}_2\wedge}} m \,. \end{aligned}$$ The gauge transformation is given by $$\begin{aligned}
\delta{{\mathscr{B}}}_2 = {{\mathrm{d}}}\chi_1\,,\qquad
\delta {{\mathscr{C}}}= {\operatorname{e}^{{{\mathscr{B}}}_2\wedge}}\,{{\mathrm{d}}}\lambda - m{\operatorname{e}^{{{\mathscr{B}}}_2\wedge}}\chi_1 \,.\end{aligned}$$
Now, let us review the uplifts of these relations to 11D. Since the R–R 1-form is contained in the 11D metric, under gauge transformations, the 11D metric is transformed as $$\begin{aligned}
\delta {{\hat{g}}}_{ij} = -m\, \bigl(\chi_i\, {{\hat{g}}}_{j{{z}}} + \chi_j\, {{\hat{g}}}_{i{{z}}}\bigr)\,,
\label{eq:g-massive}\end{aligned}$$ where the coordinate $x^{{{z}}}$ is also transformed as $\delta x^{{{z}}} = - \lambda_0$. The gauge transformations for the R–R 3-form and the $B$-field are uplifted as $$\begin{aligned}
\delta {{\hat{A}}}_{\hat{3}} &= {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{2}} + m\,\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}} {{\hat{\lambda}}}_{\hat{2}} \,,\qquad {{\hat{\lambda}}}_{\hat{2}} \equiv \lambda_2 + \chi_1\wedge{{\mathrm{d}}}x^{{{z}}}\,,\end{aligned}$$ Under these transformations, the field strength, $$\begin{aligned}
{{\hat{F}}}_{\hat{4}} \equiv {{\mathrm{d}}}{{\hat{A}}}_{\hat{3}} + \tfrac{m}{2}\,\iota_{{{z}}}{{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}}{{\hat{A}}}_{\hat{3}}
\equiv {\mathcal G}_4 + {{\mathscr{H}}}_3 \wedge ({{\mathrm{d}}}x^{{{z}}} +{{\mathscr{C}}}_1)\,,\end{aligned}$$ transforms as $$\begin{aligned}
\delta {{\hat{F}}}_{\hat{4}} = m\,\iota_{{{z}}} {{\hat{\lambda}}}_2 \wedge \iota_{{{z}}} {{\hat{F}}}_{\hat{4}} \,. \end{aligned}$$ The non-invariance is due to $\delta({{\mathrm{d}}}x^{{{z}}} +{{\mathscr{C}}}_1)=m\,\iota_{{{z}}}{{\hat{\lambda}}}_2$, although ${\mathcal G}_4$ and ${{\mathscr{H}}}_3$ are invariant. From a similar consideration, the gauge transformations for the R–R 5-, 7-, and 9-forms are also uplifted as[^11] $$\begin{aligned}
\delta {{\hat{A}}}_{\hat{6}} &= {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{5}} - \tfrac{1}{2}\,{{\hat{A}}}_{\hat{3}}\wedge {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{2}} - m\,\bigl(\iota_{{{z}}}{{\hat{\lambda}}}_{7,{{z}}} + \iota_{{{z}}} {{\hat{A}}}_{\hat{6}} \wedge \iota_{{{z}}} {{\hat{\lambda}}}_{\hat{2}} \bigr)\,,
\\
\delta \bigl(\iota_{{{z}}} {{\hat{A}}}_{\hat{8},{{z}}}\bigr) &= \iota_{{{z}}} {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{7},{{z}}} + \iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}}{{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{5}} - \tfrac{2}{3!}\, \iota_{{{z}}} {{\hat{A}}}_{\hat{3}}\wedge\iota_{{{z}}} \bigl({{\hat{A}}}_{\hat{3}}\wedge {{\mathrm{d}}}{{\hat{\lambda}}}_2\bigr)\,,
\label{eq:delta-M81}
\\
\delta \bigl(\iota_{{{z}}} {{\hat{A}}}_{\hat{10},{{z}},{{z}}}\bigr) &= \iota_{{{z}}} {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{9},{{z}},{{z}}} + \iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}}{{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{7},{{z}}}
{\nonumber}\\
&\quad - \tfrac{1}{2}\, \iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}}{{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{5}} - \tfrac{3}{4!}\, \iota_{{{z}}} {{\hat{A}}}_{\hat{3}}\wedge\iota_{{{z}}} \bigl({{\hat{A}}}_{\hat{3}}\wedge {{\mathrm{d}}}{{\hat{\lambda}}}_2\bigr)\,,\end{aligned}$$ and the associated field strengths are defined by $$\begin{aligned}
{{\hat{F}}}_{\hat{7}} &\equiv {{\mathrm{d}}}{{\hat{A}}}_{\hat{6}} + \tfrac{1}{2}\,{{\hat{A}}}_{\hat{3}}\wedge {{\hat{F}}}_{\hat{4}} -m\, \bigl(\iota_{{{z}}} {{\hat{A}}}_{\hat{8},{{z}}} - \iota_{{{z}}} {{\hat{A}}}_{\hat{3}}\wedge \iota_{{{z}}} {{\hat{A}}}_{\hat{6}} + \tfrac{1}{2\cdot 3!}\,\iota_{{{z}}} {{\hat{A}}}_{\hat{3}}\wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}}\wedge {{\hat{A}}}_{\hat{3}}\bigr)\,,
\\
\iota_{{{z}}} {{\hat{F}}}_{\hat{9},{{z}}} &\equiv \iota_{{{z}}} {{\mathrm{d}}}{{\hat{A}}}_{\hat{8},{{z}}} + \iota_{{{z}}} {{\hat{A}}}_{\hat{6}}\wedge \iota_{{{z}}} {{\hat{F}}}_{\hat{4}} + \tfrac{1}{3!}\,\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}}\bigl({{\hat{F}}}_{\hat{4}}\wedge {{\hat{A}}}_{\hat{3}} \bigr)
{\nonumber}\\
&\quad - \tfrac{2m}{2\cdot 4!}\,\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \,,
\\
\iota_{{{z}}} {{\hat{F}}}_{\hat{11},{{z}},{{z}}} &\equiv \iota_{{{z}}} {{\mathrm{d}}}{{\hat{A}}}_{\hat{10},{{z}},{{z}}} + \iota_{{{z}}} {{\hat{A}}}_{\hat{8},{{z}}}\wedge \iota_{{{z}}} {{\hat{F}}}_{\hat{4}} + \tfrac{1}{4!}\,\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge \iota_{{{z}}}\bigl({{\hat{F}}}_{\hat{4}}\wedge {{\hat{A}}}_{\hat{3}} \bigr)
{\nonumber}\\
&\quad - \tfrac{3m}{2\cdot 5!}\,\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}} \wedge\iota_{{{z}}} {{\hat{A}}}_{\hat{3}}\,.\end{aligned}$$ We note that the 7-form field strength is not invariant similar to the 4-form, $$\begin{aligned}
\delta {{\hat{F}}}_{\hat{7}} = m\,\iota_{{{z}}} {{\hat{\lambda}}}_2\wedge \iota_{{{z}}} {{\hat{F}}}_{\hat{7}} \,, \end{aligned}$$ while the projections of the 9-form and the 11-form are invariant as it is clear from $$\begin{aligned}
\iota_{{{z}}} {{\hat{F}}}_{\hat{9},{{z}}} = {\mathcal G}_8\,,\qquad
\iota_{{{z}}} {{\hat{F}}}_{\hat{11},{{z}},{{z}}} = {\mathcal G}_{10} \,.
\label{eq:cG9-cG11}\end{aligned}$$ The above gauge transformations have been discussed in [@hep-th/9712115; @hep-th/9802199; @hep-th/9806120; @hep-th/9912030] and they are gauge symmetry of the “massive 11D supergravity” [@hep-th/9712115; @hep-th/9806120; @hep-th/9912030], which reproduce the massive type IIA supergravity after the dimensional reduction.
By using the above setup, we can easily consider the 11D uplift of the electric-magnetic duality ${\mathcal G}_8 = * {\mathcal G}_2$. For this purpose, we also introduce the 2-form field strength associated with a Killing vector $k\equiv \partial_{{{z}}}$ as [@hep-th/9912030] (see also [@0907.3614]) $$\begin{aligned}
{{\hat{F}}}_{\hat{2}} \equiv {{\mathrm{d}}}k_{\hat{1}} + m\,{\lvert {k} \rvert}^2\,\iota_k{{\hat{A}}}_{\hat{3}}\,,\end{aligned}$$ where $k_{\hat{1}} \equiv k^i\, {{\hat{g}}}_{ij} \,{{\mathrm{d}}}x^j$. This transforms as $$\begin{aligned}
\delta {{\hat{F}}}_{\hat{2}} = m\, \iota_k{{\hat{\lambda}}}_{\hat{2}} \wedge \iota_k {{\mathrm{d}}}k_{\hat{1}} = m\, \iota_k{{\hat{\lambda}}}_{\hat{2}} \wedge \iota_k {{\hat{F}}}_{\hat{2}} \,. \end{aligned}$$ In terms of the type IIA fields, we have $k_{\hat{1}}={\operatorname{e}^{\frac{4}{3}{{\varphi}}}}({{\mathrm{d}}}x^{{{z}}}+{{\mathscr{C}}}_1)$, and this gives $$\begin{aligned}
{{\hat{F}}}_{\hat{2}} \equiv {\operatorname{e}^{\frac{4}{3}{{\varphi}}}} {\mathcal G}_2 + \tfrac{4}{3}{\operatorname{e}^{\frac{4}{3}{{\varphi}}}} {{\mathrm{d}}}{{\varphi}}\wedge \bigl({{\mathrm{d}}}x^{{{z}}} + {{\mathscr{C}}}_1\bigr)\,.\end{aligned}$$ Then, the electric-magnetic duality ${\mathcal G}_8 = * {\mathcal G}_2$ becomes $$\begin{aligned}
\iota_{{{z}}} {{\hat{F}}}_{\hat{9},{{z}}} = \iota_{{{z}}} \hat{*}{{\hat{F}}}_2 \,,\end{aligned}$$ which shows that the dual graviton is electric-magnetic dual to the Killing vector. In this paper, only the restricted component that couple to supersymmetric branes are considered, and the restriction corresponds to the projection $\iota_{{{z}}}$ in front of the field strength.[^12] We can similarly consider the 11D uplift of the electric-magnetic duality $m = {\mathcal G}_{0} = * {\mathcal G}_{10}$ [@hep-th/9912030] $$\begin{aligned}
m\,{\lvert {k} \rvert}^4 = \hat{*} {{\hat{F}}}_{\hat{11},{{z}},{{z}}}\,,\end{aligned}$$ where $$\begin{aligned}
{{\hat{F}}}_{\hat{11},{{z}},{{z}}} \equiv \iota_{{{z}}} {{\hat{F}}}_{\hat{11},{{z}},{{z}}}\wedge \bigl({{\mathrm{d}}}x^{{{z}}}+{{\mathscr{C}}}_1\bigr)\,,\end{aligned}$$ is the gauge-invariant field strength.
Now, we consider the relation to the recent studies on mixed-symmetry potentials in DFT. If we decompose the 7-form field strength as $$\begin{aligned}
{{\hat{F}}}_{\hat{7}} = {{\mathscr{H}}}_7 + {\mathcal G}_6 \wedge ({{\mathrm{d}}}x^{{{z}}} + {{\mathscr{C}}}_1) \,,\end{aligned}$$ the 7-form field strength ${{\mathscr{H}}}_7$ becomes $$\begin{aligned}
{{\mathscr{H}}}_7 = {{\mathrm{d}}}{{D}}_6 - \tfrac{1}{2}\,\bigl({\mathcal G}_6 \wedge {{\mathscr{C}}}_1 - {\mathcal G}_4\wedge {{\mathscr{C}}}_3 + {\mathcal G}_2 \wedge {{\mathscr{C}}}_5 - {\mathcal G}_0\,{{\mathscr{C}}}_7\bigr) + \tfrac{m}{2}\, {{A}}_7\,,\end{aligned}$$ which coincides with the expression given in [@1903.05601], and is invariant under $$\begin{aligned}
\delta {{D}}_6 ={{\mathrm{d}}}\chi_5 + \tfrac{1}{2}\,\bigl({{A}}_5\wedge {{\mathrm{d}}}\lambda_0 - {{A}}_3\wedge {{\mathrm{d}}}\lambda_2 + {{A}}_1\wedge {{\mathrm{d}}}\lambda_4 \bigr) - m\,\bigl(\lambda_6 +\tfrac{1}{2}\,{{A}}_5\wedge \chi_1\bigr)\,.\end{aligned}$$ Here, we have parameterized $$\begin{aligned}
{{\hat{\lambda}}}_{\hat{5}} = \chi_5 + \lambda_4\wedge{{\mathrm{d}}}x^{{{z}}}\,,\qquad
{{\hat{\lambda}}}_{\hat{7},{{z}}} = \overline{\lambda}_7 + \lambda_6 \wedge{{\mathrm{d}}}x^{{{z}}}\,.\end{aligned}$$
Let us also clarify the relation between the field strength of the dual graviton and the field strength $\iota_n {{\mathscr{H}}}_{8, n}$ defined in . To this end, we assume the existence of a Killing direction denoted by $n$ (i.e. ${\pounds}_n \equiv \iota_n {{\mathrm{d}}}+ {{\mathrm{d}}}\iota_n=0$) other than the M-theory circle. For simplicity, we turn off the mass parameter. Then, as before, we can easily show that the field strength $$\begin{aligned}
\iota_n {{\hat{F}}}_{\hat{9},n} &\equiv \iota_n {{\mathrm{d}}}{{\hat{A}}}_{\hat{8},n} + \iota_n {{\hat{A}}}_{\hat{6}}\wedge \iota_n {{\hat{F}}}_{\hat{4}} + \tfrac{1}{3!}\,\iota_n {{\hat{A}}}_{\hat{3}} \wedge \iota_n\bigl({{\hat{F}}}_{\hat{4}}\wedge {{\hat{A}}}_{\hat{3}} \bigr) \,,\end{aligned}$$ is invariant under $$\begin{aligned}
\begin{split}
\delta {{\hat{A}}}_{\hat{3}} &= {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{2}}\,,\qquad
\delta {{\hat{A}}}_{\hat{6}} = {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{5}} - \tfrac{1}{2}\,{{\hat{A}}}_{\hat{3}}\wedge {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{2}} \,,
\\
\delta \bigl(\iota_n {{\hat{A}}}_{\hat{8},n}\bigr) &= \iota_n {{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{7},n} + \iota_n {{\hat{A}}}_{\hat{3}} \wedge \iota_n{{\mathrm{d}}}{{\hat{\lambda}}}_{\hat{5}} - \tfrac{2}{3!}\, \iota_n {{\hat{A}}}_{\hat{3}}\wedge\iota_n \bigl({{\hat{A}}}_{\hat{3}}\wedge {{\mathrm{d}}}{{\hat{\lambda}}}_2\bigr)\,.
\end{split}\end{aligned}$$ Now, we consider the reduction to type IIA theory. We define the field strength of the dual graviton in type IIA theory as $$\begin{aligned}
\iota_n {\mathcal G}_{8,n} \equiv \iota_{{{z}}} \iota_n \bigl({{\hat{F}}}_{\hat{9},n} - {{\hat{F}}}_{\hat{9},{{z}}} {{\mathscr{C}}}_n\bigr) = \iota_{{{z}}} \iota_n {{\hat{F}}}_{\hat{9},n} + \iota_n {\mathcal G}_8 \,\iota_n {{\mathscr{C}}}_1 \,,\end{aligned}$$ which is also gauge invariant because $\iota_n {{\mathscr{C}}}_1$ is gauge invariant due to the Killing equation, ${\pounds}_{n}\lambda_0= \iota_n{{\mathrm{d}}}\lambda_0=0$. Then, a straightforward but a bit long computation gives $$\begin{aligned}
\iota_n {\mathcal G}_{8,n} &= {{\mathrm{d}}}\iota_n A_{7,n} + \iota_n {{\mathscr{C}}}_5\wedge\iota_n {{\mathrm{d}}}{{\mathscr{C}}}_3 -\iota_n {{\mathscr{B}}}_6\wedge \iota_n {{\mathscr{H}}}_3
- \tfrac{1}{3}\,\iota_n{{\mathscr{B}}}_2\wedge{{\mathscr{C}}}_3\wedge\iota_n{{\mathrm{d}}}{{\mathscr{C}}}_3
- \tfrac{2}{3}\,{{\mathscr{B}}}_2\wedge\iota_n{{\mathscr{C}}}_3\wedge\iota_n{{\mathrm{d}}}{{\mathscr{C}}}_3
{\nonumber}\\
&\quad - \tfrac{1}{3}\,\iota_n{{\mathscr{B}}}_2\wedge{{\mathrm{d}}}{{\mathscr{C}}}_3\wedge\iota_n {{\mathscr{C}}}_3
+ \tfrac{1}{6}\,\iota_n{{\mathscr{H}}}_3\wedge\iota_n{{\mathscr{C}}}_3\wedge {{\mathscr{C}}}_3
- \tfrac{1}{6}\,{{\mathscr{H}}}_3\wedge\iota_n{{\mathscr{C}}}_3\wedge \iota_n{{\mathscr{C}}}_3
+ \iota_n {\mathcal G}_8 \,\iota_n {{\mathscr{C}}}_1
{\nonumber}\\
&= \iota_n {{\mathscr{H}}}_{8, n} - \iota_n {{\mathscr{H}}}_7\wedge \iota_n {{\mathscr{B}}}_2 \,,\end{aligned}$$ where we have used . This shows that the invariant field strength $\iota_k {\mathcal G}_{8,n}$ is the component of the untwisted tensor, $$\begin{aligned}
\hat{H}_{M_1M_2M_3} \equiv ({\operatorname{e}^{\bm{{{\mathscr{B}}}}}})_{M_1}{}^{N_1}\,({\operatorname{e}^{\bm{{{\mathscr{B}}}}}})_{M_2}{}^{N_2}\,({\operatorname{e}^{\bm{{{\mathscr{B}}}}}})_{M_3}{}^{N_3}\,\hat{H}_{N_1N_2N_3}\,,\qquad ({\operatorname{e}^{\bm{{{\mathscr{B}}}}}})_M{}^N \equiv \begin{pmatrix}
\delta_m^n & {{\mathscr{B}}}_{mn} \\ 0 & \delta^m_n \end{pmatrix}.\end{aligned}$$ Indeed, we can easily check $$\begin{aligned}
\iota_n {\mathcal G}_{8,n} &= \tfrac{1}{7!\,2!}\,\epsilon_{m_1\cdots m_7 a_1a_2 n}\, \hat{H}^{a_1a_2}{}_n \,{{\mathrm{d}}}x^{m_1}\wedge\cdots\wedge{{\mathrm{d}}}x^{m_7}
= \iota_n {{\mathscr{H}}}_{8,n} - \iota_n {{\mathscr{H}}}_7\wedge \iota_n {{\mathscr{B}}}_2 \,.\end{aligned}$$ Namely, the field strength $H_{MNP}$ is similar to the R–R field strength $F$; $F$ is invariant under gauge transformations of the R–R potentials, but not under $B$-field gauge transformations, and it becomes invariant after untwisting the field strength as ${\mathcal G}={\operatorname{e}^{{{\mathscr{B}}}_2}}F$.
For completeness, let us also show that the gauge transformation reproduces [@hep-th/9802199] $$\begin{aligned}
\delta(\iota_n{{\mathscr{A}}}_{7,n}) &= \iota_n {{\mathrm{d}}}\lambda_{6,n} + \iota_n{{\mathscr{C}}}_3\wedge\iota_n{{\mathrm{d}}}\lambda_4
- \iota_n{{\mathscr{B}}}_2\wedge\iota_n{{\mathrm{d}}}\chi_5
- \tfrac{1}{3}\,\iota_n{{\mathscr{C}}}_3\wedge \bigl(\iota_n{{\mathscr{C}}}_3\wedge {{\mathrm{d}}}\chi_1
+ 2\,\iota_n{{\mathscr{B}}}_2\wedge{{\mathrm{d}}}\lambda_2\bigr)
{\nonumber}\\
&\quad + \tfrac{1}{3}\,\bigl(\iota_n{{\mathscr{C}}}_3\wedge {{\mathscr{B}}}_2+{{\mathscr{C}}}_3\wedge \iota_n{{\mathscr{B}}}_2\bigr)\wedge\iota_n{{\mathrm{d}}}\lambda_2
+ \tfrac{1}{3}\,{{\mathscr{C}}}_3\wedge{{\mathrm{d}}}\iota_n{{\mathscr{C}}}_3\wedge \iota_n{{\mathrm{d}}}\chi_1\,,\end{aligned}$$ where we have parameterized $$\begin{aligned}
{{\hat{\lambda}}}_{\hat{7}, n} = \lambda_{7,n} + \lambda_{6,n} \wedge{{\mathrm{d}}}x^{{{z}}}\,.\end{aligned}$$ In terms of the $T$-duality-covariant tensor, we have $$\begin{aligned}
\delta(\iota_n{{D}}_{7,n}) &= \iota_n {{\mathrm{d}}}\lambda_{6,n} -\tfrac{1}{2}\,\bigl(\iota_n{{A}}_5 \wedge\iota_n{{\mathrm{d}}}\lambda_2 - \iota_n{{A}}_3 \wedge\iota_n{{\mathrm{d}}}\lambda_4 + \iota_n{{A}}_1 \wedge\iota_n{{\mathrm{d}}}\lambda_6 \bigr)
{\nonumber}\\
&\quad - \iota_n {{D}}_6 \wedge\iota_n{{\mathrm{d}}}\chi_1\,.
\label{eq:D71-delta-A}\end{aligned}$$
The field strength $\iota_n {{\hat{F}}}_{\hat{9},n}$ also contains the field strength of the potential ${{\mathscr{A}}}_{8,1}$. Since we have established the 11D–10D map, it is a straightforward task to compute the field strength. The relation between ${{\mathscr{A}}}_{8,1}$ and the potential ${{E}}_{8,1}$ is also given, and it will not be difficult to rewrite the field strength in a manifestly $T$-duality-covariant form. Similarly, we can also consider the reduction of the field strength $\iota_n {{\hat{F}}}_{\hat{11},n,n}$, which gives the field strengths of ${{\mathscr{A}}}_{9,1,1}$ and ${{\mathscr{A}}}_{10,1,1}$. The former can be expressed in the $T$-duality-covariant form by rewriting ${{\mathscr{A}}}_{9,1,1}$ into ${{E}}_{9,1,1}$. Since this is a 9-form, we need to introduce another deformation parameter associated with the non-geometric $R$-flux $R^{1,1}$ (with non-vanishing component $R^{n,n}=m$), which is the magnetic flux of the exotic $7^{(1,0)}_3$-brane (see [@1805.12117]). On the other hand, the field strength of ${{\mathscr{A}}}_{10,1,1}$ gives the field strength of the $T$-duality-covariant potential ${{F}}_{10,1,1}$, although the field strength automatically vanishes in 10D.
For the mixed-symmetry potential ${{\hat{A}}}_{\hat{9},\hat{3}}$, it is not straightforward to define the field strength because it is not related to the standard fields. However, since this is the 11D uplift of the type IIA potential ${{D}}_{8,2}$, it may be useful to define the type IIA field strength by using the component $\hat{H}^m{}_{n_1n_2}$ and uplift this to 11D.
Type IIB supergravity {#sec:gauge-IIB}
---------------------
In type IIB supergravity, the gauge transformations are given as follows [@hep-th/0506013; @hep-th/0602280; @hep-th/0611036; @1004.1348]: $$\begin{aligned}
\delta {{\mathsf{A}}}_2^{{{\alpha}}}&={{\mathrm{d}}}\Lambda^{{{\alpha}}}_1\,,
\\
\delta {{\mathsf{A}}}_4&={{\mathrm{d}}}\Lambda_3 - \tfrac{1}{2!}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\mathsf{A}}}^{{{\gamma}}}_2\wedge {{\mathrm{d}}}\Lambda^{{{\delta}}}_1 \,,
\\
\delta {{\mathsf{A}}}_6^{{{\alpha}}}&={{\mathrm{d}}}\Lambda^{{{\alpha}}}_5 + {{\mathsf{A}}}_2^{{{\alpha}}} \wedge {{\mathrm{d}}}\Lambda_3 - \tfrac{2}{3!}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\mathsf{A}}}^{{{\alpha}}}_2\wedge {{\mathsf{A}}}^{{{\gamma}}}_2 \wedge {{\mathrm{d}}}\Lambda^{{{\delta}}}_1 \,,
\\
\delta {{\mathsf{A}}}_8^{{{\alpha}}{{\beta}}}&={{\mathrm{d}}}\Lambda^{{{\alpha}}{{\beta}}}_7 + {{\mathsf{A}}}_2^{({{\alpha}}} \wedge {{\mathrm{d}}}\Lambda^{{{\beta}})}_5 + \tfrac{1}{2!}\,{{\mathsf{A}}}_2^{{{\alpha}}} \wedge{{\mathsf{A}}}_2^{{{\beta}}} \wedge {{\mathrm{d}}}\Lambda_3 - \tfrac{3}{4!}\,\epsilon_{{{\gamma}}{{\delta}}}\, {{\mathsf{A}}}^{{{\alpha}}}_2\wedge {{\mathsf{A}}}^{{{\beta}}}_2\wedge {{\mathsf{A}}}^{{{\gamma}}}_2 \wedge {{\mathrm{d}}}\Lambda^{{{\delta}}}_1 \,,
\\
\delta {{\mathsf{A}}}_{10}^{{{\alpha}}{{\beta}}{{\gamma}}}&={{\mathrm{d}}}\Lambda^{{{\alpha}}{{\beta}}{{\gamma}}}_9 + {{\mathsf{A}}}_2^{({{\alpha}}} \wedge {{\mathrm{d}}}\Lambda^{{{\beta}}{{\gamma}})}_7 + \tfrac{1}{2!}\,{{\mathsf{A}}}_2^{({{\alpha}}} \wedge{{\mathsf{A}}}_2^{{{\beta}}} \wedge {{\mathrm{d}}}\Lambda^{{{\gamma}})}_5
{\nonumber}\\
&\quad + \tfrac{1}{3!}\,{{\mathsf{A}}}_2^{{{\alpha}}} \wedge{{\mathsf{A}}}_2^{{{\beta}}}\wedge{{\mathsf{A}}}_2^{{{\gamma}}} \wedge {{\mathrm{d}}}\Lambda_3 - \tfrac{4}{5!}\,\epsilon_{{{\delta}}{{\epsilon}}}\, {{\mathsf{A}}}^{{{\alpha}}}_2\wedge {{\mathsf{A}}}^{{{\beta}}}_2\wedge {{\mathsf{A}}}^{{{\gamma}}}_2\wedge {{\mathsf{A}}}^{{{\delta}}}_2 \wedge {{\mathrm{d}}}\Lambda^{{{\epsilon}}}_1 \,.\end{aligned}$$ In terms of the component fields, we find $$\begin{aligned}
\delta {{\mathsf{B}}}_2 &= {{\mathrm{d}}}\chi_1\,, \qquad
\delta {{\mathsf{C}}}= {\operatorname{e}^{{{\mathsf{B}}}_2\wedge}}{{\mathrm{d}}}\lambda = {{\mathrm{d}}}\hat{\lambda} - {{\mathsf{H}}}_3\wedge \hat{\lambda} \,,
\\
\delta {{\mathsf{B}}}_6 &= {{\mathrm{d}}}\chi_5 + {{\mathsf{C}}}_2\wedge \bigl({{\mathrm{d}}}\lambda_3 + {{\mathsf{B}}}_2\wedge {{\mathrm{d}}}\lambda_1\bigr) \,,
\\
\delta {{\mathsf{E}}}_8 &= {{\mathrm{d}}}\zeta_7 +{{\mathsf{C}}}_2\wedge {{\mathrm{d}}}\chi_5 + \tfrac{1}{2!}\,{{\mathsf{C}}}_2\wedge{{\mathsf{C}}}_2\wedge\bigl({{\mathrm{d}}}\lambda_3 + {{\mathsf{B}}}_2\wedge {{\mathrm{d}}}\lambda_1\bigr)\,,
\\
\delta {{\mathsf{F}}}_{10} &= {{\mathrm{d}}}\eta_9 +{{\mathsf{C}}}_2\wedge {{\mathrm{d}}}\zeta_7 + \tfrac{1}{2!}\,{{\mathsf{C}}}_2\wedge{{\mathsf{C}}}_2\wedge {{\mathrm{d}}}\chi_5 + \tfrac{1}{3!}\,{{\mathsf{C}}}_2\wedge{{\mathsf{C}}}_2\wedge{{\mathsf{C}}}_2\wedge \bigl({{\mathrm{d}}}\lambda_3 + {{\mathsf{B}}}_2\wedge {{\mathrm{d}}}\lambda_1\bigr)\,, \end{aligned}$$ where the gauge parameters are parameterized as $$\begin{aligned}
\begin{split}
&(\Lambda^{{{\alpha}}}_1) = \begin{pmatrix} \chi_1 \\ - \lambda_1 \end{pmatrix},\quad
\Lambda_3 = \lambda_3\,,\quad
(\Lambda^{{{\alpha}}}_5) = \begin{pmatrix} \lambda_5 \\ - \chi_5 \end{pmatrix},
\\
&\begin{pmatrix} \Lambda^{{\bm{1}}{\bm{1}}}_7 \\ \Lambda^{{\bm{2}}{\bm{2}}}_7 \end{pmatrix}= \begin{pmatrix} \lambda_7 \\ \zeta_7 \end{pmatrix},\quad
\begin{pmatrix} \Lambda^{{\bm{1}}{\bm{1}}{\bm{1}}}_9 \\ \Lambda^{{\bm{2}}{\bm{2}}{\bm{2}}}_9 \end{pmatrix}
= \begin{pmatrix} \lambda_9 \\ -\eta_9 \end{pmatrix} ,
\end{split}\end{aligned}$$ and we have defined $$\begin{aligned}
\lambda \equiv \lambda_1+\lambda_3+\lambda_5+\lambda_7+\lambda_9\,,\qquad
\hat{\lambda} \equiv \hat{\lambda}_1+\hat{\lambda}_3+\hat{\lambda}_5+\hat{\lambda}_7+\hat{\lambda}_9 \equiv {\operatorname{e}^{{{\mathsf{B}}}_2\wedge}}\lambda \,.\end{aligned}$$
Let us also consider the field strength for the dual graviton ${{\mathsf{A}}}_{7,1}$. Similar to the case of type IIA supergravity, by introducing the untwisted tensor, $$\begin{aligned}
\hat{H}_{M_1M_2M_3} \equiv ({\operatorname{e}^{\bm{{{\mathsf{B}}}}}})_{M_1}{}^{N_1}\,({\operatorname{e}^{\bm{{{\mathsf{B}}}}}})_{M_2}{}^{N_2}\,({\operatorname{e}^{\bm{{{\mathsf{B}}}}}})_{M_3}{}^{N_3}\,\hat{H}_{N_1N_2N_3}\,,\qquad ({\operatorname{e}^{\bm{{{\mathsf{B}}}}}})_M{}^N \equiv \begin{pmatrix}
\delta_m^n & {{\mathsf{B}}}_{mn} \\ 0 & \delta^m_n \end{pmatrix}.\end{aligned}$$ we define the field strength as $$\begin{aligned}
\iota_n {\mathcal G}_{8,n} &\equiv \tfrac{1}{7!\,2!}\,\epsilon_{m_1\cdots m_7 a_1a_2 n}\, \hat{H}^{a_1a_2}{}_n \,{{\mathrm{d}}}x^{m_1}\wedge\cdots\wedge{{\mathrm{d}}}x^{m_7}
= \iota_n {{\mathsf{H}}}_{8, n} - \iota_n {{\mathsf{H}}}_7\wedge \iota_n {{\mathsf{B}}}_2 \,.\end{aligned}$$ By assuming ${\pounds}_n=0$ for an arbitrary field, it is invariant under the gauge transformation $$\begin{aligned}
\delta(\iota_n{{D}}_{7,n}) &= \iota_n {{\mathrm{d}}}\lambda_{6,n} +\tfrac{1}{2}\,\bigl(\iota_n{{A}}_6 \wedge\iota_n{{\mathrm{d}}}\lambda_1 - \iota_n{{A}}_4 \wedge\iota_n{{\mathrm{d}}}\lambda_3 + \iota_n{{A}}_2 \wedge\iota_n{{\mathrm{d}}}\lambda_5 \bigr)
{\nonumber}\\
&\quad - \iota_n {{D}}_6 \wedge\iota_n{{\mathrm{d}}}\chi_1\,,
\label{eq:D71-delta-B-D}\end{aligned}$$ which is the $T$-dual counterpart of . Since the dual graviton ${{\mathsf{A}}}_{7,n}$ is $S$-duality invariant, it is natural to expect that this field strength is invariant under $S$-duality. Indeed, we can express the field strength in a manifestly $S$-duality-invariant form, $$\begin{aligned}
\iota_n {\mathcal G}_{8,n}
&= {{\mathrm{d}}}\iota_n {{\mathsf{A}}}_{7,n} -\epsilon_{{{\gamma}}{{\delta}}}\,\iota_n{{\mathrm{d}}}{{\mathsf{A}}}_6^{{{\gamma}}}\wedge\iota_n{{\mathsf{A}}}^{{{\delta}}}_2
- \tfrac{1}{2}\, \iota_n{{\mathsf{A}}}_4\wedge\iota_n{{\mathrm{d}}}{{\mathsf{A}}}_4
{\nonumber}\\
&\quad +\tfrac{1}{2}\,\epsilon_{{{\gamma}}{{\delta}}}\,\iota_n\bigl({{\mathrm{d}}}{{\mathsf{A}}}_4\wedge {{\mathsf{A}}}^{{{\gamma}}}_2\bigr)\wedge\iota_n{{\mathsf{A}}}^{{{\delta}}}_2
+ \epsilon_{{{\gamma}}{{\delta}}}\,\iota_n \bigl({{\mathsf{A}}}_4\wedge {{\mathrm{d}}}{{\mathsf{A}}}^{{{\gamma}}}_2\bigr)\wedge\iota_n{{\mathsf{A}}}^{{{\delta}}}_2
{\nonumber}\\
&\quad - \tfrac{1}{16}\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{A}}}^{{{\alpha}}}_2\wedge{{\mathrm{d}}}{{\mathsf{A}}}^{{{\beta}}}_2\wedge \iota_n{{\mathsf{A}}}^{{{\gamma}}}_2\wedge\iota_n{{\mathsf{A}}}^{{{\delta}}}_2
+ \tfrac{1}{24}\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{A}}}^{{{\alpha}}}_2\wedge\iota_n{{\mathrm{d}}}{{\mathsf{A}}}^{{{\beta}}}_2\wedge {{\mathsf{A}}}^{{{\gamma}}}_2\wedge\iota_n{{\mathsf{A}}}^{{{\delta}}}_2 \,.\end{aligned}$$ The gauge transformation also can be expressed as $$\begin{aligned}
\delta(\iota_n {{\mathsf{A}}}_{7,n}) &= \iota_n{{\mathrm{d}}}\lambda_{6,n}
- \epsilon_{{{\gamma}}{{\delta}}}\,\iota_n{{\mathsf{A}}}^{{{\gamma}}}_6 \wedge\iota_n{{\mathrm{d}}}\Lambda^{{{\delta}}}_1
-\tfrac{1}{2}\,\iota_n{{\mathsf{A}}}_4 \wedge\iota_n{{\mathrm{d}}}\Lambda_3
{\nonumber}\\
&\quad + \epsilon_{{{\gamma}}{{\delta}}}\, \iota_n\bigl({{\mathsf{A}}}_4\wedge {{\mathsf{A}}}^{{{\gamma}}}_2\bigr)\wedge\iota_n{{\mathrm{d}}}\Lambda^{{{\delta}}}_1
-\tfrac{1}{4}\,\epsilon_{{{\gamma}}{{\delta}}}\,\iota_n{{\mathsf{A}}}_4 \wedge \iota_n\bigl({{\mathsf{A}}}^{{{\gamma}}}_2\wedge{{\mathrm{d}}}\Lambda^{{{\delta}}}_1\bigr)
{\nonumber}\\
&\quad - \tfrac{1}{16}\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{A}}}^{{{\alpha}}}_2\wedge{{\mathrm{d}}}\Lambda^{{{\beta}}}_1\wedge \iota_n{{\mathsf{A}}}^{{{\gamma}}}_2\wedge\iota_n{{\mathsf{A}}}^{{{\delta}}}_2
+ \tfrac{1}{24}\,\epsilon_{{{\alpha}}{{\beta}}}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{A}}}^{{{\alpha}}}_2\wedge\iota_n {{\mathsf{A}}}^{{{\beta}}}_2\wedge {{\mathsf{A}}}^{{{\gamma}}}_2\wedge\iota_n {{\mathrm{d}}}\Lambda^{{{\delta}}}_1 \,.
\label{eq:D71-delta-B}\end{aligned}$$
Here, we do not study additional potentials, but one can obtain their field strengths as follows. As mentioned in the type IIA case, the field strength of ${{D}}_{8,2}$ will be obtained by computing $\hat{H}^m{}_{n_1n_2}$. Then, rewriting the field strength in terms of the $S$-duality covariant potentials, we can obtain the field strength of ${{\mathsf{A}}}^{{{\alpha}}}_{8,2}$. In order to define the field strength of the potentials ${{E}}_{8+n,p,n}$, it is useful to find the $T$-duality-covariant expression for ${\lvert {{{E}}_{MN}} \rangle}$ that reproduce the field strength ${{\mathsf{H}}}_9$ as the particular component. This covariant field strength contains various field strengths, and by using the relation between ${{E}}_{8+n,p,n}$ and our mixed-symmetry potentials, we can obtain the $S$-duality-covariant expressions for field strengths.
Conclusions {#sec:conclusions}
===========
In this paper, we have provided explicit definitions of mixed-symmetry potentials by finding their relation to the standard supergravity fields under $T$- and $S$-duality transformations. The obtained $T$-duality rules are generally very complicated, but by performing certain field redefinitions, they are considerably simplified. The redefined fields ${{A}}_{p}$, ${{D}}_{6+n,n}$, ${{E}}_{8+n,p,n}$, and ${{F}}_{10,p,p}$ are identified with certain components of the ${\text{O}}(10,10)$-covariant tensors ${{A}}_{\dot{a}}$, ${{D}}_{M_1\cdots M_4}$, ${{E}}_{MN\dot{a}}$, and ${{F}}^+_{M_1\cdots M_{10}}$. These ${\text{O}}(10,10)$-covariant tensors have been studied in the literature, but their relation to the standard supergravity fields have not discussed enough. For example, the potential ${{E}}_8$ has been expected to the $S$-dual of the R–R 8-form ${{\mathsf{C}}}_8$, but the $S$-duality rule , including the non-linear terms, is newly determined in this paper. The $S$-duality rule for ${{F}}_{10}$ and more mixed-symmetry potentials are also newly determined. Additionally, we have also studied the field strengths of mixed-symmetry potentials. Most of the results has been known in the literature (where the mixed-symmetry potentials are treated as $p$-forms), but here we have clarified the relation to the field strength $H_{MNP}$ studied in DFT. We have also provided the $S$-duality-invariant expression for the field strength of the dual graviton.
The linear map has been originally studied for the generalized metric [@hep-th/0402140; @1701.07819; @1909.01335]. By comparing two parameterizations, duality rules for potentials that appear in the $E_{n(n)}$ generalized metric ($n\leq 8$) are determined. The linear map for the 1-form ${\mathcal A}_\mu^I$ studied here is more efficient to find the duality rules, but the parameterization is involved and the obtained $T$-duality rules are rather long. In the linear map for the generalized metric, the parameterization is more systematic, and by considering the linear map for the $E_{11}$ generalized metric, we may find a better definition of mixed-symmetry potentials which simplify the duality rules.
As we have demonstrated, the linear map works well in finding the duality rules for mixed-symmetry potentials, and it is a straightforward to consider more mixed-symmetry potentials. In addition, having clarified the definitions of mixed-symmetry potentials, it is important to consider the application to the worldvolume theories of exotic branes. It is also interesting to study the supersymmetry transformations for mixed-symmetry potentials (where Killing vectors should be involved) by extending the series of works [@hep-th/0506013; @hep-th/0602280; @hep-th/0611036; @1004.1348].
Before closing this paper, let us comment on a relevant open issue, called the exotic duality [@1109.4484; @1009.4657; @1102.0934; @1309.2653; @1412.8769], which is the electric-magnetic duality for exotic branes with co-dimension equal to (or higher than) two. As we have already mentioned, the mixed-symmetry potentials couple to various exotic branes electrically. On the other hand, the exotic branes (with co-dimension two) magnetically couple to certain dual fields [@1303.1413; @1402.5972; @1411.1043; @1412.8769; @1612.08738], such as the $\beta$-fields[^13] or the $\gamma$-fields [@1007.5509; @1411.6640; @1412.0635; @1412.8769], which are associated with the non-geometric $Q$-flux [@hep-th/0508133] or the $P$-fluxes [@hep-th/0602089; @0811.2900; @1007.5509].[^14] Thus, the exotic duality is the electric-magnetic duality between the mixed-symmetry potentials and the dual potentials. An example is given in , $$\begin{aligned}
\tfrac{1}{9!}\,\epsilon^{m n_1\cdots n_9}\, {{\mathsf{H}}}_{n_1\cdots n_9} = - {\operatorname{e}^{-2\tilde{\phi}}} \sqrt{{\lvert {{{\mathsf{g}}}} \rvert}}\, {{\mathsf{g}}}^{mn}\, \partial_n \tilde{\gamma} \,,
\label{eq:EM-7_3-component}\end{aligned}$$ where $\tilde{\gamma}$ roughly corresponds to the $\gamma$-field (as we explain below). Recently, the $T$-duality-covariant expression of the exotic duality has been investigated in [@1508.00780; @1603.07380; @1612.02691] but it has not yet fully succeeded. For the $T$-duality-covariant exotic duality, it will be important to establish the $T$-duality-covariant description of the dual potentials, and we make a small attempt below.
First of all, let us explain the definition of the dual fields, such as the $\beta$- and $\gamma$-fields. In the $U$-duality formulations, the supergravity fields are embedded into the generalized metric ${\mathcal M}_{IJ}$, which is defined as (see [@1111.0459] and references therein) $$\begin{aligned}
{\mathcal M}_{IJ} \equiv ({\mathcal E}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}})_I{}^K\,\delta_{KL}\,{\mathcal E}^L{}_J\,,\end{aligned}$$ where the generalized vielbein ${\mathcal E}^I{}_J$ is the matrix representation of an $E_{n(n)}$ element in the vector representation. The identity matrix $\delta_{IJ}$ is invariant under the maximal compact subgroup, and the generalized vielbein ${\mathcal E}^I{}_J$ is generally parameterized by the Borel subalgebra. For example, in the type IIB theory, the Borel subalgebra is generated by $$\begin{aligned}
\{ {\mathsf{K}}^m{}_n\ (m\leq n),\ 2\,{\mathsf{R}}_{{\bm{1}}{\bm{2}}},\ {\mathsf{R}}_{{\bm{2}}{\bm{2}}},\ {\mathsf{R}}_{{{\alpha}}}^{m_1m_2},\ {\mathsf{R}}^{m_1\cdots m_4},\ {\mathsf{R}}^{m_1\cdots m_6}_{{{\alpha}}},\ {\mathsf{R}}^{m_1\cdots m_7,m},\ \cdots \}\,,\end{aligned}$$ and the generalized vielbein can be parameterized as [@hep-th/0107181] $$\begin{aligned}
{\mathcal E}= {\operatorname{e}^{\sum h_m{}^n\,{\mathsf{K}}^m{}_n}} {\operatorname{e}^{2\,{{\Phi}}\,{\mathsf{R}}_{{\bm{1}}{\bm{2}}}}} {\operatorname{e}^{-{{\mathsf{C}}}_0\,{\mathsf{R}}_{{\bm{2}}{\bm{2}}}}} {\operatorname{e}^{\frac{1}{2!}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_1m_2}\,{\mathsf{R}}_{{{\alpha}}}^{m_1m_2}}} {\operatorname{e}^{\frac{1}{4!}\,{{\bm{\mathsf{A}}}}_{m_1\cdots m_4}\, {\mathsf{R}}^{m_1\cdots m_4}}} {\operatorname{e}^{\frac{1}{6!}\,{{\bm{\mathsf{A}}}}^{{{\alpha}}}_{m_1\cdots m_6}\,{\mathsf{R}}^{m_1\cdots m_6}_{{{\alpha}}}}} \cdots \,,\end{aligned}$$ where the standard vielbein corresponds to $(e^a{}_m)={\operatorname{e}^{-h^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}}}$. On the other hand, we can also consider the negative Borel subalgebra, spanned by [@1612.08738] $$\begin{aligned}
\{ {\mathsf{K}}^m{}_n\ (m\geq n),\ 2\,{\mathsf{R}}_{{\bm{1}}{\bm{2}}},\ -{\mathsf{R}}_{{\bm{1}}{\bm{1}}},\ {\mathsf{R}}^{{{\alpha}}}_{m_1m_2},\ {\mathsf{R}}_{m_1\cdot m_4},\ {\mathsf{R}}_{m_1\cdots m_6}^{{{\alpha}}},\ {\mathsf{R}}_{m_1\cdots m_7,m},\ \cdots \}\,,\end{aligned}$$ and introduce the dual parameterization as[^15] $$\begin{aligned}
\tilde{{\mathcal E}} = {\operatorname{e}^{\sum \tilde{h}_m{}^n\,{\mathsf{K}}^m{}_n}} {\operatorname{e}^{2\,\tilde{{{\Phi}}}\,{\mathsf{R}}_{{\bm{1}}{\bm{2}}}}} {\operatorname{e}^{-\gamma\,{\mathsf{R}}_{{\bm{1}}{\bm{1}}}}} {\operatorname{e}^{-\frac{1}{2!}\,\tilde{{{\bm{\mathsf{A}}}}}_{{{\alpha}}}^{m_1m_2}\,{\mathsf{R}}^{{{\alpha}}}_{m_1m_2}}} {\operatorname{e}^{-\frac{1}{4!}\,\tilde{{{\bm{\mathsf{A}}}}}^{m_1\cdots m_4}\, {\mathsf{R}}_{m_1\cdot m_4}}} {\operatorname{e}^{-\frac{1}{6!}\,\tilde{{{\bm{\mathsf{A}}}}}_{{{\alpha}}}^{m_1\cdots m_6}\,{\mathsf{R}}_{m_1\cdots m_6}^{{{\alpha}}}}} \cdots \,.\end{aligned}$$ The dual vielbein is similarly defined by $(\tilde{e}^a{}_m)={\operatorname{e}^{-\tilde{h}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}}}$ and the dual metric is $\tilde{{{\mathsf{g}}}}_{mn}\equiv (\tilde{e}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,\tilde{e})_{mn}$. Then, by comparing the two parameterizations of the generalized metric, $$\begin{aligned}
({\mathcal E}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,{\mathcal E})_{IJ} = {\mathcal M}_{IJ} = (\tilde{{\mathcal E}}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,\tilde{{\mathcal E}})_{IJ}\,,
\label{eq:standard-M-dual}\end{aligned}$$ we can obtain the dual fields as a local redefinitions of the standard fields [@1612.08738]. For example, if we consider only the NS–NS fields, the relation is simplified as $$\begin{aligned}
\begin{split}
\begin{pmatrix} g_{mn} - B_{mp}\,g^{pq}\,B_{qn} & -B_{mp}\,g^{pn} \\ g^{mp}\,B_{pn} & g^{pq}
\end{pmatrix} &= \begin{pmatrix} \tilde{g}_{mn} & \tilde{g}_{mp}\,\beta^{pn} \\ -\beta^{mp}\,\tilde{g}_{pn} & \tilde{g}^{mn} - \beta^{mp}\,\tilde{g}_{pq}\,\beta^{qn}
\end{pmatrix} ,
\\
{\operatorname{e}^{-2{{\Phi}}}}\sqrt{{\lvert {g} \rvert}} &= {\operatorname{e}^{-2\tilde{{{\Phi}}}}}\sqrt{{\lvert {\tilde{g}} \rvert}}\,,
\end{split}
\label{eq:DFT-non-geometric}\end{aligned}$$ where $g_{mn}\equiv {\operatorname{e}^{{{\Phi}}/2}}{{\mathsf{g}}}_{mn}$ is the standard string-frame metric and $\tilde{g}_{mn}\equiv {\operatorname{e}^{\tilde{{{\Phi}}}/2}}\tilde{{{\mathsf{g}}}}_{mn}$ is the dual-string-frame metric. They are precisely the relations studied in the $\beta$-supergravity [@1106.4015; @1202.3060; @1204.1979; @1306.4381; @1411.6640]. On the other hand, if we only keep the metric ${{\mathsf{g}}}_{mn}$ and ${{\mathsf{m}}}_{{{\alpha}}{{\beta}}}$, the relation reduces to $$\begin{aligned}
{{\mathsf{g}}}_{mn} = \tilde{{{\mathsf{g}}}}_{mn} \,,\qquad \tilde{\phi} = \tilde{{{\Phi}}} \,,\qquad \tilde{\gamma} = \gamma \,,
\label{eq:S-dual-non-geometric}\end{aligned}$$ and this shows that the $\gamma$-field is similar to the $\tilde{\gamma}$ appearing in . In general, without any truncations, these relations receive non-linear corrections.
Secondly, let us explain the $T$-duality rules for the dual fields. By following the discussion of [@1701.07819], the $T$-duality rules for the dual fields are determined in the same manner as the standard potentials. To this end, we parameterize the dual fields in the same manner as –, for example, $$\begin{aligned}
\bigl(\tilde{{{\bm{\mathsf{A}}}}}_{{{\alpha}}}^{m_1m_2}\bigr) \equiv \begin{pmatrix} \beta^{m_1m_2} \\\ -\gamma^{m_1m_2} \end{pmatrix} ,\qquad
\tilde{{{\bm{\mathsf{A}}}}}^{m_1\cdots m_4} \equiv \gamma^{m_1\cdots m_4} - 3\,\gamma^{[m_1m_2}\,\beta^{m_3m_4]} \,,\quad \cdots \,.\end{aligned}$$ Then, we find that the gamma fields $\gamma^{m_1\cdots m_p}$ follow the same $T$-duality rules as those of the R–R field ${{\mathsf{C}}}_{m_1\cdots m_p}$, although the position of the indices are opposite:[^16] $$\begin{aligned}
\begin{split}
\gamma'^{a_1\cdots a_{n-1}{{y}}}&= \gamma^{a_1\cdots a_{n-1}} - \tfrac{(n-1)\,\gamma^{[a_1\cdots a_{n-2}|y|}\,\tilde{{{\mathsf{g}}}}^{a_{n-1}]y}}{\tilde{{{\mathsf{g}}}}^{yy}}\,,
\\
\gamma'^{a_1\cdots a_n} &= \gamma^{a_1\cdots a_ny} - n\, \gamma^{[a_1\cdots a_{n-1}}\, \beta^{a_n]y} - \tfrac{n\,(n-1)\,\gamma^{[a_1\cdots a_{n-2}|y|}\, \beta^{a_{n-1}|y|}\,\tilde{{{\mathsf{g}}}}^{a_n]y}}{\tilde{{{\mathsf{g}}}}^{yy}}\,.
\end{split}\end{aligned}$$ The dual fields in the NS–NS sector also transform as $$\begin{aligned}
\begin{split}
\tilde{g}'^{ab} &= \tilde{g}^{ab} - \tfrac{\tilde{g}^{a y}\,\tilde{g}^{b y}-\beta^{a y}\,\beta^{b y}}{\tilde{g}^{yy}}\,,\qquad
\tilde{g}'^{a {{y}}}=-\tfrac{\beta^{a y}}{\tilde{g}^{yy}}\,,\qquad
\tilde{g}'^{{{y}}{{y}}}=\tfrac{1}{\tilde{g}^{yy}}\,,
\\
\beta'^{ab} &= \beta^{ab} - \tfrac{\beta^{ay}\,\beta^{by}-\tilde{g}^{ay}\,\beta^{by}}{\tilde{g}^{yy}}\,,\qquad
\beta'^{ay} = -\tfrac{\tilde{g}^{ay}}{\tilde{g}^{yy}} \,, \qquad
{\operatorname{e}^{2\tilde{{{\Phi}}}'}}= \tfrac{{\operatorname{e}^{2\tilde{{{\Phi}}}}}}{\tilde{g}^{yy}} \,.
\end{split}\end{aligned}$$ Then, we find that $$\begin{aligned}
\tilde{{\mathcal H}}_{MN} \equiv
\begin{pmatrix} \tilde{g}_{mn} & \tilde{g}_{mp}\,\beta^{pn} \\ -\beta^{mp}\,\tilde{g}_{pn} & \tilde{g}^{mn} - \beta^{mp}\,\tilde{g}_{pq}\,\beta^{qn}
\end{pmatrix} ,\qquad
{\operatorname{e}^{-2\tilde{d}}} \equiv {\operatorname{e}^{-2\tilde{{{\Phi}}}}}\sqrt{{\lvert {\tilde{g}} \rvert}}\,,\end{aligned}$$ transform covariantly under $T$-duality, which reduce to when only the NS–NS fields are present. Similarly, we can show that the $\gamma$-field also transform covariantly under $T$-duality. For this purpose, we define the dual field $\alpha$ associated with ${{A}}= {\operatorname{e}^{-{{\mathscr{B}}}_2\wedge}}{{\mathscr{C}}}$ (or ${{A}}= {\operatorname{e}^{-{{\mathsf{B}}}_2\wedge}}{{\mathsf{C}}}$) as $$\begin{aligned}
\alpha \equiv {\operatorname{e}^{-\beta \wedge}} \gamma \qquad \bigl(\beta\equiv \tfrac{1}{2!}\,\beta^{mn}\,\partial_m\wedge\partial_n\bigr)\,,\end{aligned}$$ where $\alpha\equiv \sum_p \frac{1}{p!}\,\alpha^{m_1\cdots m_p}\,\partial_{m_1}\wedge\cdots\wedge\partial_{m_p}$ and $\gamma\equiv \sum_p \frac{1}{p!}\,\gamma^{m_1\cdots m_p}\,\partial_{m_1}\wedge\cdots\wedge\partial_{m_p}$ are poly-vectors and $\wedge$ is the wedge product for poly-vectors. This $\alpha^{m_1\cdots m_p}$ transforms as $$\begin{aligned}
\alpha^{a_1\cdots a_p} = \alpha^{a_1\cdots a_py}\,,\qquad
\alpha^{a_1\cdots a_{p-1}y} = \alpha^{a_1\cdots a_{p-1}}\,,\end{aligned}$$ under the $T$-duality along the $x^y$-direction. In other words, $$\begin{aligned}
{\lvert {\alpha} \rangle} \equiv \sum_p \frac{1}{p!}\,\alpha^{m_1\dots m_p}\,\tilde{\Gamma}_{m_1\cdots m_p}{\lvert {\tilde{0}} \rangle}\,, \end{aligned}$$ transforms as an ${\text{O}}(10,10)$ spinor,[^17] where we have defined a new vacuum annihilated by $\Gamma^m$, $$\begin{aligned}
{\lvert {\tilde{0}} \rangle} \equiv C\,{\lvert {0} \rangle} = \Gamma^{0\cdots 9}\,{\lvert {0} \rangle} \,,\qquad \tilde{\Gamma}^M \equiv \Gamma^{11}\,\Gamma^M\,. \end{aligned}$$ Other dual fields, such as $\beta^{m_1\cdots m_6}$ [@1412.8769; @1612.08738], also can be embedded into $T$-duality tensors.
Finally, let us consider the exotic duality, in particular . The left-hand side is the field strength of the potential ${{E}}_8$. Since ${{E}}_8$ is a component of ${\lvert {{{E}}_{MN}} \rangle}$, the field strength will be also defined covariantly. The field strength has been discussed in [@1903.05601], although the explicit form has not yet been determined, $$\begin{aligned}
{\lvert {K^M} \rangle} \sim \partial_N {\lvert {{{E}}^{MN}} \rangle} + \cdots \,.\end{aligned}$$ On the other hand, the right-hand side contains ${{\mathrm{d}}}\tilde{\gamma}$, which is roughly equal to the $P$-flux $P_1 \equiv {{\mathrm{d}}}\gamma$, as we have seen in . The $P$-flux may be also defined $T$-duality covariantly, $$\begin{aligned}
{\lvert {P_M} \rangle} \equiv \partial_M {\lvert {\gamma} \rangle} + \cdots \,, \end{aligned}$$ and the exotic duality will be a covariant relation connecting ${\lvert {K^M} \rangle}$ and ${\lvert {P_M} \rangle}$. The non-trivial point is that although the field strength ${\lvert {K^M} \rangle}$ is defined in the standard parameterization, the $P$-flux ${\lvert {P_M} \rangle}$ is defined in the dual parameterization, and it is not easy to find the relation. In order to find the covariant expression for the exotic duality, it may be useful to consider the supergravity action for the dual fields. As it has been (partially) worked out in [@1612.08738], by substituting the dual parameterization into the action of the $U$-duality-covariant supergravity, known as the exceptional field theory [@1308.1673; @1312.0614; @1312.4542; @1406.3348] (which is based on DFT and earlier works [@hep-th/0104081; @hep-th/0307098; @0712.1795; @0901.1581; @1008.1763; @1110.3930; @1111.0459; @1208.5884]), we obtain the action for the dual fields, $$\begin{aligned}
{\mathcal L}&= \tilde{*} \tilde{{\mathsf{R}}} - \tfrac{1}{2}\,{{\mathrm{d}}}\tilde{\varphi}\wedge \tilde{*} {{\mathrm{d}}}\tilde{\varphi} - \tfrac{1}{2}{\operatorname{e}^{-2\tilde{\varphi}}}{{\mathrm{d}}}\gamma\wedge \tilde{*} {{\mathrm{d}}}\gamma + \cdots \,.\end{aligned}$$ Then, the equations of motion for the dual fields $\gamma^{m_1\cdots m_p}$ are precisely the exotic duality, as discussed in [@1412.8769]. Thus, in order to find the $T$-duality-covariant exotic duality, it will be useful to find the $T$-duality-covariant action for the dual fields, by defining the dual fields as $T$-duality-covariant tensors. From the covariant action, we obtain the $T$-duality-covariant equations of motion for the dual fields, and they will correspond to the exotic duality.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank José J. Fernández-Melgarejo and Shozo Uehara for useful discussions. This work is supported by JSPS Grant-in-Aids for Scientific Research (C) 18K13540 and (B) 18H01214.
Notations {#app:notation}
=========
Differential forms and mixed-symmetry potentials
------------------------------------------------
We employ the following convention for differential forms: $$\begin{aligned}
\begin{split}
&(*\alpha_p)_{m_1\cdots m_{10-p}} =\tfrac{1}{(10-p)!}\,\varepsilon^{n_1\cdots n_p}{}_{m_1\cdots m_{10-p}}\,\alpha_{n_1\cdots n_p} \,,
\\
&*({{\mathrm{d}}}x^{m_1}\wedge \cdots \wedge {{\mathrm{d}}}x^{m_p}) = \tfrac{1}{(10-p)!}\,\varepsilon^{m_1\cdots m_p}{}_{n_1\cdots n_{10-p}}\,{{\mathrm{d}}}x^{n_1}\wedge \cdots \wedge {{\mathrm{d}}}x^{n_{10-p}} \,,
\\
&(\iota_v \alpha_p) = \tfrac{1}{(p-1)!}\,v^n\,\alpha_{n m_1\cdots m_{p-1}}\,{{\mathrm{d}}}x^{m_1}\wedge\cdots\wedge {{\mathrm{d}}}x^{m_{p-1}}\,,
\end{split}\end{aligned}$$ where $$\begin{aligned}
\varepsilon_{m_1\cdots m_{10}}\equiv \sqrt{{\lvert {g} \rvert}}\, \epsilon_{m_1\cdots m_{10}}\,,\quad
\varepsilon^{m_1\cdots m_{10}}\equiv \tfrac{1}{\sqrt{{\lvert {g} \rvert}}}\,\epsilon^{m_1\cdots m_{10}}\,,\quad
\epsilon_{0\cdots 9}= 1 \,,\quad
\epsilon^{0\cdots 9}= -1\,.\end{aligned}$$ The symmetrization and antisymmetrization are normalized as $$\begin{aligned}
A_{(m_1\cdots m_n)} \equiv \tfrac{1}{n!}\,\bigl(A_{m_1\cdots m_n} + \cdots \bigr) \,,\qquad
A_{[m_1\cdots m_n]} \equiv \tfrac{1}{n!}\,\bigl(A_{m_1\cdots m_n} \pm \cdots \bigr) \,.\end{aligned}$$ Indices separated by “$|$” are not (anti-)symmetrized. For example, $$\begin{aligned}
3\,{{\hat{A}}}_{[i_1|k_1k_2|}\,{{\hat{A}}}_{i_2i_3]j} = {{\hat{A}}}_{i_1 k_1k_2}\,{{\hat{A}}}_{i_2i_3j} + {{\hat{A}}}_{i_2 k_1k_2}\,{{\hat{A}}}_{i_3i_1j} + {{\hat{A}}}_{i_3 k_1k_2}\,{{\hat{A}}}_{i_1i_2j} \,.\end{aligned}$$ When two groups of indices are antisymmetrized, we have used overlines. For example, $$\begin{aligned}
3\,{{\hat{A}}}_{[i_1|\bar{k}_1\bar{k}_2|}\,{{\hat{A}}}_{i_2\cdots i_6]\bar{k}_3}
= {{\hat{A}}}_{[i_1|k_1k_2|}\,{{\hat{A}}}_{i_2\cdots i_6]k_3}
+ {{\hat{A}}}_{[i_1|k_2k_3|}\,{{\hat{A}}}_{i_2\cdots i_6]k_1}
+ {{\hat{A}}}_{[i_1|k_3k_1|}\,{{\hat{A}}}_{i_2\cdots i_6]k_2} \,.\end{aligned}$$
In this paper, we consider only the components of the mixed-symmetry potentials that satisfy the restriction rule or . For convenience, by using the equality $\simeq$, we have expressed various equations without making the restriction rule manifest. However, we can always convert the equality $\simeq$ into the exact equality $=$ by making the restriction rule manifest. For example, let us consider the equation , $$\begin{aligned}
{{D}}_{m_1\cdots m_7, n}
&\simeq {{\mathscr{A}}}_{m_1\cdots m_7, n}
+ 7\, {{D}}_{[m_1\cdots m_6} \,{{\mathscr{B}}}_{m_7]n}
- \tfrac{1}{2}\, {{\mathscr{C}}}_{m_1\cdots m_7} \,{{\mathscr{C}}}_{n}
- \tfrac{21}{2}\, {{\mathscr{C}}}_{[m_1\cdots m_5}\,{{\mathscr{C}}}_{m_6m_7]n}
{\nonumber}\\
&\quad + 70\, {{\mathscr{C}}}_{[m_1m_2m_3}\, {{\mathscr{C}}}_{m_4m_5 |n|}\, {{\mathscr{B}}}_{m_6m_7]} \,.\end{aligned}$$ In this example, the restriction rule is $\{m_1,\dotsc, m_7\}\ni n$, and this is automatically satisfied by choosing $m_7=n$. We then obtain $$\begin{aligned}
{{D}}_{m_1\cdots m_6 n, n}
&= {{\mathscr{A}}}_{m_1\cdots m_6 n, n}
- 6\, {{D}}_{[m_1\cdots m_5 |n|} \,{{\mathscr{B}}}_{m_6]n}
- \tfrac{1}{2}\, {{\mathscr{C}}}_{m_1\cdots m_6 n} \,{{\mathscr{C}}}_{n}
- \tfrac{15}{2}\, {{\mathscr{C}}}_{[m_1\cdots m_4|n|}\,{{\mathscr{C}}}_{m_5m_6]n}
{\nonumber}\\
&\quad
+ 20\, {{\mathscr{C}}}_{[m_1m_2m_3}\, {{\mathscr{C}}}_{m_4m_5 |n|}\, {{\mathscr{B}}}_{m_6]n}
+ 30\, {{\mathscr{C}}}_{[m_1m_2|n|}\, {{\mathscr{C}}}_{m_3m_4 |n|}\, {{\mathscr{B}}}_{m_5m_6]} \,.
\label{eq:D6-A-explicit}\end{aligned}$$ In general, this makes the expressions longer, and that is the reason why we are using $\simeq$.
In order to simplify expressions, it is also useful to use the notation of the differential form. For example, several relations for the $T$-duality-covariant potentials become $$\begin{aligned}
{{D}}_6 &= {{\mathscr{B}}}_6 - \tfrac{1}{2} \,{{\mathscr{C}}}_5\wedge {{\mathscr{C}}}_1\,,
\\
\iota_n {{D}}_{7 , n}
&= \iota_n {{\mathscr{A}}}_{7, n}
- \iota_n {{D}}_6 \wedge \iota_n {{\mathscr{B}}}_2
- \tfrac{1}{2}\, \bigl(\iota_n {{\mathscr{C}}}_7 \, \iota_n {{\mathscr{C}}}_1 + \iota_n {{\mathscr{C}}}_5 \wedge \iota_n {{\mathscr{C}}}_3\bigr)
{\nonumber}\\
&\quad
+ \tfrac{1}{3}\, \bigl({{\mathscr{B}}}_{2}\wedge \iota_n {{\mathscr{C}}}_3 - {{\mathscr{C}}}_{3} \wedge \iota_n {{\mathscr{B}}}_2 \bigr) \wedge \iota_n {{\mathscr{C}}}_3 \,,
\label{eq:A-D71-form}
\\
\iota_n {{E}}_{8 , n} &= \iota_n {{\mathscr{A}}}_{8, n} + \iota_n {{\mathscr{B}}}_6 \wedge \iota_n {{\mathscr{C}}}_3
- \tfrac{1}{3} \, \iota_n {{\mathscr{C}}}_5 \wedge \bigl({{\mathscr{C}}}_3\, \iota_n {{\mathscr{C}}}_1 - \iota_n {{\mathscr{C}}}_3 \wedge {{\mathscr{C}}}_1\bigr)\,,
\\
\iota_n{{F}}_{10, n , n}
&= \iota_n{{\mathscr{A}}}_{10, n , n} - \iota_n {{\mathscr{A}}}_{8, n}\wedge \iota_n{{\mathscr{C}}}_3
+ \tfrac{1}{2}\,\iota_n {{\mathscr{B}}}_6 \wedge \iota_n {{\mathscr{C}}}_3\wedge \iota_n {{\mathscr{C}}}_3
{\nonumber}\\
&\quad
+\tfrac{1}{20}\, \iota_n{{\mathscr{C}}}_7 \wedge\bigl(\iota_n{{\mathscr{C}}}_3 \wedge {{\mathscr{C}}}_1 - {{\mathscr{C}}}_3 \, \iota_n{{\mathscr{C}}}_1 \bigr)\,\iota_n{{\mathscr{C}}}_1
{\nonumber}\\
&\quad
+\tfrac{1}{20}\, \iota_n{{\mathscr{C}}}_5\wedge \bigl(\iota_n{{\mathscr{C}}}_5\wedge {{\mathscr{C}}}_1 +2\,{{\mathscr{C}}}_3\wedge \iota_n {{\mathscr{C}}}_3\bigr)\,\iota_n{{\mathscr{C}}}_1
{\nonumber}\\
&\quad
-\tfrac{1}{20}\, {{\mathscr{C}}}_5\wedge \bigl(\iota_n{{\mathscr{C}}}_5\wedge \iota_n{{\mathscr{C}}}_1 -\tfrac{1}{2}\,\iota_n{{\mathscr{C}}}_3\wedge \iota_n {{\mathscr{C}}}_3\bigr)\,\iota_n{{\mathscr{C}}}_1
{\nonumber}\\
&\quad
-\tfrac{1}{8}\,\iota_n{{\mathscr{C}}}_5\wedge \iota_n{{\mathscr{C}}}_3\wedge \iota_n{{\mathscr{C}}}_3\wedge{{\mathscr{C}}}_1 \,,
\\
{{D}}_6 &= {{\mathsf{B}}}_6 - \tfrac{1}{2}\,\bigl({{\mathsf{C}}}_6\,{{\mathsf{C}}}_0 + {{\mathsf{C}}}_4 \wedge {{\mathsf{C}}}_2 \bigr) \,,
\\
\iota_n{{D}}_{7, n}
&= \iota_n{{\mathsf{A}}}_{7, n}
- \tfrac{1}{2}\, \iota_n{{\mathsf{C}}}_6\wedge \iota_n {{\mathsf{C}}}_2
+ \tfrac{1}{2}\, {{\mathsf{C}}}_0\, \iota_n{{\mathsf{C}}}_6\wedge\iota_n {{\mathsf{B}}}_2
+ \tfrac{1}{2}\, {{\mathsf{C}}}_4\wedge\iota_n{{\mathsf{B}}}_2\wedge \iota_n {{\mathsf{C}}}_2
{\nonumber}\\
&\quad
+ \tfrac{1}{4}\, \iota_n{{\mathsf{C}}}_4\wedge\bigl({{\mathsf{B}}}_2\wedge \iota_n {{\mathsf{C}}}_2 - {{\mathsf{C}}}_2\wedge \iota_n {{\mathsf{B}}}_2\bigr)
- \tfrac{1}{4}\, {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2\wedge \iota_n{{\mathsf{B}}}_2\wedge\iota_n {{\mathsf{C}}}_2\,,
\\
{{E}}_8 &= {{\mathsf{E}}}_8 - {{\mathsf{B}}}_6\wedge {{\mathsf{C}}}_2 +\tfrac{1}{3}\,{{\mathsf{C}}}_4 \wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2 + \tfrac{1}{6}\,{{\mathsf{C}}}_4 \wedge {{\mathsf{C}}}_4 \,{{\mathsf{C}}}_0 \,,
\\
{{F}}_{10}
&= {{\mathsf{F}}}_{10} - {{\mathsf{E}}}_8\wedge {{\mathsf{C}}}_2
+\tfrac{1}{2}\,{{\mathsf{B}}}_6 \wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2
+ \tfrac{1}{20}\, {{\mathsf{C}}}_6\wedge {{\mathsf{C}}}_4\,{{\mathsf{C}}}_0^2
{\nonumber}\\
&\quad
- \tfrac{1}{40}\, {{\mathsf{C}}}_6\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2 \,{{\mathsf{C}}}_0
- \tfrac{1}{20}\, {{\mathsf{C}}}_4\wedge {{\mathsf{C}}}_4\wedge{{\mathsf{C}}}_2\,{{\mathsf{C}}}_0
- \tfrac{1}{8}\, {{\mathsf{C}}}_4\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2
{\nonumber}\\
&\quad
+ \tfrac{1}{30}\, {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2 \wedge {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2 \,.\end{aligned}$$
Supergravity fields {#app:sugra}
-------------------
Our gauge potentials are related to those used in [@hep-th/9802199; @hep-th/9812188; @hep-th/9806169; @hep-th/9908094] as follows. In type IIA theory, their fields (left) and our fields (right) are related as $$\begin{aligned}
B = {{\mathscr{B}}}_2 \,,\quad
C^{(p)} = {{\mathscr{C}}}_p\,,\quad
B^{(6)} = - {{\mathscr{B}}}_6\,, \quad
N^{(7)} = {{\mathscr{A}}}_{7,n} \,,\quad
N^{(8)} = -{{\mathscr{A}}}_{8,n}\,,\end{aligned}$$ where $n$ represents a Killing direction. In type IIB theory, the relation is summarized as $$\begin{aligned}
{\mathcal B}=& {{\mathsf{B}}}_2\,,\quad
C^{(0)} = - {{\mathsf{C}}}_0\,,\quad
C^{(2)} = - {{\mathsf{C}}}_2\,,\quad
C^{(4)} = - {{\mathsf{A}}}_4\,,\quad
C^{(6)} = -\bigl( {{\mathsf{C}}}_6-\tfrac{1}{4}\, {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{C}}}_2\bigr)\,,\quad
{\nonumber}\\
C^{(8)} &= -\bigl({{\mathsf{C}}}_8 - \tfrac{1}{3!}\,{{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2\wedge {{\mathsf{B}}}_2\bigr)\,,\quad
B^{(6)} = -\bigl( {{\mathsf{B}}}_6-\tfrac{1}{4}\, {{\mathsf{C}}}_2\wedge {{\mathsf{C}}}_2\wedge {{\mathsf{B}}}_2\bigr) \,,\quad
\widetilde{C}^{(8)} = {{\mathsf{E}}}_8\,.\end{aligned}$$ Although the full $T$-duality rule for the dual graviton $N^{(7)}$ has not been obtained there, by comparing the gauge transformation with Eq. (B.4) of [@hep-th/9806169], we find $$\begin{aligned}
N^{(7)} = \iota_n {{\mathsf{A}}}_{7,n} - \tfrac{1}{4}\,\epsilon_{{{\gamma}}{{\delta}}}\,{{\mathsf{A}}}^{{{\gamma}}}_2\wedge\iota_n {{\mathsf{A}}}^{{{\delta}}}_2 \,.\end{aligned}$$ In addition, $N^{(8)}$ and ${\mathcal N}^{(8)}$ correspond to our ${{\bm{\mathsf{D}}}}_{8,2}$ and ${{\bm{\mathsf{E}}}}_{8,2}$ at least under ${{\mathsf{B}}}_2=0$ and ${{\mathsf{C}}}_2=0$. We have not identified the precise relation between their $N^{(9)}$ and our ${{\bm{\mathsf{A}}}}_{9,2,1}$.
Our 11D fields ${{\hat{A}}}_{\hat{3}}$, ${{\hat{A}}}_{\hat{6}}$, and ${{\hat{A}}}_{\hat{8},\hat{1}}$ are the same as those used in [@hep-th/9802199; @hep-th/9912030; @hep-th/0003240], where ${{\hat{A}}}_{\hat{8},n}$ is denoted as $\hat{N}^{(8)}$. The 9-form $\iota_n\hat{C}^{(10)}$ used in [@hep-th/9912030; @hep-th/0003240] can be defined as $$\begin{aligned}
\iota_n \hat{C}^{(10)} = \iota_n {{\hat{A}}}_{\hat{10},n,n} + \tfrac{1}{4!}\,{{\hat{A}}}_{\hat{3}}\wedge \iota_n {{\hat{A}}}_{\hat{3}}\wedge \iota_n {{\hat{A}}}_{\hat{3}}\wedge \iota_n {{\hat{A}}}_{\hat{3}}\,.\end{aligned}$$
Let us also identify the relation between our type IIB fields and those used in [@hep-th/0506013; @hep-th/0602280; @hep-th/0611036; @1004.1348]. For this purpose, it is useful to perform a redefinition, $$\begin{aligned}
\begin{split}
\begin{alignedat}{2}
\tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2 &\equiv {{\mathsf{A}}}^{{{\alpha}}}_2\,,\qquad\quad
\tilde{{{\mathsf{A}}}}_4 \equiv {{\mathsf{A}}}_4\,,\qquad &
\tilde{{{\mathsf{A}}}}^{{{\alpha}}}_6 &\equiv {{\mathsf{A}}}^{{{\alpha}}}_6 - \tfrac{1}{3}\,{{\mathsf{A}}}_4\wedge {{\mathsf{A}}}_2^{{{\alpha}}} \,,
\\
\tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}}_8 &\equiv {{\mathsf{A}}}^{{{\alpha}}{{\beta}}}_8 - \tfrac{1}{4}\,{{\mathsf{A}}}^{({{\alpha}}}_6\wedge {{\mathsf{A}}}^{{{\beta}})}_2 \,,\qquad&
\tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10} &\equiv {{\mathsf{A}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10} - \tfrac{1}{5}\,{{\mathsf{A}}}^{({{\alpha}}{{\beta}}}_8\wedge {{\mathsf{A}}}^{{{\gamma}})}_2\,,
\end{alignedat}
\end{split}\end{aligned}$$ which makes the field strengths to have the schematic form ${{\mathsf{F}}}\sim {{\mathrm{d}}}\tilde{{{\mathsf{A}}}} + \sum {{\mathsf{F}}}\wedge \tilde{{{\mathsf{A}}}}$, $$\begin{aligned}
\begin{split}
{{\mathsf{F}}}^{{{\alpha}}}_3 &= {{\mathrm{d}}}\tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2 \,,\qquad
{{\mathsf{F}}}_5 = {{\mathrm{d}}}\tilde{{{\mathsf{A}}}}_4 + \tfrac{1}{2}\, \epsilon_{{{\alpha}}{{\beta}}}\, {{\mathsf{F}}}^{{{\alpha}}}_3\wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}}_2\,,
\\
{{\mathsf{F}}}^{{{\alpha}}}_7 &= {{\mathrm{d}}}\tilde{{{\mathsf{A}}}}^{{{\alpha}}}_6 + \tfrac{1}{3}\,{{\mathsf{F}}}_5 \wedge \tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2 - \tfrac{2}{3}\,{{\mathsf{F}}}^{{{\alpha}}}_3\wedge \tilde{{{\mathsf{A}}}}_4\,,
\\
{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_9 &= {{\mathrm{d}}}\tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}}_8 + \tfrac{1}{4}\,{{\mathsf{F}}}^{({{\alpha}}}_7 \wedge \tilde{{{\mathsf{A}}}}^{{{\beta}})}_2 - \tfrac{3}{4}\,{{\mathsf{F}}}^{({{\alpha}}}_3 \wedge \tilde{{{\mathsf{A}}}}^{{{\beta}})}_6\,,
\\
{{\mathsf{F}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{11} &= {{\mathrm{d}}}\tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10} + \tfrac{1}{5}\,{{\mathsf{F}}}^{({{\alpha}}{{\beta}}}_9 \wedge \tilde{{{\mathsf{A}}}}^{{{\gamma}})}_2 - \tfrac{4}{5}\,{{\mathsf{F}}}^{({{\alpha}}}_3 \wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}{{\gamma}})}_8 = 0\,.
\end{split}\end{aligned}$$ The gauge transformation also can be expressed as $\delta \tilde{{{\mathsf{A}}}} \sim {{\mathrm{d}}}\tilde{\Lambda} + \sum {{\mathsf{F}}}\wedge \tilde{\Lambda}$, $$\begin{aligned}
\begin{split}
\delta \tilde{{{\mathsf{A}}}}_2^{{{\alpha}}}&={{\mathrm{d}}}\tilde{\Lambda}^{{{\alpha}}}_1\,, \qquad
\delta \tilde{{{\mathsf{A}}}}_4 ={{\mathrm{d}}}\tilde{\Lambda}_3 + \tfrac{1}{2}\,\epsilon_{{{\alpha}}{{\beta}}}\, {{\mathsf{F}}}^{{{\alpha}}}_3\wedge \tilde{\Lambda}^{{{\beta}}}_1 \,,
\\
\delta \tilde{{{\mathsf{A}}}}_6^{{{\alpha}}} &={{\mathrm{d}}}\tilde{\Lambda}_5^{{{\alpha}}} + \tfrac{1}{3}\, {{\mathsf{F}}}_5\wedge \tilde{\Lambda}^{{{\alpha}}}_1 - \tfrac{2}{3}\, {{\mathsf{F}}}^{{{\alpha}}}_3 \wedge \tilde{\Lambda}_3\,,
\\
\delta \tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}}_8 &= {{\mathrm{d}}}\tilde{\Lambda}^{{{\alpha}}{{\beta}}}_7 - \tfrac{3}{4}\,{{\mathsf{F}}}_3^{({{\alpha}}}\wedge\tilde{\Lambda}^{{{\beta}})}_5 + \tfrac{1}{4}\,{{\mathsf{F}}}^{({{\alpha}}}_7\wedge \tilde{\Lambda}^{{{\beta}})}_1\,,
\\
\delta \tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10} &= {{\mathrm{d}}}\tilde{\Lambda}^{{{\alpha}}{{\beta}}{{\gamma}}}_9 + \tfrac{1}{5}\, {{\mathsf{F}}}^{({{\alpha}}{{\beta}}}_9\wedge \tilde{\Lambda}^{{{\gamma}})}_1 - \tfrac{4}{5}\,{{\mathsf{F}}}^{({{\alpha}}}_3\wedge \tilde{\Lambda}_7^{{{\beta}}{{\gamma}})} \,.
\end{split}\end{aligned}$$ by considering a field-dependent redefinitions of gauge parameters: $$\begin{aligned}
\begin{split}
\tilde{\Lambda}^{{{\alpha}}}_1 &\equiv \Lambda^{{{\alpha}}}_1\,,\qquad
\tilde{\Lambda}_3 \equiv \Lambda_3 -\tfrac{1}{2!}\,\epsilon_{{{\gamma}}{{\delta}}}\,\tilde{{{\mathsf{A}}}}_2^{{{\gamma}}}\wedge \Lambda^{{{\delta}}}_1\,,
\\
\tilde{\Lambda}^{{{\alpha}}}_5 &\equiv \Lambda^{{{\alpha}}}_5 + \tfrac{2}{3}\, \tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2 \wedge \Lambda_3 - \tfrac{1}{3}\, \tilde{{{\mathsf{A}}}}_4\wedge \Lambda^{{{\alpha}}}_1 - \tfrac{1}{3!}\, \epsilon_{{{\gamma}}{{\delta}}}\, \tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\gamma}}}_2 \wedge \Lambda^{{{\delta}}}_1 \,,
\\
\tilde{\Lambda}^{{{\alpha}}{{\beta}}}_7 &\equiv \Lambda^{{{\alpha}}{{\beta}}}_7 + \tfrac{3}{4}\, \tilde{{{\mathsf{A}}}}^{({{\alpha}}}_2\wedge \Lambda^{{{\beta}})}_5 + \tfrac{1}{4}\, \tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}}_2\wedge \Lambda_3
\\
&\quad - \tfrac{1}{4}\, \tilde{{{\mathsf{A}}}}^{({{\alpha}}}_2\wedge \Lambda^{{{\beta}})}_1 - \tfrac{1}{4!}\, \epsilon_{{{\gamma}}{{\delta}}}\, \tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}}_2 \wedge \tilde{{{\mathsf{A}}}}^{{{\gamma}}}_2 \wedge \Lambda^{{{\delta}}}_1 \,,
\\
\tilde{\Lambda}^{{{\alpha}}{{\beta}}{{\gamma}}}_9 &\equiv \Lambda^{{{\alpha}}{{\beta}}{{\gamma}}}_9 + \tfrac{4}{5}\, \tilde{{{\mathsf{A}}}}^{({{\alpha}}}_2\wedge \Lambda^{{{\beta}}{{\gamma}})}_7 + \tfrac{3}{10}\, \tilde{{{\mathsf{A}}}}^{({{\alpha}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}}_2\wedge \Lambda^{{{\gamma}})}_5 + \tfrac{1}{15}\,\tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\gamma}}}_2\wedge \Lambda_3
\\
&\quad -\tfrac{1}{5}\,\tilde{{{\mathsf{A}}}}^{({{\alpha}}{{\beta}}}_8\wedge\Lambda^{{{\gamma}})}_1 + \tfrac{1}{5}\,\tilde{{{\mathsf{A}}}}^{({{\alpha}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}}_6\wedge\Lambda^{{{\gamma}})}_1 - \tfrac{1}{5!}\, \epsilon_{{{\delta}}{{\epsilon}}}\, \tilde{{{\mathsf{A}}}}^{{{\alpha}}}_2\wedge \tilde{{{\mathsf{A}}}}^{{{\beta}}}_2 \wedge \tilde{{{\mathsf{A}}}}^{{{\gamma}}}_2 \wedge \tilde{{{\mathsf{A}}}}^{{{\delta}}}_2 \wedge \Lambda^{{{\epsilon}}}_1 \,.
\end{split}\end{aligned}$$ Then, the tilde potentials and the field strengths are related to the fields used in [@hep-th/0506013; @hep-th/0602280; @hep-th/0611036; @1004.1348] (appearing on the right-hand sides) as follows: $$\begin{aligned}
\begin{split}
\begin{alignedat}{5}
\tilde{{{\mathsf{A}}}}_2 &= A_2\,,\quad&
\tilde{{{\mathsf{A}}}}_4 &= -4 A_4 \,,\quad&
\tilde{{{\mathsf{A}}}}^{{{\alpha}}}_6 &= -A^{{{\alpha}}}_6\,,\quad&
\tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}}_8 &= -4 A^{{{\alpha}}{{\beta}}}_8 \,,\quad&
\tilde{{{\mathsf{A}}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{10} &=12 A^{{{\alpha}}{{\beta}}{{\gamma}}}_{10} \,,
\\
{{\mathsf{F}}}^{{{\alpha}}}_3 &= F^{{{\alpha}}}_3\,,\quad&
{{\mathsf{F}}}_5 &= - 4 F_5\,,\quad&
{{\mathsf{F}}}^{{{\alpha}}}_7 &= -F^{{{\alpha}}}_7\,,\quad&
{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_9 &= - 4 F^{{{\alpha}}{{\beta}}}_9 \,,\quad&
{{\mathsf{F}}}^{{{\alpha}}{{\beta}}{{\gamma}}}_{11} &= 12 F^{{{\alpha}}{{\beta}}{{\gamma}}}_{11}\,,
\\
\tilde{\Lambda}^{{{\alpha}}}_1 &= \Lambda^{{{\alpha}}}_1\,,\quad&
\tilde{\Lambda}_3 &= - 4 \Lambda_3\,,\quad&
\tilde{\Lambda}^{{{\alpha}}}_5 &= -\Lambda^{{{\alpha}}}_5\,,\quad&
\tilde{\Lambda}^{{{\alpha}}{{\beta}}}_7 &= -4 \Lambda^{{{\alpha}}{{\beta}}}_7 \,,\quad&
\tilde{\Lambda}^{{{\alpha}}{{\beta}}{{\gamma}}}_9 &=12 \Lambda^{{{\alpha}}{{\beta}}{{\gamma}}}_9 \,.
\end{alignedat}
\end{split}\end{aligned}$$
[99]{} P. C. West, “$E_{11}$ and M theory,” Class. Quant. Grav. [**18**]{}, 4443 (2001) \[hep-th/0104081\].
P. C. West, “$E_{11}$, SL(32) and central charges,” Phys. Lett. B [**575**]{}, 333 (2003) \[hep-th/0307098\].
F. Riccioni and P. C. West, “The $E_{11}$ origin of all maximal supergravities,” JHEP [**0707**]{}, 063 (2007) \[arXiv:0705.0752 \[hep-th\]\].
J. J. Fernández-Melgarejo, Y. Sakatani and S. Uehara, “Exotic branes and mixed-symmetry potentials I: predictions from $E_{11}$ symmetry,” arXiv:1907.07177 \[hep-th\].
S. Elitzur, A. Giveon, D. Kutasov and E. Rabinovici, “Algebraic aspects of matrix theory on $T^d$,” Nucl. Phys. B [**509**]{}, 122 (1998) \[hep-th/9707217\].
M. Blau and M. O’Loughlin, “Aspects of U duality in matrix theory,” Nucl. Phys. B [**525**]{}, 182 (1998) \[hep-th/9712047\].
C. M. Hull, “U duality and BPS spectrum of super Yang-Mills theory and M theory,” JHEP [**9807**]{}, 018 (1998) \[hep-th/9712075\].
N. A. Obers, B. Pioline and E. Rabinovici, “M theory and U duality on $T^d$ with gauge backgrounds,” Nucl. Phys. B [**525**]{}, 163 (1998) \[hep-th/9712084\].
N. A. Obers and B. Pioline, “U duality and M theory,” Phys. Rept. [**318**]{}, 113 (1999) \[hep-th/9809039\].
E. Eyras and Y. Lozano, “Exotic branes and nonperturbative seven-branes,” Nucl. Phys. B [**573**]{}, 735 (2000) \[hep-th/9908094\].
E. Lozano-Tellechea and T. Ortin, “7-branes and higher Kaluza-Klein branes,” Nucl. Phys. B [**607**]{}, 213 (2001) \[hep-th/0012051\].
P. P. Cook and P. C. West, “Charge multiplets and masses for $E_{11}$,” JHEP [**0811**]{}, 091 (2008) \[arXiv:0805.4451 \[hep-th\]\].
J. de Boer and M. Shigemori, “Exotic branes and non-geometric backgrounds,” Phys. Rev. Lett. [**104**]{}, 251603 (2010) \[arXiv:1004.2521 \[hep-th\]\].
J. de Boer and M. Shigemori, “Exotic Branes in String Theory,” Phys. Rept. [**532**]{}, 65 (2013) \[arXiv:1209.6056 \[hep-th\]\].
E. Bergshoeff, E. Eyras and Y. Lozano, “The Massive Kaluza-Klein monopole,” Phys. Lett. B [**430**]{}, 77 (1998) \[hep-th/9802199\].
E. Eyras, B. Janssen and Y. Lozano, “Five-branes, K K monopoles and T duality,” Nucl. Phys. B [**531**]{}, 275 (1998) \[hep-th/9806169\].
E. Eyras and Y. Lozano, “The Kaluza-Klein monopole in a massive IIA background,” Nucl. Phys. B [**546**]{}, 197 (1999) \[hep-th/9812188\].
J. J. Fernández-Melgarejo, Y. Sakatani and S. Uehara, “Exotic branes and mixed-symmetry potentials II: duality rules and exceptional $p$-form gauge fields,” arXiv:1909.01335 \[hep-th\].
A. K. Callister and D. J. Smith, “Topological BPS charges in 10-dimensional and 11-dimensional supergravity,” Phys. Rev. D [**78**]{}, 065042 (2008) \[arXiv:0712.3235 \[hep-th\]\].
A. K. Callister and D. J. Smith, “Topological charges in SL(2,R) covariant massive 11-dimensional and Type IIB SUGRA,” Phys. Rev. D [**80**]{}, 125035 (2009) \[arXiv:0907.3614 \[hep-th\]\].
Y. Sakatani and S. Uehara, “Connecting M-theory and type IIB parameterizations in Exceptional Field Theory,” PTEP [**2017**]{}, no. 4, 043B05 (2017) \[arXiv:1701.07819 \[hep-th\]\].
P. Meessen and T. Ortin, “An Sl(2,Z) multiplet of nine-dimensional type II supergravity theories,” Nucl. Phys. B [**541**]{}, 195 (1999) \[hep-th/9806120\].
E. Bergshoeff and J. P. van der Schaar, “On M nine-branes,” Class. Quant. Grav. [**16**]{}, 23 (1999) \[hep-th/9806069\].
E. Eyras and Y. Lozano, “Brane actions and string dualities,” Lect. Notes Phys. [**525**]{}, 356 (1999) \[hep-th/9812225\].
T. Sato, “A Ten form gauge potential and an M9 brane Wess-Zumino action in massive 11-D theory,” Phys. Lett. B [**477**]{}, 457 (2000) \[hep-th/9912030\].
T. Sato, “On dimensional reductions of the M-nine-brane,” hep-th/0003240.
T. Kimura, S. Sasaki and M. Yata, “World-volume Effective Action of Exotic Five-brane in M-theory,” JHEP [**1602**]{}, 168 (2016) \[arXiv:1601.05589 \[hep-th\]\].
E. A. Bergshoeff, M. de Roo, S. F. Kerstan and F. Riccioni, “IIB supergravity revisited,” JHEP [**0508**]{}, 098 (2005) \[hep-th/0506013\].
E. A. Bergshoeff, M. de Roo, S. F. Kerstan, T. Ortin and F. Riccioni, “IIA ten-forms and the gauge algebras of maximal supergravity theories,” JHEP [**0607**]{}, 018 (2006) \[hep-th/0602280\].
E. A. Bergshoeff, M. de Roo, S. F. Kerstan, T. Ortin and F. Riccioni, “SL(2,R)-invariant IIB Brane Actions,” JHEP [**0702**]{}, 007 (2007) \[hep-th/0611036\].
E. A. Bergshoeff, J. Hartong, P. S. Howe, T. Ortin and F. Riccioni, “IIA/IIB Supergravity and Ten-forms,” JHEP [**1005**]{}, 061 (2010) \[arXiv:1004.1348 \[hep-th\]\].
P. C. West, “$E_{11}$, ten forms and supergravity,” JHEP [**0603**]{}, 072 (2006) \[hep-th/0511153\].
E. A. Bergshoeff and F. Riccioni, “D-Brane Wess-Zumino Terms and U-Duality,” JHEP [**1011**]{}, 139 (2010) \[arXiv:1009.4657 \[hep-th\]\].
I. Schnakenburg and P. C. West, “Kac-Moody symmetries of 2B supergravity,” Phys. Lett. B [**517**]{}, 421 (2001) \[hep-th/0107181\].
E. Bergshoeff, J. Hartong and D. Sorokin, “Q7-branes and their coupling to IIB supergravity,” JHEP [**0712**]{}, 079 (2007) \[arXiv:0708.2287 \[hep-th\]\].
E. A. Bergshoeff and F. Riccioni, “Branes and wrapping rules,” Phys. Lett. B [**704**]{}, 367 (2011) \[arXiv:1108.5067 \[hep-th\]\].
A. Kleinschmidt, “Counting supersymmetric branes,” JHEP [**1110**]{}, 144 (2011) \[arXiv:1109.2025 \[hep-th\]\].
P. C. West, “The IIA, IIB and eleven-dimensional theories and their common $E_{11}$ origin,” Nucl. Phys. B [**693**]{}, 76 (2004) \[hep-th/0402140\].
W. Siegel, “Two vierbein formalism for string inspired axionic gravity,” Phys. Rev. D [**47**]{}, 5453 (1993) \[hep-th/9302036\].
W. Siegel, “Superspace duality in low-energy superstrings,” Phys. Rev. D [**48**]{}, 2826 (1993) \[hep-th/9305073\].
W. Siegel, “Manifest duality in low-energy superstrings,” hep-th/9308133.
C. Hull and B. Zwiebach, “Double Field Theory,” JHEP [**0909**]{}, 099 (2009) \[arXiv:0904.4664 \[hep-th\]\].
O. Hohm, C. Hull and B. Zwiebach, “Generalized metric formulation of double field theory,” JHEP [**1008**]{}, 008 (2010) \[arXiv:1006.4823 \[hep-th\]\].
E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest and A. Van Proeyen, “New formulations of D = 10 supersymmetry and D8 - O8 domain walls,” Class. Quant. Grav. [**18**]{}, 3359 (2001) \[hep-th/0103233\].
S. F. Hassan, “Supersymmetry and the systematics of T duality rotations in type II superstring theories,” Nucl. Phys. Proc. Suppl. [**102**]{}, 77 (2001) \[hep-th/0103149\].
M. Fukuma, T. Oota and H. Tanaka, “Comments on T dualities of Ramond-Ramond potentials on tori,” Prog. Theor. Phys. [**103**]{}, 425 (2000) \[hep-th/9907132\].
O. Hohm, S. K. Kwak and B. Zwiebach, “Unification of Type II Strings and T-duality,” Phys. Rev. Lett. [**107**]{}, 171603 (2011) \[arXiv:1106.5452 \[hep-th\]\].
O. Hohm, S. K. Kwak and B. Zwiebach, “Double Field Theory of Type II Strings,” JHEP [**1109**]{}, 013 (2011) \[arXiv:1107.0008 \[hep-th\]\].
E. A. Bergshoeff and F. Riccioni, “String Solitons and T-duality,” JHEP [**1105**]{}, 131 (2011) \[arXiv:1102.0934 \[hep-th\]\].
E. Bergshoeff, A. Kleinschmidt, E. T. Musaev and F. Riccioni, “The different faces of branes in Double Field Theory,” JHEP [**1909**]{}, 110 (2019) \[JHEP [**2019**]{}, 110 (2020)\] \[arXiv:1903.05601 \[hep-th\]\].
T. Damour, M. Henneaux and H. Nicolai, “E(10) and a ’small tension expansion’ of M theory,” Phys. Rev. Lett. [**89**]{}, 221601 (2002) \[hep-th/0207267\].
P. C. West, “Very extended E(8) and A(8) at low levels, gravity and supergravity,” Class. Quant. Grav. [**20**]{}, 2393 (2003) \[hep-th/0212291\].
E. A. Bergshoeff, A. Marrani and F. Riccioni, “Brane orbits,” Nucl. Phys. B [**861**]{}, 104 (2012) \[arXiv:1201.5819 \[hep-th\]\].
E. A. Bergshoeff and F. Riccioni, “Heterotic wrapping rules,” JHEP [**1301**]{}, 005 (2013) \[arXiv:1210.1422 \[hep-th\]\].
L. J. Romans, “Massive N=2a Supergravity in Ten-Dimensions,” Phys. Lett. [**169B**]{}, 374 (1986).
E. Bergshoeff, Y. Lozano and T. Ortin, “Massive branes,” Nucl. Phys. B [**518**]{}, 363 (1998) \[hep-th/9712115\].
J. J. Fernández-Melgarejo, T. Kimura and Y. Sakatani, “Weaving the Exotic Web,” JHEP [**1809**]{}, 072 (2018) \[arXiv:1805.12117 \[hep-th\]\].
E. A. Bergshoeff, T. Ortin and F. Riccioni, “Defect Branes,” Nucl. Phys. B [**856**]{}, 210 (2012) \[arXiv:1109.4484 \[hep-th\]\].
A. Chatzistavrakidis, F. F. Gautason, G. Moutsopoulos and M. Zagermann, “Effective actions of nongeometric five-branes,” Phys. Rev. D [**89**]{}, no. 6, 066004 (2014) \[arXiv:1309.2653 \[hep-th\]\].
Y. Sakatani, “Exotic branes and non-geometric fluxes,” JHEP [**1503**]{}, 135 (2015) \[arXiv:1412.8769 \[hep-th\]\].
F. Hassler and D. Lust, “Non-commutative/non-associative IIA (IIB) Q- and R-branes and their intersections,” JHEP [**1307**]{}, 048 (2013) \[arXiv:1303.1413 \[hep-th\]\].
D. Andriot and A. Betz, “NS-branes, source corrected Bianchi identities, and more on backgrounds with non-geometric fluxes,” JHEP [**1407**]{}, 059 (2014) \[arXiv:1402.5972 \[hep-th\]\].
T. Okada and Y. Sakatani, “Defect branes as Alice strings,” JHEP [**1503**]{}, 131 (2015) \[arXiv:1411.1043 \[hep-th\]\].
K. Lee, S. J. Rey and Y. Sakatani, “Effective action for non-geometric fluxes duality covariant actions,” JHEP [**1707**]{}, 075 (2017) \[arXiv:1612.08738 \[hep-th\]\].
A. D. Shapere and F. Wilczek, “Selfdual Models with Theta Terms,” Nucl. Phys. B [**320**]{}, 669 (1989).
A. Giveon, E. Rabinovici and G. Veneziano, “Duality in String Background Space,” Nucl. Phys. B [**322**]{}, 167 (1989).
M. J. Duff, “Duality Rotations in String Theory,” Nucl. Phys. B [**335**]{}, 610 (1990).
A. A. Tseytlin, “Duality Symmetric Formulation of String World Sheet Dynamics,” Phys. Lett. B [**242**]{}, 163 (1990).
A. Giveon, M. Porrati and E. Rabinovici, “Target space duality in string theory,” Phys. Rept. [**244**]{}, 77 (1994) \[hep-th/9401139\].
D. Andriot, M. Larfors, D. Lust and P. Patalong, “A ten-dimensional action for non-geometric fluxes,” JHEP [**1109**]{}, 134 (2011) \[arXiv:1106.4015 \[hep-th\]\].
D. Andriot, O. Hohm, M. Larfors, D. Lust and P. Patalong, “A geometric action for non-geometric fluxes,” Phys. Rev. Lett. [**108**]{}, 261602 (2012) \[arXiv:1202.3060 \[hep-th\]\].
D. Andriot, O. Hohm, M. Larfors, D. Lust and P. Patalong, “Non-Geometric Fluxes in Supergravity and Double Field Theory,” Fortsch. Phys. [**60**]{}, 1150 (2012) \[arXiv:1204.1979 \[hep-th\]\].
D. Andriot and A. Betz, “$\beta$-supergravity: a ten-dimensional theory with non-geometric fluxes, and its geometric framework,” JHEP [**1312**]{}, 083 (2013) \[arXiv:1306.4381 \[hep-th\]\].
D. Andriot and A. Betz, “Supersymmetry with non-geometric fluxes, or a $\beta$-twist in Generalized Geometry and Dirac operator,” JHEP [**1504**]{}, 006 (2015) \[arXiv:1411.6640 \[hep-th\]\].
G. Aldazabal, E. Andres, P. G. Camara and M. Grana, “U-dual fluxes and Generalized Geometry,” JHEP [**1011**]{}, 083 (2010) \[arXiv:1007.5509 \[hep-th\]\].
C. D. A. Blair and E. Malek, “Geometry and fluxes of SL(5) exceptional field theory,” JHEP [**1503**]{}, 144 (2015) \[arXiv:1412.0635 \[hep-th\]\].
J. Shelton, W. Taylor and B. Wecht, “Nongeometric flux compactifications,” JHEP [**0510**]{}, 085 (2005) \[hep-th/0508133\].
G. Aldazabal, P. G. Camara, A. Font and L. E. Ibanez, “More dual fluxes and moduli fixing,” JHEP [**0605**]{}, 070 (2006) \[hep-th/0602089\].
G. Aldazabal, P. G. Camara and J. A. Rosabal, “Flux algebra, Bianchi identities and Freed-Witten anomalies in F-theory compactifications,” Nucl. Phys. B [**814**]{}, 21 (2009) \[arXiv:0811.2900 \[hep-th\]\].
M. Ihl, D. Robbins and T. Wrase, “Toroidal orientifolds in IIA with general NS-NS fluxes,” JHEP [**0708**]{}, 043 (2007) \[arXiv:0705.3410 \[hep-th\]\].
D. Robbins and T. Wrase, “D-terms from generalized NS-NS fluxes in type II,” JHEP [**0712**]{}, 058 (2007) \[arXiv:0709.2186 \[hep-th\]\].
M. P. Hertzberg, S. Kachru, W. Taylor and M. Tegmark, “Inflationary Constraints on Type IIA String Theory,” JHEP [**0712**]{}, 095 (2007) \[arXiv:0711.2512 \[hep-th\]\].
U. H. Danielsson, S. S. Haque, G. Shiu and T. Van Riet, “Towards Classical de Sitter Solutions in String Theory,” JHEP [**0909**]{}, 114 (2009) \[arXiv:0907.2041 \[hep-th\]\].
B. de Carlos, A. Guarino and J. M. Moreno, “Flux moduli stabilisation, Supergravity algebras and no-go theorems,” JHEP [**1001**]{}, 012 (2010) \[arXiv:0907.5580 \[hep-th\]\].
B. de Carlos, A. Guarino and J. M. Moreno, “Complete classification of Minkowski vacua in generalised flux models,” JHEP [**1002**]{}, 076 (2010) \[arXiv:0911.2876 \[hep-th\]\].
U. Danielsson and G. Dibitetto, “On the distribution of stable de Sitter vacua,” JHEP [**1303**]{}, 018 (2013) \[arXiv:1212.4984 \[hep-th\]\].
J. Blåbäck, U. Danielsson and G. Dibitetto, “Fully stable dS vacua from generalised fluxes,” JHEP [**1308**]{}, 054 (2013) \[arXiv:1301.7073 \[hep-th\]\].
C. Damian, L. R. Diaz-Barron, O. Loaiza-Brito and M. Sabido, “Slow-Roll Inflation in Non-geometric Flux Compactification,” JHEP [**1306**]{}, 109 (2013) \[arXiv:1302.0529 \[hep-th\]\].
C. Damian and O. Loaiza-Brito, “More stable de Sitter vacua from S-dual nongeometric fluxes,” Phys. Rev. D [**88**]{}, no. 4, 046008 (2013) \[arXiv:1304.0792 \[hep-th\]\].
F. Hassler, D. Lust and S. Massai, “On Inflation and de Sitter in Non‐Geometric String Backgrounds,” Fortsch. Phys. [**65**]{}, no. 10-11, 1700062 (2017) \[arXiv:1405.2325 \[hep-th\]\].
P. Shukla, “A symplectic rearrangement of the four dimensional non-geometric scalar potential,” JHEP [**1511**]{}, 162 (2015) \[arXiv:1508.01197 \[hep-th\]\].
R. Blumenhagen, C. Damian, A. Font, D. Herschmann and R. Sun, “The Flux-Scaling Scenario: De Sitter Uplift and Axion Inflation,” Fortsch. Phys. [**64**]{}, no. 6-7, 536 (2016) \[arXiv:1510.01522 \[hep-th\]\].
P. Shukla, “Reading off the nongeometric scalar potentials via the topological data of the compactifying Calabi-Yau manifolds,” Phys. Rev. D [**94**]{}, no. 8, 086003 (2016) \[arXiv:1603.01290 \[hep-th\]\].
P. Shukla, “Revisiting the two formulations of Bianchi identities and their implications on moduli stabilization,” JHEP [**1608**]{}, 146 (2016) \[arXiv:1603.08545 \[hep-th\]\].
X. Gao, P. Shukla and R. Sun, “Symplectic formulation of the type IIA nongeometric scalar potential,” Phys. Rev. D [**98**]{}, no. 4, 046009 (2018) \[arXiv:1712.07310 \[hep-th\]\].
X. Gao, P. Shukla and R. Sun, “On Missing Bianchi Identities in Cohomology Formulation,” Eur. Phys. J. C [**79**]{}, no. 9, 781 (2019) \[arXiv:1805.05748 \[hep-th\]\].
P. Shukla, “A dictionary for the type II non-geometric flux compactifications,” arXiv:1909.07391 \[hep-th\].
P. Shukla, “$T$-dualizing the de-Sitter no-go scenarios,” arXiv:1909.08630 \[hep-th\].
P. Shukla, “Rigid non-geometric orientifolds and the swampland,” arXiv:1909.10993 \[hep-th\].
E. A. Bergshoeff, V. A. Penas, F. Riccioni and S. Risoli, “Non-geometric fluxes and mixed-symmetry potentials,” JHEP [**1511**]{}, 020 (2015) \[arXiv:1508.00780 \[hep-th\]\].
E. A. Bergshoeff, O. Hohm, V. A. Penas and F. Riccioni, “Dual Double Field Theory,” JHEP [**1606**]{}, 026 (2016) \[arXiv:1603.07380 \[hep-th\]\].
E. A. Bergshoeff, O. Hohm and F. Riccioni, “Exotic Dual of Type II Double Field Theory,” Phys. Lett. B [**767**]{}, 374 (2017) \[arXiv:1612.02691 \[hep-th\]\].
D. S. Berman, H. Godazgar, M. J. Perry and P. West, “Duality Invariant Actions and Generalised Geometry,” JHEP [**1202**]{}, 108 (2012) \[arXiv:1111.0459 \[hep-th\]\].
O. Hohm and H. Samtleben, “Exceptional Form of D=11 Supergravity,” Phys. Rev. Lett. [**111**]{}, 231601 (2013) \[arXiv:1308.1673 \[hep-th\]\].
O. Hohm and H. Samtleben, “Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB,” Phys. Rev. D [**89**]{}, no. 6, 066016 (2014) \[arXiv:1312.0614 \[hep-th\]\].
O. Hohm and H. Samtleben, “Exceptional field theory. II. E$_{7(7)}$,” Phys. Rev. D [**89**]{}, 066017 (2014) \[arXiv:1312.4542 \[hep-th\]\].
O. Hohm and H. Samtleben, “Exceptional field theory. III. E$_{8(8)}$,” Phys. Rev. D [**90**]{}, 066002 (2014) \[arXiv:1406.3348 \[hep-th\]\].
F. Riccioni and P. C. West, “$E_{11}$-extended spacetime and gauged supergravities,” JHEP [**0802**]{}, 039 (2008) \[arXiv:0712.1795 \[hep-th\]\].
C. Hillmann, “Generalized $E_{7(7)}$ coset dynamics and D=11 supergravity,” JHEP [**0903**]{}, 135 (2009) \[arXiv:0901.1581 \[hep-th\]\].
D. S. Berman and M. J. Perry, “Generalized Geometry and M theory,” JHEP [**1106**]{}, 074 (2011) \[arXiv:1008.1763 \[hep-th\]\].
D. S. Berman, H. Godazgar, M. Godazgar and M. J. Perry, “The Local symmetries of M-theory and their formulation in generalised geometry,” JHEP [**1201**]{}, 012 (2012) \[arXiv:1110.3930 \[hep-th\]\].
D. S. Berman, M. Cederwall, A. Kleinschmidt and D. C. Thompson, “The gauge structure of generalised diffeomorphisms,” JHEP [**1301**]{}, 064 (2013) \[arXiv:1208.5884 \[hep-th\]\].
[^1]: In [@0712.3235; @0907.3614], $T$-duality rules for some additional potentials, which are not studied here, have been studied.
[^2]: We do not consider overlined potentials, such as $\overline{{{\mathscr{A}}}}_{8}$, which do not couple to supersymmetric branes.
[^3]: We here ignore the component ${{\mathscr{A}}}_{9, 3}$, which couples to the $5^3_4$-brane.
[^4]: As noted in [@hep-th/9908094; @hep-th/0506013], the triplet ${{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_1$ has only two independent components because it satisfies ${{\mathsf{m}}}_{{{\alpha}}{{\beta}}}\,{{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_1 = 0$. Then, the duality shows that the triplet ${{\mathsf{F}}}^{{{\alpha}}{{\beta}}}_9$ also has only two independent components.
[^5]: Additional 10-form potential $\overline{{{\mathsf{A}}}}^{{{\alpha}}}_{10}$ was also introduced there, but here we do not consider this potential because this does not couple to supersymmetric branes.
[^6]: This property can be spoiled by a redefinition; e.g. ${{\hat{A}}}_{i_1\cdots i_9, k_1k_2k_3}\to {{\hat{A}}}_{i_1\cdots i_9, k_1k_2k_3}+ {{\hat{A}}}_{[i_1i_2i_3}\, {{\hat{A}}}_{i_4\cdots i_9]\bar{k}_1\bar{k}_2, \bar{k}_3}$.
[^7]: For example, the restriction rule for ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8({{\alpha}}_1,{{\alpha}}_2,{{\alpha}}_3)}$ is ${{\alpha}}_1={{\alpha}}_2={{\alpha}}_3$ and it gives two linear maps; ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8({{y}},{{y}},{{y}})} \overset{\tiny\textcircled{z}}{=} -\bm{{\mathcal A}}^{{\bm{222}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8{{\mathsf{y}}}}$ and ${\mathcal A}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8({{z}},{{z}},{{z}})} \overset{\tiny\textcircled{m}}{=} \bm{{\mathcal A}}^{{\bm{111}}}_{\mu;{\mathsf{a}}_1\cdots {\mathsf{a}}_8{{\mathsf{y}}}}$.
[^8]: In the literature (e.g. [@hep-th/9806169; @0712.3235]), the last index of the dual graviton is supposed to be a particular isometry direction, and it is not written down explicitly. Accordingly, the dual graviton is treated as a 7-form.
[^9]: In terms of DFT, by supposing that $D_{M_1\cdots M_4}$ is a generalized tensor with weight 1, we can show $\delta_V H_{MNP}= \hat{{\pounds}}_V H_{MNP} + 2\,\partial^R\partial_{[M}V^S\,D_{NP]RS}$ under generalized diffeomorphisms. The anomalous term vanishes under the assumptions that lower indices of $H^{mn}{}_p$, $H^m{}_{np}$, and $H_{mnp}$ are associated with Killing directions.
[^10]: The correspondent of has been given in Eq. (3.3) of [@hep-th/9908094], but there seems to be a small discrepancy regarding the terms including ${{\mathsf{B}}}_2\,({{\mathsf{C}}}_2)^2$.
[^11]: In our convention, the gauge transformations of $\iota_{{{z}}} {{\hat{A}}}_{\hat{8},{{z}}}$ and $\iota_{{{z}}} {{\hat{A}}}_{\hat{10},{{z}},{{z}}}$ does not include the mass deformation because the R–R potentials are included there such that the mass dependence is canceled out \[see \]. The R–R 6-form potential is also contained in ${{\hat{A}}}_{\hat{6}}$ such that the mass dependence is canceled out, but the gauge transformation of the potential ${{\mathscr{B}}}_6$ gives the mass dependence of $\delta {{\hat{A}}}_{\hat{6}}$.
[^12]: The full definition of the field strength ${{\hat{F}}}_{\hat{9},{{z}}}$ without the projection has been proposed in [@0907.3614]. It can be found by regarding the dilaton equations of motion as the Bianchi identity, and by uplifting this to 11D.
[^13]: This is originally introduced for example in [@Shapere:1988zv; @Giveon:1988tt; @Duff:1989tf; @Tseytlin:1990nb; @hep-th/9401139] and utilized more recently in the $\beta$-supergravity [@1106.4015; @1202.3060; @1204.1979; @1306.4381; @1411.6640].
[^14]: See [@0705.3410; @0709.2186; @0711.2512; @0907.2041; @0907.5580; @0911.2876; @1212.4984; @1301.7073; @1302.0529; @1304.0792; @1405.2325; @1508.01197; @1510.01522; @1603.01290; @1603.08545; @1712.07310; @1805.05748; @1909.07391; @1909.08630; @1909.10993] for an incomplete list of references utilizing non-geometric fluxes.
[^15]: Our dual fields have opposite sign compared to those introduced in [@1701.07819].
[^16]: The 11D uplifts also have the same form as the standard potentials.
[^17]: To be more precise, it is a spinor density. The ${\text{O}}(10,10)$ spinor ${\lvert {{{A}}} \rangle}$ has weight $1/2$ and the weight can be removed by considering ${\operatorname{e}^{d}}{\lvert {{{A}}} \rangle}$. On the other hand, ${\lvert {\alpha} \rangle}$ has weight $-1/2$ and ${\operatorname{e}^{-\tilde{d}}}{\lvert {\alpha} \rangle}$ is weightless.
|
---
abstract: 'We present optical spectra of the nuclei of seven luminous ($P_{\mbox{\scriptsize 178MHz}} \gsim 10^{25}$ W Hz$^{-1}$ Sr$^{-1}$) nearby ($z<0.08$) radiogalaxies, which mostly correspond to the FR II class. In two cases, Hydra A and 3C 285, the Balmer and break indices constrain the spectral types and luminosity classes of the stars involved, revealing that the blue spectra are dominated by blue supergiant and/or giant stars. The ages derived for the last burst of star formation in Hydra A are between 7 and 40 Myr, and in 3C 285 about 10 Myr. The rest of the narrow-line radiogalaxies (four) have break and metallic indices consistent with those of elliptical galaxies. The only broad-line radiogalaxy in our sample, 3C 382, has a strong featureless blue continuum and broad emission lines that dilute the underlying blue stellar spectra. We are able to detect the Ca II triplet in absorption in the seven objects, with good quality data for only four of them. The strengths of the absorptions are similar to those found in normal elliptical galaxies, but these values are both consistent with single stellar populations of ages as derived from the Balmer absorption and break strengths, and, also, with mixed young$+$old populations.'
author:
- |
Itziar Aretxaga$^1$, Elena Terlevich$^{1}$[^1], Roberto J. Terlevich$^{2}$[^2], Garret Cotter$^3$,\
\
[Ángeles I. Díaz$^{4}$]{}\
$^1$ Instituto Nacional de Astrofísica, Óptica y Electrónica, Apdo. Postal 25 y 216, 72000 Puebla, Pue., Mexico\
$^2$ Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, U.K.\
$^3$ Cavendish Laboratory, Univ. of Cambridge, Madingley Road, Cambridge CB3 0HE, U.K.\
$^4$ Dept. Física Teórica C-XI, Univ. Autómoma de Madrid, Cantoblanco, Madrid, Spain.\
title: Stellar populations in the nuclear regions of nearby radiogalaxies
---
epsf
galaxies: active – galaxies: starbursts – galaxies: stellar content
Introduction
============
In recent years new evidence that [*star formation plays an important role in Active Galactic Nuclei*]{} (AGN) has been gathered:
$\bullet$
: The presence of strong Ca II $\lambda\lambda$8494,8542,8662Å triplet (CaT) absorptions in a large sample of Seyfert 2 nuclei has provided direct evidence for a population of red supergiant stars that dominates the near-IR light (Terlevich, Díaz & Terlevich 1990). The values found in Seyfert 1 nuclei are also consistent with this idea if the dilution produced by a nuclear non-stellar source is taken into account (Terlevich, Díaz & Terlevich 1990, Jiménez-Benito et al. 2000). The high mass-to-light ratios $L(1.6 \mu \mbox{m})/M$ inferred in Seyfert 2 nuclei also indicate that red supergiants dominate the nuclear light (Oliva et al. 1995), but a similar conclusion does not hold for Seyfert 1 nuclei.
$\bullet$
: The absence of broad emission lines in the direct optical spectra of Seyfert 2 nuclei which show broad lines in polarized light can be understood only if there is an additional central source of continuum, most probably blue stars (Cid Fernandes & Terlevich 1995, Heckman et al. 1995). This conclusion is further supported by the detection of polarization levels which are lower in the continuum than in the broad lines (Miller & Goodrich 1990, Tran, Miller & Kay 1992).
$\bullet$
: Hubble Space Telescope imaging of the Seyfert Mrk 447 reveals that the central UV light arises in a [*resolved*]{} region of a few hundred pc, in which prominent CaT absorption and broad He II emission lines reveal the red supergiant and Wolf Rayet stars of a powerful starburst. The stars dominate the UV to near-IR light directly received from the nucleus (Heckman et al. 1997). At least 50 per cent of the light emitted by the nucleus is stellar, as a conservative estimate. Mrk 447 is not a rare case: a large sample of nearby bright Seyfert 2s and LINERs show similar resolved starburst nuclei of 80 to a few hundred pc in size (Colina et al. 1997, González-Delgado et al. 1998, Maoz et al. 1995, 1998), with some of the Seyfert 2 containing dominant Wolf-Rayet populations (Kunth & Contini 1999, Cid Fernandes et al. 1999).
A starburst–AGN connection has been proposed in at least three scenarios: starbursts giving birth to massive black holes (e.g. Scoville & Norman 1988); black holes being fed by surrounding stellar clusters (e.g. Perry & Dyson 1985, Peterson 1992); and also pure starbursts without black holes (e.g. Terlevich & Melnick 1985, Terlevich et al. 1992). The evidence for starbursts in Seyfert nuclei strongly supports some kind of connection. However, it is still to be demonstrated that starbursts play a key role in [*all*]{} kinds of AGN.
One of the most stringent tests [*to assess if all AGN have associated enhanced nuclear star formation*]{} is the case of lobe-dominated radio-sources, whose host galaxies have relatively red colours when compared to other AGN varieties. In this paper we address the stellar content associated with the active nuclei of a sample of FR II radiogalaxies, the most luminous class of radiogalaxies (Fanaroff & Riley 1974) which possess the most powerful central engines and radio-jets (Rawlings & Saunders 1991). The presence of extended collimated radio-jets, which fuel the extended radio structure over $\gsim 10^8$ yr, strongly suggests the existence of a supermassive accreting black hole in the nuclei of these radiogalaxies. This test addresses the question of whether AGN that involve conspicuous black holes and accretion processes also contain enhanced star formation.
In section 2 we introduce the sample and detail the data acquisition and reduction processes. In section 3 we provide continuum and line measurements of the most prominent features of the optical spectra of the radiogalaxies. In section 4 we discuss the main stellar populations responsible for the absorption and continuum spectra. In section 5 we offer notes on individual objects. A sumary of the main conclussions from this work is presented in section 6.
Data acquisition and reduction
==============================
Data acquisition and reduction
==============================
Our sample of radiogalaxies was extracted from the 3CRR catalogue (Laing, Riley and Longair 1983) with the only selection criteria being edge-brightened morphology, which defines the FR II class of radiogalaxies (Fanaroff & Riley 1974), and redshift $z < 0.08$. This last condition was imposed in order to be able to observe the redshifted CaT at wavelengths shorter than , where the atmospheric bands are prominent. Six out of a complete sample of ten FR II radiogalaxies that fulfill these requirements were randomly chosen. In addition to this sub-sample of FR IIs, we observed the unusually luminous FR I radiogalaxy Hydra A (3C 218). This has a radio luminosity of , which is an order of magnitude above the typical FR I/FR II dividing luminosity.
Spectroscopic observations of a total of seven radiogalaxies, one normal elliptical galaxy to serve as reference and five K III stars to serve as velocity calibrators were performed using the double-arm spectrograph ISIS mounted in the Cassegrain Focus of the 4.2m William Herschel Telescope[^3] in La Palma during two observing runs, in 1997 November 7–8 and 1998 February 19–20. The first run was photometric but the second was not, being partially cloudy on the 20th. The seeing, as measured from the spatial dimension of spectrophotometric stars, was between $0.7$ and $0.8$ arcsec throughout the nights.
A slit width of 1.2 arcsec centered on the nucleus of galaxies and stars was used. We oriented the slit along the radio-axis for all radiogalaxies, except for Hydra A, for which the orientation was perpendicular to the radio-axis.
An R300B grating centered at with a 2148x4200 pixel EEV CCD and an R316R grating centered at with a 1024x1024 pixel TEK CCD were used in the 1998 run. The projected area on these chips is 0.2 arcsec/pixel and 0.36 arcsec/pixel respectively. This configuration provides the spectral resolution necessary to resolve the Mg b and CaT features and, at the same time, offers a wide spectral span: — at 5.1Å resolution in the blue and — at 3.5Å resolution in the red. In the 1997 run, in which we assessed the viability of the project, we used the R600B and R600R gratings instead. This setup covers the — and — range in the blue and red arm, at 2.6 and 1.7Å resolution respectively. Just one radiogalaxy (DA 240) was observed with this alternative setup. The dichroics 5700 and 6100 were used in 1997 November and 1998 February, and in both runs we used a filter to avoid second order contamination in the spectra.
We obtained flux standards (HZ 44 and G191-B2B) for the four nights and gratings, except in 1998 Feb 20, when we were unable to acquire the red spectrum of the corresponding standard due to a technical failure. One calibration lamp CuAr$+$CuNe exposure per spectral region and telescope position was also obtained for all objects.
The total integration times for the radiogalaxies (from 1 to 3 hr) were split into time intervals of about 1200 or 1800 s in order to diminish the effect of cosmic rays on individual frames and allow to take lamp flat-fields with the red arm of the spectrograph between science exposures. The TEK CCD has a variable fringing pattern at the wavelengths of interest, such that the variations are correlated with the position at which the telescope is pointing. Since flat-fielding is crucial for the reddest wavelengths, where the sky lines are most prominent, after every exposure of 20 to 30 min we acquired a flat-field in the same position of the telescope as the one for which the galaxies were being observed. We followed this procedure with all galaxies except with DA 240. The same procedure was also used in the case of the elliptical galaxy, splitting its total integration time in two.
Table \[obs\] summarizes the journal of observations, where column 1 gives the name of the object; column 2 the radio-power at 178 MHz; column 3 the redshift; column 4 the integrated $V$ magnitude of the galaxy; column 5 identifies whether the object is a radiogalaxy (RG), a normal elliptical (E) or a star (S); column 6 gives the date of the beginning of the night in which the observations were carried out; column 7 the position angle (PA) of the slit; column 8 the total exposure time; column 9 the grating used; and column 10 the corresponding linear size to 1 arcsec at the redshift of the galaxies (for 50). The data for the radiogalaxies were extracted from the 3C Atlas (Leahy, Bridle & Strom, [http://www.jb.man.ac.uk/atlas/]{}) and for the host galaxy of Hydra A from the 3CR Catalogue (de Vaucouleurs et al. 1991).
The data were reduced using the IRAF software package. The frames were first bias subtracted and then flat-field corrected. In the case of the red arm spectra, the different flats obtained for a single object were combined when no significant differences were detected between them. However, in several cases the fringing pattern shifted positions that accounted for differences of up to 20 per cent. In these cases we corrected each science frame with the flat-field acquired immediately before and/or afterwards. Close inspection of the faintest levels of the flat-fielded frames showed that the fringing had been successfully eliminated. Wavelength and flux calibration were then performed, and the sky was subtracted by using the outermost parts of the slit.
The atmospheric bands, mainly water absorption at —, affect the redshifted CaT region of several radiogalaxies. The bands have been extracted using a template constructed from the stellar spectra obtained each night. The template was built averaging the normalized flux of spectrophotometric and velocity standard stars, once the stellar absorption lines had been removed. The atmospheric bands were eliminated from the spectra of the galaxies dividing by the flux-scaled template. This reduces the S/N of the region under consideration, especially since the bands are variable in time and one of our observing nights was partially cloudy. However, the technique allows the detection of the stellar atmospheric features. The CaT of the elliptical galaxy is not affected by atmospheric absorption.
Figure 1 shows the line spectrum of the sky and, as an example, the atmospheric absorption template of 1998 Feb 19. Water-band correction proved to be critical for the detection of the CaT lines when the atmospheric conditions were most adverse.
Figure 2 shows extractions of the nuclear 2 arcsec of the spectra of the galaxies. This corresponds to 844 to 2020 pc for the radiogalaxies, and 98 pc for the normal elliptical galaxy.
Line and continuum measurements
===============================
Line and continuum measurements
===============================
CaT index
---------
The CaT was detected in all of the objects, although in three cases (3C 285, 3C 382 and 4C 73.08) it was totally or partially affected by residuals left by the atmospheric band corrections and the measurement of its strength was thus precluded. For the remaining cases, the strength was measured in the rest-frame of the galaxies against a pseudo-continuum, following the definition of the CaT index of Díaz, Terlevich & Terlevich (1989). In Hydra A, 3C 285 and 3C 382, the red continuum band is seriously affected by residuals left from the atmospheric absorption removal. We defined two alternative continuum bands, and , that substitute the red-most continuum window of the CaT index. We checked this new definition against the original one in the elliptical galaxy, which doesn’t have residuals in its continuum bands, and the agreement between the two systems was good within 5 per cent.
Velocity dispersions were measured by cross-correlating the galaxy spectra with the stellar spectra obtained with the same setup. The errors in the velocity dispersions calculated in this way were less than 10 per cent.
A high velocity dispersion tends to decrease the measured values of indices based on EW measurements. The CaT index has to be corrected from broadening of the absorption lines by the corresponding velocity dispersion. In order to calculate the correction we convolved stellar profiles with gaussian functions of increasing width, and measured the CaT index in them. A good description of the correction found for our data is given by the functional form . The corrections were applied to the values measured in the galaxies, and converted into unbroadened indices.
The values of velocity dispersions ($\sigma$), uncorrected EW (CaT$_{\mbox{\scriptsize u}}$) and corrected EW (CaT), are listed in Table \[EW\].
and Balmer Break indices
-------------------------
Stellar populations can be dated through the measurement of the or Balmer breaks. In intermediate to old populations the discontinuity at results from a combination of the accumulation of the Balmer lines towards the limit of the Balmer absorption continuum at (the Balmer break) and the increase in stellar opacity caused by metal lines shortwards of .
Table 3 lists the values of the break index, $\Delta$4000Å, measured in the spectra of the 6 narrow-line radiogalaxies and the elliptical galaxy in our sample. This excludes 3C 382, which has a spectrum dominated by a strong blue continuum and broad-emission lines, and shows very weak stellar atmospheric features and no break. We adopted the definition given by Hamilton (1985), which quantifies the ratio of the average flux-level of two broad bands, one covering the break (3750-3950) and one bluewards of the break (4050-4250). Both bands contain strong metallic and Balmer absorption lines in the case of normal galaxies. In active galaxies, the measurement can be contaminated by emission of \[Ne III\], which in our case is weak. The contamination by high-order Balmer lines in emission is negligible. The net effect of emission contamination is to decrease the Balmer break index. In the radio-galaxies, we have estimated this effect by interpolating the continuum levels below the \[Ne III\] emission, and we estimate that the ratio can be affected by 6 per cent at worst, in the case of 3C 192, and by less than 3 per cent for the rest of the objects. Table 3 lists emission-devoid indices.
Hydra A and 3C 285 have spectra which are much bluer than those of normal elliptical galaxies. In order to quantify better the strength of the break and the ages of the populations derived, we have performed a bulge subtraction using as template the spectrum of NGC 4374, scaled to eliminate the G-band absorption of the radiogalaxies. Since the velocity dispersion of the stars in NGC 4374 and in the radiogalaxies are comparable inside the spectral resolution of our data, no further corrections were needed. The G-band absorption is prominent in stars of spectral types later than F5 and it is especially strong in types K. NGC 4374 is a normal elliptical galaxy, with a spectral shape which compares well with those of other normal ellipticals in the spectrophotometric atlas of galaxies of Kennicutt (1992). Thus, by removing a scaled template of NGC 4374, we are isolating the most massive stars ($M \gsim 1$) in the composite stellar population of the radiogalaxies. Figure 3 shows the bulge subtractions obtained on these two radiogalaxies.
We measured on the bulge-subtracted spectra $\Delta$4000Åand also the Balmer break index as defined by the classical $D\lambda_1$ method of stellar classification designed by Barbier and Chalonge (Barbier 1955, Chalonge 1956, see Strömgren 1963). The latter quantifies the Balmer discontinuity in terms of the logarithmic difference of the continuum levels ($D$) and the effective position of the break ($\lambda_1$). The method places a pseudo-continuum on top of the higher order terms of the Balmer series in order to measure the effective position of the discontinuity. Figure 4 shows the placement of continua, pseudo-continua and the measurements of $D$ and $\lambda_1$ for an A2I star from the stellar library of Jacoby, Hunter & Christian (1984). The functional dependences on the effective temperature and gravity of the stars are sufficiently different for $D$ and $\lambda_1$ to satisfy a two-dimensional classification.
The $D\lambda_1$ method could only be reliably applied in the cases of Hydra A and 3C 285. For the other radiogalaxies, the bulge-subtractions led to results that did not allow the identification of the absorption features and/or the break in an unambiguous way due to the resulting poor S/N. Figure 5 shows the $D\lambda_1$ measurements performed on the bulge-subtracted spectra of Hydra A and 3C 285. We have placed different continuum levels to estimate the maximum range of acceptable parameters of the stellar populations that are involved.
Table \[break\] lists the and Balmer break indices measured in both the bulge-subtracted and the original spectra of the radiogalaxies.
Lick indices
------------
The presence of prominent Balmer absorption lines, from H$\gamma$ up to H12 , is one of the most remarkable features of the blue spectra of two of the seven radiogalaxies, while H$\beta$ and H$\alpha$ are filled up by conspicuous emission lines. A clear exception to the presence of the Balmer series in absorption is the broad-line radiogalaxy 3C 382.
In order to estimate the Balmer strength, crucial to date any young stellar population involved, we use the EW of the H10 line, which appears only weakly contaminated by emission in the radiogalaxies. H10 is chosen as a compromise of an easily detectable Balmer line that shows both a minimum of emission contamination and clear wings to measure the adjacent continuum. The Balmer lines from to H9 are contaminated by prominent emission, which in Case B recombination comes in decreasing emission ratios to of 1, 0.458, 0.251, 0.154, 0.102, 0.0709 (Osterbrock 1989); H10 has an emission contamination of $0.0515 \times \Hb$. At the same time, the absorption strengths are quite similar from to H10, although the EW(H10) is actually systematically smaller than EW() in young to intermediate-age populations. González-Delgado, Leitherer & Heckman (2000) obtain, in their population synthesis models, ratios of EW()/EW(H10) between 1.3 and 1.6 for bursts with ages 0 to 1 Gyr and constant or coeval star formation histories. Lines of order higher than 10 have decreasing emission contamination, but they also increasingly merge towards the Balmer continuum limit.
A caveat in the use of H10 as an age calibrator comes from the fact that this line might be contaminated by metallic lines in old populations. Although our measurements of H10 in NGC 4374 are around 1.5Å, an inspection of the spectra of three elliptical galaxies (NGC 584, NC 720, NGC 821) observed in the same wavelegth range (but with lower S/N) and archived in the Isaac Newton Group database, indicates that a wide range of EW(H10), from 2 to 4Å, could characterize elliptical galaxies, while their indices are in the 1 to 2Å regime. If confirmed by better data, these results could indicate that although the upper Balmer series is detected in elliptical galaxies, it could indeed be contaminated by the absorptions of other species. Clearly, more work needs to be done in the near-UV spectra of elliptical galaxies before conclusive evidence can be derived for the behaviour of EW(H10) in old stellar systems, and its contamination by metallic lines.
In all the radio-galaxies observed in this work, the H10 profile is narrow and reproduces the shape of the wings of the lower-order Balmer absorption lines. Hydra A and 3C 285 clearly provide the best fittings. As an illustration, Figure 6 shows the estimated absorption line profiles for the H$\beta$, H$\gamma$ and H$\delta$ lines, assuming a constant ratio between their EWs and that of H10, and also a scaled ($\times 1.4$) H10 profile for the case of in Hydra A.
We also measured indices that are mostly sensitive to the metal content of the stellar populations involved. The Lick indices of Mg and Fe (e.g. Worthey et al. 1994) serve this purpose. In order to avoid the contribution of to the continuum measurement for the molecular index Mg$_2$, we have displaced the lower continuum band of this index to . This redefinition does not alter the value of the index in the elliptical galaxy, which shows no \[O III\] emission.
Table \[lick\] compiles the EW of H10, and the metallic indices Mg b, Fe5270, Fe5335, \[MgFe\], Mg$_2$ of the Lick system, measured in the rest-frame of the galaxies in our sample. The atomic indices are affected by broadening, like the CaT index, while Mg$_2$ is only affected by lamp contributions in the original IDS Lick system (Worthey et al. 1994, Longhetti et al. 1998). We have calculated broadening corrections as in section 3.1 for the atomic lines, and adopted the corrections calculated by Longhetti (1998) for the molecular lines. The uncorrected values of these indices are denoted with a subindex ${\mbox{\small u}}$ in Table \[lick\]. The errors of the individual line and molecular indices were estimated adopting continua shifted from the best fit continua by $\pm 1\sigma$. This lead to average errors between an 8 and a 10 per cent for individual line and molecular indices, and $\sim 6$ per cent for \[MgFe\].
The agreement between our measurements of Lick indices and those carried out by other authors (González 1993, Davies et al. 1987, Trager et al. 2000a) on our galaxy in common, NGC 4374, is better than 10 per cent.
Discussion
==========
Discussion
==========
Comparison with elliptical galaxies and population synthesis models
-------------------------------------------------------------------
The analysis of the spectral energy distributions and colours of elliptical galaxies suffers from a well known age-metallicity degeneracy (Aaronson et al. 1978). However, this is broken down when the strengths of suitable stellar absorption lines are taken into account (e.g. Heckman 1980). The plane composed by the \[\] and \[MgFe\] indices, in this sense, can discriminate the ages and metallicities of stellar systems. It is on the basis of this plot, that a large spread of ages in elliptical galaxies has been suggested (González 1993). Bressan, Chiosi & Tantalo (1996) claim, however, that when the UV emission and velocity dispersion of the galaxies are taken into account, the data are only compatible with basically old systems that have experienced different star formation histories (see also Trager et al. 2000a, 2000b). A recent burst of star formation that involves only a tiny fraction of the whole elliptical mass in stars, would rise the \[\] index to values characteristic of single stellar populations which are 1 to 2 Gyr old (Bressan et al. 1996).
Most likely, the stellar populations of radiogalaxies are also the combination of different generations. Direct support for this interpretation in the case of Hydra A comes from the fact that the stellar populations responsible for the strong Balmer lines are dynamically decoupled from those responsible for the metallic lines (Melnick, Gopal-Krishna & Terlevich 1997).
This interpretation is also consistent with the modest $\Delta$4000Å measurements we have obtained. Figure 7 shows a comparison of the values found in radiogalaxies, with those of normal elliptical, spiral and irregular galaxies, including starbursts, from the atlas of Kennicutt (1992). The radiogalaxies 3C 98, 3C 192, 4C 73.08 and DA 240 have indices of the order of 1.9 to 2.3, which overlap with those of normal E galaxies, . These values correspond to populations dominated by stars of ages 1 to 10 Gyr old, if one assumes the coeval population synthesis models of Longhetti et al. (1999). However, Hydra A and 3C 285 have indices in the range 1.4 to 1.6, typical of coeval populations which are 200 to 500 Myr old. Once the bulge population is subtracted, the $\Delta$4000Å indices of Hydra A and 3C 285 decrease to 1.2 and 1.0 respectively, which are typical of systems younger than about 60 Myr.
Hamilton (1985) measured the $\Delta$4000Å index in a sample of stars covering a wide range of spectral types and luminosity classes. He found a sequence of increasing $\Delta$4000Å from B0 to M5 stars, with values from 1 to 4 mag respectively. A comparison with the sequence he found leads us to conclude that the break in the bulge subtracted spectrum of Hydra A is dominated by B or earlier type stars while that of 3C 285 is dominated by A type stars. The index $\Delta$4000Å does not clearly discriminate luminosity classes for stars with spectral types earlier than G0.
The equivalent width of the H10 absorption line in these two radiogalaxies give further support to the interpretation of the Balmer break as produced by a young stellar population. In Hydra A we find after bulge subtraction EW(H10)$\approx 3.9$Å, which, according to the synthesis models of González-Delgado et al. (2000) gives ages of 7 to 15 Myr for an instantaneous burst of star formation, and 40 to 60 Myr for a continuous star formation mode, in solar metallicity environments. In the case of 3C 285, EW(H10)$\approx 6$Å would imply an age older than about 25 Myr for a single-population burst of solar metallicity.
The metallic indices of normal elliptical galaxies range between the values $0.56 \lsim \log {\rm [MgFe]} \lsim 0.66$ (González 1993), which characterizes oversolar metallicites for ages larger than about 5 Gyr. This is also the typical range of our radiogalaxies, although 3C 285 shows a clear departure with $\log {\rm [MgFe]} \approx 0.4$. However, \[MgFe\] tends to be smaller for populations younger than a few Gyr and similar oversolar metallic content. Since 3C 285 has a clear burst of recent star-formation, we conclude that its overall abundance is also most probably solar or oversolar.
The blue stellar content
------------------------
A better estimate of the spectral type and luminosity class of the stars that dominate the break in Hydra A and 3C 285 comes from the two-dimensional classification of Barbier and Chalonge. In Figure 8 the solid squares connected by lines represent the maximum range of possible $D\lambda_1$ values measured in these radiogalaxies.
The Balmer break index is sensitive to the positioning of the pseudo-continuum on top of the higher order Balmer series lines, which in turn is sensitive to the merging of the wings of the lines, enhanced at large velocity dispersions. In order to assign spectral types and luminosity classes to the stars that dominate the break, therefore, it is not sufficient to compare the values we have obtained with those measured in stellar catalogues. The values measured for the radiogalaxies can be corrected for their intrinsic velocity dispersions; we have chosen instead to recalibrate the index using template stars of different spectral types and luminosity classes convolved with gaussian functions, until they reproduce the width of the Balmer lines observed in the radiogalaxies (FWHM$\approx 12.5$Å). We used the B0 to A7 stars from the stellar library of Jacoby et al. (1984), which were observed with 4.5Åresolution. The values of the $D\lambda_1$ indices measured in these broadened stars are represented in Figure 8 by their respective classification. By comparison we also plot the grid traced by the locus of unbroadened stars, as published by Strömgren (1963). The broadening of the lines shifts the original locus of supergiant stars from the $\lambda_1 \lsim 3720$Årange (Chalonge 1956) to the $3720 \lsim \lambda_1 \lsim 3740$Å range, occupied by giant stars in the original (unbroadened) classification. Giant stars, in turn, shift to positions first occupied by dwarfs. Most dwarfs have Balmer line widths comparable to those of the radiogalaxies, and thus their locus in the diagram is mostly unchanged.
The value of the $D$ index indicates that the recent burst in Hydra A is dominated by B3 to B5 stars, and the effective position of the Balmer break ($\lambda_1$) indicates that these are giant or supergiant stars, respectively. These stars have masses of 7 and 20 (Schmidt-Kaler 1982). From the stellar evolutionary tracks of massive stars with standard mass-loss rate at or 2(Schaller et al. 1992, Schaerer et al. 1993, Meynet et al. 1994) we infer that these stars must have ages between 7 to 8 Myr (B3I) and 40 Myr (B5III). Note that the B4V stars in Figure 8, near the location of Hydra A, cannot originate the break and at the same time follow the kinematics of the nucleus (see section 5.3). Any dwarf star located in the stellar disk of Hydra A would show absorption lines that have been broadened beyond the 12.5Å of FWHM we measure in this radiogalaxy, and its position would have been shifted further into larger values of $\lambda_1$.
The location in the $D-\lambda_1$ plane of 3C 285 indicates that the break is produced by A2I stars. These are 15 stars. Again, ages of 10 to 12 Myr are found for the last burst of star formation in this radiogalaxy. The interpretation of the blue excess in terms of A type stars is further supported by the detection of the Ca II H line in the bulge-subtracted spectrum.
The red stellar content
-----------------------
The CaT index in the radiogalaxies has values between 6 and 7Å. Díaz et al. (1989) find that at solar or oversolar metallicities red supergiant stars have CaT indices ranging from 8.5 to 13Å, red giant stars from 6 to 9Å and dwarfs from 4.5 to 8.5Å. The values we find are thus compatible with both giant or dwarf stars. However, we favour the interpretation of giant stars since the old bulge population will be dominated by red giants, as in the case of elliptical galaxies.
We have measured the CaT in a control sample of elliptical galaxies observed by J. Gorgas and collaborators (priv. communication) and combined these measurements with those found by Terlevich et al (1990) in a sample of elliptical, lenticular, spiral, and active galaxies of different kinds. We find that the range of CaT in elliptical galaxies, 5 to 7.5Å, comprises the range of values of the radiogalaxies (see Figure 9).
García-Vargas, Mollá & Bressan (1998) find in their population synthesis models values of the CaT between 6 and 7Å for ages ranging from 100 Myr to 1 Gyr, and larger afterwards, assuming coeval star formation and solar or oversolar metallicity. A revised version of these models by Mollá & García-Vargas (2000) which includes extended libraries of M-type stars, predicts for ages between 2 and 20 Gyr a constant value of 7Å at solar metallicity, and 8.5Å at 2. These synthesis models are based on parametric fittings of the CaT strength in NLTE model atmospheres (Jørgensen, Carlsson & Johnson 1992) and in fittings of empirical values measured in stellar libraries (e.g. Díaz et al. 1989; Zhou 1991). The fits work well in the low metallicity regime. However, at metallicities higher than solar, the relationship between metallicity and the CaT index shows a big scatter, and the linear fittings loose any predictive power (see Figure 4 of Díaz et al. 1989).
Red supergiant stars appear in coeval population synthesis models between 5 and 30 Myr, and create a maximum strength of the CaT index (CaT $\gsim 9$Å) around 6 to 16 Myr for and 5 to 30 Myr for 2(García-Vargas et al. 1993, 1998; Mayya 1997). Strengths of CaT $\gsim 7$Å are characteristic of bursts with ages between 5 and 40 Myr. Leitherer et al. (1999) also find that the total strength of the population depends dramatically on the slope of the initial mass function (IMF) and star formation history. While a coeval burst with a complete Salpeter IMF yields values surpassing 7Å between 6 and 12 Myr, the same IMF in a continuous star formation mode yields values of only 5.5Å maximum. The CaT strength values for coeval star formation derived by Leitherer et al. (1999) differ substantially from those derived by García-Vargas and coworkers (1993, 1998) and by Mayya (1997), most probably due to a different calibration of the CaT index.
Mixed populations of young bursts which contain red supergiants, superposed on old populations can also yield values of the CaT between 4 and 8Å (García-Vargas et al. 1998). Since metal rich giant stars have CaT values ranging from 6 to 9Å we regard our observations of the CaT index in radiogalaxies, as being compatible with ages 1 to 15 Gyr.
Notes on individual objects
===========================
Notes on individual objects
===========================
3C 98
-----
3C 98 shows a double-lobe radio structure which spans 216 arcsec at 8.35 GHz, with a radio-jet that crosses the northern lobe to hit into a bright hotspot. There is little evidence of a southern jet, but a twin hotspot in the southern lobe is also present (Baum et al. 1988, Leathy et al. 1997, Hardcastle et al. 1998).
Broad band imaging of the host of 3C 98 reveals a smooth and slightly elongated elliptical galaxy located in a sparse environment (Baum et al. 1988, Martel et al. 1999). Deeper images reveal a faint shell as a sign of a past disturbance (Smith & Heckman 1989). However, the rotation curves of 3C 98 show that the stellar kinematics has negligible rotation $< 25$ and no anomalies (Smith, Heckman & Illingworth 1990). Although the optical colours of 3C 98 are similar to those of normal elliptical galaxies (Zibel 1996), one should note that the colours are modified by the high Galactic extiction towards the source, (Schlegel et al. 1998). The $\Delta 4000$Å and \[MgFe\] indices found in this radiogalaxy are characteristic of old metal-rich populations.
An extended narrow line region with a wealth of structure, and no particular orientation with respect to the radio-axis, is also detected in direct narrow-band images (Baum et al. 1988). The narrow emission lines detected in the optical spectra correspond to highly ionized plasma (Baldwin, Phillips & Terlevich 1981, Saunders et al. 1989, Baum, Heckman & van Breugel 1992).
3C 98 has been detected in X-rays by the Einstein satellite at a flux level $f(\mbox{0.5--3keV}) = 1\times10^{-13}$ or $L_X = 4.2 \times 10^{41}$ (Fabbiano et al. 1984). The source detection was too weak to look for any extension to a point source.
3C 192
------
3C 192 has an ‘X’ symmetric double-lobe structure which extends 212 arcsec at 8.35 GHz, showing bright hotspots at the end of the lobes (Hogbom 1979, Baum et al. 1988, Hardcastle et al. 1998).
According to Sandage (1972), 3C 192 is a member of a small group of galaxies. Broad band imaging reveals the host of 3C 192 to be a round elliptical galaxy with a companion of similar size 70 arcsec away, and no obvious signs of interaction (Baum et al. 1988, Baum & Heckman 1988). The stellar kinematics shows negligible rotation, $< 30$ , and no disturbances (Smith et al. 1990). The spectral shape of 3C 192 also shows a blue excess with respect to our template elliptical galaxy.
Extended narrow line emission is detected, with structures which are co-spatial with bridges and cocoons detected in radio-emission (Baum et al. 1988). The narrow emission lines are highly ionized (Baldwin et al. 1981, Saunders et al. 1989, Baum, Heckman & van Breugel 1992).
The Einstein satellite detected 3C 192 in X-rays at a flux level $f(\mbox{0.5--3keV}) = 1.1\times10^{-13}$ , or $1.8 \times 10^{42}$ . The source is extended at a 97% confidence level, $0.8^{+1.7}_{-0.3}$ arcmin, but a background object might be contaminating the map (Fabbiano et al. 1984).
3C 218 or Hydra A
-----------------
3C 218 is one of the most luminous radiosources in the local ($z<0.1$) Universe, only surpassed by Cygnus-A. Although the radio-luminosity of 3C 218 exceeds by an order of magnitude the characteristic FR I/FR II break luminosity, it has an edge-darkened FR I morphology (Ekers & Simkin 1983, Baum et al. 1988, Taylor et al. 1990). The total size of the radio structure extends for about 7 arcmin, such that the radio-jets, which flare at 5 arcsec, are curved and display ’S’ symetry.
3C 218 has been optically identified with the cD2 galaxy Hydra A (Simkin 1979), which dominates the poor cluster Abell 780. This however is an X-ray bright cluster with $L_{\mbox{\scriptsize X}} \approx 2 \times 10^{44}$ in the 0.5 – 4.5 keV range, as seen by the Einstein satellite (David et al. 1990). The total bolometric luminosity has been estimated to be $5\times10^{44}$ from 0.4–2 keV ROSAT observations (Peres et al. 1998). The thermal model that best fits the data supports the existence of a cooling flow which is depositing mass in the central regions of the cluster at a rate of $264^{+81}_{-60}$ yr$^{-1}$.
Hydra A has an associated type II cooling flow nebula (Heckman et al. 1989), characterized by high and X-ray luminosities, but relatively weak \[N II\] and \[S II\] and strong \[O I\] emission lines, usually found in LINERs. The extended narrow line emission (Baum et al. 1988) actually fills the gap between the radiolobes.
The , , $B$-band, $B-V$ and, also, the $U-I$ continuum colours of the center of Hydra A have been attributed to a $\sim$ 10 Myr burst of star formation involving $10^8$ to $10^9$ (Hansen, Jørgensen & Nørgaard-Nielsen 1995, McNamara 1995). This is further supported by the detection of strong absorption lines of the Balmer series in the near-UV spectrum of the nucleus (Hansen et al. 1995, Melnick et al. 1997). We also find strong absorption Balmer lines, which we identify as originating in blue supergiant or giant B stars. One of our best two matches in the $D\lambda_1$ classification we use in this work, B3I stars, also indicates ages 7 to 8 Myr.
Heckman et al. (1985) found that the stellar kinematics has negligible rotation ($13 \pm 18$ ), but their observations were limited to the region, and did not include the higher Balmer lines in absorption. On the other hand, Ekers & Simkin (1983) report very fast rotating stars in the central 20 kpc of the radiogalaxy. A two dimensional analysis of the blue spectrum shows a tilt of the Balmer absorption lines of $450 \pm 130$ in the central 3 arcsec, while the Ca II H,K lines do not show any displacement (Melnick et al. 1997). This tilt is further confirmed by our data. The conclusion derived by Melnick et al. (1997) is that the young stars have formed a disk which is rotating perpendicular to the position of the radio-axis. The star-formation disk has also been detected in $U$-band images (McNamara 1995).
3C 285
------
The host galaxy of 3C 285 has been identified with the brightest member of a group of galaxies (Sandage 1972). Optical imaging of the galaxy reveals an elliptical main body and a distorted S shape envelope aligned with a companion galaxy $\sim 40$ arcsec to the northwest (Heckman et al. 1986). Narrow band imaging shows that the S-shaped extension is due to continuum emitting structures (Heckman et al. 1986, Baum et al. 1988). The narrow emission lines are originated by photoionization with a high ionization parameter (Saunders et al. 1989, Baum et al. 1992).
Sandage (1972) found that the $B-V$ colour of 3C 285 is much bluer than that of a normal elliptical galaxy. Our observations show that the blue light of the nucleus (inner 2 arcsec) is dominated by a burst which contains A2I stars, and thus has an age of 10 to 12 Myr.
Saslaw, Tyson and Crane (1978) identified a bright blue slightly resolved object halfway between the nucleus and the eastern radio-lobe, which they denoted 3C 285/09.6. Optical spectra and imaging obtained by van Breugel & Dey (1993), showed that the knot is at the same redshift as the galaxy, and its $UBV$ colours and 4000Å break are consistent with a burst of 70 Myr, which they interpreted as induced by the radio-jet.
3C 285 is a classical double-lobed radiogalaxy of 190 arcsec total extension at 4.86 GHz, with two hotspots and an eastern ridge showing curvature roughly along the line to the optical companion (Leahy & Williams 1984, Hardcastle et al. 1998).
The source has not been detected by the Einstein satellite in X-rays, at a flux level $f(\mbox{0.5--3keV}) <1.5\times10^{-13}$ or $L_X = 4.4 \times 10^{42}$ (Fabbiano et al. 1984).
3C 382
------
3C 382 has a double-lobe structure, with a clear jet in the northern lobe that ends in a hotspot. A hotspot in the southern lobe is also detected, but a counterpart jet is not clear, although a trail of low fractional polarization is detected (Black et al. 1992). The total 3.85 GHz size between hotspots is 179 arcsec (Hardcastle et al. 1998).
Optically, the radiosource is identified with a disturbed elliptical galaxy dominated by a very bright and unresolved nucleus (Mathews, Morgan & Schmidt 1964, Martel et al. 1999), located in a moderately rich environment (Longair & Seldner 1979). The optical spectra shows a strong continuum and prominent broad-lines photoinized by a power-law type of spectrum (Saunders et al. 1989, Tadhunter, Fosbury & Quinn 1989). The stellar population of the host galaxy, as we show in our study, is barely detected in the nuclear regions.
The Einstein satellite detected 3C 382 in X-rays at a flux level $f(\mbox{0.5--3keV}) = 1.3\times10^{-13}$ , or $2 \times 10^{44}$ (Fabbiano et al. 1984). The source is resolved in ROSAT/HRI observations but its interpretation is debateable since the luminosity is too strong for a galaxy environment which is only moderately rich (Prieto 2000).
3C 382 is a variable source at X-ray (Dower et al. 1980, Barr & Giommi 1992), radio (Strom, Willis & Willis 1978), optical and UV frequencies (Puschell 1981, Tadhunter, Pérez & Fosbury 1986)
DA 240
------
This is a double-lobed giant radio-galaxy of 34 arcmin angular size between hotspots and ongoing nuclear activity at 2.8cm (Laing et al. 1983, Nilsson et al. 1993, Klein et al. 1994).
The amplitude of the angular cross-correlation of sources found in optical plates around the position of the radio source is weak, $A_{\mbox{\scriptsize gg}}=0.101\pm0.118$ (Prestage & Peacock 1988). Abell clusters at the same redshift have values $A_{\mbox{\scriptsize gg}} \gsim 0.3$.
The optical spectra shows weak and \[Ne III\] and \[O III\] narrow emission lines, compatible with a higly ionized medium which is obscured. The $\Delta 4000$Å and \[MgFe\] indices found in this radiogalaxy are characteristic of old metal-rich populations.
4C 73.08
--------
4C 73.08 is a giant double-lobed radio-galaxy, with 13 arcmin angular size between hotspots (Meyer 1979, Nilsson et al. 1993).
The environment of the radiogalaxy is also weak, with amplitude of the angular cross-correlation of optical galaxies around the radiosource of $A_{\mbox{\scriptsize gg}}=0.203 $ (Prestage & Peacock 1988).
4C 73.08 shows a high excitation spectrum typical of narrow line radio galaxies. The colours of the radiogalaxy and the $\Delta 4000$Å and \[MgFe\] indices are comparable to those of our reference elliptical galaxy.
Conclusions
===========
Conclusions
===========
We have presented spectra of 7 radiogalaxies in the – and – range. All radiogalaxies show either a clear detection or an indication of detection of the Ca II $\lambda\lambda$8494,8542,8662Å triplet in absorption, and in 6 of them we detected Balmer absorption lines.
On the basis of the $\Delta 4000$Å break measurements, we conclude that 4 of these radiogalaxies contain populations which are typical of normal elliptical galaxies, 2 have populations younger than a few hundrer Myr, and in one its stellar population cannot be characterized.
In the two cases with young bursts, Hydra A and 3C 285, we subtracted the bulge population using a normal elliptical galaxy as a template in order to characterize better the young burst. The and Balmer break index measurements indicate that the young population is dominated by blue giant and/or blue supergiant stars: B3I or B5III for Hydra A, and A2I for 3C 285. The derived age of the burst is beween 7 and 40 Myr for Hydra A, and 10 to 12 Myr for 3C 285.
The CaT strength, invoked to support the detection of young stellar populations in active galaxies, fails to provide a clear conclusion on the nature of the stars that dominate the red light in these radiogalaxies. The CaT could either be due to the red giant stars that dominate old bulge populations, or to the red dwarfs of a young starburst ($t \lsim 7$Myr), or the red giants and supergiants of a post-starburst ($t\gsim 30$ Myr), or a combination of a bulge population and a recent burst of star formation. A mixed population is again favoured as the interpretation of the red spectra.
It is known that although the hosts of FR II sources look like ellipticals, few of them have true elliptical galaxy properties: magnitudes, colours, and structural parameters show a wider dispersion than in normal ellipticals (Baum et al. 1988, Zirbel 1996). Most of the radiogalaxies in our sample have reported structural disturbances in their optical morphologies, show signs of interactions, have close companions, belong to rich environments and/or have signatures of cooling flows. These are phenomena that facilitate carrying large quantities of gas to the centers of the galaxies and can power the AGN and/or provoke bursts of star formation.
Good quality data in the blue region of this sample is necessary in order to constrain the ages of the young populations involved, especially in the cases of 3C 98, 3C 192, DA 210, 3C 382 and 4C 73.08, where our bulge subtractions led to poor signal-to-noise and therefore unreliable results.
A detailed analysis of the ages of the last burst of star formation will set the relative chronology of the onset of the radio and starburst activity in these galaxies, and shed new light into the relationships between jets, AGN and star formation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported in part by the ‘Formation and Evolution of Galaxies’ Network set up by the European Commission under contract ERB FMRX-CT96-086 of its TMR programme. We thank PATT for awarding observing time. IA, ET and RJT also thank the Guillermo Haro Programme for Advanced Astrophysics of INAOE for the oportunity it gave us to meet and make progress on the project during the 1998 workshop ’The Formation and Evolution of Galaxies’. GC acknowledges a PPARC Postdoctoral Research Fellowship, and ET an IBERDROLA Visiting Professorship to the Universidad Autónoma de Madrid. We thank J. Gorgas for providing the CaT spectra of the sample of comparison elliptical galaxies prior to publication, M. García-Vargas for suggestions on how to improve the fringing removal, and an anonymous referee for crucial comments on the relevance of Balmer indices in old populations.
[99]{}
Baldwin J.A., Phillips M.M., Terlevich R.J., 1981, PASP, 93, 5 Barbier D., 1955, in ‘Principes fondamentaux de classification stellaire’, Coll. Int. Centre Nat. Rech. Sc., vol. 55, p. 47. Barr P., Giommi P., 1992, MNRAS, 255, 495 Baum S., Heckman T,M., 1989, ApJ, 336, 681 Baum S.A, Heckman T.M., Bridle A.H., van Breugel W.J.M., Miley G.K.M., 1988, ApJS, 68, 643. Baum S., Heckman T,M., van Breugel W., 1992, ApJ, 389, 208 Black A.R.S., Baum S.A., Leahy J.P., Perley R.A., Riley J.M., Scheuer P.A.G., 1992, MNRAS, 256, 186. Bressan A., Chiosi C., Tantalo R., 1996, AA, 311, 425 van Breugel W.J.M, Dey A., ApJ, 414, 563 Burstein D., Faber S.M., Gaskell C.M., Krumm N., 1984, ApJ, 287, 586
Chalonge D., 1956, Ann. d’ap., 19, 258. Cid Fernandes R., Rodrigues-Lacerda R., Schmitt H.R., Storchi-Bergmann T., 1999, in IAU Symp. 193 ‘Wolf-Rayet Phenomena in Massive Stars and Starburst Galaxies’, eds. K.A. van der Hucht, G. Koenigsberger & P.R.J. Eenens, ASP, p. 590.
Colina, L., Garcia Vargas, M. L. , González Delgado, R. M., Mas-Hesse, J. M., Perez, E., Alberdi, A., & Krabbe, A. 1997, ApJ, 488, 71
David L.P., Arnaud K.A., Forman W., Jones C., 1990, ApJ, 356, 32. Dower R.G., Griffiths R.E., Bradt H.V., Doxsey R.E., Johnson M.D., 1980, ApJ, 235, 355
Ekers R.D., Simkin S.M., 1983, ApJ, 265, 85.
García-Vargas M.L., Díaz A.I., Terlevich E. & Terlevich R.J., 1993, ApSS, 205, 85.
García-Vargas M.L., Mollá M. & Bressan S., 1998, AAS, 130, 513.
González J., 1993, PhD thesis, Univ. Santa Cruz, USA.
González Delgado R. M., Heckman, T., Leitherer, C., Meurer, G., Krolik J., Wilson, A. S., & Kinney A. 1998, ApJ, 505, 174
González Delgado R. M., Leitherer, C., Heckman, T., 2000, ApJS, in press.
Hansen L., Jørgensen H.E., Nørgaard-Nielsen H.U., 1995, AA, 297, 13. Hardcastle M., Alexander P., Pooley G.G., Riley J.M., 1998, MNRAS, 296, 445. Heckman T.M., 1980, AA, 87, 142 Heckman T.M., Illingworth G.D., Miley G.K., van Breugel W.J.M., 1985, ApJ, 299, 41. Heckman T.M., Smith E.P., Baum S.A., van Breugel W.J.M., Miley G.K., Illingworth G.D., Bothun G.D., Balick B., 1986, ApJ, 311, 526. Heckman T.M., Baum S.A, van Breugel W.J.M., McCarthy P.A., 1989, ApJ, 338, 48. Heckman T.M. et al. 1995 ApJ, 452, 549 Hogbom J.A. 1979, ApJS, 36, 173
Kunth D., Contini T. 1999, in IAU Symp. 193 ‘Wolf-Rayet Phenomena in Massive Stars and Starburst Galaxies’, eds. K.A. van der Hucht, G. Koenigsberger & P.R.J. Eenens, ASP, p. 725.
Leahy J.P., Williams A.G., 1984, MNRAS, 210, 929. Leahy J.P et al. 1997, MNRAS, 291, 20 Longair M.S., Seldner R., 1979, MNRAS, 189, 433.
Maoz D., Filippenko A.V., Ho L.C., Rix H.-W., Bahcall J.N., Schneider D.P., Macchetto F.D., 1995, ApJ, 440, 91 Maoz D., Koratkar A., Shields J.C., Ho L.C., Filippenko A.V., Stenberg A., 1998, AJ, 116, 55 Matthews T.A., Morgan W.M., Schmidt M., 1964, ApJ, 272, 400 Meynet G., Maeder A., Schaller G., Schaerer D., 1994, AAS, 103, 97. Melnick J., Gopal-Krishna, Terlevich R.J., 1997, AA, 318, 337. Mollá M. & García-Vargas M. L., 2000, A&A, in press.
Peres C.B., Fabian A.C., Edge A.C., Allen S.W., Johnstone R.M., White D.A., 1998, 298, 416. Peterson, B.M., 1992, in ’Relationships between AGN and SB galaxies’, Ed. Filippenko. ASP Conf. Ser. 31, 29. Prestage R.M., Peacock J.A., 1988, MNRAS, 230, 131 Prieto M.A., 2000, MNRAS, in press Puschell J.J., 1981, AJ, 86, 16
Rawlings S., Saunders R., 1991, Nat., 349, 138.
Sandage A., 1972, Ap, 178, 25. Saunders R., Baldwin J.E., Rawlings S., Warner P.J., Miller L., 1989, MNRAS, 238, 777. Saslaw W.C., Tyson J.A., Crane P., 1978, ApJ, 222, 335. Schaller G., Schaerer D., Meynet G., Maeder A., 1992, AAS, 96, 269. Schaerer D., Meynet G., Maeder A., Schaller G., 1993, AAS, 98, 523. Schlegel et al., 1998, ApJ, 500, 525 Schmidt-Kaler T., 1982, in ‘Landolt-Börnstein’’, New Series, Group VI, vol. 2b, ed. K. Schaifers & H.H. Voigt, Springer. Simkin S.M., 1979, ApJ, 234, 56. Smith E.P., Heckman T.M., 1989, ApJ, 341, 658 Smith E.P., Heckman T.M., Illingworth G.D., 1990, ApJ, 356, 399 Strom R.G., Willis A.G., Willis A. S., 1978, AA, 68, 367 Strömgren B., in ‘Basic Astronomical Data’, Stars and Stellar Systems, vol. 3, p. 123.
Tadhunter C.N., Fosbury R.A.E., Quinn P.J., 1989, MNRAS, 240, 225 Tadhunter C.N., Pérez E. & Fosbury R.A.E., 1986, MNRAS, 219, 555 Taylor G.B., Perley R.A., Inoue M., Kato T., Tabara H., Aizu K., 1990, ApJ, 360, 41. Trager S.C., Faber S.M., Worthey G & González J.J., 2000a, AJ, in press Trager S.C., Faber S.M., Worthey G & González J.J., 2000b, AJ, in press Tran, H. D., Miller J.S., Kay L. E. 1992, ApJ, 397, 452
de Vaucouleurs G., de Vaucouleurs A., Corwin H.G., Buta R.J., Paturel G., Fouque P., 1991, ‘Third Reference Catalogue of Bright Galaxies’, Springer.
------------------- ---------------------------------------- -------- ------ ------ -------------- ------ ------- --------- ---------------
object $P_{{\mbox{\scriptsize 178MHz}}}$ $z$ $V$ type date PA exp. grating $l$(1 arcsec)
$\times 10^{25}$ W Hz$^{-1}$ Sr$^{-1}$ mag $^o$ s pc
3C 98 1.5 0.0306 15.9 RG 1998 Feb. 19 221 8290 R300B 422
8304 R316R
3C 192 2.6 0.0598 16.1 RG 1998 Feb 19 133 7200 R300B 786
7200 R316R
3C 218 (Hydra A) 22.0 0.0533 12.6 RG 1998 Feb 20 113 10800 R300B 708
10800 R316R
3C 285 2.5 0.0794 15.9 RG 1998 Feb 20 66 10800 R300B 1010
7200 R316R
3C 382 2.2 0.0578 15.4 RG 1998 Feb 19 47 6200 R300B 762
6200 R316R
4C 73.08 1.6 0.0581 16.0 RG 1998 Feb 19 68 7200 R300B 765
7200 R316R
DA 240 0.9 0.0350 15.1 RG 1997 Nov 7 61 3600 R600B 479
3900 R600R
NGC 4374 — 0.0034 10.1 E 1998 Feb 19 135 1000 R300B
1000 R316R 49
G191-B2B — — S 1997 Nov 7 — 300 R600B
300 R600R
1997 Nov 8 300 R600B
300 R600R
HR 1908 — — S 1998 Feb 20 — 2 R300B
3 R316R
HR 4575 — — S 1998 Feb 19 — 1 R300B
1 R316R
HR 4672 — — S 1998 Feb 19 — 1 R300B
1 R316R
HR 5826 — — S 1998 Feb 19 — 1 R300B
1 R316R
HR 8150 — — S 1997 Nov 8 — 1 R600B
1 R600R
HZ 44 — — S 1998 Feb 19 — 60 R300B
180 R316R
1998 Feb 20 60 R300B
------------------- ---------------------------------------- -------- ------ ------ -------------- ------ ------- --------- ---------------
---------- ---------- ------------------------------ -----
object $\sigma$ CaT$_{\mbox{\scriptsize u}}$ CaT
km/s Å Å
3C 98 250 5.9 6.5
3C 192 241 5.4 6.0
Hydra A 292 6.1 7.0
3C 285 202 $>2.4$ —
4C 73.08 — $>5.2$ —
DA 240 265 5.5 6.2
NGC 4374 361 5.3 6.8
---------- ---------- ------------------------------ -----
---------- ---------------------- --------------- ----------- -------------
[original spectra]{}
object $\Delta$4000Å $\Delta$4000Å $D$ $\lambda_1$
mag mag Å
3C 98 2.1
3C 192 1.9
Hydra A 1.4 1.2 0.26–0.28 3746–3752
3C 285 1.6 1.0 0.48–0.53 3740–3746
4C 73.08 2.2
DA 240 2.2
NGC 4374 2.3
---------- ---------------------- --------------- ----------- -------------
---------- -------- ---------------- ------ ------------------ -------- ------------------ -------- ---------- --------------- ---------- --
object Balmer Mg b$_{\rm u}$ Mg b Fe5270$_{\rm u}$ Fe5270 Fe5335$_{\rm u}$ Fe5335 \[MgFe\] Mg$_{2\rm u}$ Mg$_{2}$
Å Å Å Å Å Å Å Å mag mag
3C 98 3.5 4.76 5.03 2.73 3.06 2.66 3.16 3.95 0.26 0.28
3C 192 $>3.6$ 5.48 5.48 2.52 2.52 2.50 2.50 3.70 0.27 0.29
Hydra A 3.5 4.78 5.60 1.66 2.48 1.48 2.80 3.84 0.20 0.22
3C 285 4.4 3.37 3.51 1.67 1.85 1.50 1.76 2.52 0.18 0.20
4C 73.08 3.8 4.82 5.28 2.80 3.39 2.27 3.21 4.17 0.23 0.25
DA 240 4.5 4.01 4.28 — — — — — — —
NGC 4374 1.4 4.58 4.80 2.48 2.70 2.12 2.39 3.49 0.30 0.32
---------- -------- ---------------- ------ ------------------ -------- ------------------ -------- ---------- --------------- ---------- --
[^1]: Visiting Fellow at IoA, UK
[^2]: Visiting Professor at INAOE, Mexico
[^3]: The William Herschel Telescope is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias
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